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Introduction to Physical Oceanography

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Introduction to Physical Oceanography
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Stewart, Robert H.
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Oceanography, ocean physics, Oceanographic Exploration, Ocean, Seas, Sea-Floor, Depth of the Ocean, Sea-Floor Charts, Atmospheric Wind Systems, Planetary Boundary Layer, Measurement of Wind, Wind Stress, Oceanic Heat Budget, Calculation of Fluxes, Meridional Heat Transport, Meridional Fresh Water Transport, Thermocline, Salinity, Light in the Ocean, Ocean …
Earth Science, Oceanography
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This book is written for college juniors and seniors and new graduate students in meteorology, ocean engineering, and oceanography. It begins with a brief overview of what is known about the ocean. This is followed by a description of the ocean basins, for the shape of the seas influences the physical processes in the water. Next, students will study the external forces, wind and heat, acting on the ocean, and the ocean’s response. It also includes the equations describing dynamic response of the ocean. For example, the equations of motion, the influence of earth’s rotation, and viscosity. Finally, students consider some particular examples: the deep circulation, the equatorial ocean and El Ni˜no, and the circulation of particular areas of the ocean. Contents: 1) A Voyage of Discovery. 2) The Historical Setting. 3) The Physical Setting. 4) Atmospheric Influences. 5) The Oceanic Heat Budget. 6) Temperature, Salinity and Density. 7) The Equations of Motion. 8) Equations of Motion with Viscosity. 9) Response of the Upper Ocean to Winds. 10) Geostrophic Currents. 11) Wind Driven Ocean Circulation. 12) Vorticity in the Ocean. 13) Deep Circulation in the Ocean. 14) Equatorial Processes. 15) Numerical Models. 16) Ocean Waves. 17) Coastal Processes and Tides.
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Community College, Higher Education
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http://www.ogtp-cart.com/product.aspx?ISBN=9781-16100452
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Robert H. Stewart, Department of Oceanography, Texas A & M University
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Textbook
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http://oceanworld.tamu.edu/resources/ocng_textbook/contents.html
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http://florida.theorangegrove.org/og/file/f5a4004c-2094-7b6d-7134-2622aa9c9513/1/IntroToPhysicalOcean.pdf

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Copyright 2008 Robert H. Stewart. Distributed by the author for free in digital format via the world-wide web. See: http://oceanworld.tamu.edu/resources/ocng_textbook/contents.html
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IntroductionTo PhysicalOceanography RobertH.Stewart DepartmentofOceanography TexasA&MUniversity Copyright2008 September2008Edition

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ii

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Contents Preface vii 1AVoyageofDiscovery 1 1.1Physicsoftheocean .........................1 1.2Goals .................................2 1.3Organization .............................3 1.4TheBigPicture ............................3 1.5FurtherReading ...........................5 2TheHistoricalSetting 7 2.1Denitions ...............................8 2.2ErasofOceanographicExploration .................8 2.3MilestonesintheUnderstandingoftheOcean ...........12 2.4EvolutionofsomeTheoreticalIdeas ................15 2.5TheRoleofObservationsinOceanography ............16 2.6ImportantConcepts .........................20 3ThePhysicalSetting 21 3.1OceanandSeas ............................22 3.2Dimensionsoftheocean .......................23 3.3Sea-FloorFeatures ..........................25 3.4MeasuringtheDepthoftheOcean .................29 3.5SeaFloorChartsandDataSets ...................33 3.6SoundintheOcean .........................34 3.7ImportantConcepts .........................37 4AtmosphericInruences 39 4.1TheEarthinSpace ..........................39 4.2AtmosphericWindSystems .....................41 4.3ThePlanetaryBoundaryLayer ...................43 4.4MeasurementofWind ........................43 4.5CalculationsofWind .........................46 4.6WindStress ..............................48 4.7ImportantConcepts .........................49 iii

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iv CONTENTS 5TheOceanicHeatBudget 51 5.1TheOceanicHeatBudget ......................51 5.2Heat-BudgetTerms ..........................53 5.3DirectCalculationofFluxes .....................57 5.4IndirectCalculationofFluxes:BulkFormulas ...........58 5.5GlobalDataSetsforFluxes .....................61 5.6GeographicDistributionofTerms ..................65 5.7MeridionalHeatTransport .....................68 5.8VariationsinSolarConstant .....................70 5.9ImportantConcepts .........................72 6Temperature,Salinity,andDensity 73 6.1DenitionofSalinity .........................73 6.2DenitionofTemperature ......................77 6.3GeographicalDistribution ......................77 6.4TheOceanicMixedLayerandThermocline ............81 6.5Density ................................83 6.6MeasurementofTemperature ....................88 6.7MeasurementofConductivityorSalinity ..............93 6.8MeasurementofPressure ......................95 6.9TemperatureandSalinityWithDepth ...............95 6.10LightintheOceanandAbsorptionofLight ............97 6.11ImportantConcepts .........................101 7TheEquationsofMotion 103 7.1DominantForcesforOceanDynamics ...............103 7.2CoordinateSystem ..........................104 7.3TypesofFlowintheocean .....................105 7.4ConservationofMassandSalt ...................106 7.5TheTotalDerivative(D/Dt) ....................107 7.6MomentumEquation .........................108 7.7ConservationofMass:TheContinuityEquation .........111 7.8SolutionstotheEquationsofMotion ................113 7.9ImportantConcepts .........................114 8EquationsofMotionWithViscosity 115 8.1TheInruenceofViscosity ......................115 8.2Turbulence ..............................116 8.3CalculationofReynoldsStress: ...................119 8.4MixingintheOcean .........................123 8.5Stability ................................127 8.6ImportantConcepts .........................131

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CONTENTS v 9ResponseoftheUpperOceantoWinds 133 9.1InertialMotion ............................133 9.2EkmanLayerattheSeaSurface ..................135 9.3EkmanMassTransport .......................143 9.4ApplicationofEkmanTheory ....................145 9.5LangmuirCirculation ........................147 9.6ImportantConcepts .........................147 10GeostrophicCurrents 151 10.1HydrostaticEquilibrium .......................151 10.2GeostrophicEquations ........................153 10.3SurfaceGeostrophicCurrentsFromAltimetry ...........155 10.4GeostrophicCurrentsFromHydrography .............158 10.5AnExampleUsingHydrographicData ...............164 10.6CommentsonGeostrophicCurrents ................164 10.7CurrentsFromHydrographicSections ...............171 10.8LagrangianMeasurementsofCurrents ...............172 10.9EulerianMeasurements .......................179 10.10ImportantConcepts .........................180 11WindDrivenOceanCirculation 183 11.1Sverdrup'sTheoryoftheOceanicCirculation ...........183 11.2WesternBoundaryCurrents .....................189 11.3Munk'sSolution ...........................190 11.4ObservedSurfaceCirculationintheAtlantic ...........192 11.5ImportantConcepts .........................197 12VorticityintheOcean 199 12.1DenitionsofVorticity ........................199 12.2ConservationofVorticity ......................202 12.3InruenceofVorticity .........................204 12.4VorticityandEkmanPumping ...................205 12.5ImportantConcepts .........................210 13DeepCirculationintheOcean 211 13.1DeningtheDeepCirculation ....................211 13.2ImportanceoftheDeepCirculation .................212 13.3TheoryfortheDeepCirculation ..................219 13.4ObservationsoftheDeepCirculation ................222 13.5AntarcticCircumpolarCurrent ...................229 13.6ImportantConcepts .........................232 14EquatorialProcesses 235 14.1EquatorialProcesses .........................236 14.2ElNi~no ................................240 14.3ElNi~noTeleconnections .......................248

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vi CONTENTS 14.4ObservingElNi~no ..........................250 14.5ForecastingElNi~no .........................251 14.6ImportantConcepts .........................254 15NumericalModels 255 15.1Introduction{SomeWordsofCaution ................255 15.2NumericalModelsinOceanography ................257 15.3GlobalOceanModels .........................258 15.4CoastalModels ............................262 15.5AssimilationModels .........................266 15.6CoupledOceanandAtmosphereModels ..............269 15.7ImportantConcepts .........................272 16OceanWaves 273 16.1LinearTheoryofOceanSurfaceWaves ...............273 16.2Nonlinearwaves ...........................278 16.3WavesandtheConceptofaWaveSpectrum ...........278 16.4Ocean-WaveSpectra .........................284 16.5WaveForecasting ...........................288 16.6MeasurementofWaves ........................289 16.7ImportantConcepts .........................292 17CoastalProcessesandTides 293 17.1ShoalingWavesandCoastalProcesses ...............293 17.2Tsunamis ...............................297 17.3StormSurges .............................299 17.4TheoryofOceanTides ........................300 17.5TidalPrediction ...........................308 17.6ImportantConcepts .........................312 References 313

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PrefaceThisbookiswrittenforupper-divisionundergraduatesand newgraduatestudentsinmeteorology,oceanengineering,andoceanography .Becausethesestudentshaveadiversebackground,Ihaveemphasizedideasand conceptsmore thanmathematicalderivations. Unlikemostbooks,Iamdistributingthisbookforfreeindig italformatvia theworld-wideweb.Iamdoingthisfortworeasons: 1.Textbooksareusuallyoutofdatebythetimetheyarepubli shed,usually ayearortwoaftertheauthornisheswritingthebook.Rando lLarson, writingin Syllabus ,states:\Inmyopinion,technologytextbooksarea wasteofnaturalresources.They'reoutofdatethemomentth eyare published.Becauseoftheirshortshelflife,studentsdon' tevenwantto holdontothem"|(Larson,2002).Bypublishinginelectroni cform,Ican makerevisionseveryyear,keepingthebookcurrent. 2.Manystudents,especiallyinless-developedcountriesc annotaordthe highcostoftextbooksfromthedevelopedworld.Thisthenis agift fromtheUSNationalAeronauticsandSpaceAdministration nasa tothe studentsoftheworld. Acknowledgements Ihavetaughtfromthebookforseveralyears,andIthankthem anystudents inmyclassesandthroughouttheworldwhohavepointedoutpo orlywritten sections,ambiguoustext,conrictingnotation,andothere rrors.Ialsothank ProfessorFredSchlemmeratTexasA&MGalvestonwho,afteru singthebook forhisclasses,hasprovidedextensivecommentsaboutthem aterial. Ialsowishtothankmanycolleaguesforprovidinggures,co mments,and helpfulinformation.IespeciallywishtothankAanderaaIn struments,BillAllison,KevinBartlett,JamesBerger,GerbendeBoer,Daniel Bourgault,Don Chambers,GregCrawford,ThierryDeMees,RichardEanes,Pe terEtnoyer, TalEzer,GreggFoti,NevinS.Fuckar,LuizAlexandredeAra ujoGuerra,Hazel Jenkins,JodyKlymak,JudithLean,ChristianLeProvost,Br ooksMartner, NikolaiMaximenko,KevinMcKone,MikeMcPhaden,ThierryDe Mees,Pim vanMeurs,GaryMitchum,JoeMurtagh,PeterNiiler,NunoNun es,Ismael Nu~nez-Riboni,AlexOrsi,KymPerkin,MarkPowell,Richar dRay,Joachim Ribbe,WillSager,DavidSandwell,Sea-BirdElectronics,A chimStoessel,David vii

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viii PREFACE Stooksbury,TomWhitworth,CarlWunschandmanyothers. Ofcourse,Iacceptresponsibilityforallmistakesinthebo ok.Pleasesend meyourcommentsandsuggestionsforimprovement. Figuresinthebookcamefrommanysources.Iparticularlywi shtothank LinkJiformanyglobalmaps,andcolleaguesattheUniversit yofTexasCenter forSpaceResearch.DonJohnsonredrewmanyguresandturne dsketchesinto gures.TreyMorristaggedthewordsusedintheindex. Iespeciallythank nasa 'sJetPropulsionLaboratoryandtheTopex/Poseidon andJasonProjectsfortheirsupportofthebookthroughcont racts960887and 1205046. CoverphotographoftheresortislandofKurumbainNorthMal eAtollin theMaldiveswastakenbyJagdishAgara(copyrightCorbis). Coverdesignis byDonJohnson. ThebookwasproducedinL A T E X2 usingTeXShop2.14onanInteliMac computerrunningOS-X10.4.11.IespeciallywishtothankGe rbenWierdafor hisveryusefuli-Installerpackagethatmadeitallpossibl e,andRichardKoch, DirkOlmesandmanyothersforwritingtheTeXShopsoftwarep ackage.Their softwareisapleasuretouse.AllguresweredrawninAdobeI llustrator.

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Chapter1AVoyageofDiscoveryTheroleoftheoceanonweatherandclimateisoftendiscusse dinthenews. WhohasnotheardofElNi~noandchangingweatherpatterns,t heAtlantic hurricaneseasonandstormsurges?Yet,whatexactlyisther oleoftheocean? And,whydowecare?1.1WhystudythePhysicsoftheocean? Theanswerdependsonourinterests,whichdevolvefromouru seofthe ocean.Threebroadthemesareimportant: 1.Wegetfoodfromtheocean.Hencewemaybeinterestedinpro cesses whichinruencetheseajustasfarmersareinterestedinthew eatherand climate.Theoceannotonlyhasweathersuchastemperaturec hanges andcurrents,buttheoceanicweatherfertilizesthesea.Th eatmospheric weatherseldomfertilizeseldsexceptforthesmallamount ofnitrogen xedbylightning. 2.Weusetheocean.Webuildstructuresontheshoreorjusto shore.We usetheoceanfortransport.Weobtainoilandgasbelowtheoc ean.And, weusetheoceanforrecreation,swimming,boating,shing, surng,and diving.Henceweareinterestedinprocessesthatinruencet heseactivities, especiallywaves,winds,currents,andtemperature. 3.Theoceaninruencetheatmosphericweatherandclimate.T heocean inruencethedistributionofrainfall,droughts,roods,re gionalclimate, andthedevelopmentofstorms,hurricanes,andtyphoons.He nceweare interestedinair-seainteractions,especiallytheruxeso fheatandwater acrosstheseasurface,thetransportofheatbytheocean,an dtheinruence oftheoceanonclimateandweatherpatterns. Thesethemesinruenceourselectionoftopicstostudy.Thet opicsthendeterminewhatwemeasure,howthemeasurementsaremade,andtheg eographic areasofinterest.Someprocessesarelocal,suchasthebrea kingofwavesona beach,someareregional,suchastheinruenceoftheNorthPa ciconAlaskan 1

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2 CHAPTER1.AVOYAGEOFDISCOVERY weather,andsomeareglobal,suchastheinruenceoftheocea nonchanging climateandglobalwarming. Ifindeed,thesereasonsforthestudyoftheoceanareimport ant,letsbegin avoyageofdiscovery.Anyvoyageneedsadestination.Whati sours? 1.2Goals Atthemostbasiclevel,Ihopeyou,thestudentswhoarereadi ngthistext, willbecomeawareofsomeofthemajorconceptualschemes(or theories)that formthefoundationofphysicaloceanography,howtheywere arrivedat,and whytheyarewidelyaccepted,howoceanographersachieveor deroutofarandomocean,andtheroleofexperimentinoceanography(topar aphraseShamos, 1995:p.89). Moreparticularly,Iexpectyouwillbeabletodescribephys icalprocesses inruencingtheoceanandcoastalregions:theinteractiono ftheoceanwiththe atmosphere,andthedistributionofoceanicwinds,current s,heatruxes,and watermasses.Thetextemphasizesideasratherthanmathema ticaltechniques. Iwilltrytoanswersuchquestionsas: 1.Whatisthebasisofourunderstandingofphysicsoftheoce an? (a)Whatarethephysicalpropertiesofseawater? (b)Whataretheimportantthermodynamicanddynamicproces sesinruencingtheocean? (c)Whatequationsdescribetheprocessesandhowweretheyd erived? (d)Whatapproximationswereusedinthederivation? (e)Dotheequationshaveusefulsolutions? (f)Howwelldothesolutionsdescribetheprocess?Thatis,w hatisthe experimentalbasisforthetheories? (g)Whichprocessesarepoorlyunderstood?Whicharewellun derstood? 2.Whatarethesourcesofinformationaboutphysicalvariab les? (a)Whatinstrumentsareusedformeasuringeachvariable? (b)Whataretheiraccuracyandlimitations? (c)Whathistoricdataexist? (d)Whatplatformsareused?Satellites,ships,drifters,m oorings? 3.Whatprocessesareimportant?Someimportantprocesswew illstudy include: (a)Heatstorageandtransportintheocean. (b)Theexchangeofheatwiththeatmosphereandtheroleofth eocean inclimate. (c)Windandthermalforcingofthesurfacemixedlayer. (d)Thewind-drivencirculationincludingtheEkmancircul ation,Ekman pumpingofthedeepercirculation,andupwelling.

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1.3.ORGANIZATION 3 (e)Thedynamicsofoceancurrents,includinggeostrophicc urrentsand theroleofvorticity. (f)Theformationofwatertypesandmasses. (g)Thedeepcirculationoftheocean. (h)Equatorialdynamics,ElNi~no,andtheroleoftheoceani nweather. (i)Numericalmodelsofthecirculation. (j)Wavesintheoceanincludingsurfacewaves,inertialosc illations, tides,andtsunamis. (k)Wavesinshallowwater,coastalprocesses,andtidepred ictions. 4.Whatareafewofthemajorcurrentsandwatermassesintheo cean,and whatgovernstheirdistribution? 1.3Organization Beforebeginningavoyage,weusuallytrytolearnaboutthep laceswewill visit.Welookatmapsandweconsulttravelguides.Inthisbo ok,ourguidewill bethepapersandbookspublishedbyoceanographers.Webegi nwithabrief overviewofwhatisknownabouttheocean.Wethenproceedtoa description oftheoceanbasins,fortheshapeoftheseasinruencestheph ysicalprocesses inthewater.Next,westudytheexternalforces,windandhea t,actingon theocean,andtheocean'sresponse.Asweproceed,Ibringin theoryand observationsasnecessary. Bythetimewereachchapter7,wewillneedtounderstandthee quations describingdynamicresponseoftheocean.Soweconsiderthe equationsof motion,theinruenceofearth'srotation,andviscosity.Th isleadstoastudyof wind-drivenoceancurrents,thegeostrophicapproximatio n,andtheusefulness ofconservationofvorticity. Towardtheend,weconsidersomeparticularexamples:thede epcirculation, theequatorialoceanandElNi~no,andthecirculationofpar ticularareasofthe ocean.Nextwelookattheroleofnumericalmodelsindescrib ingtheocean. Attheend,westudycoastalprocesses,waves,tides,wavean dtidalforecasting, tsunamis,andstormsurges.1.4TheBigPicture Theoceanisonepartoftheearthsystem.Itmediatesprocess esinthe atmospherebythetransfersofmass,momentum,andenergyth roughthesea surface.Itreceiveswateranddissolvedsubstancesfromth eland.And,itlays downsedimentsthateventuallybecomerocksonland.Hencea nunderstanding oftheoceanisimportantforunderstandingtheearthasasys tem,especiallyfor understandingimportantproblemssuchasglobalchangeorg lobalwarming.At alowerlevel,physicaloceanographyandmeteorologyareme rging.Theocean providesthefeedbackleadingtoslowchangesintheatmosph ere. Aswestudytheocean,Ihopeyouwillnoticethatweusetheory ,observations,andnumericalmodelstodescribeoceandynamics. Noneissucientby itself

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4 CHAPTER1.AVOYAGEOFDISCOVERY 1.Oceanprocessesarenonlinearandturbulent.Yetwedon't reallyunderstandthetheoryofnon-linear,turbulentrowincomplexbas ins.Theories usedtodescribetheoceanaremuchsimpliedapproximation storeality. 2.Observationsaresparseintimeandspace.Theyprovidear oughdescriptionofthetime-averagedrow,butmanyprocessesinmanyreg ionsare poorlyobserved. 3.Numericalmodelsincludemuch-more-realistictheoreti calideas,theycan helpinterpolateoceanicobservationsintimeandspace,an dtheyareused toforecastclimatechange,currents,andwaves.Nonethele ss,thenumericalequationsareapproximationstothecontinuousanalyt icequations thatdescriberuidrow,theycontainnoinformationaboutro wbetween gridpoints,andtheycannotyetbeusedtodescribefullythe turbulent rowseenintheocean. Bycombiningtheoryandobservationsinnumericalmodelswe avoidsomeof thedicultiesassociatedwitheachapproachusedseparate ly(gure1.1).Continuedrenementsofthecombinedapproachareleadingtoev er-more-precise descriptionsoftheocean.Theultimategoalistoknowtheoc eanwellenough topredictthefuturechangesintheenvironment,including climatechangeor theresponseofsheriestoovershing. Numerical Models Data UnderstandingPrediction Theory Figure1.1Data,numericalmodels,andtheoryareallnecess arytounderstandtheocean. Eventually,anunderstandingoftheocean-atmosphere-lan dsystemwillleadtopredictions offuturestatesofthesystem. Thecombinationoftheory,observations,andcomputermode lsisrelatively new.Fourdecadesofexponentialgrowthincomputingpowerh asmadeavailabledesktopcomputerscapableofsimulatingimportantphy sicalprocessesand oceanicdynamics. Allofuswhoareinvolvedinthesciencesknowthatthecomput erhasbecomeanessentialtoolforresearch...scienticcomputati onhasreached thepointwhereitisonaparwithlaboratoryexperimentandm athematicaltheoryasatoolforresearchinscienceandengineer ing|Langer (1999). Thecombinationoftheory,observations,andcomputermode lsalsoimplies anewwayofdoingoceanography.Inthepast,anoceanographe rwoulddevise

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1.5.FURTHERREADING 5 atheory,collectdatatotestthetheory,andpublishtheres ults.Now,thetasks havebecomesospecializedthatfewcandoitall.Fewexcelin theory,collecting data,andnumericalsimulations.Instead,theworkisdonem oreandmoreby teamsofscientistsandengineers.1.5FurtherReading Ifyouknowlittleabouttheoceanandoceanography,Isugges tyoubegin byreadingMacLeish's(1989)book TheGulfStream:EncountersWiththe BlueGod ,especiallyhisChapter4on\Readingtheocean."Inmyopini on,it isthebestoverall,non-technical,descriptionofhowocea nographerscameto understandtheocean. Youmayalsobenetfromreadingpertinentchaptersfromany introductory oceanographictextbook.ThosebyGross,Pinet,orSegarare especiallyuseful.ThethreetextsproducedbytheOpenUniversityprovide aslightlymore advancedtreatment.Gross, M.GrantandElizabethGross(1996) Oceanography|AViewofEarth. 7thedition.PrenticeHall. MacLeish, William(1989) TheGulfStream:EncountersWiththeBlueGod. HoughtonMiinCompany. Pinet, PaulR.(2006) InvitationtoOceanography. 4ndedition.Jonesand BartlettPublishers. OpenUniversity (2001) OceanCirculation. 2ndedition.PergamonPress. OpenUniversity (1995) Seawater:ItsComposition,PropertiesandBehavior. 2ndedition.PergamonPress. OpenUniversity (1989) Waves,TidesandShallow-WaterProcesses. PergamonPress. Segar, DouglasA.(2007) IntroductiontoOceanSciences. 2ndedition.W.W. Norton.

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6 CHAPTER1.AVOYAGEOFDISCOVERY

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Chapter2TheHistoricalSettingOurknowledgeofoceaniccurrents,winds,waves,andtidesg oesbackthousands ofyears.Polynesiannavigatorstradedoverlongdistances inthePacicasearly as4000 bc (Service,1996).PytheasexploredtheAtlanticfromItalyt oNorway in325 bc .Arabictradersusedtheirknowledgeofthereversingwinds and currentsintheIndianOceantoestablishtraderoutestoChi naintheMiddle AgesandlatertoZanzibarontheAfricancoast.And,theconn ectionbetween tidesandthesunandmoonwasdescribedintheSamavedaofthe IndianVedic periodextendingfrom2000to1400 bc (Pugh,1987).Thoseoceanographers whotendtoacceptastrueonlythatwhichhasbeenmeasuredby instruments, havemuchtolearnfromthosewhoearnedtheirlivingontheoc ean. ModernEuropeanknowledgeoftheoceanbeganwithvoyagesof discoveryby BartholomewDias(1487{1488),ChristopherColumbus(1492 {1494),Vascoda Gama(1497{1499),FerdinandMagellan(1519{1522),andman yothers.They laidthefoundationforglobaltraderoutesstretchingfrom SpaintothePhilippinesintheearly16thcentury.Therouteswerebasedonagoo dworking knowledgeoftradewinds,thewesterlies,andwesternbound arycurrentsinthe AtlanticandPacic(Couper,1983:192{193). TheearlyEuropeanexplorersweresoonfollowedbyscienti cvoyagesof discoveryledby(amongmanyothers)JamesCook(1728{1779) onthe Endeavour Resolution ,and Adventure ,CharlesDarwin(1809{1882)onthe Beagle SirJamesClarkRossandSirJohnRosswhosurveyedtheArctic andAntarcticregionsfromthe Victory ,the Isabella ,andthe Erebus ,andEdwardForbes (1815{1854)whostudiedtheverticaldistributionoflifei ntheocean.Others collectedoceanicobservationsandproducedusefulcharts ,includingEdmond HalleywhochartedthetradewindsandmonsoonsandBenjamin Franklinwho chartedtheGulfStream. Slowshipsofthe19thand20thcenturiesgavewaytosatellit es,drifters, andautonomousinstrumentstowardtheendofthe20thcentur y.Satellites nowobservetheocean,air,andland.Thousandsofdrifterso bservetheupper twokilometersoftheocean.Datafromthesesystems,whenfe dintonumericalmodelsallowsthestudyofearthasasystem.Fortherst time,wecan 7

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8 CHAPTER2.THEHISTORICALSETTING -60o-40o-20o0o20o40o60o180o60o-60o0o120o-120o Figure2.1Examplefromtheeraofdeep-seaexploration:Tra ckofH.M.S. Challenger duringtheBritishChallengerExpedition1872{1876.After Wust(1964). studyhowbiological,chemical,andphysicalsystemsinter acttoinruenceour environment.2.1Denitions Thelonghistoryofthestudyoftheoceanhasledtothedevelo pmentof various,specializeddisciplineseachwithitsowninteres tsandvocabulary.The moreimportantdisciplinesinclude: Oceanography isthestudyoftheocean,withemphasisonitscharacteras anenvironment.Thegoalistoobtainadescriptionsucient lyquantitativeto beusedforpredictingthefuturewithsomecertainty. Geophysics isthestudyofthephysicsoftheearth. PhysicalOceanography isthestudyofphysicalpropertiesanddynamicsof theocean.Theprimaryinterestsaretheinteractionoftheo ceanwiththeatmosphere,theoceanicheatbudget,watermassformation,cu rrents,andcoastal dynamics.PhysicalOceanographyisconsideredbymanytobe asubdiscipline ofgeophysics. GeophysicalFluidDynamics isthestudyofthedynamicsofruidmotionon scalesinruencedbytherotationoftheearth.Meteorologya ndoceanography usegeophysicalruiddynamicstocalculateplanetaryrowe lds. Hydrography isthepreparationofnauticalcharts,includingchartsofo cean depths,currents,internaldensityeldoftheocean,andti des. Earth-systemScience isthestudyofearthasasinglesystemcomprising manyinteractingsubsystemsincludingtheocean,atmosphe re,cryosphere,and biosphere,andchangesinthesesystemsduetohumanactivit y. 2.2ErasofOceanographicExploration Theexplorationoftheseacanbedivided,somewhatarbitrar ily,intovarious eras(Wust,1964).Ihaveextendedhisdivisionsthroughthe endofthe20th century.

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2.2.ERASOFOCEANOGRAPHICEXPLORATION 9 -40o -20o-60o -80o0o20o40o0o20o40o60o-20o-40o-60oStationsAnchoredStations Meteor 19251927 XII XIV XII IX X XI VII VI VII IV II I III V Figure2.2Exampleofasurveyfromtheeraofnationalsystem aticsurveys.Trackofthe R/V Meteor duringtheGermanMeteorExpedition.RedrawnfromWust(196 4). 1.EraofSurfaceOceanography:Earliesttimesto1873.Thee raischaracterizedbysystematiccollectionofmariners'observationsof winds,currents, waves,temperature,andotherphenomenaobservablefromth edeckof sailingships.NotableexamplesincludeHalley'schartsof thetradewinds, Franklin'smapoftheGulfStream,andMatthewFontaineMaur y's PhysicalGeographyoftheSea 2.EraofDeep-SeaExploration:1873{1914.Characterizedb yafew,widerangingoceanographicexpeditionstosurveysurfaceandsu bsurfacecondi-

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10 CHAPTER2.THEHISTORICALSETTING 60o40o20o0o-20o-40o20o0o-20o-40o -60o -80o -100o Figure2.3Examplefromtheeraofnewmethods.Thecruisesof theR/V Atlantis outof WoodsHoleOceanographicInstitution.AfterWust(1964). tions,especiallynearcolonialclaims.Themajorexamplei sthe Challenger Expedition(gure2.1),butalsothe Gazelle and Fram Expeditions. 3.EraofNationalSystematicSurveys:1925{1940.Characte rizedbydetailed surveysofcolonialareas.Examplesinclude Meteor surveysoftheAtlantic (gure2.2),andthe Discovery Expeditions. 4.EraofNewMethods:1947{1956.Characterizedbylongsurv eysusing newinstruments(gure2.3).Examplesincludeseismicsurv eysofthe Atlanticby Vema leadingtoHeezen'smapsofthesearoor. 5.EraofInternationalCooperation:1957{1978.Character izedbymultinationalsurveysofoceanandstudiesofoceanicprocesses.Ex amplesinclude theAtlanticPolarFrontProgram,the norpac cruises,theInternational GeophysicalYearcruises,andtheInternationalDecadeofO ceanExploration(gure2.4).Multishipstudiesofoceanicprocesses include mode polymode norpax ,and jasin experiments.

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2.2.ERASOFOCEANOGRAPHICEXPLORATION 11 Crawford Crawford Crawford Crawford Crawford Crawford Crawford Crawford Chain Discovery IIDiscovery IIDiscovery II Discovery IIAtlantisAtlantisDiscovery II Atlantis Capt. Canepa Capt. Canepa20o40o-0o-20o-40o -60o -80o-60o-40o-20o0o20o40o60oAtlantic I.G.Y. Program 19571959 Figure2.4Examplefromtheeraofinternationalcooperatio n.Sectionsmeasuredbythe InternationalGeophysicalYearAtlanticProgram1957-195 9.AfterWust(1964). 6.EraofSatellites:1978{1995.Characterizedbyglobalsu rveysofoceanic processesfromspace.ExamplesincludeSeasat, noaa 6{10, nimbus {7, Geosat,Topex/Poseidon,and ers {1&2. 7.EraofEarthSystemScience:1995{Characterizedbygloba lstudiesof theinteractionofbiological,chemical,andphysicalproc essesintheocean andatmosphereandonlandusing insitu (whichmeansfrommeasurementsmadeinthewater)andspacedatainnumericalmodels.O ceanic examplesincludetheWorldOceanCirculationExperiment( woce )(gure

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12 CHAPTER2.THEHISTORICALSETTING 60o80o40o0o-20o-40o-60o-80o0o60o100o180o-140o Committed/completedS4 15Atlantic Indian Pacific S4 23 16 1 3 5 7 9 10 11 12 21 1 18 2 20 4 22 8 13 14 15 17 1 3 4 19 17 16 14 13 S4 10 5 6 4 1 2 3 7S 8S9S 9N 8N 7N 10 2 5 21 7 12 14S 17 31 20 18 11 8 9 30 25 28 26 27 29 6 -100o 140o 20o -80o -40o6 20o 11S Figure2.5WorldOceanCirculationExperiment:Tracksofre searchshipsmakingaone-time globalsurveyoftheoceanoftheworld.FromWorldOceanCirc ulationExperiment. 2.5)andTopex/Poseidon(gure2.6),theJointGlobalOcean FluxStudy ( jgofs ),theGlobalOceanDataAssimilationExperiment( godae ),and theSeaWiFS,Aqua,andTerrasatellites. 2.3MilestonesintheUnderstandingoftheOcean Whathavealltheseprogramsandexpeditionstaughtusabout theocean? Let'slookatsomemilestonesinoureverincreasingunderst andingoftheocean -6 0o-4 0o-2 0o0o20o40o60o120o160o-160o-120o-80o-40o180o Figure2.6Examplefromtheeraofsatellites.Topex/Poseid ontracksinthePacic Oceanduringa10-dayrepeatoftheorbit.FromTopex/Poseid onProject.

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2.3.MILESTONESINTHEUNDERSTANDINGOFTHEOCEAN 13 Figure2.7The1786versionofFranklin-FolgermapoftheGul fStream. beginningwiththerstscienticinvestigationsofthe17t hcentury.Initially progresswasslow.Firstcameverysimpleobservationsoffa rreachingimportancebyscientistswhoprobablydidnotconsiderthemselve soceanographers,if thetermevenexisted.Latercamemoredetaileddescription sandoceanographic experimentsbyscientistswhospecializedinthestudyofth eocean. 1685 EdmondHalley,investigatingtheoceanicwindsystemsandc urrents, published\AnHistoricalAccountoftheTradeWinds,andMon soons, observableintheSeasbetweenandneartheTropicks,withan attemptto assignthePhysicalcauseofthesaidWinds" PhilosophicalTransactions 1735 GeorgeHadleypublishedhistheoryforthetradewindsbased onconservationofangularmomentumin\ConcerningtheCauseofth eGeneral Trade-Winds" PhilosophicalTransactions ,39:58-62. 1751 HenriEllismadetherstdeepsoundingsoftemperatureinth etropics, ndingcoldwaterbelowawarmsurfacelayer,indicatingthe watercame fromthepolarregions. 1769 BenjaminFranklin,aspostmaster,madetherstmapoftheGu lfStream usinginformationfrommailshipssailingbetweenNewEngla ndandEnglandcollectedbyhiscousinTimothyFolger(gure2.7). 1775 Laplace'spublishedhistheoryoftides.

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14 CHAPTER2.THEHISTORICALSETTING 1800 CountRumfordproposedameridionalcirculationoftheocea nwithwater sinkingnearthepolesandrisingneartheEquator. 1847 MatthewFontaineMaurypublishedhisrstchartofwindsand currents basedonshipslogs.Mauryestablishedthepracticeofinter nationalexchangeofenvironmentaldata,tradinglogbooksformapsand chartsderivedfromthedata. 1872{1876 ChallengerExpeditionmarksthebeginningofthesystemati cstudy ofthebiology,chemistry,andphysicsoftheoceanofthewor ld. 1885 PillsburymadedirectmeasurementsoftheFloridaCurrentu singcurrent metersdeployedfromashipmooredinthestream. 1903 FoundingoftheMarineBiologicalLaboratoryoftheUnivers ityofCalifornia.ItlaterbecametheScrippsInstitutionofOceanogr aphy. 1910{1913 VilhelmBjerknespublished DynamicMeteorologyandHydrography whichlaidthefoundationofgeophysicalruiddynamics.Ini the developedtheideaoffronts,thedynamicmeter,geostrophi crow,air-sea interaction,andcyclones. 1930 FoundingoftheWoodsHoleOceanographicInstitution. 1942 Publicationof Theocean bySverdrup,Johnson,andFleming,acomprehensivesurveyofoceanographicknowledgeuptothattime. PostWW2 Theneedtodetectsubmarinesledthenaviesoftheworldto greatlyexpandtheirstudiesofthesea.Thisledtothefound ingof oceanographydepartmentsatstateuniversities,includin gOregonState, TexasA&MUniversity,UniversityofMiami,andUniversityo fRhodeIsland,andthefoundingofnationaloceanlaboratoriessucha sthevarious InstitutesofOceanographicScience. 1947{1950 Sverdrup,Stommel,andMunkpublishtheirtheoriesofthewi nddrivencirculationoftheocean.Togetherthethreepapersl aythefoundationforourunderstandingoftheocean'scirculation. 1949 StartofCaliforniaCooperativeFisheriesInvestigationo ftheCalifornia Current.Themostcompletestudyeverundertakenofacoasta lcurrent. 1952 CromwellandMontgomeryrediscovertheEquatorialUndercu rrentinthe Pacic. 1955 BruceHamonandNeilBrowndeveloptheCTDformeasuringcond uctivityandtemperatureasafunctionofdepthintheocean. 1958 Stommelpublisheshistheoryforthedeepcirculationofthe ocean. 1963 SippicanCorporation(TimFrancis,WilliamVanAllenClark ,Graham Campbell,andSamFrancis)inventstheExpendableBathyThe rmograph xbt nowperhapsthemostwidelyusedoceanographicinstrumentd eployed fromships. 1969 KirkBryanandMichaelCoxdeveloptherstnumericalmodelo fthe oceaniccirculation.

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2.4.EVOLUTIONOFSOMETHEORETICALIDEAS 15 1978 nasa launchestherstoceanographicsatellite,Seasat.Thepro jectdevelopedtechniquesusedbygenerationsofremotessensings atellites. 1979{1981 TerryJoyce,RobPinkel,LloydRegier,F.RoweandJ.W.Young developtechniquesleadingtotheacoustic-dopplercurren tprolerformeasuringocean-surfacecurrentsfrommovingships,aninstru mentwidely usedinoceanography. 1988 nasa EarthSystemScienceCommitteeheadedbyFrancisBretherto n outlineshowallearthsystemsareinterconnected,thusbre akingdownthe barriersseparatingtraditionalsciencesofastrophysics ,ecology,geology, meteorology,andoceanography. 1991 WallyBroeckerproposesthatchangesinthedeepcirculatio noftheocean modulatetheiceages,andthatthedeepcirculationintheAt lanticcould collapse,plungingthenorthernhemisphereintoanewiceag e. 1992 RussDavisandDougWebbinventtheautonomous,pop-updrift erthat continuouslymeasurescurrentsatdepthsto2km. 1992 nasa and cnes developandlaunchTopex/Poseidon,asatellitethatmaps oceansurfacecurrents,waves,andtideseverytendays,rev olutionizingour understandingofoceandynamicsandtides. 1993 Topex/Poseidonscience-teammemberspublishrstaccurat eglobalmaps ofthetides. Moreinformationonthehistoryofphysicaloceanographyca nbefoundinAppendixAofW.S.vonArx(1962): AnIntroductiontoPhysicalOceanography Datacollectedfromthecenturiesofoceanicexpeditionsha vebeenused todescribetheocean.Mostoftheworkwenttowarddescribin gthesteady stateoftheocean,itscurrentsfromtoptobottom,anditsin teractionwith theatmosphere.Thebasicdescriptionwasmostlycompleteb ytheearly1970s. Figure2.8showsanexamplefromthattime,thesurfacecircu lationoftheocean. Morerecentworkhassoughttodocumentthevariabilityofoc eanicprocesses, toprovideadescriptionoftheoceansucienttopredictann ualandinterannual variability,andtounderstandtheroleoftheoceaningloba lprocesses. 2.4EvolutionofsomeTheoreticalIdeas Atheoreticalunderstandingofoceanicprocessesisbasedo nclassicalphysics coupledwithanevolvingunderstandingofchaoticsystemsi nmathematicsand theapplicationtothetheoryofturbulence.Thedatesgiven belowareapproximate.19thCentury Developmentofanalytichydrodynamics.Lamb's Hydrodynamics isthepinnacleofthiswork.Bjerknesdevelopsgeostrophic method widelyusedinmeteorologyandoceanography. 1925{40 Developmentoftheoriesforturbulencebasedonaerodynami csand mixing-lengthideas.WorkofPrandtlandvonKarman.

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16 CHAPTER2.THEHISTORICALSETTING Arctic Circle East Australia Alaska California Gulf Stream Labrador Florida Equator Brazil Peru or Humboldt Greenland Guinea Somali Benguala Agulhas Canaries Norway Oyeshio North Pacific Kuroshio North Equatorial Equatorial Countercurrent South Equatorial West wind drift or Antarctic Circumpolar West wind drift or Antarctic Circumpolar Falkland S. Eq. C. Eq.C.C. N. Eq. C. S. Eq. C. West Australia Murman Irminger NorthAtlanticdrift N. Eq. C.60o45o30o15o-15o-30o-45o-60o0o C.C.warm currents N. north S. south Eq. equatorialcool currents C. current C.C. counter current Figure2.8Thetime-averaged,surfacecirculationoftheoc eanduringnorthernhemisphere winterdeducedfromacenturyofoceanographicexpeditions .AfterTolmazin(1985:16). 1940{1970 Renementoftheoriesforturbulencebasedonstatisticalc orrelationsandtheideaofisotropichomogeneousturbulence.Boo ksbyBatchelor(1967),Hinze(1975),andothers. 1970{ Numericalinvestigationsofturbulentgeophysicalruiddy namicsbased onhigh-speeddigitalcomputers. 1985{ Mechanicsofchaoticprocesses.Theapplicationtohydrody namicsis justbeginning.Mostmotionintheatmosphereandoceanmayb einherentlyunpredictable. 2.5TheRoleofObservationsinOceanography Thebrieftouroftheoreticalideassuggeststhatobservati onsareessential forunderstandingtheocean.Thetheorydescribingaconvec ting,wind-forced, turbulentruidinarotatingcoordinatesystemhasneverbee nsucientlywell knownthatimportantfeaturesoftheoceaniccirculationco uldbepredicted beforetheywereobserved.Inalmostallcases,oceanograph ersresorttoobservationstounderstandoceanicprocesses. Atrstglance,wemightthinkthatthenumerousexpeditions mounted since1873wouldgiveagooddescriptionoftheocean.Theres ultsareindeed impressive.Hundredsofexpeditionshaveextendedintoall ocean.Yet,much oftheoceanispoorlyexplored. Bytheyear2000,mostareasoftheoceanwillhavebeensample dfrom toptobottomonlyonce.Someareas,suchastheAtlantic,wil lhavebeen sparselysampledthreetimes:duringtheInternationalGeo physicalYearin 1959,duringtheGeochemicalSectionscruisesintheearly1 970s,andduring theWorldOceanCirculationExperimentfrom1991to1996.Al lareaswillbe

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2.5.THEROLEOFOBSERVATIONSINOCEANOGRAPHY 17 vastlyundersampled.Thisisthesamplingproblem(Seeboxo nnextpage). Oursamplesoftheoceanareinsucienttodescribetheocean wellenoughto predictitsvariabilityanditsresponsetochangingforcin g. Lackofsucient samplesisthelargestsourceoferrorinourunderstandingo ftheocean. Thelackofobservationshasledtoaveryimportantandwides preadconceptualerror: \Theabsenceofevidencewastakenasevidenceofabsence." Thegreat dicultyofobservingtheoceanmeantthatwhenaphenomenon wasnot observed,itwasassumeditwasnotpresent.Themoreoneisab leto observetheocean,themorethecomplexityandsubtletythat appears| Wunsch(2002a). Asaresult,ourunderstandingoftheoceanisoftentoosimpl etobecorrect. SelectingOceanicDataSets Muchoftheexistingoceanicdatahavebeen organizedintolargedatasets.Forexample,satellitedata areprocessedand distributedbygroupsworkingwith nasa .Datafromshipshavebeencollected andorganizedbyothergroups.Oceanographersnowrelymore andmoreon suchcollectionsofdataproducedbyothers. Theuseofdataproducedbyothersintroducesproblems:i)Ho waccurate arethedataintheset?ii)Whatarethelimitationsofthedat aset?And,iii) Howdoesthesetcomparewithothersimilarsets?Anyonewhou sespublicor privatedatasetsiswisetoobtainanswerstosuchquestions Ifyouplantousedatafromothers,herearesomeguidelines. 1. Usewelldocumenteddatasets .Doesthedocumentationcompletelydescribethesourcesoftheoriginalmeasurements,allstepsu sedtoprocess thedata,andallcriteriausedtoexcludedata?Doesthedata setinclude versionnumberstoidentifychangestotheset? 2. Usevalidateddata .Hasaccuracyofdatabeenwelldocumented?Was accuracydeterminedbycomparingwithdierentmeasuremen tsofthe samevariable?Wasvalidationglobalorregional? 3. Usesetsthathavebeenusedbyothersandreferencedinscien ticpapers Somedatasetsarewidelyusedforgoodreason.Thosewhoprod ucedthe setsusedthemintheirownpublishedworkandotherstrustth edata. 4. Conversely,don'tuseadatasetjustbecauseitishandy .Canyoudocumentthesourceoftheset?Forexample,manyversionsofthe digital, 5-minutemapsofthesearoorarewidelyavailable.Somedate backto therstsetsproducedbytheU.S.DefenseMappingAgency,ot hersare fromthe etopo-5 set.Don'trelyonacolleague'sstatementaboutthe source.Findthedocumentation.Ifitismissing,ndanothe rdataset. DesigningOceanicExperiments Observationsareexceedinglyimportant foroceanography,yetobservationsareexpensivebecauses hiptimeandsatellitesareexpensive.Asaresult,oceanographicexperiment smustbecarefully planned.Whilethedesignofexperimentsmaynottwellwith inanhistorical

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18 CHAPTER2.THEHISTORICALSETTING SamplingError Samplingerroristhelargestsourceoferrorinthegeoscien ces.Itiscaused byasetofsamplesnotrepresentingthepopulationofthevar iablebeing measured.Apopulationisthesetofallpossiblemeasuremen ts,andasampleisthesampledsubsetofthepopulation.Weassumeeachme asurement isperfectlyaccurate. Todetermineifyourmeasurementhasasamplingerror,youmu strst completelyspecifytheproblemyouwishtostudy.Thisdene sthepopulation.Then,youmustdetermineifthesamplesrepresentth epopulation. Bothstepsarenecessary. Supposeyourproblemistomeasuretheannual-meansea-surf acetemperatureoftheoceantodetermineifglobalwarmingisoccur ring.Forthis problem,thepopulationisthesetofallpossiblemeasureme ntsofsurface temperature,inallregionsinallmonths.Ifthesamplemean istoequal thetruemean,thesamplesmustbeuniformlydistributedthr oughoutthe yearandoveralltheareaoftheocean,andsucientlydenset oincludeall importantvariabilityintimeandspace.Thisisimpossible .Shipsavoid stormyregionssuchashighlatitudesinwinter,soshipsamp lestendnotto representthepopulationofsurfacetemperatures.Satelli tesmaynotsample uniformlythroughoutthedailycycle,andtheymaynotobser vetemperatureathighlatitudesinwinterbecauseofpersistentcloud s,althoughthey tendtosampleuniformlyinspaceandthroughouttheyearinm ostregions. Ifdailyvariabilityissmall,thesatellitesampleswillbe morerepresentative ofthepopulationthantheshipsamples. Fromtheabove,itshouldbeclearthatoceanicsamplesrarel yrepresent thepopulationwewishtostudy.Wealwayshavesamplingerro rs. Indeningsamplingerror,wemustclearlydistinguishbetw eeninstrumenterrorsandsamplingerrors.Instrumenterrorsareduet otheinaccuracyoftheinstrument.Samplingerrorsareduetoafailur etomake ameasurement.Considertheexampleabove:thedeterminati onofmean sea-surfacetemperature.Ifthemeasurementsaremadebyth ermometers onships,eachmeasurementhasasmallerrorbecausethermom etersarenot perfect.Thisisaninstrumenterror.Iftheshipsavoidshig hlatitudesin winter,theabsenceofmeasurementsathighlatitudeinwint erisasampling error. MeteorologistsdesigningtheTropicalRainfallMappingMi ssionhave beeninvestigatingthesamplingerrorinmeasurementsofra in.Theirresults aregeneralandmaybeappliedtoothervariables.Foragener aldescription oftheproblemseeNorth&Nakamoto(1989). chapter,perhapsthetopicmeritsafewbriefcommentsbecau seitisseldom mentionedinoceanographictextbooks,althoughitispromi nentlydescribedin textsforotherscienticelds.Thedesignofexperimentsi sparticularlyimportantbecausepoorlyplannedexperimentsleadtoambiguousr esults,theymay measurethewrongvariables,ortheymayproducecompletely uselessdata.

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2.5.THEROLEOFOBSERVATIONSINOCEANOGRAPHY 19 Therstandmostimportantaspectofthedesignofanyexperi mentisto determine why youwishtomakeameasurementbeforedecidinghowyouwill makethemeasurementorwhatyouwillmeasure. 1.Whatisthepurposeoftheobservations?Doyouwishtotest hypotheses ordescribeprocesses? 2.Whataccuracyisrequiredoftheobservation?3.Whatresolutionintimeandspaceisrequired?Whatisthed urationof measurements? Consider,forexample,howthepurposeofthemeasurementch angeshowyou mightmeasuresalinityortemperatureasafunctionofdepth : 1.Ifthepurposeistodescribewatermassesinanoceanbasin ,thenmeasurementswith20{50mverticalspacingand50{300kmhorizontal spacing, repeatedonceper20{50yearsindeepwaterarerequired. 2.Ifthepurposeistodescribeverticalmixingintheopeneq uatorialPacic,then0.5{1.0mmverticalspacingand50{1000kmspacin gbetween locationsrepeatedonceperhourformanydaysmayberequire d. Accuracy,Precision,andLinearity Whileweareonthetopicofexperiments,nowisagoodtimetointroducethreeconceptsneededt hroughoutthe bookwhenwediscussexperiments:precision,accuracy,and linearityofameasurement. Accuracy isthedierencebetweenthemeasuredvalueandthetruevalu e. Precision isthedierenceamongrepeatedmeasurements. Thedistinctionbetweenaccuracyandprecisionisusuallyi llustratedbythe simpleexampleofringarireatatarget.Accuracyistheave ragedistance fromthecenterofthetargettothehitsonthetarget.Precis ionistheaverage distancebetweenthehits.Thus,tenrireshotscouldbeclus teredwithinacircle 10cmindiameterwiththecenteroftheclusterlocated20cmf romthecenter ofthetarget.Theaccuracyisthen20cm,andtheprecisionis roughly5cm. Linearity requiresthattheoutputofaninstrumentbealinearfunctio nof theinput.Nonlineardevicesrectifyvariabilitytoaconst antvalue.Soanonlinearresponseleadstowrongmeanvalues.Non-linearityc anbeasimportant asaccuracy.Forexample,let Output = Input +0 : 1( Input ) 2 Input = a sin !t then Output = a sin !t +0 : 1( a sin !t ) 2 Output = Input + 0 : 1 2 a 2 0 : 1 2 a 2 cos2 !t Notethatthemeanvalueoftheinputiszero,yettheoutputof thisnonlinearinstrumenthasameanvalueof0 : 05 a 2 plusanequallylargetermat

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20 CHAPTER2.THEHISTORICALSETTING twicetheinputfrequency.Ingeneral,if input hasfrequencies 1 and 2 ,then output ofanon-linearinstrumenthasfrequencies 1 2 .Linearityofan instrumentisespeciallyimportantwhentheinstrumentmus tmeasurethemean valueofaturbulentvariable.Forexample,werequirelinea rcurrentmeterswhen measuringcurrentsneartheseasurfacewherewindandwaves producealarge variabilityinthecurrent.Sensitivitytoothervariablesofinterest. Errorsmaybecorrelatedwith othervariablesoftheproblem.Forexample,measurementso fconductivity aresensitivetotemperature.So,errorsinthemeasurement oftemperaturein salinometersleadstoerrorsinthemeasuredvaluesofcondu ctivityorsalinity. 2.6ImportantConcepts Fromtheabove,Ihopeyouhavelearned: 1.Theoceanisnotwellknown.Whatweknowisbasedondatacol lected fromonlyalittlemorethanacenturyofoceanographicexped itionssupplementedwithsatellitedatacollectedsince1978. 2.Thebasicdescriptionoftheoceanissucientfordescrib ingthetimeaveragedmeancirculationoftheocean,andrecentworkisbe ginningto describethevariability. 3.Observationsareessentialforunderstandingtheocean. Fewprocesses havebeenpredictedfromtheorybeforetheywereobserved. 4.Lackofobservationshasledtoconceptualpicturesofoce anicprocesses thatareoftentoosimpliedandoftenmisleading. 5.Oceanographersrelymoreandmoreonlargedatasetsprodu cedbyothers. Thesetshaveerrorsandlimitationswhichyoumustundersta ndbefore usingthem. 6.Theplanningofexperimentsisatleastasimportantascon ductingthe experiment. 7.Samplingerrorsarisewhentheobservations,thesamples ,arenotrepresentativeoftheprocessbeingstudied.Samplingerrorsare thelargest sourceoferrorinoceanography. 8.Almostallourobservationsoftheoceannowcomefromsate llites,drifters, andautonomousinstruments.Fewerandfewerobservationsc omefrom shipsatsea.

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Chapter3ThePhysicalSettingEarthisanoblateellipsoid,anellipserotatedaboutitsmi noraxis,withan equatorialradiusof R e =6 ; 378 : 1349km(West,1982)slightlygreaterthan thepolarradiusof R p =6 ; 356 : 7497km.Thesmallequatorialbulgeisdueto earth'srotation. Distancesoneartharemeasuredinmanydierentunits,them ostcommon aredegreesoflatitudeorlongitude,meters,miles,andnau ticalmiles. Latitude istheanglebetweenthelocalverticalandtheequatorialpl ane.Ameridianisthe intersectionatearth'ssurfaceofaplaneperpendicularto theequatorialplane andpassingthroughearth'saxisofrotation. Longitude istheanglebetween thestandardmeridianandanyothermeridian,wherethestan dardmeridianis theonethatpassesthroughapointattheRoyalObservatorya tGreenwich, England.ThuslongitudeismeasuredeastorwestofGreenwic h. Adegreeoflatitudeisnotthesamelengthasadegreeoflongi tudeexcept attheequator.Latitudeismeasuredalonggreatcircleswit hradius R ,where R isthemeanradiusofearth.Longitudeismeasuredalongcirc leswithradius R cos ,where islatitude.Thus1 latitude=111km,and1 longitude =111cos km. Becausedistanceindegreesoflongitudeisnotconstant,oc eanographers measuredistanceonmapsusingdegreesoflatitude. Nauticalmilesandmetersareconnectedhistoricallytothe sizeofearth. GabrielMoutonproposedin1670adecimalsystemofmeasurem entbasedon thelengthofanarcthatisoneminuteofagreatcircleofeart h.Thiseventually becamethenauticalmile.Mouton'sdecimalsystemeventual lybecamethe metricsystembasedonadierentunitoflength,themeter,w hichwasoriginally intendedtobeoneten-millionththedistancefromtheEquat ortothepolealong theParismeridian.Althoughthetiebetweennauticalmiles ,meters,andearth's radiuswassoonabandonedbecauseitwasnotpractical,thea pproximationsare verygood.Forexample,earth'spolarcircumferenceisappr oximately40,008 km.Thereforeoneten-millionthofaquadrantis1.0002m.Si milarly,anautical mileshouldbe1.8522km,whichisveryclosetotheocialde nitionofthe internationalnauticalmile :1nm 1.8520km. 21

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22 CHAPTER3.THEPHYSICALSETTING -80o-40o0o40o-90o-60o-3000030o60o90o -4000 -3000 -1000 -200 0 Figure3.1TheAtlanticOceanviewedwithanEckertVIequalareaprojection.Depths,in meters,arefromthe etopo 30 0 dataset.The200mcontouroutlinescontinentalshelves. 3.1OceanandSeas Thereisonlyoneocean.Itisdividedintothreenamedpartsb yinternational agreement:theAtlantic,Pacic,andIndianocean(Interna tionalHydrographic Bureau,1953).Seas,whicharepartoftheocean,aredenedi nseveralways.I considertwo. TheAtlanticOcean extendsnorthwardfromAntarcticaandincludesall oftheArcticSea,theEuropeanMediterranean,andtheAmeri canMediterraneanmorecommonlyknownastheCaribbeansea(gure3.1). Theboundary betweentheAtlanticandIndianOceanisthemeridianofCape Agulhas(20 E). TheboundarybetweentheAtlanticandPacicisthelineform ingtheshortestdistancefromCapeHorntotheSouthShetlandIslands.In thenorth,the ArcticSeaispartoftheAtlanticOcean,andtheBeringStrai tistheboundary betweentheAtlanticandPacic. ThePacicOcean extendsnorthwardfromAntarcticatotheBeringStrait (gure3.2).TheboundarybetweenthePacicandIndianOcea nfollowsthe

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3.2.DIMENSIONSOFTHEOCEAN 23 120o160o-160o-120o-80o-90o-60o-30o0o30o60o90o -4000 -3000 -1000 -200 0 Figure3.2ThePacicOceanviewedwithanEckertVIequal-ar eaprojection.Depths,in meters,arefromthe etopo 30 0 dataset.The200mcontouroutlinescontinentalshelves. linefromtheMalayPeninsulathroughSumatra,Java,Timor, AustraliaatCape Londonderry,andTasmania.FromTasmaniatoAntarcticaiti sthemeridian ofSouthEastCapeonTasmania147 E. TheIndianOcean extendsfromAntarcticatothecontinentofAsiaincludingtheRedSeaandPersianGulf(gure3.3).Someauthor susethename SouthernOceantodescribetheoceansurroundingAntarctic a. MediterraneanSeas aremostlysurroundedbyland.Bythisdenition, theArcticandCaribbeanSeasarebothMediterraneanSeas,t heArcticMediterraneanandtheCaribbeanMediterranean. MarginalSeas aredenedbyonlyanindentationinthecoast.TheArabian SeaandSouthChinaSeaaremarginalseas.3.2Dimensionsoftheocean Theoceanandseascover70.8%ofthesurfaceofearth,whicha mountsto 361,254,000km 2 .Theareasofthenamedpartsvaryconsiderably(table3.1).

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24 CHAPTER3.THEPHYSICALSETTING 40o80o120o-90o-60o-30o0o30o -4000 -3000 -1000 -200 0 Figure3.3TheIndianOceanviewedwithanEckertVIequal-ar eaprojection.Depths,in meters,arefromthe etopo 30 0 dataset.The200mcontouroutlinescontinentalshelves. Oceanicdimensionsrangefromaround1500kmfortheminimum widthof theAtlantictomorethan13,000kmforthenorth-southexten toftheAtlantic andthewidthofthePacic.Typicaldepthsareonly3{4km.So horizontal dimensionsofoceanbasinsare1,000timesgreaterthanthev erticaldimension. AscalemodelofthePacic,thesizeofan8 : 5 11insheetofpaper,would havedimensionssimilartothepaper:awidthof10,000kmsca lesto10in,and adepthof3kmscalesto0.003in,thetypicalthicknessofapi eceofpaper. Becausetheoceanissothin,cross-sectionalplotsofocean basinsmusthavea greatlyexaggeratedverticalscaletobeuseful.Typicalpl otshaveaverticalscale thatis200timesthehorizontalscale(gure3.4).Thisexag gerationdistorts ourviewoftheocean.Theedgesoftheoceanbasins,theconti nentalslopes, arenotsteepclisasshowninthegureat41 Wand12 E.Rather,theyare gentleslopesdroppingdown1meterforevery20metersinthe horizontal. Thesmallratioofdepthtowidthoftheoceanbasinsisveryim portant forunderstandingoceancurrents.Verticalvelocitiesmus tbemuchsmaller Table3.1SurfaceAreaoftheocean y PacicOcean181 : 34 10 6 km 2 AtlanticOcean106 : 57 10 6 km 2 IndianOcean74 : 12 10 6 km 2 y FromMenardandSmith(1966)

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3.3.SEA-FLOORFEATURES 25 Depth (km) -45o-30o-15o0o15o -6 -4 -2 0 Longitude 6 km6 km-45o-30o-15o0o15o Figure3.4Cross-sectionofthesouthAtlanticalong25 Sshowingthecontinentalshelf oshoreofSouthAmerica,aseamountnear35 W,themid-AtlanticRidgenear14 W,the WalvisRidgenear6 E,andthenarrowcontinentalshelfoSouthAfrica. Upper Vertical exaggerationof180:1. Lower Verticalexaggerationof30:1.Ifshownwithtrueaspectrat io, theplotwouldbethethicknessofthelineattheseasurfacei nthelowerplot. thanhorizontalvelocities.Evenoverdistancesofafewhun dredkilometers,the verticalvelocitymustbelessthan1%ofthehorizontalvelo city.Iwillusethis informationlatertosimplifytheequationsofmotion. Therelativelysmallverticalvelocitieshavegreatinruen ceonturbulence. Threedimensionalturbulenceisfundamentallydierentth antwo-dimensional turbulence.Intwodimensions,vortexlinesmustalwaysbev ertical,andthere canbelittlevortexstretching.Inthreedimensions,vorte xstretchingplaysa fundamentalroleinturbulence.3.3Sea-FloorFeatures Earth'srockysurfaceisdividedintotwotypes:oceanic,wi thathindense crustabout10kmthick,andcontinental,withathicklightc rustabout40km thick.Thedeep,lightercontinentalcrustroatshigheront hedensermantle thandoestheoceaniccrust,andthemeanheightofthecrustr elativetosea levelhastwodistinctvalues:continentshaveameanelevat ionof1100m,the oceanhasameandepthof-3400m(gure3.5). Thevolumeofthewaterintheoceanexceedsthevolumeoftheo ceanbasins, andsomewaterspillsoverontothelowlyingareasofthecont inents.These shallowseasarethecontinentalshelves.Some,suchastheS outhChinaSea, aremorethan1100kmwide.Mostarerelativelyshallow,with typicaldepths of50{100m.Afewofthemoreimportantshelvesare:theEastC hinaSea,the BeringSea,theNorthSea,theGrandBanks,thePatagonianSh elf,theArafura SeaandGulfofCarpentaria,andtheSiberianShelf.Theshal lowseashelp dissipatetides,theyareoftenareasofhighbiologicalpro ductivity,andtheyare usuallyincludedintheexclusiveeconomiczoneofadjacent countries.

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26 CHAPTER3.THEPHYSICALSETTING 0-10,000 -8,000 -6,000 -4,000 -2,000 2,000 4,000 6,000 8,000 10,000Eleva tio n ( meters) 0%10%20%30%40%50% 60%70%80%90%100% Cumulative (% Are a) 0.00%0.50%1.00%1.50%2.00%2.50% 3.00%3.50%4.00%4.50%5.00% Frequency (% Area) Figure3.5Histogramofheightoflandanddepthoftheseaasp ercentageofareaofearth in100mintervals,showingthecleardistinctionbetweenco ntinentsandsearoor.The cumulativefrequencycurveistheintegralofthehistogram .Thecurvesarecalculatedfrom the etopo 2datasetbyGeorgeSharmanofthe noaa NationalGeophysicalDataCenter. Thecrustisbrokenintolargeplatesthatmoverelativetoea chother.New crustiscreatedatthemid-oceanridges,andoldcrustislos tattrenches.The relativemotionofcrust,duetoplatetectonics,producest hedistinctivefeatures ofthesearoorsketchedingure3.6,includingmid-oceanri dges,trenches,isShore High Water Low Water Sea Level OCEAN SHELF(Gravel,SandAv slope1 in 500)SLOPE(Mudav slope1 in 20)CONTINENTRISEBASIN MID-OCEAN RIDGE DEEP SEA (Clay & Oozes) Mineral OrganicSEAMOUNTTRENCH ISLAND ARC Figure3.6Schematicsectionthroughtheoceanshowingprin cipalfeaturesofthesearoor. Notethattheslopeofthesearoorisgreatlyexaggeratedint hegure.

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3.3.SEA-FLOORFEATURES 27 landarcs,andbasins.Thenamesofthesub-seafeatureshave beendenedby theInternationalHydrographicOrganization(1953),andt hefollowingdenitionsaretakenfromSverdrup,Johnson,andFleming(1942), Shepard(1963), andDietrichetal.(1980). Basins aredeepdepressionsofthesearoorofmoreorlesscircularo roval form. Canyons arerelativelynarrow,deepfurrowswithsteepslopes,cutt ingacross thecontinentalshelfandslope,withbottomsslopingconti nuouslydownward. Continentalshelves arezonesadjacenttoacontinent(oraroundanisland) andextendingfromthelow-waterlinetothedepth,usuallya bout120m,where thereisamarkedorrathersteepdescenttowardgreatdepths .(gure3.7) Continentalslopes arethedeclivitiesseawardfromtheshelfedgeintogreater depth. Plains areveryratsurfacesfoundinmanydeepoceanbasins. Ridges arelong,narrowelevationsofthesearoorwithsteepsidesa ndrough topography. Figure3.7Anexampleofacontinentalshelf,theshelfosho reofMontereyCalifornia showingtheMontereyandothercanyons.Canyonsarecommono nshelves,oftenextending acrosstheshelfanddownthecontinentalslopetodeepwater .FigurecopyrightMonterey BayAquariumResearchInstitute( mbari ).

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28 CHAPTER3.THEPHYSICALSETTING 21.4o21.3o21.2o21.1o21.0o20.9o20.8o163.0o163.1o163.2o163.3o163.4o163.5o 163.6o40 20 14 40 40 20 40 48 30 30 Figure3.8Anexampleofaseamount,theWildeGuyot.Aguyoti saseamountwitharat topcreatedbywaveactionwhentheseamountextendedaboves ealevel.Astheseamountis carriedbyplatemotion,itgraduallysinksdeeperbelowsea level.Thedepthwascontoured fromechosounderdatacollectedalongtheshiptrack(thins traightlines)supplemented withside-scansonardata.Depthsareinunitsof100m.FromW illiamSager,TexasA&M University. Seamounts areisolatedorcomparativelyisolatedelevationsrising1 000mor morefromthesearoorandwithsmallsummitarea(gure3.8). Sills arethelowpartsoftheridgesseparatingoceanbasinsfromo neanother orfromtheadjacentsearoor. Trenches arelong,narrow,anddeepdepressionsofthesearoor,withr elativelysteepsides(gure3.9). Sub-seafeaturesstronglyinruencestheoceancirculation .Ridgesseparate deepwatersoftheoceanintodistinctbasins.Waterdeepert hanthesillbetween twobasinscannotmovefromonetotheother.Tensofthousand sofseamounts arescatteredthroughouttheoceanbasins.Theyinterrupto ceancurrents,and produceturbulenceleadingtoverticalmixingintheocean.

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3.4.MEASURINGTHEDEPTHOFTHEOCEAN 29 Longitude (West)Latitude (North)-5000 -5000 -5000 -5000 -4000 -4000 -3000 -3000 -2000 -2000 -1000 -1000 -500 -500 -200 -200 -50 -50 -200 0 0-50 -200 -500 0 -6000 167o 165o 163o 161o 159o 157o155o 51o 52o 53o 54o 55o 56o57o 51 o52 o53 o54 o55 o56 o 57 oLatitude (North) -6000 -4000 -2000 0 Depth (m) Section A:B A BAlaskan PeninsulaBering SeaAleutian TrenchPacific Ocean Figure3.9Anexampleofatrench,theAleutianTrench;anisl andarc,theAlaskanPeninsula; andacontinentalshelf,theBeringSea.Theislandarciscom posedofvolcanosproduced whenoceaniccrustcarrieddeepintoatrenchmeltsandrises tothesurface. Top: Mapof theAleutianregionoftheNorthPacic. Bottom: Cross-sectionthroughtheregion. 3.4MeasuringtheDepthoftheOcean Thedepthoftheoceanisusuallymeasuredtwoways:1)usinga coustic echo-soundersonships,or2)usingdatafromsatellitealti meters. EchoSounders Mostmapsoftheoceanarebasedonmeasurementsmade byechosounders.Theinstrumenttransmitsaburstof10{30k Hzsoundand listensfortheechofromthesearoor.Thetimeintervalbetw eentransmission ofthepulseandreceptionoftheecho,whenmultipliedbythe velocityofsound, givestwicethedepthoftheocean(gure3.10). ThersttransatlanticechosoundingsweremadebytheU.S.N avyDestroyer Stewart in1922.Thiswasquicklyfollowedbytherstsystematicsur veyofan oceanbasin,madebytheGermanresearchandsurveyship Meteor duringits

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30 CHAPTER3.THEPHYSICALSETTING Transmitter transducer Receiver transducer Oscillator Electromechanical drive Electronics Bottom Transmitter transducer Receiver transducer AmplifierOscillator Time-interval Measurment, Display, Recording Strip chart Surface Contact bankZero-contactswitchSlidingcontact Endlessribbon 33 kHz sound pulse Figure3.10 Left: Echosoundersmeasuredepthoftheoceanbytransmittingpul sesofsound andobservingthetimerequiredtoreceivetheechofromtheb ottom. Right: Thetimeis recordedbyasparkburningamarkonaslowlymovingrollofpa per.AfterDietrichetal. (1980:124).expeditiontothesouthAtlanticfrom1925to1927.Sincethe n,oceanographic andnavalshipshaveoperatedechosoundersalmostcontinuo uslywhileatsea. Millionsofmilesofship-trackdatarecordedonpaperhaveb eendigitizedto producedatabasesusedtomakemaps.Thetracksarenotwelld istributed. Trackstendtobefarapartinthesouthernhemisphere,evenn earAustralia (gure3.11)andclosertogetherinwellmappedareassuchas theNorthAtlantic. Echosoundersmakethemostaccuratemeasurementsofoceand epth.Their accuracyis 1%. SatelliteAltimetry Gapsinourknowledgeofoceandepthsbetweenship trackshavenowbeenlledbysatellite-altimeterdata.Alt imetersprolethe shapeoftheseasurface,anditsshapeisverysimilartothes hapeofthesea roor(TapleyandKim,2001;CazenaveandRoyer,2001;Sandwe llandSmith, 2001).Toseethis,wemustrstconsiderhowgravityinruenc essealevel. TheRelationshipBetweenSeaLevelandtheOcean'sDepth Excessmassat thesearoor,forexamplethemassofaseamount,increaseslo calgravitybecause themassoftheseamountislargerthanthemassofwateritdis places.Rocks aremorethanthreetimesdenserthanwater.Theexcessmassi ncreaseslocal gravity,whichattractswatertowardtheseamount.Thischa ngestheshapeof theseasurface(gure3.12). Let'smaketheconceptmoreexact.Toaverygoodapproximati on,thesea surfaceisaparticular levelsurface calledthe geoid (seebox).Bydenitiona levelsurfaceisasurfaceofconstantgravitationalpotent ial,anditiseverywhere

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3.4.MEASURINGTHEDEPTHOFTHEOCEAN 31 90o 100o 110o 120o 130o 140o 150o 160o 170o 180o-40o-30o-20o-10o0o Walter H. F. Smith and David T. Sandwell, Ship Tracks, Version 4.0, SIO, September 26, 1996Copyright 1996, Walter H. F. Smith and David T. Sandwell Figure3.11Locationsofecho-sounderdatausedformapping theoceanroornearAustralia. Notethelargeareaswheredepthshavenotbeenmeasuredfrom ships.FromDavidSandwell, ScrippsInstitutionofOceanography.perpendiculartogravity.Inparticular,itmustbeperpend iculartothelocal verticaldeterminedbyaplumbline,whichis\alineorcordh avingatoneend ametalweightfordeterminingverticaldirection"(Oxford EnglishDictionary). Theexcessmassoftheseamountattractstheplumbline'swei ght,causing theplumblinetopointalittletowardtheseamountinsteado ftowardearth's centerofmass.Becausetheseasurfacemustbeperpendicula rtogravity,itmust haveaslightbulgeaboveaseamountasshowningure3.12.If therewereno bulge,theseasurfacewouldnotbeperpendiculartogravity .Typicalseamounts produceabulgethatis1{20mhighoverdistancesof100{200k ilometers.This bulgeisfartoosmalltobeseenfromaship,butitiseasilyme asuredby satellitealtimeters.Oceanictrencheshaveadecitofmas s,andtheyproduce adepressionoftheseasurface. Thecorrespondencebetweentheshapeoftheseasurfaceandt hedepthof thewaterisnotexact.Itdependsonthestrengthofthesearo or,theageof thesea-roorfeature,andthethicknessofsediments.Ifase amountroatsonthe searoorlikeiceonwater,thegravitationalsignalismuchw eakerthanitwould beiftheseamountrestedonthesearoorlikeicerestingonat abletop.As aresult,therelationshipbetweengravityandsea-roortop ographyvariesfrom regiontoregion. Depthsmeasuredbyacousticechosoundersareusedtodeterm inetheregionalrelationships.Hence,altimetryisusedtointerpol atebetweenacoustic echosoundermeasurements(SmithandSandwell,1994). Satellite-altimetersystems Nowlet'sseehowaltimetersmeasuretheshape

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32 CHAPTER3.THEPHYSICALSETTING TheGeoid Thelevelsurfacethatcorrespondstothesurfaceofanocean atrest isaspecialsurface,the geoid .Toarstapproximation,thegeoidisan ellipsoidthatcorrespondstothesurfaceofarotating,hom ogeneousruid insolid-bodyrotation,whichmeansthattheruidhasnointe rnalrow. Toasecondapproximation,thegeoiddiersfromtheellipso idbecause oflocalvariationsingravity.Thedeviationsarecalled geoidundulations Themaximumamplitudeoftheundulationsisroughly 60m.Toathird approximation,thegeoiddeviatesfromtheseasurfacebeca usetheocean isnotatrest.Thedeviationofsealevelfromthegeoidisde nedtobethe topography .Thedenitionisidenticaltothedenitionforlandtopogr aphy, forexampletheheightsgivenonatopographicmap. Theocean'stopographyiscausedbytides,heatcontentofth ewater,and oceansurfacecurrents.Iwillreturntotheirinruenceinch apters10and 17.Themaximumamplitudeofthetopographyisroughly 1m,soitis smallcomparedtothegeoidundulations. Geoidundulationsarecausedbylocalvariationsingravity duetothe unevendistributionofmassatthesearoor.Seamountshavea nexcessof massbecausetheyaremoredensethanwater.Theyproduceanu pward bulgeinthegeoid(seebelow).Trencheshaveadeciencyofm ass.They produceadownwardderectionofthegeoid.Thusthegeoidisc loselyrelatedtosea-roortopography.Mapsoftheoceanicgeoidhave aremarkable resemblancetothesea-roortopography. sea surface sea floor 10 m 2 km 200 km Figure3.12Seamountsaremoredensethanseawater.Theyinc reaselocalgravity, causingaplumblineattheseasurface(arrows)tobederecte dtowardtheseamount. Becausethesurfaceofanoceanatrestmustbeperpendicular togravity,theseasurface andthelocalgeoidmusthaveaslightbulgeasshown.Suchbul gesareeasilymeasured bysatellitealtimeters.Asaresult,satellitealtimeterd atacanbeusedtomapthesea roor.Note,thebulgeattheseasurfaceisgreatlyexaggerat ed,atwo-kilometerhigh seamountwouldproduceabulgeofapproximately10m. oftheseasurface.Satellitealtimetersystemsincludeara dartomeasurethe heightofthesatelliteabovetheseasurfaceandatrackings ystemtodetermine theheightofthesatelliteingeocentriccoordinates.Thes ystemmeasuresthe

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3.5.SEAFLOORCHARTSANDDATASETS 33 SatellitesOrbit Geoid Geoid Undulation Sea Surface Topography(not to scale)ReferenceEllipsoidCenter of Mass{ }rh Figure3.13Asatellitealtimetermeasurestheheightofthe satelliteabovetheseasurface. Whenthisissubtractedfromtheheight r ofthesatellite'sorbit,thedierenceissealevel relativetothecenterofearth.Theshapeofthesurfaceisdu etovariationsingravity,which producethegeoidundulations,andtooceancurrentswhichp roducetheoceanictopography, thedepartureoftheseasurfacefromthegeoid.Thereferenc eellipsoidisthebestsmooth approximationtothegeoid.Thevariationsinthegeoid,geo idundulations,andtopography aregreatlyexaggeratedinthegure.FromStewart(1985).heightoftheseasurfacerelativetothecenterofmassofear th(gure3.13). Thisgivestheshapeoftheseasurface. Manyaltimetricsatelliteshaverowninspace.Allobserved themarinegeoid andtheinruenceofsea-roorfeaturesonthegeoid.Thealtim etersthatproduced themostusefuldataincludeSeasat(1978), geosat (1985{1988), ers {1(1991{ 1996), ers {2(1995{),Topex/Poseidon(1992{2006),Jason(2002{),an dEnvisat (2002).Topex/PoseidonandJasonwerespeciallydesignedt omakeextremely accuratemeasurementsofsea-surfaceheight.Theymeasure sea-surfaceheight withanaccuracyof 0 : 05m. SatelliteAltimeterMapsoftheSea-roorTopography Seasat, geosat ers { 1,and ers {2wereoperatedinorbitswithgroundtracksspaced3{10kma part, whichwassucienttomapthegeoid.Bycombiningdatafromec ho-sounders withdatafrom geosat and ers {1altimetersystems,SmithandSandwell (1997)producedmapsofthesearoorwithhorizontalresolut ionof5{10km andaglobalaveragedepthaccuracyof 100m. 3.5SeaFloorChartsandDataSets Almostallecho-sounderdatahavebeendigitizedandcombin edtomakesearoorcharts.Datahavebeenfurtherprocessedandeditedtop roducedigital datasetswhicharewidelydistributedin cd-rom format.Thesedatahavebeen supplementedwithdatafromaltimetricsatellitestoprodu cemapsofthesea roorwithhorizontalresolutionaround3km. TheBritishOceanographicDataCentrepublishestheGenera lBathymetric Chartoftheocean( gebco )DigitalAtlasonbehalfoftheIntergovernmentalOceanographicCommissionof unesco andtheInternationalHydrographic

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34 CHAPTER3.THEPHYSICALSETTING 60o0o30o-30o-60o180o120o60o-120o-60o0o Walter H. F. Smith and David T. Sandwell Seafloor Topography Version 4.0 SIO September 26, 1996 1996 Walter H. F. Smith and David T. Sandwell 0o Figure3.14Thesea-roortopographyoftheoceanwith3kmres olutionproducedfrom satellitealtimeterobservationsoftheshapeoftheseasur face.FromSmithandSandwell. Organization.Theatlasconsistsprimarilyofthelocation ofdepthcontours, coastlines,andtracklinesfromthe gebco 5thEditionpublishedatascaleof 1:10million.Theoriginalcontoursweredrawnbyhandbased ondigitized echo-sounderdataplottedonbasemaps. TheU.S.NationalGeophysicalDataCenterpublishesthe etopo-2cd-rom containingdigitalvaluesofoceanicdepthsfromechosound ersandaltimetry andlandheightsfromsurveys.Dataareinterpolatedtoa2-m inute(2nautical mile)grid.Oceandatabetween64 Nand72 SarefromtheworkofSmith andSandwell(1997),whocombinedecho-sounderdatawithal timeterdatafrom geosat and ers{1 .Searoordatanorthwardof64 NarefromtheInternational BathymetricChartoftheArcticOcean.Searoordatasouthwa rdof72 Sare fromarefromtheUSNavalOceanographicOce'sDigitalBath ymetricData BaseVariableResolution.Landdataarefromthe globe Project,thatproduced adigitalelevationmodelwith0.5-minute(0.5nauticalmil e)gridspacingusing datafrommanynations. Nationalgovernmentspublishcoastalandharbormaps.Inth eUSA,the noaa NationalOceanServicepublishesnauticalchartsusefulfo rnavigationof shipsinharborsandoshorewaters.3.6SoundintheOcean Soundprovidestheonlyconvenientmeansfortransmittingi nformationover greatdistancesintheocean.Soundisusedtomeasurethepro pertiesofthe searoor,thedepthoftheocean,temperature,andcurrents. Whalesandother oceananimalsusesoundtonavigate,communicateovergreat distances,and ndfood.

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3.6.SOUNDINTHEOCEAN 35 Salinity 0 o5 o10 o15 o20 o -6 -5 -4 -3 -2 -1 0Depth (k m) 33.033.534.034.535.0 Speed Corrections (m/s) 0204060 80 100 020406080100 Sound Speed (m/s) 1500 1520 1540 1560 1500 1520 1540 1560 -6 -5 -4 -3 -2 -1 -0 tS C T oC DCt DCP DCS Figure3.15Processesproducingthesoundchannelintheoce an. Left: Temperature T and salinity S measuredasafunctionofdepthduringtheR.V. HakuhoMaru cruiseKH-87-1, stationJT,on28January1987atLatitude33 52.90 0 N,Long141 55.80 0 EintheNorth Pacic. Center: Variationsinsoundspeedduetovariationsintemperature, salinity,and depth. Right: Soundspeedasafunctionofdepthshowingthevelocityminim umnear1 kmdepthwhichdenesthesoundchannelintheocean.(Datafr om jpots EditorialPanel, 1991).SoundSpeed Thesoundspeedintheoceanvarieswithtemperature,salini ty, andpressure(MacKenzie,1981;Munketal.1995:33): C =1448 : 96+4 : 591 t 0 : 05304 t 2 +0 : 0002374 t 3 +0 : 0160 Z (3.1) +(1 : 340 0 : 01025 t )( S 35)+1 : 675 10 7 Z 2 7 : 139 10 13 tZ 3 where C isspeedinm/s, t istemperatureinCelsius, S issalinity(seeChapter 6foradenitionofsalinity),and Z isdepthinmeters.Theequationhasan accuracyofabout0.1m/s(Dushawetal.1993).Othersound-s peedequations havebeenwidelyused,especiallyanequationproposedbyWi lson(1960)which hasbeenwidelyusedbytheU.S.Navy. Fortypicaloceanicconditions, C isusuallybetween1450m/sand1550m/s (gure3.15).Using(3.1),wecancalculatethesensitivity of C tochangesof temperature,depth,andsalinitytypicaloftheocean.Thea pproximatevalues are:40m/sper10 Criseoftemperature,16m/sper1000mincreaseindepth, and1.5m/sper1increaseinsalinity.Thustheprimarycause sofvariabilityof soundspeedistemperatureanddepth(pressure).Variation sofsalinityaretoo smalltohavemuchinruence. Ifweplotsoundspeedasafunctionofdepth,wendthatthesp eedusually

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36 CHAPTER3.THEPHYSICALSETTING +9 +10 -9 ray +8 Range (km) 0 100200 1.501.55 -0-1-2-3-4Depth (km)C (km/s)axis Figure3.16Raypathsofsoundintheoceanforasourcenear theaxisofthesoundchannel.AfterMunketal.(1995). hasaminimumatadeptharound1000m(gure3.16).Thedeptho fminimum speediscalledthe soundchannel .Itoccursinallocean,anditusuallyreaches thesurfaceatveryhighlatitudes. Thesoundchannelisimportantbecausesoundinthechannelc antravel veryfar,sometimeshalfwayaroundtheearth.Hereishowthe channelworks: Soundraysthatbegintotraveloutofthechannelarerefract edbacktoward thecenterofthechannel.Rayspropagatingupwardatsmalla nglestothe horizontalarebentdownward,andrayspropagatingdownwar datsmallangles tothehorizontalarebentupward(gure3.16).Typicaldept hsofthechannel varyfrom10mto1200mdependingongeographicalarea.AbsorptionofSound Absorptionofsoundperunitdistancedependsonthe intensity I ofthesound: dI = kI 0 dx (3.2) where I 0 istheintensitybeforeabsorptionand k isanabsorptioncoecient whichdependsonfrequencyofthesound.Theequationhasthe solution: I = I 0 exp( kx )(3.3) Typicalvaluesof k (indecibelsdBperkilometer)are:0.08dB/kmat1000Hz, and50dB/kmat100,000Hz.Decibelsarecalculatedfrom: dB =10log( I=I 0 ), where I 0 istheoriginalacousticpower, I istheacousticpowerafterabsorption. Forexample,atarangeof1kma1000Hzsignalisattenuatedby only1.8%: I =0 : 982 I 0 .Atarangeof1kma100,000Hzsignalisreducedto I =10 5 I 0 The30,000Hzsignalusedbytypicalechosounderstomaptheo cean'sdepths arelittleattenuatedgoingfromthesurfacetothebottoman dback. Verylowfrequencysoundsinthesoundchannel,thosewithfr equencies below500Hzhavebeendetectedatdistancesofmegameters.I n196015-Hz soundsfromexplosionssetointhesoundchanneloPerthAu straliawere

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3.7.IMPORTANTCONCEPTS 37 heardinthesoundchannelnearBermuda,nearlyhalfwayarou ndtheworld. Laterexperimentshowedthat57-Hzsignalstransmittedint hesoundchannel nearHeardIsland(75 E,53 S)couldbeheardatBermudaintheAtlanticand atMonterey,CaliforniainthePacic(Munketal.1994).UseofSound Becauselowfrequencysoundcanbeheardatgreatdistances, theUSNavy,inthe1950s,placedarraysofmicrophonesonthe searoorin deepandshallowwaterandconnectedthemtoshorestations. TheSound SurveillanceSystem sosus ,althoughdesignedtotracksubmarines,hasfound manyotheruses.Ithasbeenusedtolistentoandtrackwhales upto1,700km away,andtondthelocationofsub-seavolcaniceruptions.3.7ImportantConcepts 1.Iftheoceanwerescaleddowntoawidthof8inchesitwouldh avedepths aboutthesameasthethicknessofapieceofpaper.Asaresult ,the velocityeldintheoceanisnearly2-dimensional.Vertica lvelocitiesare muchsmallerthanhorizontalvelocities. 2.Thereareonlythreeocialocean.3.Thevolumeofoceanwaterexceedsthecapacityoftheocean basins,and theoceanoverrowsontothecontinentscreatingcontinenta lshelves. 4.Thedepthsoftheoceanaremappedbyechosounderswhichme asurethe timerequiredforasoundpulsetotravelfromthesurfacetot hebottom andback.Depthsmeasuredbyship-basedechosoundershaveb eenusedto producemapsofthesearoor.Themapshavepoorhorizontalre solution insomeregionsbecausetheregionswereseldomvisitedbysh ipsandship tracksarefarapart. 5.Thedepthsoftheoceanarealsomeasuredbysatellitealti metersystems whichproletheshapeoftheseasurface.Thelocalshapeoft hesurface isinruencedbychangesingravityduetosub-seafeatures.R ecentmaps basedonsatellitealtimetermeasurementsoftheshapeofth eseasurface combinedwithshipdatahavedepthaccuracyof 100mandhorizontal resolutionsof 3km. 6.Typicalsoundspeedintheoceanis1480m/s.Speeddepends primarily ontemperature,lessonpressure,andverylittleonsalinit y.Thevariabilityofsoundspeedasafunctionofpressureandtemperaturep roducesa horizontalsoundchannelintheocean.Soundinthechannelc antravel greatdistances.Low-frequencysoundsbelow500Hzcantrav elhalfway aroundtheworldprovidedthepathisnotinterruptedbyland

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38 CHAPTER3.THEPHYSICALSETTING

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Chapter4AtmosphericInruencesThesunandtheatmospheredrivedirectlyorindirectlyalmo stalldynamical processesintheocean.Thedominantexternalsourcesandsi nksofenergy aresunlight,evaporation,infraredemissionsfromthesea surface,andsensible heatingoftheseabywarmorcoldwinds.Windsdrivetheocean 'ssurface circulationdowntodepthsofaroundakilometer.Windandti dalmixingdrive thedeepercurrentsintheocean. Theocean,inturn,isthedominantsourceofheatthatdrives theatmosphericcirculation.Theunevendistributionofheatlossa ndgainbytheocean leadstowindsintheatmosphere.Sunlightwarmsthetropica locean,which evaporate,transferringheatintheformofwatervaportoth eatmosphere.The heatisreleasedwhenthevaporcondensesasrain.Windsando ceancurrents carryheatpoleward,whereitislosttospace. Becausetheatmospheredrivestheocean,andtheoceandrive stheatmosphere,wemustconsidertheoceanandtheatmosphereasacou pleddynamic system.Inthischapterwewilllookmainlyattheexchangeof momentumbetweentheatmosphereandtheocean.Inthenextchapter,wewi lllookatheat exchanges.Inchapter14wewilllookathowtheoceanandthea tmosphere interactinthePacictoproduceElNi~no.4.1TheEarthinSpace Earth'sorbitaboutthesunisnearlycircularatameandista nceof1 : 5 10 8 km.Theeccentricityoftheorbitissmall,0.0168.Thuseart his3.4%further fromtheSunataphelionthanatperihelion,thetimeofclose stapproachtothe sun.PerihelionoccurseveryyearinJanuary,andtheexactt imechangesby about20minutesperyear.In1995,itoccurredon3January.E arth'saxisof rotationisinclined23.45 totheplaneofearth'sorbitaroundthesun(gure 4.1).Theorientationissuchthatthesunisdirectlyoverhe adattheEquator onthevernalandautumnalequinoxes,whichoccuronorabout 21Marchand 21Septembereachyear. Thelatitudesof23.45 NorthandSoutharetheTropicsofCancerand Capricornrespectively.Thetropicslieequatorwardofthe selatitudes.Asa 39

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40 CHAPTER4.ATMOSPHERICINFLUENCES Sun Tropic ofCapricorn Wintersolstice Arctic Circle Autumnalequinox Antarctic Circle Vernal equinox Tropic ofCancer Summersolstice 23 1 2operihelion aphelion Figure4.1Theearthinspace.Theellipticityofearth'sorb itaroundthesunandthetiltof earth'saxisofrotationrelativetotheplaneofearthorbit leadstoanunequaldistributionof heatingandtotheseasons.Earthisclosesttothesunatperi helion. resultoftheeccentricityofearth'sorbit,maximumsolari nsolationaveraged overthesurfaceoftheearthoccursinearlyJanuaryeachyea r.Asaresult oftheinclinationofearth'saxisofrotation,themaximumi nsolationatany locationoutsidethetropicsoccursaround21Juneinthenor thernhemisphere, andaround21Decemberinthesouthernhemisphere. Annual Wind Speed and Sea Level Pressure (hPa) For 1989 1012 1012 1014 1014 1014 1014 1014 1012 1014 1014 1014 1014 1014 1014 1014 1012 1012 1012 1012 1012 1018 1018 1018 1012 1018 1018 1018 1012 1014 1018 1018 1000 990 990 1000 1000 1010 1010 1010 1010 1010 1010 1010 1020 980 1010 1020 1010 1020 980 1010 1020 1020 1010 101210141012101099099099020o60o100o140o180o-140o-100o-60o-20o20o-90o -60o-30o0o30o60o90o 0o Figure4.2MapofmeanannualwindvelocitycalculatedfromT renberthetal.(1990)and sea-levelpressurefor1989fromthe nasa GoddardSpaceFlightCenter'sDataAssimilation Oce(Schubertetal.1993).Thewindsnear140 WintheequatorialPacicareabout8 m/s.

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4.2.ATMOSPHERICWINDSYSTEMS 41 Ifsolarheatwasrapidlyredistributedoverearth,maximum temperature wouldoccurinJanuary.Conversely,ifheatwerepoorlyredi stributed,maximum temperatureinthenorthernhemispherewouldoccurinsumme r.Soitisclear thatheatisnotrapidlyredistributedbywindsandcurrents 4.2AtmosphericWindSystems Figure4.2showsthedistributionofsea-levelwindsandpre ssureaveraged overtheyear1989.Themapshowsstrongwindsfromthewestbe tween40 to 60 latitude,theroaringforties,weakwindsinthesubtropics near30 latitude, tradewindsfromtheeastinthetropics,andweakerwindsfro mtheeastalong theEquator.Thestrengthanddirectionofwindsintheatmos phereisthe resultofunevendistributionofsolarheatingandcontinen tallandmassesand thecirculationofwindsinaverticalplaneintheatmospher e. Acartoonofthedistributionofwindsintheatmosphere(gu re4.3)shows Equatorial Low or Doldrums Subtropical High or Horse latitudes Subpolar Low Polar High Subpolar Low Subtropical High or Horse latitudes Polar High easterlies westerlies North-East Trades South-East Trades westerlies easterlies60o-60o-60o-30o-30o30o30o0o0o-90o90o heavy precipitation variable winds and calms large evaporation variable winds and calms very heavy precipitation variable winds and calms large evaporation sinking air rising air upper westerlies sinking air rising airHadley cellsupper westerlies sinking air rising air sinking air heavy precipitation60o moist air 10 20 0 Pole easterlies 60owesterlies 30oTrade Winds Polar High Subpolar Low Subtropical HighIntertropical Convergence ZoneEquatorial Lowtropospherepolar fronttropopausetemperature inversion cumulonimbus activityPOLAR SUB TROPICAL Height (km) Figure4.3Sketchofearth'satmosphericcirculationdrive nbysolarheatinginthetropicsand coolingathighlatitudes. Upper: Themeridionalcellsintheatmosphereandtheinruence ofearth'srotationonthewinds. Bottom: Cross-sectionthroughtheatmosphereshowing thetwomajorcellsofmeridionalcirculation.AfterTheOpe nUniversity(1989a:14).

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42 CHAPTER4.ATMOSPHERICINFLUENCES thatthesurfacewindsareinruencedbyequatorialconvecti onandotherprocesseshigherintheatmosphere.Themeanvalueofwindsover theoceanis (Wentzetal.1984): U 10 =7 : 4m/s(4.1) Mapsofsurfacewindschangesomewhatwiththeseasons.Thel argest changesareintheIndianOceanandthewesternPacicOcean( gure4.4). BothregionsarestronglyinruencedbytheAsianmonsoon.In winter,thecold airmassoverSiberiacreatesaregionofhighpressureatthe surface,andcold July Wind Speed January Wind Speed 20o60o100o140o180o-140o-100o-60o-20o20o-90o -60o-30o0o30o60o90o 0o 20o60o100o140o180o-140o-100o-60o-20o20o-90o -60o-30o0o30o60o90o 0o Figure4.4Mean,sea-surfacewindsforJulyandJanuarycalc ulatedfromtheTrenberthetal. (1990)dataset,whichisbasedonthe ecmwf reanalysesofweatherdatafrom1980to1989. Thewindsnear140 WintheequatorialPacicareabout8m/s.

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4.3.THEPLANETARYBOUNDARYLAYER 43 airblowssoutheastwardacrossJapanandonacrossthehotKu roshio,extractingheatfromtheocean.Insummer,thethermallowoverTibet drawswarm, moistairfromtheIndianOceanleadingtotherainyseasonov erIndia. 4.3ThePlanetaryBoundaryLayer Theatmospherewithin100moftheseasurfaceisinruencedby theturbulent dragofthewindontheseaandtheruxesofheatthroughthesur face.This isthe atmosphericboundarylayer .It'sthickness Z i variesfromafewtens ofmetersforweakwindsblowingoverwatercolderthantheai rtoarounda kilometerforstrongerwindsblowingoverwaterwarmerthan theair. Thelowestpartoftheatmosphericboundarylayeristhesurf acelayer. Withinthislayer,whichhasthicknessof 0 : 1 Z i ,verticalruxesofheatand momentumarenearlyconstant. Windspeedvariesasthelogarithmofheightwithinthesurfa celayerfor neutralstability.See\TheTurbulentBoundaryLayerOvera FlatPlate"in Chapter8.Hence,theheightofawindmeasurementisimporta nt.Usually, windsarereportedasthevalueofwindataheight10maboveth esea U 10 4.4MeasurementofWind Windatseahasbeenmeasuredforcenturies.Maury(1855)was therstto systematicallycollectandmapwindreports.Recently,the USNationalAtmosphericandOceanicAdministration noaa hascollected,edited,anddigitized millionsofobservationsgoingbackoveracentury.Theresu lting International ComprehensiveOcean,AtmosphereDataSet icoads discussedin x 5.5iswidely usedforstudyingatmosphericforcingoftheocean. Ourknowledgeofwindsattheseasurfacecomefrommanysourc es.Here arethemoreimportant,listedinacrudeorderofrelativeim portance: BeaufortScale Byfarthemostcommonsourceofwinddataupto1991 havebeenreportsofspeedbasedontheBeaufortscale.Thesc aleisbasedon features,suchasfoamcoverageandwaveshape,seenbyanobs erveronaship (table4.1). ThescalewasoriginallyproposedbyAdmiralSirF.Beaufort in1806togive theforceofthewindonaship'ssails.ItwasadoptedbytheBr itishAdmiralty in1838anditsooncameintogeneraluse. TheInternationalMeteorologicalCommitteeadoptedthefo rcescaleforinternationalusein1874.In1926theyadoptedarevisedscale givingthewind speedataheightof6meterscorrespondingtotheBeaufortNu mber.Thescale wasrevisedagainin1946toextendthescaletohigherwindsp eedsandtogive theequivalentwindspeedataheightof10meters.The1946sc alewasbased ontheequation U 10 =0 : 836 B 3 = 2 ,where B =BeaufortNumberand U 10 isthe windspeedinmeterspersecondataheightof10meters(List, 1966).More recently,variousgroupshaverevisedtheBeaufortscaleby comparingBeaufort forcewithshipmeasurementsofwinds.KentandTaylor(1997 )comparedthe variousrevisionsofthescalewithwindsmeasuredbyshipsh avinganemometers atknownheights.Theirrecommendedvaluesaregivenintabl e4.1.

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44 CHAPTER4.ATMOSPHERICINFLUENCES Table4.1BeaufortWindScaleandStateoftheSea BeaufortDescriptivem/sAppearanceoftheSea Numberterm 0Calm0Sealikeamirror.1LightAir1.2Rippleswithappearanceofscales;nofoamcre sts. 2LightBreeze2.8Smallwavelets;crestsofglassyappearan ce, notbreaking. 3Gentlebreeze4.9Largewavelets;crestsbegintobreak;sc attered whitecaps. 4Moderatebreeze7.7Smallwaves,becominglonger;numerou swhitecaps. 5Freshbreeze10.5Moderatewaves,takinglongertoform;ma ny whitecaps;somespray. 6Strongbreeze13.1Largewavesforming;whitecapseverywh ere; morespray. 7Neargale15.8Seaheapsup;whitefoamfrombreakingwavesb egins tobeblownintostreaks. 8Gale18.8Moderatelyhighwavesofgreaterlength;edgesof crestsbegintobreakintospindrift;foamisblowninwell-markedstreaks. 9Stronggale22.1Highwaves;seabeginstoroll;densestrea ksoffoam; spraymayreducevisibility. 10Storm25.9Veryhighwaveswithoverhangingcrests;seata kes whiteappearanceasfoamisblowninverydensestreaks;rollingisheavyandvisibilityreduced. 11Violentstorm30.2Exceptionallyhighwaves;seacovered withwhite foampatches;visibilitystillmorereduced. 12Hurricane35.2Airislledwithfoam;seacompletelywhit e withdrivingspray;visibilitygreatlyreduced. FromKentandTaylor(1997) Observersonshipseverywhereintheworldusuallyreportwe atherobservations,includingBeaufortforce,atthesamefourtimesever yday.Thetimesare at0000Z,0600Z,1200Zand1800Z,whereZindicatesGreenwic hMeanTime. Thereportsarecodedandreportedbyradiotonationalmeteo rologicalagencies.Thebiggesterrorinthereportsisthesamplingerror. Shipsareunevenly distributedovertheocean.Theytendtoavoidhighlatitude sinwinterand hurricanesinsummer,andfewshipscrossthesouthernhemis phere(gure4.5). Overall,theaccuracyisaround10%.Scatterometers Observationsofwindsatseanowcomemostlyfromscatterometersonsatellites(Liu,2002).Thescatterometeris ainstrumentvery muchlikearadarthatmeasuresthescatterofcentimeter-wa velengthradio wavesfromsmall,centimeter-wavelengthwavesontheseasu rface.Thearea oftheseacoveredbysmallwaves,theiramplitude,andtheir orientation,dependonwindspeedanddirection.Thescatterometermeasure sscatterfrom 2{4directions,fromwhichwindspeedanddirectionarecalc ulated. Thescatterometerson ers-1 and2havemadeglobalmeasurementsofwinds fromspacesince1991.The nasa scatterometeron adeos measuredwindsfor asix-monthperiodbeginningNovember1996andendingwitht hepremature failureofthesatellite.Itwasreplacedbyanotherscatter ometeronQuikScat, launchedon19June1999.Quikscatviews93%oftheoceanever y24hrwitha resolutionof25km.

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4.4.MEASUREMENTOFWIND 45 Volunteer Observing Ship Data September 1997 90o130o170o-150o-110o-70o-30o10o50o10o-10o-30o-50o-70o30o50o70o90o 0o Figure4.5Locationofsurfaceobservationsmadefromvolun teerobservingshipsand reportedtonationalmeteorologicalagencies.From noaa ,NationalOceanService. FreilichandDunbar(1999)reportthat,overall,the nasa scatterometeron adeos measuredwindspeedwithanaccuracyof 1 : 3m/s.Theerrorinwind directionwas 17 .Spatialresolutionwas25km.DatafromQuikScathasan accuracyof 1m/s. Becausescatterometersviewaspecicoceanicareaonlyonc eaday,the datamustbeusedwithnumericalweathermodelstoobtain6-h ourlywind mapsrequiredforsomestudies.Windsat Windsatisanexperimental,polarimetric,microwaveradio meterdevelopedbytheUSNavythatmeasurestheamountandpolarizat ionofmicrowaveradiationemittedfromtheseaatanglesbetween50 to55 relativeto theverticalandatveradiofrequencies.Itwaslaunchedon 6January2003on theCoriolissatellite.Thereceivedradiosignalisafunct ionofwindspeed,seasurfacetemperature,watervaporintheatmosphere,rainra te,andtheamount ofwaterinclouddrops.Byobservingseveralfrequenciessi multaneously,data fromtheinstrumentareusedforcalculatingthesurfacewin dspeedanddirection,sea-surfacetemperature,totalprecipitablewater, integratedcloudliquid water,andrainrateovertheoceanregardlessoftimeofdayo rcloudiness. Windsarecalculatedovermostoftheoceanona25-kmgridonc eaday. WindsmeasuredbyWindsathaveanaccuracyof 2m/sinspeedand 20 in directionovertherangeof5{25m/s.SpecialSensorMicrowaveSSM/I Anothersatelliteinstrumentthatisused tomeasurewindspeedistheSpecial-SensorMicrowave/Imag er( ssm/i )carried since1987onthesatellitesoftheU.S.DefenseMeteorologi calSatelliteProgram inorbitssimilartothe noaa polar-orbitingmeteorologicalsatellites.Theinstrumentmeasuresthemicrowaveradiationemittedfromthe seaatanangle

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46 CHAPTER4.ATMOSPHERICINFLUENCES near60 fromthevertical.Theradiosignalisafunctionofwindspee d,water vaporintheatmosphere,andtheamountofwaterinclouddrop s.Byobservingseveralfrequenciessimultaneously,datafromtheinst rumentareusedfor calculatingthesurfacewindspeed,watervapor,cloudwate r,andrainrate. Windsmeasuredby ssm/i haveanaccuracyof 2m/sinspeed.When combinedwith ecmwf 1000mbwindanalyses,winddirectioncanbecalculated withanaccuracyof 22 (Atlas,Homan,andBloom,1993).Global,gridded dataareavailablesinceJuly1987ona0.25 gridevery6hours.Butremember, theinstrumentviewsaspecicoceanicareaonlyonceaday,a ndthegridded, 6-hourlymapshavebiggaps.AnemometersonShips Satelliteobservationsaresupplementedbywinds reportedtometeorologicalagenciesbyobserversreadinga nemometersonships. TheanemometerisreadfourtimesadayatthestandardGreenw ichtimesand reportedviaradiotometeorologicalagencies. Again,thebiggesterroristhesamplingerror.Veryfewship scarrycalibrated anemometers.Thosethatdotendtobecommercialshipsparti cipatinginthe VolunteerObservingShipprogram(gure4.5).Theseshipsa remetinportby scientistswhochecktheinstrumentsandreplacethemifnec essary,andwho collectthedatameasuredatsea.Theaccuracyofwindmeasur ementsfrom theseshipsisabout 2m/s. CalibratedAnemometersonWeatherBuoys Themostaccuratemeasurementsofwindsatseaaremadebycalibratedanemometersonmo oredweather buoys.Unfortunatelytherearefewsuchbuoys,perhapsonly ahundredscatteredaroundtheworld.Some,suchasTropicalAtmosphereOc ean tao array inthetropicalPacic(gure14.14)providedatafromremot eareasrarelyvisitedbyships,butmosttendtobelocatedjustoshoreofcoas talareas. noaa operatesbuoysoshoreoftheUnitedStatesandthe tao arrayinthePacic. Datafromthecoastalbuoysareaveragedforeightminutesbe forethehour,and theobservationsaretransmittedtoshoreviasatellitelin ks. Thebestaccuracyofanemometersonbuoysoperatedbythe us National DataBuoyCenteristhegreaterof 1m/sor10%forwindspeedand 10 for winddirection(Beardsleyetal.1997).4.5CalculationsofWind Satellites,ships,andbuoysmeasurewindsatvariouslocat ionsandtimes oftheday.Ifyouwishtousetheobservationstocalculatemo nthlyaveraged windsoverthesea,thentheobservationscanbeaveragedand gridded.Ifyou wishtousewinddatainnumericalmodelsoftheocean'scurre nts,thenthe datawillbelessuseful.Youarefacedwithaverycommonprob lem:Howto takeallobservationsmadeinasix-hourperiodanddetermin ethewindsover theoceanonaxedgrid? Onesourceofgriddedwindsovertheoceanisthe surfaceanalysis calculated bynumericalweathermodels.Thestrategyusedtoproduceth esix-hourly griddedwindsiscalled sequentialestimationtechniques or dataassimilation

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4.5.CALCULATIONSOFWIND 47 \Measurementsareusedtoprepareinitialconditionsforth emodel,whichis thenintegratedforwardintimeuntilfurthermeasurements areavailable.The modelisthereuponre-initialized"(Bennett,1992:67).Th einitialconditionis calledthe analysis Usually,allavailablemeasurementsareusedintheanalysi s,includingobservationsfromweatherstationsonland,pressureandtemp eraturereportedby shipsandbuoys,windsfromscatterometersinspace,anddat afrommeteorologicalsatellites.Themodelinterpolatesthemeasurementst oproduceananalysis consistentwithpreviousandpresentobservations.Daley( 1991)describesthe techniquesinconsiderabledetail.SurfaceAnalysisfromNumericalWeatherModels Perhapsthemost widelyusedweathermodelisthatrunbytheEuropeanCentref orMediumrangeWeatherForecasts ecmwf .Itcalculatesasurfaceanalysis,including surfacewindsandheatruxes(seeChapter5)everysixhourso na1 1 gridfromanexplicitboundary-layermodel.Calculatedval uesarearchivedon a2.5 grid.Thusthewindmapsfromthenumericalweathermodelsla ckthe detailseeninmapsfromscatterometerdata,whichhavea1/4 grid. ecmwf calculationsofwindshaverelativelygoodaccuracy.Freil ichand Dunbar(1999)estimatedthattheaccuracyforwindspeedat1 0metersis 1 : 5 m/s,and 18 fordirection. Accuracyinthesouthernhemisphereisprobablyasgoodasin thenorthern hemispherebecausecontinentsdonotdisrupttherowasmuch asinthenorthern hemisphere,andbecausescatterometersgiveaccurateposi tionsofstormsand frontsovertheocean. The noaa NationalCentersforEnvironmentalPredictionandtheUSNa vy alsoproducesglobalanalysesandforecastseverysixhours ReanalyzedDatafromNumericalWeatherModels Surfaceanalyses ofweatheroversomeregionshavebeenproducedformorethan ahundred years,andoverthewholeearthsinceabout1950.Surfaceana lysescalculatedbynumericalmodelsoftheatmosphericcirculationha vebeenavailablefor decades.Throughoutthisperiod,themethodsforcalculati ngsurfaceanalyses haveconstantlychangedasmeteorologistsworkedtomakeev ermoreaccurate forecasts.Fluxescalculatedfromtheanalysesaretherefo renotconsistentin time.Thechangescanbelargerthantheinterannualvariabi lityoftheruxes (White,1996).Tominimizethisproblem,meteorologicalag encieshavetaken allarchivedweatherdataandreanalyzedthemusingthebest numericalmodels toproduceauniform,internally-consistent,surfaceanal ysis. Thereanalyzeddataareusedtostudyoceanicandatmospheri cprocesses inthepast.Surfaceanalysesissuedeverysixhoursfromwea theragenciesare usedonlyforproblemsthatrequireup-to-dateinformation .Forexample,ifyou aredesigninganoshorestructure,youwillprobablyusede cadesofreanalyzed data.Ifyouareoperatinganoshorestructure,youwillwat chthesurface analysisandforecastsputouteverysixhoursbymeteorolog icalagencies.

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48 CHAPTER4.ATMOSPHERICINFLUENCES SourcesofReanalyzedData Reanalyzedsurfaceruxdataareavailablefrom nationalmeteorologicalcentersoperatingnumericalweat herpredictionmodels. 1.TheU.S.NationalCentersforEnvironmentalPredictions ,workingwith theNationalCenterforAtmosphericResearchhaveproduced the ncep/ ncar reanalysisbasedon51yearsofweatherdatafrom1948to2005 using the25January1995versionoftheirforecastmodel.Therean alysisperiod isbeingextendedforwardtoincludealldateuptothepresen twithabout athree-daydelayinproducingdatasets.Thereanalysisuse ssurfaceand shipobservationsplussounderdatafromsatellites.Reana lysisproducts areavailableeverysixhoursonaT62gridhaving192 94gridpoints withaspatialresolutionof209kmandwith28verticallevel s.Important subsetsofthereanalysis,includingsurfaceruxes,areava ilableon cd{rom (Kalnayetal.1996;Kistleretal.2000). 2.TheEuropeanCentreforMedium-rangeWeatherForecasts ecmwf has reanalyzed45yearsofweatherdatafromSeptember1957toAu gust 2002( era -40)usingtheirforecastmodelof2001(Uppalaetal.2005). Thereanalysisusesmostlythesamesurfaceandshipdatause dbythe ncep/ncar reanalysisplusdatafromthe ers -1and ers -2satellitesand ssm/i .The era -40full-resolutionproductsareavailableeverysixhours onaN80gridhaving160 320gridpointswithaspatialresolutionof 1.125 andwith60verticallevels.The era -40basic-resolutionproducts areavailableeverysixhourswithaspatialresolutionof2. 5 andwith 23verticallevels.Thereanalysisincludesanocean-wavem odelthatcalculatesoceanwaveheightsandwavespectraeverysixhourso na1.5 grid. 4.6WindStress Thewind,byitself,isusuallynotveryinteresting.Oftenw earemuchmore interestedintheforceofthewind,ortheworkdonebythewin d.Thehorizontal forceofthewindontheseasurfaceiscalledthe windstress .Putanotherway,it istheverticaltransferofhorizontalmomentum.Thusmomen tumistransferred fromtheatmospheretotheoceanbywindstress. Windstress T iscalculatedfrom: T = a C D U 2 10 (4.2) where a =1 : 3kg/m 3 isthedensityofair, U 10 iswindspeedat10meters,and C D isthe dragcoecient C D ismeasuredusingthetechniquesdescribedin x 5.6.Fastresponseinstrumentsmeasurewindructuationswi thin10{20mof theseasurface,fromwhich T isdirectlycalculated.Thecorrelationof T with U 2 10 gives C D (gure4.6). Variousmeasurementsof C D havebeenpublishedbasedoncarefulmeasurementsofturbulenceinthemarineboundarylayer.Trenberth etal.(1989)and Harrison(1989)discusstheaccuracyofaneectivedragcoe cientrelatingwind stresstowindvelocityonaglobalscale.Perhapsthebestof therecentlypub-

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4.7.IMPORTANTCONCEPTS 49 0 0.001 0.002 0.003 CDU10 (m/s) 5101520 0 303540 45 25 55 50 Figure4.6Thedragcoecientasafunctionofwindspeed U 10 tenmetersabovethesea. Circles:MeasuredvaluesfromSmith(1980).Triangles:Mea suredvaluesfromPowell, Vickery,andReinhold(2003).Thesolidlineisfromeq(4.3) proposedbyYellandandTaylor (1996).ThedashedlineisfromJarosz(2007).lishedvaluesarethoseofYellandandTaylor(1996)andYell andetal.(1998) whogive: 1000 C D =0 : 29+ 3 : 1 U 10 + 7 : 7 U 2 10 (3 U 10 6m/s)(4.3a) 1000 C D =0 : 60+0 : 071 U 10 (6 U 10 26m/s)(4.3b) forneutrallystableboundarylayer.Othervaluesareliste dintheirtable1and ingure4.6.4.7ImportantConcepts 1.Sunlightistheprimaryenergysourcedrivingtheatmosph ereandocean. 2.Thereisaboundarylayeratthebottomoftheatmospherewh erewind speeddecreaseswithastheboundaryisapproached,andinwh ichruxes ofheatandmomentumareconstantinthelower10{20meters. 3.Windismeasuredmanydierentways.Themostcommonuntil 1995was fromobservationsmadeatseaoftheBeaufortforceofthewin d. 4.Since1995,themostimportantsourceofwindmeasurement sisfrom scatterometersonsatellites.Theyproducedailyglobalma pswith25 kmresolution. 5.Thesurfaceanalysisfromnumericalmodelsoftheatmosph ereisthemost usefulsourceofglobal,griddedmapsofwindvelocityforda tesbefore 1995.Italsoisausefulsourcefor6-hourlymaps.Resolutio nis100-250 km. 6.Theruxofmomentumfromtheatmospheretotheocean,thewi ndstress, iscalculatedfromwindspeedusingadragcoecient.

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50 CHAPTER4.ATMOSPHERICINFLUENCES

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Chapter5TheOceanicHeatBudgetAbouthalfthesolarenergyreachingearthisabsorbedbythe oceanandland, whereitistemporarilystorednearthesurface.Onlyabouta fthoftheavailablesolarenergyisdirectlyabsorbedbytheatmosphere.Of theenergyabsorbed bytheocean,mostisreleasedlocallytotheatmosphere,mos tlybyevaporation andinfraredradiation.Theremainderistransportedbycur rentstootherareas especiallymidlatitudes. Heatlostbythetropicaloceanisthemajorsourceofheatnee dedtodrive theatmosphericcirculation.And,solarenergystoredinth eoceanfromsummer towinterhelpsameliorateearth'sclimate.Thethermalene rgytransportedby oceancurrentsisnotsteady,andsignicantchangesinthet ransport,particularlyintheAtlantic,mayhavebeenimportantforthedevelo pmentoftheice ages.Forthesereasons,oceanicheatbudgetsandtransport sareimportantfor understandingearth'sclimateanditsshortandlongtermva riability. 5.1TheOceanicHeatBudget Changesinenergystoredintheupperoceanresultfromanimb alancebetweeninputandoutputofheatthroughtheseasurface.Thist ransferofheat acrossorthroughasurfaceiscalleda heatrux .Theruxofheatandwateralso changesthedensityofsurfacewaters,andhencetheirbuoya ncy.Asaresult, thesumoftheheatandwaterruxesisoftencalledthe buoyancyrux Theruxofenergytodeeperlayersisusuallymuchsmallertha ntherux throughthesurface.And,thetotalruxofenergyintoandout oftheocean mustbezero,otherwisetheoceanasawholewouldheatuporco oldown.The sumoftheheatruxesintooroutofavolumeofwateristhe heatbudget Themajortermsinthebudgetattheseasurfaceare: 1. Insolation Q SW ,theruxofsolarenergyintothesea; 2. NetInfraredRadiation Q LW ,netruxofinfraredradiationfromthesea; 3. SensibleHeatFlux Q S ,theruxofheatoutoftheseaduetoconduction; 4. LatentHeatFlux Q L ,theruxofenergycarriedbyevaporatedwater;and 5. Advection Q V ,heatcarriedawaybycurrents. 51

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52 CHAPTER5.THEOCEANICHEATBUDGET 4.00 4.05 4.10 4.15 4.20 0o10o20o30oTemperature (Celsius) 010203040 Salinity Figure5.1Specicheatofseawateratatmosphericpressure C p injoulespergramperdegree CelsiusasafunctionoftemperatureinCelsiusandsalinity ,calculatedfromtheempirical formulagivenbyMilleroetal.(1973)usingalgorithmsinFo fonoandMillard(1983).The lowerlineisthefreezingpointofsaltwater. Conservationofheatrequires: Q = Q SW + Q LW + Q S + Q L + Q V (5.1) where Q istheresultantheatgainorloss.Unitsforheatruxesarewa tts/m 2 Theproductofruxtimessurfaceareatimestimeisenergyinj oules.Thechange intemperature t ofthewaterisrelatedtochangeinenergy E through: E = C p m t (5.2) where m isthemassofwaterbeingwarmedorcooled,and C p isthespecic heatofseawateratconstantpressure. C p 4 : 0 10 3 J kg 1 C 1 (5.3) Thus,4,000joulesofenergyarerequiredtoheat1.0kilogra mofseawaterby 1.0 C(gure5.1). ImportanceoftheOceaninEarth'sHeatBudget Tounderstandthe importanceoftheoceaninearth'sheatbudget,let'smakeac omparisonofthe heatstoredintheoceanwithheatstoredonlandduringanann ualcycle.During thecycle,heatisstoredinsummerandreleasedinthewinter .Thepointisto showthattheoceanstoreandreleasemuchmoreheatthanthel and. Tobegin,use(5.3)andtheheatcapacityofsoilandrocks C p ( rock ) =800J kg 1 C 1 (5.4) toobtain C p ( rock ) 0 : 2 C p ( water )

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5.2.HEAT-BUDGETTERMS 53 Thevolumeofwaterwhichexchangesheatwiththeatmosphere onaseasonal cycleis100m 3 persquaremeterofsurface,i.e.thatmassfromthesurfacet o adepthof100meters.Thedensityofwateris1000kg/m 3 ,andthemassin contactwiththeatmosphereisdensity volume= m water =100 ; 000kg.The volumeoflandwhichexchangesheatwiththeatmosphereonas easonalcycle is1m 3 .Becausethedensityofrockis3,000kg/m 3 ,themassofthesoiland rockincontactwiththeatmosphereis3,000kg. Theseasonalheatstoragevaluesfortheoceanandlandareth erefore: E ocean = C p ( water ) m water t t =10 C =(4000)(10 5 )(10 )Joules =4 : 0 10 9 Joules E land = C p ( rock ) m rock t t =20 C =(800)(3000)(20 )Joules =4 : 8 10 7 Joules E ocean E land =100 where t isthetypicalchangeintemperaturefromsummertowinter. Thelargestorageofheatintheoceancomparedwiththelandh asimportant consequences.Theseasonalrangeofairtemperaturesonlan dincreaseswith distancefromtheocean,anditcanexceed40 Cinthecenterofcontinents, reaching60 CinSiberia.Typicalrangeoftemperatureovertheoceanand alongcoastsislessthan10 C.Thevariabilityofwatertemperaturesisstill smaller(seegure6.3,bottom).5.2Heat-BudgetTerms Let'slookatthefactorsinruencingeachtermintheheatbud get. FactorsInruencingInsolation Incomingsolarradiationisprimarilydeterminedbylatitude,season,timeofday,andcloudiness.Thep olarregionsare heatedlessthanthetropics,areasinwinterareheatedless thanthesamearea insummer,areasinearlymorningareheatedlessthanthesam eareaatnoon, andcloudydayshavelesssunthansunnydays. Thefollowingfactorsareimportant: 1.Theheightofthesunabovethehorizon,whichdependsonla titude,season,andtimeofday.Don'tforget,thereisnoinsolationatn ight! 2.Thelengthofday,whichdependsonlatitudeandseason.3.Thecross-sectionalareaofthesurfaceabsorbingsunlig ht,whichdepends onheightofthesunabovethehorizon. 4.Attenuation,whichdependson:i)Clouds,whichabsorban dscatterradiation.ii)Pathlengththroughtheatmosphere,whichvari esascsc where isangleofthesunabovethehorizon.iii)Gasmoleculeswhic h absorbradiationinsomebands(gure5.2).H 2 O,O 3 ,andCO 2 areall

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54 CHAPTER5.THEOCEANICHEATBUDGET Solar Radiationat Surface (m=2) Blackbody Radiation(5900 K) Solar Radiationabove Atmosphere(m=0) 0.51.01.5 2.01.51.00.5 Wavelength (mm)Spectral Irradiance El (kWm-2mm-1) 0 0 Figure5.2Insolation(spectralirradiance)ofsunlightat topoftheatmosphereandatthe seasurfaceonaclearday.Thedashedlineisthebest-tting curveofblackbodyradiation thesizeanddistanceofthesun.Thenumberofstandardatmos phericmassesisdesignated by m .Thus m =2isapplicableforsunlightwhenthesunis30 abovethehorizon.After Stewart(1985:43). important.iv)Aerosolswhichscatterandabsorbradiation .Bothvolcanic andmarineaerosolsareimportant.Andv)dust,whichscatte rsradiation, especiallySaharandustovertheAtlantic. 5.Rerectivityofthesurface,whichdependsonsolarelevat ionangleand roughnessofseasurface. Solarinclinationandcloudinessdominate.Absorptionbyo zone,watervapor, aerosols,anddustaremuchweaker. Theaverageannualvalueforinsolation(gure5.3)isinthe range: 30W/m 2
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5.2.HEAT-BUDGETTERMS 55 150 150 300300 300 300 300 300 150 150 150 350 350 0 0 50 350 Clear-Sky Downward Insolation (W/m2) MarAprMayJuneJulyAugSepOctNovDecJan Feb -90o -60o -30o 0o 30o 60o90o Figure5.3Monthlyaverageofdownwardruxofsunlightthrou ghacloud-freeskyandinto theoceaninW/m 2 during1989calculatedbytheSatelliteDataAnalysisCente ratthe nasa LangleyResearchCenter(Darnelletal.1992)usingdatafro mtheInternationalSatellite CloudClimatologyProject.atmospherebutwithvaryingamountsofwatervapor(gure5. 4)showsthe atmosphereisnearlytransparentinsomewavelengthbandsc alledwindows. Thetransmittanceonacloud-freedaythroughthewindowfro m8 mto13 misdeterminedmostlybywatervapor.Absorptioninotherba nds,suchas thoseat3.5 mto4.0 mdependsonCO 2 concentrationintheatmosphere. AstheconcentrationofCO 2 increases,thesewindowscloseandmoreradiation istrappedbytheatmosphere. Becausetheatmosphereismostlytransparenttoincomingsu nlight,and somewhatopaquetooutgoinginfraredradiation,theatmosp heretrapsradiation.Thetrappedradiation,coupledwithconvectioninthe atmosphere,keeps earth'ssurface33 warmerthanitwouldbeintheabsenceofaconvecting,wet atmospherebutinthermalequilibriumwithspace.Theatmos phereactslike thepanesofglassonagreenhouse,andtheeectisknownasth e greenhouse eect .SeeHartmann(1994:24{26)forasimplediscussionofthera diative balanceofaplanet.CO 2 ,watervapor,methane,andozoneareallimportant greenhousegasses. Thenetinfraredruxdependson: 1.Cloudsthickness.Thethickertheclouddeck,thelesshea tescapesto space. 2.Cloudheight,whichdeterminesthetemperatureatwhicht hecloudradiatesheatbacktotheocean.Therateisproportionalto t 4 ,where t is thetemperatureoftheradiatingbodyinKelvins.Highcloud sarecolder thanlowclouds. 3.Atmosphericwater-vaporcontent.Themorehumidtheatmo spherethe lessheatescapestospace.

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56 CHAPTER5.THEOCEANICHEATBUDGET Wavlength (mm)Transmittance0 1 0 12345 Wavlength (mm) 68101214161820222428 26 Transmittance1 0 Subarctic Winter 62 U.S. Standard Tropical Subarctic Winter Midlatitude Winter 1962 U.S. StandardSubarctic SummerMidlatitude SummerTropical Figure5.4Atmospherictransmittanceforaverticalpathto spacefromsealevelforsix modelatmosphereswithveryclear,23 km ,visibility,includingtheinruenceofmolecular andaerosolscattering.Noticehowwatervapormodulatesth etransparencyofthe10-14 m atmosphericwindow,henceitmodulates Q LW ,whichisamaximumatthesewavelengths. AfterSelbyandMcClatchey(1975). 4.WaterTemperature.Thehotterthewaterthemoreheatisra diated. Again,radiationdependsof t 4 5.Iceandsnowcover.Iceemitsasablackbody,butitcoolsmu chfaster thanopenwater.Ice-coveredseasareinsulatedfromtheatm osphere. Watervaporandcloudsinruencethenetlossofinfraredradi ationmore thansurfacetemperature.Hottropicalregionsloselesshe atthancoldpolar regions.Thetemperaturerangefrompolestoequatoris0 C < t < 25 Cor 273K < t < 298K,andtheratioofmaximumtominimumtemperaturein Kelvinsis298/273=1.092.Raisedtothefourthpowerthisis 1.42.Thusthere isa42%increaseinemittedradiationfrompoletoequator.O verthesame distancewatervaporcanchangethenetemittedradianceby2 00%. Theaverageannualvaluefornetinfraredruxisinthenarrow range: 60W/m 2
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5.3.DIRECTCALCULATIONOFFLUXES 57 Theaverageannualvalueforlatent-heatruxisintherange: 130W/m 2
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58 CHAPTER5.THEOCEANICHEATBUDGET Table5.1NotationDescribingFluxes SymbolVariable ValueandUnits C p Specicheatcapacityofair1030J kg 1 K 1 C D Dragcoecient(see4.3)(0 : 50+0 : 071 U 10 ) 10 3 C L Latentheattransfercoecient1 : 2 10 3 C S Sensibleheattransfercoecient1 : 0 10 3 L E Latentheatofevaporation2 : 5 10 6 J/kg q Specichumidityofairkg(watervapor)/kg(air) q a Specichumidityofair10mabovetheseakg(watervapor)/kg (air) q s Specichumidityofairattheseasurfacekg(watervapor)/k g(air) Q S Sensibleheatrux W/m 2 Q L Latentheatrux W/m 2 T Windstress Pascals t a Temperatureoftheair10mabovetheseaKor C t s Sea-surfacetemperatureKor C t 0 Temperatureructuation C u 0 Horizontalcomponentofructuationofwindm/s u Frictionvelocity m/s U 10 Windspeedat10mabovetheseam/s w 0 Verticalcomponentofwindructuationm/s a Densityofair 1.3kg/m 3 T Vectorwindstress Pa C S and C L fromSmith(1988). RadiometerMeasurementsofRadiativeFluxes Radiometersonships, oshoreplatforms,andevensmallislandsareusedtomakedi rectmeasurements ofradiativeruxes.Widebandradiometerssensitivetoradi ationfrom0.3 mto 50 mcanmeasureincomingsolarandinfraredradiationwithana ccuracyof around3%providedtheyarewellcalibratedandmaintained. Other,specialized radiometerscanmeasuretheincomingsolarradiation,thed ownwardinfrared radiation,andtheupwardinfraredradiation.5.4IndirectCalculationofFluxes:BulkFormulas Theuseofgust-probesisveryexpensive,andradiometersmu stbecarefully maintained.Neithercanbeusedtoobtainlong-term,global valuesofruxes.To calculatetheseruxesfrompracticalmeasurements,weuseo bservedcorrelations betweenruxesandvariablesthatcanbemeasuredglobally. Forruxesofsensibleandlatentheatandmomentum,thecorre lationsare called bulkformulas .Theyare: T = a C D U 2 10 (5.10a) Q S = a C p C S U 10 ( t s t a )(5.10b) Q L = a L E C L U 10 ( q s q a )(5.10c) Airtemperature t a ismeasuredusingthermometersonships.Itcannotbe measuredfromspaceusingsatelliteinstruments. t s ismeasuredusingthermometersonshipsorfromspaceusinginfraredradiometerss uchasthe avhrr

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5.4.INDIRECTCALCULATIONOFFLUXES:BULKFORMULAS 59 Table5.2AccuracyofWindandFluxesObservedGloballyFrom Space VariableAccuracyComments WindSpeed 1.5m/sInstrumentError 1.5m/sSamplingError(MonthlyAverage) WindStress 10%DragCoecientError 14PaAssuming10m/sWindSpeed Insolation 5%MonthlyAverage 15W/m 2 MonthlyAverage 10%DailyAverage RainRate 50% Rainfall 10%5 5 areafor trmm NetLongWaveRadiation 4{8%DailyAverage 15{27W/m 2 LatentHeatFlux 35W/m 2 DailyAverage 15W/m 2 MonthlyAverage Thespecichumidityofairat10mabovetheseasurface q a iscalculated frommeasurementsofrelativehumiditymadefromships.Gil l(1982:pp:39{ 41,43{44,&605{607)describesequationsrelatingwaterva porpressure,vapor density,andspecicheatcapacityofwetair.Thespecichu midityatthesea surface q s iscalculatedfrom t s assumingtheairatthesurfaceissaturatedwith watervapor. U 10 ismeasuredorcalculatedusingtheinstrumentsortechniqu es describedinChapter4.Notethatwindstressisavectorwith magnitudeand direction.Itisparalleltothesurfaceinthedirectionoft hewind. Theproblemnowbecomes:Howtocalculatetheruxesacrossth eseasurfacerequiredforstudiesofoceandynamics?Theruxesinclu de:1)stress;2) solarheating;3)evaporation;4)netinfraredradiation;5 )rain;5)sensibleheat; and6)otherssuchasCO 2 andparticles(whichproducemarineaerosols).Furthermore,theruxesmustbeaccurate.Weneedanaccuracyofa pproximately 15W/m 2 .Thisisequivalenttotheruxofheatwhichwouldwarmorcool a columnofwater100mdeepbyroughly1 Cinoneyear.Table5.2liststypicalaccuraciesofruxesmeasuredgloballyfromspace.Now,l et'slookateach variable.WindSpeedandStress Stressiscalculatedfromwindobservationsmade fromshipsatseaandfromscatterometersinspaceasdescrib edinthelast chapter.Insolation iscalculatedfromcloudobservationsmadefromshipsandfr om visible-lightradiometersonmeteorologicalsatellites. Satellitemeasurements arefarmoreaccuratethantheshipdatabecauseit'sveryhar dtomeasure cloudinessfrombelowtheclouds.Satellitemeasurementsp rocessedbythe InternationalSatelliteCloudClimatologyProject isccp arethebasisformaps ofinsolationanditsvariabilityfrommonthtomonth(Darne lletal.1988; RossowandSchier1991). Thebasicideabehindthecalculationofinsolationisthis. Sunlightatthe

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60 CHAPTER5.THEOCEANICHEATBUDGET topoftheatmosphereisaccuratelyknownfromthesolarcons tant,latitude, longitude,andtime.Sunlightiseitherrerectedbacktospa cebyclouds,orit eventuallyreachestheseasurface.Onlyasmallandnearlyc onstantfraction isabsorbedintheatmosphere.But,recentworkbyCessetal. (1995)and Ramanathanetal.(1995)suggestthatthisbasicideamaybei ncomplete, andthatatmosphericabsorptionmaybeafunctionofcloudin ess.Assuming atmosphericabsorptionisconstant,insolationiscalcula tedfrom: Insolation= S (1 A ) C where S =1365W/m 2 isthesolarconstant, A isalbedo,theratioofincidentto rerectedsunlight,and C isaconstantwhichincludesabsorptionbyozoneand otheratmosphericgasesandbyclouddroplets.Insolationi scalculatedfrom clouddata(whichalsoincludesrerectionfromaerosols)co llectedfrominstrumentssuchasthe avhrr onmeteorologicalsatellites.Ozoneandgasabsorption arecalculatedfromknowndistributionsofthegasesinthea tmosphere.Q SW iscalculatedfromsatellitedatawithanaccuracyof5{7%.WaterFluxIn(Rainfall) Rainrateisanothervariablethatisverydicult tomeasurefromships.Raincollectedfromgaugesatdieren tlocationsonships andfromgaugesonnearbydocksalldierbymorethanafactor oftwo.Rain atseafallsmostlyhorizontallybecauseofwind,andtheshi p'ssuperstructure distortsthepathsofraindrops.Raininmanyareasfallsmos tlyasdrizzle,and itisdiculttodetectandmeasure. Themostaccuratemeasurementsofrainrateinthetropics( 35 )arecalculatedfrommicrowaveradiometersandradarobservations ofrainatseveral frequenciesusinginstrumentsontheTropicalRainMeasuri ngMission trmm launchedin1997.Rainforothertimesandlatitudescanbeca lculatedaccuratelybycombiningmicrowavedatawithinfraredobservati onsoftheheightof cloudtopsandwithraingaugedata(gure5.5).Rainisalsoc alculatedfrom thereanalysesweatherdatabynumericalmodelsoftheatmos phericcirculation (Schubert,Rood,andPfaendtner,1993),andbycombiningsh ipandsatellite observationswithanalysesfromnumericalweather-predic tionmodels(Xieand Arkin,1997). Thelargestsourceoferrorisduetoconversionofrainratet ocumulative rainfall,asamplingerror.Rainisveryrare,itislog-norm allydistributed,and mostraincomesfromafewstorms.Satellitestendtomisssto rms,anddata mustbeaveragedoverareasupto5 onasidetoobtainusefulvaluesofrainfall. NetLong-WaveRadiation NetLong-waveradiationisnoteasilycalculated becauseitdependsontheheightandthicknessofclouds,and thevertical distributionofwatervaporintheatmosphere.Itiscalcula tedbynumerical weather-predictionmodelsorfromobservationsofthevert icalstructureofthe atmospherefromatmosphericsounders.WaterFluxOut(LatentHeatFlux) Latentheatruxiscalculatedfrom shipobservationsofrelativehumidity,watertemperature ,andwindspeedusing bulkformulas(5.10c)andshipdataaccumulatedinthe icoads describedbelow.

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5.5.GLOBALDATASETSFORFLUXES 61 Global Precipitation for 1995 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1.0 1.5 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.5 1.5 1.5 1.5 1.5 1.5 2.5 0.5 2.5 2.5 1.5 0.5 0.5 0.5 0.5 1.5 2.5 1.5 1.5 20o60o100o140o180o-140o-100o-60o-20o20o-90o -60o-30o0o30o60o90o 0o Figure5.5Rainfallinm/yearcalculatedfromdatacompiled bytheGlobalPrecipitation ClimatologyProjectat nasa 'sGoddardSpaceFlightCenterusingdatafromraingauges, infraredradiometersongeosynchronousmeteorologicalsa tellites,andthe ssm/i .Contour intervalis0.5m/yr,lightshadedareasexceed2m/yr,heavy shadedareasexceed3m/yr. Theruxesarenotcalculatedfromsatellitedatabecausesat elliteinstruments arenotverysensitivetowatervaporclosetothesea.Perhap sthebestruxes arethosecalculatedfromnumericalweathermodels.SensibleHeatFlux Sensibleheatruxiscalculatedfromobservationsofairseatemperaturedierenceandwindspeedmadefromships,or bynumerical weathermodels.Sensibleruxesaresmallalmosteverywhere exceptoshoreof theeastcoastsofcontinentsinwinterwhencold,Arcticair massesextractheat fromwarm,western,boundarycurrents.Intheseareas,nume ricalmodelsgive perhapsthebestvaluesoftheruxes.Historicalshipreport givethelong-term meanvaluesoftheruxes.5.5GlobalDataSetsforFluxes Shipandsatellitedatahavebeenprocessedtoproducegloba lmapsofruxes. Shipmeasurementsmadeoverthepast150yearsyieldmapsoft helong-term meanvaluesoftheruxes,especiallyinthenorthernhemisph ere.Shipdata, however,aresparseintimeandspace,andtheyarebeingrepl acedmoreand morebyruxescalculatedbynumericalweathermodelsandbys atellitedata. Themostusefulmapsarethosemadebycombininglevel3and4s atellite datasetswithobservationsfromships,usingnumericalwea thermodels.Let's lookrstatthesourcesofdata,thenatafewofthemorewidel yuseddatasets. InternationalComprehensiveOcean-AtmosphereDataSet Datacollectedbyobserversonshipsaretherichestsourceofmarine information.Slutz etal.(1985)describingtheireortstocollect,edit,andp ublishallmarine observationswrite:

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62 CHAPTER5.THEOCEANICHEATBUDGET Since1854,shipsofmanycountrieshavebeentakingregular observationsoflocalweather,seasurfacetemperature,andmanyot hercharacteristicsneartheboundarybetweentheoceanandtheatmosp here.The observationsbyonesuchship-of-opportunityatonetimean dplace,usuallyincidentaltoitsvoyage,makeupamarinereport.Inlat eryears xedresearchvessels,buoys,andotherdeviceshavecontri buteddata. Marinereportshavebeencollected,ofteninmachine-reada bleform,by variousagenciesandcountries.Thatvastcollectionofdat a,spanningthe oceanfromthemid-nineteenthcenturytodate,isthehistor icaloceanatmosphererecord. Thesemarinereportshavebeeneditedandpublishedasthe InternationalComprehensiveOcean-AtmosphereDataSet icoads (Woodruetal.1987)available throughtheNationalOceanicandAtmosphericAdministrati on. The icoads release2.3includes213millionreportsofmarinesurfacec onditionscollectedfrom1784{2005bybuoys,otherplatformtyp es,andbyobservers onmerchantships.Thedatasetincludefullyquality-contr olled(trimmed)reportsandsummaries.Eachuniquereportcontains22observe dandderivedvariables,aswellasragsindicatingwhichobservationswerest atisticallytrimmed orsubjectedtoadaptivequalitycontrol.Here,statistica llytrimmedmeans outlierswereremovedfromthedataset.Thesummariesinclu dedinthedata setgive14statistics,suchasthemedianandmean,foreacho feightobserved variables:airandseasurfacetemperatures,windvelocity ,sea-levelpressure, humidity,andcloudiness,plus11derivedvariables. Thedatasetconsistsofaneasily-useddatabaseatthreepri ncipalresolutions:1)individualreports,2)year-monthsummariesofth eindividualreports in2 latitudeby2 longitudeboxesfrom1800to2005and1 latitudeby1 longitudeboxesfrom1960to2005,and3)decade-monthsumma ries.Notethat datafrom1784throughtheearly1800sareextremelysparse{ basedonscattered shipvoyages. Duplicatereportsjudgedinferiorbyarstqualitycontrol processdesigned bytheNationalClimaticDataCenter ncdc wereeliminatedorragged,and \untrimmed"monthlyanddecadalsummarieswerecomputedfo racceptable datawithineach2 latitudeby2 longitudegrid.Tighter,median-smoothed limitswereusedascriteriaforstatisticalrejectionofap parentoutliersfrom thedatausedforseparatesetsof trimmed monthlyanddecadalsummaries. Individualobservationswereretainedinreportformbutra ggedduringthis secondqualitycontrolprocessiftheyfelloutside2.8or3. 5estimatedstandarddeviationsaboutthesmoothedmedianapplicabletotheir2 latitudeby2 longitudebox,month,and56{,40{,or30{yearperiod( i.e. ,1854{1990,1910{ 1949,or1950{1979). Thedataaremostusefulinthenorthernhemisphere,especia llytheNorth Atlantic.Dataaresparseinthesouthernhemisphereandthe yarenotreliable southof30 S.GlecklerandWeare(1997)analyzedtheaccuracyofthe icoads dataforcalculatingglobalmapsandzonalaveragesoftheru xesfrom55 N to40 S.Theyfoundthatsystematicerrorsdominatedthezonalmea ns.Zonal averagesofinsolationwereuncertainbyabout10%,ranging from 10W/m 2

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5.5.GLOBALDATASETSFORFLUXES 63 inhighlatitudesto 25W/m 2 inthetropics.Longwaveruxeswereuncertain byabout 7W/m 2 .Latentheatruxuncertaintiesrangedfrom 10W/m 2 in someareasofthenorthernoceanto 30W/m 2 inthewesterntropicalocean to 50W/m 2 inwesternboundarycurrents.Sensibleheatruxuncertaint ies tendtobearound 5 10W/m 2 Joseyetal(1999)comparedaveragedruxescalculatedfrom icoads with ruxescalculatedfromobservationsmadebycarefullycalib ratedinstrumentson someshipsandbuoys.Theyfoundthatmeanruxintotheocean, whenaveraged overalltheseassurfacehaderrorsof 30W/m 2 .Errorsvaryseasonallyand byregion,andglobalmapsofruxesrequirecorrectionssuch asthoseproposed byDaSilva,Young,andLevitus(1995)showningure5.7.SatelliteData Rawdataareavailablefromsatelliteprojects,butweneed processeddata.Variouslevelsofprocesseddatafromsatel liteprojectsare produced(table5.3): Table5.3LevelsofProcessedSatelliteData LevelLevelofProcessing Level1Datafromthesatelliteinengineeringunits(volts)Level2Dataprocessedintogeophysicalunits(windspeed)a tthetimeandplace thesatelliteinstrumentmadetheobservation Level3Level2datainterpolatedtoxedcoordinatesintime andspace Level4Level3dataaveragedintimeandspaceorfurtherproc essed Theoperationalmeteorologicalsatellitesthatobserveth eoceaninclude: 1. noaa seriesofpolar-orbiting,meteorologicalsatellites; 2.U.S.DefenseMeteorologicalSatelliteProgram dmsp polar-orbitingsatellites,whichcarrytheSpecialSensorMicrowave/Imager (ssm/i) ; 3.Geostationarymeteorologicalsatellitesoperatedby noaa ( goes ),Japan ( gms )andtheEuropeanSpaceAgency( meteosats ). Dataarealsoavailablefrominstrumentsonexperimentalsa tellitessuchas: 1.Nimbus-7,EarthRadiationBudgetInstruments;2.EarthRadiationBudgetSatellite,EarthRadiationBudge tExperiment; 3.TheEuropeanSpaceAgency's ers {1&2; 4.TheJapaneseADvancedEarthObservingSystem( adeos )andMidori; 5.QuikScat;6.TheEarth-ObservingSystemsatellitesTerra,Aqua,andE nvisat; 7.TheTropicalRainfallMeasuringMission( trmm );and, 8.Topex/PoseidonanditsreplacementJason-1. Satellitedataarecollected,processed,andarchivedbygo vernmentorganizations.Archiveddataarefurtherprocessedtoproduceuse fulruxdatasets.

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64 CHAPTER5.THEOCEANICHEATBUDGET InternationalSatelliteCloudClimatologyProject TheInternationalSatelliteCloudClimatologyProjectisanambitiousprojectto collectobservations ofcloudsmadebydozensofmeteorologicalsatellitesfrom1 983to2000,to calibratethethesatellitedata,tocalculatecloudcoveru singcarefullyveried techniques,andtocalculatesurfaceinsolationandnetsur faceinfraredruxes (RossowandSchier,1991).Thecloudswereobservedwithvi sible-lightinstrumentsonpolar-orbitingandgeostationarysatellites.GlobalPrecipitationClimatologyProject Thisprojectusesthreesources ofdatatocalculaterainrate(Human,etal.1995,1997): 1.Infraredobservationsoftheheightofcumuluscloudsfro m goes satellites. Thebasicideaisthatthemorerainproducedbycumuluscloud s,the higherthecloudtop,andthecolderthetopappearsintheinf rared.Thus rainrateatthebaseofthecloudsisrelatedtoinfraredtemp erature. 2.Measurementsbyraingaugesonislandsandland.3.Radioemissionsfromwaterdropsintheatmosphereobserv edby ssm{i Accuracyisabout1mm/day.Datafromtheprojectareavailab leona2.5 latitudeby2.5 longitudegridfromJuly1987toDecember1995fromtheGloba l LandOceanPrecipitationAnalysisatthe nasa GoddardSpaceFlightCenter. XieandArkin(1997)produceda17-yeardatasetbasedonseve ntypes ofsatelliteandrain-gaugedatacombinedwiththeraincalc ulatedfromthe ncep/ncar reanalyzeddatafromnumericalweathermodels.Thedataset has thesamespatialandtemporalresolutionastheHumandatas et. ReanalyzedOutputFromNumericalWeatherModels Surfaceheatrux hasbeencalculatedfromweatherdatausingnumericalweath ermodelsbyvariousreanalysisprojectsdescribedin x 4.5.Theruxesareconsistentwithatmosphericdynamics,theyareglobal,theyarecalculatedever ysixhours,andthey areavailableformanyyearsonauniformgrid.Forexample,t he ncar/ncep reanalysis,availableona cd-rom ,includedailyaveragesofwindstress,sensible andlatentheatruxes,netlongandshortwaveruxes,near-su rfacetemperature, andprecipitation.AccuracyofCalculatedFluxes Recentstudiesoftheaccuracyofruxes computedbynumericalweathermodelsandreanalysisprojec tssuggest: 1.Heatruxesfromthe ncep and ecmwf reanalyseshavesimilarglobalaveragevalues,buttheruxeshaveimportantregionaldieren ces.Fluxes fromtheGoddardEarthObservingSystemreanalysisaremuch lessaccurate(Taylor,2000:258).Chouetal(2004)ndslargedie rencesin ruxescalculatedbydierentgroups. 2.Theruxesarebiasedbecausetheywerecalculatedusingnu mericalmodels optimizedtoproduceaccurateweatherforecasts.Thetimemeanvalues oftheruxesmaynotbeasaccurateasthetime-meanvaluescal culated directlyfromshipobservations.

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5.6.GEOGRAPHICDISTRIBUTIONOFTERMS 65 3.Thesimulationofboundary-layercloudsisasignicants ourceoferrorin calculatedruxes.Thepoorverticalresolutionofthenumer icalmodels doesnotadequatelyresolvethelow-levelcloudstructure( Taylor,2001). 4.Theruxeshavezonalmeansthatdiersignicantlyfromth esamezonal meanscalculatedfrom icoads data.Thedierencescanexceed40W/m 2 5.Theatmosphericmodelsdonotrequirethatthenetheatrux averaged overtimeandearth'ssurfacebezero.The ecmwf datasetaveraged overfteenyearsgivesanetruxof3.7W/m 2 intotheocean.The ncep reanalysisgivesanetruxof5.8W/m 2 outoftheocean(Taylor,2000: 206). Icoads datagiveanetruxof16W/m 2 intotheocean(gure5.7). Thusreanalyzedruxesaremostusefulforforcingclimatemo delsneedingactual heatruxesandwindstress. icoads dataaremostusefulforcalculatingtimemeanruxesexceptperhapsinthesouthernhemisphere.Overa ll,Taylor(2000) notesthattherearenoidealdatasets,allhavesignicanta ndunknownerrors. OutputFromNumericalWeatherModels Someprojectsrequireruxes afewhoursafterafterobservationsarecollected.Thesurf aceanalysisfrom numericalweathermodelsisagoodsourceforthistypeofrux 5.6GeographicDistributionofTermsintheHeatBudget Variousgroupshaveusedshipandsatellitedatainnumerica lweathermodelstocalculategloballyaveragedvaluesofthetermsforea rth'sheatbudget. Thevaluesgiveanoverallviewoftheimportanceofthevario usterms(gure 5.6).Noticethatinsolationbalancesinfraredradiationa tthetopoftheatmosphere.Atthesurface,latentheatruxandnetinfraredra diationtendto balanceinsolation,andsensibleheatruxissmall. Notethatonly20%ofinsolationreachingearthisabsorbedd irectlybythe atmospherewhile49%isabsorbedbytheoceanandland.Whatt henwarms theatmosphereanddrivestheatmosphericcirculation?The answerisrainand infraredradiationfromtheoceanabsorbedbythemoisttrop icalatmosphere. Here'swhathappens.Sunlightwarmsthetropicaloceanwhic hevaporateswater tokeepfromwarmingup.Theoceanalsoradiatesheattotheat mosphere,but thenetradiationtermissmallerthantheevaporativeterm. Tradewindscarry theheatintheformofwatervaportothetropicalconvergenc ezone.Therethe vaporcondensesasrain,releasingitslatentheat,andheat ingtheatmosphere byasmuchas125W/m 2 averagedoverayear(Seegure14.1). Atrstitmayseemstrangethatrainheatstheair.Afterall, wearefamiliar withsummertimethunderstormscoolingtheairatgroundlev el.Thecoolair fromthunderstormsisduetodowndrafts.Higherinthecumul uscloud,heat releasedbyrainwarmsthemid-levelsoftheatmospherecaus ingairtorise rapidlyinthestorm.Thunderstormsarelargeheatenginesc onvertingthe energyoflatentheatintokineticenergyofwinds. Thezonalaverageoftheoceanicheat-budgetterms(gure5. 7)showsthat insolationisgreatestinthetropics,thatevaporationbal ancesinsolation,and thatsensibleheatruxissmall. Zonalaverage isanaveragealonglinesof

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66 CHAPTER5.THEOCEANICHEATBUDGET Reflected by Clouds Aerosol and Atmosphere 77 Reflected SolarRadiation107 W m-2Incoming Solar Radiation 342 W m-2107 3427767 Absorbed byAtmosphere 30 40 AtmosphericWindow Greenhouse Gases Surface Radia tionEvapo-transpiration Thermals 24 78 350 40 324 Back Radiation 168 Absorbed by Surface 324 Absorbed by Surface Outgoing Longwave Radiation 235 W m-2235 Reflected by Surface 30 24 Latent Heat 78 Emitted by Atmosphere 165 390 Figure5.6Themeanannualradiationandheatbalanceofthee arth. AfterHoughtonetal.(1996:58),whichuseddatafromKiehla ndTrenberth(1996). 90o60o30o0o-30o-60o-90o250200150100 50 0 50 100150 100 50 0 50 QSWQLQSQLWTotal Heat Flux Heat Flux ComponentsW/m2W/m2 Figure5.7 Upper: Zonalaveragesofheattransfertotheoceanbyinsolation Q SW ,andloss byinfraredradiation Q LW ,sensibleheatrux Q S ,andlatentheatrux Q L ,calculatedby DaSilva,Young,andLevitus(1995)usingthe icoads dataset. Lower: Netheatruxthrough theseasurfacecalculatedfromthedataabove(solidline)a ndnetheatruxconstrainedto giveheatandothertransportsthatmatchindependentcalcu lationsofthesetransports.The areaunderthelowercurvesoughttobezero,butitis16W/m 2 fortheunconstrainedcase and-3W/m 2 fortheconstrainedcase.

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5.6.GEOGRAPHICDISTRIBUTIONOFTERMS 67 Total Sky Net Insolation (W/m2) 100 100 100 150 150 100 100 200 200 200 200 200 250 200 250 20o60o100o140o180o-140o-100o-60o-20o20o-90o -60o-30o0o30o60o90o 0o Net Infrared Flux (W/m2) -40 -40 -40 -40 -30 -40 -40 -40 -50 -50 -50 -50 -50 -50 -70 -40 -40 -40 -40 -40 -40 -40 -30 -30 -50 -50-60 -40 -50 -50 -60 -30 -50 -40 20o60o100o140o180o-140o-100o-60o-20o20o-90o -60o-30o0o30o60o90o 0o Figure5.8Annual-meaninsolation Q SW ( top )andinfraredradiation Q LW ( bottom ) throughtheseasurfaceduring1989calculatedbytheSatell iteDataAnalysisCenterat the nasa LangleyResearchCenter(Darnelletal.,1992)usingdatafr omtheInternational SatelliteCloudClimatologyProject.UnitsareW/m 2 ,contourintervalis10W/m 2 constantlatitude.Notethatthetermsingure5.7don'tsum tozero.The areal-weightedintegralofthecurvefortotalheatruxisno tzero.Becausethe netheatruxintotheoceanaveragedoverseveralyearsmustb elessthanafew wattspersquaremeter,thenon-zerovaluemustbeduetoerro rsinthevarious termsintheheatbudget. Errorsintheheatbudgettermscanbereducedbyusingadditi onalinformation.Forexample,weknowroughlyhowmuchheatandotherq uantities aretransportedbytheoceanandatmosphere,andtheknownva luesforthese transportscanbeusedtoconstrainthecalculationsofneth eatruxes(gure

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68 CHAPTER5.THEOCEANICHEATBUDGET Surface Evaporation For 1989 (W/m2) 50 50 50 50 50 50 100 100 100 100 100 50 20o60o100o140o180o-140o-100o-60o-20o20o-90o -60o-30o0o30o60o90o 0o Figure5.9Annual-meanlatentheatruxfromtheseasurface Q L inW/m 2 during1989 calculatedfromdatacompiledbytheDataAssimilationOce of nasa 'sGoddardSpace FlightCenterusingreanalyzeddatafromthe ecmwf numericalweatherpredictionmodel. Contourintervalis10W/m 2 5.7).Theconstrainedruxesshowthattheheatgainedbytheo ceaninthe tropicsisbalancedbyheatlostbytheoceanathighlatitude s. Mapsoftheregionaldistributionofruxesgivecluestothep rocessesproducingtheruxes.Cloudsregulatetheamountofsunlightreachi ngtheseasurface (gure5.8top),andsolarheatingiseverywherepositive.T henetinfraredheat rux(gure5.8bottom)islargestinregionswiththeleastcl ouds,suchasthe centeroftheoceanandtheeasterncentralPacic.Thenetin fraredruxis everywherenegative.Latentheatruxes(gure5.9)aredomi natedbyevaporationinthetradewindregionsandtheoshorerowofcoldairm assesbehind coldfrontsinwinteroshoreofJapanandNorthAmerica.Sen sibleheatruxes (gure5.10top)aredominatedbycoldairblowingocontine nts.Thenet heatinggain(gure5.10bottom)islargestinequatorialre gionsandnetheat lossislargestdownwindonAsiaandNorthAmerica. Heatruxeschangesubstantiallyfromyeartoyear,especial lyinthetopics, especiallyduetoElNi~no.SeeChapter14formoreontropica lvariability. 5.7MeridionalHeatTransport Overall,earthgainsheatatthetopofthetropicalatmosphe re,anditloses heatatthetopofthepolaratmosphere.Theatmosphericando ceaniccirculationtogethermusttransportheatfromlowtohighlatitudes tobalancethegains andlosses.Thisnorth-southtransportiscalledthe meridionalheattransport Thesumofthemeridionalheattransportintheoceanandatmo sphereis calculatedfromthezonalaverageofthenetheatruxthrough thetopofthe atmospheremeasuredbysatellites.Inmakingthecalculati on,weassumethat transportsaveragedoverafewyearsaresteady.Thusanylon g-term,netheat

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5.7.MERIDIONALHEATTRANSPORT 69 Corrected Sensible Heat Flux (W/m2) 0 0 0 10 0 1010100 20 20 20 20 30 0 0 10 20 10 0 0 40 10 30 10 101010101030202020 403030020o60o100o140o180o-140o-100o-60o-20o20o-90o -60o-30o0o30o60o90o 0o Constrained Net Heat Flux Annual Mean (W/m2) 20 20 20 -20 20 20 80 -20 20 80 80 -20 20 20 -20 -20 20 -20 80 20 80 20o60o100o140o180o-140o-100o-60o-20o20o-90o -60o-30o0o30o60o90o 0o Figure5.10Annual-meanupwardsensibleheatrux Q S ( top )andconstrained,net,downward heatrux( bottom )throughtheseasurfaceinW/m 2 calculatedbyDaSilva,Young,and Levitus(1995)usingthe icoads datasetfrom1945to1989.Contourintervalis2W/m 2 (top)and20W/m 2 (bottom). gainorlossthroughthetopoftheatmospheremustbebalance dbyameridional transportandnotbyheatstorageintheoceanoratmosphere.NetHeatFluxattheTopoftheAtmosphere Heatruxthroughthetop oftheatmosphereismeasuredveryaccuratelybyradiometer sonsatellites. 1.Insolationiscalculatedfromthesolarconstantandobse rvationsofrerectedsunlightmadebymeteorologicalsatellitesandbysp ecialsatellites oftheEarthRadiationBudgetExperiment. 2.Infraredradiationismeasuredbyinfraredradiometerso nthesatellites.

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70 CHAPTER5.THEOCEANICHEATBUDGET 3.Thedierencebetweeninsolationandnetinfraredradiat ionisthenet heatruxacrossthetopoftheatmosphere. NetMeridionalHeatTransport Tocalculatethemeridionalheattransport intheatmosphereandtheocean,werstaveragethenetheatr uxthroughthe topoftheatmosphereinazonalband.Becausethemeridional derivativeofthe transportisthezonal-meanrux,wecalculatethetransport fromthemeridional integralofthezonal-meanrux.Theintegralmustbebalance dbytheheat transportedbytheatmosphereandtheoceanacrosseachlati tudeband. CalculationsbyTrenberthandCaron(2001)showthatthetot al,annualmean,meridionalheattransportbytheoceanandatmosphere peaksat6PW towardeachpoleat35 latitude. OceanicHeatTransport Themeridionalheattransportintheoceancanbe calculatedthreeways: 1. SurfaceFluxMethod calculatestheheatruxthroughtheseasurfacefrom measurementsofwind,insolation,air,andseatemperature ,andcloudiness.Theruxesareintegratedtoobtainthezonalaverageof theheat rux(gure5.7).Finally,wecalculatethetransportfromth emeridional integralofthezonal-meanruxjustaswedidatthetopofthea tmosphere. 2. DirectMethod calculatestheheattransportfromvaluesofcurrentvelocityandtemperaturemeasuredfromtoptobottomalongazonal section spanninganoceanbasin.Theruxistheproductofnorthwardv elocity andheatcontentderivedfromthetemperaturemeasurement. 3. ResidualMethod rstcalculatestheatmosphericheattransportfromatmosphericmeasurementsortheoutputofnumericalweatherm odels.This isthedirectmethodappliedtotheatmosphere.Theatmosphe rictransportissubtractedfromthetotalmeridionaltransportcalc ulatedfromthe top-of-the-atmosphereheatruxtoobtaintheoceaniccontr ibutionasa residual(gure5.11). Variouscalculationsofoceanicheattransports,suchasth oseshowningure 5.11,tendtobeinagreement,andtheerrorbarsshowninthe gurearerealistic. Thetotalmeridionaltransportofheatbytheoceanissmallc omparedwiththe totalmeridionalheattransportbytheatmosphereexceptin thetropics.At35 wherethetotalmeridionalheattransportisgreatest,theo ceancarriesonly22% oftheheatinthenorthernhemisphere,and8%inthesouthern (Trenberthand Caron,2001).5.8VariationsinSolarConstant Wehaveassumedsofarthatthesolarconstant,theoutputofl ightandheat fromthesun,issteady.Recentevidencebasedonvariabilit yofsunspotsand faculae(brightspots)showsthattheoutputvariedby 0 : 2%overcenturies (Lean,Beer,andBradley,1995),andthatthisvariabilityi scorrelatedwith changesinglobalmeantemperatureofearth'ssurfaceof 0 : 4 C.(gure5.12). Inaddition,WhiteandCayan(1998)foundasmall12yr,22yr, andlongerperiodvariationsofsea-surfacetemperaturemeasuredbyb athythermographs

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5.8.VARIATIONSINSOLARCONSTANT 71 PacificAtlanticIndian Total Global Heat Transport Northward Heat Transport (PW) 2.52.01.51.00.5 0 -0.5-1.0-1.5-2.0-2.5 Latitude -80o-60o-40o-20o0o20o40o60o80o Figure5.11Northwardheattransportfor1988ineachoceana ndthetotaltransportsummed overalloceancalculatedbytheresidualmethodusingatmos phericheattransportfrom ecmwf andtopoftheatmosphereheatruxesfromtheEarthRadiation BudgetExperiment satellite.AfterHoughtonetal.(1996:212),whichuseddat afromTrenberthandSolomon (1994).1PW=1petawatt=10 15 W. Surface Temperature solar plus manmade forcing mostly volcanic forcing mostly solar forcing Paleo reconstructionNH instrumentalReconstructed solartotal irradiance Year 16001700180019002000Surface Temperature Anomalies (Celsius)0.2o0.0o-0.2o-0.4o-0.6oSolar Total Irradiance (W/m2)136913681367136613651364 Figure5.12Changesinsolarconstant(totalsolarirradian ce)andglobalmeantemperature ofearth'ssurfaceoverthepast400years.Exceptforaperio dofenhancedvolcanicactivity intheearly19thcentury,surfacetemperatureiswellcorre latedwithsolarvariability.After Lean,personalcommunication.

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72 CHAPTER5.THEOCEANICHEATBUDGET andship-boardthermometersoverthepastcentury.Theobse rvedresponseof earthtosolarvariabilityisaboutthatcalculatedfromnum ericalmodelsofthe coupledocean-atmosphereclimatesystem.Manyotherchang esinclimateand weatherhavebeenattributedtosolarvariability.Thecorr elationsaresomewhat controversial,andmuchmoreinformationcanbefoundinHoy tandSchatten's (1997)bookonthesubject.5.9ImportantConcepts 1.Sunlightisabsorbedprimarilyinthetropicalocean.The amountofsunlightchangeswithseason,latitude,timeofday,andcloudc over. 2.Mostoftheheatabsorbedbytheoceaninthetropicsisrele asedaswater vaporwhichheatstheatmospherewhenwateriscondensesasr ain.Most oftherainfallsinthetropicalconvergencezones,lessera mountsfallin mid-latitudesnearthepolarfront. 3.Heatreleasedbyrainandabsorbedinfraredradiationfro mtheoceanare theprimarydriversfortheatmosphericcirculation. 4.Thenetheatruxfromtheoceanislargestinmid-latitudes andoshore ofJapanandNewEngland. 5.Heatruxescanbemeasureddirectlyusingfastresponsein strumentson low-ryingaircraft,butthisisnotusefulformeasuringhea truxesover largeoceanicregions. 6.Heatruxesthroughlargeregionsoftheseasurfacecanbec alculatedfrom bulkformula.Seasonal,regional,andglobalmapsofruxesa reavailable basedonshipandsatelliteobservations. 7.Themostwidelyuseddatasetsforstudyingheatruxesaret heInternationalComprehensiveOcean-AtmosphereDataSetandtherea nalysisof meteorologicaldatabynumericalweatherpredictionmodel s. 8.Theatmospheretransportsmostoftheheatneededtowarml atitudes higherthan35 .Theoceanicmeridionaltransportiscomparabletothe atmosphericmeridionaltransportonlyinthetropics. 9.Solaroutputisnotconstant,andtheobservedsmallvaria tionsinoutput ofheatandlightfromthesunseemtoproducethechangesingl obal temperatureobservedoverthepast400years.

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Chapter6Temperature,Salinity,andDensityHeatruxes,evaporation,rain,riverinrow,andfreezingan dmeltingofseaice allinruencethedistributionoftemperatureandsalinitya ttheocean'ssurface. Changesintemperatureandsalinitycanincreaseordecreas ethedensityofwateratthesurface,whichcanleadtoconvection.Ifwaterfro mthesurfacesinks intothedeeperocean,itretainsadistinctiverelationshi pbetweentemperature andsalinitywhichhelpsoceanographerstrackthemovement ofdeepwater.In addition,temperature,salinity,andpressureareusedtoc alculatedensity.The distributionofdensityinsidetheoceanisdirectlyrelate dtothedistributionof horizontalpressuregradientsandoceancurrents.Forallt hesereasons,weneed toknowthedistributionoftemperature,salinity,anddens ityintheocean. Beforediscussingthedistributionoftemperatureandsali nity,let'srstdenewhatwemeanbytheterms,especiallysalinity.6.1DenitionofSalinity Atthesimplestlevel,salinityisthetotalamountofdissol vedmaterialin gramsinonekilogramofseawater.Thussalinityisadimensi onlessquantity. Ithasnounits.Thevariabilityofdissolvedsaltisverysma ll,andwemustbe verycarefultodenesalinityinwaysthatareaccurateandp ractical.Tobetter understandtheneedforaccuracy,lookatgure6.1.Noticet hattherangeof salinityformostoftheocean'swaterisfrom34.60to34.80p artsperthousand, whichis200partspermillion.ThevariabilityinthedeepNo rthPaciciseven smaller,about20partspermillion.Ifwewanttoclassifywa terwithdierent salinity,weneeddenitionsandinstrumentsaccuratetoab outonepartper million.Noticethattherangeoftemperatureismuchlarger ,about1 C,and temperatureiseasiertomeasure. Writingapracticaldenitionofsalinitythathasusefulac curacyisdicult (seeLewis,1980,forthedetails),andvariousdenitionsh avebeenused. 73

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74 CHAPTER6.TEMPERATURE,SALINITY,ANDDENSITY 34.4034.5034.7034.8034.9034.6035.00 Potent ial Temperature (Celsius)Salinity4o3o2o1o0oWorld Ocean Figure6.1Histogramoftemperatureandsalinityofoceanwa tercolderthan4 C.Heightis proportionaltovolume.Heightofhighestpeakcorresponds toavolumeof26millioncubic kilometersperbivariateclassof0.1 Cand0.01.AfterWorthington(1981:47). ASimpleDenition Originallysalinitywasdenedtobethe\Totalamount ofdissolvedmaterialingramsinonekilogramofseawater." Thisisnotuseful becausethedissolvedmaterialisalmostimpossibletomeas ureinpractice.For example,howdowemeasurevolatilemateriallikegasses?No rcanweevaporatesea-watertodrynessbecausechloridesarelostinthel aststagesofdrying (Sverdrup,Johnson,andFleming,1942:50).AMoreCompleteDenition Toavoidthesediculties,theInternational CouncilfortheExplorationoftheSeasetupacommissionin1 889whichrecommendedthatsalinitybedenedasthe\Totalamountofsoli dmaterialsin gramsdissolvedinonekilogramofseawaterwhenallthecarb onatehasbeen convertedtooxide,thebromineandiodinereplacedbychlor ineandallorganic mattercompletelyoxidized."Thedenitionwaspublishedi n1902. Thisis usefulbutdiculttouseroutinely SalinityBasedonChlorinity Becausetheabovedenitionwasdicultto implementinpractice,becausesalinityisdirectlypropor tionaltotheamount ofchlorineinseawater,andbecausechlorinecanbemeasure daccuratelybya simplechemicalanalysis,salinity S wasredenedusingchlorinity: S =0 : 03+1 : 805 Cl (6.1) where chlorinity Cl isdenedas\themassofsilverrequiredtoprecipitate completelythehalogensin0.3285234kgofthesea-watersam ple." Asmoreandmoreaccuratemeasurementsweremade,(6.1)turn edoutto betooinaccurate.In1964 unesco andotherinternationalorganizationsappointedaJointPanelonOceanographicTablesandStandards toproducea

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6.1.DEFINITIONOFSALINITY 75 moreaccuratedenition.TheJointPanelrecommendedin196 6(Wooster,Lee, andDietrich,1969)thatsalinityandchlorinityberelated using: S =1 : 80655 Cl (6.2) Thisisthesameas(6.1)for S =35. SalinityBasedonConductivity Atthesametime(6.2)wasadopted,oceanographershadbeganusingconductivitymeterstomeasuresa linity.Themeters wereverypreciseandrelativelyeasytousecomparedwithth echemicaltechniquesusedtomeasurechlorinity.Asaresult,theJointPan elalsorecommended thatsalinityberelatedtoconductivityofseawaterusing: S = 0 : 08996+28 : 29729 R 15 +12 : 80832 R 2 15 10 : 67869 R 3 15 +5 : 98624 R 4 15 1 : 32311 R 5 15 (6.3a) R 15 = C ( S; 15 ; 0) =C (35 ; 15 ; 0)(6.3b) where C ( S; 15 ; 0)istheconductivityofthesea-watersampleat15 Candatmosphericpressure,havingasalinity S derivedfrom(6.4),and C (35 ; 15 ; 0)is theconductivityofstandard\Copenhagen"seawater.Mille ro(1996)points outthat(6.3)isnotanewdenitionofsalinity,itmerelygi veschlorinityasa functionofconductivityofseawaterrelativetostandards eawater. PracticalSalinityScaleof1978 Bytheearly1970s,accurateconductivity meterscouldbedeployedfromshipstomeasureconductivity atdepth.The needtore-evaluatethesalinityscaleledtheJointPanelto recommendin1981 ( jpots ,1981;Lewis,1980)thatsalinitybedenedusingonlycondu ctivity, breakingthelinkwithchlorinity.Allwatersampleswithth esameconductivity ratiohavethesamesalinityeventhoughthetheirchlorinit ymaydier. The PracticalSalinityScaleof1978 isnowtheocialdenition: S =0 : 0080 0 : 1692 K 1 = 2 15 +25 : 3851 K 15 +14 : 0941 K 3 = 2 15 7 : 0261 K 2 15 +2 : 7081 K 5 = 2 15 (6.4a) K 15 = C ( S; 15 ; 0) =C ( KCl; 15 ; 0)(6.4b) 2 S 42 where C ( S; 15 ; 0)istheconductivityofthesea-watersampleatatemperatu reof 14.996 ContheInternationalTemperatureScaleof1990( its -90,see x 6.2)and standardatmosphericpressureof101325Pa. C ( KCl; 15 ; 0)istheconductivity ofthestandardpotassiumchloride(KCl)solutionatatempe ratureof15 Cand standardatmosphericpressure.ThestandardKClsolutionc ontainsamassof 32 : 4356gramsofKClinamassof1 : 000000kgofsolution.Millero(1996:72) andLewis(1980)givesequationsforcalculatingsalinitya totherpressuresand temperatures.Comments Thevariousdenitionsofsalinityworkwellbecausetherat iosof thevariousionsinseawaterarenearlyindependentofsalin ityandlocationin

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76 CHAPTER6.TEMPERATURE,SALINITY,ANDDENSITY theocean(table6.1).Onlyveryfreshwaters,suchasarefou ndinestuaries,have signicantlydierentratios.TheresultisbasedonDittma r's(1884)chemical analysisof77samplesofseawatercollectedbythe Challenger Expeditionand furtherstudiesbyCarrittandCarpenter(1959). Theimportanceofthisresultcannotbeoveremphasized,asu ponitdependsthevalidityofthechlorinity:salinity:densityrel ationshipsand, hence,theaccuracyofallconclusionsbasedonthedistribu tionofdensity wherethelatterisdeterminedbychemicalorindirectphysi calmethods suchaselectricalconductivity ::: |Sverdrup,Johnson,Fleming(1942). Therelationshipbetweenconductivityandsalinityhasana ccuracyofaround 0 : 003insalinity.Theverysmallerroriscausedbyvariations inconstituents suchasSiO 2 whichcausesmallchangesindensitybutnochangeinconduct ivity. Table6.1MajorConstituentsofSeaWater IonAtoms 55.3%Chlorine55.3%Chlorine30.8%Sodium30.8%Sodium 7.7%Sulfate3.7%Magnesium3.7%Magnesium2.6%Sulfur1.2%Calcium1.2%Calcium1.1%Potassium1.1%Potassium ReferenceSeawaterandSalinity ThePracticalSalinityScaleof1978introducedseveralsmallproblems.Itledtoconfusionaboutu nitsandtothe useof\practicalsalinityunits"thatarenotpartofthede nitionofPractical Salinity.Inaddition,absolutesalinitydiersfromsalin itybyabout0.5%.And, thecompositionofseawaterdiersslightlyfromplacetopl aceintheocean, leadingtosmallerrorsinmeasuringsalinity. Toavoidtheseandotherproblems,Milleroetal(2008)dene danewmeasureofsalinity,theReferenceSalinity,thataccuratelyr epresentstheAbsolute Salinityofanarticialseawatersolution.ItisbasedonaR eferenceComposition ofseawaterthatismuchmoreaccuratethanthevaluesinTabl e6.1above.The ReferenceComposition ofthearticialseawaterisdenedbyalistofsolutes andtheirmolefractionsgiveninTable4oftheirpaper.From this,theydened articial ReferenceSeawater tobeseawaterhavingaReferenceCompositionsolutedissolvedinpurewaterasthesolvent,andadjustedtoi tsthermodynamic equilibriumstate.Finally,the ReferenceSalinity ofReferenceSeawaterwas denedtobeexactly35.16504gkg 1 Withthesedenitions,plusmanydetailsdescribedintheir paper,Millero etal(2008)showReferenceSalinityisrelatedtoPractical Salinityby: S R (35 : 16504 = 35)gkg 1 S (6.5) Theequationisexactat S =35.ReferenceSalinityisapproximately0.47% largerthanPracticalSalinity.ReferenceSalinity S R isintendedtobeusedas anSI-basedextensionofPracticalSalinity.

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6.2.DEFINITIONOFTEMPERATURE 77 6.2DenitionofTemperature Manyphysicalprocessesdependontemperature.Afewcanbeu sedtodene absolutetemperature T .Theunitof T isthekelvin,whichhasthesymbolK. Thefundamentalprocessesusedfordeninganabsolutetemp eraturescaleover therangeoftemperaturesfoundintheoceaninclude(Soulen andFogle,1997): 1)thegaslawsrelatingpressuretotemperatureofanidealg aswithcorrections forthedensityofthegas;and2)thevoltagenoiseofaresist ance R Themeasurementoftemperatureusinganabsolutescaleisdi cultandthe measurementisusuallymadebynationalstandardslaborato ries.Theabsolute measurementsareusedtodeneapracticaltemperaturescal ebasedonthe temperatureofafewxedpointsandinterpolatingdevicesw hicharecalibrated atthexedpoints. Fortemperaturescommonlyfoundintheocean,theinterpola tingdeviceis aplatinum-resistancethermometer.Itconsistsofaloosel ywound,strain-free, pureplatinumwirewhoseresistanceisafunctionoftempera ture.Itiscalibrated atxedpointsbetweenthetriplepointofequilibriumhydro genat13.8033K andthefreezingpointofsilverat961.78K,includingthetr iplepointofwater at0.060 C,themeltingpointofGalliumat29.7646 C,andthefreezingpoint ofIndiumat156.5985 C(Preston-Thomas,1990).Thetriplepointofwateris thetemperatureatwhichice,water,andwatervaporareineq uilibrium.The temperaturescaleinkelvin T isrelatedtothetemperaturescaleindegrees Celsius t [ C]by: t [ C]= T [K] 273 : 15(6.6) Thepracticaltemperaturescalewasrevisedin1887,1927,1 948,1968,and 1990asmoreaccuratedeterminationsofabsolutetemperatu rebecameaccepted. ThemostrecentscaleistheInternationalTemperatureScal eof1990( its -90). ItdiersslightlyfromtheInternationalPracticalTemper atureScaleof1968 ipts -68.At0 Ctheyarethesame,andabove0 C its -90isslightlycooler. t 90 t 68 = 0 : 002at10 C, 0 : 005at20 C, 0 : 007at30 Cand 0 : 010at 40 C. Noticethatwhileoceanographersusethermometerscalibra tedwithanaccuracyofamillidegree,whichis0.001 C,thetemperaturescaleitselfhasuncertaintiesofafewmillidegrees.6.3GeographicalDistributionofSurfaceTemperatureandS alinity Thedistributionoftemperatureattheseasurfacetendstob e zonal ,thatis, itisindependentoflongitude(gure6.2).Warmestwateris neartheequator, coldestwaterisnearthepoles.Thedeviationsfromzonalar esmall.Equatorwardof40 ,coolerwaterstendtobeontheeasternsideofthebasin.Nor thof thislatitude,coolerwaterstendtobeonthewesternside. The anomalies ofsea-surfacetemperature,thedeviationfromalongterm average,aresmall,lessthan1.5 C(HarrisonandLarkin,1998)exceptinthe equatorialPacicwherethedeviationscanbe3 C(gure6.3:top). Theannualrangeofsurfacetemperatureishighestatmid-la titudes,especiallyonthewesternsideoftheocean(gure6.3:bottom).I nthewest,coldair

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78 CHAPTER6.TEMPERATURE,SALINITY,ANDDENSITY Average Sea-Surface Temperature for January 0 0 5 5 10 10 15 15 20 20 25 25 25 25 28 28 28 25 25 20 20 15 15 10 10 5 5 5 5 0 0 0 0 20o60o100o140o180o-140o-100o-60o-20o20o-90o -60o-30o0o30o60o90o 0o Average Sea-Surface Temperature for July 0 0 5 5 10 10 15 15 20 20 20 25 25 25 25 25 25 25 20 20 15 15 10 10 5 5 5 0 0 0 28 28 28 28 20o60o100o140o180o-140o-100o-60o-20o20o-90o -60o-30o0o30o60o90o 0o Figure6.2Meansea-surfacetemperaturecalculatedfromth eoptimalinterpolationtechnique (ReynoldsandSmith,1995)usingshipreportsand avhrr measurementsoftemperature. Contourintervalis1 Cwithheavycontoursevery5 C.Shadedareasexceed29 C. blowsothecontinentsinwinterandcoolstheocean.Thecoo lingdominates theheatbudget.Inthetropicsthetemperaturerangeismost lylessthan2 C. Thedistributionofsea-surfacesalinityalsotendstobezo nal.Thesaltiest watersareatmid-latitudeswhereevaporationishigh.Less saltywatersarenear theequatorwhererainfreshensthesurface,andathighlati tudeswheremelted seaicefreshensthesurface(gure6.4).Thezonal(east-we st)averageofsalinity showsaclosecorrelationbetweensalinityandevaporation minusprecipitation plusriverinput(gure6.5). BecausemanylargeriversdrainintotheAtlanticandtheArc ticSea,why istheAtlanticsaltierthanthePacic?Broecker(1997)sho wedthat0.32Svof

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6.3.GEOGRAPHICALDISTRIBUTION 79 thewaterevaporatedfromtheAtlanticdoesnotfallasraino nland.Instead, itiscarriedbywindsintothePacic(gure6.6).Broeckerp ointsoutthatthe quantityissmall,equivalenttoalittlemorethantherowin theAmazonRiver, but\werethisruxnotcompensatedbyanexchangeofmoresalt yAtlantic watersforlesssaltyPacicwaters,thesalinityoftheenti reAtlanticwouldrise about1gramperliterpermillennium." MeanTemperatureandSalinityoftheOcean Themeantemperatureofthe 1 1 2 2 2 3 3 3 3 4 4 4 4 4 4 6 5 3 3 3 3 3 3 2 4 4 4 4 5 5 4 4 4 4 1 1 1 1 2 2 2 2 3 3 3 3 9 10 8 6 6 7 3 2 4 4 4 3 3 5 15 12 4 5 5 5 5 5 5 5 6 6 6 6 6 7 11 12 14 9 8 6 4 7 8 9 10 11 12 13 14 14 15 16 17 2 2 2 2 5 5 5 5 5 6 2 1 7 8 9 6 3 2 1 2 2 5 6 2 7 7 8 8 8 9 9 10 11 12 13 14 15 16 17 18 19 20 7 10 11 5 6 16 17 Annual Range of Sea-Surface Temperature 20o60o100o140o180o-140o-100o-60o-20o20o-90o -60o-30o0o30o60o90o 0o 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 1 1 1 0 0 0 1 0 0 0 1 1 Optimal Interpolation Monthly SST Anomalies for Jan. 1996 20o60o100o140o180o-140o-100o-60o-20o20o-90o -60o-30o0o30o60o90o 0o Figure6.3 Top: Sea-surfacetemperatureanomalyforJanuary1996relative tomean temperatureshowningure6.2usingdatapublishedbyReyno ldsandSmith(1995)inthe ClimateDiagnosticsBulletin forFebruary1995.Contourintervalis1 C.Shadedareas arepositive. Bottom: Annualrangeofsea-surfacetemperaturein Ccalculatedfromthe ReynoldsandSmith(1995)meansea-surfacetemperaturedat aset.Contourintervalis1 C withheavycontoursat4 Cand8 C.Shadedareasexceed8 C.

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80 CHAPTER6.TEMPERATURE,SALINITY,ANDDENSITY Annual Mean Sea Surface Salinity 34 34 35 35 35 35 35 35 36 37 34 34 34 34 34 33 33 33 33 33 32 32 32 32 32 36 37 35 35 34 34 31 30 30 30 30 30 31 37 33 32 36 34 34 34 36 34 35 34 33 32 31 35 29 39 20o60o100o140o180o-140o-100o-60o-20o20o-90 o -60o-30o0o30o60o90o 0 o 0 .5 -0. 5 -0.5 -0. 5 -1.0 0 5 0.5 1.0 1.5 -1.0 1.5 2.5 0.5 -0.5 2.0 2.5 3.0 0.5 0.5 0.5 -0 5 -1.0 -0 5 0.5 1.5 2.0 2.0 -0.5 -1. 0 -0. 5 0.5 -1. 0 0.5 1.0 0.5 1.0 1.5 0. 5 0.5 Annual Mean Precipitation Evaporation (m/yr) 20o60o100o140o180o-140o-100o-60o-20o20o-90 o -60o-30o0o30o60o90o 0 o Figure6.4 Top :Meansea-surfacesalinity.Contourintervalis0.25.Shad edareasexceed asalinityof36.FromLevitus(1982). Bottom :Precipitationminusevaporationinmeters peryearcalculatedfromglobalrainfallbytheGlobalPreci pitationClimatologyProjectand latentheatruxcalculatedbytheDataAssimilationOce,bo that nasa 'sGoddardSpace FlightCenter.Precipitationexceedsevaporationinthesh adedregions,contourintervalis 0.5m.ocean'swatersis:t=3.5 C.ThemeansalinityisS=34.7.Thedistribution aboutthemeanissmall:50%ofthewaterisintherange: 1 : 3 C
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6.4.THEOCEANICMIXEDLAYERANDTHERMOCLINE 81 evaporation precipitation salinity Evaporation precipitation (m/yr) 0.5 1.0 -1.5 -1.0 -0.5 0.0 -90o-60o-30o0o30o60o90o30 32 34 36 Salinity (p s u) Figure6.5Zonalaverageofsea-surfacesalinitycalculate dforalltheoceanfromLevitus (1982)andthedierencebetweenevaporationandprecipita tion( E P )calculatedfrom datashowningure6.4(bottom).6.4TheOceanicMixedLayerandThermocline Windblowingontheoceanstirstheupperlayersleadingtoat hin mixed layer attheseasurfacehavingconstanttemperatureandsalinity fromthe surfacedowntoadepthwherethevaluesdierfromthoseatth esurface.The magnitudeofthedierenceisarbitrary,buttypicallythet emperatureatthe bottomofthelayermustbenomorethan0.02{0.1 colderthanatthesurface. 0.23 +0.03 -0.32 Sv 0.36 0.07 0.20 0.19 -0.17 -0.18 0.25 Desert Desert 0o3 0o6 0o90o120o150o180o-30o-60o-90o-120o-150o18 0o 75o60o45o30o15o0o-15o-30o-45o-60o Figure6.6Watertransportedbytheatmosphereintoandouto ftheAtlantic.Basins drainingintotheAtlanticareblack,desertsarewhite,and otherdrainagebasinsareshaded. Arrowsgivedirectionofwatertransportbytheatmosphere, andvaluesareinSverdrups. BoldnumbersgivethenettransportfortheAtlanticateachl atitudeband.Overall,the Atlanticloses0.32Sv,anamountapproximatelyequaltothe rowintheAmazonRiver. AfterBroecker(1997).

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82 CHAPTER6.TEMPERATURE,SALINITY,ANDDENSITY Sept 90 Aug 90 Nov 89 Jan 90 April 90 30o28o26o24o22o20o18o0 50 100150200 Temperature (Celsius)Pressure (decibars) Figure6.7Growthanddecayofthemixedlayerandseasonalth ermoclinefromNovember 1989toSeptember1990attheBermudaAtlanticTime-seriesS tation( bats )at31.8 N 64.1 W.DatawerecollectedbytheBermudaBiologicalStationfor Research,Inc.Note thatpressureindecibarsisnearlythesameasdepthinmeter s(see x 6.8foradenitionof decibars).Notethatbothtemperatureandsalinitymustbeconstantint hemixedlayer. Wewillseelaterthatmeanvelocityisnotconstant.Themixe dlayerisroughly 10{200mthickovermostofthetropicalandmid-latitudebel ts. Thedepthandtemperatureofthemixedlayervariesfromdayt odayand fromseasontoseasoninresponsetotwoprocesses: 1.Heatruxesthroughthesurfaceheatandcoolthesurfacewa ters.Changes intemperaturechangethedensitycontrastbetweenthemixe dlayerand deeperwaters.Thegreaterthecontrast,themoreworkisnee dedtomix thelayerdownwardandvisaversa. 2.Turbulenceinthemixedlayermixesheatdownward.Thetur bulence dependsonthewindspeedandontheintensityofbreakingwav es.Turbulencemixeswaterinthelayer,anditmixesthewaterinthe layerwith waterinthethermocline. Themid-latitudemixedlayeristhinnestinlatesummerwhen windsare weak,andsunlightwarmsthesurfacelayer(gure6.7).Atti mes,theheating issostrong,andthewindssoweak,thatthelayerisonlyafew metersthick.In fall,therststormsoftheseasonmixtheheatdownintotheo ceanthickening themixedlayer,butlittleheatislost.Inwinter,heatislo st,andthemixed layercontinuestothicken,becomingthickestinlatewinte r.Inspring,winds weaken,sunlightincreases,andanewmixedlayerforms. Belowthemixedlayer,watertemperaturedecreasesrapidly withdepthexceptathighlatitudes.Therangeofdepthswheretherateofc hange,the gradientoftemperature,islargeiscalledthe thermocline .Becausedensityis

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6.5.DENSITY 83 AAC Winter Summer Warm Pool BATS AAC Warm Pool BATS Salinity Temperature (Celsius) 34353637 0o10o20o30o0 500 10001500Pressure (decibars) Figure6.8Typicaltemperatureandsalinityprolesintheo penocean.AAC:At62.0 S, 170.0 EintheAntarcticCircumpolarCurrenton16January1969asm easuredbythe R/V HakuhoMaru .WarmPool:At9.5 N176.3 EinthetropicalwestPacicwarmpoolon12 March1989asmeasuredbyBrydenandHallonthe R/VMoanaWave bats :At31.8 N 64.1 WnearBermudaon17Apriland10September1990asmeasuredby theBermuda BiologicalStationforResearch,Inc.Dataareincludedwit hJavaOceanAtlas. closelyrelatedtotemperature,thethermoclinealsotends tobethelayerwhere densitygradientisgreatest,the pycnocline Theshapeofthethermoclinevariesslightlywiththeseason s(gure6.7). Thisisthe seasonalthermocline .The permanentthermocline extendsfrom belowtheseasonalthermoclinetodepthsof1500{2000meter s(gure6.8).At highlatitudes,suchasatthe aac stationinthegure,theremaybeacooler, fresherlayerabovethepermanentthermocline. Themixedlayertendstobesaltierthanthethermoclinebetw een10 and 40 latitude,whereevaporationexceedsprecipitation.Athig hlatitudesthe mixedlayerisfresherbecauserainandmeltingicereducesa linity.Insome tropicalregions,suchasthewarmpoolinthewesterntropic alPacic,rainalso producesathinfreshermixedlayer.6.5Density,PotentialTemperature,andNeutralDensity Duringwinter,coldwaterformedatthesurfacesinkstoadep thdetermined byitsdensityrelativetothedensityofthedeeperwater.Cu rrentsthencarry thewatertootherpartsoftheocean.Atalltimes,thewaterp arcelmoves tostaybelowlessdensewaterandabovemoredensewater.The distribution ofcurrentswithintheoceandependsonthedistributionofp ressure,which dependsonthevariationsofdensityinsidetheoceanasoutl inedin x 10.4.So,

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84 CHAPTER6.TEMPERATURE,SALINITY,ANDDENSITY ifwewanttofollowwatermovementwithintheocean,weneedt oknowthe distributionofdensitywithintheocean.Densityandsigma-t Thecalculationofwatermovementrequiresmeasurementsofdensitywithanaccuracyofafewpartspermillion.T hisisnoteasy. AbsoluteDensity ofwatercanonlybemeasuredinspeciallaboratories,and onlywithdiculty.Thebestaccuracyis1:2 : 5 10 5 =4partspermillion. Toavoidthedicultyofworkingwithabsolutedensity,ocea nographersuse densityrelativetodensityofpurewater.Density ( S;t;p )isnowdenedusing StandardMeanOceanWaterofknownisotopiccomposition,as sumingsaturationofdissolvedatmosphericgasses.Here S;t;p referstosalinity,temperature, andpressure. Inpractice,densityisnotmeasured,itiscalculatedfrom insitu measurementsofpressure,temperature,andconductivityusingthe equationofstate forseawater.Thiscanbedonewithanaccuracyoftwopartspe rmillion. Densityofwaterattheseasurfaceistypically1027kg/m 3 .Forsimplication,physicaloceanographersoftenquoteonlythelast2di gitsofthedensity, aquantitytheycall densityanomaly or Sigma(S,t,p) : ( S;t;p )= ( S;t;p ) 1000kg/m 3 (6.7) TheWorkingGrouponSymbols,UnitsandNomenclatureinPhys icalOceanography( sun ,1985)recommendsthat bereplacedby r because wasoriginally denedrelativetopurewateranditwasdimensionless.Here ,however,Iwill followcommonpracticeanduse Ifwearestudyingsurfacelayersoftheocean,wecanignorec ompressibility, andweuseanewquantitysigma-t(written t ): t = ( S;t; 0)(6.8) Thisisthedensityanomalyofawatersamplewhenthetotalpr essureonit hasbeenreducedtoatmosphericpressure( i.e. zerowaterpressure),butthe temperatureandsalinityare insitu values. PotentialTemperature Asawaterparcelmoveswithintheoceanbelowthe mixedlayer,itssaltandheatcontentcanchangeonlybymixi ngwithother water.Thuswecanusemeasurementsoftemperatureandsalin itytotracethe pathofthewater.Thisisbestdoneifweremovetheeectofco mpressibility. Aswatersinks,pressureincreases,thewateriscompressed ,andthecompressiondoesworkonthewater.Thisincreasestheinternal energyofthewater. Tounderstandhowcompressionincreasesenergy,considera cubecontaininga xedmassofwater.Asthecubesinks,itssidesmoveinwardas thecubeis compressed.Recallingthatworkisforcetimesdistance,th eworkisthedistance thesidemovestimestheforceexertedonthesidebypressure .Thechangein internalenergymayormaynotresultinachangeintemperatu re(McDougall andFeistel,2003).Theinternalenergyofaruidisthesumof molecularkinetic energy(temperature)andmolecularpotentialenergy.Inse awater,thelater termdominates,andthechangeofinternalenergyproducest hetemperature

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6.5.DENSITY 85 st t q s q Density 28 27.5 27.6 27.7 27.9 27.8 -8000 -7000 -6000 -5000 -3000 -2000 -1000 5o0oTemperature (Celsius) 1o2o3o4o -4000Depth (m) Figure6.9Prolesof Left insitu t andpotential temperatureand Right sigma-tand sigma-thetaintheKermadecTrenchinthePacicmeasuredby theR/VEltaninduringthe ScorpioExpeditionon13July1967at175.825 Eand28.258 S.DatafromWarren(1973). changeshowningure6.9.Atadepthof8km,theincreaseinte mperatureis almost0.9 C. Toremovetheinruenceofcompressibilityfrommeasurement softemperature,oceanographers(andmeteorologistswhohavethesam eprobleminthe atmosphere)usetheconceptofpotentialtemperature. Potentialtemperature isdenedasthetemperatureofaparcelofwaterattheseasu rfaceafterit hasbeenraisedadiabaticallyfromsomedepthintheocean.R aisingtheparcel adiabatically meansthatitisraisedinaninsulatedcontainersoitdoesno texchangeheatwithitssurroundings.Ofcourse,theparcelisn otactuallybrought tothesurface.Potentialtemperatureiscalculatedfromth etemperatureinthe wateratdepth,the insitu temperature. PotentialDensity Ifwearestudyingintermediatelayersoftheocean,say atdepthsnearakilometer,wecannotignorecompressibilit y.Becausechanges inpressureprimarilyinruencethetemperatureofthewater ,theinruenceof pressurecanberemoved,toarstapproximation,byusingth e potentialdensity Potentialdensity isthedensityaparcelofwaterwouldhaveifitwere

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86 CHAPTER6.TEMPERATURE,SALINITY,ANDDENSITY raisedadiabaticallytothesurfacewithoutchangeinsalin ity.Writtenassigma, = ( S; ; 0)(6.9) isespeciallyusefulbecauseitisaconservedthermodynami cproperty. Potentialdensityisnotusefulforcomparingdensityofwat eratgreatdepths. Ifwebringwaterparcelstothesurfaceandcomparetheirden sities,thecalculationofpotentialdensityignorestheeectofpressureon thecoecientsfor thermalandsaltexpansion.Asaresult,twowatersamplesha vingthesame densitybutdierenttemperatureandsalinityatadepthoff ourkilometerscan havenoticeablydierentpotentialdensity.Insomeregion stheuseof ()can leadtoanapparentdecreaseofdensitywithdepth(gure6.1 0)althoughwe knowthatthisisnotpossiblebecausesuchacolumnofwaterw ouldbeunstable. Tocomparesamplesfromgreatdepths,itisbettertobringbo thsamplesto anearbydepthinsteadoftothesurface p =0.Forexample,wecanbringboth parcelstoapressureof4,000decibars,whichisnearadepth of4km: 4 = ( S; ; 4000)(6.10) where 4 isthedensityofaparcelofwaterbroughtadiabaticallytoa pressure 7000 6000 5000 4000 3000 2000 1000 0m -80oANTARCTICA GREENLAND-ICELAND RIDGE-60o-40o-20o0o20o40o60o80o28.08 28.0 27.7 27.9 27.7 27.8 26.0 25.0 26.0 24.0 26.0 27.0 27.4 27.6 27.7 27.8 27.227.0 27.88 27.85 27.88 27.88 27.9 27.92 27.85 27.9 27.94 46.0 46.1 45.6 45.4 45.0 43.0 41.0 45.2 45.6 45.6 45.7 45.4 45.0 44.0 45.7 45.6 45.6 45.5 45.4 45.4 45.2 46.0 46.4 >46.4 46.4 46.2 46.1 45.9 45.8 46.0 45.95 45.9 45.8 45.93 46.0 43.0 42.0 40.0 7000 6000 5000 4000 3000 2000 1000 0m -80oANTARCTICA GREENLAND-ICELAND RIDGE-60o-40o-20o0o20o40o60o80o Pressure (decibars)s4000stheta Figure6.10VerticalsectionsofdensityinthewesternAtla ntic.Notethatthedepthscale changesat1000mdepth. Upper : ,showinganapparentdensityinversionbelow3,000m. Lower : 4 showingcontinuousincreaseindensitywithdepth.AfterLy nnandReid(1968).

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6.5.DENSITY 87 of4,000decibars.Moregenerally,oceanographerssometim esuse r r = ( S; ;p;p r )(6.11) where p ispressure,and p r ispressureatsomereferencelevel.In(6.8)thelevel is p r =0decibars,andin(6.9) p r =4000decibars. Theuseof r leadstoproblems.Ifwewishtofollowparcelsofwaterdeep intheocean,wemightuse 3 insomeareas,and 4 inothers.Butwhat happenswhenaparcelmovesfromadepthof3kminoneareatoad epth of4kminanother?Thereisasmalldiscontinuitybetweenthe densityofthe parcelexpressedas 3 comparedwithdensityexpressedas 4 .Toavoidthis diculty,JackettandMcDougall(1997)proposedanewvaria bletheycalled neutraldensity.NeutralSurfacesandDensity Aparcelofwatermoveslocallyalongapath ofconstantdensitysothatitisalwaysbelowlessdensewate randabovemore densewater.Moreprecisely,itmovesalongapathofconstan tpotentialdensity r referencedtothelocaldepth r .Suchapathiscalleda neutralpath (Eden andWillebrand,1999).A neutralsurfaceelement isthesurfacetangenttothe neutralpathsthroughapointinthewater.Noworkisrequire dtomovea parcelonthissurfacebecausethereisnobuoyancyforceact ingontheparcel asitmoves(ifweignorefriction). Nowlet'sfollowtheparcelasitmovesawayfromalocalregio n.Atrstwe mightthinkthatbecauseweknowthetangentstothesurfacee verywhere,we candeneasurfacethatistheenvelopeofthetangents.Buta nexactsurfaceis notmathematicallypossibleintherealocean,althoughwec ancomeveryclose. JackettandMcDougall(1997)developedapracticalneutral densityvariable r n andsurfacethatstayswithinafewtensmetersofanidealsur faceanywhere intheworld.Theyconstructedtheirvariablesusingdatain theLevitus(1982) atlas.Theneutraldensityvalueswerethenusedtolabelthe dataintheLevitus atlas.Thisprelabeleddatasetisusedtocalculate r n atnewlocationswhere t;S aremeasuredasafunctionofdepthbyinterpolationtothefo urclosestpoints intheLevitusatlas.Throughthispractice,neutraldensit y r n isafunctionof salinity S insitu temperature t ,pressure p ,longitude,andlatitude. Theneutralsurfacedenedabovediersonlyslightlyfroma nidealneutral surface.Ifaparcelmovesaroundagyreontheneutralsurfac eandreturnsto itsstartinglocation,itsdepthattheendwilldierbyarou nd10metersfrom thedepthatthestart.Ifpotentialdensitysurfacesareuse d,thedierencecan behundredsofmeters,afarlargererror.Equationofstateofseawater Densityofseawaterisrarelymeasured. Densityiscalculatedfrommeasurementsoftemperature,co nductivity,orsalinity,andpressureusingtheequationofstateofseawater .The equationofstate isanequationrelatingdensitytotemperature,salinity,a ndpressure. Theequationisderivedbyttingcurvesthroughlaboratory measurements ofdensityasafunctionoftemperature,pressure,andsalin ity,chlorinity,or conductivity.TheInternationalEquationofState(1980)p ublishedbytheJoint

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88 CHAPTER6.TEMPERATURE,SALINITY,ANDDENSITY PanelonOceanographicTablesandStandards(1981)isnowus ed.Seealso MilleroandPoisson(1981)andMilleroetal(1980).Theequa tionhasan accuracyof10partspermillion,whichis0.01unitsof (). Ihavenotactuallywrittenouttheequationofstatebecause itconsistsof threepolynomialswith41constants( jpots ,1991). AccuracyofTemperature,Salinity,andDensity Ifwewanttodistinguishbetweendierentwatermassesintheocean,andifthet otalrangeof temperatureandsalinityisassmallastherangeingure6.1 ,thenwemust measuretemperature,salinity,andpressureverycarefull y.Wewillneedan accuracyofafewpartspermillion. Suchaccuracycanbeachievedonlyifallquantitiesarecare fullydened,if allmeasurementsaremadewithgreatcare,ifallinstrument sarecarefullycalibrated,andifallworkisdoneaccordingtointernationally acceptedstandards. Thestandardsarelaidoutin ProcessingofOceanographicStationData ( jpots 1991)publishedby unesco .Thebookcontainsinternationallyaccepteddenitionsofprimaryvariablessuchastemperatureandsalinity andmethodsforthe measuringtheprimaryvariables.Italsodescribesaccepte dmethodsforcalculatingquantitiesderivedfromprimaryvariables,suchasp otentialtemperature, density,andstability.6.6MeasurementofTemperature Temperatureintheoceanismeasuredmanyways.Thermistors andmercury thermometersarecommonlyusedonshipsandbuoys.Theseare calibratedin thelaboratorybeforebeingused,andafteruseifpossible, usingmercuryorplatinumthermometerswithaccuracytraceabletonationalstan dardslaboratories. Infraredradiometersonsatellitesmeasuretheocean'ssur facetemperature. MercuryThermometer Thisisthemostwidelyused,non-electronicthermometer.Itwaswidelyusedinbucketsdroppedoverthesideo fashiptomeasurethetemperatureofsurfacewaters,onNansenbottlesto measuresub-sea temperatures,andinthelaboratorytocalibrateotherther mometers.Accuracy ofthebestthermometersisabout 0.001 Cwithverycarefulcalibration. Oneveryimportantmercurythermometeristhereversingthe rmometer(gure6.11)carriedonNansenbottles,whicharedescribedint henextsection.It isathermometerthathasaconstrictioninthemercurycapil larythatcauses thethreadofmercurytobreakatapreciselydeterminedpoin twhenthethermometeristurnedupsidedown.Thethermometerisloweredde epintothe oceaninthenormalposition,anditisallowedtocometoequi libriumwith thewater.Mercuryexpandsintothecapillary,andtheamoun tofmercuryin thecapillaryisproportionaltotemperature.Thethermome teristhenripped upsidedown,thethreadofmercurybreakstrappingthemercu ryinthecapillary,andthethermometerisbroughtback.Themercuryinthe capillaryofthe reversedthermometerisreadondeckalongwiththetemperat ureofanormal thermometer,whichgivesthetemperatureatwhichtherever sedthermometer isread.Thetworeadingsgivethetemperatureofthewaterat thedepthwhere

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6.6.MEASUREMENTOFTEMPERATURE 89 2 1 0 1 2 3 4 5 6 7 8 9 1 110 4 1 12 13 15 0 6 0 5 0 4 0 3 0 0 2 0 1 10 2 1 0 1 2 3 4 5 6 7 8 9 1 110 4 1 12 13 15 0 6 0 5 0 4 0 3 0 0 2 0 1 10 Figure6.11 Left :Protectedandunprotectedreversingthermometersissetp osition,before reversal. Right :Theconstrictedpartofthecapillaryinsetandreversedpo sitions.After vonArx(1962:259).thethermometerwasreversed. Thereversingthermometeriscarriedinsideaglasstubewhi chprotects thethermometerfromtheocean'spressurebecausehighpres surecansqueeze additionalmercuryintothecapillary.Ifthethermometeri sunprotected,the apparenttemperaturereadondeckisproportionaltotemper atureandpressure atthedepthwherethethermometerwasripped.Apairofprote ctedand unprotectedthermometersgivestemperatureandpressureo fthewateratthe depththethermometerwasreversed. PairsofreversingthermometerscarriedonNansenbottlesw eretheprimary sourceofsub-seameasurementsoftemperatureasafunction ofpressurefrom

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90 CHAPTER6.TEMPERATURE,SALINITY,ANDDENSITY around1900to1970.PlatinumResistanceThermometer Thisisthestandardfortemperature. Itisusedbynationalstandardslaboratoriestointerpolat ebetweendened pointsonthepracticaltemperaturescale.Itisusedprimar ilytocalibrateother temperaturesensors.Thermistor Athermistorisasemiconductorhavingresistancethatvari es rapidlyandpredictablywithtemperature.Ithasbeenwidel yusedonmoored instrumentsandoninstrumentsdeployedfromshipssinceab out1970.Ithas highresolutionandanaccuracyofabout 0.001 Cwhencarefullycalibrated. Buckettemperatures Thetemperatureofsurfacewatershasbeenroutinely measuredatseabyputtingamercurythermometerintoabucke twhichis loweredintothewater,lettingitsitatadepthofaboutamet erforafew minutesuntilthethermometercomestoequilibrium,thenbr ingingitaboard andreadingthetemperaturebeforewaterinthebuckethasti metochange temperature.Theaccuracyisaround0.1 C.Thisisaverycommonsourceof directsurfacetemperaturemeasurements.ShipInjectionTemperature Thetemperatureofthewaterdrawnintothe shiptocooltheengineshasbeenrecordedroutinelyfordeca des.Theserecorded valuesoftemperaturearecalledinjectiontemperatures.E rrorsareduetoship's structurewarmingwaterbeforeitisrecorded.Thishappens whenthetemperaturerecorderisnotplacedclosetothepointonthehullwher ewaterisbrought in.Accuracyis0.5 {1 C. AdvancedVeryHighResolutionRadiometer Themostcommonlyused instrumenttomeasuresea-surfacetemperaturefromspacei stheAdvancedVery HighResolutionRadiometer avhrr .Theinstrumenthasbeencarriedonall polar-orbitingmeteorologicalsatellitesoperatedby noaa sinceTiros-Nwas launchedin1978. Theinstrumentwasoriginallydesignedtomeasurecloudtem peraturesand hencecloudheight.Theinstrumenthad,however,sucienta ccuracyandprecisionthatitwassoonusedtomeasureregionalandglobaltemp eraturepatterns attheseasurface. Theinstrumentisaradiometerthatconvertsinfraredradia tionintoanelectricalvoltage.Itincludesamirrorthatscansfromsidetos ideacrossthe sub-satellitetrackandrerectsradiancefromthegroundin toatelescope,a telescopethatfocusestheradianceondetectors,detector ssensitivetodierentwavelengthsthatconverttheradianceatthosewaveleng thsintoelectrical signals,andelectroniccircuitrytodigitizeandstorethe radiancevalues.The instrumentsobservesa2700-kmwideswathcenteredonthesu b-satellitetrack. Eachobservationalongthescanisfromapixelthatisroughl yonekilometer indiameternearthecenterofthescanandthatincreasesins izewithdistance fromthesub-satellitetrack. Theradiometersmeasuresinfraredradiationemittedfromt hesurfacein vewavelengthbands:threeinfraredbands:3.55{3.99 m,10.3{11.3 m,and

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6.6.MEASUREMENTOFTEMPERATURE 91 11.5{12.5 m;anear-infraredbandat0.725{1.10 m;andavisible-lightband at0.55{0.90 m.Allinfraredbandsincluderadiationemittedfromthesea and fromwatervaporintheairalongthepathfromthesatellitet otheground.The 3.7 mbandisleastsensitivetowatervaporandothererrors,but itworksonly atnightbecausesunlighthasradianceinthisband.Thetwol ongestwavelength bandsat10.8 mand12.0 mareusedtoobservesea-surfacetemperatureand watervaporalongthepathindaylight. Datawith1-kmresolutionaretransmitteddirectlytogroun dstationsthat viewthesatelliteasitpassesthestation.ThisistheLocal AreaCoverage mode.Dataarealsoaveragedtoproduceobservationsfrom4 4kmpixels. Thesedataarestoredbythesatelliteandlatertransmitted to noaa receiving stations.ThisistheGlobalAreaCoveragemode. Theswathwidthissucientlywidethatthesatelliteviewst heentireearth twiceperday,atapproximately09:00AMand9:00PMlocaltim e.Areasat highlatitudesmaybeobservedasoftenaseightormoretimes perday. Themostimportanterrorsaredueto: 1.Unresolvedorundetectedclouds:Large,thickcloudsare obviousinthe imagesofwatertemperatureThincloudssuchaslowstratusa ndhigh cirrusproducemuchsmallerrorsthataredicultoralmosti mpossibleto detect.Cloudssmallerindiameterthan1km,suchastrade-w indcumuli, arealsodiculttodetect.Specialtechniqueshavebeendev elopedfor detectingsmallclouds(gure6.12). 2.Watervapor,whichabsorbspartoftheenergyradiatedfro mthesea surface:Watervaporreducestheapparenttemperatureofth eseasurface. Theinruenceisdierentinthe10.8 mand12.0 mchannels,allowing thedierenceinthetwosignalstobeusedtoreducetheerror 4T11 T3.7T116543210 0 Local Mean Temperature Difference (K) 3 2 1Local Maximum Difference10 50 270275280285290295Local Standard DeviationLocal Mean Temperature (K) Figure6.12Theinruenceofcloudsoninfraredobservations Left: Thestandarddeviation oftheradiancefromsmall,partlycloudyareaseachcontain ing64pixels.Thefeetofthe arch-likedistributionofpointsarethesea-surfaceandcl oud-toptemperatures.AfterCoakley andBretherton(1982). Right: Themaximumdierencebetweenlocalvaluesof T 11 T 3 : 7 andthelocalmeanvaluesofthesamequantity.Valuesinside thedashedboxindicate cloud-freepixels. T 11 and T 3 : 7 aretheapparenttemperaturesat11.0and3.7 m(datafrom K.Kelly).AfterStewart(1985:137).

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92 CHAPTER6.TEMPERATURE,SALINITY,ANDDENSITY 3.Aerosols,whichabsorbinfraredradiation.Theyradiate attemperatures foundhighintheatmosphere.Stratosphericaerosolsgener atedbyvolcaniceruptionscanlowertheobservedtemperaturesbyupto afewdegreesCelsius.DustparticlescarriedovertheAtlanticfro mSaharandust stormscanalsocauseerrors. 4.Skintemperatureerrors.Theinfraredradiationseenbyt heinstrument comesfromalayerattheseasurfacethatisonlyafewmicrome tersthick. Thetemperatureinthislayerisnotquitethesameastempera tureameter belowtheseasurface.Theycandierbyseveraldegreeswhen windsare light(EmeryandSchussel,1989).Thiserrorisgreatlyredu cedwhen avhrr dataareusedtointerpolatebetweenshipmeasurementsofsu rface temperature. MapsoftemperatureprocessedfromLocalAreaCoverageofcl oud-freeregionsshowvariationsoftemperaturewithaprecisionof0.1 C.Thesemaps areusefulforobservinglocalphenomenaincludingpattern sproducedbylocal currents.Figure10.16showssuchpatternsotheCaliforni acoast. GlobalmapsaremadebytheU.S.NavalOceanographicOce,wh ichreceivestheglobal avhrr datadirectlyfrom noaa 'sNationalEnvironmental Satellite,DataandInformationServiceinnear-realtimee achday.Thedata arecarefullyprocessedtoremovetheinruenceofclouds,wa tervapor,aerosols, andothersourcesoferror.Dataarethenusedtoproduceglob almapsbetween 70 withanaccuracyof 0 : 6 C(Mayetal1998).Themapsofsea-surface temperaturearesenttotheU.S.Navyandto noaa 'sNationalCentersforEnvironmentalPrediction.Inaddition,theoceproducesdai ly100-kmglobal and14-kmregionalmapsoftemperature.GlobalMapsofSea-SurfaceTemperature Global,monthlymapsofsurfacetemperatureareproducedbytheNationalCentersforEn vironmentalPredictionusingReynoldsetal(2002)optimal-interpolation method.Thetechniqueblendsshipandbuoymeasurementsofsea-surfacetemp eraturewith avhrr dataprocessedbytheNavalOceanographicOcein1 areasfora month.Essentially, avhrr dataareinterpolatedbetweenbuoyandshipreportsusingpreviousinformationaboutthetemperatureel d.Overallaccuracy rangesfromapproximately 0 : 3 Cinthetropicsto 0 : 5 Cnearwestern boundarycurrentsinthenorthernhemispherewheretempera turegradientsare large.MapsareavailablefromNovember1981.Figures6.2{6 .4weremadeby noaa usingReynolds'technique.Otherdatasetshavebeenproduc edbythe noaa/nasa Pathnderprogram(Kilpatrick,Podesta,andEvans,2001). Mapsofmeantemperaturehavealsobeenmadefrom icoads data(Smith andReynolds,2004).Becausethedataarepoorlydistribute dintimeandspace, errorsalsovaryintimeandspace.SmithandReynolds(2004) estimatedthe errorintheglobalmeantemperatureandfoundthe95%conde nceuncertainty forthenear-globalaverageis0.48 Cormoreinthenineteenthcentury,near 0.28 Cforthersthalfofthetwentiethcentury,and0.18 Corlessafter 1950.Anomaliesofsea-surfacetemperaturewerecalculate dusingmeansea-

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6.7.MEASUREMENTOFCONDUCTIVITYORSALINITY 93 Cell Terminals SeawaterFlow Out Current Field between Electrodes Borosilicate Glass Cell Platinum Electrodes (3 places) Seawater Flow In Figure6.13Aconductivitycell.Currentrowsthroughthese awaterbetweenplatinum electrodesinacylinderofborosilicateglass191mmlongwi thaninsidediameterbetweenthe electrodesof4mm.Theelectriceldlines(solidlines)are connedtotheinteriorofthecell inthisdesignmakingthemeasuredconductivity(andinstru mentcalibration)independent ofobjectsnearthecell.Thisisthecellusedtomeasurecond uctivityandsalinityshownin gure6.15.FromSea-BirdElectronics.surfacetemperaturefromtheperiod1854{1997using icoads supplemented withsatellitedatasince1981.6.7MeasurementofConductivityorSalinity Conductivityismeasuredbyplacingplatinumelectrodesin seawaterand measuringthecurrentthatrowswhenthereisaknownvoltage betweenthe electrodes.Thecurrentdependsonconductivity,voltage, andvolumeofsea waterinthepathbetweenelectrodes.Iftheelectrodesarei natubeofnonconductingglass,thevolumeofwaterisaccuratelyknown,a ndthecurrentis independentofotherobjectsneartheconductivitycell(g ure6.13).Thebest measurementsofsalinityfromconductivitygivesalinityw ithanaccuracyof 0.005. Beforeconductivitymeasurementswerewidelyused,salini tywasmeasured usingchemicaltitrationofthewatersamplewithsilversal ts.Thebestmeasurementsofsalinityfromtitrationgivesalinitywithana ccuracyof 0.02. Individualsalinitymeasurementsarecalibratedusingsta ndardseawater. Long-termstudiesofaccuracyusedatafrommeasurementsof deepwatermasses ofknown,stable,salinity.Forexample,Saunders(1986)no tedthattemperature isveryaccuratelyrelatedtosalinityforalargevolumeofw atercontainedinthe deepbasinofthenorthwestAtlanticundertheMediterranea noutrow.Heused theconsistencyofmeasurementsoftemperatureandsalinit ymadeatmanyhydrographicstationsintheareatoestimatetheaccuracyoft emperature,salinity andoxygenmeasurements.Heconcludedthatthemostcareful measurements madesince1970haveanaccuracyof0.005forsalinityand0.0 05 Cfortemperature.Thelargestsourceofsalinityerrorwastheerrorind eterminationofthe standardwaterusedforcalibratingthesalinitymeasureme nts. GouretskiandJancke(1995)estimatedaccuracyofsalinity measurementsas afunctionoftime.Usinghighqualitymeasurementsfrom16, 000hydrographic stationsinthesouthAtlanticfrom1912to1991,theyestima tedaccuracyby plottingsalinityasafunctionoftemperatureusingalldat acollectedbelow1500 mintwelveregionsforeachdecadefrom1920to1990.Aplotof accuracyas afunctionoftimesince1920showsconsistentimprovementi naccuracysince 1950(gure6.14).Recentmeasurementsofsalinityarethem ostaccurate.The

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94 CHAPTER6.TEMPERATURE,SALINITY,ANDDENSITY Year0.000 0.005 0.010 0.015 0.020 0.025192019301940195019601970198019902000 Salinity Accuracy Figure6.14.Standarddeviationofsalinitymeasurementsb elow1500minthesouthAtlantic. Eachpointistheaverageforthedecadecenteredonthepoint .Thevaluefor1995isan estimateoftheaccuracyofrecentmeasurements.FromGoure tskiandJancke(1995). standarddeviationofsalinitydatacollectedfromallarea sinthesouthAtlantic from1970to1993adjustedasdescribedbyGouretskiandJanc ke(1995)was 0.0033.RecentinstrumentssuchastheSea-BirdElectronic sModel911Plus haveanaccuracyofbetterthan0.005withoutadjustments.A comparisonof salinitymeasuredat43 10N,14 4.5Wbythe911Pluswithhistoricdata collectedbySaunders(1986)givesanaccuracyof0.002(gu re6.15). 5262 m 5000 m depth (dbar) 3000400050005262 911 CTD (PSU) 34.9503 34.9125 34.8997 34.8986 Autosal (PSU) 34.9501 34.9134 34.8995 34.8996 911-Autosal (PSU) +0.0002 -0.0009 +0.0002 -0.0010 No. Bottles, (sample range) 3 (.0012)4 (.0013)4 (.0011)3 (.0004) Saunders, P. (1986) S = 34.698 + 0.098 q PSU Valid: q < 2.5o C. 4000 m 3000 m Salinity, PSS-78 34.89 34.96 2.0o2.5oPotential Temperature (Celsius)2.7o 2.6o2.1o2.2o2.3o2.4o Figure6.15.ResultsfromatestoftheSea-BirdElectronics 911PlusCTDintheNorth AtlanticDeepWaterin1992.Datawerecollectedat43.17 Nand14.08 WfromtheR/V Poseidon.FromSea-BirdElectronics(1992).

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6.8.MEASUREMENTOFPRESSURE 95 6.8MeasurementofPressure Pressureisroutinelymeasuredbymanydierenttypesofins truments.The SIunitofpressureisthepascal(Pa),butoceanographersno rmallyreportpressureindecibars(dbar),where: 1dbar=10 4 Pa(6.12) becausethepressureindecibarsisalmostexactlyequaltot hedepthinmeters. Thus1000dbaristhepressureatadepthofabout1000m.StrainGage Thisisthesimplestandcheapestinstrument,anditiswidel y used.Accuracyisabout 1%. Vibratron Muchmoreaccuratemeasurementsofpressurecanbemadeby measuringthenaturalfrequencyofavibratingtungstenwir estretchedina magneticeldbetweendiaphragmsclosingtheendsofacylin der.Pressuredistortsthediaphragm,whichchangesthetensiononthewirean ditsfrequency. Thefrequencycanbemeasuredfromthechangingvoltageindu cedasthewire vibratesinthemagneticeld.Accuracyisabout 0.1%,orbetterwhentemperaturecontrolled.Precisionis100{1000timesbetterth anaccuracy.The instrumentisusedtodetectsmallchangesinpressureatgre atdepths.Snodgrass(1964)obtainedaprecisionequivalenttoachangeind epthof 0.8mm atadepthof3km.Quartzcrystal Veryaccuratemeasurementsofpressurecanalsobemadeby measuringthenaturalfrequencyofaquartzcrystalcutform inimumtemperaturedependence.Thebestaccuracyisobtainedwhenthetem peratureofthe crystalisheldconstant.Theaccuracyis 0.015%,andprecisionis 0.001%of full-scalevalues.QuartzBourdonGage hasaccuracyandstabilitycomparabletoquartzcrystals.Ittooisusedforlong-termmeasurementsofpressurei nthedeepsea. 6.9MeasurementofTemperatureandSalinitywithDepth Temperature,salinity,andpressurearemeasuredasafunct ionofdepth usingvariousinstrumentsortechniques,anddensityiscal culatedfromthe measurements.Bathythermograph(BT) wasamechanicaldevicethatmeasuredtemperaturevsdepthonasmokedglassslide.Thedevicewaswidelyus edtomapthe thermalstructureoftheupperocean,includingthedepthof themixedlayer beforebeingreplacedbytheexpendablebathythermographi nthe1970s. ExpendableBathythermograph(XBT) isanelectronicdevicethatmeasurestemperaturevsdepthusingathermistoronafree-fall ingstreamlined weight.Thethermistorisconnectedtoanohm-meteronthesh ipbyathin copperwirethatisspooledoutfromthesinkingweightandfr omthemovingship.The xbt isnowthemostwidelyusedinstrumentformeasuringthe thermalstructureoftheupperocean.Approximately65,000 areusedeachyear.

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96 CHAPTER6.TEMPERATURE,SALINITY,ANDDENSITY Before turning While it turns After turning Figure6.16 Left ACTDreadytobeloweredoverthesideofaship.FromDavis(19 87). Right Nansenwaterbottlesbefore(I),during(II),andafter(III )reversing.Bothinstrumentsare shownatclosetothesamescale.AfterDefant(1961:33). Thestreamlinedweightfallsthroughthewaterataconstant velocity.So depthcanbecalculatedfromfalltimewithanaccuracyof 2%.Temperature accuracyis 0.1 C.And,verticalresolutionistypically65cm.Probesreach to depthsof200mto1830mdependingonmodel.NansenBottles (gure6.16)weredeployedfromshipsstoppedathydrographicstations. Hydrographicstations areplaceswhereoceanographersmeasurewaterpropertiesfromthesurfacetosomedepth,ortoth ebottom,using instrumentsloweredfromaship.Usually20bottleswereatt achedatintervals ofafewtenstohundredsofmeterstoawireloweredoverthesi deoftheship. Thedistributionwithdepthwasselectedsothatmostbottle sareintheupper layersofthewatercolumnwheretherateofchangeoftempera tureintheverticalisgreatest.Aprotectedreversingthermometerforme asuringtemperature wasattachedtoeachbottlealongwithanunprotectedrevers ingthermometer formeasuringdepth.Thebottlecontainsatubewithvalveso neachendto collectseawateratdepth.Salinitywasdeterminedbylabor atoryanalysisof watersamplecollectedatdepth. Afterbottleshadbeenattachedtothewireandallhadbeenlo weredto theirselecteddepths,aleadweightwasdroppeddownthewir e.Theweight

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6.10.LIGHTINTHEOCEANANDABSORPTIONOFLIGHT 97 trippedamechanismoneachbottle,andthebottlerippedove r,reversingthe thermometers,shuttingthevalvesandtrappingwaterinthe tube,andreleasing anotherweight.Whenallbottleshadbeentripped,thestrin gofbottleswas recovered.Thedeploymentandretrievaltypicallytooksev eralhours. CTD MechanicalinstrumentsonNansenbottleswerereplacedbeg inninginthe 1960sbyanelectronicinstrument,calleda ctd ,thatmeasuredconductivity, temperature,anddepth(gure6.16).Themeasurementsarer ecordedindigital formeitherwithintheinstrumentasitisloweredfromaship orontheship. Temperatureisusuallymeasuredbyathermistor.Conductiv ityismeasured byaconductivitycell.Pressureismeasuredbyaquartzcrys tal.Modern instrumentshaveaccuracysummarizedintable6.2. Table6.2SummaryofMeasurementAccuracy VariableRangeBestAccuracy Temperature42 C 0.001 C Salinity1 0.02bytitration 0.005byconductivity Pressure10,00dbar 0.65dbar Density2kg/m 3 0.005kg/m 3 EquationofState 0.005kg/m 3 CTDonDrifters Perhapsthemostcommonsourceoftemperatureandsalinityasafunctionofdepthintheuppertwokilometersoftheoc eanisthesetof proling argo roatsdescribedin x 11.8.Theroatsdriftatadepthof1km,sink to2km,thenrisetothesurface.Theyproletemperatureand salinitywhile changingdepthusinginstrumentsverysimilartothoseona ctd .Dataaresent toshoreviatheArgossystemonthe noaa polar-orbitingsatellites.In2006, nearly2500roatswereproducingoneproleevery10daysthr oughoutmostof theocean.Theaccuracyofdatafromtheroatsis0.005 Cfortemperature,5 decibarsforpressure,and0.01forsalinity(Riseretal(20 08). DataSets DataareintheMarineEnvironmentandSecurityForEuropean Area mersea Enact/Ensembles( en 3QualityControlledinsituOceanTemparatureandSalinityProlesdatabase.Asof2008thedatab asecontained aboutonemillion xbt proles,700,000 ctd proles,60,000 argos proles, 1,100,000Nansenbottledataofhighqualityintheupper700 moftheocean (Dominguesetal,2008).6.10LightintheOceanandAbsorptionofLight Sunlightintheoceanisimportantformanyreasons:Itheats seawater, warmingthesurfacelayers;itprovidesenergyrequiredbyp hytoplankton;itis usedfornavigationbyanimalsnearthesurface;andrerecte dsubsurfacelight isusedformappingchlorophyllconcentrationfromspace. Lightintheoceantravelsatavelocityequaltothevelocity oflightin avacuumdividedbytheindexofrefraction( n ),whichistypically n =1 : 33. Hencethevelocityinwaterisabout2 : 25 10 8 m/s.Becauselighttravelsslower

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98 CHAPTER6.TEMPERATURE,SALINITY,ANDDENSITY 200500 1000 1500 20002500l nm 10-210410310210110010-110 -3 Absorbtion coefficient (m-1) Lenoble-Saint Guily (1955), path length: 400 cm;Hulburt (1934)(1945), path length: 364 cm;Sullivan (1963), path length: 132 cm;Clarke-James (1939), path length: 97 cm (Ceresin lined tube);James-Birge (1938), path length: 97 cm (Silver lined tube). Infra-red Visible UVviolet blue greenyellow orange red Figure6.17Absorptioncoecientforpurewaterasafunctio nofwavelength ofthe radiation.RedrawnfromMorel(1974:18,19).SeeMorel(197 4)forreferences. inwaterthaninair,somelightisrerectedattheseasurface .Forlightshining straightdownonthesea,thererectivityis( n 1) 2 = ( n +1) 2 .Forseawater, thererectivityis0 : 02=2%.Hencemostsunlightreachingtheseasurfaceis transmittedintothesea,littleisrerected.Thismeanstha tsunlightincident ontheoceaninthetropicsismostlyabsorbedbelowtheseasu rface. Therateatwhichsunlightisattenuateddeterminesthedept hwhichis lightedandheatedbythesun.Attenuationisduetoabsorpti onbypigments andscatteringbymoleculesandparticles.Attenuationdep endsonwavelength. Bluelightisabsorbedleast,redlightisabsorbedmoststro ngly.Attenuation perunitdistanceisproportionaltotheradianceortheirra dianceoflight: dI dx = cI (6.13) where x isthedistancealongbeam, c isanattenuationcoecient(gure6.17), and I isirradianceorradiance. Radiance isthepowerperunitareapersolidangle.Itisusefulfordes cribing theenergyinabeamoflightcomingfromaparticulardirecti on.Sometimes wewanttoknowhowmuchlightreachessomedepthintheoceanr egardlessof

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6.10.LIGHTINTHEOCEANANDABSORPTIONOFLIGHT 99 100 80604020 0 3004005006007000.525102050100 -0 -40 -80-120-160 1Depth (m)Wavelength (nm)PercentageTransmittance (%/m) I II III 13579 1 III II IB I IA 5 9 Figure6.18 Left: Transmittanceofdaylightintheoceanin%permeterasafunc tion ofwavelength.I:extremelypureoceanwater;II:turbidtro pical-subtropicalwater;III: mid-latitudewater;1-9:coastalwatersofincreasingturb idity.Incidenceangleis90 for therstthreecases,45 fortheothercases. Right: Percentageof465nmlightreaching indicateddepthsforthesametypesofwater.AfterJerlov(1 976). whichdirectionitisgoing.Inthiscaseweuse irradiance ,whichisthepower perunitareaofsurface. Iftheabsorptioncoecientisconstant,thelightintensit ydecreasesexponentiallywithdistance. I 2 = I 1 exp( cx )(6.14) where I 1 istheoriginalradianceorirradianceoflight,and I 2 istheradianceor irradianceoflightafterabsorption. ClarityofOceanWater Seawaterinthemiddleoftheoceanisveryclear| clearerthandistilledwater.Thesewatersareaverydeep,c obalt,blue|almost black.Thusthestrongcurrentwhichrowsnorthwardoshore ofJapancarrying veryclearwaterfromthecentralPacicintohigherlatitud esisknownasthe BlackCurrent,orKuroshioinJapanese.Theclearestoceanw ateriscalled TypeIwatersbyJerlov(gure6.18).Thewaterissocleartha t10%ofthe lighttransmittedbelowtheseasurfacereachesadepthof90 m. Inthesubtropicsandmid-latitudesclosertothecoast,sea watercontains morephytoplanktonthantheveryclearcentral-oceanwater s.Chlorophyllpigmentsinphytoplanktonabsorblight,andtheplantsthemsel vesscatterlight. Together,theprocesseschangethecoloroftheoceanasseen byobserverlooking downwardintothesea.Veryproductivewaters,thosewithhi ghconcentrations ofphytoplankton,appearblue-greenorgreen(gure6.19). Oncleardaysthe colorcanbeobservedfromspace.Thisallowsocean-colorsc anners,suchas thoseonSeaWiFS,tomapthedistributionofphytoplanktono verlargeareas. Astheconcentrationofphytoplanktonincreases,thedepth wheresunlight isabsorbedintheoceandecreases.Themoreturbidtropical andmid-latitude watersareclassiedastypeIIandIIIwatersbyJerlov(gur e6.18).Thus thedepthwheresunlightwarmsthewaterdependsontheprodu ctivityofthe waters.Thiscomplicatesthecalculationofsolarheatingo fthemixedlayer. Coastalwatersaremuchlessclearthanwatersoshore.Thes earethetype 1{9watersshowningure6.18.Theycontainpigmentsfromla nd,sometimes

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100 CHAPTER6.TEMPERATURE,SALINITY,ANDDENSITY ( < 0.1) 0.3 1.3 0.6 3.0 43210 0.4000.5000.6000.700 Wavelength (mm)Reflectance (%) Figure6.19Spectralrerectanceofseawaterobservedfroma naircraftryingat305mover watersofdierentcolorsintheNorthwestAtlantic.Thenum ericalvaluesaretheaverage chlorophyllconcentrationintheeuphotic(sunlit)zonein unitsofmg/m 3 .Thererectance isforverticallypolarizedlightobservedatBrewster'san gleof53 .Thisangleminimizes rerectedskylightandemphasizesthelightfrombelowthese asurface.AfterClarke,Ewing, andLorenzen(1970).calledgelbstoe,whichjustmeansyellowstu,muddywater fromrivers,and mudstirredupbywavesinshallowwater.Verylittlelightpe netratesmore thanafewmetersintothesewaters.MeasurementofChlorophyllfromSpace Thecoloroftheocean,and hencethechlorophyllconcentrationintheupperlayersoft heoceanhasbeen measuredbytheCoastalZoneColorScannercarriedontheNim bus-7satellite launchedin1978,bytheSea-viewingWideField-of-viewSen sor(SeaWiFS) carriedonSeaStar,launchedin1997,andontheModerateRes olutionImaging Spectrometer( modis )carriedontheTerraandAquasatelliteslaunchedin1999 and2002respectively. modis measuresupwellingradiancein36wavelength bandsbetween405nmand14,385nm. Mostoftheupwellingradianceseenbythesatellitecomesfr omtheatmosphere.Onlyabout10%comesfromtheseasurface.Bothairmo leculesand aerosolsscatterlight,andveryaccuratetechniqueshaveb eendevelopedtoremovetheinruenceoftheatmosphere.

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6.11.IMPORTANTCONCEPTS 101 Thetotalradiance L t receivedbyaninstrumentinspaceis: L t ( i )= t ( i ) L W ( i )+ L r ( i )+ L a ( i )(6.15) where i isthewavelengthoftheradiationinthebandmeasuredbythe instrument, L W istheradianceleavingtheseasurface, L r isradiancescatteredby molecules,calledtheRayleighradiance, L a isradiancescatteredfromaerosols, and t isthetransmittanceoftheatmosphere. L r canbecalculatedfromtheory, and L a canbecalculatedfromtheamountofredlightreceivedatthe instrumentbecauseverylittleredlightisrerectedfromthewater .Therefore L W can becalculatedfromtheradiancemeasuredatthespacecraft. Chlorophyllconcentrationinthewatercolumniscalculate dfromtheratio of L W attwofrequencies.UsingdatafromtheCoastalZoneColorSc anner, Gordonetal.(1983)proposed C 13 =1 : 1298 L W (443) L W (550) 1 : 71 (6.16a) C 23 =3 : 3266 L W (520) L W (550) 2 : 40 (6.16b) where C isthechlorophyllconcentrationinthesurfacelayersinmg pigment/m 3 and L W (443) ;L W (520) ;andL W (550)istheradianceatwavelengthsof443,520, and550nm. C 13 isusedwhen C 13 1 : 5mg/m 3 ,otherwise C 23 isused. Thetechniqueisusedtocalculatechlorophyllconcentrati onwithinafactor of50%overawiderangeofconcentrationsfrom0.01to10mg/m 3 6.11ImportantConcepts 1.Densityintheoceanisdeterminedbytemperature,salini ty,andpressure. 2.Densitychangesintheoceanareverysmall,andstudiesof watermasses andcurrentsrequiredensitywithanaccuracyof10partsper million. 3.Densityisnotmeasured,itiscalculatedfrommeasuremen tsoftemperature,salinity,andpressureusingtheequationofstateofs eawater. 4.Accuratecalculationsofdensityrequireaccuratedeni tionsoftemperatureandsalinityandanaccurateequationofstate. 5.Salinityisdiculttodeneandtomeasure.Toavoidthedi culty, oceanographersuseconductivityinsteadofsalinity.They measureconductivityandcalculatedensityfromtemperature,conductivi ty,andpressure. 6.Amixedlayerofconstanttemperatureandsalinityisusua llyfoundinthe top1{100metersoftheocean.Thedepthisdeterminedbywind speed andtheruxofheatthroughtheseasurface. 7.Tocomparetemperatureanddensityofwatermassesatdie rentdepthsin theocean,oceanographersusepotentialtemperatureandpo tentialdensity whichremovemostoftheinruenceofpressureondensity. 8.Waterparcelsbelowthemixedlayermovealongneutralsur faces.

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102 CHAPTER6.TEMPERATURE,SALINITY,ANDDENSITY 9.Surfacetemperatureoftheoceanwasusuallymeasuredats eausingbucket orinjectiontemperatures.Globalmapsoftemperaturecomb inethese observationswithobservationsofinfraredradiancefromt heseasurface measuredbyan avhrr inspace. 10.Temperatureandconductivityareusuallymeasureddigi tallyasafunction ofpressureusinga ctd .Before1960{1970thesalinityandtemperature weremeasuredatroughly20depthsusingNansenbottleslowe redonaline fromaship.Thebottlescarriedreversingthermometerswhi chrecorded temperatureanddepthandtheyreturnedawatersamplefromt hatdepth whichwasusedtodeterminesalinityonboardtheship. 11.Lightisrapidlyabsorbedintheocean.95%ofsunlightis absorbedinthe upper100moftheclearestseawater.Sunlightrarelypenetr atesdeeper thanafewmetersinturbidcoastalwaters. 12.Phytoplanktonchangethecolorofseawater,andthechan geincolorcan beobservedfromspace.Watercolorisusedtomeasurephytop lankton concentrationfromspace.

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Chapter7SomeMathematics:TheEquationsofMotionInthischapterIconsidertheresponseofaruidtointernala ndexternalforces. Thisleadstoaderivationofsomeofthebasicequationsdesc ribingoceandynamics.Inthenextchapter,wewillconsidertheinruenceof viscosity,andin chapter12wewillconsidertheconsequencesofvorticity. FluidmechanicsusedinoceanographyisbasedonNewtonianm echanics modiedbyourevolvingunderstandingofturbulence.Conse rvationofmass, momentum,angularmomentum,andenergyleadtoparticulare quationshaving namesthathidetheirorigins(table7.1).Table7.1ConservationLawsLeadingtoBasicEquationsofFl uidMotion ConservationofMass:LeadstoContinuityEquation.ConservationofEnergy:ConservationofheatleadstoHeatB udgets. Conservationofmechanicalenergyleadsto WaveEquation. ConservationofMomentum:LeadstoMomentum(Navier-Stoke s)Eq. ConservationofAngularMomentum:LeadstoConservationof Vorticity. 7.1DominantForcesforOceanDynamics Onlyafewforcesareimportantinphysicaloceanography:gr avity,friction, andCoriolis(table7.2).Rememberthatforcesarevectors. Theyhavemagnitudeanddirection. 1. Gravity isthedominantforce.Theweightofthewaterintheocean producespressure.Changesingravity,duetothemotionofs unand moonrelativetoearthproducestides,tidalcurrents,andt idalmixingin theinterioroftheocean.Buoyancy istheupwardordownwardforceduetogravityactingona parcelofwaterthatismoreorlessdensethanotherwaterati tslevel.For example,coldairblowingovertheseacoolssurfacewatersc ausingthem 103

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104 CHAPTER7.THEEQUATIONSOFMOTION tobemoredensethanthewaterbeneath.Gravityactingonthe dierence indensityresultsinaforcethatcausesthewatertosink.Horizontalpressuregradients areduetothevaryingweightofwaterin dierentregionsoftheocean. 2. Friction istheforceactingonabodyasitmovespastanotherbodywhil e incontactwiththatbody.Thebodiescanbeparcelsofwatero rair. Windstress isthefrictionduetowindblowingacrosstheseasurface. Ittransfershorizontalmomentumtothesea,creatingcurre nts.Wind blowingoverwavesontheseasurfaceleadstoanunevendistr ibutionof pressureoverthewaves.Thepressuredistributiontransfe rsenergytothe waves,causingthemtogrowintobiggerwaves. 3. Pseudo-forces areapparentforcesthatarisefrommotionincurvilinearor rotatingcoordinatesystems.Forexample,Newton'srstla wstatesthat thereisnochangeinmotionofabodyunlessaresultantforce actsonit. Yetabodymovingatconstantvelocityseemstochangedirect ionwhen viewedfromarotatingcoordinatesystem.Thechangeindire ctionisdue toapseudo-force,theCoriolisforce.CoriolisForce isthedominantpseudo-forceinruencingmotioninacoordinatesystemxedtotheearth. Table7.2ForcesinGeophysicalFluidDynamics DominantForcesGravityGivesrisetopressuregradients,buoyancy,andtid es. CoriolisResultsfrommotioninarotatingcoordinatesyste m FrictionIsduetorelativemotionbetweentworuidparcels. Windstressisanimportantfrictionalforce. OtherForcesAtmosphericPressureResultsininvertedbarometereect.SeismicResultsin tsunamis drivenbyearthquakes. Notethatthelasttwoforcesaremuchlessimportantthanthe rstthree. 7.2CoordinateSystem Coordinatesystemsallowustondlocationsintheoryandpr actice.Varioussystemsareuseddependingonthesizeofthefeaturesto bedescribedor mapped.Iwillrefertothesimplestsystems;descriptionso fothersystemscan befoundingeographyandgeodesybooks. 1. CartesianCoordinateSystem istheoneIwillusemostcommonlyinthe followingchapterstokeepthediscussionassimpleaspossi ble.WecandescribemostprocessesinCartesiancoordinateswithoutthe mathematical complexityofsphericalcoordinates.Thestandardconvent ioningeophysicalruidmechanicsis x istotheeast, y istothenorth,and z isup. f-Plane isaCartesiancoordinatesysteminwhichtheCoriolisforce is assumedconstant.Itisusefulfordescribingrowinregions smallcompared withtheradiusoftheearthandlargerthanafewtensofkilom eters.

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7.3.TYPESOFFLOWINTHEOCEAN 105 -plane isaCartesiancoordinatesysteminwhichtheCoriolisforce is assumedtovarylinearlywithlatitude.Itisusefulfordesc ribingrowover areasaslargeasoceanbasins. 2. Sphericalcoordinates areusedtodescriberowsthatextendoverlarge distancesandinnumericalcalculationsofbasinandglobal scalerows. 7.3TypesofFlowintheocean Manytermsareusedfordescribingtheoceancirculation.He reareafewof themorecommonlyusedtermsfordescribingcurrentsandwav es. 1. GeneralCirculation isthepermanent,time-averagedcirculation. 2. Abyssal alsocalledthe DeepCirculation isthecirculationofmass,inthe meridionalplane,inthedeepocean,drivenbymixing. 3. Wind-DrivenCirculation isthecirculationintheupperkilometerofthe oceanforcedbythewind.Thecirculationcanbecausedbyloc alwinds orbywindsinotherregions. 4. Gyres arewind-drivencyclonicoranticycloniccurrentswithdim ensions nearlythatofoceanbasins. 5. BoundaryCurrents arecurrentsrowingparalleltocoasts.Twotypesof boundarycurrentsareimportant: Westernboundarycurrentsonthewesternedgeoftheoceante ndto befast,narrowjetssuchastheGulfStreamandKuroshio. Easternboundarycurrentsareweak, e.g .theCaliforniaCurrent. 6. Squirts or Jets arelongnarrowcurrents,withdimensionsofafewhundred kilometers,thatarenearlyperpendiculartowestcoasts. 7. MesoscaleEddies areturbulentorspinningrowsonscalesofafewhundred kilometers. Inadditiontorowduetocurrents,therearemanytypesofosc illatoryrows duetowaves.Normally,whenwethinkofwavesintheocean,we visualize wavesbreakingonthebeachorthesurfacewavesinruencings hipsatsea.But manyothertypesofwavesoccurintheocean. 1. PlanetaryWaves dependontherotationoftheearthforarestoringforce, andtheyincludingRossby,Kelvin,Equatorial,andYanaiwa ves. 2. SurfaceWaves sometimescalledgravitywaves,arethewavesthateventuallybreakonthebeach.Therestoringforceisduetothela rgedensity contrastbetweenairandwaterattheseasurface. 3. InternalWaves aresub-seawavesimilarinsomerespectstosurfacewaves. Therestoringforceisduetochangeindensitywithdepth. 4. Tsunamis aresurfacewaveswithperiodsnear15minutesgeneratedby earthquakes. 5. TidalCurrents arehorizontalcurrentsandcurrentsassociatedwithinter nalwavesdrivenbythetidalpotential. 6. EdgeWaves aresurfacewaveswithperiodsofafewminutesconnedto shallowregionsnearshore.Theamplitudeofthewavesdrops oexponentiallywithdistancefromshore.

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106 CHAPTER7.THEEQUATIONSOFMOTION 7.4ConservationofMassandSalt Conservationofmassandsaltcanbeusedtoobtainveryusefu linformation aboutrowsintheocean.Forexample,supposewewishtoknowt henetloss offreshwater,evaporationminusprecipitation,fromtheM editerraneanSea. Wecouldcarefullycalculatethelatentheatruxoverthesur face,butthereare probablytoofewshipreportsforanaccurateapplicationof thebulkformula. Orwecouldcarefullymeasurethemassofwaterrowinginando utofthe seathroughtheStraitofGibraltar,butthedierenceissma llandperhaps impossibletomeasureaccurately. Wecan,however,calculatethenetevaporationknowingthes alinityofthe rowin S i andout S o ,togetherwitharoughestimateofthevolumeofwater V o rowingout,where V o isavolumerowinunitsofm 3 /s(gure7.1). Precipitation In Evaporation Out PE Mediterranean VoSill 330 m River Flow In R Atlantic Ocean ViSi = 36.2 So = 38.3 0.79 Sv Figure7.1Schematicdiagramofrowintoandoutofabasin. ValuesfromBrydenandKinder(1991). Themassrowingoutis,bydenition, o V o .Ifthevolumeoftheseadoes notchange,conservationofmassrequires: i V i = o V o (7.1) where, i ; o arethedensitiesofthewaterrowinginandout.Wecanusuall y assume,withlittleerror,that i = o Ifthereisprecipitation P andevaporation E atthesurfaceofthebasinand riverinrow R ,conservationofmassbecomes: V i + R + P = V o + E (7.2) Solvingfor( V o V i ): V o V i =( R + P ) E (7.3) whichstatesthatthenetrowofwaterintothebasinmustbala nceprecipitation plusriverinrowminusevaporationwhenaveragedoverasuc ientlylongtime. Becausesaltisnotdepositedorremovedfromthesea,conser vationofsalt requires: i V i S i = o V o S o (7.4) Where i S i arethedensityandsalinityoftheincomingwater,and o S o are densityandsalinityoftheoutrow.Withlittleerror,wecan againassumethat i = o

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7.5.THETOTALDERIVATIVE(D/DT) 107 AnExampleofConservationofMassandSalt Usingthevaluesforthe rowattheStraitofGibraltarmeasuredbyBrydenandKinder( 1991)andshown ingure7.1,solving(7.4)for V i assumingthat i = o ,andusingtheestimated valueof V o ,gives V i =0 : 836Sv=0 : 836 10 6 m 3 /s,whereSv=Sverdrup=10 6 m 3 /sistheunitofvolumetransportusedinoceanography.Usin g V i and V o in (7.3)gives( R + P E )= 4 : 6 10 4 m 3 /s. Knowing V i wecanalsocalculateaminimumrushingtimeforreplacing waterintheseabyincomingwater.Theminimumrushingtime T m isthe volumeoftheseadividedbythevolumeofincomingwater.The Mediterranean hasavolumeofaround4 10 6 km 3 .Converting0 : 836 10 6 m 3 /stokm 3 /yr weobtain2 : 64 10 4 km 3 /yr.Then, T m =4 10 6 km 3 /2 : 64 10 4 km 3 /yr =151yr.Theactualtimedependsonmixingwithinthesea.Ift hewatersare wellmixed,therushingtimeisclosetotheminimumtime,ift heyarenotwell mixed,therushingtimeislonger. OurexampleofrowintoandoutoftheMediterraneanSeaisane xample ofa boxmodel .Aboxmodelreplaceslargesystems,suchastheMediterrane an Sea,withboxes.Fluidsorchemicalsororganismscanmovebe tweenboxes,and conservationequationsareusedtoconstraintheinteracti onswithinsystems. 7.5TheTotalDerivative(D/Dt) Ifthenumberofboxesinasystemincreasestoaverylargenum beras thesizeofeachboxshrinks,weeventuallyapproachlimitsu sedindierential calculus.Forexample,ifwesubdividetherowofwaterintob oxesafewmeters onaside,andifweuseconservationofmass,momentum,oroth erproperties withineachbox,wecanderivethedierentialequationsgov erningruidrow. Considertheexampleofaccelerationofrowinasmallboxofr uid.The resultingequationiscalledthe totalderivative .Itrelatestheaccelerationofa particle Du=Dt toderivativesofthevelocityeldataxedpointintheruid Wewillusetheequationtoderivetheequationsforruidmoti onfromNewton's SecondLawwhichrequirescalculatingtheaccelerationofa particlespassinga xedpointintheruid. Webeginbyconsideringtherowofaquantity q in intoand q out outofthe smallboxsketchedingure7.2.If q canchangecontinuouslyintimeandspace, therelationshipbetween q in and q out is: q out = q in + @q @t t + @q @x x (7.5) qin Particle path qoutdt + t =x qdx + qin y,v x,u z,wq Figure7.2Sketchofrowusedforderivingthetotalderivati ve.

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108 CHAPTER7.THEEQUATIONSOFMOTION Therateofchangeofthequantity q withinthevolumeis: Dq Dt = q out q in t = @q @t + @q @x x t (7.6) But x=t isthevelocity u ,andtherefore: Dq Dt = @q @t + u @q @x Inthreedimensions,thetotalderivativebecomes: D Dt = @ dt + u @ @x + v @ @y + w @ @z (7.7a) D Dt = @ dt + u r ()(7.7b) where u isthevectorvelocityand r istheoperator del ofvectoreldtheory (SeeFeynman,Leighton,andSands1964:2{6). Thisisanamazingresult.Transformingcoordinatesfromon efollowinga particletoonexedinspaceconvertsasimplelinearderiva tiveintoanonlinearpartialderivative.Nowlet'susetheequationtocal culatethechangeof momentumofaparcelofruid.7.6MomentumEquation Newton'sSecondLawrelatesthechangeofthemomentumofaru idmass duetoanappliedforce.Thechangeis: D ( m v ) Dt = F (7.8) where F isforce, m ismass,and v isvelocity.Ihaveemphasizedtheneedto usethetotalderivativebecausewearecalculatingtheforc eonaparticle.We canassumethatthemassisconstant,and(7.8)canbewritten : D v Dt = F m = f m (7.9) where f m isforceperunitmass. Fourforcesareimportant:pressuregradients,Coriolisfo rce,gravity,and friction.Withoutderivingtheformoftheseforces(theder ivationsaregivenin thenextsection),wecanwrite(7.9)inthefollowingform. D v Dt = 1 r p 2 n v + g + F r (7.10) Accelerationequalsthenegativepressuregradientminust heCoriolisforceplus gravityplusotherforces.Here g isaccelerationofgravity, F r isfriction,and themagnitudenof n isthe RotationRateofearth ,2 radianspersiderealday or n=7 : 292 10 5 radians/s (7.11)

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7.6.MOMENTUMEQUATION 109 MomentumEquationinCartesiancoordinates: Expandingthederivative in(7.10)andwritingthecomponentsinaCartesiancoordina tesystemgivesthe MomentumEquation : @u @t + u @u @x + v @u @y + w @u @z = 1 @p @x +2n v sin + F x (7.12a) @v @t + u @v @x + v @v @y + w @v @z = 1 @p @y 2n u sin + F y (7.12b) @w @t + u @w @x + v @w @y + w @w @z = 1 @p @z +2n u cos g + F z (7.12c) where F i arethecomponentsofanyfrictionalforceperunitmass,and is latitude.Inaddition,wehaveassumedthat w<
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110 CHAPTER7.THEEQUATIONSOFMOTION Dividingbythemassoftheruid m inthebox,theaccelerationoftheruidin the x directionis: a x = F x m = @p @x V m a x = 1 @p @x (7.13) Thepressureforcesandtheaccelerationduetothepressure forcesinthe y and z directionsarederivedinthesameway. TheCoriolisTermintheMomentumEquation TheCoriolistermexists becausewedescribecurrentsinareferenceframexedonear th.Thederivation oftheCoriolistermsisnoteasy.HenryStommel,thenotedoc eanographerat theWoodsHoleOceanographicInstitutionevenwroteabooko nthesubject withDennisMoore(Stommel&Moore,1989). Usually,wejuststatethattheforceperunitmass,theaccel erationofa parcelofruidinarotatingsystem,canbewritten: a fixed = D v Dt fixed = D v Dt rotating +(2 n v )+ n ( n R )(7.14) where R isthevectordistancefromthecenterofearth, n istheangularvelocity vectorofearth,and v isthevelocityoftheruidparcelincoordinatesxedto earth.Theterm2 n v istheCoriolisforce,andtheterm n ( n R )isthe centrifugalacceleration.Thelattertermisincludedingr avity(gure7.4). TheGravityTermintheMomentumEquation Thegravitationalattractionoftwomasses M 1 and m is: F g = GM 1 m R 2 where R isthedistancebetweenthemasses,and G isthegravitationalconstant. Thevectorforce F g isalongthelineconnectingthetwomasses. Theforceperunitmassduetogravityis: F g m = g f = GM E R 2 (7.15) where M E isthemassofearth.Addingthecentrifugalaccelerationto (7.15) givesgravity g (gure7.4): g = g f n ( n R )(7.16) Notethatgravitydoesnotpointtowardearth'scenterofmas s.Thecentrifugalaccelerationcausesaplumbbobtopointatasmalla ngletotheline directedtoearth'scenterofmass.Asaresult,earth'ssurf aceincludingthe ocean'ssurfaceisnotsphericalbutitisaoblateellipsoid .Arotatingruid planethasanequatorialbulge.

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7.7.CONSERVATIONOFMASS:THECONTINUITYEQUATION 111 gfg ( x R)x Figure7.4Downwardacceleration g ofabodyatrestonearth'ssurfaceisthesumof gravitationalaccelerationbetweenthebodyandearth'sma ss g f andthecentrifugal accelerationduetoearth'srotationn (n R ).Thesurfaceofanoceanatrestmustbe perpendicularto g ,andsuchasurfaceisclosetoanellipsoidofrotation.eart h'sellipticityis greatlyexaggeratedhere.7.7ConservationofMass:TheContinuityEquation Nowlet'sderivetheequationfortheconservationofmassin aruid.We beginbywritingdowntherowofmassintoandoutofasmallbox (gure7.5). u + du zdx ydy xdz u, r r + dr Figure7.5Sketchofrowusedforderivingthecontinuityequ ation. Massrowin= uzy Massrowout=( + )( u + u ) zy Themassruxintothevolumemustbe(massrowout) (massrowin).Therefore, Massrux=( u + u + u ) zy But u = @u @x x ; = @ @x x Therefore Massrux= @u @x + u @ @x + @ @x @u @x x xyz Thethirdterminsidetheparenthesesbecomesmuchsmallert hanthersttwo termsas x 0,and Massrux= @ ( u ) @x xyz

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112 CHAPTER7.THEEQUATIONSOFMOTION Inthreedimensions: Massrux= @ ( u ) @x + @ ( v ) @y + @ ( w ) @z xyz Themassruxmustbebalancedbyachangeofmassinsidethevol ume,which is: @ @t xyz andconservationofmassrequires: @ @t + @ ( u ) @x + d ( v ) @y + @ ( w ) @z =0(7.17) Thisisthe continuityequation forcompressiblerow,rstderivedbyLeonhard Euler(1707{1783). Theequationcanbeputinanalternateformbyexpandingthed erivatives ofproductsandrearrangingtermstoobtain: @ @t + u @ @x + v @ @y + w @ @z + @u @x + @v @y + @w @z =0 Therstfourtermsconstitutethetotalderivativeofdensi ty D=Dt from(7.7), andwecanwrite(7.17)as: 1 D Dt + @u @x + @v @y + @w @z =0 (7.18) Thisisthealternateformforthecontinuityequationforac ompressibleruid. TheBoussinesqApproximation Densityisnearlyconstantintheocean, andJosephBoussinesq(1842{1929)notedthatwecansafelya ssumedensity isconstantexceptwhenitismultipliedby g incalculationsofpressureinthe ocean.Theassumptiongreatlysimpliestheequationsofmo tion. Boussinesq'sassumptionrequiresthat: 1.Velocitiesintheoceanmustbesmallcomparedtothespeed ofsound c Thisensuresthatvelocitydoesnotchangethedensity.Asve locityapproachesthespeedofsound,thevelocityeldcanproducesl argechanges ofdensitysuchasshockwaves. 2.Thephasespeedofwavesordisturbancesmustbesmallcomp aredwith c .Soundspeedinincompressiblerowsisinnite,andwemusta ssume theruidiscompressiblewhendiscussingsoundintheocean. Thusthe approximationisnottrueforsound.Allotherwavesintheoc eanhave speedssmallcomparedtosound. 3.Theverticalscaleofthemotionmustbesmallcomparedwit h c 2 / g ,where g isgravity.Thisensuresthataspressureincreaseswithdep thinthe ocean,theincreaseinpressureproducesonlysmallchanges indensity. Theapproximationsaretrueforoceanicrows,andtheyensur ethatoceanic rowsareincompressible.SeeKundu(1990:79and112),Gill( 1982:85),Batchelor(1967:167),orothertextsonruiddynamicsforamoreco mpletedescription oftheapproximation.

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7.8.SOLUTIONSTOTHEEQUATIONSOFMOTION 113 Compressibility TheBoussinesqapproximationisequivalenttoassumingsea waterisincompressible.Nowlet'sseehowtheassumptionsi mpliesthecontinuityequation.Wedenethe coecientofcompressibility 1 V @V @p = 1 V dV dt dp dt where V isvolume,and p ispressure.Forincompressiblerows, =0,and: 1 V dV dt =0 because dp / dt 6 =0.Rememberingthatdensityismass m perunitvolume V andthatmassisconstant: 1 V dV dt = V d dt 1 V = V m d dt m V = 1 d dt = 1 D Dt =0 Iftherowisincompressible,(7.18)becomes: @u @x + @v @y + @w @z =0 (7.19) Thisisthe ContinuityEquationforIncompressibleFlows 7.8SolutionstotheEquationsofMotion Equations(7.12)and(7.19)arefourequations,thethreeco mponentsofthe momentumequationplusthecontinuityequation,withfouru nknowns: u v w p .Note,however,thatthesearenon-linearpartialdierent ialequations. Conservationofmomentum,whenappliedtoaruid,converted asimple,rstorder,ordinary,dierentialequationforvelocity(Newto n'sSecondLaw),which isusuallyeasytosolve,intoanon-linearpartialdierent ialequation,whichis almostimpossibletosolve.BoundaryConditions: Inruidmechanics,wegenerallyassume: 1.Novelocitynormaltoaboundary,whichmeansthereisnoro wthrough theboundary;and 2.Norowparalleltoasolidboundary,whichmeansnoslipatt hesolid boundary. Solutions Weexpectthatfourequationsinfourunknownsplusboundary conditionsgiveasystemofequationsthatcanbesolvedinprinc iple.Inpractice, solutionsarediculttondevenforthesimplestrows.Firs t,asfarasIknow, therearenoexactsolutionsfortheequationswithfriction .Thereareveryfew exactsolutionsfortheequationswithoutfriction.Thosew hoareinterestedin oceanwavesmightnotethatonesuchexactsolutionisGerstn er'ssolutionfor waterwaves(Lamb,1945:251).Becausetheequationsarealm ostimpossible tosolve,wewilllookforwaystogreatlysimplifytheequati ons.Later,wewill ndthatevennumericalcalculationsaredicult.

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114 CHAPTER7.THEEQUATIONSOFMOTION Analyticalsolutionscanbeobtainedformuchsimpliedfor msoftheequationsofmotion.Suchsolutionsareusedtostudyprocessesi ntheocean,includingwaves.Solutionsforoceanicrowswithrealisticco astsandbathymetric featuresmustbeobtainedfromnumericalsolutions.Inthen extfewchapters weseeksolutionstosimpliedformsoftheequations.InCha pter15wewill considernumericalsolutions.7.9ImportantConcepts 1.Gravity,buoyancy,andwindarethedominantforcesactin gontheocean. 2.Earth'srotationproducesapseudoforce,theCoriolisfo rce. 3.Conservationlawsappliedtorowintheoceanleadtoequat ionsofmotion. Conservationofsalt,volumeandotherquantitiescanleadt odeepinsights intooceanicrow. 4.Thetransformationfromequationsofmotionappliedtoru idparcelsto equationsappliedataxedpointinspacegreatlycomplicat estheequationsofmotion.Thelinear,rst-order,ordinarydierent ialequations describingNewtoniandynamicsofamassacceleratedbyafor cebecome nonlinear,partialdierentialequationsofruidmechanic s. 5.Flowintheoceancanbeassumedtobeincompressibleexcep twhendescribingsound.Densitycanbeassumedtobeconstantexcept whendensityismultipliedbygravity g .TheassumptioniscalledtheBoussinesq approximation. 6.Conservationofmassleadstothecontinuityequation,wh ichhasanespeciallysimpleformforanincompressibleruid.

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Chapter8EquationsofMotionWithViscosityThroughoutmostoftheinterioroftheoceanandatmospheref rictionisrelativelysmall,andwecansafelyassumethattherowisfricti onless.Atthe boundaries,friction,intheformofviscosity,becomesimp ortant.Thisthin, viscouslayeriscalleda boundarylayer .Withinthelayer,thevelocityslows fromvaluestypicaloftheinteriortozeroatasolidboundar y.Iftheboundary isnotsolid,thentheboundarylayerisathinlayerofrapidl ychangingvelocity wherebyvelocityononesideoftheboundarychangestomatch thevelocityon theothersideoftheboundary.Forexample,thereisabounda rylayeratthe bottomoftheatmosphere,theplanetaryboundarylayerIdes cribedinChapter 3.Intheplanetaryboundarylayer,velocitygoesfrommanym eterspersecond inthefreeatmospheretotensofcentimeterspersecondatth eseasurface. Belowtheseasurface,anotherboundarylayer,theEkmanlay erdescribedin Chapter9,matchestherowattheseasurfacetothedeeperrow InthischapterIconsidertheroleoffrictioninruidrows,a ndthestability oftherowstosmallchangesinvelocityordensity.8.1TheInruenceofViscosity Viscosityisthetendencyofaruidtoresistshear.Inthelas tchapterIwrote the x {componentofthemomentumequationforaruidintheform(7: 12a): @u @t + u @u @x + v @u @y + w @u @z = 1 @p @x +2n v sin # + F x (8.1) where F x wasafrictionalforceperunitmass.Nowwecanconsiderthef ormof thistermifitisduetoviscosity. Moleculesinaruidclosetoasolidboundarysometimestrike theboundary andtransfermomentumtoit(gure8.1).Moleculesfurtherf romtheboundary collidewithmoleculesthathavestrucktheboundary,furth ertransferringthe changeinmomentumintotheinterioroftheruid.Thistransf erofmomentum 115

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116 CHAPTER8.EQUATIONSOFMOTIONWITHVISCOSITY z x Molecules carry horizontalmomentum perpendicular towall through perpendicular velocity and collisions with other molecules Velocity Wall Figure8.1Moleculescollidingwiththewallandwitheachot hertransfer momentumfromtheruidtothewall,slowingtheruidvelocity is molecularviscosity .Molecules,however,travelonlymicrometersbetween collisions,andtheprocessisveryinecientfortransferr ingmomentumevena fewcentimeters.Molecularviscosityisimportantonlywit hinafewmillimeters ofaboundary. Molecularviscosity istheratioofthestress T tangentialtotheboundary ofaruidandthevelocityshearattheboundary.Sothestress hastheform: T xz = @u @z (8.2) forrowinthe( x;z )planewithinafewmillimetresofthesurface,where is thekinematicmolecularviscosity.Typically =10 6 m 2 /sforwaterat20 C. Generalizing(8.2)tothreedimensionsleadstoastressten sorgivingthe ninecomponentsofstressatapointintheruid,includingpr essure,whichis anormalstress,andshearstresses.Aderivationofthestre sstensorisbeyond thescopeofthisbook,butyoucanndthedetailsinLamb(194 5: x 328)or Kundu(1990:p.93).Foranincompressibleruid,thefrictio nalforceperunit massin(8.1)takestheform: F x = @ @x @u @x + @ @y @u @y + @ @z @u @z = 1 @T xx @x + @T xy @y + @T xz @z (8.3) 8.2Turbulence Ifmolecularviscosityisimportantonlyoverdistancesofa fewmillimeters, andifitisnotimportantformostoceanicrows,unlessofcou rseyouarea zooplanktertryingtoswimintheocean,howthenistheinrue nceofaboundary transferredintotheinterioroftherow?Theansweris:thro ughturbulence. Turbulencearisesfromthenon-lineartermsinthemomentum equation ( u@u=@x etc .).Theimportanceofthesetermsisgivenbyanon-dimension al number,theReynoldsNumber Re ,whichistheratioofthenon-lineartermsto theviscousterms: Re = Non-linearTerms ViscousTerms = u @u @x @ 2 u @x 2 U U L U L 2 = UL (8.4)

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8.2.TURBULENCE 117 Water Valve Glass Tube Dye Figure8.2Reynoldsapparatusforinvestigatingthetransi tiontoturbulenceinpiperow, withphotographsofnear-laminarrow(left)andturbulentr ow(right)inaclearpipemuch liketheoneusedbyReynolds.AfterBinder(1949:88-89).where, U isatypicalvelocityoftherowand L isatypicallengthdescribing therow.Youarefreetopickwhatever U;L mightbetypicaloftherow.For example L canbeeitheratypicalcross-streamdistance,oranalong-s treamdistance.Typicalvaluesintheopenoceanare U =0 : 1m/sand L =1megameter, so Re =10 11 .Becausenon-lineartermsareimportantifRe > 10{1000,they arecertainlyimportantintheocean.Theoceanisturbulent TheReynoldsnumberisnamedafterOsborneReynolds(1842{1 912)who conductedexperimentsinthelate19thcenturytounderstan dturbulence.In onefamousexperiment(Reynolds1883),heinjecteddyeinto waterrowingat variousspeedsthroughatube(gure8.2).Ifthespeedwassm all,therow wassmooth.Thisiscalled laminarrow .Athigherspeeds,therowbecame irregularandturbulent.ThetransitionoccurredatRe= VD= 2000,where V istheaveragespeedinthepipe,and D isthediameterofthepipe. AsReynoldsnumberincreasesabovesomecriticalvalue,the rowbecomes moreandmoreturbulent.Notethatrowpatternisafunctiono fReynold's number.AllrowswiththesamegeometryandthesameReynolds numberhave thesamerowpattern.Thusrowaroundallcircularcylinders ,whether1mm or1mindiameter,lookthesameastherowatthetopofgure8. 3ifthe Reynoldsnumberis20.Furthermore,theboundarylayerisco nnedtoavery thinlayerclosetothecylinder,inalayertoothintoshowin thegure. TurbulentStresses:TheReynoldsStress Prandtl,Karmanandothers whostudiedruidmechanicsintheearly20thcentury,hypoth esizedthatparcels ofruidinaturbulentrowplayedthesameroleintransferrin gmomentum withintherowthatmoleculesplayedinlaminarrow.Thework ledtotheidea ofturbulentstresses. Toseehowthesestressesmightarise,considerthemomentum equationfor

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118 CHAPTER8.EQUATIONSOFMOTIONWITHVISCOSITY S A B <1 20 S S S80,000 1,000,000 E F Total Head -2-10 C 174 A A 0.5 0 1 Width YD A 5,00014,480 10 0.5 1 D X/D Width D 0 Figure8.3FlowpastacircularcylinderasafunctionofReyn oldsnumberbetweenoneand amillion.FromRichardson(1961).Theappropriaterowsare :A|atoothpickmovingat1 mm/s;B|ngermovingat2cm/s;F|handoutacarwindowat60mp h.Allrowatthe sameReynoldsnumberhasthesamestreamlines.Flowpasta10 cmdiametercylinderat1 cm/slooksthesameas10cm/srowpastacylinder1cmindiamet erbecauseinbothcases Re=1000.arowwithmean( U;V;W )andturbulent( u 0 ;v 0 ;w 0 )components: u = U + u 0 ; v = V + v 0 ; w = W + w 0 ; p = P + p 0 (8.5) wherethemeanvalue U iscalculatedfromatimeorspaceaverage: U = h u i = 1 T Z T 0 u ( t ) dt or U = h u i = 1 X Z X 0 u ( x ) dx (8.6) Thenon-lineartermsinthemomentumequationcanbewritten : ( U + u 0 ) @ ( U + u 0 ) @x = U @U @x + U @u 0 @x + u 0 @U @x + u 0 @u 0 @x ( U + u 0 ) @ ( U + u 0 ) @x = U @U @x + u 0 @u 0 @x (8.7)

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8.3.CALCULATIONOFREYNOLDSSTRESS: 119 Thesecondequationfollowsfromtherstsince h U@u 0 =@x i =0and h u 0 @U=@x i =0,whichfollowfromthedenitionof U : h U@u 0 =@x i = U@ h u 0 i =@x =0. Using(8.5)in(7.19)gives: @U @x + @V @y + @W @z + @u 0 @x + @v 0 @y + @w 0 @z =0(8.8) Subtractingthemeanof(8.8)from(8.8)splitsthecontinui tyequationintotwo equations: @U @x + @V @y + @W @z =0(8.9a) @u 0 @x + @v 0 @y + @w 0 @z =0(8.9b) Using(8.5)in(8.1)takingthemeanvalueoftheresultingeq uation,then simplifyingusing(8.7),thex-componentofthemomentumeq uationforthe meanrowbecomes: DU Dt = 1 @P @x +2n V sin + @ @x @U @x h u 0 u 0 i + @ @y @U @y h u 0 v 0 i + @ @z @U @z h u 0 w 0 i (8.10) Thederivationisnotassimpleasitseems.SeeHinze(1975:2 2)fordetails. Thustheadditionalforceperunitmassduetotheturbulence is: F x = @ @x h u 0 u 0 i @ @y h u 0 v 0 i @ @z h u 0 w 0 i (8.11) Theterms h u 0 u 0 i h u 0 v 0 i ,and h u 0 w 0 i transfereastwardmomentum( u 0 ) inthe x y ,and z directions.Forexample,theterm h u 0 w 0 i givesthedownwardtransportofeastwardmomentumacrossahorizontalpla ne.Becausethey transfermomentum,andbecausetheywererstderivedbyOsb orneReynolds, theyarecalled ReynoldsStresses 8.3CalculationofReynoldsStress: TheReynoldsstressessuchas @ h u 0 w 0 i =@z arecalledvirtualstresses(cf. Goldstein,1965: x 69& x 81)becauseweassumethattheyplaythesameroleas theviscoustermsintheequationofmotion.Toproceedfurth er,weneedvalues orfunctionalformfortheReynoldsstress.Severalapproac hesareused. FromExperiments WecancalculateReynoldsstressesfromdirectmeasurementsof( u 0 ;v 0 ;w 0 )madeinthelaboratoryorocean.Thisisaccurate,buthard togeneralizetootherrows.Soweseekmoregeneralapproach es.

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120 CHAPTER8.EQUATIONSOFMOTIONWITHVISCOSITY ByAnalogywithMolecularViscosity Let'sreturntotheexampleingure8.1,whichshowsrowaboveasurfaceinthe x y plane.Prandtl,ina revolutionarypaperpublishedin1904,statedthatturbule ntviscouseectsare onlyimportantinaverythinlayerclosetothesurface,theb oundarylayer. Prandtl'sinventionoftheboundarylayerallowsustodescr ibeveryaccurately turbulentrowofwindabovetheseasurface,orrowatthebott omboundary layerintheocean,orrowinthemixedlayerattheseasurface .Seethebox TurbulentBoundaryLayerOveraFlatPlate Tocalculaterowinaboundarylayer,weassumethatrowiscon stantin the x y direction,thatthestatisticalpropertiesoftherowvaryo nlyinthe z direction,andthatthemeanrowissteady.Therefore @=@t = @=@x = @=@y =0, and(8.10)canbewritten: 2n V sin + @ @z @U @z h u 0 w 0 i =0(8.12) Wenowassume,inanalogywith(8.2) h u 0 w 0 i = T xz = A z @U @z (8.13) where A z isan eddyviscosity or eddydiusivity whichreplacesthemolecular viscosity in(8.2).Withthisassumption, @T xz @z = @ @z A z @U @z A z @ 2 U @z 2 (8.14) whereIhaveassumedthat A z iseitherconstantorthatitvariesmoreslowly inthe z directionthan @U=@z .Later,Iwillassumethat A z z Becauseeddiescanmixheat,salt,orotherpropertiesaswel lasmomentum, Iwillusethetermeddydiusivity.Itismoregeneralthaned dyviscosity,which isthemixingofmomentum. The x and y momentumequationsforahomogeneous,steady-state,turbu lentboundarylayeraboveorbelowahorizontalsurfaceare: fV + @T xz @z =0(8.15a) fU @T yz @z =0(8.15b) where f =2 sin istheCoriolisparameter,andIhavedroppedthemolecularviscositytermbecauseitismuchsmallerthantheturbul enteddyviscosity. Note,(8.15b)followsfromasimilarderivationfromthe y -componentofthemomentumequation.Wewillneed(8.15)whenwedescriberownea rthesurface. Theassumptionthataneddyviscosity A z canbeusedtorelatetheReynolds stresstothemeanrowworkswellinturbulentboundarylayer s.However A z cannotbeobtainedfromtheory.Itmustbecalculatedfromda tacollectedin windtunnelsormeasuredinthesurfaceboundarylayeratsea .SeeHinze(1975,

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8.3.CALCULATIONOFREYNOLDSSTRESS: 121 TheTurbulentBoundaryLayerOveraFlatPlate Therevolutionaryconceptofaboundarylayerwasinventedb yPrandtlin1904 (Anderson,2005).Later,theconceptwasappliedtorowover aratplatebyG.I. Taylor(1886{1975),L.Prandtl(1875{1953),andT.vonKarm an(1881{1963)who workedindependentlyonthetheoryfrom1915to1935.Theire mpiricaltheory, sometimescalledthe mixing-lengththeory predictswellthemeanvelocityprole closetotheboundary.Ofinteresttous,itpredictsthemean rowofairabove thesea.Here'sasimpliedversionofthetheoryappliedtoa smoothsurface. Webeginbyassumingthatthemeanrowintheboundarylayeris steadyand thatitvariesonlyinthe z direction.Withinafewmillimetersoftheboundary, frictionisimportantand(8.2)hasthesolution U = T x z (8.16) andthemeanvelocityvarieslinearlywithdistanceaboveth eboundary.Usually (8.16)iswrittenindimensionlessform: U u = u z (8.17) where u 2 T x = isthe frictionvelocity Furtherfromtheboundary,therowisturbulent,andmolecul arfrictionisnot important.Inthisregime,wecanuse(8.13),and A z @U @z = u 2 (8.18) PrandtlandTaylorassumedthatlargeeddiesaremoreeecti veinmixing momentumthansmalleddies,andtherefore A z oughttovarywithdistancefrom thewall.Karmanassumedthatithadtheparticularfunction alform A z = zu where isadimensionlessconstant.Withthisassumption,theequa tionforthe meanvelocityprolebecomes zu @U @z = u 2 (8.19) Because U isafunctiononlyof z ,wecanwrite dU = u = ( z ) dz ,whichhasthe solution U = u ln z z 0 (8.20) where z 0 isdistancefromtheboundaryatwhichvelocitygoestozero. Forairrowoverthesea, =0 : 4and z o isgivenbyCharnock's(1955)relation z 0 =0 : 0156 u 2 =g .Themeanvelocityintheboundarylayerjustabovethesea surfacedescribedin x 4.3tswellthelogarithmicproleof(8.20),asdoesthe meanvelocityintheupperfewmetersoftheseajustbelowthe seasurface. Furthermore,using(4.2)inthedenitionofthefrictionve locity,thenusing(8.20) givesCharnock'sformofthedragcoecientasafunctionofw indspeed. x 5{2and x 7{5)andGoldstein(1965: x 80)formoreonthetheoryofturbulence rowneararatplate.

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122 CHAPTER8.EQUATIONSOFMOTIONWITHVISCOSITY Prandtl'stheorybasedonassumption(8.13)workswellonly wherefriction ismuchlargerthantheCoriolisforce.Thisistrueforairro wwithintensof metersoftheseasurfaceandforwaterrowwithinafewmeters ofthesurface. Theapplicationofthetechniquetootherrowsintheoceanis lessclear.For example,therowinthemixedlayeratdepthsbelowabouttenm etersisless welldescribedbytheclassicalturbulenttheory.Tennekes andLumley(1990: 57)write: Mixing-lengthandeddyviscositymodelsshouldbeusedonly togenerateanalyticalexpressionsfortheReynoldsstressandme an-velocity proleifthosearedesiredforcurvettingpurposesinturb ulentrows characterizedbyasinglelengthscaleandasinglevelocity scale.The useofmixing-lengththeoryinturbulentrowswhosescaling lawsarenot knownbeforehandshouldbeavoided. Problemswiththeeddy-viscosityapproach: 1.Exceptinboundarylayersafewmetersthick,geophysical rowsmaybe inruencedbyseveralcharacteristicscales.Forexample,i ntheatmospheric boundarylayerabovethesea,atleastthreescalesmaybeimp ortant:i) theheightabovethesea z ,ii)theMonin-Obukhovscale L discussedin x 4.3,andiii)thetypicalvelocity U dividedbytheCoriolisparameter U=f 2.Thevelocities u 0 ;w 0 areapropertyofthe ruid ,while A z isapropertyof the row ; 3.Eddyviscositytermsarenotsymmetric: h u 0 v 0 i = h v 0 u 0 i ;but A x @V @x 6 = A y @U @y FromaStatisticalTheoryofTurbulence TheReynoldsstressescanbe calculatedfromvarioustheorieswhichrelate h u 0 u 0 i tohigherordercorrelations oftheform h u 0 u 0 u 0 i .Theproblemthenbecomes:Howtocalculatethehigher orderterms?Thisisthe closureproblem inturbulence.Thereisnogeneral solution,buttheapproachleadstousefulunderstandingof someformsofturbulencesuchasisotropicturbulencedownstreamofagridin awindtunnel (Batchelor1967). Isotropicturbulence isturbulencewithstatisticalproperties thatareindependentofdirection. Theapproachcanbemodiedsomewhatforrowintheocean.Int heidealizedcaseofhighReynoldsrow,wecancalculatethestatis ticalpropertiesof arowinthermodynamicequilibrium.Becausetheactualrowi ntheoceanis farfromequilibrium,weassumeitwillevolvetowardsequil ibrium.Holloway (1986)providesagoodreviewofthisapproach,showinghowi tcanbeusedto derivetheinruenceofturbulenceonmixingandheattranspo rts.Oneinterestingresultoftheworkisthatzonalmixingoughttobelargert hanmeridional mixing.

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8.4.MIXINGINTHEOCEAN 123 Summary Theturbulenteddyviscosities A x A y ,and A z cannotbecalculated accuratelyformostoceanicrows. 1.Theycanbeestimatedfrommeasurementsofturbulentrows .Measurementsintheocean,however,aredicult.Measurementsinth elab,althoughaccurate,cannotreachReynoldsnumbersof10 11 typicalofthe ocean. 2.Thestatisticaltheoryofturbulencegivesusefulinsigh tintotheroleof turbulenceintheocean,andthisisanareaofactiveresearc h. Table8.1SomeValuesforViscosity water =10 6 m 2 /s tarat 15 C =10 6 m 2 /s glacierice =10 10 m 2 /s A yocean =10 4 m 2 /s A zocean =(10 5 10 3 )m 2 /s 8.4MixingintheOcean Turbulenceintheoceanleadstomixing.Becausetheoceanha sstablestratication,verticaldisplacementmustworkagainstthebuoy ancyforce.Vertical mixingrequiresmoreenergythanhorizontalmixing.Asares ult,horizontal mixingalongsurfacesofconstantdensityismuchlargertha nverticalmixing acrosssurfacesofconstantdensity.Thelatter,however,u suallycalled diapycnal mixing ,isveryimportantbecauseitchangestheverticalstructur eoftheocean, anditcontrolstoalargeextenttherateatwhichdeepwatere ventuallyreaches thesurfaceinmidandlowlatitudes. Theequationsdescribingmixingdependonmanyprocesses.S eeGarrett (2006)foragoodoverviewofthesubject.HereIconsidersom esimplerows. Asimpleequationforverticalmixingbyeddiesofatracers uchassaltor temperatureis: @ @t + W @ @z = @ @z A z @ @z + S (8.21) where A z istheverticaleddydiusivity, W isameanverticalvelocity,and S isasourceterm.AverageVerticalMixing WalterMunk(1966)usedaverysimpleobservationtocalculateverticalmixingintheocean.Heobservedt hattheoceanhas athermoclinealmosteverywhere,andthedeeperpartofthet hermoclinedoes notchangeevenoverdecades(gure8.4).Thiswasaremarkab leobservation becauseweexpectdownwardmixingwouldcontinuouslydeepe nthethermocline.Butitdoesn't.Therefore,asteady-statethermocli nerequiresthatthe downwardmixingofheatbyturbulencemustbebalancedbyanu pwardtransportofheatbyameanverticalcurrent W .Thisfollowsfrom(8.21)forsteady

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124 CHAPTER8.EQUATIONSOFMOTIONWITHVISCOSITY 19661985 -5000 -4000 -3000 -2000 -1000 1o2o3o4o5o6oPressure (decibars) Potential Temperature (Celsius) Figure8.4Potentialtemperaturemeasuredasafunctionofd epth(pressure)near24.7 N, 161.4 WinthecentralNorthPacicbythe Yaquina in1966( ),andbythe Thompson in 1985( ).Datafrom AtlasofOceanSections producedbySwift,Rhines,andSchlitzer. statewithnosourcesorsinksofheat: W @T @z = A z @ 2 T @z 2 (8.22) where T istemperatureasafunctionofdepthinthethermocline. Theequationhasthesolution: T T 0 exp( z=H )(8.23) where H = A z =W isthescaledepthofthethermocline,and T 0 isthetemperaturenearthetopofthethermocline.Observationsofthesh apeofthedeep thermoclineareindeedveryclosetoaexponentialfunction .Munkusedan exponentialfunctiontthroughtheobservationsof T ( z )toget H Munkcalculated W fromtheobservedverticaldistributionof 14 C,aradioactiveisotopeofcarbon,toobtainaverticaltimescale .Inthiscase, S = 1 : 24 10 4 years 1 .Thelengthandtimescalesgave W =1 : 2cm/dayand h A z i =1 : 3 10 4 m 2 /sAverageVerticalEddyDiusivity(8.24) wherethebracketsdenoteaverageeddydiusivityinthethe rmocline. Munkalsoused W tocalculatetheaverageverticalruxofwaterthroughthe thermoclineinthePacic,andtheruxagreedwellwiththera teofformation ofbottomwaterassumingthatbottomwaterupwellsalmostev erywhereata constantrateinthePacic.Globally,histheoryrequiresu pwardmixingof25 to30Sverdrupsofwater,whereoneSverdrupis10 6 cubicmeterspersecond.

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8.4.MIXINGINTHEOCEAN 125 MeasuredVerticalMixing Directobservationsofverticalmixingrequired thedevelopmentoftechniquesformeasuring:i)thenestru ctureofturbulence, includingprobesabletomeasuretemperatureandsalinityw ithaspatialresolutionofafewcentimeters(Gregg1991),andii)thedistrib utionoftracerssuch assulphurhexaruoride(SF 6 )whichcanbeeasilydetectedatconcentrationsas smallasonegraminacubickilometerofseawater. Directmeasurementsofopen-oceanturbulenceandthedius ionofSF 6 yield aneddydiusivity: A z 1 10 5 m 2 /sOpen-OceanVerticalEddyDiusivity(8.25) Forexample,Ledwell,Watson,andLaw(1998)injected139kg ofSF 6 inthe Atlanticnear26 N,29 W1200kmwestoftheCanaryIslandsatadepthof 310m.Theythenmeasuredtheconcentrationforvemonthsas itmixed overhundredsofkilometerstoobtainadiapycnaleddydius ivityof A z = 1 : 2 0 : 2 10 5 m 2 /s. ThelargediscrepancybetweenMunk'scalculationoftheave rageeddydiffusivityforverticalmixingandthesmallvaluesobservedi ntheopenoceanhas beenresolvedbyrecentstudiesthatshow: A z 10 3 10 1 m 2 /sLocalVerticalEddyDiusivity(8.26) Polzinetal.(1997)measuredtheverticalstructureoftemp eratureinthe BrazilBasininthesouthAtlantic.Theyfound A z > 10 3 m 2 /sclosetothe bottomwhenthewaterrowedoverthewesternrankofthemid-A tlanticridgeat theeasternedgeofthebasin.KunzeandToole(1997)calcula tedenhancededdy diusivityaslargeas A =10 3 m 2 /saboveFieberlingGuyotintheNorthwest Pacicandsmallerdiusivityalongtherankoftheseamount .And,Garbatoet al(2004)calculatedevenstrongermixingintheScotiaSeaw heretheAntarctic CircumpolarCurrentrowsbetweenAntarcticaandSouthAmer ica. Theresultsoftheseandotherexperimentsshowthatmixingo ccursmostly bybreakinginternalwavesandbyshearatoceanicboundarie s:alongcontinentalslopes,aboveseamountsandmid-oceanridges,atfronts ,andinthemixed layerattheseasurface.Toalargeextent,themixingisdriv enbydeep-ocean tidalcurrents,whichbecometurbulentwhentheyrowpastob staclesonthesea roor,includingseamountsandmid-oceanridges(Jayneetal ,2004). Becausewaterismixedalongboundariesorinotherregions( Gnadadesikan, 1999),wemusttakecareininterpretingtemperatureprole ssuchasthatin gure8.4.Forexample,waterat1200minthecentralnorthAt lanticcould movehorizontallytotheGulfStream,whereitmixeswithwat erfrom1000 m.Themixedwatermaythenmovehorizontallybackintothece ntralnorth Atlanticatadepthof1100m.Thusparcelsofwaterat1200man dat1100m atsomelocationmayreachtheirpositionalongentirelydi erentpaths. MeasuredHorizontalMixing Eddiesmixruidinthehorizontal,andlarge eddiesmixmoreruidthansmalleddies.Eddiesrangeinsizef romafewmetersduetoturbulenceinthethermoclineuptoseveralhundr edkilometersfor geostrophiceddiesdiscussedinChapter10.

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126 CHAPTER8.EQUATIONSOFMOTIONWITHVISCOSITY Ingeneral,mixingdependsonReynoldsnumber R (Tennekes1990:p.11) A r A UL = R (8.27) where r isthemoleculardiusivityofheat, U isatypicalvelocityinaneddy, and L isthetypicalsizeofaneddy.Furthermore,horizontaleddy diusivity aretenthousandtotenmilliontimeslargerthantheaverage verticaleddy diusivity. Equation(8.27)implies A x UL .Thisfunctionalformagreeswellwith JosephandSender's(1958)analysis,asreportedin(Bowden 1962)ofspreading ofradioactivetracers,opticalturbidity,andMediterran eanSeawaterinthe northAtlantic.Theyreport A x = PL (8.28) 10km
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8.5.STABILITY 127 3.Observationsofmixingintheoceanimplythatnumericalm odelsofthe oceaniccirculationshouldusemixingschemesthathavedi erenteddy diusivityparallelandperpendiculartosurfacesofconst antdensity,not parallelandperpendiculartolevelsurfacesofconstant z asIusedabove. Horizontalmixingalongsurfacesofconstant z leadstomixingacrosslayers ofconstantdensitybecauselayersofconstantdensityarei nclinedtothe horizontalbyabout10 3 radians(see x 10.7andgure10.13). StudiesbyDanabasoglu,McWilliams,andGent(1994)showth atnumericalmodelsusingisopycnalanddiapycnalmixingleadstomu chmore realisticsimulationsoftheoceaniccirculation. 4.Mixingishorizontalandtwodimensionalforhorizontals calesgreaterthan NH= (2 f )where H isthewaterdepth, N isthestabilityfrequency(8.36), and f istheCoriolisparameter(Dritschel,Juarez,andAmbaum(1 999). 8.5Stability Wesawinsection8.2thatruidrowwithasucientlylargeRey noldsnumberisturbulent.Thisisoneformofinstability.Manyother typesofinstability occurintheintheocean.Here,let'sconsiderthreeofthemo reimportantones: i) staticstability associatedwithchangeofdensitywithdepth,ii) dynamicstability associatedwithvelocityshear,andiii) double-diusion associatedwith salinityandtemperaturegradientsintheocean.StaticStabilityandtheStabilityFrequency Considerrststaticstability. Ifmoredensewaterliesabovelessdensewater,theruidisun stable.Themore densewaterwillsinkbeneaththelessdense.Conversely,if lessdensewater liesabovemoredensewater,theinterfacebetweenthetwois stable.Buthow stable?Wemightguessthatthelargerthedensitycontrasta crosstheinterface, themorestabletheinterface.Thisisanexampleofstaticst ability.Static stabilityisimportantinany stratied rowwheredensityincreaseswithdepth, andweneedsomecriterionfordeterminingtheimportanceof thestability. Consideraparcelofwaterthatisdisplacedverticallyanda diabaticallyina stratiedruid(gure8.5).Thebuoyancyforce F actingonthedisplacedparcel isthedierencebetweenitsweight Vg 0 andtheweightofthesurroundingwater Vg 2 ,where V isthevolumeoftheparcel: F = Vg ( 2 0 ) Displaced Volume of Water V @ r2Displacement Distance dz Parcel with Density r Figure8.5Sketchforcalculatingstaticstabilityandstab ilityfrequency.

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128 CHAPTER8.EQUATIONSOFMOTIONWITHVISCOSITY Theaccelerationofthedisplacedparcelis: a = F m = g ( 2 0 ) 0 (8.31) but 2 = + d dz water z (8.32) 0 = + d dz parcel z (8.33) Using(8.32)and(8.33)in(8.31),ignoringtermsproportio nalto z 2 ,weobtain: E = 1 d dz water d dz parcel # (8.34) where E a= ( gz )isthe stability ofthewatercolumn(McDougall,1987; Sverdrup,Johnson,andFleming,1942:416;orGill,1982:50 ). Intheupperkilometeroftheoceanstabilityislarge,andth ersttermin (8.34)ismuchlargerthanthesecond.Thersttermispropor tionaltothe rateofchangeofdensityofthewatercolumn.Thesecondterm isproportional tothecompressibilityofseawater,whichisverysmall.Neg lectingthesecond term,wecanwritethe stabilityequation : E 1 d dz (8.35) Theapproximationusedtoderive(8.35)isvalidfor E> 50 10 8 /m. Belowaboutakilometerintheocean,thechangeindensitywi thdepthis sosmallthatwemustconsiderthesmallchangeindensityoft heparceldueto changesinpressureasitismovedvertically. Stabilityisdenedsuchthat E> 0Stable E =0NeutralStability E< 0Unstable Intheupperkilometeroftheocean, z< 1 ; 000m, E =(50|1000) 10 8 /m, andindeeptrencheswhere z> 7 ; 000m, E =1 10 8 /m. Theinruenceofstabilityisusuallyexpressedbya stabilityfrequency N : N 2 gE (8.36) Thestabilityfrequencyisoftencalledthe buoyancyfrequency orthe BruntVaisalafrequency .Thefrequencyquantiestheimportanceofstability,andi t isafundamentalvariableinthedynamicsofstratiedrow.I nsimplestterms, thefrequencycanbeinterpretedastheverticalfrequencye xcitedbyavertical displacementofaruidparcel.Thus,itisthemaximumfreque ncyofinternal wavesintheocean.Typicalvaluesof N areafewcyclesperhour(gure8.6).

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8.5.STABILITY 129 7.5 N, 137.0 E 16 June 1974 35.0 N, 151.9 E 24 April 1976 0 1000200030004000 0123Depth (decibars)-0-100-200-300-400 051015 Stability Frequency (cycles per hour) Figure8.6.ObservedstabilityfrequencyinthePacic. Left: Stabilityofthedeep thermoclineeastoftheKuroshio. Right: Stabilityofashallowthermoclinetypicalofthe tropics.Notethechangeofscales.DynamicStabilityandRichardson'sNumber Ifvelocitychangeswith depthinastable,stratiedrow,thentherowmaybecomeunst ableifthe changeinvelocitywithdepth,the currentshear ,islargeenough.Thesimplest exampleiswindblowingovertheocean.Inthiscase,stabili tyisverylargeacross theseasurface.Wemightsayitisinnitebecausethereisas tepdiscontinuity in ,and(8.36)isinnite.Yet,windblowingontheoceancreate swaves,andif thewindisstrongenough,thesurfacebecomesunstableandt hewavesbreak. Thisisanexampleof dynamicinstability inwhichastableruidismade unstablebyvelocityshear.Anotherexampleofdynamicinst ability,theKelvinHelmholtzinstability,occurswhenthedensitycontrastin ashearedrowismuch lessthanattheseasurface,suchasinthethermoclineoratt hetopofastable, atmosphericboundarylayer(gure8.7). Therelativeimportanceofstaticstabilityanddynamicins tabilityisexpressedbythe RichardsonNumber : R i gE ( @U=@z ) 2 (8.37) wherethenumeratoristhestrengthofthestaticstability, andthedenominator isthestrengthofthevelocityshear. R i > 0 : 25Stable R i < 0 : 25VelocityShearEnhancesTurbulence NotethatasmallRichardsonnumberisnottheonlycriterion forinstability. TheReynoldsnumbermustbelargeandtheRichardsonnumberm ustbeless than0.25forturbulence.Thesecriteriaaremetinsomeocea nicrows.The turbulencemixesruidinthevertical,leadingtoavertical eddyviscosityand eddydiusivity.Becausetheoceantendstobestronglystra tiedandcurrents tendtobeweak,turbulentmixingisintermittentandrare.M easurementsof

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130 CHAPTER8.EQUATIONSOFMOTIONWITHVISCOSITY Figure8.7BillowcloudsshowingaKelvin-Helmholtzinstab ilityatthetopofastable atmosphericlayer.Somebillowscanbecomelargeenoughtha tmoredenseairoverliesless denseair,andthenthebillowscollapseintoturbulence.Ph otographycopyrightBrooks Martner, noaa EnvironmentalTechnologyLaboratory. densityasafunctionofdepthrarelyshowmoredenseruidove rlessdenseruid asseeninthebreakingwavesingure8.7(MoumandCaldwell1 985). DoubleDiusionandSaltFingers Insomeregionsoftheocean,lessdense wateroverliesmoredensewater,yetthewatercolumnisunst ableevenifthere arenocurrents.Theinstabilityoccursbecausethemolecul ardiusionofheat isabout100timesfasterthanthemoleculardiusionofsalt .Theinstability wasrstdiscoveredbyMelvinSternin1960whoquicklyreali zeditsimportance inoceanography. Warm, Salty r1 Cold, Less Salty r2Warm, Salty r1 Cold, Salty r > r2Cold, Less Salty r2 Initial DensityDensity after a few minutes Figure8.8 Left: Initialdistributionofdensityinthevertical. Right: Aftersometime,the diusionofheatleadstoathinunstablelayerbetweenthetw oinitiallystablelayers.The thinunstablelayersinksintothelowerlayerassaltynger s.Theverticalscaleinthegures isafewcentimeters. Considertwothinlayersafewmetersthickseparatedbyasha rpinterface (gure8.8).Iftheupperlayeriswarmandsalty,andifthelo weriscolderand lesssaltythantheupperlayer,theinterfacebecomesunsta bleeveniftheupper layerislessdensethanthelower. Here'swhathappens.Heatdiusesacrosstheinterfacefast erthansalt, leadingtoathin,cold,saltylayerbetweenthetwoinitiall ayers.Thecold saltylayerismoredensethanthecold,less-saltylayerbel ow,andthewaterin thelayersinks.Becausethelayeristhin,theruidsinksin ngers1{5cmin

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8.6.IMPORTANTCONCEPTS 131 diameterand10sofcentimeterslong,notmuchdierentinsi zeandshapefrom ourngers.Thisis saltngering .Becausetwoconstituentsdiuseacrossthe interface,theprocessiscalled doublediusion Therearefourvariationsonthistheme.Twovariablestaken twoatatime leadstofourpossiblecombinations: 1. Warmsaltyovercolderlesssalty .Thisprocessiscalled saltngering .It occursinthethermoclinebelowthesurfacewatersofsub-tr opicalgyres andthewesterntropicalnorthAtlantic,andintheNorth-ea stAtlantic beneaththeoutrowfromtheMediterraneanSea.Saltngerin geventuallyleadstodensityincreasingwithdepthinaseriesofste ps.Layersof constant-densityareseparatedbythinlayerswithlargech angesindensity,andtheproleofdensityasafunctionofdepthlooksli kestairsteps. Schmittetal(1987)observed5{30mthickstepsinthewester n,tropical northAtlanticthatwerecoherentover200{400kmandthatla stedfor atleasteightmonths.Kerr(2002)reportsarecentexperime ntbyRaymondSchmitt,JamesLeswell,JohnToole,andKurtPolzinsho wedsalt ngeringoBarbadosmixedwater10timesfasterthanturbul ence. 2. Colderlesssaltyoverwarmsalty .Thisprocessiscalled diusiveconvection .Itismuchlesscommonthansaltngering,anditusmostlyfo und athighlatitudes.Diusiveconvectionalsoleadstoastair stepofdensity asafunctionofdepth.Here'swhathappensinthiscase.Doub lediusion leadstoathin,warm,less-saltylayeratthebaseoftheuppe r,colder, less-saltylayer.Thethinlayerofwaterrisesandmixeswit hwaterinthe upperlayer.Asimilarprocessesoccursinthelowerlayerwh ereacolder, saltylayerformsattheinterface.Asaresultoftheconvect ioninthe upperandlowerlayers,theinterfaceissharpened.Anysmal lgradients ofdensityineitherlayerarereduced.Nealetal(1969)obse rved2{10m thicklayersintheseabeneaththeArcticice. 3. Coldsaltyoverwarmerlesssalty .Alwaysstaticallyunstable. 4. Warmerlesssaltyovercoldsalty .Alwaysstableanddoublediusion diusestheinterfacebetweenthetwolayers. Doublediusionmixesoceanwater,anditcannotbeignored. Merryeld etal(1999),usinganumericalmodeloftheoceancirculatio nthatincluded doublediusion,foundthatdouble-diusivemixingchange dtheregionaldistributionsoftemperatureandsalinityalthoughithadlittlei nruenceonlarge-scale circulationoftheocean.8.6ImportantConcepts 1.Frictionintheoceanisimportantonlyoverdistancesofa fewmillimeters. Formostrows,frictioncanbeignored. 2.Theoceanisturbulentforallrowswhosetypicaldimensio nexceedsa fewcentimeters,yetthetheoryforturbulentrowintheocea nispoorly understood.

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132 CHAPTER8.EQUATIONSOFMOTIONWITHVISCOSITY 3.TheinruenceofturbulenceisafunctionoftheReynoldsnu mberofthe row.FlowswiththesamegeometryandReynoldsnumberhaveth esame streamlines. 4.Oceanographersassumethatturbulenceinruencesrowsov erdistances greaterthanafewcentimetersinthesamewaythatmolecular viscosity inruencesrowovermuchsmallerdistances. 5.TheinruenceofturbulenceleadstoReynoldsstressterms inthemomentumequation. 6.Theinruenceofstaticstabilityintheoceanisexpressed asafrequency, thestabilityfrequency.Thelargerthefrequency,themore stablethe watercolumn. 7.TheinruenceofshearstabilityisexpressedthroughtheR ichardsonnumber.Thegreaterthevelocityshear,andtheweakerthestati cstability, themorelikelytherowwillbecometurbulent. 8.Moleculardiusionofheatismuchfasterthanthediusio nofsalt.This leadstoadouble-diusioninstabilitywhichmodiesthede nsitydistributioninthewatercolumninmanyregionsoftheocean. 9.Instabilityintheoceanleadstomixing.Mixingacrosssu rfacesofconstant densityismuchsmallerthanmixingalongsuchsurfaces. 10.Horizontaleddydiusivityintheoceanismuchgreatert hanverticaleddy diusivity. 11.Measurementsofeddydiusivityindicatewaterismixed verticallynear oceanicboundariessuchasaboveseamountsandmid-oceanri dges.

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Chapter9ResponseoftheUpperOceantoWindsIfyouhavehadachancetotravelaroundtheUnitedStates,yo umayhave noticedthattheclimateoftheeastcoastdiersconsiderab lyfromthatonthe westcoast.Why?WhyistheclimateofCharleston,SouthCaro linasodierent fromthatofSanDiego,althoughbotharenear32 N,andbothareonornear theocean?Charlestonhas125{150cmofrainayear,SanDiego has25{50 cm,Charlestonhashotsummers,SanDiegohascoolsummers.O rwhyisthe climateofSanFranciscosodierentfromthatofNorfolk,Vi rginia? Ifwelookcloselyatthecharacteristicsoftheatmospherea longthetwo coastsnear32 N,wendmoredierencesthatmayexplaintheclimate.For example,whenthewindblowsonshoretowardSanDiego,itbri ngsacool, moist,marine,boundarylayerafewhundredmetersthickcap pedbymuch warmer,dryair.Ontheeastcoast,whenthewindblowsonshor e,itbringsa warm,moist,marine,boundarylayerthatismuchthicker.Co nvection,which producesrain,ismucheasierontheeastcoastthanonthewes tcoast.Why thenistheatmosphericboundarylayeroverthewatersodie rentonthetwo coasts?Theanswercanbefoundinpartbystudyingtheocean' sresponseto localwinds,thesubjectofthischapter.9.1InertialMotion Tobeginourstudyofcurrentsneartheseasurface,let'scon siderrsta verysimplesolutiontotheequationsofmotion,therespons eoftheoceantoan impulsethatsetsthewaterinmotion.Forexample,theimpul secanbeastrong windblowingforafewhours.Thewaterthenmovesonlyundert heinruence ofCoriolisforce.Nootherforceactsonthewater. Suchmotionissaidtobeinertial.Themassofwatercontinue stomove duetoitsinertia.Ifthewaterwereinspace,itwouldmovein astraightline accordingtoNewton'ssecondlaw.Butonarotatingearth,th emotionismuch dierent. 133

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134 CHAPTER9.RESPONSEOFTHEUPPEROCEANTOWINDS From(7.12)theequationsofmotionforaparcelofwatermovi nginthe oceanwithoutfrictionare: du dt = 1 @p @x +2n v sin (9.1a) dv dt = 1 @p @y 2n u sin (9.1b) dw dt = 1 @p @z +2n u cos g (9.1c) where p ispressure,n=2 /(siderealday)=7 : 292 10 5 rad/sistherotation oftheearthinxedcoordinates,and islatitude. Let'snowlookforsimplesolutionstotheseequations.Todo thiswemust simplifythemomentumequations.First,ifonlytheCorioli sforceactsonthe water,theremustbenohorizontalpressuregradient: @p @x = @p @y =0 Furthermore,wecanassumethattherowishorizontal,and(9 .1)becomes: du dt =2n v sin = fv (9.2a) dv dt = 2n u sin = fu (9.2b) where: f =2nsin (9.3) isthe CoriolisParameter andn=7 : 292 10 5 /sistherotationrateofearth. Equations(9.2)aretwocoupled,rst-order,linear,dier entialequations whichcanbesolvedwithstandardtechniques.Ifwesolvethe secondequation for u ,andinsertitintotherstequationweobtain: du dt = 1 f d 2 v dt 2 = fv Rearrangingtheequationputsitintoastandardformweshou ldrecognize,the equationfortheharmonicoscillator: d 2 v dt 2 + f 2 v =0(9.4) whichhasthesolution(9.5).Thiscurrentiscalledan inertialcurrent or inertial oscillation : u = V sin ft v = V cos ft V 2 = u 2 + v 2 (9.5)

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9.2.EKMANLAYERATTHESEASURFACE 135 100 50 0 km Inertial Currents 142o 140o 138o136o 46o47o Longitude (West)Latitude (North) Figure9.1InertialcurrentsintheNorthPacicinOctober1 987(days275{300)measured byholey-sockdriftingbuoysdroguedatadepthof15meters. Positionswereobserved10{12 timesperdaybytheArgossystemon noaa polar-orbitingweathersatellitesandinterpolated topositionseverythreehours.Thelargestcurrentswerege neratedbyastormonday 277.Notethesearenotindividualeddies.Theentiresurfac eisrotating.Adrogueplaced anywhereintheregionwouldhavethesamecircularmotion.A ftervanMeurs(1998). Notethat(9.5)aretheparametricequationsforacirclewit hdiameter D i = 2 V=f andperiod T i =(2 ) =f = T sd = (2sin )where T sd isasiderealday. T i isthe inertialperiod .Itisonehalfthetimerequiredfortherotation ofalocalplaneonearth'ssurface(Table9.1).Therotation is anti-cyclonic : clockwiseinthenorthernhemisphere,counterclockwisein thesouthern.Inertial currentsarethefreemotionofparcelsofwateronarotating plane. Table9.1InertialOscillations Latitude( ) T i (hr)D(km) forV=20cm/s 90 11.972.7 35 20.874.8 10 68.9315.8 Inertialcurrentsarethemostcommoncurrentsintheocean( gure9.1). Webster(1968)reviewedmanypublishedreportsofinertial currentsandfound thatcurrentshavebeenobservedatalldepthsintheoceanan datalllatitudes. Themotionsaretransientanddecayinafewdays.Oscillatio nsatdierent depthsoratdierentnearbysitesareusuallyincoherent. Inertialcurrentsarecausedbyrapidchangesofwindatthes easurface,with rapidchangesofstrongwindsproducingthelargestoscilla tions.Althoughwe havederivedtheequationsfortheoscillationassumingfri ctionlessrow,friction cannotbecompletelyneglected.Withtime,theoscillation sdecayintoother surfacecurrents.(See,forexample,Apel,1987: x 6.3formoreinformation.) 9.2EkmanLayerattheSeaSurface Steadywindsblowingon(theseasurfaceproduceathin,hori zontalboundarylayer,the Ekmanlayer .Bythin,Imeanalayerthatisatmostafew-hundred metersthick,whichisthincomparedwiththedepthofthewat erinthedeep

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136 CHAPTER9.RESPONSEOFTHEUPPEROCEANTOWINDS Table9.2ContributionstotheTheoryoftheWind-DrivenCir culation FridtjofNansen(1898)Qualitativetheory,currentstrans portwateratan angletothewind. VagnWalfridEkman(1902)Quantitativetheoryforwind-dri ventransportat theseasurface. HaraldSverdrup(1947)Theoryforwind-drivencirculation intheeastern Pacic. HenryStommel(1948)Theoryforwestwardintensicationof wind-driven circulation(westernboundarycurrents). WalterMunk(1950)Quantitativetheoryformainfeaturesof thewinddrivencirculation. KirkBryan(1963)Numericalmodelsoftheoceaniccirculati on. BertSemtner(1988)Global,eddy-resolving,realisticmod elofthe andRobertChervinocean'scirculation. ocean.Asimilarboundarylayerexistsatthebottomoftheoc ean,the bottom Ekmanlayer ,andatthebottomoftheatmospherejustabovetheseasurfac e, theplanetaryboundarylayerorfrictionallayerdescribed in x 4.3.TheEkman layerisnamedafterProfessorWalfridEkman,whoworkedout itsdynamicsfor hisdoctoralthesis. Ekman'sworkwastherstofaremarkableseriesofstudiesco nductedduring thersthalfofthetwentiethcenturythatledtoanundersta ndingofhowwinds drivetheocean'scirculation(Table9.1).Inthischapterw econsiderNansen andEkman'swork.Therestofthestoryisgiveninchapters11 and13. Nansen'sQualitativeArguments FridtjofNansennoticedthatwindtended toblowiceatanangleof20 {40 totherightofthewindintheArctic,by whichhemeantthatthetrackoftheicebergwastotherightof thewindlooking downwind(Seegure9.2).Helaterworkedoutthebalanceoff orcesthatmust existwhenwindtriedtopushicebergsdownwindonarotating earth. Nansenarguedthatthreeforcesmustbeimportant: 1.WindStress, W ; 2.Friction F (otherwisetheicebergwouldmoveasfastasthewind); 3.CoriolisForce, C Nansenarguedfurtherthattheforcesmusthavethefollowin gattributes: 1.Dragmustbeoppositethedirectionoftheice'svelocity;2.Coriolisforcemustbeperpendiculartothevelocity;3.Theforcesmustbalanceforsteadyrow. W + F + C =0 Ekman'sSolution NansenaskedVilhelmBjerknestoletoneofBjerknes' studentsmakeatheoreticalstudyoftheinruenceofearth's rotationonwinddrivencurrents.WalfridEkmanwaschosen,andhepresented theresultsin histhesisatUppsala(Kullenberg,1954).Ekmanlaterexpan dedthestudyto

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9.2.EKMANLAYERATTHESEASURFACE 137 Wind W Velocity ofIceberg CoriolisForce Drag (Friction) F C Coriolis Drag Wind Coriolis Drag Wind All forces about equal Weak Coriolis force Figure9.2Thebalanceofforcesactingonaniceberginawind onarotatingearth. includetheinruenceofcontinentsanddierencesofdensit yofwater(Ekman, 1905).ThefollowingfollowsEkman'slineofreasoninginth atpaper. Ekmanassumedasteady,homogeneous,horizontalrowwithfr ictionona rotatingearth.Thushorizontalandtemporalderivativesa rezero: @ @t = @ @x = @ @y =0(9.6) Forrowonarotatingearth,thisleavesabalancebetweenfri ctionalandCoriolis forces(8.15).Ekmanfurtherassumedaconstantverticaled dyviscosityofthe form(8:13): T xz = A z @u @z ;T yz = A z @v @z (9.7) where T xz T yz arethecomponentsofthewindstressinthe x y directions,and isthedensityofseawater. Using(9.7)in(8.15),the x and y momentumequationsare: fv + A z @ 2 u @z 2 =0(9.8a) fu + A z @ 2 v @z 2 =0(9.8b) where f istheCoriolisparameter. Itiseasytoverifythattheequations(9.9)havesolutions: u = V 0 exp( az )cos( = 4+ az )(9.9a) v = V 0 exp( az )sin( = 4+ az )(9.9b) whenthewindisblowingtothenorth( T = T yz ).Theconstantsare V 0 = T p 2w fA z and a = r f 2 A z (9.10)

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138 CHAPTER9.RESPONSEOFTHEUPPEROCEANTOWINDS and V 0 isthevelocityofthecurrentattheseasurface. Nowlet'slookattheformofthesolutions.Attheseasurface z =0, exp( z =0)=1,and u (0)= V 0 cos( = 4)(9.11a) v (0)= V 0 sin( = 4)(9.11b) Thecurrenthasaspeedof V 0 tothenortheast.Ingeneral,thesurfacecurrentis 45 totherightofthewindwhenlookingdownwindinthenorthern hemisphere. Thecurrentis45 totheleftofthewindinthesouthernhemisphere.Below thesurface,thevelocitydecaysexponentiallywithdepth( gure9.3): u 2 ( z )+ v 2 ( z ) 1 = 2 = V 0 exp( az )(9.12) 0 -20-40-60-80 -100-120-140Depth (m) Wind Direction 45oVo = 6.4 cm/s Figure9.3.Ekmancurrentgeneratedbya10m/swindat35 N. ValuesforEkman'sConstants Toproceedfurther,weneedvaluesforany twoofthefreeparameters:thevelocityatthesurface, V 0 ;thecoecientof eddyviscosity, A z ;orthewindstress T Thewindstressiswellknown,andEkmanusedthebulkformula (4.2): T yz = T = air C D U 2 10 (9.13) where air isthedensityofair, C D isthedragcoecient,and U 10 isthewind speedat10mabovethesea.Ekmanturnedtotheliteraturetoo btainvalues

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9.2.EKMANLAYERATTHESEASURFACE 139 for V 0 asafunctionofwindspeed.Hefound: V 0 = 0 : 0127 p sin j j U 10 ; j j 10(9.14) Withthisinformation,hecouldthencalculatethevelocity asafunctionof depthknowingthewindspeed U 10 andwinddirection. EkmanLayerDepth ThethicknessoftheEkmanlayerisarbitrarybecause theEkmancurrentsdecreaseexponentiallywithdepth.Ekma nproposedthat thethicknessbethedepth D E atwhichthecurrentvelocityisoppositethe velocityatthesurface,whichoccursatadepth D E = =a ,andthe Ekman layerdepth is: D E = s 2 2 A z f (9.15) Using(9.13)in(9.10),dividingby U 10 ,andusing(9.14)and(9.15)gives: D E = 7 : 6 p sin j j U 10 (9.16) inSIunits.Windinmeterspersecondgivesdepthinmeters.T heconstantin (9.16)isbasedon =1027kg/m 3 air =1 : 25kg/m 3 ,andEkman'svalueof C D =2 : 6 10 3 forthedragcoecient. Using(9.16)withtypicalwinds,thedepthoftheEkmanlayer variesfrom about45to300meters(Table9.3),andthevelocityofthesur facecurrentvaries from2.5%to1.1%ofthewindspeeddependingonlatitude. Table9.3TypicalEkmanDepths Latitude U 10 [m/s] 15 45 5 75m45m 10 150m90m 20 300m180m TheEkmanNumber:CoriolisandFrictionalForces Thedepthofthe Ekmanlayeriscloselyrelatedtothedepthatwhichfriction alforceisequalto theCoriolisforceinthemomentumequation(9.9).TheCorio lisforceis fu andthefrictionalforceis A z @ 2 U=@z 2 .Theratiooftheforces,whichisnon dimensional,iscalledthe EkmanNumber E z : E z = FrictionForce CoriolisForce = A z @ 2 u @z 2 fu = A z u d 2 fu E z = A z fd 2 (9.17)

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140 CHAPTER9.RESPONSEOFTHEUPPEROCEANTOWINDS wherewehaveapproximatedthetermsusingtypicalvelociti es u ,andtypical depths d .Thesubscript z isneededbecausetheoceanisstratiedandmixing intheverticalismuchlessthanmixinginthehorizontal.No tethatasdepth increases,frictionbecomessmall,andeventually,onlyth eCoriolisforceremains. Solving(9.17)for d gives d = s A z fE z (9.18) whichagreeswiththefunctionalform(9.15)proposedbyEkm an.Equating (9.18)and(9.15)requires E z =1 = (2 2 )=0 : 05attheEkmandepth.ThusEkmanchoseadepthatwhichfrictionalforcesaremuchsmaller thantheCoriolis force.BottomEkmanLayer TheEkmanlayeratthebottomoftheoceanand theatmospherediersfromthelayerattheoceansurface.Th esolutionfora bottomlayerbelowaruidwithvelocity U inthe x -directionis: u = U [1 exp( az )cos az ](9.19a) v = U exp( az )sin az (9.19b) Thevelocitygoestozeroattheboundary, u = v =0at z =0.Thedirection oftherowclosetotheboundaryis45 totheleftoftherow U outsidethe boundarylayerinthenorthernhemisphere,andthedirectio noftherowrotates withdistanceabovetheboundary(gure9.4).Thedirection ofrotationisanticyclonicwithdistanceabovethebottom. Windsabovetheplanetaryboundarylayerareperpendicular tothepressure gradientintheatmosphereandparalleltolinesofconstant surfacepressure. Windsatthesurfaceare45 totheleftofthewindsaloft,andsurfacecurrents are45 totherightofthewindatthesurface.Thereforeweexpectcu rrents attheseasurfacetobenearlyinthedirectionofwindsabove theplanetary boundarylayerandparalleltolinesofconstantpressure.O bservationsofsurface driftersinthePacictendtoconrmthehypothesis(gure9 .5). 40 4812 u (m/s)v (m/s) 200 300 600 1000 30 305 610 910 100 0 16 Figure9.4Ekmanlayerinthelowestkilometeroftheatmosph ere(solidline),withwind velocitymeasuredbyDobson(1914)---.Thenumbersgivehei ghtabovethesurfacein meters.TheEkmanlayeratthesearoorhasasimilarshape.Af terHoughton(1977:107).

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9.2.EKMANLAYERATTHESEASURFACE 141 60o50o40o30o20o10o0o-10o120o140o160o180o-160o-140o-120o-100oMean Sea Level Pressure in April 1978 (mb)1010 1010 1010 5 5 5 5 5 4 4 4 4 4 4 4 3 3 3 3 3 3 6 6 6 6 3 5 4 6 5 4 5 6 6 4 6 5 5 5 5 5 4 3 6 3 5 4 3 4 5 6 3 4 3 5 6 6 5 4 4 3 5 6 5 4 3 5 5 4 6 3 6 6 1015 1015 1012.5 1012.5 1012.5 1017.5 1012.5 Figure9.5TrajectoriesofsurfacedriftersinApril1978to getherwithsurfacepressurein theatmosphereaveragedforthemonth.Notethatdrifterste ndtofollowlinesofconstant pressureexceptintheKuroshiowhereoceancurrentsarefas tcomparedwithvelocitiesin theEkmanlayerintheocean.AfterMcNallyetal.(1983).ExaminingEkman'sAssumptions BeforeconsideringthevalidityofEkman'stheoryfordescribingrowinthesurfaceboundarylaye roftheocean, let'srstexaminethevalidityofEkman'sassumptions.Hea ssumed: 1.Noboundaries.Thisisvalidawayfromcoasts.2.Deepwater.Thisisvalidifdepth 200m. 3. f -plane.Thisisvalid. 4.Steadystate.Thisisvalidifwindblowsforlongerthanap endulumday. NotehoweverthatEkmanalsocalculatedatime-dependentso lution,as didHasselmann(1970). 5. A z isafunctionof U 2 10 only.Itisassumedtobeindependentofdepth. Thisisnotagoodassumption.Themixedlayermaybethinnert hanthe Ekmandepth,and A z willchangerapidlyatthebottomofthemixedlayer becausemixingisafunctionofstability.Mixingacrossast ablelayeris muchlessthanmixingthroughalayerofaneutralstability. Morerealistic prolesforthecoecientofeddyviscosityasafunctionofd epthchange theshapeofthecalculatedvelocityprole.Ireconsiderth isproblem below. 6.Homogeneousdensity.Thisisprobablygood,exceptasite ectsstability.

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142 CHAPTER9.RESPONSEOFTHEUPPEROCEANTOWINDS ObservationsofFlowNeartheSeaSurface Doestherowclosetothe seasurfaceagreewithEkman'stheory?Measurementsofcurr entsmadeduring several,verycarefulexperimentsindicatethatEkman'sth eoryisremarkably good.Thetheoryaccuratelydescribestherowaveragedover manydays. WellerandPlueddmann(1996)measuredcurrentsfrom2mto13 2musing14vector-measuringcurrentmetersdeployedfromtheFl oatingInstrument Platform flip inFebruaryandMarch1990500kmwestofpointConception, California.Thiswasthelastofaremarkableseriesofexper imentscoordinated byWellerusinginstrumentson flip Davis,DeSzoeke,andNiiler(1981)measuredcurrentsfrom2 mto175m using19vector-measuringcurrentmetersdeployedfromamo oringfor19days inAugustandSeptember1977at50 N,145 WinthenortheastPacic. RalphandNiiler(2000)tracked1503driftersdroguedto15m depthinthe PacicfromMarch1987toDecember1994.Windvelocitywasob tainedevery6 hoursfromtheEuropeanCentreforMedium-RangeWeatherFor ecasts ecmwf Theresultsoftheexperimentsindicatethat: 1.Inertialcurrentsarethelargestcomponentoftherow.2.Therowisnearlyindependentofdepthwithinthemixedlay erforperiodsneartheinertialperiod.Thusthemixedlayermovesli keaslab attheinertialperiod.Currentshearisconcentratedatthe topofthe thermocline. 3.Therowaveragedovermanyinertialperiodsisalmostexac tlythatcalculatedfromEkman'stheory.TheshearoftheEkmancurrents extends throughtheaveragedmixedlayerandintothethermocline.R alphand Niilerfound(usingSIunits, U inm/s): D E = 7 : 12 p sin j j U 10 (9.20) V 0 = 0 : 0068 p sin j j U 10 (9.21) TheEkman-layerdepth D E isalmostexactlythatproposedbyEkman (9.16),butthesurfacecurrent V 0 ishalfhisvalue(9.14). 4.Theanglebetweenthewindandtherowatthesurfacedepend sonlatitude,anditisnear45 atmidlatitudes(gure9.6). 5.Thetransportis90 totherightofthewindinthenorthernhemisphere. ThetransportdirectionagreeswellwithEkman'stheory. InruenceofStabilityintheEkmanLayer RalphandNiiler(2000)point outthatEkman'schoiceofanequationforsurfacecurrents( 9.14),whichleads to(9.16),isconsistentwiththeoriesthatincludetheinru enceofstabilityinthe upperocean.Currentswithperiodsneartheinertialperiod produceshearinthe thermocline.TheshearmixesthesurfacelayerswhentheRic hardsonnumber

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9.3.EKMANMASSTRANSPORT 143 806040200 20406080 Latit ude (de grees) 0 20 40 60 80 20406080Angle to the Wi nd (de grees)New Data Ralp h & Niiler (2000) Figure9.6Anglebetweenthewindandrowatthesurfacecalcu latedbyMaximenkoand Niilerusingpositionsfromdriftersdroguedat15mwithsat ellite-altimeter,gravity,and grace dataandwindsfromthe ncar/ncep reanalysis. fallsbelowthecriticalvalue(Pollardetal.1973).Thisid ea,whenincludedin mixed-layertheories,leadstoasurfacecurrent V 0 thatisproportionalto p N=f V 0 U 10 p N=f (9.22) where N isthestabilityfrequencydenedby(8.36).Furthermore A z U 2 10 =N and D E U 10 = p Nf (9.23) Noticethat(9.22)and(9.23)arenowdimensionallycorrect .Theequationsused earlier,(9.14),(9.16),(9.20),and(9.21)allrequiredad imensionalcoecient. 9.3EkmanMassTransport FlowintheEkmanlayerattheseasurfacecarriesmass.Forma nyreasons wemaywanttoknowthetotalmasstransportedinthelayer.Th e Ekman masstransport M E isdenedastheintegraloftheEkmanvelocity U E ;V E fromthesurfacetoadepth d belowtheEkmanlayer.Thetwocomponentsof thetransportare M Ex M Ey : M Ex = Z 0 d U E dz;M Ey = Z 0 d V E dz (9.24) Thetransporthasunitskg/(m s).Itisthemassofwaterpassingthrough averticalplaneonemeterwidethatisperpendiculartothet ransportand extendingfromthesurfacetodepth d (gure9.7). WecalculatetheEkmanmasstransportsbyintegrating(8.15 )in(9.24): f Z 0 d V E dz = fM Ey = Z 0 d dT xz fM Ey = T xz z =0 + T xz z = d (9.25)

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144 CHAPTER9.RESPONSEOFTHEUPPEROCEANTOWINDS } U U z y QX = Y MXr Y 1m x Sea Surface } Figure9.7Sketchfordening Left: masstransports,and Right: volumetransports. AfewhundredmetersbelowthesurfacetheEkmanvelocitiesa pproachzero, andthelasttermof(9.25)iszero.Thusmasstransportisdue onlytowind stressattheseasurface( z =0).Inasimilarway,wecancalculatethetransport inthe x directiontoobtainthetwocomponentsofthe Ekmantransport : fM Ey = T xz (0)(9.26a) fM Ex = T yz (0)(9.26b) where T xz (0) ;T yz (0)arethetwocomponentsofthestressattheseasurface. Noticethatthetransportisperpendiculartothewindstres s,andtotheright ofthewindinthenorthernhemisphere.Ifthewindistotheno rthinthepositive y direction(asouthwind),then T xz (0)=0, M Ey =0,and M Ex = T yz (0) =f Inthenorthernhemisphere, f ispositive,andthemasstransportisinthe x direction,totheeast. Itmayseemstrangethatthedragofthewindonthewaterleads toacurrent atrightanglestothedrag.Theresultfollowsfromtheassum ptionthatfriction isconnedtoathinsurfaceboundarylayer,thatitiszeroin theinteriorof theocean,andthatthecurrentisinequilibriumwiththewin dsothatitisno longeraccelerating. Volumetransport Q isthemasstransportdividedbythedensityofwater andmultipliedbythewidthperpendiculartothetransport. Q x = YM x ;Q y = XM y (9.27) where Y isthenorth-southdistanceacrosswhichtheeastwardtrans port Q x is calculated,and X intheeast-westdistanceacrosswhichthenorthwardtransport Q y iscalculated.Volumetransporthasdimensionsofcubicmet ersper second.Aconvenientunitforvolumetransportintheoceani samillioncubic meterspersecond.Thisunitiscalleda Sverdrup ,anditisabbreviatedSv. RecentobservationsofEkmantransportintheoceanagreewi ththetheoreticalvalues(9.26).ChereskinandRoemmich(1991)meas uredtheEkman volumetransportacross11 NintheAtlanticusinganacousticDopplercurrent prolerdescribedinChapter10.Theycalculatedatranspor tof Q y =12 : 0 5 : 5 Sv(northward)fromdirectmeasurementsofcurrent, Q y =8 : 8 1 : 9Svfrom

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9.4.APPLICATIONOFEKMANTHEORY 145 measuredwindsusing(9.26)and(9.27),and Q y =13 : 5 0 : 3Svfrommean windsaveragedovermanyyearsat11 N. UseofTransports Masstransportsarewidelyusedfortwoimportantreasons.First,thecalculationismuchmorerobustthancalcul ationsofvelocities intheEkmanlayer.Byrobust,Imeanthatthecalculationisb asedonfewer assumptions,andthattheresultsaremorelikelytobecorre ct.Thusthecalculatedmasstransportdoesnotdependonknowingthedistri butionofvelocity intheEkmanlayerortheeddyviscosity. Second,thevariabilityoftransportinspacehasimportant consequences. Let'slookatafewapplications.9.4ApplicationofEkmanTheory BecausesteadywindsblowingontheseasurfaceproduceanEk manlayer thattransportswateratrightanglestothewinddirection, anyspatialvariability ofthewind,orwindsblowingalongsomecoasts,canleadtoup welling.And upwellingisimportant: 1.Upwellingenhancesbiologicalproductivity,whichfeed ssheries. 2.Coldupwelledwateralterslocalweather.Weatheronshor eofregionsof upwellingtendtohavefog,lowstratusclouds,astablestra tiedatmosphere,littleconvection,andlittlerain. 3.Spatialvariabilityoftransportsintheopenoceanleads toupwellingand downwelling,whichleadstoredistributionofmassintheoc ean,which leadstowind-drivengeostrophiccurrentsviaEkmanpumpin g. CoastalUpwelling Toseehowwindsleadtoupwelling,considernorthwinds blowingparalleltotheCaliforniaCoast(gure9.8left).T hewindsproduce amasstransportawayfromtheshoreeverywherealongthesho re.Thewater pushedoshorecanbereplacedonlybywaterfrombelowtheEk manlayer. Thisis upwelling (gure9.8right).Becausetheupwelledwateriscold,the upwellingleadstoaregionofcoldwateratthesurfacealong thecoast.Figure 10.16showsthedistributionofcoldwaterothecoastofCal ifornia. Upwelledwateriscolderthanwaternormallyfoundonthesur face,and itisricherinnutrients.Thenutrientsfertilizephytopla nktoninthemixed layer,whichareeatenbyzooplankton,whichareeatenbysma llsh,which areeatenbylargershandsoontoinnity.Asaresult,upwel lingregions areproductivewaterssupportingtheworld'smajorsherie s.Theimportant regionsareoshoreofPeru,California,Somalia,Morocco, andNamibia. NowIcananswerthequestionIaskedatthebeginningofthech apter: WhyistheclimateofSanFranciscosodierentfromthatofNo rfolk,Virginia? Figures4.2or9.8showthatwindalongtheCaliforniaandOre goncoastshasa strongsouthwardcomponent.Thewindcausesupwellingalon gthecoast,which leadstocoldwaterclosetoshore.Theshorewardcomponento fthewindbrings warmerairfromfaroshoreoverthecolderwater,whichcool stheincoming airclosetothesea,leadingtoathin,coolatmosphericboun darylayer.Asthe aircools,fogformsalongthecoast.Finally,thecoollayer ofairisblownover

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146 CHAPTER9.RESPONSEOFTHEUPPEROCEANTOWINDS UpwellingWaterMEME 100 300 m Col dUpwe lledWaterWindLand (California)Ekman Transpor t 100 km Figure9.8SketchofEkmantransportalongacoastleadingto upwellingofcoldwateralong thecoast. Left: Planview.Northwindsalongawestcoastinthenorthernhemi sphere causeEkmantransportsawayfromtheshore. Right: Crosssection.Thewatertransported oshoremustbereplacedbywaterupwellingfrombelowthemi xedlayer. SanFrancisco,coolingthecity.Thewarmerairabovethebou ndarylayer,due todownwardvelocityoftheHadleycirculationintheatmosp here(seegure 4.3),inhibitsverticalconvection,andrainisrare.Rainf ormsonlywhenwinter stormscomingashorebringstrongconvectionhigherupinth eatmosphere. Inadditiontoupwelling,otherprocessesinruenceweather inCaliforniaand Virginia. 1.Theoceanicmixedlayertendstobethinontheeasternside ofocean,and upwellingcaneasilybringupcoldwater. 2.Currentsalongtheeasternsideoftheoceanatmid-latitu destendtobring colderwaterfromhigherlatitudes. Alltheseprocessesarereversedoshoreofeastcoasts,lea dingtowarmwater closetoshore,thickatmosphericboundarylayers,andfreq uentconvectiverain. ThusNorfolkismuchdierentthatSanFranciscoduetoupwel lingandthe directionofthecoastalcurrents.EkmanPumping Thehorizontalvariabilityofthewindblowingonthesea surfaceleadstohorizontalvariabilityoftheEkmantransp orts.Becausemass mustbeconserved,thespatialvariabilityofthetransport smustleadtovertical velocitiesatthetopoftheEkmanlayer.Tocalculatethisve locity,werst integratethecontinuityequation(7.19)inthevertical: Z 0 d @u @x + @v @y + @w @z dz =0 @ @x Z 0 d udz + @ @y Z 0 d vdz = Z 0 d @w @z dz @M Ex @x + @M Ey @y = [ w (0) w ( d )] Bydenition,theEkmanvelocitiesapproachzeroatthebase oftheEkman layer,andtheverticalvelocityatthebaseofthelayer w E ( d )duetodivergence

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9.5.LANGMUIRCIRCULATION 147 oftheEkmanrowmustbezero.Therefore: @M Ex @x + @M Ey @y = w E (0)(9.28a) r H M E = w E (0) (9.28b) Where M E isthevectormasstransportduetoEkmanrowintheupperboun darylayeroftheocean,and r H isthehorizontaldivergenceoperator.(9.28) statesthatthehorizontaldivergenceoftheEkmantranspor tsleadstoaverticalvelocityintheupperboundarylayeroftheocean,aproce sscalled Ekman Pumping IfweusetheEkmanmasstransports(9.26)in(9.28)wecanrel ateEkman pumpingtothewindstress. w E (0)= 1 @ @x T yz (0) f @ @y T xz (0) f (9.29a) w E (0)= curl z T f (9.29b) where T isthevectorwindstressandthesubscript z indicatesthevertical componentofthecurl. Theverticalvelocityattheseasurface w (0)mustbezerobecausethesurface cannotriseintotheair,so w E (0)mustbebalancedbyanotherverticalvelocity. WewillseeinChapter12thatitisbalancedbyageostrophicv elocity w G (0) atthetopoftheinteriorrowintheocean. NotethatthederivationabovefollowsPedlosky(1996:13), anditdiers fromthetraditionalapproachthatleadstoaverticalveloc ityatthebaseofthe Ekmanlayer.PedloskypointsoutthatiftheEkmanlayerisve rythincompared withthedepthoftheocean,itmakesnodierencewhetherthe velocityis calculatedatthetoporbottomoftheEkmanlayer,butthisis usuallynottrue fortheocean.Hence,wemustcomputeverticalvelocityatth etopofthelayer. 9.5LangmuirCirculation Measurementsofsurfacecurrentsshowthatwindsgeneratem orethanEkmanandinertialcurrentsattheseasurface.Theyalsogener ateaLangmuir circulation(Langmuir,1938),acurrentthatspiralaround anaxisparalleltothe winddirection.Welleretal.(1985)observedsucharowduri nganexperiment tomeasurethewind-drivencirculationintheupper50meter softhesea.They foundthatduringaperiodwhenthewindspeedwas14m/s,surf acecurrents wereorganizedintoLangmuircellsspaced20mapart,thecel lswerealignedat anangleof15 totherightofthewind,andverticalvelocityat23mdepthwa s concentratedinnarrowjetsundertheareasofsurfaceconve rgence(gure9.9). Maximumverticalvelocitywas 0 : 18m/s.Theseasonalthermoclinewasat50 m,andnodownwardvelocitywasobservedinorbelowthetherm ocline. 9.6ImportantConcepts 1.Changesinwindstressproducetransientoscillationsin theoceancalled inertialcurrents

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148 CHAPTER9.RESPONSEOFTHEUPPEROCEANTOWINDS -10 -20 10 V 10 10 20 0 W U T Figure9.9Athree-dimensionalviewoftheLangmuircircula tionatthesurfaceofthePacic observedfromtheFloatingInstrumentPlatform flip .Theheavydashedlineonthesea surfaceindicatelinesofconvergencemarkedbycardsonthe surface.Verticalarrowsare individualvaluesofverticalvelocitymeasuredevery14se condsat23mdepthastheplatform driftedthroughtheLangmuircurrents.Horizontalarrows, whicharedrawnonthesurface forclarity,arevaluesofhorizontalvelocityat23m.Thebr oadarrowgivesthedirectionof thewind.AfterWelleretal.(1985). (a)Inertialcurrentsareverycommonintheocean. (b)Theperiodofthecurrentis(2 ) =f 2.Steadywindsproduceathinboundarylayer,theEkmanlaye r,atthetop oftheocean.Ekmanboundarylayersalsoexistatthebottomo ftheocean andtheatmosphere.TheEkmanlayerintheatmosphereabovet hesea surfaceiscalledtheplanetaryboundarylayer. 3.TheEkmanlayerattheseasurfacehasthefollowingcharac teristics: (a) Direction :45 totherightofthewindlookingdownwindinthe NorthernHemisphere. (b) SurfaceSpeed :1{2.5%ofwindspeeddependingonlatitude. (c) Depth :approximately40{300mdependingonlatitudeandwind velocity. 4.Carefulmeasurementsofcurrentsneartheseasurfacesho wthat: (a)Inertialoscillationsarethelargestcomponentofthec urrentinthe mixedlayer. (b)Therowisnearlyindependentofdepthwithinthemixedla yerfor periodsneartheinertialperiod.Thusthemixedlayermoves likea slabattheinertialperiod. (c)AnEkmanlayerexistsintheatmospherejustabovethesea (and land)surface.

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9.6.IMPORTANTCONCEPTS 149 (d)Surfacedrifterstendtodriftparalleltolinesofconst antatmospheric pressureattheseasurface.ThisisconsistentwithEkman's theory. (e)Therowaveragedovermanyinertialperiodsisalmostexa ctlythat calculatedfromEkman'stheory. 5.Transportis90 totherightofthewindinthenorthernhemisphere. 6.SpatialvariabilityofEkmantransport,duetospatialva riabilityofwinds overdistancesofhundredsofkilometersanddays,leadstoc onvergence anddivergenceofthetransport. (a)Windsblowingtowardtheequatoralongwestcoastsofcon tinents producesupwellingalongthecoast.Thisleadstocold,prod uctive waterswithinabout100kmoftheshore. (b)Upwelledwateralongwestcoastsofcontinentsmodiest heweather alongthewestcoasts. 7.Ekmanpumping,whichisdrivenbyspatialvariabilityofw inds,drives averticalcurrent,whichdrivestheinteriorgeostrophicc irculationofthe ocean.

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150 CHAPTER9.RESPONSEOFTHEUPPEROCEANTOWINDS

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Chapter10GeostrophicCurrentsWithintheocean'sinteriorawayfromthetopandbottomEkma nlayers,for horizontaldistancesexceedingafewtensofkilometers,an dfortimesexceeding afewdays,horizontalpressuregradientsintheoceanalmos texactlybalance theCoriolisforceresultingfromhorizontalcurrents.Thi sbalanceisknownas the geostrophicbalance Thedominantforcesactingintheverticalaretheverticalp ressuregradient andtheweightofthewater.Thetwobalancewithinafewparts permillion. Thuspressureatanypointinthewatercolumnisduealmosten tirelytothe weightofthewaterinthecolumnabovethepoint.Thedominan tforcesinthe horizontalarethepressuregradientandtheCoriolisforce .Theybalancewithin afewpartsperthousandoverlargedistancesandtimes(SeeB ox). Bothbalancesrequirethatviscosityandnonlineartermsin theequations ofmotionbenegligible.Isthisreasonable?Considervisco sity.Weknowthat arowboatweighingahundredkilogramswillcoastformaybet enmetersafter therowerstops.Asupertankermovingatthespeedofarowboa tmaycoast forkilometers.Itseemsreasonable,thereforethatacubic kilometerofwater weighing10 15 kgwouldcoastforperhapsadaybeforeslowingtoastop.And oceanicmesoscaleeddiescontainperhaps1000cubickilome tersofwater.Hence, ourintuitionmayleadustoconcludethatneglectofviscosi tyisreasonable.Of course,intuitioncanbewrong,andweneedtoreferbacktosc alingarguments. 10.1HydrostaticEquilibrium Beforedescribingthegeostrophicbalance,let'srstcons iderthesimplest solutionofthemomentumequation,thesolutionforanocean atrest.Itgives thehydrostaticpressurewithintheocean.Toobtainthesol ution,weassume theruidisstationary: u = v = w =0;(10.1) theruidremainsstationary: du dt = dv dt = dw dt =0;(10.2) 151

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152 CHAPTER10.GEOSTROPHICCURRENTS ScalingtheEquations:TheGeostrophicApproximation Wewishtosimplifytheequationsofmotiontoobtainsolutio nsthatdescribethe deep-seaconditionswellawayfromcoastsandbelowtheEkma nboundarylayerat thesurface.Tobegin,let'sexaminethetypicalsizeofeach termintheequations intheexpectationthatsomewillbesosmallthattheycanbed roppedwithout changingthedominantcharacteristicsofthesolutions.Fo rinterior,deep-sea conditions,typicalvaluesfordistance L ,horizontalvelocity U ,depth H ,Coriolis parameter f ,gravity g ,anddensity are: L 10 6 m H 1 10 3 m f 10 4 s 1 10 3 kg/m 3 U 10 1 m/s H 2 1m 10 3 kg/m 3 g 10m/s 2 where H 1 and H 2 aretypicaldepthsforpressureintheverticalandhorizont al. Fromthesevariableswecancalculatetypicalvaluesforver ticalvelocity W pressure P ,andtime T : @W @z = @U @x + @v @y W H 1 = U L ; W = UH 1 L = 10 1 10 3 10 6 m/s=10 4 m/s P = gH 1 =10 3 10 1 10 3 =10 7 Pa; @p=@x = gH 2 =L =10 2 Pa/m T = L=U =10 7 s Themomentumequationforverticalvelocityistherefore: @w @t + u @w @x + v @w @y + w @w @z = 1 @p @z +2n u cos g W T + UW L + UW L + W 2 H = P H 1 + fU g 10 11 +10 11 +10 11 +10 11 =10+10 5 10 andtheonlyimportantbalanceintheverticalishydrostati c: @p @z = g Correctto1:10 6 : Themomentumequationforhorizontalvelocityinthe x directionis: @u @t + u @u @x + v @u @y + w @u @z = 1 @p @x + fv 10 8 +10 8 +10 8 +10 8 =10 5 +10 5 ThustheCoriolisforcebalancesthepressuregradientwith inonepartperthousand.Thisiscalledthe geostrophicbalance ,andthe geostrophicequations are: 1 @p @x = fv ; 1 @p @y = fu ; 1 @p @z = g Thisbalanceappliestooceanicrowswithhorizontaldimens ionslargerthan roughly50kmandtimesgreaterthanafewdays.

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10.2.GEOSTROPHICEQUATIONS 153 and,thereisnofriction: f x = f y = f z =0 : (10.3) Withtheseassumptionsthemomentumequation(7.12)become s: 1 @p @x =0; 1 @p @y =0; 1 @p @z = g ( ';z )(10.4) whereIhaveexplicitlynotedthatgravity g isafunctionoflatitude andheight z .IwillshowlaterwhyIhavekeptthisexplicit. Equations(10.4)requiresurfacesofconstantpressuretob elevelsurface (seepage30).Asurfaceofconstantpressureisan isobaricsurface .Thelast equationcanbeintegratedtoobtainthepressureatanydept h h .Recalling that isafunctionofdepthforanoceanatrest. p = Z 0 h g ( ';z ) ( z ) dz (10.5) Formanypurposes, g and areconstant,and p = gh .Later,Iwillshowthat (10.5)applieswithanaccuracyofaboutonepartpermillion eveniftheocean isnotatrest. TheSIunitforpressureisthepascal(Pa).Abarisanotherun itofpressure. Onebarisexactly10 5 Pa(table10.1).Becausethedepthinmetersandpressure indecibarsarealmostthesamenumerically,oceanographer sprefertostate pressureindecibars. Table10.1UnitsofPressure 1pascal(Pa)=1N/m 2 =1kg s 2 m 1 1bar=10 5 Pa 1decibar=10 4 Pa 1millibar=100Pa 10.2GeostrophicEquations ThegeostrophicbalancerequiresthattheCoriolisforceba lancethehorizontalpressuregradient.Theequationsforgeostrophicbalan cearederivedfrom theequationsofmotionassumingtherowhasnoacceleration du=dt = dv=dt = dw=dt =0;thathorizontalvelocitiesaremuchlargerthanvertica l, w u;v ; thattheonlyexternalforceisgravity;andthatfrictionis small.Withthese assumptions(7.12)become @p @x = fv ; @p @y = fu ; @p @z = g (10.6) where f =2nsin istheCoriolisparameter.Thesearethe geostrophicequations Theequationscanbewritten: u = 1 f @p @y ; v = 1 f @p @x (10.7a)

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154 CHAPTER10.GEOSTROPHICCURRENTS p = p 0 + Z h g ( ';z ) ( z ) dz (10.7b) where p 0 isatmosphericpressureat z =0,and istheheightoftheseasurface. NotethatIhaveallowedfortheseasurfacetobeaboveorbelo wthesurface z =0;andthepressuregradientattheseasurfaceisbalancedb yasurface current u s Substituting(10.7b)into(10.7a)gives: u = 1 f @ @y Z 0 h g ( ';z ) ( z ) dz g f @ @y u = 1 f @ @y Z 0 h g ( ';z ) ( z ) dz u s (10.8a) whereIhaveusedtheBoussinesqapproximation,retainingf ullaccuracyfor onlywhencalculatingpressure. Inasimilarway,wecanderivetheequationfor v v = 1 f @ @x Z 0 h g ( ';z ) ( z ) dz + g f @ @x v = 1 f @ @x Z 0 h g ( ';z ) ( z ) dz + v s (10.8b) Iftheoceanishomogeneousanddensityandgravityareconst ant,therst termontheright-handsideof(10.8)isequaltozero;andthe horizontalpressure gradientswithintheoceanarethesameasthegradientat z =0.Thisis barotropicrowdescribedin x 10.4. Iftheoceanisstratied,thehorizontalpressuregradient hastwoterms,one duetotheslopeattheseasurface,andanadditionaltermdue tohorizontal densitydierences.Theseequationsincludebaroclinicro walsodiscussedin x 10.4.Thersttermontheright-handsideof(10.8)isduetov ariationsin density ( z ),anditiscalledtherelativevelocity.Thuscalculationo fgeostrophic currentsfromthedensitydistributionrequirestheveloci ty( u 0 ;v 0 )atthesea surfaceoratsomeotherdepth. 1m 100km z x -r z=-r 0 Sea Surface ( z=z) z Figure10.1Sketchdening and r ,usedforcalculatingpressurejustbelowtheseasurface.

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10.3.SURFACEGEOSTROPHICCURRENTSFROMALTIMETRY 155 10.3SurfaceGeostrophicCurrentsFromAltimetry Thegeostrophicapproximationappliedat z =0leadstoaverysimple relation:surfacegeostrophiccurrentsareproportionalt osurfaceslope.Consider alevelsurfaceslightlybelowtheseasurface,saytwometer sbelowthesea surface,at z = r (gure10.1). Thepressureonthelevelsurfaceis: p = g ( + r )(10.9) assuming and g areessentiallyconstantintheupperfewmetersoftheocean Substitutingthisinto(10.7a),givesthetwocomponents( u s ;v s )ofthesurfacegeostrophiccurrent: u s = g f @ @y ; v s = g f @ @x (10.10) where g isgravity, f istheCoriolisparameter,and istheheightofthesea surfaceabovealevelsurface.TheOceanicTopography In x 3.4wedenethetopographyoftheseasurface tobetheheightoftheseasurfacerelativetoaparticularle velsurface,the geoid;andwedenedthegeoidtobethelevelsurfacethatcoi ncidedwiththe surfaceoftheoceanatrest.Thus,accordingto(10.10)thes urfacegeostrophic currentsareproportionaltotheslopeofthetopography(g ure10.2),aquantity thatcanbemeasuredbysatellitealtimetersifthegeoidisk nown. z=zVs =g f dzdx Sea SurfaceGeoid Vs 1m 100kmx Figure10.2Theslopeoftheseasurfacerelativetothegeoid ( @=@x )isdirectlyrelatedto surfacegeostrophiccurrents v s .Theslopeof1meterper100kilometers(10 rad)istypical ofstrongcurrents. V s isintothepaperinthenorthernhemisphere. Becausethegeoidisalevelsurface,itisasurfaceofconsta ntgeopotential.Toseethis,considertheworkdoneinmovingamass m byadistance h perpendiculartoalevelsurface.Theworkis W = mgh ,andthechangeof potentialenergyperunitmassis gh .Thuslevelsurfacesaresurfacesofconstant geopotential,wherethe geopotential = gh Topographyisduetoprocessesthatcausetheoceantomove:t ides,ocean currents,andthechangesinbarometricpressurethatprodu cetheinverted barometereect.Becausetheocean'stopographyisduetody namicalprocesses, itisusuallycalled dynamictopography .Thetopographyisapproximatelyone hundredthofthegeoidundulations.Thustheshapeofthesea surfaceisdominatedbylocalvariationsofgravity.Theinruenceofcurren tsismuchsmaller.

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156 CHAPTER10.GEOSTROPHICCURRENTS mean sea surface mean geoid Gulf Stream Geoid Error Cold Core Rings Warm Core Rings 40o 38o 36o 34o 32o30o 28o 26o 24o 22o 20o -0.5 0.0 0.5 1.0 -60 -55 -50 -45 -40 -35SST (m) Height (m)North Latitude Figure10.3Topex/PoseidonaltimeterobservationsoftheG ulfStream.Whenthealtimeter observationsaresubtractedfromthelocalgeoid,theyyiel dtheoceanictopography,whichis dueprimarilytooceancurrentsinthisexample.Thegravime tricgeoidwasdeterminedby theOhioStateUniversityfromshipandothersurveysofgrav ityintheregion.FromCenter forSpaceResearch,UniversityofTexas.Typically,sea-surfacetopographyhasamplitudeof 1m(gure10.3).Typical slopesare @=@x 1{10microradiansfor v =0.1{1.0m/satmidlatitude. Theheightofthegeoid,smoothedoverhorizontaldistances greaterthan roughly400km,isknownwithanaccuracyof 1mmfromdatacollectedby theGravityRecoveryandClimateExperiment grace satellitemission. SatelliteAltimetry Veryaccurate,satellite-altimetersystemsareneededfor measuringtheoceanictopography.Therstsystems,carrie donSeasat,Geosat, ers {1,and ers {2weredesignedtomeasureweek-to-weekvariabilityofcur rents. Topex/Poseidon,launchedin1992,wastherstsatellitede signedtomakethe muchmoreaccuratemeasurementsnecessaryforobservingth epermanent(timeaveraged)surfacecirculationoftheocean,tides,andthev ariabilityofgyre-scale currents.Itwasfollowedin2001byJasonandin2008byJason -2. Becausethegeoidwasnotwellknownlocallybeforeabout200 4,altimeters wereusuallyrowninorbitsthathaveanexactlyrepeatinggr oundtrack.Thus Topex/PoseidonandJasonryoverthesamegroundtrackevery 9.9156days. Bysubtractingsea-surfaceheightfromonetraverseoftheg roundtrackfrom heightmeasuredonalatertraverse,changesintopographyc anbeobserved withoutknowingthegeoid.Thegeoidisconstantintime,and thesubtraction removesthegeoid,revealingchangesduetochangingcurren ts,suchasmesoscale eddies,assumingtideshavebeenremovedfromthedata(gur e10.4).Mesoscale variabilityincludeseddieswithdiametersbetweenroughl y20and500km.

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10.3.SURFACEGEOSTROPHICCURRENTSFROMALTIMETRY 157 15 15 10 8 8 6 6 15 151515Topography Variability (cm) 20o60o100o140o180o-140o-100o-60o-20o20o -64o-32o0o32o64o15 0o1515 Figure10.4Globaldistributionofstandarddeviationofto pographyfromTopex/Poseidon altimeterdatafrom10/3/92to10/6/94.Theheightvariance isanindicatorofvariabilityof surfacegeostrophiccurrents.FromCenterforSpaceResear ch,UniversityofTexas. ThegreataccuracyandprecisionoftheTopex/PoseidonandJ asonaltimeter systemsallowthemtomeasuretheoceanictopographyoveroc eanbasinswith anaccuracyof (2{5)cm(Cheltonetal,2001).Thisallowsthemtomeasure: 1.Changesinthemeanvolumeoftheoceanandsea-levelrisew ithanaccuracyof 0 : 4mm/yrsince1993(Neremetal,2006); 2.Seasonalheatingandcoolingoftheocean(Chambersetal1 998); 3.Openoceantideswithanaccuracyof (1{2)cm(Shumetal,1997); 4.Tidaldissipation(EgbertandRay,1999;Rudnicketal,20 03); 5.Thepermanentsurfacegeostrophiccurrentsystem(gure 10.5); 6.Changesinsurfacegeostrophiccurrentsonallscales(g ure10.4);and 7.Variationsintopographyofequatorialcurrentsystemss uchasthoseassociatedwithElNi~no(gure10.6). AltimeterErrors(Topex/PoseidonandJason) Themostaccurateobservationsofthesea-surfacetopographyarefromTopex/Po seidonandJason. Errorsforthesesatellitealtimetersystemaredueto(Chel tonetal,2001): 1.Instrumentnoise,oceanwaves,watervapor,freeelectro nsintheionosphere,andmassoftheatmosphere.Bothsatellitescarried aprecise altimetersystemabletoobservetheheightofthesatellite abovethesea surfacebetween 66 latitudewithaprecisionof (1{2)cmandanaccuracyof (2{5)cm.Thesystemsconsistofatwo-frequencyradaraltim eter tomeasureheightabovethesea,theinruenceoftheionosphe re,andwave height,andathree-frequencymicrowaveradiometerableto measurewater vaporinthetroposphere.

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158 CHAPTER10.GEOSTROPHICCURRENTS Four-Year Mean Sea-Surface Topography (cm) -100 -100 -100 -150 -150 -150 120 0 20 40 40 40 40 40 0 0 20 20 20 20 -50 120 60 60 60 60 60 80 80 80 80 80 80 100 60 80 40 60 0 60 40 60 40 40 20 40 40 80 -50 -100 -150 20o60o100o140o180o-140o-100o-60o-20o20o 0o -66o-44o-22o0o22o44o66o Figure10.5Globaldistributionoftime-averagedtopograp hyoftheoceanfromTopex/Poseidonaltimeterdatafrom10/3/92to10/6/99relativetothe jgm {3geoid.Geostrophic currentsattheoceansurfaceareparalleltothecontours.C omparewithgure2.8calculated fromhydrographicdata.FromCenterforSpaceResearch,Uni versityofTexas. 2.Trackingerrors.Thesatellitescarriedthreetrackings ystemsthatenable theirpositioninspace,theephemeris,tobedeterminedwit hanaccuracy of (1{3.5)cm. 3.Samplingerror.Thesatellitesmeasureheightalongagro undtrackthat repeatswithin 1kmevery9.9156days.Eachrepeatisacycle.Becausecurrentsaremeasuredonlyalongthesub-satellitetr ack,thereisa samplingerror.Thesatellitecannotmapthetopographybet weenground tracks,norcantheyobservechangeswithperiodslessthan2 9 : 9156d (see x 16.3). 4.Geoiderror.Thepermanenttopographyisnotwellknownov erdistances shorterthanahundredkilometersbecausegeoiderrorsdomi nateforshort distances.Mapsoftopographysmoothedovergreaterdistan cesareused tostudythedominantfeaturesofthepermanentgeostrophic currentsat theseasurface(gure10.5).Newsatellitesystems grace and champ are measuringearth'sgravityaccuratelyenoughthatthegeoid errorisnow smallenoughtoignoreoverdistancesgreaterthan100km. Takentogether,themeasurementsofheightabovetheseaand thesatellitepositiongivesea-surfaceheightingeocentriccoordinatesw ithin (2{5)cm.Geoid erroraddsfurthererrorsthatdependonthesizeoftheareab eingmeasured. 10.4GeostrophicCurrentsFromHydrography Thegeostrophicequationsarewidelyusedinoceanographyt ocalculatecurrentsatdepth.Thebasicideaistousehydrographicmeasure mentsoftemperature,salinityorconductivity,andpressuretocalcul atethedensityeldof theoceanusingtheequationofstateofseawater.Densityis usedin(10.7b)

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10.4.GEOSTROPHICCURRENTSFROMHYDROGRAPHY 159 150 155 160 165 170 175 180 185 190 195 200 205 210 215 220 225 230 235 240 245 250 255 260 265 270 275 280 285 290 295 300 305 310 315 320 140 160 180 200 220 240 260 280 10/9/96 11/28/96 1/16/97 3/7/97 4/25/97 6/14/97 8/3/97 9/21/97 11/10/97 12/29/97 2/17/98 4/7/98 5/27/98 7/16/98 9/3/98 10/23/98 12/11/98 1/30/99 3/20/99 5/9/99 6/28/99 8/4/99 10/5/99 11/23/99 1/22/00 3/2/00 4/20/00 6/9/00 7/28/00 9/16/00 11/4/00 12/24/00 2/12/01 4/2/01 5/22/01-25 -20 -15 -10 -505 1015 20 25 30 cm Figure10.6Time-longitudeplotofsea-levelanomaliesint heEquatorialPacicobserved byTopex/Poseidonduringthe1997{1998ElNi~no.Warmanoma liesarelightgray,cold anomaliesaredarkgray.Theanomaliesarecomputedfrom10daydeviationsfroma three-yearmeansurfacefrom3Oct1992to8Oct1995.Thedata aresmoothedwith aGaussianweightedlterwithalongitudinalspanof5 andalatitudinalspanof2 Theannotationsontheleftarecyclesofsatellitedata.Fro mCenterforSpaceResearch, UniversityofTexas.

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160 CHAPTER10.GEOSTROPHICCURRENTS tocalculatetheinternalpressureeld,fromwhichthegeos trophiccurrentsare calculatedusing(10.8a,b).Usually,however,theconstan tofintegrationin (10.8)isnotknown,andonlytherelativevelocityeldcanb ecalculated. Atthispoint,youmayask,whynotjustmeasurepressuredire ctlyasisdone inmeteorology,wheredirectmeasurementsofpressureareu sedtocalculate winds.And,aren'tpressuremeasurementsneededtocalcula tedensityfrom theequationofstate?Theansweristhatverysmallchangesi ndepthmake largechangesinpressurebecausewaterissoheavy.Errorsi npressurecaused byerrorsindeterminingthedepthofapressuregaugearemuc hlargerthan thepressureduetocurrents.Forexample,using(10.7a),we calculatethatthe pressuregradientduetoa10cm/scurrentat30 latitudeis7 : 5 10 3 Pa/m, whichis750Pain100km.Fromthehydrostaticequation(10.5 ),750Pais equivalenttoachangeofdepthof7.4cm.Therefore,forthis example,wemust knowthedepthofapressuregaugewithanaccuracyofmuchbet terthan7.4 cm.Thisisnotpossible.GeopotentialSurfacesWithintheOcean Calculationofpressuregradients withintheoceanmustbedonealongsurfacesofconstantgeop otentialjustaswe calculatedsurfacepressuregradientsrelativetothegeoi dwhenwecalculated surfacegeostrophiccurrents.Aslongagoas1910,VilhelmB jerknes(Bjerknes andSandstrom,1910)realizedthatsuchsurfacesarenotat xedheightsinthe atmospherebecause g isnotconstant,and(10.4)mustincludethevariability ofgravityinboththehorizontalandverticaldirections(S aundersandFofono, 1976)whencalculatingpressureintheocean. The geopotential is: = Z z 0 gdz (10.11) Because = 9 : 8inSIunitshasalmostthesamenumericalvalueasheightinm eters,themeteorologicalcommunityacceptedBjerknes'pro posalthatheightbe replacedby dynamicmeters D = = 10toobtainanaturalverticalcoordinate. Later,thiswasreplacedbythe geopotentialmeter (gpm) Z = = 9 : 80.The geopotentialmeterisameasureoftheworkrequiredtolifta unitmassfrom sealeveltoaheight z againsttheforceofgravity.HaraldSverdrup,Bjerknes' student,carriedtheconcepttooceanography,anddepthsin theoceanareoften quotedingeopotentialmeters.Thedierencebetweendepth sofconstantverticaldistanceandconstantpotentialcanberelativelylar ge.Forexample,the geometricdepthofthe1000dynamicmetersurfaceis1017.40 matthenorth poleand1022.78mattheequator,adierenceof5.38m. Notethatdepthingeopotentialmeters,depthinmeters,and pressurein decibarsarealmostthesamenumerically.Atadepthof1mete rthepressure isapproximately1.007decibarsandthedepthis1.00geopot entialmeters. EquationsforGeostrophicCurrentsWithintheOcean Tocalculategeostrophiccurrents,weneedtocalculatethehorizontalpres suregradientwithin theocean.Thiscanbedoneusingeitheroftwoapproaches: 1.Calculatetheslopeofaconstantpressuresurfacerelati vetoasurfaceof

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10.4.GEOSTROPHICCURRENTSFROMHYDROGRAPHY 161 constantgeopotential.Weusedthisapproachwhenweusedse a-surface slopefromaltimetrytocalculatesurfacegeostrophiccurr ents.Thesea surfaceisaconstant-pressuresurface.Theconstantgeopo tentialsurface wasthegeoid. 2.Calculatethechangeinpressureonasurfaceofconstantg eopotential. Suchasurfaceiscalleda geopotentialsurface FB-FA} FA = F( P1A) F(P2A)FB = F(P1B) F(P2B) P2P1A B L b Figure10.7.Sketchofgeometryusedforcalculatinggeostr ophiccurrentfromhydrography. Oceanographersusuallycalculatetheslopeofconstant-pr essuresurfaces. Theimportantstepsare: 1.Calculatedierencesingeopotential( A B )betweentwoconstantpressuresurfaces( P 1 ;P 2 )athydrographicstationsAandB(gure10.7). Thisissimilartothecalculationof ofthesurfacelayer. 2.Calculatetheslopeoftheupperpressuresurfacerelativ etothelower. 3.Calculatethegeostrophiccurrentattheuppersurfacere lativetothecurrentatthelower.Thisisthecurrentshear. 4.Integratethecurrentshearfromsomedepthwherecurrent sareknown toobtaincurrentsasafunctionofdepth.Forexample,fromt hesurfacedownward,usingsurfacegeostrophiccurrentsobserve dbysatellite altimetry,orupwardfromanassumedlevelofnomotion. Tocalculategeostrophiccurrentsoceanographersuseamod iedformofthe hydrostaticequation.Theverticalpressuregradient(10. 6)iswritten p = p = gz (10.12a) p = (10.12b) where = ( S;t;p )isthe specicvolume ,and(10.12b)followsfrom(10.11). Dierentiating(10.12b)withrespecttohorizontaldistan ce x allowsthegeostrophicbalancetobewrittenintermsoftheslopeoftheconstan t-pressuresurface

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162 CHAPTER10.GEOSTROPHICCURRENTS using(10.6)with f =2nsin : @p @x = 1 @p @x = 2n v sin (10.13a) @ ( p = p 0 ) @x = 2n v sin (10.13b) whereisthegeopotentialattheconstant-pressuresurfac e. Nowlet'sseehowhydrographicdataareusedforevaluating @ =@x ona constant-pressuresurface.Integrating(10.12b)between twoconstant-pressure surfaces( P 1 ;P 2 )intheoceanasshowningure10.7givesthegeopotential dierencebetweentwoconstant-pressuresurfaces.Atstat ionAtheintegration gives: ( P 1 A ) ( P 2 A )= Z P 2 A P 1 A ( S;t;p ) dp (10.14) Thespecicvolumeanomalyiswrittenasthesumoftwoparts: ( S;t;p )= (35 ; 0 ;p )+ (10.15) where (35 ; 0 ;p )isthespecicvolumeofseawaterwithsalinityof35,tempe ratureof0 C,andpressure p .Thesecondterm isthe specicvolumeanomaly Using(10.15)in(10.14)gives: ( P 1 A ) ( P 2 A )= Z P 2 A P 1 A (35 ; 0 ;p ) dp + Z P 2 A P 1 A dp ( P 1 A ) ( P 2 A )=( 1 2 ) std + A where( 1 2 ) std isthe standardgeopotentialdistance betweentwoconstantpressuresurfaces P 1 and P 2 ,and A = Z P 2 A P 1 A dp (10.16) istheanomalyofthegeopotentialdistancebetweenthesurf aces.Itiscalledthe geopotentialanomaly .Thegeometricdistancebetween 2 and 1 isnumerically approximately( 2 1 ) =g where g =9 : 8m/s 2 istheapproximatevalueof gravity.Thegeopotentialanomalyismuchsmaller,beingap proximately0.1% ofthestandardgeopotentialdistance. Considernowthegeopotentialanomalybetweentwopressure surfaces P 1 and P 2 calculatedattwohydrographicstationsAandBadistance L meters apart(gure10.7).Forsimplicityweassumethelowerconst ant-pressuresurfaceisalevelsurface.Hencetheconstant-pressureandgeo potentialsurfaces coincide,andthereisnogeostrophicvelocityatthisdepth .Theslopeofthe uppersurfaceis B A L =slopeofconstant-pressuresurface P 2

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10.4.GEOSTROPHICCURRENTSFROMHYDROGRAPHY 163 becausethestandardgeopotentialdistanceisthesameatst ationsAandB. Thegeostrophicvelocityattheuppersurfacecalculatedfr om(10.13b)is: V = ( B A ) 2n L sin (10.17) where V isthevelocityattheuppergeopotentialsurface.Theveloc ity V is perpendiculartotheplaneofthetwohydrographicstations anddirectedinto theplaneofgure10.7iftherowisinthenorthernhemispher e. Ausefulruleof thumbisthattherowissuchthatwarmer,lighterwateristot herightlooking downstreaminthenorthernhemisphere. NotethatIcouldhavecalculatedtheslopeoftheconstant-p ressuresurfaces usingdensity insteadofspecicvolume .Iused becauseitisthecommon practiceinoceanography,andtablesofspecicvolumeanom aliesandcomputer codetocalculatetheanomaliesarewidelyavailable.Theco mmonpracticefollowsfromnumericalmethodsdevelopedbeforecalculatorsa ndcomputerswere available,whenallcalculationsweredonebyhandorbymech anicalcalculators withthehelpoftablesandnomograms.Becausethecomputati onmustbedone withanaccuracyofafewpartspermillion,andbecauseallsc ienticeldstend tobeconservative,thecommonpracticehascontinuedtouse specicvolume anomaliesratherthandensityanomalies.BarotropicandBaroclinicFlow: Iftheoceanwerehomogeneouswithconstantdensity,thenconstant-pressuresurfaceswouldalwa ysbeparalleltothe seasurface,andthegeostrophicvelocitywouldbeindepend entofdepth.In thiscasetherelativevelocityiszero,andhydrographicda tacannotbeused tomeasurethegeostrophiccurrent.Ifdensityvarieswithd epth,butnotwith horizontaldistance,theconstant-pressuresurfacesarea lwaysparalleltothesea surfaceandthelevelsofconstantdensity,the isopycnalsurfaces .Inthiscase, therelativerowisalsozero.Bothcasesareexamplesof barotropicrow Barotropicrow occurswhenlevelsofconstantpressureintheoceanare alwaysparalleltothesurfacesofconstantdensity.Note,s omeauthorscallthe verticallyaveragedrowthebarotropiccomponentoftherow .Wunsch(1996: 74)pointsoutthatbarotropicisusedinsomanydierentway sthattheterm ismeaninglessandshouldnotbeused. Baroclinicrow occurswhenlevelsofconstantpressureareinclinedtosurfacesofconstantdensity.Inthiscase,densityvarieswith depthandhorizontal position.Agoodexampleisseeningure10.8whichshowslev elsofconstant densitychangingdepthbymorethan1kmoverhorizontaldist ancesof100km attheGulfStream.Baroclinicrowvarieswithdepth,andthe relativecurrentcanbecalculatedfromhydrographicdata.Note,consta nt-densitysurfaces cannotbeinclinedtoconstant-pressuresurfacesforaruid atrest. Ingeneral,thevariationofrowintheverticalcanbedecomp osedintoa barotropiccomponentwhichisindependentofdepth,andaba rocliniccomponentwhichvarieswithdepth.

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164 CHAPTER10.GEOSTROPHICCURRENTS 10.5AnExampleUsingHydrographicData Let'snowconsideraspecicnumericalcalculationofgeost rophicvelocity usinggenerallyacceptedproceduresfrom ProcessingofOceanographicStation Data ( jpots EditorialPanel,1991).Thebookhasworkedexamplesusing hydrographicdatacollectedbythe r/v Endeavor inthenorthAtlantic.Data werecollectedonCruise88along71 WacrosstheGulfStreamsouthofCape Cod,Massachusettsatstations61and64.Station61isonthe SargassoSea sideoftheGulfStreaminwater4260mdeep.Station64isnort hoftheGulf Streaminwater3892mdeep.ThemeasurementsweremadebyaCo nductivityTemperature-Depth-OxygenProler,MarkIIICTD/02,madeb yNeilBrown InstrumentsSystems. The ctd sampledtemperature,salinity,andpressure22timesperse cond, andthedigitaldatawereaveragedover2dbarintervalsasth e ctd waslowered inthewater.Dataweretabulatedat2dbarpressureinterval scenteredon oddvaluesofpressurebecausetherstobservationisatthe surface,andthe rstaveragingintervalextendsto2dbar,andthecenteroft herstinterval isat1dbar.Datawerefurthersmoothedwithabinomiallter andlinearly interpolatedtostandardlevelsreportedintherstthreec olumnsoftables10.2 and10.3.Allprocessingwasdonebycomputer. ( S;t;p )inthefthcolumnoftables10.2and10.3iscalculatedfrom thevaluesof t;S;p inthelayer. <> istheaveragevalueofspecicvolumeanomaly forthelayerbetweenstandardpressurelevels.Itistheave rageofthevaluesof ( S;t;p )atthetopandbottomofthelayer( cf. themean-valuetheoremofcalculus).Thelastcolumn(10 5 )istheproductoftheaveragespecicvolume anomalyofthelayertimesthethicknessofthelayerindecib ars.Therefore,the lastcolumnisthegeopotentialanomalycalculatedbyint egrating(10.16) between P 1 atthebottomofeachlayerand P 2 atthetopofeachlayer. Thedistancebetweenthestationsis L =110 ; 935m;theaverageCoriolis parameteris f =0 : 88104 10 4 ;andthedenominatorin(10.17)is0.10231s/m. Thiswasusedtocalculatethegeostrophiccurrentsrelativ eto2000decibars reportedintable10.4andplottedingure10.8. NoticethattherearenoEkmancurrentsingure10.8.Ekmanc urrents arenotgeostrophic,sotheydon'tcontributedirectlytoth etopography.They contributeonlyindirectlythroughEkmanpumping(seegur e12.7). 10.6CommentsonGeostrophicCurrents Nowthatweknowhowtocalculategeostrophiccurrentsfromh ydrographic data,let'sconsidersomeofthelimitationsofthetheoryan dtechniques. ConvertingRelativeVelocitytoVelocity Hydrographicdatagivegeostrophiccurrentsrelativetogeostrophiccurrentsatsomerefe rencelevel.Howcan weconverttherelativegeostrophicvelocitiestovelociti esrelativetotheearth? 1. AssumeaLevelofnoMotion :Traditionally,oceanographersassumethere isa levelofnomotion ,sometimescalleda referencesurface ,roughly2,000 mbelowthesurface.Thisistheassumptionusedtoderivethe currents intable10.4.Currentsareassumedtobezeroatthisdepth,a ndrelative currentsareintegrateduptothesurfaceanddowntothebott omto

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10.6.COMMENTSONGEOSTROPHICCURRENTS 165 Table10.2ComputationofRelativeGeostrophicCurrents. DatafromEndeavorCruise88,Station61(36 40.03'N,70 59.59'W;23August1982;1102Z) PressuretS ( ) ( S;t;p ) <> 10 5 decibar Ckg/m 3 10 8 m 3 /kg10 8 m 3 /kgm 2 /s 2 025.69835.22123.296457.24 457.260.046 125.69835.22123.296457.28 440.220.396 1026.76336.10623.658423.15 423.410.423 2026.67836.10623.658423.66 423.820.424 3026.67636.10723.659423.98 376.230.752 5024.52836.56124.670328.48 302.070.755 7522.75336.61425.236275.66 257.410.644 10021.42736.63725.630239.15 229.610.574 12520.63336.62725.841220.06 208.840.522 15019.52236.55826.086197.62 189.650.948 20018.79836.55526.273181.67 178.720.894 25018.43136.53726.354175.77 174.120.871 30018.18936.52626.408172.46 170.381.704 40017.72636.47726.489168.30 166.761.668 50017.16536.38126.557165.22 158.781.588 60015.95236.10526.714152.33 143.181.432 70013.45835.77626.914134.03 124.201.242 80011.10935.43727.115114.36 104.481.045 9008.79835.17827.30694.60 80.840.808 10006.29235.04427.56267.07 61.890.619 11005.24935.00427.66056.70 54.640.546 12004.81334.99527.70552.58 51.740.517 13004.55434.98627.72750.90 50.400.504 14004.35734.97727.74349.89 49.730.497 15004.24534.97527.75349.56 49.301.232 17504.02834.97327.77749.03 48.831.221 20003.85234.97527.79948.62 47.772.389 25003.42434.96827.83946.92 45.942.297 30002.96334.94627.86844.96 43.402.170 35002.46234.92027.89441.84 41.932.097 40002.25934.90427.90142.02 obtaincurrentvelocityasafunctionofdepth.Thereissome experimental evidencethatsuchalevelexistsonaverageformeancurrent s(seefor example,Defant,1961:492).Defantrecommendschoosingareferencelevelwherethecurr entshearin theverticalissmallest.Thisisusuallynear2km.Thislead stouseful mapsofsurfacecurrentsbecausesurfacecurrentstendtobe fasterthan deepercurrents.Figure10.9showsthegeopotentialanomal yandsurface currentsinthePacicrelativetothe1,000dbarpressurele vel. 2. Useknowncurrents: Theknowncurrentscouldbemeasuredbycurrent

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166 CHAPTER10.GEOSTROPHICCURRENTS Table10.3ComputationofRelativeGeostrophicCurrents. DatafromEndeavorCruise88,Station64(37 39.93'N,71 0.00'W;24August1982;0203Z) PressuretS ( ) ( S;t;p ) <> 10 5 decibar Ckg/m 3 10 8 m 3 /kg10 8 m 3 /kgm 2 /s 2 026.14834.64622.722512.09 512.150.051 126.14834.64622.722512.21 512.610.461 1026.16334.64522.717513.01 512.890.513 2026.16734.65522.724512.76 466.290.466 3025.64035.73323.703419.82 322.380.645 5018.96735.94425.755224.93 185.560.464 7515.37135.90426.590146.19 136.180.340 10014.35635.89726.809126.16 120.910.302 12513.05935.69626.925115.66 111.930.280 15012.13435.56727.008108.20 100.190.501 20010.30735.36027.18592.17 87.410.437 2508.78335.16827.29082.64 79.400.397 3008.04635.11727.36476.16 66.680.667 4006.23535.05227.56857.19 52.710.527 5005.23035.01827.66748.23 46.760.468 6005.00535.04427.71045.29 44.670.447 7004.75635.02727.73144.04 43.690.437 8004.39934.99227.74443.33 43.220.432 9004.29134.99127.75643.11 43.120.431 10004.17934.98627.76443.12 43.100.431 11004.07734.98227.77343.07 43.120.431 12003.96934.97527.77943.17 43.280.433 13003.90934.97427.78643.39 43.380.434 14003.83134.97327.79343.36 43.310.433 15003.76734.97527.80243.26 43.201.080 17503.60034.97527.82143.13 43.001.075 20003.40134.96827.83742.86 42.132.106 25002.94234.94827.86741.39 40.332.016 30002.47534.92327.89139.26 39.221.961 35002.21934.90427.90039.17 40.082.004 40002.17734.89627.90140.98 metersorbysatellitealtimetry.Problemsariseifthecurr entsarenot measuredatthesametimeasthehydrographicdata.Forexamp le,the hydrographicdatamayhavebeencollectedoveraperiodofmo nthsto decades,whilethecurrentsmayhavebeenmeasuredoveraper iodof onlyafewmonths.Hence,thehydrographymaynotbeconsiste ntwith thecurrentmeasurements.Sometimescurrentsandhydrogra phicdata aremeasuredatnearlythesametime(gure10.10).Inthisex ample, currentsweremeasuredcontinuouslybymooredcurrentmete rs(points) inadeepwesternboundarycurrentandcalculatedfrom ctd datataken justafterthecurrentmetersweredeployedandjustbeforet heywere

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10.6.COMMENTSONGEOSTROPHICCURRENTS 167 Table10.4ComputationofRelativeGeostrophicCurrents. DatafromEndeavorCruise88,Station61and64 Pressure10 5 61 10 5 64 V decibarm 2 /s 2 at61 m 2 /s 2 at64 (m/s) 02.18721.25830.95 0.0460.051 12.18261.25320.95 0.3960.461 102.14301.20700.96 0.4230.513 202.10061.15570.97 0.4240.466 302.05831.10910.97 0.7520.645 501.98301.04460.96 0.7550.464 751.90750.99820.93 0.6440.340 1001.84310.96420.90 0.5740.302 1251.78570.93400.87 0.5220.280 1501.73350.90600.85 0.9480.501 2001.63870.85590.80 0.8940.437 2501.54930.81220.75 0.8710.397 3001.46230.77250.71 1.7040.667 4001.29190.70580.60 1.6680.527 5001.12520.65310.48 1.5880.468 6000.96640.60630.37 1.4320.447 7000.82320.56170.27 1.2420.437 8000.69900.51800.19 1.0450.432 9000.59450.47480.12 0.8080.431 10000.51370.43170.08 0.6190.431 11000.45180.38860.06 0.5460.431 12000.39720.34540.05 0.5170.433 13000.34540.30220.04 0.5040.434 14000.29500.25880.04 0.4970.433 15000.24530.21550.03 1.2321.080 17500.12210.10750.01 1.2211.075 20000.00000.00000.00 2.3892.106 2500-0.2389-0.2106-0.03 2.2972.016 3000-0.4686-0.4123-0.06 2.1701.961 3500-0.6856-0.6083-0.08 2.0972.004 4000-0.8952-0.8087-0.09 Geopotentialanomalyintegratedfrom2000dbarlevel. Velocityiscalculatedfrom(10.17) recovered(smoothcurves).Thesolidlineisthecurrentass umingalevel ofnomotionat2,000m,thedottedlineisthecurrentadjuste dusingthe currentmeterobservationssmoothedforvariousintervals beforeorafter the ctd casts. 3. UseConservationEquations :Linesofhydrographicstationsacrossastrait oranoceanbasinmaybeusedwithconservationofmassandsal tto calculatecurrents.Thisisanexampleofaninverseproblem (Wunsch, 1996describestheapplicationofinversemethodsinoceano graphy).See Mercieretal.(2003)foradescriptionofhowtheydetermine dthecir-

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168 CHAPTER10.GEOSTROPHICCURRENTS 26.50 26.60 27.00 27.60 27.70 27.80 27.82 27.84 27.86 7969 89 42o40o38oNorth Latitude Station Number baroclinic barotropic -4000 -3500 -3000 -2500 -2000 -1500 -1000 -500 0 -0.5 0 0.5 1Depth (decibars)Speed (m/s) Figure10.8 Left Relativecurrentasafunctionofdepthcalculatedfromhydr ographicdata collectedbythe Endeavor cruisesouthofCapeCodinAugust1982.TheGulfStreamis thefastcurrentshallowerthan1000decibars.Theassumedd epthofnomotionisat2000 decibars. Right Crosssectionofpotentialdensity acrosstheGulfStreamalong63.66 W calculatedfrom ctd datacollectedfrom Endeavor on25{28April1986.TheGulfStreamis centeredonthesteeplyslopingcontoursshallowerthan100 0mbetween40 and41 .Notice thattheverticalscaleis425timesthehorizontalscale.(D atacontouredbyLynnTalley, ScrippsInstitutionofOceanography). culationintheupperlayersoftheeasternbasinsofthesout hAtlantic usinghydrographicdatafromtheWorldOceanCirculationEx periment anddirectmeasurementsofcurrentinaboxmodelconstraine dbyinverse theory. DisadvantageofCalculatingCurrentsfromHydrographicDa ta Currentscalculatedfromhydrographicdatahavebeenusedtoma kemapsofocean currentssincetheearly20thcentury.Nevertheless,itisi mportanttoreview thelimitationsofthetechnique. 1.Hydrographicdatacanbeusedtocalculateonlythecurren trelativetoa currentatanotherlevel.

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10.6.COMMENTSONGEOSTROPHICCURRENTS 169 110110130150180190170170150150130180200190200190210 22017017090 70500o0o2 0o2 0o40o40o60o60o80o80o-20o-20o-40o-40o-6 0o-6 0o-80o-80o Figure10.9.Meangeopotentialanomalyrelativetothe1,00 0dbarsurfaceinthePacic basedon36,356observations.Heightoftheanomalyisingeo potentialcentimeters.Ifthe velocityat1,000dbarwerezero,themapwouldbethesurface topographyofthePacic. AfterWyrtki(1979). 2.Theassumptionofalevelofnomotionmaybesuitableinthe deepocean, butitisusuallynotausefulassumptionwhenthewaterissha llowsuch asoverthecontinentalshelf. 3.Geostrophiccurrentscannotbecalculatedfromhydrogra phicstationsthat areclosetogether.Stationsmustbetensofkilometersapar t. LimitationsoftheGeostrophicEquations Ibeganthissectionbyshowing thatthegeostrophicbalanceapplieswithgoodaccuracytor owsthatexceeda fewtensofkilometersinextentandwithperiodsgreatertha nafewdays.The balancecannot,however,beperfect.Ifitwere,therowinth eoceanwould neverchangebecausethebalanceignoresanyaccelerationo ftherow.The importantlimitationsofthegeostrophicassumptionare: 1.Geostrophiccurrentscannotevolvewithtimebecausethe balanceignores

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170 CHAPTER10.GEOSTROPHICCURRENTS -6.99 MOORING 6(206-207)recovery 18 XI 1200 1-day2-day3-day5-day *7-day rel 2000best fit Northward Speed (cm/s)Depth (m)-6000 -5500 -5000 -4500 -4000 -3500 -3000 -2500 -2000 -15-10-505 Figure10.10Currentmetermeasurementscanbeusedwith ctd measurementstodetermine currentasafunctionofdepthavoidingtheneedforassuming adepthofnomotion.Solid line:proleassumingadepthofnomotionat2000decibars.D ashedline:proleadjusted toagreewithcurrentsmeasuredbycurrentmeters1{7daysbe forethe ctd measurements. (PlotsfromTomWhitworth,TexasA&MUniversity) accelerationoftherow.Accelerationdominatesifthehori zontaldimensionsarelessthanroughly50kmandtimesarelessthanafewd ays. Accelerationisnegligible,butnotzero,overlongertimes anddistances. 2.Thegeostrophicbalancedoesnotapplywithinabout2 oftheequator wheretheCoriolisforcegoestozerobecausesin 0. 3.Thegeostrophicbalanceignorestheinruenceoffriction Accuracy Strubetal.(1997)showedthatcurrentscalculatedfromsat ellite altimetermeasurementsofsea-surfaceslopehaveanaccura cyof 3{5cm/s. Uchida,Imawaki,andHu(1998)comparedcurrentsmeasuredb ydriftersin theKuroshiowithcurrentscalculatedfromsatellitealtim eterdataassuming geostrophicbalance.Usingslopesoverdistancesof12.5km ,theyfoundthe dierencebetweenthetwomeasurementswas 16cm/sforcurrentsupto150 cm/s,orabout10%.Johns,Watts,andRossby(1989)measured thevelocityof theGulfStreamnortheastofCapeHatterasandcomparedthem easurements withvelocitycalculatedfromhydrographicdataassumingg eostrophicbalance. Theyfoundthatthemeasuredvelocityinthecoreofthestrea m,atdepths lessthan500m,was10{25cm/sfasterthanthevelocitycalcu latedfromthe geostrophicequationsusingmeasuredvelocitiesatadepth of2000m.The maximumvelocityinthecorewasgreaterthan150cm/s,sothe errorwas 10%.WhentheyaddedtheinruenceofthecurvatureoftheGulf Stream, whichaddsanaccelerationtermtothegeostrophicequation s,thedierencein thecalculatedandobservedvelocitydroppedtolessthan5{ 10cm/s( 5%).

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10.7.CURRENTSFROMHYDROGRAPHICSECTIONS 171 NorthernHemispherep1p 2 p 3 p 4 v1v2r 1 r2Level Surface Interfaceb 2 b 1 g x z Figure10.11Slopes oftheseasurfaceandtheslope r oftheinterfacebetweentwo homogeneous,movinglayers,withdensity 1 and 2 inthenorthernhemisphere.After NeumannandPierson(1966:166)10.7CurrentsFromHydrographicSections Linesofhydrographicdataalongshiptracksareoftenusedt oproducecontourplotsofdensityinaverticalsectionalongthetrack.C ross-sectionsof currentssometimesshowsharplydippingdensitysurfacesw ithalargecontrast indensityoneithersideofthecurrent.Thebarocliniccurr entsinthesection canbeestimatedusingatechniquerstproposedbyMargules (1906)anddescribedbyDefant(1961:453).Thetechniqueallowsoceanog rapherstoestimate thespeedanddirectionofcurrentsperpendiculartothesec tionbyaquicklook atthesection. ToderiveMargules'equation,considertheslope @z=@x ofastationaryinterfacebetweentwowatermasseswithdensities 1 and 2 (seegure10.11). Tocalculatethechangeinvelocityacrosstheinterfacewea ssumehomogeneous layersofdensity 1 < 2 bothofwhichareingeostrophicequilibrium.Although theoceandoesnothaveanidealizedinterfacethatweassume d,andthewater massesdonothaveuniformdensity,andtheinterfacebetwee nthewatermasses isnotsharp,theconceptisstillusefulinpractice. Thechangeinpressureontheinterfaceis: p = @p @x x + @p @z z; (10.18) andtheverticalandhorizontalpressuregradientsareobta inedfrom(10.6): @p @z = 1 g + 1 fv 1 (10.19) Therefore: p 1 = 1 fv 1 x + 1 gz (10.20a) p 2 = 2 fv 2 x + 2 gz (10.20b) Theboundaryconditionsrequire p 1 = p 2 ontheinterfaceiftheinterfaceis notmoving.Equating(10.20a)with(10.20b),dividingby x ,andsolvingfor

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172 CHAPTER10.GEOSTROPHICCURRENTS z=x gives: z x tan r = f g 2 v 2 1 v 1 2 1 Because 1 2 ,andforsmall and r tan r f g 1 2 1 ( v 2 v 1 )(10.21a) tan 1 = f g v 1 (10.21b) tan 2 = f g v 2 (10.21c) where istheslopeoftheseasurface,and r istheslopeoftheinterfacebetween thetwowatermasses.Becausetheinternaldierencesinden sityaresmall,the slopeisapproximately1000timeslargerthantheslopeofth econstantpressure surfaces. ConsidertheapplicationofthetechniquetotheGulfStream (gure10.8). Fromthegure: =36 1 =1026 : 7kg/m 3 ,and 2 =1027 : 5kg/m 3 ata depthof500decibars.Ifweusethe t =27 : 1surfacetoestimatetheslope betweenthetwowatermasses,weseethatthesurfacechanges fromadepth of350mtoadepthof650moveradistanceof70km.Therefore,t an r = 4300 10 6 =0 : 0043,and v = v 2 v 1 = 0 : 38m/s.Assuming v 2 =0,then v 1 =0 : 38m/s.ThisroughestimateofthevelocityoftheGulfStream compares wellwithvelocityatadepthof500mcalculatedfromhydrogr aphicdata(table 10.4)assumingalevelofnomotionat2,000decibars. Theslopeoftheconstant-densitysurfacesareclearlyseen ingure10.8. Andplotsofconstant-densitysurfacescanbeusedtoquickl yestimatecurrent directionsandaroughvalueforthespeed.Incontrast,thes lopeofthesea surfaceis8 : 4 10 6 or0.84min100kmifweusedatafromtable10.4. Notethatconstant-densitysurfacesintheGulfStreamslop edownwardto theeast,andthatsea-surfacetopographyslopesupwardtot heeast.Constant pressureandconstantdensitysurfaceshaveoppositeslope Ifthesharpinterfacebetweentwowatermassesreachesthes urface,itisan oceanicfront,whichhaspropertiesthatareverysimilarto atmosphericfronts. EddiesinthevicinityoftheGulfStreamcanhavewarmorcold cores(gure10.12).ApplicationofMargules'methodtothesemesosc aleeddiesgives thedirectionoftherow.Anticycloniceddies(clockwisero tationinthenorthernhemisphere)havewarmcores( 1 isdeeperinthecenteroftheeddythan elsewhere)andtheconstant-pressuresurfacesbowupward. Inparticular,the seasurfaceishigheratthecenterofthering.Cycloniceddi esarethereverse. 10.8LagrangianMeasurementsofCurrents Oceanographyandruidmechanicsdistinguishbetweentwote chniquesfor measuringcurrents:LagrangianandEulerian.Lagrangiant echniquesfollowa waterparticle.Euleriantechniquesmeasurethevelocityo fwaterataxed position.

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10.8.LAGRANGIANMEASUREMENTSOFCURRENTS 173 r1p0r2r2p4p3p2p5p1p0p1p2p3p4p5r1Warm-core ring Cold-core ring Figure10.12Shapeofconstant-pressuresurfaces p i andtheinterfacebetweentwowater massesofdensity 1 ; 2 iftheupperisrotatingfasterthanthelower. Left: Anticyclonic motion,warm-coreeddy. Right: Cyclonic,cold-coreeddy.Notethattheseasurface p 0 slopesuptowardthecenterofthewarm-corering,andthecon stant-densitysurfacesslope downtowardthecenter.Circlewithdotiscurrenttowardthe reader,circlewithcrossis currentawayfromthereader.AfterDefant(1961:466).BasicTechnique Lagrangiantechniquestrackthepositionofadrifterdesignedtofollowawaterparceleitheronthesurfaceordeepe rwithinthewater column.Themeanvelocityoversomeperiodiscalculatedfro mthedistance betweenpositionsatthebeginningandendoftheperioddivi dedbytheperiod. Errorsaredueto: 1.Thefailureofthedriftertofollowaparcelofwater.Weas sumethedrifter staysinaparcelofwater,butwindblowingonthesurfaceroa tofasurface driftercancausethedriftertomoverelativetothewater. 2.Errorsindeterminingthepositionofthedrifter.3.Samplingerrors.Driftersgoonlywheredrifterswanttog o.Anddrifters wanttogotoconvergentzones.Hencedrifterstendtoavoida reasof divergentrow. SatelliteTrackedSurfaceDrifters Surfacedriftersconsistofadrogueplusa roat.ItspositionisdeterminedbytheArgossystemonmeteo rologicalsatellites (SwensonandShaw,1990)orcalculatedfrom gps datarecordedcontinuously bythebuoyandrelayedtoshore. Argos-trackedbuoyscarryaradiotransmitterwithaveryst ablefrequency F 0 .Areceiveronthesatellitereceivesthesignalanddetermi nestheDoppler shift F asafunctionoftime t (gure10.13).TheDopplerfrequencyis F = dR dt F 0 c + F 0 where R isthedistancetothebuoy, c isthevelocityoflight.Thecloserthe buoytothesatellitethemorerapidlythefrequencychanges .When F = F 0 the rangeisaminimum.Thisisthetimeofclosestapproach,andt hesatellite's velocityvectorisperpendiculartothelinefromthesatell itetothebuoy.The timeofclosestapproachandthetimerateofchangeofDopple rfrequencyat thattimegivesthebuoy'spositionrelativetotheorbitwit ha180 ambiguity (BandBBinthegure).Becausetheorbitisaccuratelyknown ,andbecause thebuoycanbeobservedmanytimes,itspositioncanbedeter minedwithout ambiguity.

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174 CHAPTER10.GEOSTROPHICCURRENTS F0F t B BB E A2 A1 K Figure10.13SystemArgosusesradiosignalstransmittedfr omsurfacebuoystodetermine thepositionofthebuoy.Asatellitereceivesthesignalfro mthebuoyB.Thetimerateof changeofthesignal,theDopplershift F ,isafunctionofbuoypositionanddistancefrom thesatellite'strack.NotethatabuoyatBBwouldproduceth esameDopplershiftasthe buoyatB.TherecordedDopplersignalistransmittedtogrou ndstationsE,whichrelaysthe informationtoprocessingcentersAviacontrolstationsK. AfterDietrichetal.(1980:149). Theaccuracyofthecalculatedpositiondependsonthestabi lityofthefrequencytransmittedbythebuoy.TheArgossystemtracksbuoy swithanaccuracyof (1{2)km,collecting1{8positionsperdaydependingonlati tude. Because1cm/s 1km/day,andbecausetypicalvaluesofcurrentsintheocean rangefromonetotwohundredcentimeterspersecond,thisis anveryuseful accuracy.Holey-SockDrifters Themostwidelyused,satellite-trackeddrifteristhe holey-sockdrifter.Itconsistsofacylindricaldrogueofc loth1mindiameter by15mlongwith14largeholescutinthesides.Theweightoft hedrogueis supportedbyaroatset3mbelowthesurface.Thesubmergedro atistethered toapartiallysubmergedsurfaceroatcarryingtheArgostra nsmitter. ThebuoywasdesignedfortheSurfaceVelocityProgramandex tensively tested.Niileretal.(1995)carefullymeasuredtherateatw hichwindblowing onthesurfaceroatpullsthedroguethroughthewater,andth eyfoundthat thebuoymoves12 9 totherightofthewindataspeed U s =(4 : 32 0 : 67 )10 2 U 10 DAR +(11 : 04 1 : 63) D DAR (10.22) where DAR isthedragarearatiodenedasthedrogue'sdragareadivide dby thesumofthetether'sdragareaandthesurfaceroat'sdraga rea,and D isthe

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10.8.LAGRANGIANMEASUREMENTSOFCURRENTS 175 dierenceinvelocityofthewaterbetweenthetopofthecyli ndricaldrogueand thebottom.Drifterstypicallyhavea DAR of40,andthedrift U s < 1cm/sfor U 10 < 10m/s. ArgoFloats ThemostwidelyusedsubsurfaceroatsaretheArgoroats.The roats(gure10.14)aredesignedtocyclebetweenthesurfac eandsomepredetermineddepth.Mostroatsdriftfor10daysatadepthof1k m,sinkto2 km,thenrisetothesurface.Whilerising,theyproletempe ratureandsalinity asafunctionofpressure(depth).Theroatsremainsonthesu rfaceforafew hours,relaysdatatoshoreviatheArgossystem,thensinkag ainto1km.Each roatcarriesenoughpowertorepeatthiscycleforseveralye ars.Theroatthus measurescurrentsat1kmdepthanddensitydistributionint heupperocean. ThreethousandArgoroatsarebeingdeployedinallpartsoft heoceanforthe GlobalOceanDataAssimilationExperiment godae From Internal Reservoir 17cm 107 cm Evacuation Port Camping Disk Internal Reservoir Argos Transmitter Controller andCircuit BoardsMicroprocessorBattery PacksPump Battery Packs MotorFilter Hydraulic Pump Latching Valve Pressure Case Argos Antenna Port External Bladder One-WayCheck Valve 70 cm Wobble Plate Self-Priming Channel Piston To Internal Reservoir Inlet Channel Motor Latching Valve Descending State Ascending State Argos Antenna Figure10.14TheAutonomousLagrangianCirculationExplor er(ALACE)roatsisthe prototypefortheArgosroats.Itmeasurescurrentsatadept hof1km. Left: Schematicof thedrifter.Toascend,thehydraulicpumpmovesoilfromani nternalreservoirtoanexternal bladder,reducingthedrifter'sdensity.Todescend,thela tchingvalveisopenedtoallowoil torowbackintotheinternalreservoir.Theantennaismount edtotheendcap. Right: Expandedschematicofthehydraulicsystem.Themotorrotat esthewobbleplateactuating thepistonwhichpumpshydraulicoil.AfterDavisetal.(199 2). LagrangianMeasurementsUsingTracers Themostcommonmethodfor measuringtherowinthedeepoceanistotrackparcelsofwate rcontaining moleculesnotnormallyfoundintheocean.Thankstoatomicb ombtestsin the1950sandtherecentexponentialincreaseofchlororuor ocarbonsintheatmosphere,suchtracershavebeenintroducedintotheoceani nlargequantities. See x 13.4foralistoftracersusedinoceanography.Thedistribu tionoftrace

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176 CHAPTER10.GEOSTROPHICCURRENTS 0.6 5 0.8 0.2 0.6 0.4 0.6 0.8 0.8 0.8 0.8 0.8 3 2 1 2 1 1 2 2 3 4 5 6 5 H H L L 4 2 2 4 1 0.2 L 0.2 0.4 0.8 2 2 2 2 4 4 4 5 1 H 3 3 3 3 3Western North Atlantic (1981) Western North Atlantic (1972) 80o70o60o50o40o30o20o10o0o10o0 -6000 -5000 -4000 -3000 -2000 -1000Depth (m) 80o70o60o50o40o30o20o10o0o10o0 -6000 -5000 -4000 -3000 -2000 -1000Depth (m) Figure10.15Distributionoftritiumalongasectionthroug hthewesternbasinsinthenorth Atlantic,measuredin1972( Top )andremeasuredin1981( Bottom ).Unitsaretritium units,whereonetritiumunitis10 18 (tritiumatoms)/(hydrogenatoms)correctedtothe activitylevelsthatwouldhavebeenobservedon1January19 81.Comparethisguretothe densityintheoceanshowningure13.10.AfterToggweiler( 1994). moleculesisusedtoinferthemovementofthewater.Thetech niqueisespeciallyusefulforcalculatingvelocityofdeepwatermasses averagedoverdecades andformeasuringturbulentmixingdiscussedin x 8.4. Thedistributionoftracemoleculesiscalculatedfromthec oncentrationof themoleculesinwatersamplescollectedonhydrographicse ctionsandsurveys. Becausethecollectionofdataisexpensiveandslow,therea refewrepeated sections.Figure10.15showstwomapsofthedistributionof tritiuminthe northAtlanticcollectedin1972{1973bytheGeosecsProgra mandin1981,a

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10.8.LAGRANGIANMEASUREMENTSOFCURRENTS 177 Figure10.16Oceantemperatureandcurrentpatternsarecom binedinthis avhrr analysis. Surfacecurrentswerecomputedbytrackingthedisplacemen tofsmallthermalorsediment featuresbetweenapairofimages.Adirectionaledge-enhan cementlterwasappliedhereto denebetterthedierentwatermasses.Warmwaterisshaded darker.FromOceanImaging, SolanaBeach,California,withpermission.decadelater.Thesectionsshowthattritium,introducedin totheatmosphere duringtheatomicbombtestsintheatmosphereinthe1950sto 1972,penetrated todepthsbelow4kmonlynorthof40 Nby1971andto35 Nby1981.This showsthatdeepcurrentsareveryslow,about1.6mm/sinthis example. Becausethedeepcurrentsaresosmall,wecanquestionwhatp rocessare responsiblefortheobserveddistributionoftracers.Both turbulentdiusion andadvectionbycurrentscanttheobservations.Hence,do esgure10.15 givemeancurrentsinthedeepAtlantic,ortheturbulentdi usionoftritium?

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178 CHAPTER10.GEOSTROPHICCURRENTS Anotherusefultraceristhetemperatureandsalinityofthe water.Iwillconsidertheseobservationsin x 13.4whereIdescribethecoremethodforstudying deepcirculation.Here,Inotethat avhrr observationsofsurfacetemperature oftheoceanareanadditionalsourceofinformationaboutcu rrents. Sequentialinfraredimagesofsurfacetemperatureareused tocalculatethe displacementoffeaturesintheimages(gure10.16).Thete chniqueisespecially usefulforsurveyingthevariabilityofcurrentsnearshore .Landprovidesreferencepointsfromwhichdisplacementcanbecalculatedacc urately,andlarge temperaturecontrastscanbefoundinmanyregionsinsomese asons. Therearetwoimportantlimitations. 1.Manyregionshaveextensivecloudcover,andtheoceancan notbeseen. 2.Flowisprimarilyparalleltotemperaturefronts,andstr ongcurrentscan existalongfrontseventhoughthefrontmaynotmove.Itisth erefore essentialtotrackthemotionofsmalleddiesembeddedinthe rownear thefrontandnotthepositionofthefront. TheRubberDuckieSpill OnJanuary10,1992a12.2-mcontainerwith 29,000bathtubtoys,includingrubberducks(calledrubber duckiesbychildren) washedoverboardfromacontainershipat44.7 N,178.1 E(gure10.17).Ten 170o-170o-150o-130o-110o60o50o40o30o20oNorth America10 o Hawaii Bering Sea 1992-941990-92 1961-63 1984-86 1959-64 North Pacific Ocean Origin of Toy Spill Sitka Figure10.17Trajectoriesthatspilledrubberduckieswoul dhavefollowedhadtheybeen spilledonJanuary10ofdierentyears.Fivetrajectoriesw ereselectedfromasetof48 simulationsofthespilleachyearbetween1946and1993.The trajectoriesbeginonJanuary 10andendtwoyearslater(solidsymbols).Greysymbolsindi catepositionsonNovember 16oftheyearofthespill.Thegreycirclegivesthelocation whererubberducksrstcame ashorenearSitkain1992.Thecodeatlowerleftgivesthedat esofthetrajectories.After EbbesmeyerandIngraham(1994).

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10.9.EULERIANMEASUREMENTS 179 monthslaterthetoysbeganwashingashorenearSitka,Alask a.AsimilaraccidentonMay27,1990released80,000Nike-brandshoesat48 N,161 Wwhen waveswashedcontainersfromthe HansaCarrier Thespillsandeventualrecoveryofthetoysandshoesproved tobegood testsofanumericalmodelforcalculatingthetrajectories ofoilspillsdeveloped byEbbesmeyerandIngraham(1992,1994).Theycalculatedth epossibletrajectoriesofthespilledtoysusingtheOceanSurfaceCurrentSi mulations oscurs numericalmodeldrivenbywindscalculatedfromtheFleetNu mericalOceanographyCenter'sdailysea-levelpressuredata.Aftermodify ingtheircalculations byincreasingthewindagecoecientby50%forthetoysandby decreasing theirangleofderectionfunctionby5 ,theircalculationsaccuratelypredicted thearrivalofthetoysnearSitka,AlaskaonNovember16,199 2,tenmonths afterthespill.10.9EulerianMeasurements Eulerianmeasurementsaremadebymanydierenttypesofins trumentson shipsandmoorings. Moorings(gure10.18)areplacedonthesearoorbyships.Th emoorings maylastformonthstolongerthanayear.Becausethemooring mustbe deployedandrecoveredbydeep-searesearchships,thetech niqueisexpensive Wire 20m Chafe Chain 5m Chafe Chain 3m Nylon Nylon Wire Wire Wire Sea Surface Chafe Chain Instrument Instrument Instrument Instrument Instrument Wire to 2000m Backup-Recovery Section 6" or 7" glass spheres inhardhats on chain(Typ. 35 Spheres) Acoustic Release Anchor Tag Line-Nylon (Typ. 20m) Anchor (Typ. 3000lb.) Light, Radio, & Float Wire 20m Wire Wire Wire 20m Wire Wire 20m Wire Sea Surface Wire Anchor Tag LineNylon(Typ. 20m) Instrument Instrument Instrument Instrument 1/2" Chain Wire (Typ. 20m) Radio, Light, & Radio Float Top Buoyancy (Typ. 20 shperes) Intermediate Buoyancy (Typ. 10 spheres) Intermediate Buoyancy (Typ. 6 spheres) Backup Recovery Bouyancy (Typ. 15 spheres) 3m Chafe Chain Release 5m Chafe Chain Anchor (Typ. 2000lb.) Figure10.18 Left: AnexampleofasurfacemooringofthetypedeployedbytheWoo ds HoleOceanographicInstitution'sBuoyGroup. Right: Anexampleofasubsurfacemooring deployedbythesamegroup.AfterBaker(1981:410{411).

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180 CHAPTER10.GEOSTROPHICCURRENTS andfewmooringsarenowbeingdeployed.Thesubsurfacemoor ingshownon therightinthegureispreferredforseveralreasons:itdo esnothaveasurface roatthatisforcedbyhighfrequency,strong,surfacecurre nts;themooringisout ofsightanditdoesnotattracttheattentionofshermen;an dtheroatationis usuallydeepenoughtoavoidbeingcaughtbyshingnets.Mea surementsmade frommooringshaveerrorsdueto: 1.Mooringmotion.Subsurfacemooringsmoveleast.Surface mooringsin strongcurrentsmovemost,andareseldomused. 2.InadequateSampling.Mooringstendnottolastlongenoug htogiveaccurateestimatesofmeanvelocityorinterannualvariabili tyofthevelocity. 3.Foulingofthesensorsbymarineorganisms,especiallyin strumentsdeployedformorethanafewweeksclosetothesurface. Acoustic-DopplerCurrentMetersandProlers ThemostcommonEulerianmeasurementsofcurrentsaremadeusingsound.Typic ally,thecurrent meterorprolertransmitssoundinthreeorfournarrowbeam spointedin dierentdirections.Planktonandtinybubblesrerectthes oundbacktotheinstrument.TheDopplershiftofthererectedsoundisproport ionaltotheradial componentofthevelocityofwhateverrerectsthesound.Byc ombiningdata fromthreeorfourbeams,thehorizontalvelocityofthecurr entiscalculated assumingthebubblesandplanktondonotmoveveryfastrelat ivetothewater. Twotypesofacousticcurrentmetersarewidelyused.TheAco ustic-Doppler CurrentProler,calledthe adcp ,measurestheDopplershiftofsoundrerected fromwateratvariousdistancesfromtheinstrumentusingso undbeamsprojectedintothewaterjustasaradarmeasuresradioscattera safunctionofrange usingradiobeamsprojectedintotheair.Datafromthebeams arecombined togiveprolesofcurrentvelocityasafunctionofdistance fromtheinstrument.Onships,thebeamsarepointeddiagonallydownwardat 3{4horizontal anglesrelativetotheship'sbow.Bottom-mountedmetersus ebeamspointed diagonallyupward. Ship-boardinstrumentsarewidelyusedtoprolecurrentsw ithin200to 300moftheseasurfacewhiletheshipsteamsbetweenhydrogr aphicstations. Becauseashipmovesrelativetothebottom,theship'sveloc ityandorientation mustbeaccuratelyknown. gps datahaveprovidedthisinformationsincethe early1990s. Acoustic-Dopplercurrentmetersaremuchsimplerthanthe adcp .They transmitcontinuousbeamsofsoundtomeasurecurrentveloc ityclosetothe meter,notasafunctionofdistancefromthemeter.Theyarep lacedonmoorings andsometimesona ctd .Instrumentsonmooringsrecordvelocityasafunction oftimeformanydaysormonths.TheAanderaacurrentmeter( gure10.19) inthegureisanexampleofthistype.Instrumentson ctd sprolecurrents fromthesurfacetothebottomathydrographicstations.10.10ImportantConcepts 1.Pressuredistributionisalmostpreciselythehydrostat icpressureobtained byassumingtheoceanisatrest.Pressureisthereforecalcu latedvery

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10.10.IMPORTANTCONCEPTS 181 Mooring LineRCM AcousticDoppler Current SensorOptional Sensors: -Oxygen -Pressure -Temperature -TurbidityElectronics andData StorageMooring Line Figure10.19Anexampleofamooredacousticcurrentmeter,t he rcm9 producedby AanderaaInstruments.Twocomponentsofhorizontalveloci tyaremeasuredbyanacoustic system,andthedirectionsarereferencedtonorthusingani nternalHall-eectcompass.The electronics,datarecorder,andbatteryareinthepressure -resistanthousing.Accuracyis 0.15cm/sand 5 .(CourtesyAanderaaInstruments) accuratelyfrommeasurementsoftemperatureandconductiv ityasafunctionofpressureusingtheequationofstateofseawater.Hyd rographic datagivetherelative,internalpressureeldoftheocean. 2.Flowintheoceanisinalmostexactgeostrophicbalanceex ceptforrow intheupperandlowerboundarylayers.Coriolisforcealmos texactly balancesthehorizontalpressuregradient. 3.Satellitealtimetricobservationsoftheoceanictopogr aphygivethesurfacegeostrophiccurrent.Thecalculationoftopographyre quiresanaccurategeoid.Ifthegeoidisnotknown,altimeterscanmeasure thechange intopographyasafunctionoftime,whichgivesthechangein surface geostrophiccurrents. 4.Topex/PoseidonandJasonarethemostaccuratealtimeter systems,and theycanmeasurethetopographyorchangesintopographywit hanaccuracyof 4cm. 5.Hydrographicdataareusedtocalculatetheinternalgeos trophiccurrents intheoceanrelativetoknowncurrentsatsomelevel.Thelev elcanbe surfacecurrentsmeasuredbyaltimetryoranassumedlevelo fnomotion atdepthsbelow1{2km. 6.Flowintheoceanthatisindependentofdepthiscalledbar otropicrow,

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182 CHAPTER10.GEOSTROPHICCURRENTS rowthatdependsondepthiscalledbaroclinicrow.Hydrogra phicdata giveonlythebaroclinicrow. 7.Geostrophicrowcannotchangewithtime,sotherowintheo ceanisnot exactlygeostrophic.Thegeostrophicmethoddoesnotapply torowsat theequatorwheretheCoriolisforcevanishes. 8.Slopesofconstantdensityortemperaturesurfacesseeni nacross-section oftheoceancanbeusedtoestimatethespeedofrowthroughth esection. 9.Lagrangiantechniquesmeasurethepositionofaparcelof waterinthe ocean.Thepositioncanbedeterminedusingsurfacedrifter sorsubsurface roats,orchemicaltracerssuchastritium. 10.Euleriantechniquesmeasurethevelocityofrowpastapo intintheocean. Thevelocityoftherowcanbemeasuredusingmooredcurrentm etersor acousticvelocityprolersonships, ctd sormoorings.

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Chapter11WindDrivenOceanCirculationWhatdrivestheoceancurrents?Atrst,wemightanswer,the winds.Butif wethinkmorecarefullyaboutthequestion,wemightnotbeso sure.Wemight notice,forexample,thatstrongcurrents,suchastheNorth EquatorialCountercurrentsintheAtlanticandPacicOceangoupwind.Span ishnavigators inthe16thcenturynoticedstrongnorthwardcurrentsalong theFloridacoast thatseemedtobeunrelatedtothewind.Howcanthishappen?A nd,whyare strongcurrentsfoundoshoreofeastcoastsbutnotoshore ofwestcoasts? Answerstothequestionscanbefoundinaseriesofthreerema rkablepapers publishedfrom1947to1951.Intherst,HaraldSverdrup(19 47)showedthat thecirculationintheupperkilometerorsooftheoceanisdi rectlyrelatedto thecurlofthewindstressiftheCoriolisforcevarieswithl atitude.Henry Stommel(1948)showedthatthecirculationinoceanicgyres isasymmetricalso becausetheCoriolisforcevarieswithlatitude.Finally,W alterMunk(1950) addededdyviscosityandcalculatedthecirculationoftheu pperlayersofthe Pacic.Togetherthethreeoceanographerslaidthefoundat ionsforamodern theoryofoceancirculation.11.1Sverdrup'sTheoryoftheOceanicCirculation WhileSverdrupwasanalyzingobservationsofequatorialcu rrents,hecame upon(11.6)belowrelatingthecurlofthewindstresstomass transportwithin theupperocean.Toderivetherelationship,Sverdrupassum edthattherowis stationary,thatlateralfrictionandmolecularviscosity aresmall,thatnon-linear termssuchas u@u=@x aresmall,andthatturbulenceneartheseasurfacecanbe describedusingaverticaleddyviscosity.Healsoassumedt hatthewind-driven circulationvanishesatsomedepthofnomotion.Withthesea ssumptions,the horizontalcomponentsofthemomentumequationfrom8.9and 8.12become: @p @x = fv + @T xz @z (11.1a) 183

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184 CHAPTER11.WINDDRIVENOCEANCIRCULATION @p @y = fu + @T yz @z (11.1b) Sverdrupintegratedtheseequationsfromthesurfacetoade pth D equal toorgreaterthanthedepthatwhichthehorizontalpressure gradientbecomes zero.Hedened: @P @x = 0 Z D @p @x dz; @P @y = 0 Z D @p @y dz; (11.2a) M x 0 Z D u ( z ) dz;M y 0 Z D v ( z ) dz; (11.2b) where M x ;M y arethemasstransportsinthewind-drivenlayerextendingd own toanassumeddepthofnomotion. Thehorizontalboundaryconditionattheseasurfaceisthew indstress.At depth D thestressiszerobecausethecurrentsgotozero: T xz (0)= T x T xz ( D )=0 T yz (0)= T y T yz ( D )=0(11.3) where T x and T y arethecomponentsofthewindstress. Usingthesedenitionsandboundaryconditions,(11.1)bec ome: @P @x = fM y + T x (11.4a) @P @y = fM x + T y (11.4b) Inasimilarway,Sverdrupintegratedthecontinuityequati on(7.19)overthe sameverticaldepth,assumingtheverticalvelocityatthes urfaceandatdepth D arezero,toobtain: @M x @x + @M y @y =0(11.5) Dierentiating(11.4a)withrespectto y and(11.4b)withrespectto x ,subtracting,andusing(11.5)gives: M y = @T y @x @T x @y M y =curl z ( T )(11.6) where @f=@y istherateofchangeofCoriolisparameterwithlatitude,an d wherecurl z ( T )istheverticalcomponentofthecurlofthewindstress. Thisisanimportantandfundamentalresult|thenorthwardm asstransport ofwinddrivencurrentsisequaltothecurlofthewindstress .NotethatSverdrup allowed f tovarywithlatitude.Wewillseelaterthatthisisessentia l.

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11.1.SVERDRUP'STHEORYOFTHEOCEANICCIRCULATION 185 Streamlines of Mass Transport Boundaries of Counter Current Values of stream function, Y, given in units of 10 metric tons/sec 25 20 15 10 5 0 0 0 0 -10 -5 10 5 5 10 0 15 -15 15 -15 -5 -5 -10 -10 -20 -160o-150o-140o-130o-120o-110o-100o-90o-80o-10o0o10o20o30o Figure11.1StreamlinesofmasstransportintheeasternPac iccalculated fromSverdrup'stheoryusingmeanannualwindstress.After Reid(1948). Wecalculate from @f @y = 2ncos R (11.7) where R isearth'sradiusand islatitude. Overmuchoftheopenocean,especiallyinthetropics,thewi ndiszonal and @T y =@x issucientlysmallthat M y 1 @T x @y (11.8) Substituting(11.8)into(11.5),assuming varieswithlatitude,Sverdrupobtained: @M x @x = 1 2ncos @T x @y tan + @ 2 T x @y 2 R (11.9) Sverdrupintegratedthisequationfromanorth-southeaste rnboundaryat x =0,assumingnorowintotheboundary.Thisrequires M x =0at x =0. Then M x = x 2ncos @T x @y tan + @ 2 T x @y 2 R (11.10) where x isthedistancefromtheeasternboundaryoftheoceanbasin, and bracketsindicatezonalaveragesofthewindstress(gure1 1.1). Totesthistheory,Sverdrupcomparedtransportscalculate dfromknown windsintheeasterntropicalPacicwithtransportscalcul atedfromhydrographicdatacollectedbythe Carnegie and Bushnell inOctoberandNovember 1928,1929,and1939between34 Nand10 Sandbetween80 Wand160 W.

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186 CHAPTER11.WINDDRIVENOCEANCIRCULATION -7-5-3-1135 -10 -5 5 10 15 20 25 Latitude My 10 30 5070 -10 -30 -50 Latitude MxNorth Equatorial CurrentWestwardEastwardSouthwardNorthward -10 -5 5 10 15 20 25 North Equatorial Counter Current South Equatorial Current Figure11.2MasstransportintheeasternPaciccalculated fromSverdrup'stheoryusing observedwindswith11.8and11.10(solidlines)andpressur ecalculatedfromhydrographic datafromshipswith11.4(dots).Transportisintonspersec ondthroughasectiononemeter wideextendingfromtheseasurfacetoadepthofonekilomete r.Notethedierenceinscale between M y and M x .AfterReid(1948). Thehydrographicdatawereusedtocompute P byintegratingfromadepth of D = 1000m.Thecomparison,gures11.2,showednotonlythatthe transportscanbeaccuratelycalculatedfromthewind,buta lsothatthetheory predictswind-drivencurrentsgoingupwind.CommentsonSverdrup'sSolutions 1.Sverdrupassumedi)Theinternalrowintheoceanisgeostr ophic;ii)there isauniformdepthofnomotion;andiii)Ekman'stransportis correct.I examinedEkman'stheoryinChapter9,andthegeostrophicba lancein Chapter10.Weknowlittleaboutthedepthofnomotioninthet ropical Pacic. 2.Thesolutionsarelimitedtotheeastsideoftheoceanbeca use M x grows with x .Theresultcomesfromneglectingfrictionwhichwouldeven tually balancethewind-drivenrow.Nevertheless,Sverdrupsolut ionshavebeen usedfordescribingtheglobalsystemofsurfacecurrents.T hesolutions areappliedthroughouteachbasinallthewaytothewesternl imitofthe basin.There,conservationofmassisforcedbyincludingno rth-south currentsconnedtoathin,horizontalboundarylayer(gur e11.3). 3.Onlyoneboundaryconditioncanbesatised,norowthroug htheeastern boundary.Morecompletedescriptionsoftherowrequiremor ecomplete equations. 4.Thesolutionsdonotgivetheverticaldistributionofthe current. 5.Resultswerebasedondatafromtwocruisesplusaveragewi nddataassumingasteadystate.LatercalculationsbyLeetmaa,McCre ary,and

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11.1.SVERDRUP'STHEORYOFTHEOCEANICCIRCULATION 187 0 0 20 0 0 0 20 0 20 0 30 20 20 10 0 60 20 40 40o20o0o-20o-40o30o60o90o120o150o180o-150o-120o-90o-60o-30o0o Figure11.3Depth-integratedSverdruptransportappliedg loballyusingthewindstressfrom HellermanandRosenstein(1983).Contourintervalis10Sve rdrups.AfterTomczakand Godfrey(1994:46). Moore(1981)usingmorerecentwinddataproducessolutions withseasonalvariabilitythatagreeswellwithobservationsprovi dedthelevelof nomotionisat500m.Ifanotherdepthwerechosen,theresult sarenot asgood. 6.Wunsch(1996: x 2.2.3)aftercarefullyexaminingtheevidenceforaSverdrupbalanceintheoceanconcludedwedonothavesucientin formation totestthetheory.Hewrites Thepurposeofthisextendeddiscussionhasnotbeentodisap prove thevalidityofSverdrupbalance.Rather,itwastoemphasiz ethegap commonlyexistinginoceanographybetweenaplausibleanda ttractivetheoreticalideaandtheabilitytodemonstrateitsqua ntitative applicabilitytoactualoceanicrowelds.|Wunsch(1996). Wunsch,however,notes Sverdrup'srelationshipissocentraltotheoriesoftheoce ancirculationthatalmostalldiscussionsassumeittobevalidwith outany commentatallandproceedtocalculateitsconsequencesfor higherorderdynamics...itisdiculttooverestimatetheimporta nceof Sverdrupbalance.|Wunsch(1996). Butthegapisshrinking.Measurementsofmeanstressinthee quatorial Pacic(YuandMcPhaden,1999)showthattherowthereisinSv erdrup balance. StreamLines,PathLines,andtheStreamFunction Beforediscussing moreabouttheocean'swind-drivencirculation,weneedtoi ntroducetheconceptofstreamlinesandthestreamfunction(seeKundu,1990 :51&66). Ateachinstantintime,wecanrepresentaroweldbyavector velocityat eachpointinspace.Theinstantaneouscurvesthatareevery wheretangentto

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188 CHAPTER11.WINDDRIVENOCEANCIRCULATION x x + dx-u dy v dx y xy+dyy Figure11.4Volumetransportbetweenstreamlinesina two-dimensional,steadyrow.AfterKundu(1990:68). thedirectionofthevectorsarecalledthe streamlines oftherow.Iftherowis unsteady,thepatternofstreamlineschangewithtime. Thetrajectoryofaruidparticle,thepathfollowedbyaLagr angiandrifter, iscalledthe pathline inruidmechanics.Thepathlineisthesameasthestream lineforsteadyrow,andtheyaredierentforanunsteadyrow Wecansimplifythedescriptionoftwo-dimensional,incomp ressiblerowsby usingthe streamfunction denedby: u @ @y ;v @ @x ; (11.11) Thestreamfunctionisoftenusedbecauseitisascalarfromw hichthevector velocityeldcanbecalculated.Thisleadstosimplerequat ionsforsomerows. Streamfunctionsarealsousefulforvisualizingtherow.At eachinstant, therowisparalleltolinesofconstant .Thusiftherowissteady,thelinesof constantstreamfunctionarethepathsfollowedbywaterpar cels. Thevolumerateofrowbetweenanytwostreamlinesofasteady rowis d ,andthevolumerateofrowbetweentwostreamlines 1 and 2 isequal to 1 2 .Toseethis,consideranarbitraryline dx =( dx;dy )betweentwo streamlines(gure11.4).Thevolumerateofrowbetweenthe streamlinesis: vdx +( u ) dy = @ @x dx @ @y dy = d (11.12) andthevolumerateofrowbetweenthetwostreamlinesisnume ricallyequal tothedierenceintheirvaluesof Now,letsapplytheconceptstosatellite-altimetermapsof theoceanictopography.In x 10.3Iwrote(10.10) u s = g f @ @y v s = g f @ @x (11.13)

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11.2.WESTERNBOUNDARYCURRENTS 189 Comparing(11.13)with(11.11)itisclearthat = g f (11.14) andtheseasurfaceisastreamfunctionscaledby g=f .Turningtogure10.5,the linesofconstantheightarestreamlines,androwisalongth elines.Thesurface geostrophictransportisproportionaltothedierenceinh eight,independentof distancebetweenthestreamlines.Thesamestatementsappl ytogure10.9, exceptthatthetransportisrelativetotransportatthe100 0decibarssurface, whichisroughlyonekilometerdeep. Inadditiontothestreamfunction,oceanographersusethem ass-transport streamfunctiondenedby: M x @ @y ;M y @ @x (11.15) Thisisthefunctionshowningures11.2and11.3.11.2Stommel'sTheoryofWesternBoundaryCurrents AtthesametimeSverdrupwasbeginningtounderstandcircul ationinthe easternPacic,Stommelwasbeginningtounderstandwhywes ternboundary currentsoccurinoceanbasins.Tostudythecirculationint henorthAtlantic, Stommel(1948)usedessentiallythesameequationsusedbyS verdrup(11.1, 11.2,and11.3)butheaddedabottomstressproportionaltov elocityto(11.3): A z @u @z 0 = T x = F cos( y=b ) A z @u @z D = Ru (11.16a) A z @v @z 0 = T y =0 A z @v @z D = Rv (11.16b) where F and R areconstants. Stommelcalculatedsteady-statesolutionsforrowinarect angularbasin 0 y b ,0 x ofconstantdepth D lledwithwaterofconstantdensity. Hisrstsolutionwasforanon-rotatingearth.Thissolutio nhadasymmetric rowpatternwithnowesternboundarycurrent(gure11.5,le ft).Next,Stommelassumedaconstantrotation,whichagainledtoasymmetr icsolutionwith nowesternboundarycurrent.Finally,heassumedthattheCo riolisforcevaries withlatitude.Thisledtoasolutionwithwesternintensic ation(gure11.5, right).Stommelsuggestedthatthecrowdingofstreamlines inthewestindicatedthatthevariationofCoriolisforcewithlatitudemay explainwhytheGulf Streamisfoundintheocean.Wenowknowthatthevariationof Coriolisforce withlatitudeisrequiredfortheexistenceofthewesternbo undarycurrent,and thatothermodelsfortherowwhichusedierentformulation sforfriction,lead towesternboundarycurrentswithdierentstructure.Pedl osky(1987,Chapter5)givesaveryuseful,succinct,andmathematicallycle ardescriptionofthe varioustheoriesforwesternboundarycurrents.

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190 CHAPTER11.WINDDRIVENOCEANCIRCULATION Inthenextchapter,wewillseethatStommel'sresultscanal sobeexplained intermsofvorticity|windproducesclockwisetorque(vort icity),whichmust bebalancedbyacounterclockwisetorqueproducedatthewes ternboundary. -80 -60 -40 -20 -10 -20 -30 -40 1000 km 1000 km Wind Stress y x Figure11.5Streamfunctionforrowinabasinascalculatedb yStommel(1948). Left: Flow fornon-rotatingbasinorrowforabasinwithconstantrotat ion. Right: Flowwhenrotation varieslinearlywithy.11.3Munk'sSolution Sverdrup'sandStommel'sworksuggestedthedominantproce ssesproducing abasin-wide,wind-drivencirculation.Munk(1950)builtu ponthisfoundation, addinginformationfromRossby(1936)onlateraleddyvisco sity,toobtaina solutionforthecirculationwithinanoceanbasin.Munkuse dSverdrup'sidea ofaverticallyintegratedmasstransportrowingoveramoti onlessdeeperlayer. Thissimpliedthemathematicalproblem,anditismorereal istic.Theocean currentsareconcentratedintheupperkilometeroftheocea n,theyarenot barotropicandindependentofdepth.Toincludefriction,M unkusedlateral eddyfrictionwithconstant A H = A x = A y .Equations(11.1)become: 1 @p @x = fv + @ @z A z @u @z + A H @ 2 u @x 2 + A H @ 2 u @y 2 (11.17a) 1 @p @y = fu + @ @z A z @v @z + A H @ 2 v @x 2 + A H @ 2 v @y 2 (11.17b) Munkintegratedtheequationsfromadepth D tothesurfaceat z = z 0 whichissimilartoSverdrup'sintegrationexceptthatthes urfaceisnotat z =0. Munkassumedthatcurrentsatthedepth D vanish,that(11.3)applyat thehorizontalboundariesatthetopandbottomofthelayer, andthat A H is constant. Tosimplifytheequations,Munkusedthemass-transportstr eamfunction (11.15),andheproceededalongthelinesofSverdrup.Heeli minatedthepressuretermbytakingthe y derivativeof(11.17a)andthe x derivativeof(11.17b) toobtaintheequationformasstransport: A H r 4 | {z } Friction @ @x = curl z T | {z } SverdrupBalance (11.18)

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11.3.MUNK'SSOLUTION 191 10o0o20o30o40o60o50o 10o0o20o30o40o60o50o-10o 1 0 -1 10 10002000300040005000600070008000900010000 0 10002000 30004000 5000600070008000 9000 10000 0 dynes cm-2and dynes cm-3X 108dTx/dy tx jb ja jb ja jb jb jb ja ja Figure11.6 Left: Meanannualwindstress T x ( y )overthePacicandthecurlofthewind stress. b arethenorthernandsouthernboundariesofthegyres,where M y =0andcurl =0. 0 isthecenterofthegyre. UpperRight: Themasstransportstreamfunctionfor arectangularbasincalculatedbyMunk(1950)usingobserve dwindstressforthePacic. Contourintervalis10Sverdrups.Thetotaltransportbetwe enthecoastandanypoint x;y is ( x;y ).Thetransportintherelativelynarrownorthernsectioni sgreatlyexaggerated. LowerRight: North-Southcomponentofthemasstransport.AfterMunk(19 50). where r 4 = @ 4 @x 4 +2 @ 4 @x 2 @y 2 + @ 4 @y 4 (11.19) isthebiharmonicoperator.Equation(11.18)isthesameas( 11.6)withthe additionofthelateralfrictionterm A H .Thefrictiontermislargeclosetoa lateralboundarywherethehorizontalderivativesoftheve locityeldarelarge, anditissmallintheinterioroftheoceanbasin.Thusinthei nterior,the balanceofforcesisthesameasthatinSverdrup'ssolution. Equation(11.18)isafourth-orderpartialdierentialequ ation,andfour boundaryconditionsareneeded.Munkassumedtherowatabou ndaryis paralleltoaboundaryandthatthereisnoslipattheboundar y: bdry =0 ; @ @n bdry =0(11.20) where n isnormaltotheboundary.Munkthensolved(11.18)with(11. 20) assumingtherowwasinarectangularbasinextendingfrom x =0to x = r andfrom y = s to y =+ s .Hefurtherassumedthatthewindstresswaszonal

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192 CHAPTER11.WINDDRIVENOCEANCIRCULATION andintheform: T = a cos ny + b sin ny + c n = j=s;j =1 ; 2 ;::: (11.21) Munk'ssolution(gure11.6)showsthedominantfeaturesof thegyre-scale circulationinanoceanbasin.Ithasacirculationsimilart oSverdrup'sinthe easternpartsoftheoceanbasinandastrongwesternboundar ycurrentinthe west.Using A H =5 10 3 m 2 /sgivesaboundarycurrentroughly225kmwide withashapesimilartotherowobservedintheGulfStreamand theKuroshio. Thetransportinwesternboundarycurrentsisindependento f A H ,andit dependsonlyon(11.6)integratedacrossthewidthoftheoce anbasin.Hence, itdependsonthewidthoftheocean,thecurlofthewindstres s,and .Using thebestavailableestimatesofthewindstress,Munkcalcul atedthattheGulf Streamshouldhaveatransportof36SvandthattheKuroshios houldhavea transportof39Sv.Thevaluesareaboutonehalfofthemeasur edvaluesof therowavailabletoMunk.Thisisverygoodagreementconsid eringthewind stresswasnotwellknown. Recentrecalculationsshowgoodagreementexceptforthere gionoshoreof CapeHatteraswherethereisastrongrecirculation.Munk's solutionwasbased onwindstressaveragedaver5 squares.Thisunderestimatedthecurlofthe stress.LeetmaaandBunker(1978)usedmoderndragcoecien tand2 5 averagesofstresstoobtain32SvtransportintheGulfStrea m,avaluevery closetothatcalculatedbyMunk.11.4ObservedSurfaceCirculationintheAtlantic ThetheoriesbySverdrup,Munk,andStommeldescribeanidea lizedrow. Buttheoceanismuchmorecomplicated.Toseejusthowcompli catedthe rowisatthesurface,let'slookatawholeoceanbasin,theno rthAtlantic.I havechosenthisregionbecauseitisthebestobserved,andb ecausemid-latitude processesintheAtlanticaresimilartomid-latitudeproce ssesintheotherocean. Thus,forexample,IusetheGulfStreamasanexampleofawest ernboundary current. Let'sbeginwiththeGulfStreamtoseehowourunderstanding ofocean surfacecurrentshasevolved.Ofcourse,wecan'tlookatall aspectsoftherow. YoucanndoutmuchmorebyreadingTomczakandGodfrey(1994 )book on RegionalOceanography:AnIntroduction NorthAtlanticCirculation ThenorthAtlanticisthemostthoroughlystudiedoceanbasin.Thereisanextensivebodyoftheorytodescr ibemostaspectsof thecirculation,includingrowatthesurface,inthethermo cline,andatdepth, togetherwithanextensivebodyofeldobservations.Byloo kingatgures showingthecirculation,wecanlearnmoreaboutthecircula tion,andbylookingattheguresproducedoverthepastfewdecadeswecantra ceanevermore completeunderstandingofthecirculation. Let'sbeginwiththetraditionalviewofthetime-averageds urfacerowinthe northAtlanticbasedmostlyonhydrographicobservationso fthedensityeld

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11.4.OBSERVEDSURFACECIRCULATIONINTHEATLANTIC 193 0o20o40o30o0o-30o-60o-90o 60o 3 3 3 6 5 6 4 -4 14 10 16 12 12 16 26 55 10 2 Figure11.7SketchofthemajorsurfacecurrentsintheNorth Atlantic.Valuesaretransport inunitsof10 6 m 3 /s.AfterSverdrup,Johnson,andFleming(1942:g.187). (gure2.7).Itisacontemporaryviewofthemeancirculatio noftheentire oceanbasedonacenturyofmoreofobservations.Becausethe gureincludes alltheocean,perhapsitisoverlysimplied.So,let'slook thenatasimilar viewofthemeancirculationofjustthenorthAtlantic(gur e11.7). Thegureshowsabroad,basin-wide,midlatitudegyreaswee xpectfrom Sverdrup'stheorydescribedin x 11.1.Inthewest,awesternboundarycurrent, theGulfStream,completesthegyre.Inthenorthasubpolarg yreincludesthe Labradorcurrent.Anequatorialcurrentsystemandcounter currentarefound atlowlatitudeswithrowsimilartothatinthePacic.Note, however,the strongcrossequatorialrowinthewestwhichrowsalongthen ortheastcoast ofBraziltowardtheCaribbean. IfwelookcloserattherowinthefarnorthAtlantic(gure11 .8)weseethat therowisstillmorecomplex.Thisgureincludesmuchmored etailofaregion importantforsheriesandcommerce.Becauseitisbasedona nextensivebase ofhydrographicobservations,isthisreality?Forexample ,ifweweretodropa LagrangianroatintotheAtlanticwoulditfollowthestream lineshowninthe gure? Toanswerthequestion,let'slookatthetracksofa110buoys driftingon theseasurfacecompiledbyPhilRichardson(gure11.9top) .Thetracksgivea verydierentviewofthecurrentsinthenorthAtlantic.Iti shardtodistinguish therowfromthejumbleoflines,sometimescalledspaghetti tracks.Clearly,the rowisveryturbulent,especiallyintheGulfStream,afast, western-boundary current.Furthermore,theturbulenteddiesseemtohaveadi ameterofafew degrees.Thisismuchdierentthanturbulenceintheatmosp here.Intheair, thelargeeddiesarecalledstorms,andstormshavediameter sof10 {20 .Thus oceanic\storms"aremuchsmallerthanatmosphericstorms. Perhapswecanseethemeanrowifweaveragethedriftertrack s.What happenswhenRichardsonaveragesthetracksthrough2 2 boxes?The averages(gure11.9bottom)begintoshowsometrends,butn otethatinsome

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194 CHAPTER11.WINDDRIVENOCEANCIRCULATION -100 o -90 o -80 o -60 o -40 o70o40o30o20o0o-20 o-10o-20o-30o-40o-50o60o50o40o 15Gu50 30 35 10 25 10 4 Po 2 NaNaIr2 2 7 1 2 EgEiNiNgEg8 4 2 3 3 2 SbNcWg1 1 4 La4 Wg80o Figure11.8Detailedschematicofnamedcurrentsinthenort hAtlantic.Thenumbersgive thetransportinunitson10 6 m 3 =s fromthesurfacetoadepthof1km. Eg :EastGreenland Current; Ei :EastIcelandCurrent; Gu :GulfStream; Ir :IrmingerCurrent; La :Labrador Current; Na :NorthAtlanticCurrent; Nc :NorthCapeCurrent; Ng :NorwegianCurrent; Ni :NorthIcelandCurrent; Po :PortugalCurrent; Sb :SpitsbergenCurrent; Wg :West GreenlandCurrent.Numberswithinsquaresgivesinkingwat erinunitson10 6 m 3 =s .Solid Lines:Warmercurrents.BrokenLines:Coldercurrents.Aft erDietrichetal.(1980:542). regions,suchaseastoftheGulfStream,adjacentboxeshave verydierent means,somehavingcurrentsgoingindierentdirections.T hisindicatesthe rowissovariable,thattheaverageisnotstable.Fortyormo reobservations donotyieldsastablemeanvalue.Overall,Richardsonndst hatthekinetic energyoftheeddiesis8to37timeslargerthanthekineticen ergyofthemean row.Thusoceanicturbulenceisverydierentthanlaborato ryturbulence.In thelab,themeanrowistypicallymuchfasterthantheeddies FurtherworkbyRichardson(1993)basedonsubsurfacebuoys freelydrifting atdepthsbetween500and3,500m,showsthatthecurrentexte ndsdeepbelow thesurface,andthattypicaleddydiameteris80km.GulfStreamRecirculationRegion Ifwelookcloselyatgure11.7we seethatthetransportintheGulfStreamincreasesfrom26Sv intheFlorida Strait(betweenFloridaandCuba)to55SvoshoreofCapeHat teras.Later measurementsshowedthetransportincreasesfrom30Svinth eFloridaStrait to150Svnear40 N. Theobservedincrease,andthelargetransportoHatteras, disagreewith transportscalculatedfromSverdrup'stheory.Theorypred ictsamuchsmaller

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11.4.OBSERVEDSURFACECIRCULATIONINTHEATLANTIC 195 200200200200200200200200200 -80o-70o-60o-50o-40o-30o-20o-10o0oSpeed (cm/sec) 050100m 20o30o40o50o60o 20o30o40o50o60o -80o-70o-60o-50o-40o-30o-20o-10o0o Figure11.9 Top Tracksof110driftingbuoysdeployedinthewesternnorthAt lantic. Bottom Meanvelocityofcurrentsin2 2 boxescalculatedfromtracksabove.Boxes withfewerthan40observationswereomitted.Lengthofarro wisproportionaltospeed. Maximumvaluesarenear0.6m/sintheGulfStreamnear37 N71 W.AfterRichardson (1981).maximumtransportof30Sv,andthatthemaximumoughttobene ar28 N. Nowwehaveaproblem:Whatcausesthehightransportsnear40 N?

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196 CHAPTER11.WINDDRIVENOCEANCIRCULATION 42o38o34o-74o-78o-70o-66o 42o38o34o-74o-78o-70o-66o 42o38o34o-74o-78o-70o-66o 42o38o34o-74o-78o-70o-66oFeb. 15 Feb. 26-27 Feb. 23-4 Mar. 9-10200 m 200 m 200 m 200 mXBT sectionAB Figure11.10GulfStreammeandersleadtotheformationofas pinningeddy,aring. Noticethatringshaveadiameterofabout1 .AfterRingGroup(1981). Niiler(1987)summarizesthetheoryandobservations.Firs t,thereisno hydrographicevidenceforalargeinruxofwaterfromtheAnt illesCurrent thatrowsnorthoftheBahamasandintotheGulfStream.Thisr ulesoutthe possibilitythattheSverdruprowislargerthanthecalcula tedvalue,andthat therowbypassestheGulfofMexico.Therowseemstocomeprim arilyfrom theGulfStreamitself.Therowbetween60 Wand55 Wistothesouth.The waterthenrowssouthandwest,andrejoinstheStreambetwee n65 Wand 75 W.Thus,therearetwosubtropicalgyres:asmallgyredirect lysouthofthe Streamcenteredon65 W,calledtheGulfStreamrecirculationregion,andthe broad,wind-drivengyrenearthesurfaceseeningure11.7t hatextendsallthe waytoEurope. TheGulfStreamrecirculationcarriestwotothreetimesthe massofthe broadergyre.Currentmetersdeployedintherecirculation regionshowthat therowextendstothebottom.Thisexplainswhytherecircul ationisweak inthemapscalculatedfromhydrographicdata.Currentscal culatedfromthe densitydistributiongiveonlythebarocliniccomponentof therow,andthey missthecomponentthatisindependentofdepth,thebarotro piccomponent. TheGulfStreamrecirculationisdrivenbythepotentialene rgyofthesteeply slopingthermoclineattheGulfStream.Thedepthofthe27.0 0sigma-theta ( )surfacedropsfrom250metersnear41 Ningure10.8to800mnear 38 NsouthoftheStream.EddiesintheStreamconvertthepotent ialenergy

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11.5.IMPORTANTCONCEPTS 197 200m200mOct 23 Dec 7 Oct 23 Oct 23 Dec 7 B B A Oct 23 Dec 7 Dec 7 Oct 25200mDec 7 Oct 27 Dec 7 Oct 23Dec 7Bermuda the mainstream and therings on October 23the rings on October 25the mainstream and therings on December 7200m depth contour 200m -85o-80o-75o-70o-65o-60o45o25o40o35o30o Key New York Boston Washington Cape Hatteras Jacksonville NOVA SCOTIA Miami Figure11.11SketchofthepositionoftheGulfStream,warmc ore,andcoldcoreeddies observedininfraredimagesoftheseasurfacecollectedbyt heinfraredradiometeron noaa -5 inOctoberandDecember1978.AfterTolmazin(1985:91).tokineticenergythroughbaroclinicinstability.Theinst abilityleadstoan interestingphenomena:negativeviscosity.TheGulfStrea macceleratesnot decelerates.Itactsasthoughitwereundertheinruenceofa negativeviscosity. Thesameprocessdrivesthejetstreamintheatmosphere.The steeplysloping densitysurfaceseparatingthepolarairmassfrommid-lati tudeairmassesat theatmosphere'spolarfrontalsoleadstobaroclinicinsta bility.Formoreon thistopicseeStarr's(1968)bookon PhysicsofNegativeViscosityPhenomena Let'slookatthisprocessintheGulfStream(gure11.10).T hestrong currentshearintheStreamcausestherowtobegintomeander .Themeander intensies,andeventuallytheStreamthrowsoaring.Thos eonthesouth sidedriftsouthwest,andeventuallymergewiththestreams everalmonthslater (gure11.11).Theprocessoccursallalongtherecirculati onregion,andsatellite imagesshownearlyadozenorsoringsoccurnorthandsouthof thestream (gure11.11).11.5ImportantConcepts 1.Thetheoryforwind-driven,geostrophiccurrentswasrs toutlinedina seriesofpapersbySverdrup,Stommel,andMunkbetween1947 and1951.

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198 CHAPTER11.WINDDRIVENOCEANCIRCULATION 2.Theyshowedthatrealisticcurrentscanbecalculatedonl yiftheCoriolis parametervarieswithlatitude. 3.Sverdrupshowedthatthecurlofthewindstressdrivesano rthwardmass transport,andthatthiscanbeusedtocalculatecurrentsin theocean awayfromwesternboundarycurrents. 4.Stommelshowedthatwesternboundarycurrentsarerequir edforrowto circulatearoundanoceanbasinwhentheCoriolisparameter varieswith latitude. 5.Munkshowedhowtocombinethetwosolutionstocalculatet hewinddrivengeostrophiccirculationinanoceanbasin.Inallcas es,thecurrent isdrivenbythecurlofthewindstress. 6.Theobservedcirculationintheoceanisveryturbulent.M anyyearsof observationsmayneedtobeaveragedtogethertoobtainasta blemapof themeanrow. 7.TheGulfStreamisaregionofbaroclinicinstabilityinwh ichturbulence acceleratesthestream.ThiscreatesaGulfStreamrecircul ation.Transportsintherecirculationregionaremuchlargerthantrans portscalculated fromtheSverdrup-Munktheory.

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Chapter12VorticityintheOceanMostoftheruidrowswithwhichwearefamiliar,frombathtub stoswimmingpools,arenotrotating,ortheyarerotatingsoslowlyt hatrotationisnot importantexceptmaybeatthedrainofabathtubaswaterisle tout.Asa result,wedonothaveagoodintuitiveunderstandingofrota tingrows.Inthe ocean,rotationandtheconservationofvorticitystrongly inruencerowover distancesexceedingafewtensofkilometers.Theconsequen cesoftherotation leadstoresultswehavenotseenbeforeinourday-to-daydea lingswithruids. Forexample,didyouaskyourselfwhythecurlofthewindstre ssleadstoa masstransportinthenorth-southdirectionandnotintheea st-westdirection? Whatisspecialaboutnorth-southmotion?Inthischapter,I willexploresome oftheconsequencesofrotationforrowintheocean.12.1DenitionsofVorticity Insimplewords,vorticityistherotationoftheruid.Thera teofrotation canbedenedvariousways.Considerabowlofwatersittingo natableina laboratory.Thewatermaybespinninginthebowl.Inadditio ntothespinning ofthewater,thebowlandthelaboratoryarerotatingbecaus etheyareona rotatingearth.Thetwoprocessesareseparateandleadtotw otypesofvorticity. PlanetaryVorticity Everythingonearth,includingtheocean,theatmosphere,andbowlsofwater,rotateswiththeearth.Thisrota tionisthe planetary vorticity f .Itistwicethelocalrateofrotationofearth: f 2nsin (radians/s)=2sin (cycles/day) (12.1) PlanetaryvorticityistheCoriolisparameterIusedearlie rtodiscussrowinthe ocean.Itisgreatestatthepoleswhereitistwicetherotati onrateofearth. Notethatthevorticityvanishesattheequatorandthatthev orticityinthe southernhemisphereisnegativebecause isnegative. RelativeVorticity Theoceanandatmospheredonotrotateatexactlythe samerateasearth.Theyhavesomerotationrelativetoearth duetocurrents 199

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200 CHAPTER12.VORTICITYINTHEOCEAN andwinds. Relativevorticity isthevorticityduetocurrentsintheocean. Mathematicallyitis: curl z V = @v @x @u @y (12.2) where V =( u;v )isthehorizontalvelocityvector,andwherewehaveassume d thattherowistwo-dimensional.Thisistrueiftherowexten dsoverdistances greaterthanafewtensofkilometers. istheverticalcomponentofthethreedimensionalvorticityvector ,anditissometimeswritten z ispositivefor counter-clockwiserotationviewedfromabove.Thisisthes amesenseasearth's rotationinthenorthernhemisphere. NoteonSymbols Symbolscommonlyusedinonepartofoceanographyoften haveverydierentmeaninginanotherpart.HereIuse forvorticity,butin Chapter10,Iused tomeantheheightoftheseasurface.Icoulduse z for relativevorticity,but isalsocommonlyusedtomeanfrequencyinradiansper second.Ihavetriedtoeliminatemostconfusinguses,butth edualuseof is onewewillhavetolivewith.Fortunately,itshouldn'tcaus emuchconfusion. Forarigidbodyrotatingatraten,curl V =2n.Ofcourse,therowdoes notneedtorotateasarigidbodytohaverelativevorticity. Vorticitycan alsoresultfromshear.Forexample,atanorth/southwester nboundaryinthe ocean, u =0, v = v ( x )and = @v ( x ) =@x isusuallymuchsmallerthan f ,anditisgreatestattheedgeoffastcurrents suchastheGulfStream.Toobtainsomeunderstandingofthes izeof ,consider theedgeoftheGulfStreamoCapeHatteraswherethevelocit ydecreasesby 1m/sin100kmattheboundary.Thecurlofthecurrentisappro ximately(1 m/s)/(100km)=0.14cycles/day=1cycle/week.Henceeventh islargerelative vorticityisstillalmostseventimessmallerthan f .Amoretypicalvaluesof relativevorticity,suchasthevorticityofeddies,isacyc lepermonth. AbsoluteVorticity Thesumoftheplanetaryandrelativevorticityiscalled absolutevorticity : AbsoluteVorticity ( + f ) (12.3) Wecanobtainanequationforabsolutevorticityintheocean bymanipulatingtheequationsofmotionforfrictionlessrow.Webegi nwith: Du Dt fv = 1 @p @x (12.4a) Dv Dt + fu = 1 @p @y (12.4b) Ifweexpandthesubstantialderivative,andifwesubtract @=@y of(12.4a)from @=@x of(12.4b)toeliminatethepressureterms,weobtainafters omealgebraic manipulations: D Dt ( + f )+( + f ) @u @x + @v @y =0 (12.5)

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12.1.DEFINITIONSOFVORTICITY 201 Surface Z H (x,y,t)b (x,y) Reference Level (z=0) Bott om Figure12.1Sketchofruidrowusedforderivingconservatio nof potentialvorticity.AfterCushman-Roisin(1994:55). Inderiving(12.5)weused: Df Dt = @f @t + u @f @x + v @f @y = v because f isindependentoftime t andeastwarddistance x PotentialVorticity Therotationrateofacolumnofruidchangesasthe columnisexpandedorcontracted.Thischangesthevorticit ythroughchanges in .Toseehowthishappens,considerbarotropic,geostrophic rowinan oceanwithdepth H ( x;y;t ),where H isthedistancefromtheseasurfacetothe bottom.Thatis,weallowthesurfacetohavetopography(gu re12.1). Integratingthecontinuityequation(7.19)fromthebottom tothetopofthe oceangives(Cushman-Roisin,1994): @u @x + @v @y Z b + H b dz + w b + H b =0(12.6) where b isthetopographyofthebottom,and H isthedepthofthewater.Notice that @u=@x and @v=@y areindependentof z becausetheyarebarotropic,and thetermscanbetakenoutsidetheintegral. Theboundaryconditionsrequirethatrowatthesurfaceandt hebottombe alongthesurfaceandthebottom.Thustheverticalvelociti esatthetopand thebottomare: w ( b + H )= @ ( b + H ) @t + u @ ( b + H ) @x + v @ ( b + H ) @y (12.7) w ( b )= u @ ( b ) @x + v @ ( b ) @y (12.8) whereweused @b=@t =0becausethebottomdoesnotmove,and @H=@z =0. Substituting(12.7)and(12.8)into(12.6)weobtain @u @x + @v @y + 1 H DH Dt =0

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202 CHAPTER12.VORTICITYINTHEOCEAN Substitutingthisinto(12.5)gives: D Dt ( + f ) ( + f ) H DH Dt =0 whichcanbewritten: D Dt + f H =0 Thequantitywithintheparenthesesmustbeconstant.Itisc alled potential vorticity .Potentialvorticityisconservedalongaruidtrajectory : PotentialVorticity= + f H (12.9) Forbaroclinicrowinacontinuouslystratiedruid,thepot entialvorticity canbewritten(Pedlosky,1987: x 2.5): = + f r (12.10) where isanyconservedquantityforeachruidelement.In,particu lar,if = then: = + f @ @z (12.11) assumingthehorizontalgradientsofdensityaresmallcomp aredwiththeverticalgradients,agoodassumptioninthethermocline.Inmost oftheinteriorof theocean, f and(12.11)iswritten(Pedlosky,1996,eq3.11.2): = f @ @z (12.12) Thisallowsthepotentialvorticityofvariouslayersofthe oceantobedetermined directlyfromhydrographicdatawithoutknowledgeoftheve locityeld. 12.2ConservationofVorticity Theangularmomentumofanyisolatedspinningbodyisconser ved.The spinningbodycanbeaneddyintheoceanortheearthinspace. Ifthethe spinningbodyisnotisolated,thatis,ifitislinkedtoanot herbody,thenangular momentumcanbetransferredbetweenthebodies.Thetwobodi esneednot beinphysicalcontact.Gravitationalforcescantransferm omentumbetween bodiesinspace.IwillreturntothistopicinChapter17when Idiscusstidesin theocean.Here,let'slookatconservationofvorticityina spinningocean. Frictionisessentialforthetransferofmomentuminaruid. Frictiontransfersmomentumfromtheatmospheretotheoceanthroughtheth in,frictional, Ekmanlayerattheseasurface.Frictiontransfersmomentum fromtheocean tothesolidearththroughtheEkmanlayeratthesearoor.Fri ctionalongthe sidesofsub-seamountainsleadstopressuredierencesone ithersideofthe mountainwhichcausesanotherkindofdragcalled formdrag .Thisisthesame dragthatcauseswindforceoncarsmovingathighspeed.Inth evastinterior oftheocean,however,therowisfrictionless,andvorticit yisconserved.Such arowissaidtobe conservative

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12.2.CONSERVATIONOFVORTICITY 203 ConservationofPotentialVorticity Theconservationofpotentialvorticity coupleschangesindepth,relativevorticity,andchangesi nlatitude.Allthree interact. 1.Changesinthedepth H oftherowchangesintherelativevorticity.The conceptisanalogouswiththewaygureskatersdecreasesth eirspinby extendingtheirarmsandlegs.Theactionincreasestheirmo mentof inertiaanddecreasestheirrateofspin(gure12.2). W1 W2 > W1U H Figure12.2Sketchoftheproductionofrelativevorticityb ythechangesintheheightofa ruidcolumn.Astheverticalruidcolumnmovesfromlefttori ght,verticalstretchingreduces themomentofinertiaofthecolumn,causingittospinfaster 2.Changesinlatituderequireacorrespondingchangein .Asacolumnof watermovesequatorward, f decreases,and mustincrease(gure12.3). Ifthisseemssomewhatmysterious,vonArx(1962)suggestsw econsidera barrelofwateratrestatthenorthpole.Ifthebarrelismove dsouthward, thewaterinitretainstherotationithadatthepole,anditw illappear torotatecounterclockwiseatthenewlatitudewhere f issmaller. Figure12.3Angularmomentumtendstobeconservedascolumn sofwaterchangelatitude. Thischangestherelativevorticityofthecolumns.Aftervo nArx(1962:110).

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204 CHAPTER12.VORTICITYINTHEOCEAN 12.3InruenceofVorticity Theconceptofconservationofpotentialvorticityhasfarr eachingconsequences,anditsapplicationtoruidrowintheoceangivesad eeperunderstandingofoceancurrents.FlowTendstobeZonal Intheocean f tendstobemuchlargerthan and thus f=H =constant.Thisrequiresthattherowinanoceanofconstant depth bezonal.Ofcourse,depthisnotconstant,butingeneral,cu rrentstendtobe east-westratherthannorthsouth.Windmakessmallchanges in ,leadingto asmallmeridionalcomponenttotherow(seegure11.3).TopographicSteering Barotropicrowsaredivertedbysearoorfeatures. Considerwhathappenswhenarowthatextendsfromthesurfac etothebottom encountersasub-searidge(gure12.4).Asthedepthdecrea ses, + f must alsodecrease,whichrequiresthat f decrease,andtherowisturnedtowardthe equator.Thisiscalled topographicsteering .Ifthechangeindepthissuciently large,nochangeinlatitudewillbesucienttoconservepot entialvorticity,and therowwillbeunabletocrosstheridge.Thisiscalled topographicblocking x z y H(x) x Figure12.4Barotropicrowoverasub-searidgeisturnedequ atorward toconservepotentialvorticity.AfterDietrichetal.(198 0:333). WesternBoundaryCurrents Thebalanceofvorticityprovidesanalternate explanationfortheexistenceofwesternboundarycurrents .Considerthegyrescalerowinanoceanbasin(gure12.5),sayinthenorthAtla nticfrom10 N to50 N.ThewindblowingovertheAtlanticaddsnegativevorticit y .As thewaterrowsaroundthegyre,thevorticityofthegyremust remainnearly constant,elsetherowwouldspinfasterorslower.Overall, thenegativevorticity inputbythewindmustbebalancedbyasourceofpositivevort icity. Throughoutmostofthebasinthenegativevorticityinputby thewindis balancedbyanincreaseinrelativevorticity.Astherowmov essouthward throughoutthebasin, f decreasesand mustincreaseaccordingto(12.9)because H ,thedepthofthewind-drivencirculation,doesnotchangem uch. Thebalancebreaksdown,however,inthewestwheretherowre turnsnorthward.Inthewest, f increases, decreases,andasourceofpositivevorticityis needed.Thepositivevorticity b isproducedbythewesternboundaryboundary current.

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12.4.VORTICITYANDEKMANPUMPING 205 Westerlies Trades z t z b Westerlies Trades z t z t + + +z z z z Figure12.5Thebalanceofpotentialvorticitycanclarifyw hywesternboundarycurrentsare necessary. Left: Vorticityinputbythewind balancesthechangeinrelativevorticity in theeastastherowmovessouthwardand f decreases.Thetwodonotbalanceinthewest where mustdecreaseastherowmovesnorthwardand f increases. Right: Vorticityinthe westisbalancedbyrelativevorticity b generatedbyshearinthewesternboundarycurrent. 12.4VorticityandEkmanPumping Rotationplacesanotherveryinterestingconstraintonthe geostrophicrow eld.Tohelpunderstandtheconstraints,let'srstconsid errowinaruid withconstantrotation.Thenwewilllookintohowvorticity constrainsthe rowofaruidwithrotationthatvarieswithlatitude.Anunde rstandingofthe constraintsleadstoadeeperunderstandingofSverdrup'sa ndStommel'sresults discussedinthelastchapter.Fluiddynamicsonthe f Plane:theTaylor-ProudmanTheorem The inruenceofvorticityduetoearth'srotationismoststriki ngforgeostrophicrow ofaruidwithconstantdensity 0 onaplanewithconstantrotation f = f 0 FromChapter10,thethreecomponentsofthegeostrophicequ ations(10.4)are: fv = 1 0 @p @x (12.13a) fu = 1 0 @p @y (12.13b) g = 1 0 @p @z (12.13c) andthecontinuityequations(7.19)is: 0= @u @x + @v @y + @w @z (12.13d) Takingthe z derivativeof(12.13a)andusing(12.13c)gives: f 0 @v @z = 1 0 @ @z @p @x = @ @x 1 0 @p @z = @g @x =0 f 0 @v @z =0 ) @v @z =0

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206 CHAPTER12.VORTICITYINTHEOCEAN Similarly,fortheu-componentofvelocity(12.13b).Thus, theverticalderivative ofthehorizontalvelocityeldmustbezero. @u @z = @v @z =0 (12.14) Thisisthe Taylor-ProudmanTheorem ,whichappliestoslowlyvaryingrows inahomogeneous,rotating,inviscidruid.Thetheoremplac esstrongconstraintsontherow: Ifthereforeanysmallmotionbecommunicatedtoarotatingr uid theresultingmotionoftheruidmustbeoneinwhichanytwopa rticles originallyinalineparalleltotheaxisofrotationmustrem ainso,except forpossiblesmalloscillationsaboutthatposition|Taylo r(1921). Hence,rotationgreatlystienstherow!Geostrophicrowca nnotgoovera seamount,itmustgoaroundit.Taylor(1921)explicitlyder ived(12.14)and (12.16)below.Proudman(1916)independentlyderivedthes ametheorembut notasexplicitly. Furtherconsequencesofthetheoremcanbeobtainedbyelimi natingthe pressuretermsfrom(12.13a&12.13b)toobtain: @u @x + @v @y = @ @x 1 f 0 0 @p @y + @ @y 1 f 0 0 @p @x (12.15a) @u @x + @v @y = 1 f 0 0 @ 2 p @x@y + @ 2 p @x@y (12.15b) @u @x + @v @y =0 (12.15c) Becausetheruidisincompressible,thecontinuityequatio n(12.13d)requires @w @z =0(12.16) Furthermore,because w =0attheseasurfaceandatthesearoor,ifthe bottomislevel,therecanbenoverticalvelocityonan f {plane.Notethatthe derivationof(12.16)didnotrequirethatdensitybeconsta nt.Itrequiresonly slowmotioninafrictionless,rotatingruid.FluidDynamicsontheBetaPlane:EkmanPumping If(12.16)istrue, therowcannotexpandorcontractintheverticaldirection, anditisindeedas rigidasasteelbar.Therecanbenogradientofverticalvelo cityinanocean withconstantplanetaryvorticity.Howthencanthediverge nceoftheEkman transportattheseasurfaceleadtoverticalvelocitiesatt hesurfaceoratthe baseoftheEkmanlayer?Theanswercanonlybethatoneofthec onstraints usedinderiving(12.16)mustbeviolated.Oneconstraintth atcanberelaxed istherequirementthat f = f 0

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12.4.VORTICITYANDEKMANPUMPING 207 Considerthenrowonabetaplane.If f = f 0 + y ,then(12.15a)becomes: @u @x + @v @y = 1 f 0 @ 2 p @x@y + 1 f 0 @ 2 p @x@y f 1 f 0 @p @x (12.17) f @u @x + @v @y = v (12.18) wherewehaveused(12.13a)toobtain v intheright-handsideof(12.18). Usingthecontinuityequation,andrecallingthat y f 0 f 0 @w G @z = v (12.19) wherewehaveusedthesubscript G toemphasizethat(12.19)appliestothe ocean'sinterior,geostrophicrow.ThusthevariationofCo riolisforcewithlatitudeallowsverticalvelocitygradientsinthegeostrophi cinterioroftheocean, andtheverticalvelocityleadstonorth-southcurrents.Th isexplainswhySverdrupandStommelbothneededtodotheircalculationsona -plane. EkmanPumpingintheOcean InChapter9,wesawthatthecurlofthe windstress T producedadivergenceoftheEkmantransportsleadingtoa verticalvelocity w E (0)atthetopoftheEkmanlayer.InChapter9wederived w E (0)= curl T f (12.20) whichis(9.30b)where isdensityand f istheCoriolisparameter.Becausethe verticalvelocityattheseasurfacemustbezero,theEkmanv erticalvelocity mustbebalancedbyaverticalgeostrophicvelocity w G (0). w E (0)= w G (0)= curl T f (12.21) Ekmanpumping( w E (0))drivesaverticalgeostrophiccurrent( w G (0))in theocean'sinterior.Butwhydoesthisproducethenorthwar dcurrentcalculatedbySverdrup(11.6)?PeterNiiler(1987:16)givesanex planation. Letuspostulatethereexistsadeeplevelwherehorizontala ndvertical motionofthewaterismuchreducedfromwhatitisjustbelowt hemixed layer[gure12.6] ::: Alsoletusassumethatvorticityisconservedthere (ormixingissmall)andtherowissoslowthataccelerations overthe earth'ssurfacearemuchsmallerthanCoriolisacceleratio ns.Insucha situationacolumnofwaterofdepth H willconserveitsspinperunit volume, f=H (relativetothesun,paralleltotheearth'saxisofrotatio n). Avortexcolumnwhichiscompressedfromthetopbywind-forc edsinking ( H decreases)andwhosebottomisinrelativelyquiescentwate rwould tendtoshortenandslowitsspin.Thusbecauseofthecurvedo cean surfaceithastomovesouthward(orextenditscolumn)toreg ainits spin.Therefore,thereshouldbeamassiverowofwateratsom edepth

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208 CHAPTER12.VORTICITYINTHEOCEAN EkmanLayer North PoleWesterliesTradeWindsGeostrophicInterior Figure12.6Ekmanpumpingthatproducesadownwardvelocity atthebaseoftheEkman layerforcestheruidintheinterioroftheoceantomovesout hward.Seetextforwhythis happens.AfterNiiler(1987). belowthesurfacetothesouthinareaswherethesurfacelaye rsproduce asinkingmotionandtothenorthwhererisingmotionisprodu ced.This phenomenonwasrstmodeledcorrectlybySverdrup(1947)(a fterhe wrote\ocean")andgivesadynamicallyplausibleexplanati onofhowwind producesdeepercirculationintheocean. PeterRhines(1984)pointsoutthattherigidcolumnofwater tryingtoescapethesqueezingimposedbytheatmosphereescapesbymovi ngsouthward. Thesouthwardvelocityisabout5,000timesgreaterthanthe verticalEkman velocity.EkmanPumping:AnExample Nowlet'sseehowEkmanpumpingdrives geostrophicrowinsaythecentralnorthPacic(gure12.7) wherethecurlof thewindstressisnegative.Westerliesinthenorthdriveas outhwardtransport, thetradesinthesouthdriveanorthwardtransport.Theconv ergingEkman transportsmustbebalancedbydownwardgeostrophicveloci ty(12.21). Becausethewaternearthesurfaceiswarmerthanthedeeperw ater,the verticalvelocityproducesapoolofwarmwater.Muchdeeper intheocean,the wind-drivengeostrophiccurrentmustgotozero(Sverdrup' shypothesis)and thedeeppressuregradientsmustbezero.Asaresult,thesur facemustdome upwardbecauseacolumnofwarmwaterislongerthanacolumno fcoldwater havingthesameweight(theymusthavethesameweight,other wise,thedeep pressurewouldnotbeconstant,andtherewouldbeadeephori zontalpressure gradient).Suchadensitydistributionproducesnorth-sou thpressuregradients atmiddepthsthatmustbebalancedbyeast-westgeostrophic currents.Inshort, thedivergenceoftheEkmantransportsredistributesmassw ithinthefrictionless interioroftheoceanleadingtothewind-drivengeostrophi ccurrents. Nowlet'scontinuetheideatoincludetheentirenorthPaci ctoseehow

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12.4.VORTICITYANDEKMANPUMPING 209 Trade WindsWesterliesNorth EastEkman Layer r1r6r2r3r5r4 Figure12.7WindsattheseasurfacedriveEkmantransportst otherightofthewindin thisnorthernhemisphereexample(boldarrowsinshadedEkm anlayer).Theconverging Ekmantransportsdrivenbythetradesandwesterliesdrives adownwardgeostrophicrow justbelowtheEkmanlayer(boldverticalarrows),leadingt odownwardbowingconstant densitysurfaces i .Geostrophiccurrentsassociatedwiththewarmwateraresh ownbybold arrows.AfterTolmazin(1985:64). Trades Westerlies Easterlies ConvergenceConvergence Divergence Convergence Divergence H L H L H NECC NEC AK Sea Surface Height and Geostrophic Currents Ekman Transports curl t > 0 curl t < 0 curl t > 0 curl t < 0 curl t < 0 Mean Wind Speed (m/s) -404 8060 4020 Figure12.8Anexampleofhowwindsproducegeostrophiccurr entsrunningupwind.Ekman transportsduetowindsinthenorthPacic( Left )leadtoEkmanpumping( Center ), whichsetsupnorth-southpressuregradientsintheupperoc ean.Thepressuregradientsare balancedbytheCoriolisforceduetoeast-westgeostrophic currents( Right ).Horizontal linesindicateregionswherethecurlofthezonalwindstres schangessign. AK :Alaskan Current, NEC :NorthEquatorialCurrent, NECC :NorthEquatorialCounterCurrent.

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210 CHAPTER12.VORTICITYINTHEOCEAN windsproducecurrentsrowingupwind.Theexamplewillgive adeeperunderstandingofSverdrup'sresultswediscussedin x 11.1. Figure12.8showsshowsthemeanzonalwindsinthePacic,to getherwith thenorth-southEkmantransportsdrivenbythezonalwinds. Noticethatconvergenceoftransportleadstodownwelling,whichproduces athicklayerof warmwaterintheupperkilometerofthewatercolumn,andhig hsealevel. Figure12.6isasketchofthecrosssectionoftheregionbetw een10 Nand 60 N,anditshowsthepoolofwarmwaterintheupperkilometerce nteredon 30 N.Conversely,divergenttransportsleadstolowsealevel. Themeannorthsouthpressuregradientsassociatedwiththehighsandlows arebalancedbythe Coriolisforceofeast-westgeostrophiccurrentsintheupp erocean(shownat therightinthegure).12.5ImportantConcepts 1.Vorticitystronglyconstrainsoceandynamics.2.Vorticityduetoearth'srotationismuchgreaterthanoth ersourcesof vorticity. 3.TaylorandProudmanshowedthatverticalvelocityisimpo ssibleina uniformlyrotatingrow.Theoceanisrigidinthedirectionp aralleltothe rotationaxis.HenceEkmanpumpingrequiresthatplanetary vorticity varywithlatitude.ThisexplainswhySverdrupandStommelf oundthat realisticoceaniccirculation,whichisdrivenbyEkmanpum ping,requires that f varywithlatitude. 4.Thecurlofthewindstressaddsrelativevorticitytocent ralgyresofeach oceanbasin.Forsteadystatecirculationinthegyre,theoc eanmustlose vorticityinwesternboundarycurrents. 5.PositivewindstresscurlleadstodivergentrowintheEkm anlayer.The ocean'sinteriorgeostrophiccirculationadjuststhrough anorthwardmass transport. 6.Conservationofabsolutevorticityinanoceanwithconst antdensityleads totheconservationofpotentialvorticity.Thuschangesin depthinan oceanofconstantdensityrequireschangesoflatitudeofth ecurrent.

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Chapter13DeepCirculationintheOceanThedirectforcingoftheoceaniccirculationbywinddiscus sedinthelastfew chaptersisstrongestintheupperkilometerofthewatercol umn.Belowa kilometerliesthevastwatermassesoftheoceanextendingt odepthsof4{5km. Thewateriseverywherecold,withapotentialtemperaturel essthan4 C.The watermassisformedwhencold,densewatersinksfromthesur facetogreat depthsathighlatitudes.Itspreadsoutfromtheseregionst olltheocean basins.Deepmixingeventuallypullsthewaterupthroughth ethermocline overlargeareasoftheocean.Itisthisupwardmixingthatdr ivesthedeep circulation.Thevastdeepoceanisusuallyreferredtoasth e abyss ,andthe circulationasthe abyssalcirculation Thedensestwaterattheseasurface,waterthatisdenseenou ghtosinkto thebottom,isformedwhenfrigidairblowsacrosstheoceana thighlatitudes inwinterintheAtlanticbetweenNorwayandGreenlandandne arAntarctica. Thewindcoolsandevaporateswater.Ifthewindiscoldenoug h,seaiceforms, furtherincreasingthesalinityofthewaterbecauseiceisf resherthanseawater. Bottomwaterisproducedonlyinthesetworegions.Cold,den sewaterisformed intheNorthPacic,butitisnotsaltyenoughtosinktothebo ttom. Atmidandlowlatitudes,thedensity,eveninwinter,issuc ientlylowthat thewatercannotsinkmorethanafewhundredmetersintotheo cean.Theonly exceptionaresomeseas,suchastheMediterraneanSea,wher eevaporationis sogreatthatthesalinityofthewaterissucientlygreatfo rthewatertosink tothebottomoftheseseas.Iftheseseascanexchangewaterw iththeopen ocean,thewatersformedinwinterintheseasmixeswiththew aterintheopen oceananditspreadsoutalongintermediatedepthsintheope nocean. 13.1DeningtheDeepCirculation Manytermshavebeenusedtodescribethedeepcirculation.T heyinclude: 1) abyssalcirculation ,2) thermohalinecirculation 3) meridionaloverturning 211

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212 CHAPTER13.DEEPCIRCULATIONINTHEOCEAN circulation ,and4) globalconveyor .Thetermthermohalinecirculationwas oncewidelyused,butithasalmostentirelydisappearedfro mtheoceanographic literature(ToggweilerandRussell,2008).Itisnolongeru sedbecauseitisnow clearthattherowisnotdensitydriven,andbecausetheconc epthasnotbeen clearlydened(Wunsch,2002b). Themeridionaloverturningcirculationisbetterdened.I tisthezonal averageoftherowplottedasafunctionofdepthandlatitude .Plotsofthe circulationshowwhereverticalrowisimportant,buttheys hownoinformation abouthowcirculationinthegyresinruencestherow. FollowingWunsch(2002b),Idenethedeepcirculationasth ecirculation ofmass.Ofcourse,themasscirculationalsocarriesheat,s alt,oxygen,and otherproperties.Butthecirculationoftheotherproperti esisnotthesameas themasstransport.Forexample,Wunschpointsoutthatthen orthAtlantic importsheatbutexportsoxygen.13.2ImportanceoftheDeepCirculation Thedeepcirculationcarriesheat,salinity,oxygen,CO 2 ,andotherproperties fromhighlatitudesinwintertolowerlatitudesthroughout theworld.Thishas veryimportantconsequences. 1.Thecontrastbetweenthecolddeepwaterandthewarmsurfa cewaters determinesthestraticationoftheocean,whichstronglyi nruencesocean dynamics. 2.Thevolumeofdeepwaterisfarlargerthanthevolumeofsur facewater.Althoughcurrentsinthedeepoceanarerelativelyweak ,theyhave transportscomparabletothesurfacetransports. 3.Theruxesofheatandothervariablescarriedbythedeepci rculation inruencesearth'sheatbudgetandclimate.Theruxesvaryfr omdecades tocenturiestomillennia,andthisvariabilitymodulatesc limateoversuch timeintervals.Theoceanmaybetheprimarycauseofvariabi lityover timesrangingfromyearstodecades,anditmayhavehelpedmo dulate ice-ageclimate. Twoaspectsofthedeepcirculationareespeciallyimportan tforunderstanding earth'sclimateanditspossibleresponsetoincreasedcarb ondioxideCO 2 in theatmosphere:i)theabilityofcoldwatertostoreCO 2 andheatabsorbed fromtheatmosphere,andii)theabilityofdeepcurrentstom odulatetheheat transportedfromthetropicstohighlatitudes.TheoceanasaReservoirofCarbonDioxide Theoceanaretheprimary reservoirofreadilyavailableCO 2 ,animportantgreenhousegas.Theocean contain40,000GtCofdissolved,particulate,andlivingfo rmsofcarbon.The landcontains2,200GtC,andtheatmospherecontainsonly75 0GtC.Thusthe oceanhold50timesmorecarbonthantheair.Furthermore,th eamountof newcarbonputintotheatmospheresincetheindustrialrevo lution,150GtC, islessthantheamountofcarboncycledthroughthemarineec osysteminve

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13.2.IMPORTANCEOFTHEDEEPCIRCULATION 213 years.(1GtC=1gigatonofcarbon=10 12 kilogramsofcarbon.)Carbonate rockssuchaslimestone,theshellsofmarineanimals,andco ralareother,much larger,reservoirs.Butthiscarbonislockedup.Itcannotb eeasilyexchanged withcarboninotherreservoirs. MoreCO 2 dissolvesincoldwaterthaninwarmwater.Justimagineshak ing andopeningahotcanofCoke TM .TheCO 2 fromahotcanwillspewoutfar fasterthanfromacoldcan.Thusthecolddeepwaterintheoce anisthemajor reservoirofdissolvedCO 2 intheocean. NewCO 2 isreleasedintotheatmospherewhenfossilfuelsandtreesa re burned.Veryquickly,48%oftheCO 2 releasedintotheatmospheredissolves intotheocean(Sabineetal,2004),muchofwhichendsupdeep intheocean. Forecastsoffutureclimatechangedependstronglyonhowmu chCO 2 is storedintheoceanandforhowlong.Iflittleisstored,orif itisstoredand laterreleasedintotheatmosphere,theconcentrationinth eatmospherewill change,modulatingearth'slong-waveradiationbalance.H owmuchandhow longCO 2 isstoredintheoceandependsonthetransportofCO 2 bythedeep circulationandonthenetruxofcarbondepositedonthesear oor.Theamount thatdissolvesdependsonthetemperatureofthedeepwater, thestoragetime inthedeepoceandependsontherateatwhichdeepwaterisrep lenished,and thedepositiondependsonwhetherthedeadplantsandanimal sthatdropto thesearoorareoxidized.Increasedventilationofdeeplay ers,andwarmingof thedeeplayerscouldreleaselargequantitiesofthegastot heatmosphere. Thestorageofcarbonintheoceanalsodependsonthedynamic sofmarine ecosystems,upwelling,andtheamountofdeadplantsandani malsstoredin sediments.ButIwon'tconsidertheseprocesses.OceanicTransportofHeat Theoceancarryabouthalftheheatoutof thetropicsneededtomaintainearth'stemperature.Heatca rriedbytheGulf StreamandthenorthAtlanticdriftkeepsthefarnorthAtlan ticicefree,andit helpswarmEurope.Norway,at60 NisfarwarmerthansouthernGreenland ornorthernLabradoratthesamelatitude.Palmtreesgrowon thewestcoast ofIreland,butnotinNewfoundlandwhichisfurthersouth. WallyBroecker(1987),workingatLamont-DohertyGeophysi calObservatoryofColumbiaUniversity,callstheoceaniccomponentof theheat-transport systemthe GlobalConveyorBelt .Thebasicideaisthatsurfacecurrentscarry heattothefarnorthAtlantic(gure13.1).Therethesurfac ewaterreleases heatandwatertotheatmosphere,anditbecomessucientlyc old,salty,and densethatitsinkstothebottomintheNorwegianandGreenla ndSeas.Itthen rowssouthwardincold,bottomcurrents.Someofthewaterre mainsonthe surfaceandreturnstothesouthincoolsurfacecurrentssuc hastheLabrador CurrentandPortugalCurrent(seegure11.8).Richardson( 2008)haswritten averyusefulpapersurveyingourunderstandingofthegloba lconveyorbelt. ThedeepbottomwaterfromthenorthAtlanticismixedupward inother regionsandocean,andeventuallyitmakesitswaybacktothe GulfStreamand theNorthAtlantic.Thusmostofthewaterthatsinksintheno rthAtlantic mustbereplacedbywaterfromthefarsouthAtlantic.Asthis surfacewater

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214 CHAPTER13.DEEPCIRCULATIONINTHEOCEAN 60 o E 70 o N 80 o N 30 o E 60 o N 50 o N 30 o W 40 o N 0 o 60 o W Greenland SeaNorwegian SeaLabrador Sea Greenland CanadaLabradorEUROPE Norway North Atlantic Current Norwegian Current Figure13.1Thesurface(narrowdashes)anddeep(widedashe s)currentsinthenorth Atlantic.TheNorthAtlanticCurrentbringswarmwaternort hwardwhereitcools.Some sinksandreturnssouthwardasacold,deep,western-bounda rycurrent.Somereturns southwardatthesurface.FromWoodsHoleOceanographicIns titution. movesnorthwardacrosstheequatorandeventuallyintotheG ulfStream,it carriesheatoutofthesouthAtlantic. SomuchheatispullednorthwardbytheformationofnorthAtl anticbottom waterinwinterthatheattransportintheAtlanticisentire lynorthward,evenin thesouthernhemisphere(gure5.11).Muchofthesolarheat absorbedbythe tropicalAtlanticisshippednorthtowarmEuropeandthenor thernhemisphere. Imaginethenwhatmighthappenifthesupplyofheatisshuto .Iwillget backtothattopicinthenextsection. WecanmakeacrudeestimateoftheimportanceofthenorthAtl anticsurface anddeepcirculationfromacalculationbasedonwhatweknow aboutwatersin theAtlanticcompiledbyBillSchmitz(1996)inhiswonderfu lsummaryofhis life'swork.TheGulfStreamcarries40Svof18 Cwaternorthward.Ofthis, 14Svreturnsouthwardinthedeepwesternboundarycurrenta tatemperature of2 C.Thewatermustthereforelose0.9petawatts(1petawatt=1 0 15 watt) inthenorthAtlanticnorthof24 N.Althoughthecalculationisverycrude,it isremarkablyclosetothevalueof1 : 2 0 : 2petawattsestimatedmuchmore carefullybyRintoulandWunsch(1991). Notethatifthewaterremainedonthesurfaceandreturnedas aneastern boundarycurrent,itwouldbefarwarmerthanthedeepcurren twhenitreturned

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13.2.IMPORTANCEOFTHEDEEPCIRCULATION 215 southward.Hence,theheattransportwouldbemuchreduceda nditwould probablynotkeepthefarnorthAtlanticicefree. Theproductionofbottomwaterisinruencedbythesurfacesa linityand windsinthenorthAtlantic(ToggweilerandRussell,2008). Itisalsoinruenced bytherateofupwellingduetomixinginotheroceanicareas. First,let'slook attheinruenceofsalinity. Saltiersurfacewatersformdenserwaterinwinterthanless saltywater.At rstyoumaythinkthattemperatureisalsoimportant,butat highlatitudes waterinalloceanbasinsgetscoldenoughtofreeze,soalloc eanproduce 2 C wateratthesurface.Ofthis,onlythemostsaltywillsink,a ndthesaltiest waterisintheAtlanticandundertheiceonthecontinentals helvesaround Antarctica. Theproductionofbottomwaterisremarkablysensitivetosm allchanges insalinity.Rahmstorf(1995),usinganumericalmodelofth emeridionaloverturningcirculation,showedthata 0.1Svvariationoftherowoffreshwater intothenorthAtlanticcanswitchonorothedeepcirculati onof14Sv.Ifthe deep-waterproductionisshutoduringtimesoflowsalinit y,the1petawattof heatmayalsobeshuto.WeaverandHillaire-Marcel(2004)p ointoutthatthe shutdownoftheproductionofbottomwaterisunlikely,andi fitdidhappen, itwouldleadtoacolderEurope,notanewiceage,becauseoft hehigher concentrationsofCO 2 nowintheatmosphere. Iwrite maybeshuto becausetheoceanisaverycomplexsystem.We don'tknowifotherprocesseswillincreaseheattransporti fthedeepcirculation isdisturbed.Forexample,thecirculationatintermediate depthsmayincrease whendeepcirculationisreduced. Theproductionofbottomwaterisalsoremarkablysensitive tosmallchanges inmixinginthedeepocean.MunkandWunsch(1998)calculate that2.1TW (terawatts=10 12 watts)arerequiredtodrivethedeepcirculation,andthatt his smallsourceofmechanicalmixingdrivesapolewardheatrux of2000TW.Some oftheenergyformixingcomesfromwindswhichcanproducetu rbulentmixing throughouttheocean.Someenergycomesfromthedissipatio noftidalcurrents, whichdependonthedistributionofthecontinents.Someoft heenergycomes fromtherowofdeepwaterpastthemid-oceanridgesystem.Th usduring thelasticeage,whensealevelwasmuchlower,tides,tidalc urrents,tidal dissipation,winds,anddeepcirculationalldieredfromp resentvalues. RoleoftheOceaninIce-AgeClimateFluctuations Whatmighthappen iftheproductionofdeepwaterintheAtlanticisshuto?Inf ormationcontained inGreenlandandAntarcticicesheets,innorthAtlanticsed iments,andinlake sedimentsprovideimportantclues. SeveralicecoresthroughtheGreenlandandAntarcticicesh eetsprovidea continuousrecordofatmosphericconditionsoverGreenlan dandAntarcticaextendingbackmorethan700,000yearsbeforethepresentinso mecores.Annual layersinthecorearecountedtogetage.Deeperinthecore,w hereannual layersarehardtosee,ageiscalculatedfromdepthandfromd ustlayersfrom well-datedvolcaniceruptions.Oxygen-isotoperatiosoft heicegiveairtemper-

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216 CHAPTER13.DEEPCIRCULATIONINTHEOCEAN atureattheglaciersurfacewhentheicewasformed.Deuteri umconcentrations giveocean-surfacetemperatureatthemoisturesourceregi on.Bubblesinthe icegiveatmosphericCO 2 andmethaneconcentration.Pollen,chemicalcomposition,andparticlesgiveinformationaboutvolcanicer uptions,windspeed, anddirection.Thicknessofannuallayersgivessnowaccumu lationrates.And isotopesofsomeelementsgivesolarandcosmicrayactivity (Alley,2000). Coresthroughdeep-seasedimentsinthenorthAtlanticmade bytheOcean DrillingProgramgiveinformationabouti)surfaceanddeep temperaturesand salinityatthelocationofabovethecore,ii)theproductio nofnorthAtlantic deepwater,iii)icevolumeinglaciers,andiv)productiono ficebergs.Ice-sheet anddeep-seacoreshaveallowedreconstructionsofclimate forthepastfew hundredthousandyears. 1.Theoxygen-isotopeanddeuteriumrecordsintheicecores showabrupt climatevariabilitymanytimesoverthepast700,000years. Manytimes duringthelasticeagetemperaturesnearGreenlandwarmedr apidlyover periodsof1{100years,followedbygradualcoolingoverlon gerperiods. (Dansgaardetal,1993).Forexample, 11 ; 500yearsago,temperaturesoverGreenlandwarmedby 8 Cin40yearsinthreesteps,each spanning5years(Alley,2000).Suchabruptwarmingiscalle daDansgaard/Oeschgerevent.Otherstudieshaveshownthatmuchof thenorthernhemispherewarmedandcooledinphasewithtemperatures calculated fromtheicecore. 2.Theclimateofthepast8,000yearswasconstantwithveryl ittlevariability.Ourperceptionofclimatechangeisthusbasedonhighly unusual circumstances.Allofrecordedhistoryhasbeenduringaper iodofwarm andstableclimate. 3.HartmutHeinrichandcolleagues(Bondetal.1992),study ingthesedimentsinthenorthAtlanticfoundperiodswhencoarsemateri alwasdepositedonthebottominmidocean.Onlyicebergscancarrysu chmaterialouttosea,andthendindicatedtimeswhenlargenumber soficebergswerereleasedintothenorthAtlantic.Thesearenowca lledHeinrich events. 4.ThecorrelationofGreenlandtemperaturewithicebergpr oductionisrelatedtothedeepcirculation.Whenicebergsmelted,thesur geoffresh waterincreasedthestabilityofthewatercolumnshuttingo theproductionofnorthAtlanticDeepWater.Theshut-oofdeep-water formation greatlyreducedthenorthwardtransportofwarmwaterintot henorth Atlantic,producingverycoldnorthernhemisphereclimate (gure13.2). Themeltingoftheicepushedthepolarfront,theboundarybe tweencold andwarmwaterinthenorthAtlanticfurthersouththanitspr esentposition.Thelocationofthefront,andthetimeitwasatdieren tpositions canbedeterminedfromanalysisofbottomsediments. 5.Whenthemeridionaloverturningcirculationshutsdown, heatnormally carriedfromthesouthAtlantictothenorthAtlanticbecome savailableto

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13.2.IMPORTANCEOFTHEDEEPCIRCULATION 217 Polarfront 3 2 4 1 Holocene Interglacial Eemian Interglacial 1100ky bp130ky bp Icemargin Glacial 20ky bp40ky bp Today 10ky bp ColderCold ColderWarmer ColderWarmer Figure13.2Periodicsurgesoficebergsduringthelasticea geappeartohavemodulated temperaturesofthenorthernhemispherebyloweringthesal inityofthefarnorthAtlanticand reducingthemeridionaloverturningcirculation.Datafro mcoresthroughtheGreenlandice sheet(1),deep-seasediments(2,3),andalpine-lakesedim ents(4)indicatethat: Left: During recenttimesthecirculationhasbeenstable,andthepolarf rontwhichseparateswarmand coldwatermasseshasallowedwarmwatertopenetratebeyond Norway. Center: During thelasticeage,periodicsurgesoficebergsreducedsalini tyandreducedthemeridional overturningcirculation,causingthepolarfronttomoveso uthwardandkeepingwarmwater southofSpain. Right: Similarructuationsduringthelastinterglacialappearto havecaused rapid,largechangesinclimate.The Bottom plotisaroughindicationoftemperaturein theregion,butthescalesarenotthesame.AfterZahn(1994) 3 4 1 2 North Atlantic SalinityNorth Atlantic Surface Temperature Figure13.3Themeridional-overturningcirculationinthe northAtlanticmaybestablenear 2 and 4 .But,theswitchingfromawarm,saltyregimetoacold,fresh regimeandback hashysteresis.Thismeansthatasthewarmsaltyoceaninani nitialstate 1 freshens,and becomesmorefreshthan 2 itquicklyswitchestoacold,freshstate 3 .Whentheareaagain becomessalty,itmustmovepaststate 4 beforeitcanswitchbackto 1

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218 CHAPTER13.DEEPCIRCULATIONINTHEOCEAN warmthesouthernhemisphere.Thisresultsinaclimate'sea -saw'between northernandsouthernhemispheres. 6.Theswitchingonandoofthedeepcirculationhaslargehy steresis(gure 13.3).Thecirculationhastwostablestates.Therstisthe present circulation.Inthesecond,deepwaterisproducedmostlyne arAntarctica, andupwellingoccursinthefarnorthPacic(asitdoestoday )andinthe farnorthAtlantic.Oncethecirculationisshuto,thesyst emswitchesto thesecondstablestate.Thereturntonormalsalinitydoesn otcausethe circulationtoturnon.Surfacewatersmustbecomesaltiert hanaverage fortherststatetoreturn(Rahmstorf,1995). 7.HeinricheventsseemtoprecedethelargestDansgaard/Oe schgerevents (StockerandMarchal,2000).Here'swhatseemstohappen.Th eHeinrich eventshutsotheAtlanticdeepcirculationwhichleadstoa verycold northAtlantic(Martratetal,2007).Thisisfollowedabout 1000years laterbyaDansgaard/Oeschgereventwithrapidwarming. 8.Dansgaard/Oeschger{Heinrichtandemeventshaveglobal inruence,and theyarerelatedtowarmingeventsseeninAntarcticicecore s.Temperatureschangesinthetwohemispheresareoutofphase.WhenGr eenland warms,Antarcticacools.RecentdatafromtheEuropeanProj ectforIce CoringinAntarctica( epica )showsthatintheperiodbetween20,000and 90,000yearsago,40%ofthevarianceintheGreenlandtemper aturedata canbeexplainedbyAntarctictemperaturedata(Steig,2006 ). 9.Aweakenedversionofthisprocesswithaperiodofabout10 00yearsmay bemodulatingpresent-dayclimateinthenorthAtlantic,an ditmayhave beenresponsiblefortheLittleIceAgefrom1100to1800. Therelationshipbetweenvariationsinsalinity,airtempe rature,deep-water formation,andtheatmosphericcirculationisnotyetunder stood.Forexample, wedon'tknowifchangesintheatmosphericcirculationtrig gerchangesinthe meridionaloverturningcirculation,orifchangesintheme ridionaloverturning circulationtriggerchangesintheatmosphericcirculatio n(Braueretal,2008). Furthermore,surgesmayresultfromwarmertemperaturesca usedbyincreased watervaporfromthetropics(agreenhousegas)orfromanint ernalinstabilityof alargeicesheet.Wedoknow,however,thatclimatecanchang everyabruptly, andthatcirculationinthenorthernhemispherehasaveryse nsitivethreshold, thatwhencrossed,causeslargechangesinthecirculationp attern. Forexample,Steensen(2008)foundthat11,704,12,896,an d14,694years before2000 ad thetemperatureofthesourcewaterforGreenlandprecipita tion warmed2{4 Cin1{3years.Thisindicatesaveryrapidreorganizationof the atmosphericcirculationathighlatitudesinthenorthernh emisphereandashift inthelocationofthesourceregion.Duringtheearliesteve ntairtemperature overGreenlandwarmedby 10 Cin3years.Atthelaterevents,airtemperatureoverGreenlandchangedmoreslowly,over60to200year s.Braueretal (2008)foundanabruptchangeinstorminessoverGermanyata lmostexactly thesametime,12,679yearsago.

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13.3.THEORYFORTHEDEEPCIRCULATION 219 13.3TheoryfortheDeepCirculation Stommel,Arons,andFallerinaseriesofpapersfrom1958to1 960described asimpletheoryoftheabyssalcirculation(Stommel1958;St ommel,Arons,and Faller,1958;StommelandArons,1960).Thetheorydiereds ogreatlyfrom whatwasexpectedthatStommelandAronsdevisedlaboratory experiments withrotatingruidstoconrmedtheirtheory.Thetheoryfor thedeepcirculationhasbeenfurtherdiscussedbyMarotzke(2000)andMun kandWunsch (1998). TheStommel,Arons,Fallertheoryisbasedonthreefundamen talideas: 1.Cold,deepwaterissuppliedbydeepconvectionatafewhig h-latitude locationsintheAtlantic,notablyintheIrmingerandGreen landSeasin thenorthandtheWeddellSeainthesouth. 2.Uniformmixingintheoceanbringsthecold,deepwaterbac ktothe surface. 3.Thedeepcirculationisstrictlygeostrophicintheinter ioroftheocean, andthereforepotentialvorticityisconserved. Noticethatthedeepcirculationisdrivenbymixing,notbyt hesinkingof coldwaterathighlatitudes.MunkandWunsch(1998)pointou tthatdeep convectionbyitselfleadstoadeep,stagnant,poolofcoldw ater.Inthiscase, thedeepcirculationisconnedtotheupperlayersoftheoce an.Mixingor upwellingisrequiredtopumpcoldwaterupwardthroughthet hermoclineand drivethedeepcirculation.Windsandtidesaretheprimarys ourceofenergy drivingthemixing. Noticealsothatconvectionandsinkingarenotthesame,and theydonot occurinthesameplace(MarotzkeandScott,1999).Convecti onoccursin smallregionsafewkilometersonaside.Sinking,drivenbyE kmanpumping andgeostrophiccurrents,canoccuroverfarlargerareas.I nthischapter,we arediscussingmostlysinkingofwater. Todescribethesimplestaspectsoftherow,webeginwiththe Sverdrup equationappliedtoabottomcurrentofthickness H inanoceanofconstant depth: v = f @w @z (13.1) where f =2nsin =(2ncos ) =R ,nisearth'srotationrate, R earth's radius,and islatitude.Integrating(13.1)fromthebottomoftheocean to thetopoftheabyssalcirculationgives: V = Z H 0 vdz = Z H 0 f @w @z dz V = R tan 'W 0 (13.2) where V istheverticalintegralofthenorthwardvelocity,and W 0 isthevelocity atthebaseofthethermocline. W 0 mustbepositive(upward)almosteverywhere tobalancethedownwardmixingofheat.Then V mustbeeverywheretoward

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220 CHAPTER13.DEEPCIRCULATIONINTHEOCEAN S2S1 Figure13.4Idealizedsketchofthedeepcirculationduetod eepconvectionintheAtlantic (darkcircles)andupwellingthroughthethermoclineelsew here.Therealcirculationismuch dierentthanthecirculationshowninthissketch.AfterSt ommel(1958). thepoles.Thisistheabyssalrowintheinterioroftheocean sketchedby Stommelingure13.4.The U componentoftherowiscalculatedfrom V and w usingthecontinuityequation. Toconnectthestreamlinesoftherowinthewest,Stommeladd edadeep westernboundarycurrent.Thestrengthofthewesternbound arycurrentdependsonthevolumeofwater S producedatthesourceregions. StommelandAronscalculatedtherowforasimpliedoceanbo undedby theEquatorandtwomeridians(apieshapedocean).Firstthe yplacedthe source S 0 nearthepoletoapproximatetherowinthenorthAtlantic.If the volumeofwatersinkingatthesourceequalsthevolumeofwat erupwelledinthe basin,andiftheupwelledvelocityisconstanteverywhere, thenthetransport T w inthewesternboundarycurrentis: T w = 2 S 0 sin (13.3) Thetransportinthewesternboundarycurrentatthepolesis twicethevolume ofthesource,andthetransportdiminishestozeroattheEqu ator(Stommel andArons,1960a:eq,7.3.15;seealsoPedlosky,1996: x 7.3).Therowdrivenby theupwellingwateraddsarecirculationequaltothesource .If S 0 exceedsthe volumeofwaterupwelledinthebasin,thenthewesternbound arycurrentcarries wateracrosstheEquator.Thisgivesthewesternboundarycu rrentsketchedin thenorthAtlanticingure13.4. Next,StommelandAronscalculatedthetransportinawester nboundary currentinabasinwithnosource.Thetransportis: T w = S [1 2sin ](13.4) where S isthetransportacrosstheEquatorfromtheotherhemispher e.Inthis basinStommelnotes:

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13.3.THEORYFORTHEDEEPCIRCULATION 221 Crozet Is. Crozet Basin Kerguelen Is. Basin 0.1 -0.6 -0.8 -0.2 0.8 0.5 0.9 0.85 0.61.231.20 1.10 0.97 0.95 0.75 0.70 0.65 0.5 0.2 -0.4 -0.2 0.96 MascareneBasin Madagascar South Australian Basin West Australian Basin North and Northwest Australian Basin Central Indian Arabian Basin Somall BasinCarlsbergRidgeMadagascarAntarctic Arabia Australia Evolution of potential temperature at depths greater than 4000m AsiaKerguelenPlateauMedian BasinNinety East Ridge0o0o80o100o120o140o60o40o20o-70o-60o-50o-40o-30o-60o-10o0o10o20o30o Figure13.5DeeprowintheIndianOceaninferredfromthetem perature,givenin C.Note thattherowisconstrainedbythedeepmid-oceanridgesyste m.AfterTchernia(1980). Acurrentofrecirculatedwaterequaltothesourcestrength startsatthe poleandrowstowardthesource ::: [and]graduallydiminishestozeroat =30 northlatitude.Anorthwardcurrentofequalstrengthstart sat theequatorialsourceandalsodiminishestozeroat30 northlatitude. Thisgivesthewesternboundarycurrentassketchedintheno rthPacicin gure13.4. NotethattheStommel-Aronstheoryassumesaratbottom.The mid-ocean ridgesystemdividesthedeepoceanintoaseriesofbasinsco nnectedbysills throughwhichthewaterrowsfromonebasintothenext.Asare sult,the rowinthedeepoceanisnotassimpleasthatsketchedbyStomm el.Boundary currentrowalongtheedgesofthebasins,androwintheeaste rnbasinsinthe Atlanticcomesthroughthemid-Atlanticridgefromthewest ernbasics.Figure 13.5showshowridgescontroltherowintheIndianOcean. Finally,Stommel-Aronstheorygivessomevaluesfortimere quiredforwater tomovefromthesourceregionstothebaseofthethermocline invariousbasins. Thetimevariesfromafewhundredyearsforbasinsneartheso urcestonearly athousandyearsforthenorthPacic,whichisfartherfromt hesources. SomeCommentsontheTheoryfortheDeepCirculation Ourunderstandingofthedeepcirculationisstillevolving. 1.MarotzkeandScott(1999)pointsoutthatdeepconvection andmixingare verydierentprocesses.Convectionreducesthepotential energyofthe

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222 CHAPTER13.DEEPCIRCULATIONINTHEOCEAN watercolumn,anditisselfpowered.Mixinginastratiedru idincreases thepotentialenergy,anditmustbedrivenbyanexternalpro cess. 2.Numericalmodelsshowthatthedeepcirculationisveryse nsitivetothe assumedvalueofverticaleddydiusivityinthethermoclin e(Gargettand Holloway,1992). 3.NumericalcalculationsbyMarotzkeandScott(1999)indi catethatthe masstransportisnotlimitedbytherateofdeepconvection, butitis sensitivetotheassumedvalueofverticaleddydiusivity, especiallynear sideboundaries. 4.Coldwaterismixedupwardattheocean'sboundaries,abov eseamounts andmid-oceanridges,alongstrongcurrentssuchastheGulf Stream, andintheAntarcticCircumpolarCurrent(ToggweilerandRu ssell,2008; Garabatoetal,2004,2007).Becausemixingisstrongovermi d-ocean ridgesandsmallinnearbyareas,rowiszonalintheoceanbas insand polewardalongtheridges(Hoggetal.2001).Amapofthecirc ulation willnotlooklikegure13.4.Numericalmodelsandmeasurem entsofdeep rowbyroatsshowtherowisindeedzonal. 5.Becausethetransportofmass,heat,andsaltarenotclose lyrelatedthe transportofheatintothenorthAtlanticmaynotbeassensit ivetosurface salinityasdescribedabove. 13.4ObservationsoftheDeepCirculation Theabyssalcirculationislesswellknownthantheupper-oc eancirculation. Directobservationsfrommooredcurrentmetersordeep-dri ftingroatswere diculttomakeuntilrecently,andtherearefewlong-termd irectmeasurements ofcurrent.Inaddition,themeasurementsdonotproduceast ablemeanvaluefor thedeepcurrents.Forexample,ifthedeepcirculationtake sroughly1,000years totransportwaterfromthenorthAtlantictotheAntarcticC ircumpolarCurrent andthentothenorthPacic,themeanrowisabout1mm/s.Obse rving thissmallmeanrowinthepresenceoftypicaldeepcurrentsh avingvariable velocitiesofupto10cm/sorgreater,isverydicult. Mostofourknowledgeofthedeepcirculationisinferredfro mmeasured distributionofwatermasseswiththeirdistinctivetemper atureandsalinityand theirconcentrationsofoxygen,silicate,tritium,ruoroc arbonsandothertracers. Thesemeasurementsaremuchmorestablethandirectcurrent measurements, andobservationsmadedecadesapartcanbeusedtotracethec irculation.Tomczak(1999)carefullydescribeshowthetechniquescanbema dequantitativeand howtheycanbeappliedinpractice.WaterMasses Theconceptofwatermassesoriginatesinmeteorology.VilhelmBjerknes,aNorwegianmeteorologist,rstdescribedt hecoldairmasses thatforminthepolarregions.Heshowedhowtheymovesouthw ard,where theycollidewithwarmairmassesatplaceshecalledfronts, justasmassesof troopscollideatfrontsinwar(Friedman,1989).Inasimila rway,watermasses

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13.4.OBSERVATIONSOFTHEDEEPCIRCULATION 223 areformedindierentregionsoftheocean,andthewatermas sesareoftenseparatedbyfronts.Note,however,thatstrongwindsareassoc iatedwithfronts intheatmospherebecauseofthelargedierenceindensitya ndtemperatureon eithersideofthefront.Frontsintheoceansometimeshavel ittlecontrastin density,andthesefrontshaveonlyweakcurrents. Tomczak(1999)denesa watermass asa bodyofwaterwithacommonformationhistory,havingitsori ginina physicalregionoftheocean.Justasairmassesintheatmosp here,watermassesarephysicalentitieswithameasurablevolumean dtherefore occupyanitevolumeintheocean.Intheirformationregion theyhave exclusiveoccupationofaparticularpartoftheocean.Else wherethey sharetheoceanwithotherwatermasseswithwhichtheymix.T hetotal volumeofawatermassisgivenbythesumofallitselementsre gardless oftheirlocation. Plotsofsalinityasafunctionoftemperature,called T-S plots,areusedto delineatewatermassesandtheirgeographicaldistributio n,todescribemixing amongwatermasses,andtoinfermotionofwaterinthedeepoc ean.Here's whytheplotsaresouseful:waterproperties,suchastemper atureandsalinity, areformedonlywhenthewaterisatthesurfaceorinthemixed layer.Heating, cooling,rain,andevaporationallcontribute.Oncethewat ersinksbelowthe mixedlayertemperatureandsalinitycanchangeonlybymixi ngwithadjacent watermasses.Thuswaterfromaparticularregionhasaparti culartemperature associatedwithaparticularsalinity,andtherelationshi pchangeslittleasthe watermovesthroughthedeepocean. Temperature (Celsius) SalinityPressure (decibars) 0 1000 2000 3000 40000o6o12o18o24o30o34.5 35 35.5 36 36.5 37Salinity Temperature 0o5o10o15o20o25o30oTemperature (Celsius) 3535.53636.537 34.5 Station 61Station 64 75 600 50 30 Figure13.6Temperatureandsalinitymeasuredathydrograp hicstationsoneithersideofthe GulfStream.Dataarefromtables10.2and10.4. Left: Temperatureandsalinityplotted asafunctionofdepth. Right: Thesamedata,butsalinityisplottedasafunctionof temperatureina T-S plot.Noticethattemperatureandsalinityareuniquelyrel atedbelow themixedlayer.Afewdepthsarenotednexttodatapoints.

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224 CHAPTER13.DEEPCIRCULATIONINTHEOCEAN SalinityTemperature(T1 ,S1 ) (T2 ,S2 ) S2S1DepthSalinity TemperatureDepthT1T2 Temperature T2T12 2 1 1DepthT3DepthS21 2 S1S3S3TemperatureSalinity (T1 ,S1 ) (T 3 ,S 3 ) (T2 ,S2 ) Salinity Figure13.7 Upper: Mixingoftwowatermassesproducesalineona T-S plot. Lower: Mixingamongthreewatermassesproducesintersectingline sona T-S plot,andtheapexat theintersectionisroundedbyfurthermixing.AfterDefant (1961:205). Thustemperatureandsalinityarenotindependentvariable s.Forexample, thetemperatureandsalinityofthewateratdierentdepths belowtheGulf Streamareuniquelyrelated(gure13.6,right),indicatin gtheycamefromthe samesourceregion,eventhoughtheydonotappearrelatedif temperatureand salinityareplottedindependentlyasafunctionofdepth( gure13.6,left). Temperatureandsalinityare conservativeproperties becausethereareno sourcesorsinksofheatandsaltintheinterioroftheocean. Otherproperties, suchasoxygenarenon-conservative.Forexample,oxygenco ntentmaychange slowlyduetooxidationoforganicmaterialandrespiration byanimals. Eachpointinthe T-S plotisa watertype .Thisisamathematicalideal. Somewatermassesmaybeveryhomogeneousandtheyarealmost pointson theplot.Otherwatermassesarelesshomogeneous,andtheyo ccupyregions ontheplot. Mixingtwowatertypesleadstoastraightlineona T-S diagram(gure 13.7).Becausethelinesofconstantdensityona T-S plotarecurved,mixing increasesthedensityofthewater.Thisiscalled densication (gure13.8).

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13.4.OBSERVATIONSOFTHEDEEPCIRCULATION 225 L M Gst = 24.525.025.526.027.027.528.026.5 0o5o10o15o32.533.033.534.034.535.0Temperature (Celsius)Salinity Figure13.8Mixingoftwowatertypesofthesamedensity(Lan dG)produceswaterthatis denser(M)thaneitherwatertype.AfterTolmazin(1985:137 ). WaterMassesandtheDeepCirculation Let'susetheseideasofwater massesandmixingtostudythedeepcirculation.Westartint hesouthAtlantic becauseithasveryclearlydenedwatermasses.A T-S plotcalculatedfrom hydrographicdatacollectedinthesouthAtlantic(gure13 .9)showsthreeimportantwatermasseslistedinorderofdecreasingdepth(ta ble13.1):Antarctic BottomWater aab ,NorthAtlanticDeepWater nadw ,andAntarcticIntermediateWater aiw .Allaredeeperthanonekilometer.Themixingamongthree watermassesshowsthecharacteristicroundedapexesshown intheidealized caseshowningure13.7. Theplotindicatesthatthesamewatermassescanbefoundthr oughoutthe westernbasinsinthesouthAtlantic.Nowlet'suseacrossse ctionofsalinityto tracethemovementofthewatermassesusingthecoremethod.CoreMethod Theslowvariationfromplacetoplaceintheoceanofatracer suchassalinitycanbeusedtodeterminethesourceofthewat ersmassessuch asthoseingure13.9.Thisiscalledthe coremethod .Themethodmayalso beusedtotracktheslowmovementofthewatermass.Note,how ever,thata slowdriftofthewaterandhorizontalmixingbothproduceth esameobserved propertiesintheplot,andtheycannotbeseparatedbytheco remethod. Table13.1WaterMassesofthesouthAtlanticbetween33 Sand11 N Temp. Salinity ( C) AntarcticwaterAntarcticIntermediateWater aiw 3.3 34.15 AntarcticBottomWater abw 0.4 34.67 NorthAtlanticwaterNorthAtlanticDeepWater nadw 4.0 35.00 NorthAtlanticBottomWater nabw 2.5 34.90 ThermoclinewaterSubtropicalLowerWater u 18.0 35.94 FromDefant(1961:table82)

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226 CHAPTER13.DEEPCIRCULATIONINTHEOCEAN AAB NADW AIW 47 S 35 S 24 S 15 S 4 S 5 N U 0o5o10o15o20o25o30o34 3534 35 34 35 34 3534 3534 35 36Temperature (Celsius)Salinity Figure13.9 T-S plotofdatacollectedatvariouslatitudesinthewesternba sinsofthe southAtlantic.Linesdrawnthroughdatafrom5 N,showingpossiblemixingbetween watermasses: nadw {NorthAtlanticDeepWater, aiw {AntarcticIntermediateWater, aab AntarcticBottomWater, u SubtropicalLowerWater. A core isalayerofwaterwithextremevalue(inthemathematicalse nse)of salinityorotherpropertyasafunctionofdepth.Anextreme valueisalocal maximumorminimumofthequantityasafunctionofdepth.The method assumesthattherowisalongthecore.Waterinthecoremixes withthewater massesaboveandbelowthecoreanditgraduallylosesitside ntity.Furthermore, therowtendstobealongsurfacesofconstantpotentialdens ity. Let'sapplythemethodtothedatafromthesouthAtlanticto ndthesource ofthewatermasses.Asyoumightexpect,thiswillexplainth eirnames. Westartwithanorth-southcrosssectionofsalinityinthew esternbasinsof theAtlantic(gure13.10).Itwelocatethemaximaandminim aofsalinityas afunctionofdepthatdierentlatitudes,wecanseetwoclea rlydenedcores. Theupperlow-salinitycorestartsnear55 Sanditextendsnorthwardatdepths near1000m.ThiswateroriginatesattheAntarcticPolarFro ntzone.Thisis theAntarcticIntermediateWater.Belowthiswatermassisa coreofsaltywater originatinginthefarnorthAtlantic.ThisistheNorthAtla nticDeepWater. Belowthisisthemostdensewater,theAntarcticBottomWate r.Itoriginates inwinterwhencold,dense,salinewaterformsintheWeddell Seaandother shallowseasaroundAntarctica.Thewatersinksalongtheco ntinentalslope andmixeswithCircumpolarDeepWater.Itthenllsthedeepb asinsofthe southPacic,Atlantic,andIndianocean. TheCircumpolarDeepWaterismostlyNorthAtlanticDeepWat erthat hasbeencarriedaroundAntarctica.Asitiscarriedalong,i tmixeswithdeep watersoftheIndianandPacicOceantoformthecircumpolar water.

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13.4.OBSERVATIONSOFTHEDEEPCIRCULATION 227 34.7 34.68 34.1 34.2 .5 .3 34.7 34.8 34.9 34.94 35 34.7 35.5 36 36.5 37 36 <34.9 <34.9 34.9 34.9 35 35 .8 34.5 34.3 36 <37 34.9 Greenland-Iceland Ridge Antarctica -7000 -6000 -5000 -4000 -3000 -2000 -1000 0 -80 o -60 o -40 o -20 o 0 o 20 o 40 o 60 o 80 oDepth (m) PFSAF Figure13.10Contourplotofsalinityasafunctionofdepthi nthewesternbasinsofthe AtlanticfromtheArcticOceantoAntarctica.Theplotclear lyshowsextensivecores,one atdepthsnear1000mextendingfrom50 Sto20 N,theotheratisatdepthsnear2000m extendingfrom20 Nto50 S.TheupperistheAntarcticIntermediateWater,theloweri s theNorthAtlanticDeepWater.Thearrowsmarktheassumeddi rectionoftherowinthe cores.TheAntarcticBottomWaterllsthedeepestlevelsfr om50 Sto30 N. pf isthepolar front, saf isthesubantarcticfront.Seealsogures10.15and6.10.Af terLynnandReid (1968). Therowisprobablynotalongthearrowsshowningure13.10. Thedistributionofpropertiesintheabysscanbeexplainedbyacombin ationofslowrow inthedirectionofthearrowsplushorizontalmixingalongs urfacesofconstant potentialdensitywithsomeweakverticalmixing.Theverti calmixingprobablyoccursattheplaceswherethedensitysurfacereachesth eseabottomata lateralboundarysuchasseamounts,mid-oceanridges,anda longthewestern boundary.Flowinaplaneperpendiculartothatofthegurem aybeatleast asstrongastherowintheplaneofthegureshownbythearrow s. Thecoremethodcanbeappliedonlytoatracerthatdoesnotin ruence density.Hencetemperatureisusuallyapoorchoice.Ifthet racercontrolsdensity,thenrowwillbearoundthecoreaccordingtoideasofge ostrophy,not alongcoreasassumedbythecoremethod. ThecoremethodworksespeciallywellinthesouthAtlanticw ithitsclearly denedwatermasses.Inotheroceanbasins,the T-S relationshipismorecomplicated.Theabyssalwatersintheotherbasinsareacomple xmixtureof waterscomingfromdierentareasintheocean(gure13.11) .Forexample, warm,saltywaterfromtheMediterraneanSeaentersthenort hAtlanticand spreadsoutatintermediatedepthsdisplacingintermediat ewaterfromAntarcticainthenorthAtlantic,addingadditionalcomplexityto therowasseenin thelowerrightpartofthegure.OtherTracers Ihaveillustratedthecoremethodusingsalinityasatracer butmanyothertracersareused.Anidealtraceriseasytomea sureevenwhen itsconcentrationisverysmall;itisconserved,whichmean sthatonlymixing changesitsconcentration;itdoesnotinruencethedensity ofthewater;itexists inthewatermasswewishtotrace,butnotinotheradjacentwa termasses;and itdoesnotinruencemarineorganisms(wedon'twanttorelea setoxictracers).

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228 CHAPTER13.DEEPCIRCULATIONINTHEOCEAN SalinityTemperature (Celsius) 15o10o5o0o34.034.535.035.536.036.5 15o10o5o0o34.034.535.035.536.036.5 15o10o5o0o34.034.535.035.536.036.5 15o10o5o0o34.034.535.035.536.036.5 (d) Atlantic Ocean (b) South Pacific Ocean (a) Indian Ocean (c) North Pacific Ocean100-200m 1000m 2000m 3000m circumpolar water 1000-4000m Antarctic bottom water5008000msubantarctic waterequatorial waterIndian centra l waterAntarctic intermdiate waterRed Sea water100-200m 1000m 2000m 3000m 1000m300-400m 400700mKey SA subarctic waterAI Arctic intermediate wa terPacificequatorialwatereastNorthPacificcentralwaterPacifi c subarctic water North Pacific intermediate waterwest North Pacific ce ntral water100-200m 100-200m 2000m 3000m 500800m 50 01000m No rth Atlantic deep and bottom water circumpolar water 1000-4000m Antarctic bottom watersubantarctic waterAntarcticintermediate wa terMediterr anean waterSouthAtlanticcentralwaterNorth Atlantic ce ntral waterSAAI100-200m circumpolar water 1000-4000m 1000m 2000m 3000meastSouthPacificcentralwaterPacificequatorialwaterwestsouthPacificcentralwaterPacific subarctic water 500800m 400600m subantarctic waterAntarctic intermediate water Figure13.11 T-S plotsofwaterinthevariousoceanbasins.AfterTolmazin(1 985:138). Varioustracersmeetthesecriteriatoagreaterorlesserex tent,andtheyare usedtofollowthedeepandintermediatewaterintheocean.H erearesomeof themostwidelyusedtracers. 1.Salinityisconserved,anditinruencesdensitymuchless thantemperature. 2.Oxygenisonlypartlyconserved.Itsconcentrationisred ucedbythe respirationbymarineplantsandanimalsandbyoxidationof organic carbon. 3.Silicatesareusedbysomemarineorganisms.Theyarecons ervedatdepths belowthesunlitzone. 4.Phosphatesareusedbyallorganisms,buttheycanprovide additional information. 5. 3 Heisconserved,buttherearefewsources,mostlyatdeep-se avolcanic areasandhotsprings. 6. 3 H(tritium)wasproducedbyatomicbombtestsintheatmosphe reinthe 1950s.Itenterstheoceanthroughthemixedlayer,anditisu sefulfor tracingtheformationofdeepwater.Itdecayswithahalflif eof12.3y anditisslowlydisappearingfromtheocean.Figure10.16sh owstheslow advectionorperhapsmixingofthetracerintothedeepnorth Atlantic. Notethatafter25yearslittletritiumisfoundsouthof30 N.Thisimplies ameanvelocityoflessthanamm/s.

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13.5.ANTARCTICCIRCUMPOLARCURRENT 229 7.Fluorocarbons(Freonusedinairconditioning)havebeen recentlyinjected intoatmosphere.Theycanbemeasuredwithverygreatsensit ivity,and theyarebeingusedfortracingthesourcesofdeepwater. 8.SulphurhexaruorideSF 6 canbeinjectedintoseawater,andtheconcentrationcanbemeasuredwithgreatsensitivityformanymont hs. Eachtracerhasitsusefulness,andeachprovidesadditiona linformationabout therow.NorthAtlanticMeridionalOverturningCirculation ThegreatimportanceofthemeridionaloverturningcirculationforEurope anclimatehasledto programstomonitorthecirculation.TheRapidClimateChan ge/Meridional OverturningCirculationandHeatFluxArray rapid/mocha deployedanarrayofinstrumentsthatmeasuredbottompressureplustempe ratureandsalinity throughoutthewatercolumnat15locationsalong24 Nnearthewesternand easternboundariesandoneithersideofthemid-Atlanticri dgebeginningin 2004(Church,2007).Atthesametime,rowoftheGulfStreamw asmeasuredthroughtheStraitofFlorida,andwindstress,whichg ivestheEkman transports,wasmeasuredalong24 Nbysatelliteinstruments.Themeasurementsshowthattransportacross24 Nwaszero,withintheaccuracyofthe measurements,asexpected.Theone-yearaverageoftheMeri dionalOverturningCirculationwas18 : 7 5 : 6Sv,withvariabilityrangingfrom4.4to35.3Sv. Accuracyofthemeasurementwas 1.5Sv. 13.5AntarcticCircumpolarCurrent TheAntarcticCircumpolarCurrentisanimportantfeatureo ftheocean's deepcirculationbecauseittransportsdeepandintermedia tewaterbetween theAtlantic,Indian,andPacicOcean,andbecauseEkmanpu mpingdriven bywesterlywindsisamajordriverofthedeepcirculation.B ecauseitisso importantforunderstandingthedeepcirculationinalloce an,let'slookatwhat isknownaboutthiscurrent. AplotofdensityacrossalineofconstantlongitudeintheDr akePassage (gure13.12)showsthreefronts.Theyare,fromnorthtosou th:1)theSubantarcticFront,2)thePolarFront,and3)theSouthern acc Front.Eachfront iscontinuousaroundAntarctica(gure13.13).Theplotals oshowsthatthe constant-densitysurfacesslopeatalldepths,whichindic atesthatthecurrents extendtothebottom. Typicalcurrentspeedsarearound10cm/swithspeedsofupto 50cm/snear somefronts.Althoughthecurrentsareslow,theytransport muchmorewater thanwesternboundarycurrentsbecausetherowisdeepandwi de.Whitworth andPeterson(1985)calculatedtransportthroughtheDrake Passageusingseveralyearsofdatafromanarrayof91currentmeterson24moor ingsspaced approximately50kmapartalongalinespanningthepassage. Theyalsoused measurementsofbottompressuremeasuredbygaugesoneithe rsideofthepassage.TheyfoundthattheaveragetransportthroughtheDrak ePassagewas 125 11Sv,andthatthetransportvariedfrom95Svto158Sv.Thema ximum transporttendedtooccurinlatewinterandearlyspring(g ure13.14).

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230 CHAPTER13.DEEPCIRCULATIONINTHEOCEAN | 27.427.8 27.92828.0528.128.1528.228.228.228.25 26.9 sACCf PF SAF27.228.0528.128.1528.228.2528.1527.427.628.128.052827.927.827.627.2 WOCE A21 (1990) Neutral Density (kg/m3) gn| | | 60S 58S 62S 27.92826.5 -4 -3 -2 -1 0 117 116 114 112 110 108 106 104 1020200400600800 Distance (km)Depth (km) 0 20 40Transport (Sv) Figure13.12CrosssectionofneutraldensityacrosstheAnt arcticCircumpolarCurrentin theDrakePassagefromtheWorldOceanCirculationExperime ntsectionA21in1990.The currenthasthreestreamsassociatedwiththethreefronts( darkshading): sf =Southern acc Front, pf =PolarFront,and saf =SubantarcticFront.Hydrographicstationnumbers aregivenatthetop,andtransportsarerelativeto3,000dba r.Circumpolardeepwateris indicatedbylightshading.DatafromAlexOrsi,TexasA&MUn iversity. Becausetheantarcticcurrentsreachthebottom,theyarein ruencedbytopographicsteering.Asthecurrentcrossesridgessuchasth eKerguelenPlateau, thePacic-AntarcticRidge,andtheDrakePassage,itisder ectedbytheridges. ThecoreofthecurrentiscomposedofCircumpolarDeepWater ,amixtureofdeepwaterfromallocean.Theupperbranchofthecurr entcontains oxygen-poorwaterfromallocean.Thelower(deeper)branch containsacore ofhigh-salinitywaterfromtheAtlantic,includingcontri butionsfromthenorth AtlanticdeepwatermixedwithsaltyMediterraneanSeawate r.AsthedierentwatermassescirculatearoundAntarcticatheymixwitho therwatermasses withsimilardensity.Inasense,thecurrentisagiant`mixmaster'takingdeep waterfromeachocean,mixingitwithdeepwaterfromotheroc ean,andthen redistributingitbacktoeachocean(Garabatoetal,2007).

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13.5.ANTARCTICCIRCUMPOLARCURRENT 231 -60o-90o-120o-20o0o20o60o90o120o-160o180o160o-40o40o140o-140o50 3 0 -70 South Atlantic Ocean Indian Ocean South Pacific Ocean New ZealandSAF PF SACC ACC boundary STF Figure13.13DistributionoffrontsaroundAntarctica: STF :SubtrobicalFront; SAF : SubantarcticFront; PF :PolarFront; SACC :SouthernAntarcticCircumpolarFront. Shadedareasareshallowerthan3km.FromOrsi(1995). 81-82 79-80 78 77July Transport (106 m3 s-1)160150140130120110100 90 Jan 1977 JulyJan 1978 July Jan 1979 JulyJan 1980 Jan 1981 JulyJan 1982 Figure13.14VariabilityofthetransportintheAntarcticC ircumpolarCurrentasmeasured byanarrayofcurrentmetersdeployedacrosstheDrakePassa ge.Theheavierlineis smoothed,time-averagedtransport.FromWhitworth(1988)

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232 CHAPTER13.DEEPCIRCULATIONINTHEOCEAN Thecoldest,saltiestwaterintheoceanisproducedontheco ntinentalshelf aroundAntarcticainwinter,mostlyfromtheshallowWeddel landRossseas. Thecoldsaltywaterdrainsfromtheshelves,entrainssomed eepwater,and spreadsoutalongthesearoor.Eventually,8{10Svofbottom waterareformed (Orsi,Johnson,andBullister,1999).Thisdensewaterthen seepsintoallthe oceanbasins.Bydenition,thiswateristoodensetocrosst hroughtheDrake Passage,soitisnotcircumpolarwater. TheAntarcticcurrentsarewinddriven.Strongwestwindswi thmaximum speednear50 Sdrivethecurrents(seegure4.2),andthenorth-southgra dientofwindspeedproducesconvergenceanddivergenceofEkm antransports. Divergencesouthofthezoneofmaximumwindspeed,southof5 0 Sleads toupwellingoftheCircumpolarDeepWater.Convergencenor thofthezone ofmaximumwindsleadstodownwellingoftheAntarcticinter mediatewater. Thesurfacewaterisrelativelyfreshbutcold,andwhenthey sinktheydene characteristicsoftheAntarcticintermediatewater. Thepositionofthecircumpolarcurrentrelativetothemaxi mumofthewesterlywindsinruencesthemeridionaloverturningcirculati onandclimate.North ofthemaximum,Ekmantransportsconverge,pushingwaterdo wnwardintothe AntarcticIntermediateWaternorthofthePolarFront.Sout hofthemaximum winds,Ekmantransportsdiverge,pullingCircumpolarAtla nticDeepWaterto thesurfacesouthofthePolarFront,whichhelpsdrivethede epcirculation (gure13.10).Whenthemaximumwindsarefurtherfromthepo le,lessdeep waterispulledupward,andthedeepcirculationisweak,asi twasduringthe lasticeage.Astheearthwarmedaftertheiceage,themaximu mwindsshifted south.ThewindsweremorealignedwiththeCircumpolarCurr ent,andthey pulledmoredeepwatertothesurface.Since1960,thewindsh avestrengthened andshiftedsouthward,furtherstrengtheningCircumpolar Currentandthedeep circulationToggweilerandRussell,2008). BecausewindconstantlytransfersmomentumtotheAntarcti cCircumpolar Current,causingittoaccelerate,theaccelerationmustbe balancedbydrag,and weareledtoask:Whatkeepstherowfromacceleratingtovery highspeeds? MunkandPalmen(1951),suggestformdragdominates.Formdr agisdueto thecurrentcrossingsub-searidges,especiallyattheDrak ePassage.Formdrag isalsothedragofthewindonafastmovingcar.Inbothcases, therowis diverted,bytheridgeorbyyourcar,creatingalowpressure zonedownstream oftheridgeordownwindofthecar.Thelowpressurezonetran sfersmomentum intothesolidearth,slowingdownthecurrent.13.6ImportantConcepts 1.Thedeepcirculationoftheoceanisveryimportantbecaus eitdetermines theverticalstraticationoftheoceanandbecauseitmodul atesclimate. 2.TheoceanabsorbsCO 2 fromtheatmospherereducingatmosphericCO 2 concentrations.ThedeepcirculationcarriestheCO 2 deepintotheocean temporarilykeepingitfromreturningtotheatmosphere.Ev entually,however,mostoftheCO 2 mustbereleasedbacktotheatmosphere.But,some

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13.6.IMPORTANTCONCEPTS 233 remainsintheocean.PhytoplanktonconvertCO 2 intoorganiccarbon, someofwhichsinkstothesearoorandisburiedinsediments. SomeCO 2 isusedtomakeseashells,andittooremainsintheocean. 3.TheproductionofdeepbottomwatersinthenorthAtlantic drawsa petawattofheatintothenorthernhemispherewhichhelpswa rmEurope. 4.VariabilityofdeepwaterformationinthenorthAtlantic hasbeentiedto largeructuationsofnorthernhemispheretemperatureduri ngthelastice ages. 5.Deepconvectionwhichproducesbottomwateroccursonlyi nthefarnorth AtlanticandatafewlocationsaroundAntarctica. 6.Thedeepcirculationisdrivenbyverticalmixing,whichi slargestabove mid-oceanridges,nearseamounts,andinstrongboundarycu rrents. 7.Thedeepcirculationistooweaktomeasuredirectly.Itis inferredfrom observationsofwatermassesdenedbytheirtemperaturean dsalinity andfromobservationoftracers. 8.TheAntarcticCircumpolarCurrentmixesdeepwaterfromt heAtlantic, Pacic,andIndianOceanandredistributesitbacktoeachoc ean.The currentisdeepandslowwithatransportof125Sv.

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234 CHAPTER13.DEEPCIRCULATIONINTHEOCEAN

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Chapter14EquatorialProcessesEquatorialprocessesareatthecenterofourunderstanding theinruenceof theoceanontheatmosphere,andtheydominatetheinterannu alructuations inglobalweatherpatterns.Thesunwarmsthevastexpanseso fthetropical PacicandIndianocean,evaporatingwater.Whenthewaterc ondensesas rainitreleasessomuchheatthattheseareasaretheprimary enginedrivingthe atmosphericcirculation(gure14.1).Rainfalloverexten siveareasexceedsthree metersperyear(gure5.5),andsomeoceanicregionsreceiv emorethanve metersofrainperyear.Toputthenumbersinperspective,v emetersofrain peryearreleasesonaverage400W/m 2 ofheattotheatmosphere.Equatorial currentsmodulatetheair-seainteractions,especiallyth roughthephenomenon knownasElNi~no,withglobalconsequences.Idescribehere rstthebasic equatorialprocesses,thentheyear-to-yearvariabilityo ftheprocessesandthe inruenceofthevariabilityonweatherpatterns. 0 60 120 180 -120 -60 0 -60 -30 0 30 60 -25 +75+25-25 +25 -25+25+75+ 25 +1 2 5 Figure14.1Averagediabaticheatingbetween700and50mbin theatmosphereduring December,JanuaryandFebruarycalculatedfrom ecmwf datafor1983{1989.Mostofthe heatingisduetothereleaseoflatentheatbyrain.AfterWeb steretal.(1992). 235

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236 CHAPTER14.EQUATORIALPROCESSES 10o15o20o25o160o180o-160o-140o-120o-100o-80oLongitude -600 -500 -400 -300 -200 -100Depth (m) 0 Figure14.2Themean,upper-ocean,thermalstructurealong theequatorinthePacicfrom northofNewGuineatoEcuadorcalculatedfromdatainLevitu s(1982). 14.1EquatorialProcesses Thetropicaloceanischaracterizedbyathin,permanent,sh allowlayerof warmwateroverdeeper,colderwater.Inthisrespect,theve rticalstratication issimilartothesummerstraticationathigherlatitudes. Surfacewatersare hottestinthewest(gure6.3)inthegreatPacicwarmpool. Themixedlayer isdeepinthewestandveryshallowintheeast(Figure14.2). Theshallowthermoclinehasimportantconsequences.Theso utheasttrade windsblowalongtheequator(gure4.2)althoughtheytendt obestrongestin theeast.Northoftheequator,Ekmantransportisnorthward .Southofthe equatoritissouthward.ThedivergenceoftheEkmanrowcaus esupwelling ontheequator.Inthewest,theupwelledwateriswarm.Butin theeastthe upwelledwateriscoldbecausethethermoclineissoshallow .Thisleadstoa coldtongueofwaterattheseasurfaceextendingfromSouthA mericatonear thedateline(gure6.3). Surfacetemperatureintheeastisabalanceamongfourproce sses: 1.Thestrengthoftheupwelling,whichisdeterminedbythew estwardcomponentofthewind. 2.Thespeedofwestwardcurrentswhichcarrycoldwaterfrom thecoastof PeruandEcuador. 3.North-southmixingwithwarmerwatersoneithersideofth eequator. 4.Heatruxesthroughtheseasurfacealongtheequator. Theeast-westtemperaturegradientontheequatordrivesaz onalcirculation intheatmosphere,theWalkercirculation.Thunderstormso verthewarmpool carryairupward,andsinkingairintheeastfeedsthereturn rowatthesurface. VariationsinthetemperaturegradientinruencestheWalke rcirculation,which, inturn,inruencesthegradient.Thefeedbackcanleadtoani nstability,theEl Ni~no-SouthernOscillation( enso )discussedinthenextsection.

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14.1.EQUATORIALPROCESSES 237 20.0 cm/s Average Velocity at 10 m Jan 1981 Dec 1994 140o160o180o-160o-140o-120o-100o-80o -20o-10o0o10o20o Figure14.3Averagecurrentsat10mcalculatedfromtheModu larOceanModeldrivenby observedwindsandmeanheatruxesfrom1981to1994.Themode l,operatedbythe noaa NationalCentersforEnvironmentalPrediction,assimilat esobservedsurfaceandsubsurface temperatures.AfterBehringer,Ji,andLeetmaa(1998).SurfaceCurrents Thestrongstraticationconnesthewind-drivencirculationtothemixedlayerandupperthermocline.Sverdrup'sth eoryandMunk's extension,describedin x 11.1and x 11.3,explainthesurfacecurrentsinthetropicalAtlantic,Pacic,andIndianocean.Thecurrentsinclu de(gure14.3): 1.TheNorthEquatorialCountercurrentbetween3 Nand10 N,whichrows eastwardwithatypicalsurfacespeedof50cm/s.Thecurrent iscentered onthebandofweakwinds,the doldrums ,around5{10 Nwherethenorth andsouthtradewindsconverge,the tropicalconvergencezone 2.TheNorthandSouthEquatorialCurrentswhichrowwestwar dinthe zonalbandoneithersideofthecountercurrent.Thecurrent sareshallow, lessthan200mdeep.Thenortherncurrentisweak,withaspee dless thanroughly20cm/s.Thesoutherncurrenthasamaximumspee dof around100cm/s,inthebandbetween3 Nandtheequator. ThecurrentsintheAtlanticaresimilartothoseinthePaci cbecausethe tradewindsinthatoceanalsoconvergenear5 {10 N.TheSouthEquatorial CurrentintheAtlanticcontinuesnorthwestalongthecoast ofBrazil,whereit isknownastheNorthBrazilCurrent.IntheIndianOcean,the doldrumsoccur inthesouthernhemisphereandonlyduringthenorthern-hem ispherewinter.In thenorthernhemisphere,thecurrentsreversewiththemons oonwinds. Thereis,however,muchmoretothestoryofequatorialcurre nts. EquatorialUndercurrent:Observations Justafewmetersbelowthesurfaceontheequatorisastrongeastwardrowingcurrent,theE quatorialUndercurrent,thelastmajoroceaniccurrenttobediscovered.He re'sthestory: InSeptember1951,aboardtheU.S.FishandWildlifeService research vessellong-lineshingontheequatorsouthofHawaii,itwa snoticedthat thesubsurfacegeardriftedsteadilytotheeast.Thenextye arCromwell, incompanywithMontgomeryandStroup,ledanexpeditiontoi nvestigatetheverticaldistributionofhorizontalvelocityatth eequator.Using

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238 CHAPTER14.EQUATORIALPROCESSES roatingdroguesatthesurfaceandatvariousdepths,theywe reableto establishthepresence,neartheequatorinthecentralPaci c,ofastrong, narroweastwardcurrentinthelowerpartofthesurfacelaye randthe upperpartofthethermocline(Cromwell, et.al. ,1954).Afewyearslater theScripps Eastropac Expedition,underCromwell'sleadership,foundthe currentextendedtowardtheeastnearlytotheGalapagosIsl andsbutwas notpresentbetweenthoseislandsandtheSouthAmericancon tinent. Thecurrentisremarkableinthat,eventhoughcomparablein transporttotheFloridaCurrent,itspresencewasunsuspectedte nyearsago. Evennow,neitherthesourcenortheultimatefateofitswate rshasbeen established.Notheoryofoceaniccirculationpredictedit sexistence,and onlynowaresuchtheoriesbeingmodiedtoaccountfortheim portant featuresofitsrow.|WarrenS.Wooster(1960). TheEquatorialUndercurrentintheAtlanticwasrstdiscov eredbyBuchanan in1886,andinthePacicbytheJapaneseNavyinthe1920sand 1930s (McPhaden,1986). However,noattentionwaspaidtotheseobservations.Other earlierhints regardingthisundercurrentwerementionedbyMatthaus(1 969).Thus theoldexperiencebecomesevenmoreobviouswhichsaysthat discoveriesnotattractingtheattentionofcontemporariessimplyd onotexist.| Dietrichetal.(1980). BobArthur(1960)summarizedthemajoraspectsoftherow: 1.Surfacerowmaybedirectedwestwardatspeedsof25{75cm/ s; 2.Currentreversesatadepthoffrom20to40m;3.Eastwardundercurrentextendstoadepthof400meterswit hatransport ofasmuchas30Sv=30 10 6 m 3 /s; 4.Coreofmaximumeastwardvelocity(0.50{1.50m/s)risesf romadepthof 100mat140 Wto40mat98 W,thendipsdown; 5.Undercurrentappearstobesymmetricalabouttheequator andbecomes muchthinnerandweakerat2 Nand2 S. Inessence,thePacicEquatorialUndercurrentisaribbonw ithdimensionsof 0 : 2km 300km 13 ; 000km(gure14.4). EquatorialUndercurrent:Theory Althoughwedonotyethaveacomplete theoryfortheundercurrent,wedohaveaclearunderstandin gofsomeofthe moreimportantprocessesatworkintheequatorialregions. Pedlosky(1996),in hisexcellentchapteronEquatorialDynamicsoftheThermoc line:TheEquatorialUndercurrent,pointsoutthatthebasicdynamicalba lanceswehaveused inmidlatitudesbreakdownnearorontheequator. Neartheequator: 1.TheCoriolisparameterbecomesverysmall,goingtozeroa ttheequator: f =2nsin = y 2n (14.1) where islatitude, = @f=@y 2n =R neartheequator,and y = R'

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14.1.EQUATORIALPROCESSES 239 u (cm/s) S035.00 35.20 35.4035.6034.800020 10 0010 30 50 0 10 0 10 20 0t (Celsius)28.026.024.022.020.018.016.014.012.010.034.6034.8034.60Depth (m)0 -100-200-300-400 -500 10o0o5o15o-5o-10o-15o Depth (m)0 -100-200-300-400 -500 10o0o5o15o-5o-10o-15o Depth (m)0 -100-200-300-400 -500 10o0o5o15o-5o-10o-15o Latitude Figure14.4CrosssectionoftheEquatorialUndercurrentin thePaciccalculatedfrom ModularOceanModelwithassimilatedsurfacedata(See x 14.5).Thesectionisanaverage from160 Eto170 EfromJanuary1965toDecember1999.Stippledareasarewest ward rowing.FromNevinS.Fuckar. 2.Planetaryvorticity f isalsosmall,andtheadvectionofrelativevorticity cannotbeneglected.ThustheSverdrupbalance(11.7)mustb emodied. 3.Thegeostrophicandvorticitybalancesfailwhenthemeri dionaldistance L totheequatoris O p U= ,where = @f=@y .If U =1m/s,then L =200kmor2 oflatitude.Lagerloeetal(1999),usingmeasured currents,showthatcurrentsneartheequatorcanbedescrib edbythe geostrophicbalancefor j j > 2 : 2 .Theyalsoshowthatrowclosertothe equatorcanbedescribedusinga -planeapproximation f = y 4.Thegeostrophicbalancefor zonal currentsworkssowellneartheequator because f and @=@y 0as 0,where isseasurfacetopography. UpwelledwateralongtheequatorproducedbyEkmanpumpingi snotpart ofatwo-dimensionalrowinanorth-south,meridionalplane .Therowisthreedimensional.Watertendstorowalongthecontoursofconsta ntdensity(isopycnalsurfaces),closetothelinesofconstanttemperaturei ngure14.2.Cold waterenterstheundercurrentinthefarwestPacic,anditm oveseastward

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240 CHAPTER14.EQUATORIALPROCESSES SeaSu rfaceW ind C A B Thermocline Mixed LayerP/x East WestThermocline Figure14.5 Left: Cross-sectionalsketchofthethermoclineandsea-surface topographyalong theequator. Right: EastwardpressuregradientinthecentralPaciccausedbyt hedensity structureatleft.andupwardalongtheequator.Forexample,the25 isothermenterstheundercurrentatadepthnear125minthewesternPacicat170 Eandeventually reachesthesurfaceat125 WintheeasternPacic. Themeridionalgeostrophicbalanceneartheequatorgivest hespeedofthe zonalcurrents,butitdoesnotexplainwhatdrivestheunder current.Avery simpliedtheoryfortheundercurrentisbasedonabalanceo fzonalpressure gradientsalongtheequator.Windstresspusheswaterwestw ard,producingthe deepthermoclineandwarmpoolinthewest.Thedeepeningoft hethermocline causesthesea-surfacetopography tobehigherinthewest,assumingthat rowbelowthethermoclineisweak.Thusthereisaneastwardp ressuregradient alongtheequatorinthesurfacelayerstoadepthofafewhund redmeters.The eastwardpressuregradientatthesurface(layerAingure1 4.5)isbalancedby thewindstress T x ,and T x =H = @p=@x ,whereHisthemixed-layerdepth BelowafewtensofmetersinlayerB,theinruenceofthewinds tressissmall, andthepressuregradientisunbalanced,leadingtoanaccel eratedrowtoward theeast,theequatorialundercurrent.Withinthislayer,t herowaccelerates untilthepressuregradientisbalancedbyfrictionalforce swhichtendtoslow thecurrent.AtdepthsbelowafewhundredmetersinlayerC,t heeastward pressuregradientistooweaktoproduceacurrent, @p=@x 0. Coriolisforceskeeptheequatorialundercurrentcentered ontheequator.If therowstraysnorthward,theCoriolisforcederectsthecur rentsouthward.The oppositeoccursiftherowstrayssouthward.14.2VariableEquatorialCirculation:ElNi~no/LaNi~na Thetradesareremarkablysteady,buttheydovaryfrommonth tomonth andyeartoyear,especiallyinthewesternPacic.Oneimpor tantsourceof variabilityareMadden-Julianwavesintheatmosphere(McP haden,1999).If thetradesinthewestweakenorreverse,theair-seasystemi ntheequatorial regionscanbethrownintoanotherstatecalledElNi~no.Thi sdisruptionof theequatorialsysteminthePacicisthemostimportantcau seofchanging weatherpatternsaroundtheglobe. AlthoughthemodernmeaningofthetermElNi~nodenotesadis ruptionof theentireequatorialsysteminthePacic,thetermhasbeen usedinthepast todescribeseveralverydierentprocesses.Thiscausesal otofconfusion.To reducetheconfusion,let'slearnalittlehistory.

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14.2.ELNI ~ NO 241 ALittleHistory Inthe19thcentury,thetermwasappliedtoconditionso thecoastofPeru.Thefollowingquotecomesfromtheintrodu ctiontoPhilander's(1990)excellentbook ElNi~no,LaNi~na,andtheSouthernOscillation : Intheyear1891,Se~norDr.LuisCarranzaoftheLimaGeograp hical Society,contributedasmallarticletotheBulletinofthat Society,calling attentiontothefactthatacounter-currentrowingfromnor thtosouth hadbeenobservedbetweentheportsofPaitaandPacasmayo. ThePaitasailors,whofrequentlynavigatealongthecoasti nsmall craft,eithertothenorthorthesouthofthatport,namethis countercurrentthecurrentof\ElNi~no"(theChildJesus)becausei thasbeen observedtoappearimmediatelyafterChristmas. Asthiscounter-currenthasbeennoticedondierentoccasi ons,and itsappearancealongthePeruviancoasthasbeenconcurrent withrainsin latitudeswhereitseldomifeverrainstoanygreatextent,I wish,onthe presentoccasion,tocalltheattentionofthedistinguishe dgeographers hereassembledtothisphenomenon,whichexercises,undoub tedly,avery greatinruenceovertheclimaticconditionsofthatpartoft heworld.| Se~norFredericoAlfonsoPezet'saddresstotheSixthInter nationalGeographicalCongressinLima,Peru1895. ThePeruviansnoticedthatinsomeyearstheElNi~nocurrent wasstronger thannormal,itpenetratedfurthersouth,anditisassociat edwithheavyrains inPeru.Thisoccurredin1891when(againquotingfromPhila nder'sbook) ...itwasthenseenthat,whereasnearlyeverysummerherean dthere thereisatraceofthecurrentalongthecoast,inthatyearit wassovisible, anditseectsweresopalpablebythefactthatlargedeadall igatorsand trunksoftreeswerebornedowntoPacasmayofromthenorth,a ndthat thewholetemperatureofthatportionofPerusueredsuchac hange owingtothehotcurrentthatbathedthecoast....|Se~norFr ederico AlfonsoPezet. ...theseaisfullofwonders,thelandevenmoreso.Firstofa llthe desertbecomesagarden....Thesoilissoakedbytheheavydo wnpour, andwithinafewweeksthewholecountryiscoveredbyabundan tpasture. Thenaturalincreaseofrocksispracticallydoubledandcot toncanbe growninplaceswhereinotheryearsvegetationseemsimposs ible.|From Mr.S.M.Scott&Mr.H.TwiddlequotedfromapaperbyMurphy,1 926. TheElNi~noof1957wasevenmoreexceptional.Somuchsothat itattracted theattentionofmeteorologistsandoceanographersthroug houtthePacicbasin. Bythefallof1957,thecoralringofCantonIsland,inthemem oryof maneverbleakanddry,waslushwiththeseedlingsofcountle sstropical treesandvines. Oneisinclinedtoselecttheeventsofthisisolatedatollas epitomizing theyear,forevenhere,ontheremoteedgesofthePacic,vas tconcerted shiftsintheoceanandatmospherehadwroughtdramaticchan ge. ElsewhereaboutthePacicitalsowascommonknowledgethat the yearhadbeenoneofextraordinaryclimaticevents.Hawaiih aditsrst recordedtyphoon;theseabird-killing ElNi~no visitedthePeruviancoast;

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242 CHAPTER14.EQUATORIALPROCESSES theicewentoutofPointBarrowattheearliesttimeinhistor y;andonthe Pacic'swesternrim,thetropicalrainyseasonlingeredsi xweeksbeyond itsappointedterm|SetteandIsaacs(1960). Justmonthsaftertheevent,in1958,adistinguishedgroupo foceanographers andmeteorologistsassembledinRanchoSantaFe,Californi atotrytounderstandthe ChangingPacicOceanin1957and1958 (SetteandIsaacs(1960). There,forperhapsthersttime,theybeganthesynthesisof atmosphericand oceaniceventsleadingtoourpresentunderstandingofElNi ~no. Whileoceanographershadbeenmostlyconcernedwiththeeas ternequatorialPacicandElNi~no,meteorologistshadbeenmostlycon cernedwiththe westerntropicalPacic,thetropicalIndianOcean,andthe SouthernOscillation.Hildebrandsson,theLockyers,andSirGilbertWalker noticedintheearly decadesofthe20thcenturythatpressureructuationsthrou ghoutthatregion arehighlycorrelatedwithpressureructuationsinmanyoth erregionsofthe world(gure14.6).Becausevariationsinpressureareasso ciatedwithwinds andrainfall,theywantedtondoutifpressureinoneregion couldbeusedto forecastweatherinotherregionsusingthecorrelations. Theearlystudiesfoundthatthetwostrongestcentersofthe variabilityare nearDarwin,AustraliaandTahiti.TheructuationsatDarwi nareopposite thoseatTahiti,andresembleanoscillation.Furthermore, thetwocentershad strongcorrelationswithpressureinareasfarfromthePaci c.Walkernamed theructuationsthe SouthernOscillation The SouthernOscillationIndex issea-levelpressureatTahitiminussea-level pressureatDarwin(gure14.7)normalizedbythestandardd eviationofthe dierence.Theindexisrelatedtothetradewinds.Whenthei ndexishigh,the pressuregradientbetweeneastandwestinthetropicalPaci cislarge,andthe tradewindsarestrong.Whentheindexisnegativetrades,ar eweak. 2 2 0 2 2 0L0 2 4 0 -2 8 6 4H-2 -4 -4 2 6 4 0 L L L2 0 -2L L -5 -6 -8 -4 -6 -2 0 H 2 -2 0 0 -2 -2 0 -2 0 0 2 2 2L -60o-40o0o-20o20o40o60o80o60o0o12 0o-12 0o18 0o-6 0o Figure14.6Correlationcoecientofannual-meansea-leve lpressurewithpressureat Darwin.{{{{Coecient < 0 : 4.AfterTrenberthandShea(1987).

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14.2.ELNI ~ NO 243 Normalized Southern Oscillation Index -4 -3 -2 -1 0 1 2 3 19501955196019651970197519801985199019952000 DateNormalized Index Figure14.7NormalizedSouthernOscillationIndexfrom195 1to1999.Thenormalizedindex issea-levelpressureanomalyatTahitidividedbyitsstand arddeviationminussea-level pressureanomalyatDarwindividedbyitsstandarddeviatio nthenthedierenceisdivided bythestandarddeviationofthedierence.Themeansarecal culatedfrom1951to1980. Monthlyvaluesoftheindexhavebeensmoothedwitha5-month runningmean.StrongEl Ni~noeventsoccurredin1957{58,1965{66,1972{73,1982{8 3,1997{98.Datafrom noaa TheconnectionbetweentheSouthernOscillationandElNi~n owasmadesoon aftertheRanchoSantaFemeeting.IchiyeandPetersen(1963 )andBjerknes (1966)noticedtherelationshipbetweenequatorialtemper aturesinthePacic duringthe1957ElNi~noandructuationsinthetradewindsas sociatedwiththe SouthernOscillation.ThetheorywasfurtherdevelopedbyW yrtki(1975). BecauseElNi~noandtheSouthernOscillationaresoclosely related,the phenomenonisoftenreferredtoasthe ElNi~no{SouthernOscillation or enso Morerecently,theoscillationisreferredtoasElNi~no/La Ni~na,whereLaNi~na referstothepositivephaseoftheoscillationwhentradewi ndsarestrong,and watertemperatureintheeasternequatorialregionisveryc old. DenitionofElNi~no Philander(1990)pointsoutthateachElNi~nois unique,withdierenttemperature,pressure,andrainfall patterns.Someare strong,someareweak.So,exactlywhateventsdeservetobec alledElNi~no? The icoads datashowthatthebestindicatorofElNi~noissea-levelpre ssure anomalyintheeasternequatorialPacicfrom4 Sto4 Nandfrom108 Wto 98 W(HarrisonandLarkin,1996).Itcorrelatesbetterwithsea -surfacetemperatureinthecentralPacicthanwiththeSouthern-Oscil lationIndex.Thus theimportanceoftheElNi~noisnotexactlyproportionalto theSouthernOscillationIndex|thestrongElNi~noof1957{58,hasaweakersig nalingure14.7 thantheweakerElNi~noof1965{66. Trenberth(1997)recommendsthatthosedisruptionsofthee quatorialsysteminthePacicshallbecalledanElNi~noonlywhenthe5-mo nthrunning meanofsea-surfacetemperatureanomaliesintheregion5 N{5 S,120 W{ 170 Wexceeds0.4 Cforsixmonthsorlonger. SoElNi~no,whichstartedlifeasachangeincurrentsoPeru eachChristmas, hasgrownintoagiant.Itnowmeansadisruptionoftheoceanatmosphere systemoverthewholeequatorialPacic.

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244 CHAPTER14.EQUATORIALPROCESSES TheoryofElNi~no Wyrtki(1975)givesacleardescriptionofElNi~no. DuringthetwoyearsprecedingElNi~no,excessivelystrong southeast tradesarepresentinthecentralPacic.Thesestrongsouth easttrades intensifythesubtropicalgyreoftheSouthPacic,strengt hentheSouth EquatorialCurrent,andincreasetheeast-westslopeofsea levelbybuildingupwaterinthewesternequatorialPacic.Assoonasthew indstress inthecentralPacicrelaxes,theaccumulatedwaterrowsea stward,probablyintheformofanequatorialKelvinwave.Thiswaveleads totheaccumulationofwarmwateroEcuadorandPeruandtoadepressi onofthe usuallyshallowthermocline.Intotal,ElNi~noistheresul toftheresponse oftheequatorialPacictoatmosphericforcingbythetrade winds. SometimesthetradesinthewesternequatorialPacicnoton lyweaken, theyactuallyreversedirectionforafewweekstoamonth,pr oducing westerly windbursts thatquicklydeepenthethermoclinethere.Thedeepeningof the thermoclinelaunchesaneastwardpropagatingKelvinwavea ndawestward propagatingRossbywave.(Ifyouareasking,WhatareKelvin andRossby waves?Iwillanswerthatinaminute.Sopleasebepatient.) TheKelvinwavedeepensthethermoclineasitmoveseastward ,anditcarries warmwatereastward.Bothprocessescauseadeepeningofthe mixedlayerin theeasternequatorialPacicafewmonthsafterthewaveisl aunchedinthe westernPacic.Thedeeperthermoclineintheeastleadstou pwellingofwarm water,andthesurfacetemperaturesoshoreofEcuadorandP eruwarmsby2{ 4 .Thewarmwaterreducesthetemperaturecontrastbetweenea standwest, furtherreducingthetrades.Thestrongpositivefeedbackb etweensea-surface temperatureandthetradewindscausesrapiddevelopmentof ElNi~no. Withtime,thewarmpoolspreadseast,eventuallyextending asfaras140 W (gure14.8).Plus,waterwarmsintheeastalongtheequator duetoupwelling ofwarmwater,andtoreducedadvectionofcoldwaterfromthe eastdueto weakertradewinds. Thewarmwatersalongtheequatorintheeastcausetheareaso fheavyrain tomoveeastwardfromMelanesiaandFijitothecentralPaci c.Essentially, amajorsourceofheatfortheatmosphericcirculationmoves fromthewestto thecentralPacic,andthewholeatmosphererespondstothe change.Bjerknes (1972),describingtheinteractionbetweentheoceanandth eatmosphereover theeasternequatorialPacicconcluded: Inthecoldoceancase(1964)theatmospherehasapronounced stable layerbetween900and800mb,preventingconvectionandrain fall,andin thewarmcase(1965)theheatsupplyfromtheoceaneliminate stheatmosphericstabilityandactivatesrainfall....Asideeecto fthewidespread warmingofthetropicalbeltoftheatmosphereshowsupinthe increase ofexchangeofangularmomentumwiththeneighboringsubtro picalbelt, wherebythesubtropicalwesterlyjetstrengthens...Theva riabilityofthe heatandmoisturesupplytotheglobalatmosphericthermale nginefrom theequatorialPaciccanbeshowntohavefar-reachinglarg e-scaleeects. KlausWyrtki(1985),drawingonextensiveobservationsofE lNi~no,writes:

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14.2.ELNI ~ NO 245 0.2 0.2 0.2 0.2 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.2 -0.2 0 0 -0.2 0 0 0 0 0 0 0 0 0 0 0.2 -0.2 -0.4 -0.4 -0.2 -0.6 0.2 0.4 0.6 0.6 0.6 0.8 0.2 0.2 -0.2 -0.2 -0.2 0.2 0.6 0.2 -0.2 0.2 0.4 0.4 -0.2 0.4 -0.2 -0.2 -0.4 0.6 0 0 0 0 0 0 0 -0.2 -0.2 1.6 1.4 1.2 1.0 0.2 0 0 0 0.2 0.2 0 0 0 -0.2 -0.2 -0.4 -0.2 -0.2 -0.4 -0.4 0.4 -0.6 1.0 -0.2 1.2 0 0.4 0.6 0.2 1.0 0.2 0.8 0.8 1.0 0 -0.2 1.4 1.2 -0.2 -0.2 0.2 0.2 0.4 0.4 0.2 0 0.2 0.2 0 0 0.2 -0.2 0 0 0 0 0 0 0.6 0.4 0.4 0.8 0.6 0.8 0.6 0.4 0.8 1.0 (a) March-May (b) August-October (b) December-February (d) May-July 30o20o10o0o-10o-20o-30o100o120o140o160o180o-160o-140o-120o-100o-80o30o20o10o0o-10o-20o-30o100o120o140o160o180o-160o-140o-120o-100o-80o30o20o10o0o-10o-20o-30o100o120o140o160o180o-160o-140o-120o-100o-80o30o20o10o0o-10o-20o-30o100o120o140o160o180o-160o-140o-120o-100o-80o Figure14.8Anomaliesofsea-surfacetemperature(in C)duringatypicalElNi~noobtained byaveragingdatafromElNi~nosbetween1950and1973.Month sareaftertheonsetofthe event.AfterRasmussonandCarpenter(1982).

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246 CHAPTER14.EQUATORIALPROCESSES AcompleteElNi~nocycleresultsinanetheatdischargefrom thetropical Pacictowardhigherlatitudes.Attheendofthecyclethetr opicalPacic isdepletedofheat,whichcanonlyberestoredbytheslowacc umulation ofwarmwaterinthewesternPacicbythenormaltradewinds. Consequently,thetimescaleoftheSouthernOscillationisgiven bythetime requiredfortheaccumulationofwarmwaterinthewesternPa cic. ItisthesefarreachingeventsthatmakeElNi~nosoimportan t.Fewpeople careaboutwarmwateroPeruaroundChristmas,manycareabo utglobal changestheweather.ElNi~noisimportantbecauseofitsatm osphericinruence. WhentheKelvinwavereachesthecoastofEcuador,partisrer ectedasan westwardpropagatingRossbywave,andpartpropagatesnort handsouthas acoastaltrappedKelvinwavecarryingwarmwatertohigherl atitudes.For example,duringthe1957ElNi~no,thenorthwardpropagatin gKelvinwave producedunusuallywarmwateroshoreofCalifornia,andit eventuallyreached Alaska.ThiswarmingofthewestcoastofNorthAmericafurth erinruences climateinNorthAmerica,especiallyinCalifornia. AstheKelvinwavemovesalongthecoast,itforcesRossbywav eswhich movewestacrossthePacicatavelocitythatdependsonthel atitude(14.4). Thevelocityisveryslowathighlatitudesandfastestonthe equator,where thererectedwavemovesbackasadeepeningofthethermoclin e,reachingthe centralequatorialPacicayearlater.Similarly,thewest wardpropagating RossbywavelaunchedatthestartoftheElNi~nointhewest,r erectsoAsia andreturnstothecentralequatorialPacicasaKelvinwave ,againabouta yearlater. ElNi~noendswhentheRossbywavesrerectedfromAsiaandEcu adormeet inthecentralPacicaboutayearaftertheonsetofElNi~no( Picaut,Masia,and duPenhoat,1997).Thewavespushthewarmpoolatthesurface towardthe west.Atthesametime,theRossbywavererectedfromthewest ernboundary causesthethermoclineinthecentralPacictobecomeshall owerwhenthe wavesreachesthecentralPacic.Thenanystrengtheningof thetradescauses upwellingofcoldwaterintheeast,whichincreasestheeast -westtemperature gradient,whichincreasesthetrades,whichincreasestheu pwelling(Takayabuet al1999).ThesystemisthenthrownintotheLaNi~nastatewit hstrongtrades, andaverycoldtonguealongtheequatorintheeast. LaNi~natendstolastlongerthanElNi~no,andthecyclefrom LaNi~natoEl Ni~noandbacktakesaboutthreeyears.Thecycleisn'texact .ElNi~nocomes backatintervalsfrom2-7years,withanaveragenearfourye ars(gure14.7). EquatorialKelvinandRossbyWaves KelvinandRossbywavesarethe ocean'swayofadjustingtochangesinforcingsuchaswester lywindbursts. Theadjustmentoccursaswavesofcurrentandsealevelthata reinruencedby gravity,Coriolisforce f ,andthenorth-southvariationofCoriolisforce @f=@y = .Therearemanykindsofthesewaveswithdierentfrequenci es,wavelengths, andvelocities.Ifgravityand f aretherestoringforces,thewavesarecalled KelvinandPoincarewaves.If istherestoringforce,thewavesarecalled planetarywaves.Oneimportanttypeofplanetarywaveisthe Rossbywave.

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14.2.ELNI ~ NO 247 H y xr1r2h O EastNorth Figure14.9Sketchofthetwo-layermodeloftheequatorialo ceanusedtocalculateplanetary wavesinthoseregions.AfterPhilander(1990:107). TwotypesofwavesareespeciallyimportantforElNi~no:int ernalKelvin wavesandRossbywaves.Bothwavescanhavemodesthatarecon nedto anarrow,north-southregioncenteredontheequator.These are equatorially trappedwaves .Bothexistinslightlydierentformsathigherlatitudes. KelvinandRossbywavetheoryisbeyondthescopeofthisbook ,soIwill justtellyouwhattheyarewithoutderivingthepropertieso fthewaves.Ifyou arecurious,youcanndthedetailsinPhilander(1990):Cha pter3;Pedlosky (1987):Chapter3;andApel(1987): x 6.10{6.12.Ifyouknowlittleaboutwaves, theirwavelength,frequency,groupandphasevelocities,s kiptoChapter16and read x 16.1. Thetheoryforequatorialwavesisbasedonatwo-layermodel oftheocean (gure14.9).Becausethetropicaloceanhaveathin,warm,s urfacelayerabove asharpthermocline,suchamodelisagoodapproximationfor thoseregions. Equatorial-trappedKelvinwavesarenon-dispersive,with groupvelocity: c Kg = c p g 0 H ;where g 0 = 2 1 1 g (14.2) g 0 is reducedgravity 1 ; 2 arethedensitiesaboveandbelowthethermocline, and g isgravity.TrappedKelvinwavespropagateonlytotheeast. Note,that c isthephaseandgroupvelocityofashallow-water,internal ,gravitywave.Itis themaximumvelocityatwhichdisturbancescantravelalong thethermocline. Typicalvaluesofthequantitiesin(14.2)are: 2 1 1 =0 : 003; H =150m; c =2 : 1m/s Attheequator,Kelvinwavespropagateeastwardatspeedsof upto3m/s,and theycrossthePacicinafewmonths.Currentsassociatedwi ththewaveare everywhereeastwardwithnorth-southcomponent(gure14. 10). Kelvinwavescanalsopropagatepolewardasatrappedwaveal onganeast coastofanoceanbasin.Theirgroupvelocityisalsogivenby (14.3),andthey areconnedtoacoastalzonewithwidth x = c= ( y )

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248 CHAPTER14.EQUATORIALPROCESSES 2.1 .3 .3 2.1 h(x,y) cm Latitude (km)1000 -1000 Longitute (km) -2000020004000 0 -2000020004000 Equator Longitute (km) Figure14.10 Left: Horizontalcurrentsassociatedwithequatoriallytrapped wavesgenerated byabell-shapeddisplacementofthethermocline. Right: Displacementofthethermocline duetothewaves.Theguresshowsthatafter20days,theinit ialdisturbancehasseparated intoanwestwardpropagatingRossbywave(left)andaneastw ardpropagatingKelvinwave (right).AfterPhilanderetal.(1984:120). TheimportantRossbywavesontheequatorhavefrequenciesm uchlessthan theCoriolisfrequency.Theycantravelonlytothewest.The groupvelocityis: c Rg = c (2 n +1) ; n =1 ; 2 ; 3 ;::: (14.3) Thefastestwavetravelswestwardatavelocitynear0.8m/s. Thecurrents associatedwiththewavearealmostingeostrophicbalancei ntwocounterrotatingeddiescenteredontheequator(gure14.10). Awayfromtheequator,low-frequency,long-wavelengthRos sbywavesalso travelonlytothewest,andthecurrentsassociatedwiththe wavesareagain almostingeostrophicbalance.Groupvelocitydependsstro nglyonlatitude: c Rg = g 0 H f 2 (14.4) Wavedynamicsintheequatorialregionsdiermarkedlyfrom wavedynamics atmid-latitudes.Baroclinicwavesaremuchfaster,andthe responseoftheocean tochangesinwindforcingismuchfasterthanatmid-latitud es.Forplanetary wavesneartheequator,wecanspeakofan equatorialwaveguide Now,let'sreturntoElNi~noandits\far-reachinglarge-sc aleeects." 14.3ElNi~noTeleconnections Teleconnections arestatisticallysignicantcorrelationsbetweenweathe reventsthatoccuratdierentplacesontheearth.Figure14.1 1showsthedominantglobalteleconnectionsassociatedwiththeElNi~no/S outhernOscillation. Theinruenceof enso isthroughitsinruenceonconvectionandassociatedlatentheatreleaseintheequatorialPacic.Astheare aofheavyrain moveseast,thesourceofatmosphericheatingmoveswiththe rain,leadingto widespreadchangesinatmosphericcirculationandweather patternsoutsidethe tropicalPacic(McPhaden,ZebiakandGlantz,2006),inclu dingperturbations inatmosphericpressure(gure14.12).Thissequenceofeve ntsleadstosome predictabilityofweatherpatternsaseasoninadvanceover NorthAmerica, Brazil,Australia,SouthAfricaandotherregions.

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14.3.ELNI ~ NOTELECONNECTIONS 249 Jun(0)-Sep(0) Sep(0)-May(+) Oct(0)-Dec(0) Nov(0)-May(+) Jun(0)-Nov(0) Sep(0)-Mar(+) Mar(0)-Feb(+) May(0)-Oct(0) Nov(0)-May(+) May(0)-Apr(+) Apr(0)-Mar(+) Jul(0)-Jun(+) Apr(0)-Oct(0) Insufficientstation datafor analysis Jul(0)-Mar(+) Oct(0)-Mar(+) Nov(0)-Feb(+) 0o40o80o120o160o-160o-120o-80o-40o60o40o20o0o-20o-40o-60o180o Figure14.11Sketchofregionsreceivingenhancedrain(das hedlines)ordrought(solidlines) duringanElNi~noevent.(0)indicatesthatrainchangeddur ingtheyearinwhichElNi~no began,(+)indicatesthatrainchangedduringtheyearafter ElNi~nobegan.AfterRopelewski andHalpert(1987). The enso perturbationstomid-latitudeandtropicalweathersystem sleads todramaticchangesinrainfallinsomeregions(gure14.11 ).Astheconvective regionsmigrateeastalongtheequator,theybringraintoth enormallyarid, central-Pacicislands.ThelackofrainthewesternPacic leadstodroughtin IndonesiaandAustralia. EQ NP GM Figure14.12Changingpatternsofconvectionintheequator ialPacicduringanElNi~no, setupapatternofpressureanomaliesintheatmosphere(sol idlines)whichinruencethe extratropicalatmosphere.AfterRasmussonandWallace(19 83).

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250 CHAPTER14.EQUATORIALPROCESSES -2-1 0 123 Southern Oscillation IndexAverage Texas Rainfall (inches) 4540353025201510 Figure14.13Correlationofyearlyaveragedrainfallavera gedoverTexasplottedasa functionoftheSouthernOscillationIndexaveragedforthe year.FromStewart(1995). AnExample:VariabilityofTexasRainfall Figure14.11showsaglobal viewofteleconnections.Let'szoomintooneregion,Texas, thatIchoseonly becauseIlivethere.Theglobalgureshowsthattheregions houldhavehigher thannormalrainfallinthewinterseasonafterElNi~nobegi ns.IthereforecorrelatedyearlyaveragedrainfallforthestateofTexastotheS outhernOscillation Index(gure14.13).WetyearscorrespondtoElNi~noyearsi ntheequatorial Pacic.DuringElNi~no,convectionnormallyfoundinthewe sternequatorial PacicmovedeastintothecentralequatorialPacic.Thesu btropicaljetalso moveseast,carryingtropicalmoistureacrossMexicotoTex asandtheMississippiValley.Coldfrontsinwinterinteractwiththeupperl evelmoistureto produceabundantwinterrainsfromTexaseastward.14.4ObservingElNi~no ThetropicalandequatorialPacicisavast,remoteareasel domvisitedby ships.Toobservetheregion noaa 'sPacicMarineEnvironmentalLaboratory inSeattledeployedanarrayofbuoystomeasureoceanograph icandmeteorologicalvariables(gure14.14).Therstbuoywassuccessf ullydeployedin1976 byDavidHalpern.Sincethen,newmooringshavebeenaddedto thearray,new instrumentshavebeenaddedtothemoorings,andthemooring shavebeenimproved.TheprogramhasnowevolvedintotheTropicalAtmosp hereOcean tao arrayofapproximately70deep-oceanmooringsspanningthe equatorialPacic Oceanbetween8 Nand8 Sfrom95 Wto137 E(McPhadenetal,1998). ThearraybeganfulloperationinDecember1994,anditconti nuestoevolve. Theworknecessarytodesignandcalibrateinstruments,dep loymoorings,and processdataiscoordinatedthroughthe tao Project.Itisamulti-national eortinvolvingtheUnitedStates,Japan,Korea,Taiwan,an dFrancewitha projectoceatthePacicMarineEnvironmentalLaboratory

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14.5.FORECASTINGELNI ~ NO 251 Tropical Atmosphere Ocean (TAO) Array AtlasCurrent Meter 180o160o140o120o-160o-140o-120o-100o-80o20o0o10o-10o-20o Figure14.14TropicalAtmosphereOcean tao arrayofmooredbuoysoperatedbythe noaa PacicMarineEnvironmentalLaboratorywithhelpfromJapa n,Korea,Taiwan,andFrance. Figurefrom noaa PacicMarineEnvironmentalLaboratory. The tao mooringsmeasureairtemperature,relativehumidity,surf acewind velocity,sea-surfacetemperatures,andsubsurfacetempe raturesfrom10meters downto500meters.Fivemooringslocatedontheequatorat11 0 W,140 W, 170 W,165 E,and147 Ealsocarryupward-lookingAcousticDopplerCurrent Prolers adcp tomeasureupper-oceancurrentsbetween10mand250m.Data aresentbackthroughtheArgossystem,anddataareprocesse dandmade availableinnearrealtime.Themooringsarerecoveredandr eplacedyearly.All sensorsarecalibratedpriortodeploymentandafterrecove ry. Datafrom tao aremergedwithaltimeterdatafromJasin,and ers -2toobtainamorecomprehensivemeasurementofElNi~no.Jasinand Topex/Poseidon datahavebeenespeciallyusefulbecausetheycouldbeusedt oproduceaccuratemapsofsealeveleverytendays.Themapsprovideddetai ledviewsof thedevelopmentofthe1997{1998ElNi~noinnearrealtimeth atwerewidely reproducedthroughouttheworld.Theobservations(gure1 0.6)showhighsea levelpropagatingfromwesttoeast,peakingintheeasterne quatorialPacic inNovember1997.Inaddition,satellitedataextendedbeyo ndthe tao data regiontoincludetheentiretropicalPacic.Thisallowedo ceanographersto lookforextra-tropicalinruencesonElNi~no. Rainratesaremeasuredby nasa 'sTropicalRainfallMeasuringMission whichwasspeciallydesignedforthispurpose.Itwaslaunch edon27November 1997,anditcarriesveinstruments:therstspacebornepr ecipitationradar, ave-frequencymicrowaveradiometer,avisibleandinfrar edscanner,acloud andearthradiantenergysystem,andalightningimagingsen sor.Working together,theinstrumentsprovidedatanecessarytoproduc emonthlymapsof tropicalrainfallaveragedover500kmby500kmareaswith15 %accuracy. Thegridisglobalbetween 35 latitude.Inaddition,thesatellitedataare usedtomeasurelatentheatreleasedtotheatmospherebyrai n,thusproviding continuousmonitoringofheatingoftheatmosphereinthetr opics. 14.5ForecastingElNi~no TheimportanceofElNi~notoglobalweatherpatternshasled tomany schemesforforecastingeventsintheequatorialPacic.Se veralgenerations ofmodelshavebeenproduced,buttheskilloftheforecastsh asnotalways

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252 CHAPTER14.EQUATORIALPROCESSES increased.Modelsworkedwellforafewyears,thenfailed.F ailurewasfollowed byimprovedmodels,andthecyclecontinued.Thus,thebestm odelsin1991 failedtopredictweakElNi~nosin1993and1994(Ji,Leetmaa ,andKousky, 1996).Thebestmodelofthemid1990sfailedtopredicttheon setofthestrong ElNi~noof1997-1998althoughanewmodeldevelopedbytheNa tionalCenters forEnvironmentalPredictionmadethebestforecastofthed evelopmentofthe event.Ingeneral,themoresophisticatedthemodel,thebet tertheforecasts (Kerr,1998). Thefollowingrecountssomeofthemorerecentworktoimprov etheforecasts.Forsimplicity,IdescribethetechniqueusedbytheN ationalCentersfor EnvironmentalPrediction(Ji,Behringer,andLeetmaa,199 8).ButChenetal. (1995),Latifetal.(1993),andBarnettetal.(1993),among others,haveall developedusefulpredictionmodels. AtmosphericModels Howwellcanwemodelatmosphericprocessesover thePacic?Tohelpanswerthequestion,theWorldClimateRe searchProgram'sAtmosphericModelIntercomparisonProject(Gates, 1992)compared outputfrom30dierentatmosphericnumericalmodelsfor19 79to1988. The VariabilityintheTropics:SynoptictoIntraseasonalTime scales subprojectis especiallyimportantbecauseitdocumentstheabilityof15 atmosphericgeneralcirculationmodelstosimulatetheobservedvariabilityin thetropicalatmosphere(Slingoetal.1995).Themodelsincludedseveralope ratedbygovernmentweatherforecastingcenters,includingthemodelu sedforday-to-day forecastsbytheEuropeanCenterforMedium-RangeWeatherF orecasts. Therstresultsindicatethatnoneofthemodelswereableto duplicateall importantinterseasonalvariabilityofthetropicalatmos phereontimescalesof 2to80days.Modelswithweakintraseasonalactivitytended tohaveaweak annualcycle.Mostmodelsseemedtosimulatesomeimportant aspectsofthe interannualvariabilityincludingElNi~no.Thelengthoft hetimeserieswas, however,tooshorttoprovideconclusiveresultsoninteran nualvariability. Theresultsofthesubstudyimplythatnumericalmodelsofth eatmospheric generalcirculationneedtobeimprovediftheyaretobeused tostudytropical variabilityandtheresponseoftheatmospheretochangesin thetropicalocean. OceanicModels OurabilitytounderstandElNi~noalsodependsonour abilitytomodeltheoceaniccirculationintheequatorialP acic.Becausethe modelsprovidetheinitialconditionsusedfortheforecast s,theymustbeableto assimilateup-to-datemeasurementsofthePacicalongwit hheatruxesandsurfacewindscalculatedfromtheatmosphericmodels.Themeas urementsinclude sea-surfacewindsfromscatterometersandmooredbuoys,su rfacetemperature fromtheoptimal-interpolationdataset(see x 6.6),subsurfacetemperaturesfrom buoysand xbt s,andsealevelfromaltimetryandtide-gaugesonislands. Ji,Behringer,andLeetmaa(1998)attheNationalCentersfo rEnvironmentalPredictionhavemodiedtheGeophysicalFluidDynam icsLaboratory's ModularOceanModelforuseinthetropicalPacic(see x 15.3formoreinformationaboutthismodel).It'sdomainisthePacicbetween4 5 Sand55 N andbetween120 Eand70 W.Thezonalresolutionis1.5 .Themeridional resolutionis 1/3 within10 oftheequator,increasingsmoothlyto1 pole-

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14.5.FORECASTINGELNI ~ NO 253 wardof20 latitude.Ithas28verticallevels,with18intheupper400m to resolvethemixedlayerandthermocline.Themodelisdriven bymeanwinds fromHellermanandRosenstein(1983),anomaliesinthewind eldfromFlorida StateUniversity,andmeanheatruxesfromOberhuber(1988) .Itassimilates subsurfacetemperaturefromthe tao arrayand xbt s,andsurfacetemperatures fromthemonthlyoptimal-interpolationdataset(Reynolds andSmith,1994). Theoutputofthemodelisanoceananalysis,thedensityandc urrenteld thatbesttsthedatausedintheanalysis(gures14.3and14 .4).Thisisused todriveacoupledocean-atmospheremodeltoproduceforeca sts. CoupledModels Coupledmodelsareseparateatmosphericandoceanic modelsthatpassinformationthroughtheircommonboundary attheseasurface,thuscouplingthetwocalculations.Thecouplingcanb eoneway,fromthe atmosphere,ortwoway,intoandoutoftheocean.Intheschem eusedbythe noaa NationalCentersforEnvironmentalPredictiontheoceanmo delisthe sameModularOceanModeldescribedabove.Itiscoupledtoal ow-resolution versionoftheglobal,medium-rangeforecastmodeloperate dbytheNational Centers(Kumar,Leetmaa,andJi,1994).Anomaliesofwindst ress,heat,and fresh-waterruxescalculatedfromtheatmosphericmodelar eaddedtothemean annualvaluesoftheruxes,andthesumsareusedtodrivetheo ceanmodel. Sea-surfacetemperaturecalculatedfromtheoceanmodelis usedtodrivethe atmosphericmodelfrom15 Nto15 S. Ascomputerpowerdecreasesincost,modelsarebecomingeve rmorecomplex.Thetrendistoglobalcoupledmodelsabletoincludeot hercoupledoceanatmospheresystemsinadditionto enso .Ireturntotheproblemin x 15.6where Idescribeglobalcoupledmodels. StatisticalModels Statisticalmodelsarebasedonananalysisofweather patternsinthePacicusingdatagoingbackmanydecades.Th ebasicideais thatifweatherpatternstodayaresimilartopatternsatsom etimeinthepast, thentodayspatternswillevolveastheydidatthatpasttime .Forexample,if windsandtemperaturesinthetropicalPacictodayaresimi lartowindand temperaturesjustbeforethe1976ElNi~no,thenwemightexp ectasimilarEl Ni~notostartinthenearfuture. Forecasts Ingeneral,thecoupledocean-atmospheremodelsproducefo recaststhatarenobetterthanthestatisticalforecasts(Jan vanOldenborgh, 2005).TheforecastsincludenotonlyeventsinthePacicbu talsotheglobal consequencesofElNi~no.Theforecastsarejudgedtwoways: 1.Usingthecorrelationbetweenthearea-averagedsea-sur face-temperature anomaliesfromthemodelandtheobservedtemperatureanoma liesinthe easternequatorialPacic.Theareaisusuallyfrom170 Wto120 W between5 Sand5 N.Usefulforecastshavecorrelationsexceeding0.6. 2.Usingtheroot-mean-squaredierencebetweentheobserv edandpredicted sea-surfacetemperatureinthesamearea. Theforecastsoftheverystrong1997ElNi~nohavebeencaref ullystudied. JanvanOldenborgetal(2005)andBarnstonetal(1999)found nomodels successfullyforecasttheearliestonsetoftheElNi~noinl ate1996andearly

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254 CHAPTER14.EQUATORIALPROCESSES 1997.TherstformalannouncementsoftheElNi~noweremade inMay1997. Nordidanymodelforecastthelargetemperatureanomalieso bservedinthe easternequatorialPacicuntiltheareahadalreadywarmed .Therewasno cleardistinctionbetweentheaccuracyofthedynamicalors tatisticalforecasts. 14.6ImportantConcepts 1.Equatorialprocessesareimportantbecauseheatrelease dbyraininthe equatorialregionhelpsdrivesmuchoftheatmosphericcirc ulation. 2.SolarenergyabsorbedbythePacicisthemostimportantd riverofatmosphericcirculation.Solarenergyislostfromtheoceanm ainlybyevaporation.Theheatwarmstheatmosphereanddrivesthecircul ationwhen thelatentheatofevaporationisreleasedinrainyareas,pr imarilyinthe westerntropicalPacicandtheIntertropicalConvergence Zone. 3.Theinterannualvariabilityofcurrentsandtemperature sintheequatorial Pacicmodulatestheoceanicforcingoftheatmosphere.Thi sinterannual variabilityisassociatedwithElNi~no/LaNi~na. 4.Changesinequatorialdynamicscausechangesinatmosphe riccirculation bychangingthelocationofrainintropicalPacicandthere forethelocationofthemajorheatsourcedrivingtheatmosphericcircul ation. 5.ElNi~nocausesthebiggestchangesinequatorialdynamic s.DuringEl Ni~no,trade-windsweakeninthewesternPacic,thethermo clinebecomes lessdeepinthewest.ThisdrivesaKelvinwaveeastwardalon gtheequator,whichdeepensthethermoclineintheeasternPacic.Th ewarmpool inthewestmoveseastwardtowardthecentralPacic,andthe intense tropicalrainareasmovewiththewarmpool. 6.ElNi ~ [n]oisthelargestsourceofyear-to-yearructuationsingl obalweather patterns. 7.AsaresultofElNi~no,droughtoccursintheIndonesianar eaandnorthern Australia,androodsoccurinwestern,tropicalSouthAmeri ca.Variations intheatmosphericcirculationinruencemoredistantareas throughteleconnections. 8.ForecastsofElNi~noaremadeusingcoupledocean-atmosp hericnumericalmodels.Forecastsappeartohaveusefulaccuracyfor3{6 monthsin advance,mostlyaftertheonsetofElNi~no.

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Chapter15NumericalModelsWesawearlierthatanalyticsolutionsoftheequationsofmo tionareimpossible toobtainfortypicaloceanicrows.Theproblemisduetononlineartermsin theequationsofmotion,turbulence,andtheneedforrealis ticshapesforthesea roorandcoastlines.Wehavealsoseenhowdicultitistodes cribetheocean frommeasurements.Satellitescanobservesomeprocessesa lmosteverywhere everyfewdays.Buttheyobserveonlysomeprocesses,andonl ynearorat thesurface.Shipsandroatscanmeasuremorevariables,and deeperintothe water,butthemeasurementsaresparse.Hence,numericalmo delsprovidethe onlyuseful,globalviewofoceancurrents.Let'slookatthe accuracyandvalidity ofthemodels,keepinginmindthatalthoughtheyareonlymod els,theyprovide aremarkablydetailedandrealisticviewoftheocean.15.1Introduction{SomeWordsofCaution Numericalmodelsofoceancurrentshavemanyadvantages.Th eysimulate rowsinrealisticoceanbasinswitharealisticsearoor.The yincludetheinruenceofviscosityandnon-lineardynamics.Andtheycanca lculatepossible futurerowsintheocean.Perhaps,mostimportant,theyinte rpolatebetween sparseobservationsoftheoceanproducedbyships,drifter s,andsatellites. Numericalmodelsarenotwithoutproblems.\Thereisaworld ofdierence betweenthecharacterofthefundamentallaws,ontheonehan d,andthenature ofthecomputationsrequiredtobreathelifeintothem,onth eother"|Berlinski (1996).Themodelscannevergivecompletedescriptionsoft heoceanicrows eveniftheequationsareintegratedaccurately.Theproble msarisefromseveral sources. Discreteequationsarenotthesameascontinuousequations InChapter7 wewrotedownthedierentialequationsdescribingthemoti onofacontinuous ruid.Numericalmodelsusealgebraicapproximationstothe dierentialequations.Weassumethattheoceanbasinsarelledwithagridof points,and timemovesforwardintinysteps.Thevalueofthecurrent,pr essure,temperature,andsalinityarecalculatedfromtheirvaluesatnearb ypointsandprevious times.IanStewart(1992),anotedmathematician,pointsou tthat 255

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256 CHAPTER15.NUMERICALMODELS Discretizationisessentialforcomputerimplementationa ndcannotbedispensedwith.Theessenceofthedicultyisthatthedynamics ofdiscrete systemsisonlylooselyrelatedtothatofcontinuoussystem s|indeedthe dynamicsofdiscretesystemsisfarricherthanthatoftheir continuous counterparts|andtheapproximationsinvolvedcancreates purioussolutions. Calculationsofturbulencearedicult. Numericalmodelsprovideinformationonlyatgridpointsofthemodel.Theyprovidenoinforma tionaboutthe rowbetweenthepoints.Yet,theoceanisturbulent,andanyo ceanicmodel capableofresolvingtheturbulenceneedsgridpointsspace dmillimetersapart, withtimestepsofmilliseconds. Practicaloceanmodelshavegridpointsspacedtenstohundr edsofkilometersapartinthehorizontal,andtenstohundredsofmetersa partinthevertical. Thismeansthatturbulencecannotbecalculateddirectly,a ndtheinruenceof turbulencemustbeparameterized.Holloway(1994)statest heproblemsuccinctly: Oceanmodelsretainfewerdegreesoffreedomthantheactual ocean (byabout20ordersofmagnitude).Wecompensatebyapplying `eddyviscousgoo'tosquashmotionatallbutthesmallestretaine dscales. (Wealsousenon-conservativenumerics.)Thisisanalogous toplacinga partitioninaboxtopreventgasmoleculesfrominvadingano therregion ofthebox.Ouroceanicmodelscannotinvademostoftherealo ceanic degreesoffreedomsimplybecausethemodelsdonotincludet hem. Giventhatwecannotdothings`right',isitbettertodonoth ing? Thatisnotanoption.`Nothing'meansapplyingviscousgooa ndwishing fortheeverbiggercomputer.Canwedobetter?Forexample,c anwe guessahigherentropycongurationtowardwhichtheeddies tendto drivetheocean(thattendencytocompetewiththeimposedfo rcingand dissipation)? By\degreesoffreedom"Hollowaymeansallpossiblemotions fromthesmallestwavesandturbulencetothelargestcurrents.Let'sdoac alculation.We knowthattheoceanisturbulentwitheddiesassmallasafewm illimeters.To completelydescribetheoceanweneedamodelwithgridpoint sspaced1mm apartandtimestepsofabout1ms.Themodelmustthereforeha ve360 180 (111km/degree) 2 10 12 (mm/km) 2 3km 10 6 (mm/km)=2 : 4 10 27 datapointsfora3kmdeepoceancoveringtheglobe.Thegloba lParallelOcean ProgramModeldescribedinthenextsectionhas2 : 2 10 7 points.Soweneed 10 20 timesmorepointstodescribetherealocean.Thesearethemi ssing10 20 degreesoffreedom. Practicalmodelsmustbesimplerthantherealocean. Modelsoftheocean mustrunonavailablecomputers.Thismeansoceanographers furthersimplify theirmodels.WeusethehydrostaticandBoussinesqapproxi mations,andwe oftenuseequationsintegratedinthevertical,theshallow -waterequations(HaidvogelandBeckmann,1999:37).Wedothisbecausewecannotye trunthemost detailedmodelsofoceaniccirculationforthousandsofyea rstounderstandthe roleoftheoceaninclimate.

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15.2.NUMERICALMODELSINOCEANOGRAPHY 257 Numericalcodehaserrors. Doyouknowofanysoftwarewithoutbugs? Numericalmodelsusemanysubroutineseachwithmanylineso fcodewhich areconvertedintoinstructionsunderstoodbyprocessorsu singothersoftware calledacompiler.Eliminatingallsoftwareerrorsisimpos sible.Withcareful testing,theoutputmaybecorrect,buttheaccuracycannotb eguaranteed. Plus,numericalcalculationscannotbemoreaccuratethant heaccuracyofthe roating-pointnumbersandintegersusedbythecomputer.Ro und-oerrorscannotbeignored.Lawrenceetal(1999),examiningtheoutputo fanatmospheric numericalmodelfoundanerrorinthecodeproducedbythe fortran-90 compilerusedonthe cray Researchsupercomputerusedtorunthecode.They alsofoundround-oerrorsintheconcentrationoftracersc alculatedfromthe model.Botherrorsproducedimportanterrorsintheoutputo fthemodel. Mostmodelsarenotwellveriedorvalidated(Post&Votta,2 005).Yet, withoutadequatevericationandvalidation,outputfromn umericalmodelsis notcredible.Summary Despitethesemanysourcesoferror,mostaresmallinpracti ce. Numericalmodelsoftheoceanaregivingthemostdetailedan dcompleteviews ofthecirculationavailabletooceanographers.Someofthe simulationscontain unprecedenteddetailsoftherow.Iincludedthewordsofwar ningnottolead youtobelievethemodelsarewrong,buttoleadyoutoacceptt heoutputwith agrainofsalt.15.2NumericalModelsinOceanography Numericalmodelsareverywidelyusedformanypurposesinoc eanography. Forourpurposewecandividemodelsintotwoclasses:Mechanisticmodels aresimpliedmodelsusedforstudyingprocesses.Becausethemodelsaresimplied,theoutputiseasiertointer pretthanoutput frommorecomplexmodels.Manydierenttypesofsimpliedm odelshavebeen developed,includingmodelsfordescribingplanetarywave s,theinteractionof therowwithsea-roorfeatures,ortheresponseoftheuppero ceantothewind. Theseareperhapsthemostusefulofallmodelsbecausetheyp rovideinsight intothephysicalmechanismsinruencingtheocean.Thedeve lopmentanduse ofmechanisticmodelsis,unfortunately,beyondthescopeo fthisbook. Simulationmodels areusedforcalculatingrealisticcirculationofoceanic regions.Themodelsareoftenverycomplexbecauseallimpor tantprocessesare included,andtheoutputisdiculttointerpret. TherstsimulationmodelwasdevelopedbyKirkBryanandMic haelCox (Bryan,1969)attheGeophysicalFluidDynamicslaboratory inPrinceton.They calculatedthe3-dimensionalrowintheoceanusingthecont inuityandmomentumequationwiththehydrostaticandBoussinesqapproxima tionsandasimple equationofstate.Suchmodelsarecalled primitiveequation modelsbecause theyusethebasic,orprimitiveformoftheequationsofmoti on.Theequation ofstateallowsthemodeltocalculatechangesindensitydue toruxesofheat andwaterthroughthesurface,sothemodelincludesthermod ynamicprocesses.

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258 CHAPTER15.NUMERICALMODELS TheBryan-Coxmodelusedlargehorizontalandverticalvisc osityanddiusiontoeliminateturbulenteddieshavingdiameterssmalle rabout500km,which isafewgridpointsinthemodel.Ithadcomplexcoastlines,s moothedsea-roor features,andarigidlid.Therigidlidwasneededtoelimina teocean-surface waves,suchastidesandtsunamis,thatmovefartoofastfort hecoarsetime stepsusedbyallsimulationmodels.Therigidlidhad,howev er,disadvantages. Islandssubstantiallyslowedthecomputation,andthesearoorfeatureswere smoothedtoeliminatesteepgradients. Therstsimulationmodelwasregional.Itwasquicklyfollo wedbyaglobal model(Cox,1975)withahorizontalresolutionof2 andwith12levelsinthe vertical.Themodelranfartooslowlyevenonthefastestcom putersoftheday, butitlaidthefoundationformorerecentmodels.Thecoarse spatialresolution requiredthatthemodelhavelargevaluesforviscosity,and evenregionalmodels weretooviscoustohaverealisticwesternboundarycurrent sormesoscaleeddies. Sincethosetimes,thegoalhasbeentoproducemodelswithev ernerresolution,morerealisticmodelingofphysicalprocesses,an dbetternumerical schemes.Computertechnologyischangingrapidly,andmode lsareevolving rapidly.TheoutputfromthemostrecentmodelsofthenorthA tlantic,which haveresolutionof0.03 lookverymuchliketherealocean.Modelsofotherareas showpreviouslyunknowncurrentsnearAustraliaandinthes outhAtlantic. OceanandAtmosphereModels useverydierentspacingofgridpoints. Asaresult,oceanmodelinglagsaboutadecadebehindatmosp heremodeling.Dominantoceaneddiesare1/30thesizeofdominantatmo sphereeddies (storms).But,oceanfeaturesevolveataratethatis1/30th erateintheatmosphere.Thusoceanmodelsrunningforsayoneyearhave(30 30)more horizontalgridpointsthantheatmosphere,buttheyhave1/ 30thenumberof timesteps.Bothhaveaboutthesamenumberofgridpointsint hevertical.As aresult,oceanmodelsrun30timesslowerthanatmospheremo delsofthesame complexity.15.3GlobalOceanModels Severaltypesofglobalmodelsarewidelyusedinoceanograp hy.Mosthave gridpointsaboutonetenthofadegreeapart,whichissucie nttoresolve mesoscaleeddies,suchasthoseseeningures11.10,11.11, and15.2,thathave adiameterlargerthantwotothreetimesthedistancebetwee ngridpoints.Verticalresolutionistypicallyaround30verticallevels.Mo delsinclude:i)realistic coastsandbottomfeatures;ii)heatandwaterruxesthought hesurface;iii) eddydynamics;andiv)themeridional-overturningcircula tion.Manyassimilate satelliteandroatdatausingtechniquesdescribedin x 15.5.Themodelsrange incomplexityfromthosethatcanrunondesktopworkstation stothosethat requiretheworld'sfastestcomputers. Allmodelsmustbeberuntocalculateonetotwodecadesofvar iability beforetheycanbeusedtosimulatetheocean.Thisiscalled spin-up .Spin-up isneededbecauseinitialconditionsfordensity,ruxesofm omentumandheat throughthesea-surface,andtheequationsofmotionarenot allconsistent.

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15.3.GLOBALOCEANMODELS 259 ModelsarestartedfromrestwithvaluesofdensityfromtheL evitus(1982) atlasandintegratedforadecadeusingmean-annualwindstr ess,heatruxes, andwaterrux.Themodelmaybeintegratedforseveralmoreye arsusing monthlywindstress,heatruxes,andwaterruxes. TheBryan-Coxmodelsevolvedintoseveralwidelyusedmodel swhichare providingimpressiveviewsoftheglobaloceancirculation GeophysicalFluidDynamicsLaboratoryModularOceanModel MOM consistsofalargesetofmodulesthatcanbeconguredtorun onmanydierent computerstomodelmanydierentaspectsofthecirculation .Thesourcecode isopenandfree,anditisinthepublicdomain.Themodeliswi delyusefor climatestudiesandforstudyingtheocean'scirculationov erawiderangeof spaceandtimescales(PacanowskiandGries,1999). Because mom isusedtoinvestigateprocesseswhichcoverawiderangeof timeandspacescales,thecodeandmanualarelengthy.Howev er,itisfar fromnecessaryforthetypicaloceanmodelertobecomeacqua intedwith allofitsaspects.Indeed, mom canbelikenedtoagrowingcitywithmany dierentneighborhoods.Someoftheneighborhoodscommuni catewith oneanother,somearemutuallyincompatible,andothersare basically independent.Thisdiversityisquiteachallengetocoordin ateandsupport. Indeed,overtheyearscertain\neighborhoods"havebeenje ttisonedor greatlyrenovatedforvariousreasons.|PacanowskiandGri es. Themodelusesthemomentumequations,equationofstate,an dthehydrostaticandBoussinesqapproximations.Subgrid-scalemoti onsarereducedby useofeddyviscosity.Version4ofthemodelhasimprovednum ericalschemes, afreesurface,realisticbottomfeatures,andmanytypesof mixingincluding horizontalmixingalongsurfacesofconstantdensity.Plus ,itcanbecoupledto atmosphericmodels.ParallelOceanProgramModel producedbySmithandcolleaguesatLos AlamosNationalLaboratory(Maltrudetal,1998)isanother widelyusedmodel growingoutoftheoriginalBryan-Coxcode.Themodelinclud esimproved numericalalgorithms,realisticcoasts,islands,andunsm oothedbottomfeatures. Ithasmodelhas1280 896equallyspacedgridpointsonaMercatorprojection extendingfrom77 Sto77 N,and20levelsinthevertical.Thusithas2 : 2 10 7 pointsgivingaresolutionof0 : 28 0 : 28 cos ,whichvariesfrom0.28 (31.25 km)attheequatorto0.06 (6.5km)atthehighestlatitudes.Theaverage resolutionisabout0 : 2 .Themodelwasisforcedby ecmwf windstressand surfaceheatandwaterruxes(Barnieretal,1995).HybridCoordinateOceanModel hycom Allthemodelsjustdescribed use x;y;z coordinates.Suchacoordinatesystemhasbothadvantagesa nd disadvantages.Itcanhavehighresolutioninthesurfacemi xedlayerandin shallowerregions.Butitislessusefulintheinteriorofth eocean.Belowthe mixedlayer,mixingintheoceaniseasyalongsurfacesofcon stantdensity,and dicultacrosssuchsurfaces.Amorenaturalcoordinatesys temintheinterior oftheoceanuses x;y; ,where isdensity.Suchamodeliscalledan isopycnal

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260 CHAPTER15.NUMERICALMODELS -80o -60o -40o -20o 0o 20o 40o 60o -100o0o Figure15.1Near-surfacegeostrophiccurrentsonOctober1 ,1995calculatedbytheParallel OceanProgramnumericalmodeldevelopedattheLosAlamosNa tionalLaboratory.The lengthofthevectoristhemeanspeedintheupper50moftheoc ean.Thedirectionisthe meandirectionofthecurrent.FromRichardSmith,LosAlamo sNationalLaboratory. model .Essentially, ( z )isreplacedwith z ( ).Becauseisopycnalsurfacesare surfacesofconstantdensity,horizontalmixingisalwayso nconstant-density surfacesinthismodel. TheHybridCoordinateOceanModel hycom modelusesdierentvertical coordinatesindierentregionsoftheocean,combiningthe bestaspectsof z coordinatemodelandisopycnal-coordinatemodel(Bleck,2 002).Thehybrid modelhasevolvedfromtheMiamiIsopycnic-CoordinateOcea nModel(g-

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15.3.GLOBALOCEANMODELS 261 Figure15.2OutputofBleck'sMiamiIsopycnalCoordinateOc eanModel micom .Itisa high-resolutionmodeloftheAtlanticshowingtheGulfStre am,itsvariability,andthe circulationofthenorthAtlantic.FromBleck.ure15.2).Itisaprimitive-equationmodeldrivenbywindst ressandheat ruxes.Ithasrealisticmixedlayerandimprovedhorizontal andverticalmixingschemesthatincludetheinruencesofinternalwaves,sh earinstability,and double-diusion(see x 8.5).Themodelresultsfromcollaborativeworkamong investigatorsatmanyoceanographiclaboratories.RegionalOceanicModelingSystem roms isaregionalmodelthatcanbe imbeddedinmodelsofmuchlargerregions.Itiswidelyusedf orstudyingcoastal currentsystemscloselytiedtorowfurtheroshore,forexa mple,theCalifornia Current. roms isahydrostatic,primitiveequation,terrain-followingm odel usingstretchedverticalcoordinates,drivenbysurfaceru xesofmomentum,heat, andwater.Ithasimprovedsurfaceandbottomboundarylayer s(Shchepetkin andMcWilliams,2004).Climatemodels areusedforstudiesoflarge-scalehydrographicstructure climatedynamics,andwater-massformation.Thesemodelsa rethesameas theeddy-admitting,primitiveequationmodelsIhavejustd escribedexcept thehorizontalresolutionismuchcoarserbecausetheymust simulateocean processesfordecadesorcenturies.Asaresult,theymustha vehighdissipation fornumericalstability,andtheycannotsimulatemesoscal eeddies.Typical horizontalresolutionsare2 to4 .Themodelstend,however,tohavehigh verticalresolutionnecessaryfordescribingthedeepcirc ulationimportantfor climate.

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262 CHAPTER15.NUMERICALMODELS 15.4CoastalModels Thegreateconomicimportanceofthecoastalzonehasledtot hedevelopmentofmanydierentnumericalmodelsfordescribingcoast alcurrents,tides, andstormsurges.Themodelsextendfromthebeachtothecont inentalslope, andtheycanincludeafreesurface,realisticcoastsandbot tomfeatures,river runo,andatmosphericforcing.Becausethemodelsdon'tex tendveryfar intodeepwater,theyneedadditionalinformationaboutdee p-watercurrentsor conditionsattheshelfbreak. Themanydierentcoastalmodelshavemanydierentgoals,a ndmanydifferentimplementations.Severalofthemodelsdescribedab ove,including mom and rom ,havebeenusedtomodelcoastalprocesses.Butmanyothersp ecializedmodelshavealsobeendeveloped.Heaps(1987),Lynchet al(1996),and HaidvogelandBeckman(1998)providegoodoverviewsofthes ubject.Rather thanlookatamenuofmodels,let'slookattwotypicalmodels PrincetonOceanModel developedbyBlumbergandMellor(1987,andMellor,1998)andiswidelyusedfordescribingcoastalcurrent s.Itincludesthermodynamicprocesses,turbulentmixing,andtheBoussinesq andhydrostatic approximations.TheCoriolisparameterisallowedtovaryu singabeta-plane approximation.Becausethemodelmustincludeawiderangeo fdepths,BlumbergandMellorusedaverticalcoordinate scaledbythedepthofthewater: = z H + (15.1) where z = ( x;y;t )istheseasurface,and z = H ( x;y )isthebottom. Sub-gridturbulenceisparameterizedusingaclosureschem eproposedby MellorandYamada(1982)wherebyeddydiusioncoecientsv arywiththe sizeoftheeddiesproducingthemixingandtheshearofthero w. Themodelisdrivenbywindstressandheatandwaterruxesfro mmeteorologicalmodels.Themodelusesknowngeostrophic,tidal,an dEkmancurrents attheouterboundary. Themodelhasbeenusedtocalculatethethree-dimensionald istributionof velocity,salinity,sealevel,temperature,andturbulenc eforupto30daysover aregionroughly100{1000kmonasidewithgridspacingof1{5 0km. DartmouthGulfofMaineModel developedbyLynchetal(1996)isa3dimensionalmodelofthecirculationusingatriangular,n ite-elementgrid.The sizeofthetrianglesisproportionaltobothdepthandthera teofchangeof depth.Thetrianglesaresmallinregionswherethebottomsl opesarelarge andthedepthisshallow,andtheyarelargeindeepwater.The variablemesh isespeciallyusefulincoastalregionswherethedepthofwa tervariesgreatly. Thusthevariablegridgiveshighestresolutionwhereitism ostneeded. Themodelusesroughly13,000trianglestocovertheGulfofM aineand nearbywatersofthenorthAtlantic(gures15.3).Minimums izeoftheelements isroughlyonekilometer.Themodelhas10to40horizontalla yers.Thevertical spacingofthelayersisnotuniform.Layersareclosertoget hernearthetopand bottomandtheyaremorewidelyspacedintheinterior.Minim umspacingis roughlyonemeterinthebottomboundarylayer.

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15.4.COASTALMODELS 263 NEW ENGLANDSHELF NANTUCKET SHOALS60m100m200m1000m300mGREAT SOUTH CHANNELGEORGES BANK GULF OF MAINEWILKONSON BASIN200m300m200m200mGEORGES BASINNORTHEAST CHANNELBROWNS BANK JORDAN BASIN100mBAY OF FUNDY60m200m100m1000mSCOTIAN SHELF100mCAPE SABLE CAPE COD60m 100 km 70o68o66o64o62o40o41o42o43o44o45o 72o Figure15.3 Top :TopographicmapoftheGulfofMaineshowingimportantfeat ures. Inset : Triangular,nite-elementgridusedtocomputerowinthegu lf.Thesizeofthetriangles varieswithdepthandrateofchangeofdepth.AfterLyncheta l,(1996). Themodelintegratesthethree-dimensional,primitiveequ ationsinshallowwaterform.Themodelhasasimpliedequationofstateandad epth-averaged continuityequation,anditusesthehydrostaticandBoussi nesqassumptions. Sub-gridmixingofmomentum,heatandmassisparameterized usingtheMellorandYamada(1982)turbulence-closureschemewhichgive sverticalmixing

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264 CHAPTER15.NUMERICALMODELS coecientsthatvarywithstraticationandvelocityshear .Horizontalmixing coecientswerecalculatedfromSmagorinski(1963).Acare fullychosen,turbulent,eddyviscosityisusedinthebottomboundarylayer.Th emodelisforced bywind,heating,andtidalforcingfromthedeepocean. Themodelisspunupfromrestforafewdaysusingaspeciedde nsityeld atallgridpoints,usuallyfromacombinationof ctd dataplushistoricaldata. Thisgivesavelocityeldconsistentwiththedensityeld. Themodelisthen forcedwithlocalwindsandheatruxestocalculatetheevolu tionofthedensity andvelocityelds.CommentsonCoastalModels Roedetal.(1995)examinedtheaccuracy ofcoastalmodelsbycomparingtheabilityofvemodels,inc ludingBlumberg andMellor'stodescribetherowintypicalcases.Theyfound thatthemodels producedverydierentresults,butthatafterthemodelswe readjusted,the dierenceswerereduced.Thedierenceswereduetodieren cesinverticaland horizontalmixingandspatialandtemporalresolution. Hackettetal.(1995)comparedtheabilityoftwoofthevemo delsto describeobservedrowontheNorwegianshelf.Theyconclude that ...bothmodelsareabletoqualitativelygeneratemanyofth eobserved featuresoftherow,butneitherisabletoquantitativelyre producedetailedcurrents...[Dierences]areprimarilyattributab letoinadequate parameterizationsofsubgridscaleturbulentmixing,tola ckofhorizontal resolutionandtoimperfectinitialandboundarycondition s. Storm-SurgeModels Stormscomingashoreacrosswide,shallow,continental shelvesdrivelargechangesofsealevelatthecoastcalleds tormsurges(see x 17.3foradescriptionofsurgesandprocessesinruencingsu rges).Thesurges cancausegreatdamagetocoastsandcoastalstructures.Int ensestormsinthe BayofBengalhavekilledhundredsofthousandsinafewdaysi nBangladesh. Becausesurgesaresoimportant,governmentagenciesinman ycountrieshave developedmodelstopredictthechangesofsealevelandthee xtentofcoastal rooding. Calculatingstormsurgesisnoteasy.Herearesomereasons, inaroughorder ofimportance. 1.Thedistributionofwindovertheoceanisnotwellknown.N umerical weathermodelscalculatewindspeedataconstantpressures urface,stormsurgemodelsneedwindataconstantheightof10m.Windsinba ysand lagoonstendtobeweakerthanwindsjustoshorebecausenea rbyland distortstheairrow,andthisisnotincludedintheweatherm odels. 2.Theshorewardextentofthemodel'sdomainchangeswithti me.Forexample,ifsealevelrises,waterwillroodinland,andthebou ndarybetween waterandseamovesinlandwiththewater. 3.Thedragcoecientofwindonwaterisnotwellknownforhur ricaneforce winds. 4.Thedragcoecientofwateronthesearoorisalsonotwellk nown.

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15.4.COASTALMODELS 265 5.Themodelsmustincludewavesandtideswhichinruencesea levelin shallowwaters. 6.Stormsurgemodelsmustincludethecurrentsgeneratedin astratied, shallowseabywind. Toreduceerrors,modelsaretunedtogiveresultsthatmatch conditionsseenin paststorms.Unfortunately,thosepastconditionsarenotw ellknown.Changes insealevelandwindspeedarerarelyrecordedaccuratelyin stormsexceptata few,widelypacedlocations.Yetstorm-surgeheightscanch angebymorethan ameteroverdistancesoftensofkilometers. Despitetheseproblems,modelsgiveveryusefulresults.Le t'slookatthe ocial noaa model,andanewexperimentalmodeldevelopedbytheCorpsof Engineers. Sea,Lake,andOverlandSurgesModel slosh isusedby noaa forforecasting stormsurgesproducedbyhurricanescomingashorealongthe AtlanticandGulf coastsoftheUnitedStates(Jelesnianski,Chen,andShaer ,1992). ThemodelistheresultofalifetimeofworkbyChesterJelesn ianski.In developingthemodel,Jelesnianskipaidcarefulattention totherelativeimportanceoferrorsinthemodel.Heworkedtoreducethelargeste rrors,andignored thesmallerones.Forexample,thedistributionofwindsina hurricaneisnot wellknown,soitmakeslittlesensetouseaspatiallyvaryin gdragcoecient forthewind.Thus,Jelesnianskiusedaconstantdragcoeci entintheair,and aconstanteddystresscoecientinthewater. slosh calculateswaterlevelfromdepth-integrated,quasi-line ar,shallowwaterequations.Thusitignoresstratication.Italsoign oresriverinrow, rain,andtides.Thelattermayseemstrange,butthemodelis designedfor forecasting.Thetimeoflandfallcannotbeforecastaccura tely,andhencethe heightofthetidesismostlyunknown.Tidescanbeaddedtoth ecalculated surge,butthenonlinearinteractionoftidesandsurgeisig nored. Themodelisforcedbyidealizedhurricanewinds.Itneedson lyatmospheric pressureatthecenterofthestorm,thedistancefromthecen tertotheareaof maximumwinds,theforecaststormtrackandspeedalongthet rack. Inpreparationforhurricanescomingashorenearpopulated areas,themodel hasbeenadaptedfor27basinsfromBostonHarborMassachuse ttstoLaguna MadreTexas.Themodelusesaxedpolarmesh.Meshspacingbe ginswith anemeshnearthepole,whichislocatednearthecoastalcit yforwhichthe modelisadapted.Thegridstretchescontinuouslytoacoars emeshatdistant boundariesofalargebasin.Suchameshgiveshighresolutio ninbaysandnear thecoastwhereresolutionismostneeded.Usingmeasuredde pthsatseaand elevationsonland,themodelallowsroodingofland,overto ppingofleveesand dunes,andsub-gridrowthroughchannelsbetweenoshoreis lands. Sealevelcalculatedfromthemodelhasbeencomparedwithhe ightsmeasuredbytidegaugesfor13storms,includingBetsy:1965,Ca mile:1969,Donna: 1960,andCarla:1961.Theoverallaccuracyis 20%. AdvancedCirculationModel adcirc isanexperimentalmodelforforecasting stormsurgesproducedbyhurricanescomingashorealongthe AtlanticandGulf

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266 CHAPTER15.NUMERICALMODELS coastsoftheUnitedStates(Graberetal,2006).Themodelus esanite-element grid,theBoussinesqapproximation,quadraticbottomfric tion,andvertically integratedcontinuityandmomentumequationsforrowonaro tatingearth. Itcanberunaseitheratwo-dimensional,depth-integrated model,orasa three-dimensionalmodel.Becausewavescontributetostor msurges,themodel includeswavescalculatedfromthe wam third-geneationwavemodel(see x 16.5). Themodelisforcedby: 1.Highresolutionwindsandsurfacepressureobtainedbyco mbiningweather forecastsfromthe noaa NationalWeatherServiceandtheNationalHurricaneCenteralongtheocialandalternateforecaststorm tracks. 2.Tidesattheopen-oceanboundariesofthemodel.3.Sea-surfaceheightandcurrentsattheopen-oceanbounda riesofthemodel. ThemodelsuccessfullyforecasttheHurricaneKatrinastor msurge,givingvalues inexcessof6.1mnearNewOrleans.15.5AssimilationModels ManyofthemodelsIhavedescribedsofarhaveoutput,suchas current velocityorsurfacetopography,constrainedbyoceanicobs ervationsofthevariablestheycalculate.Suchmodelsarecalled assimilationmodels .Inthissection, Iwillconsiderhowdatacanbeassimilatedintonumericalmo dels. Let'sbeginwithaprimitive-equation,eddy-admittingnum ericalmodelused tocalculatethepositionoftheGulfStream.Let'sassumeth atthemodelis drivenwithreal-timesurfacewindsfromthe ecmwf weathermodel.Using themodel,wecancalculatethepositionofthecurrentandal sothesea-surface topographyassociatedwiththecurrent.Wendthattheposi tionoftheGulf StreamwigglesoshoreofCapeHatterasduetoinstabilitie s,andtheposition calculatedbythemodelisjustoneofmanypossibleposition sforthesamewind forcing.Whichpositioniscorrect,thatis,whatistheposi tionofthecurrent today?Weknow,fromsatellitealtimetry,thepositionofth ecurrentatafew pointsafewdaysago.Canweusethisinformationtocalculat ethecurrent's positiontoday?Howdoweassimilatethisinformationintot hemodel? Manydierentapproachesarebeingexplored(Malanotte-Ri zzoli,1996). RogerDaley(1991)givesacompletedescriptionofhowdataa reusedwith atmosphericmodels.AndrewBennet(1992)andCarlWunsch(1 996)describe oceanicapplications. Thedierentapproachesarenecessarybecauseassimilatio nofdatainto modelsisnoteasy. 1.Dataassimilationisan inverseproblem :Anitenumberofobservations areusedtoestimateacontinuouseld|afunction,whichhas aninnite numberofpoints.Thecalculatedelds,thesolutiontothei nverseproblem,arecompletelyunder-determined.Therearemanyelds thattthe observationsandthemodelprecisely,andthesolutionsare notunique.In ourexample,thepositionoftheGulfStreamisafunction.We maynot needaninnitenumberofvaluestospecifythepositionofth estreamif

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15.5.ASSIMILATIONMODELS 267 weassumethepositionissomewhatsmoothinspace.Butwecer tainly needhundredsofvaluesalongthestream'saxis.Yet,wehave onlyafew satellitepointstoconstrainthepositionoftheStream.Tolearnmoreaboutinverseproblemsandtheirsolution,rea dParker (1994)whogivesaverygoodintroductionbasedongeophysic alexamples. 2.Oceandynamicsarenon-linear,whilemostmethodsforcal culatingsolutionstoinverseproblemsdependonlinearapproximations. Forexample thepositionoftheGulfStreamisaverynonlinearfunctiono ftheforcing bywindandheatruxesoverthenorthAtlantic. 3.Boththemodelandthedataareincompleteandbothhaveerr ors.For example,wehavealtimetermeasurementsonlyalongthetrac kssuchas thoseshowningure2.6,andthemeasurementshaveerrorsof 2cm. 4.Mostdataavailableforassimilationintodatacomesfrom thesurface,such as avhrr andaltimeterdata.Surfacedataobviouslyconstrainthesu rface geostrophicvelocity,andsurfacevelocityisrelatedtode epervelocities. Thetrickistocouplethesurfaceobservationstodeepercur rents. Whilevarioustechniquesareusedtoconstrainnumericalmo delsinoceanography,perhapsthemostpracticalaretechniquesborrowedf rommeteorology. Mostmajoroceancurrentshavedynamicswhicharesignican tlynonlinear.Thisprecludesthereadydevelopmentofinversemethod s...Accordingly,mostattemptstocombineoceanmodelsandmeasuremen tshavefollowedthepracticeinoperationalmeteorology:measuremen tsareusedto prepareinitialconditionsforthemodel,whichistheninte gratedforward intimeuntilfurthermeasurementsareavailable.Themodel isthereupon re-initialized.Suchastrategymaybedescribedassequent ial.|Bennet (1992). Let'sseehowProfessorAllanRobinsonandcolleaguesatHar vardUniversity usedsequentialestimationtechniquestomaketherstfore castsoftheGulf Streamusingaverysimplemodel. TheHarvardOpen-OceanModel wasaneddy-admitting,quasi-geostropic modeloftheGulfStreameastofCapeHatteras(Robinsonetal .1989).Ithad sixlevelsinthevertical,15kmresolution,andone-hourti mesteps.Ituseda ltertosmoothhigh-frequencyvariabilityandtodampgrid -scalevariability. By quasi-geostrophic wemeanthattheroweldisclosetogeostrophic balance.Theequationsofmotionincludetheaccelerationt erms D=Dt ,where D=Dt isthesubstantialderivativeand t istime.Therowcanbestratied, butthereisnochangeindensityduetoheatruxesorvertical mixing.Thus thequasi-geostrophicequationsaresimplerthantheprimi tiveequations,and theycouldbeintegratedmuchfaster.Cushman-Roisin(1994 :204)givesagood descriptionofthedevelopmentofquasi-geostrophicequat ionsofmotion. ThemodelreproducestheimportantfeaturesoftheGulfStre amandit's extension,includingmeanders,cold-andwarm-corerings, theinteractionof ringswiththestream,andbaroclinicinstability(gure15 .4).Becausethe

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268 CHAPTER15.NUMERICALMODELS 76o72o68o74o60o56o52o 76o72o68o74o60o56o52o 42o40o44o38o36o34o32o 42o40o44o38o36o34o32o ColdRing H Cold RingA Nowcast 2 March 1988Cold Ring Cold Ring33L 3 Warm Ring C Forcast 9 March 1988Cold Ring ColdRing Vertical Section Location33333H B A Warm Ring B Data 2 March 1988 D Actual 9 March 19883H33333/7L3/6 3/8 3/5 3/2 3/7 3/4 2/28 2/25 2/26 2/29333333 IR Fronts AXBT Locations2/24 2/27 3/1GEOSAT Track Figure15.4OutputfromtheHarvardOpen-OceanModel: A theinitialstateofthemodel, theanalysis,and B Datausedtoproducetheanalysisfor2March1988. C Theforecastfor 9March1988. D Theanalysisfor9March.AlthoughtheGulfStreamchangedsu bstantially inoneweek,themodelforecaststhechangeswell.AfterRobi nsonetal.(1989). modelwasdesignedtoforecastthedynamicsoftheGulfStrea m,itmustbe constrainedbyoceanicmeasurements: 1.Dataprovidetheinitialconditionsforthemodel.Satell itemeasurements ofsea-surfacetemperaturefromthe avhrr andtopographyfromanaltimeterareusedtodeterminethelocationoffeaturesinthe region.Expendablebathythermograph, axbt measurementsofsubsurfacetemperature,andhistoricalmeasurementsofinternaldensityarea lsoused.The featuresarerepresentedbyananalyticfunctionsinthemod el. 2.Thedataareintroducedintothenumericalmodel,whichin terpolatesand smoothesthedatatoproducethebestestimateoftheinitial eldsof densityandvelocity.Theresultingeldsarecalledan analysis 3.Themodelisintegratedforwardforoneweek,whennewdata areavailable, toproduceaforecast. 4.Finally,thenewdataareintroducedintothemodelasinth erststep above,andtheprocessesisrepeated. Themodelmadeuseful,one-weekforecastsoftheGulfStream region.Much moreadvancedmodelswithmuchhigherresolutionarenowbei ngusedtomake globalforecastsofoceancurrentsuptoonemonthinadvance insupportofthe GlobalOceanDataAssimilationExperiment godae thatstartedin2003.The goalof godae isproduceroutineoceanicforecastssimilartotodaysweat her forecasts.

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15.6.COUPLEDOCEANANDATMOSPHEREMODELS 269 Anexampleofa godae modelistheglobalUSNavyLayeredOceanModel. Itisaprimitiveequationmodelwith1/32 resolutioninthehorizontaland sevenlayersinthevertical.Itassimilatesaltimeterdata fromJason,Geosat Follow-on( gfo ),and ers -2satellitesandsea-surfacetemperaturefrom avhrr on noaa satellites.Themodelisforcedwithwindsandheatruxesfor uptove daysinthefutureusingoutputfromtheNavyOperationalGlo balAtmospheric PredictionSystem.Beyondvedays,seasonalmeanwindsand ruxesareused. Themodelisrundaily(gure15.5)andproducesforecastsfo ruptoonemonth inthefuture.Themodelhasusefulskillouttoabout20days. 45N 40N 35N 30N 75W65W55W45W 1/16 Global Navy Layed Ocean Model Sea-Surface Height and Current Analysis for 25 June 2003 -62.5-50.8-38.4-22-7.68.821.235.850 Sea-Surface Height (cm) 0.80 m/s Figure15.5AnalysisoftheGulfStreamregionfromtheNavyL ayeredOceanModel. FromtheU.S.NavalOceanographicOce. AgroupofFrenchlaboratoriesandagenciesoperatesasimil aroperational forecastingsystem,Mercator,basedonassimilationofalt imetermeasurements ofsea-surfaceheight,satellitemeasurementsofsea-surf acetemperature,and internaldensityeldsintheocean,andcurrentsat1000mfr omthousandsof Argoroats.Theirmodelhas1/15 resolutionintheAtlanticand2 globally. 15.6CoupledOceanandAtmosphereModels Couplednumericalmodelsoftheatmosphereandoceanareuse dtostudythe climate,itsvariability,anditsresponsetoexternalforc ing.Themostimportant useofthemodelshasbeentostudyhowearth'sclimatemightr espondtoa doublingof CO 2 intheatmosphere.Muchoftheliteratureonclimatechange isbasedonstudiesusingsuchmodels.Otherimportantuseso fcoupledmodels includestudiesofElNi~noandthemeridionaloverturningc irculation.The formervariesoverafewyears,thelattervariesoverafewce nturies. Developmentofthecoupledmodelstendstobecoordinatedth roughthe WorldClimateResearchProgramoftheWorldMeteorological Organization

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270 CHAPTER15.NUMERICALMODELS wcrp/wmo ,andrecentprogressissummarizedinChapter8ofthe Climate Change2001:TheScienticBasis reportbytheIntergovernmentalPanelon ClimateChange( ipcc ,2007). Manycoupledoceanandatmospheremodelshavebeendevelope d.Some includeonlyphysicalprocessesintheocean,atmosphere,a ndtheice-covered polarseas.Othersaddtheinruenceoflandandbiologicalac tivityintheocean. Let'slookattheoceaniccomponentsofafewmodels.ClimateSystemModel TheClimateSystemModeldevelopedbytheNationalCenterforAtmosphericResearch ncar includesphysicalandbiogeochemicalinruenceontheclimatesystem(BovilleandGent,1998) .Ithasatmosphere, ocean,land-surface,andsea-icecomponentscoupledbyrux esbetweencomponents.Theatmosphericcomponentisthe ncar CommunityClimateModel,the oceaniccomponentisamodiedversionofthePrincetonModu larOceanModel, usingtheGentandMcWilliams(1990)schemeforparameteriz ingmesoscaleeddies.Resolutionisapproximately2 2 with45verticallevelsintheocean. Themodelhasbeenspunupandintegratedfor300years,there sultsare realistic,andthereisnoneedforaruxadjustment.(Seethe specialissueof JournalofClimate ,June1998). PrincetonCoupledModel Themodelconsistsofanatmosphericmodelwith ahorizontalresolutionof7.5 longitudeby4.5 latitudeand9levelsinthevertical,anoceanmodelwithahorizontalresolutionof4 and12levelsinthevertical,andaland-surfacemodel.Theoceanandatmospherear ecoupledthrough heat,water,andmomentumruxes.Landandoceanarecoupledt hroughriver runo.Andlandandatmospherearecoupledthroughwaterand heatruxes. HadleyCenterModel Thisisanatmosphere-ocean-icemodelthatminimizes theneedforruxadjustments(Johnsetal,1997).Theoceanco mponentisbased ontheBryan-Coxprimitiveequationmodel,withrealisticb ottomfeatures, verticalmixingcoecientsfromPacanowskiandPhilander( 1981).Boththe oceanandtheatmosphericcomponenthaveahorizontalresol utionof96 73 gridpoints,theoceanhas20levelsinthevertical. Incontrasttomostcoupledmodels,thisoneisspunupasacou pledsystem withruxadjustmentsduringspinuptokeepseasurfacetempe ratureandsalinityclosetoobservedmeanvalues.Thecoupledmodelwasinte gratedfromrest usingLevitusvaluesfortemperatureandsalinityforSepte mber.Theinitial integrationwasfrom1850to1940.Themodelwasthenintegra tedforanother 1000years.Noruxadjustmentwasnecessaryaftertheinitia l140-yearintegrationbecausedriftofglobal-averagedairtemperaturewas 0 : 016K/century. CommentsonAccuracyofCoupledModels Modelsofthecoupled,landair-ice-oceanclimatesystemmustsimulatehundredstotho usandsofyears.Yet, Itwillbeveryhardtoestablishanintegrationframework,p articularly onaglobalscale,aspresentcapabilitiesformodellingthe earthsystem areratherlimited.Adualapproachisplanned.Ontheonehan d,the relativelyconventionalapproachofimprovingcoupledatm osphere-oceanland-icemodelswillbepursued.Ingenuityaside,thecompu tationalde-

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15.6.COUPLEDOCEANANDATMOSPHEREMODELS 271 mandsareextreme,asisborneoutbytheEarthSystemSimulat or|640 linkedsupercomputersproviding40terarops[10 12 roating-pointoperationspersecond]andacoolingsystemfromhellunderoneroo f|tobe builtinJapanby2003.|Newton,1999. Becausemodelsmustbesimpliedtorunonexistingcomputer s,themodels mustbesimplerthanmodelsthatsimulaterowforafewyears( wcrp ,1995). Inaddition,thecoupledmodelmustbeintegratedformanyye arsforthe oceanandatmospheretoapproachequilibrium.Astheintegr ationproceeds, thecoupledsystemtendstodriftawayfromrealityduetoerr orsincalculating ruxesofheatandmomentumbetweentheoceanandatmosphere. Forexample, verysmallerrorsinprecipitationovertheAntarcticCircu mpolarCurrentleads tosmallchangesthesalinityofthecurrent,whichleadstol argechangesindeep convectionintheWeddellSea,whichgreatlyinruencesthev olumeofdeepwater masses. Somemodelersallowthesystemtodrift,othersadjustsea-s urfacetemperatureandthecalculatedruxesbetweentheoceanandatmosph ere.Returning totheexample,theruxoffreshwaterinthecircumpolarcurr entcouldbeadjustedtokeepsalinityclosetotheobservedvalueinthecur rent.Thereisno goodscienticbasisfortheadjustmentsexceptthedesiret oproducea\good" coupledmodel.Hence,theadjustmentsareadhocandcontrov ersial.Such adjustmentsarecalled ruxadjustments or ruxcorrections Fortunately,asmodelshaveimproved,theneedforadjustme ntorthemagnitudeoftheadjustmenthasbeenreduced.Forexample,usin gtheGentMcWilliamsschemeformixingalongconstant-densitysurfa cesinacoupled ocean-atmospheremodelgreatlyreducedclimatedriftinac oupledocean-atmospheremodelbecausethemixingschemereduceddeepconvecti onintheAntarcticCircumpolarCurrentandelsewhere(Hirst,O'Farrell,a ndGordon,2000). Grassl(2000)listsfourcapabilitiesofacrediblecoupled generalcirculation model: 1.\Adequaterepresentationofthepresentclimate.2.\Reproduction(withintypicalinterannualanddecadest ime-scaleclimate variability)ofthechangessincethestartoftheinstrumen talrecordfora givenhistoryofexternalforcing; 3.\Reproductionofadierentclimateepisodeinthepastas derivedfrom paleoclimaterecordsforgivenestimatesofthehistoryofe xternalforcing; and 4.\Successfulsimulationofthegrossfeaturesofanabrupt climatechange eventfromthepast." McAvaneyetal.(2001)comparedtheoceaniccomponentoftwe nty-four coupledmodels,includingmodelswithandwithoutruxadjus tments.They foundsubstantialdierencesamongthemodels.Forexample ,onlyvemodels calculatedameridionaloverturningcirculationwithin10 %theobservedvalueof 20Sv.Somehadvaluesaslowas3Sv,othershadvaluesaslarge as36Sv.Most

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272 CHAPTER15.NUMERICALMODELS modelscouldnotcalculatearealistictransportfortheAnt arcticCircumpolar Current. Grassl(2000)foundthatmanycoupledclimatemodels,inclu dingmodels withandwithoutruxadjustment,meettherstcriterion.So memodelsmeet thesecondcriterion(Smithetal.2002,Stottetal.2000),b utexternalsolar forcingisstillnotwellknownandmoreworkisneeded.Andaf ewmodelsare startingtoreproducesomeaspectsofthewarmeventof6,000 yearsago. Buthowusefularethesemodelsinmakingprojectionsoffutu reclimate? Opinionispolarized.Atoneextremearethosewhotakethemo delresults asgospel.Attheotherarethosewhodenigrateresultssimpl ybecause theydistrustmodels,oronthegroundsthatthemodelperfor manceis obviouslywronginsomerespectsorthataprocessisnotadeq uatelyincluded.Thetruthliesinbetween.Allmodelsareofcoursewr ongbecause, bydesign,theygiveasimpliedviewofthesystembeingmode lled.Nevertheless,many|butnotall|modelsareveryuseful.|Trenb erth,1997. 15.7ImportantConcepts 1.Numericalmodelsareusedtosimulateoceanicrowswithre alisticand usefulresults.Themostrecentmodelsincludeheatruxesth roughthe surface,windforcing,mesoscaleeddies,realisticcoasts andsea-roorfeatures,andmorethan20levelsinthevertical. 2.Recentmodelsarenowsogood,withresolutionnear0.1 ,thattheyshow previouslyunknownaspectsoftheoceancirculation, 3.Numericalmodelsarenotperfect.Theysolvediscreteequ ations,which arenotthesameastheequationsofmotiondescribedinearli erchapters. And, 4.Numericalmodelscannotreproduceallturbulenceoftheo ceanbecause thegridpointsaretenstohundredsofkilometersapart.The inruenceof turbulentmotionoversmallerdistancesmustbecalculated fromtheory, andthisintroduceserrors. 5.Numericalmodelscanbeforcedbyreal-timeoceanographi cdatafrom shipsandsatellitestoproduceforecastsofoceaniccondit ions,including ElNi~nointhePacic,andthepositionoftheGulfStreamint heAtlantic. 6.Coupledocean-atmospheremodelshavemuchcoarserspati alresolution sothatthattheycanbeintegratedforhundredsofyearstosi mulatethe naturalvariabilityoftheclimatesystemanditsresponset oincreasedCO 2 intheatmosphere.

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Chapter16OceanWavesLookingouttoseafromtheshore,wecanseewavesontheseasu rface.Looking carefully,wenoticethewavesareundulationsoftheseasur facewithaheightof aroundameter,whereheightistheverticaldistancebetwee nthebottomofa troughandthetopofanearbycrest.Thewavelength,whichwe mighttaketo bethedistancebetweenprominentcrests,isaround50-100m eters.Watching thewavesforafewminutes,wenoticethatwaveheightandwav elengthare notconstant.Theheightsvaryrandomlyintimeandspace,an dthestatistical propertiesofthewaves,suchasthemeanheightaveragedfor afewhundred waves,changefromdaytoday.Theseprominentoshorewaves aregenerated bywind.Sometimesthelocalwindgeneratesthewaves,other timesdistant stormsgeneratewaveswhichultimatelyreachthecoast.For example,waves breakingontheSouthernCaliforniacoastonasummerdaymay comefrom vaststormsoshoreofAntarctica10,000kmaway. Ifwewatchcloselyforalongtime,wenoticethatsealevelch angesfrom hourtohour.Overaperiodofaday,sealevelincreasesandde creasesrelativeto apointontheshorebyaboutameter.Theslowriseandfallofs ealevelisdue tothetides,anothertypeofwaveontheseasurface.Tidesha vewavelengthsof thousandsofkilometers,andtheyaregeneratedbytheslow, verysmallchanges ingravityduetothemotionofthesunandthemoonrelativeto earth. Inthischapteryouwilllearnhowtodescribeocean-surface wavesquantitatively.InthenextchapterIwilldescribetidesandwavesal ongcoasts. 16.1LinearTheoryofOceanSurfaceWaves Surfacewavesareinherentlynonlinear:Thesolutionofthe equationsof motiondependsonthesurfaceboundaryconditions,butthes urfaceboundary conditionsarethewaveswewishtocalculate.Howcanweproc eed? Webeginbyassumingthattheamplitudeofwavesonthewaters urface isinnitelysmallsothesurfaceisalmostexactlyaplane.T osimplifythe mathematics,wecanalsoassumethattherowis2-dimensiona lwithwaves travelinginthe x -direction.WealsoassumethattheCoriolisforceandvisco sity canbeneglected.Ifweretainrotation,wegetKelvinwavesd iscussedin x 14.2. 273

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274 CHAPTER16.OCEANWAVES Withtheseassumptions,thesea-surfaceelevation ofawavetravelingin the x directionis: = a sin( kx !t )(16.1) with =2 f = 2 T ; k = 2 L (16.2) where iswavefrequencyinradianspersecond, f isthewavefrequencyin Hertz(Hz), k iswavenumber, T iswaveperiod, L iswavelength,andwhere weassume,asstatedabove,that ka = O (0). The waveperiod T isthetimeittakestwosuccessivewavecrestsortroughs topassaxedpoint.The wavelength L isthedistancebetweentwosuccessive wavecrestsortroughsataxedtime.DispersionRelation Wavefrequency isrelatedtowavenumber k bythe dispersionrelation (Lamb,1945 x 228): 2 = gk tanh( kd )(16.3) where d isthewaterdepthand g istheaccelerationofgravity. Twoapproximationsareespeciallyuseful. 1. Deep-waterapproximation isvalidifthewaterdepth d ismuchgreater thanthewavelength L .Inthiscase, d L kd 1,andtanh( kd )=1. 2. Shallow-waterapproximation isvalidifthewaterdepthismuchlessthan awavelength.Inthiscase, d L kd 1,andtanh( kd )= kd Forthesetwolimitsofwaterdepthcomparedwithwavelength thedispersion relationreducesto: 2 = gk Deep-waterdispersionrelation(16.4) d>L= 4 2 = gk 2 d Shallow-waterdispersionrelation(16.5) d
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16.1.LINEARTHEORYOFOCEANSURFACEWAVES 275 Thedirectionofpropagationisperpendiculartothewavecr estandtowardthe positive x direction. Thedeep-andshallow-waterapproximationsforthedispers ionrelationgive: c = r g k = g Deep-waterphasevelocity(16.7) c = p gd Shallow-waterphasevelocity(16.8) Theapproximationsareaccuratetoabout5%forlimitsstate din(16.4,16.5). Indeepwater,thephasespeeddependsonwavelengthorwavef requency. Longerwavestravelfaster.Thus,deep-waterwavesaresaid tobedispersive. Inshallowwater,thephasespeedisindependentofthewave; itdependsonly onthedepthofthewater.Shallow-waterwavesarenon-dispe rsive. GroupVelocity Theconceptofgroupvelocity c g isfundamentalforunderstandingthepropagationoflinearandnonlinearwaves.Fir st,itisthevelocity atwhichagroupofwavestravelsacrosstheocean.Moreimpor tantly,itisalso thepropagationvelocityofwaveenergy.Whitham(1974, x 1.3and x 11.6)gives aclearderivationoftheconceptandthefundamentalequati on(16.9). Thedenitionofgroupvelocityintwodimensionsis: c g @! @k (16.9) Usingtheapproximationsforthedispersionrelation: c g = g 2 = c 2 Deep-watergroupvelocity(16.10) c g = p gd = c Shallow-watergroupvelocity(16.11) Forocean-surfacewaves,thedirectionofpropagationispe rpendiculartothe wavecrestsinthepositive x direction.Inthemoregeneralcaseofothertypes ofwaves,suchasKelvinandRossbywavesthatwemetin x 14.2,thegroup velocityisnotnecessarilyinthedirectionperpendicular towavecrests. Noticethatagroupofdeep-waterwavesmovesathalfthephas espeedofthe wavesmakingupthegroup.Howcanthishappen?Ifwecouldwat chcloselya groupofwavescrossingthesea,wewouldseewavescrestsapp earattheback ofthewavetrain,movethroughthetrain,anddisappearatth eleadingedgeof thegroup.Eachwavecrestmovesattwicethespeedofthegrou p. Dorealoceanwavesmoveingroupsgovernedbythedispersion relation?Yes. WalterMunkandcolleagues(1963)inaremarkableseriesofe xperimentsinthe 1960sshowedthatoceanwavespropagatingovergreatdistan cesaredispersive, andthatthedispersioncouldbeusedtotrackstorms.Theyre cordedwaves formanydaysusinganarrayofthreepressuregaugesjustos horeofSan ClementeIsland,60milesduewestofSanDiego,California. Wavespectrawere

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276 CHAPTER16.OCEANWAVES 9111315171921 September 1959 10 10 10 10 .01 .01 .01 .01 0.1 1.0Frequency (mHz)0.01 0 0.07 0.08D=11 5oq=205oD=126oq=200oD=135oq=200o0.02 0.03 0.04 0.05 0.06D=12 1oq=205o Figure16.1Contoursofwaveenergyonafrequency-timeplot calculatedfromspectraof wavesmeasuredbypressuregaugesoshoreofsouthernCalif ornia.Theridgesofhighwave energyshowthearrivalofdispersedwavetrainsfromdistan tstorms.Theslopeoftheridge isinverselyproportionaltodistancetothestorm.isdist anceindegrees, isdirectionof arrivalofwavesatCalifornia.AfterMunketal.(1963).calculatedforeachday'sdata.(Theconceptofaspectraisd iscussedbelow.) Fromthespectra,theamplitudesandfrequenciesofthelowfrequencywaves andthepropagationdirectionofthewaveswerecalculated. Finally,theyplotted contoursofwaveenergyonafrequency-timediagram(gure1 6.1). Tounderstandthegure,consideradistantstormthatprodu ceswavesof manyfrequencies.Thelowest-frequencywaves(smallest )travelthefastest (16.11),andtheyarrivebeforeother,higher-frequencywa ves.Thefurtheraway thestorm,thelongerthedelaybetweenarrivalsofwavesofd ierentfrequencies. Theridgesofhighwaveenergyseeninthegureareproducedb yindividual storms.Theslopeoftheridgegivesthedistancetothestorm indegrees alongagreatcircle;andthephaseinformationfromthearra ygivestheangleto thestorm .Thetwoanglesgivethestorm'slocationrelativetoSanCle mente. Thuswavesarrivingfrom15to18Septemberproducearidgein dicatingthe stormwas115 awayatanangleof205 whichissouthofnewZealandnear Antarctica. Thelocationsofthestormsproducingthewavesrecordedfro mJunethrough October1959werecomparedwiththelocationofstormsplott edonweather mapsandinmostcasesthetwoagreedwell.

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16.1.LINEARTHEORYOFOCEANSURFACEWAVES 277 WaveEnergy Waveenergy E inJoulespersquaremeterisrelatedtothe varianceofsea-surfacedisplacement by: E = w g n 2 (16.12) where w iswaterdensity, g isgravity,andthebracketsdenoteatimeorspace average. 020406080100120 -200 -100 0 100 200 Time (s) Wave Amplitude (m) Figure16.2Ashortrecordofwaveamplitudemeasured byawavebuoyinthenorthAtlantic. SignicantWaveHeight Whatdowemeanbywaveheight?Ifwelookat awind-drivensea,weseewavesofvariousheights.Somearem uchlargerthan most,othersaremuchsmaller(gure16.2).Apracticalden itionthatisoften usedistheheightofthehighest1/3ofthewaves, H 1 = 3 .Theheightiscomputed asfollows:measurewaveheightforafewminutes,pickoutsa y120wavecrests andrecordtheirheights.Pickthe40largestwavesandcalcu latetheaverage heightofthe40values.Thisis H 1 = 3 forthewaverecord. Theconceptofsignicantwaveheightwasdevelopedduringt heWorldWar IIaspartofaprojecttoforecastoceanwaveheightsandperi ods.Wiegel(1964: p.198)reportsthatworkattheScrippsInstitutionofOcean ographyshowed ...waveheightestimatedbyobserverscorrespondstotheav erageofthe highest20to40percentofwaves...Originally,thetermsig nicantwave heightwasattachedtotheaverageoftheseobservations,th ehighest30 percentofthewaves,buthasevolvedtobecometheaverageof thehighest one-thirdofthewaves,(designated H S or H 1 = 3 ) Morerecently,signicantwaveheightiscalculatedfromme asuredwavedisplacement.Iftheseacontainsanarrowrangeofwavefrequen cies, H 1 = 3 is relatedtothestandarddeviationofsea-surfacedisplacem ent( nas ,1963:22; HomanandKarst,1975) H 1 = 3 =4 n 2 1 = 2 (16.13) where n 2 1 = 2 isthestandarddeviationofsurfacedisplacement.Thisrel ationshipismuchmoreuseful,anditisnowtheacceptedwaytocalc ulatewave heightfromwavemeasurements.

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278 CHAPTER16.OCEANWAVES 16.2Nonlinearwaves Wederivedthepropertiesofanoceansurfacewaveassumingt hewaveamplitudewasinnitelysmall ka = O (0).If ka 1butnotinnitelysmall,the wavepropertiescanbeexpandedinapowerseriesof ka (Stokes,1847).He calculatedthepropertiesofawaveofniteamplitudeandfo und: = a cos( kx !t )+ 1 2 ka 2 cos2( kx !t )+ 3 8 k 2 a 3 cos3( kx !t )+ (16.14) ThephasesofthecomponentsfortheFourierseriesexpansio nof in(16.14)are suchthatnon-linearwaveshavesharpenedcrestsandratten edtroughs.The maximumamplitudeoftheStokeswaveis a max =0 : 07 L ( ka =0 : 44).Such steepwavesindeepwaterarecalledStokeswaves(SeealsoLa mb,1945, x 250). Knowledgeofnon-linearwavescameslowlyuntilHasselmann (1961,1963a, 1963b,1966),usingthetoolsofhigh-energyparticlephysi cs,workedoutto6th ordertheinteractionsofthreeormorewavesontheseasurfa ce.He,Phillips (1960),andLonguet-HigginsandPhillips(1962)showedtha t n freewaveson theseasurfacecaninteracttoproduceanotherfreewaveonl yifthefrequencies andwavenumbersoftheinteractingwavessumtozero: 1 2 3 n =0(16.15a) k 1 k 2 k 3 k n =0(16.15b) 2 i = gk i (16.15c) whereweallowwavestotravelinanydirection,and k i isthevectorwave numbergivingwavelengthanddirection.(16.15)aregenera lrequirementsfor anyinteractingwaves.Thefewestnumberofwavesthatmeett heconditionsof (16.15)arethreewaveswhichinteracttoproduceafourth.T heinteractionis weak;wavesmustinteractforhundredsofwavelengthsandpe riodstoproduce afourthwavewithamplitudecomparabletotheinteractingw aves.TheStokes wavedoesnotmeetthecriteriaof(16.15)andthewavecompon entsarenotfree waves;thehigherharmonicsareboundtotheprimarywave.WaveMomentum Theconceptofwavemomentumhascausedconsiderable confusion(McIntyre,1981).Ingeneral,wavesdonothavemo mentum,amass rux,buttheydohaveamomentumrux.Thisistrueforwavesont hesea surface.Ursell(1950)showedthatoceanswellonarotating earthhasnomass transport.Hisproofseemstocontradicttheusualtextbook discussionsofsteep, non-linearwavessuchasStokeswaves.WaterparticlesinaS tokeswavemove alongpathsthatarenearlycircular,butthepathsfailtocl ose,andtheparticles moveslowlyinthedirectionofwavepropagation.Thisisama sstransport,and thephenomenaiscalledStokesdrift.Buttheforwardtransp ortnearthesurface isbalancedbyanequaltransportintheoppositedirectiona tdepth,andthere isnonetmassrux.16.3WavesandtheConceptofaWaveSpectrum Ifwelookouttosea,wenoticethatwavesontheseasurfacear enotsinusoids.Thesurfaceappearstobecomposedofrandomwavesofv ariouslengths

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16.3.WAVESANDTHECONCEPTOFAWAVESPECTRUM 279 andperiods.Howcanwedescribethissurface?Theansweris, Notveryeasily. Wecanhowever,withsomesimplications,comeclosetodesc ribingthesurface. Thesimplicationsleadtotheconceptofthespectrumofoce anwaves.The spectrumgivesthedistributionofwaveenergyamongdiere ntwavefrequencies orwavelengthsontheseasurface. TheconceptofaspectrumisbasedonworkbyJosephFourier(1 768{1830), whoshowedthatalmostanyfunction ( t )(or ( x )ifyoulike),canberepresentedovertheinterval T= 2 t T= 2asthesumofaninniteseriesofsine andcosinefunctionswithharmonicwavefrequencies: ( t )= a 0 2 + 1 X n =1 ( a n cos2 nft + b n sin2 nft )(16.16) where a n = 2 T Z T= 2 T= 2 ( t )cos2 nftdt; ( n =0 ; 1 ; 2 ;::: )(16.17a) b n = 2 T Z T= 2 T= 2 ( t )sin2 nftdt; ( n =0 ; 1 ; 2 ;::: )(16.17b) f =1 =T isthefundamentalfrequency,and nf areharmonicsofthefundamental frequency.Thisformof ( t )iscalleda Fourierseries (Bracewell,1986:204; WhittakerandWatson,1963: x 9.1).Noticethat a 0 isthemeanvalueof ( t ) overtheinterval. Equations(16.18and16.19)canbesimpliedusing exp(2 inft )=cos(2 nft )+ i sin(2 nft )(16.18) where i = p 1.Equations(16.18and16.19)thenbecome: ( t )= 1 X n = 1 Z n exp i 2 nft (16.19) where Z n = 1 T Z T= 2 T= 2 ( t )exp i 2 nft dt; ( n =0 ; 1 ; 2 ;::: )(16.20) Z n iscalledthe Fouriertransform of ( t ). Thespectrum S ( f )of ( t )is: S ( nf )= Z n Z n (16.21) where Z isthecomplexconjugateof Z .WewillusetheseformsfortheFourier seriesandspectrawhenwedescribingthecomputationofoce anwavespectra. WecanexpandtheideaofaFourierseriestoincludeseriesth atrepresent surfaces ( x;y )usingsimilartechniques.Thus,anysurfacecanbereprese nted asaninniteseriesofsineandcosinefunctionsorientedin allpossibledirections.

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280 CHAPTER16.OCEANWAVES Now,let'sapplytheseideastotheseasurface.Supposefora momentthat theseasurfacewerefrozenintime.UsingtheFourierexpans ion,thefrozen surfacecanberepresentedasaninniteseriesofsineandco sinefunctionsof dierentwavenumbersorientedinallpossibledirections. Ifweunfreezethe surfaceandletitmove,wecanrepresenttheseasurfaceasan inniteseries ofsineandcosinefunctionsofdierentwavelengthsmoving inalldirections. Becausewavelengthsandwavefrequenciesarerelatedthrou ghthedispersion relation,wecanalsorepresenttheseasurfaceasaninnite sumofsineand cosinefunctionsofdierentfrequenciesmovinginalldire ctions. NoteinourdiscussionofFourierseriesthatweassumetheco ecients ( a n ;b n ;Z n )areconstant.Fortimesofperhapsanhour,anddistancesof perhapstensofkilometers,thewavesontheseasurfacearesuc ientlyxedthat theassumptionistrue.Furthermore,non-linearinteracti onsamongwavesare veryweak.Therefore,wecanrepresentalocalseasurfaceby alinearsuperpositionofreal,sinewaveshavingmanydierentwavelengt hsorfrequencies anddierentphasestravelinginmanydierentdirections. TheFourierseries innotjustaconvenientmathematicalexpression,itstates thattheseasurface isreally,trulycomposedofsinewaves,eachonepropagatin gaccordingtothe equationsIwrotedownin x 16.1. Theconceptoftheseasurfacebeingcomposedofindependent wavescanbe carriedfurther.SupposeIthrowarockintoacalmocean,mak ingabigsplash. AccordingtoFourier,thesplashcanberepresentedasasupe rpositionofcosine wavesallofnearlyzerophasesothewavesadduptoabigsplas hattheorigin. EachindividualFourierwavebeginstotravelawayfromthes plash.Thelongest wavestravelfastest,andeventually,farfromthesplash,t heseaconsistsofa dispersedtrainofwaveswiththelongestwavesfurtherfrom thesplashandthe shortestwavesclosest.Thisisexactlywhatweseeingure1 6.1.Thestorm makesthesplash,andthewavesdisperseasseeninthegure.SamplingtheSeaSurface CalculatingtheFourierseriesthatrepresentsthe seasurfaceisperhapsimpossible.Itrequiresthatwemeasu retheheightofthe seasurface ( x;y;t )everywhereinanareaperhapstenkilometersonasidefor perhapsanhour.So,let'ssimplify.Supposeweinstallawav estasomewherein theoceanandrecordtheheightoftheseasurfaceasafunctio noftime ( t ).We wouldobtainarecordlikethatingure16.2.Allwavesonthe seasurfacewill bemeasured,butwewillknownothingaboutthedirectionoft hewaves.This isamuchmorepracticalmeasurement,anditwillgivethefre quencyspectrum ofthewavesontheseasurface. Workingwithatraceofwaveheightonsayapieceofpaperisdi cult,so let'sdigitizetheoutputofthewavestatoobtain j ( t j ) ;t j j (16.22) j =0 ; 1 ; 2 ; ;N 1 whereisthetimeintervalbetweenthesamples,and N isthetotalnumber ofsamples.Thelength T oftherecordis T = N .Figure16.3showstherst 20secondsofwaveheightfromgure16.2digitizedatinterv alsof=0 : 32s.

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16.3.WAVESANDTHECONCEPTOFAWAVESPECTRUM 281 Time, jD (s) 051015 20 -200 -100 0 100 200 Wave Height, zj (m) Figure16.3Therst20secondsofdigitizeddatafromgure1 6.2.=0 : 32s. Noticethat j isnotthesameas ( t ).Wehaveabsolutelynoinformation abouttheheightoftheseasurfacebetweensamples.Thusweh aveconverted fromaninnitesetofnumberswhichdescribes ( t )toanitesetofnumbers whichdescribe j .Byconvertingfromacontinuousfunctiontoadigitized function,wehavegivenupaninniteamountofinformationa boutthesurface. Thesamplingintervaldenesa Nyquistcriticalfrequency (Pressetal, 1992:494) Ny 1 = (2)(16.23) TheNyquistcriticalfrequencyisimportantfortworelated ,butdistinct,reasons.Oneisgoodnews,theotherisbadnews.First thegood news.Itistheremarkablefactknownasthe samplingtheorem :Ifacontinuousfunction ( t ),sampledataninterval,happenstobe bandwidth limited tofrequenciessmallerinmagnitudethan Ny ,i.e.,if S ( nf )=0 forall j nf j Ny ,thenthefunction ( t )is completelydetermined byits samples j ...Thisisaremarkabletheoremformanyreasons,amongthem thatitshowsthatthe\informationcontent"ofabandwidthl imitedfunctionis,insomesense,innitelysmallerthanthatofagener alcontinuous function... Nowthebadnews.Thebadnewsconcernstheeectofsamplinga continuousfunctionthatis not bandwidthlimitedtolessthantheNyquist criticalfrequency.Inthatcase,itturnsoutthatallofthe powerspectral densitythatliesoutsidethefrequencyrange Ny nf Ny isspuriouslymovedintothatrange.Thisphenomenoniscalled aliasing .Any frequencycomponentoutsideoftherange( Ny;Ny )is aliased (falsely translated)intothatrangebytheveryactofdiscretesampl ing...There islittlethatyoucandotoremovealiasedpoweronceyouhave discretely sampledasignal.Thewaytoovercomealiasingisto(i)knowt henatural bandwidthlimitofthesignal|orelseenforceaknownlimitb yanalog lteringofthecontinuoussignal,andthen(ii)sampleatar atesuciently rapidtogiveatleasttwopointspercycleofthehighestfreq uencypresent. |Pressetal1992,butwithnotationchangedtoournotation. Figure16.4illustratesthealiasingproblem.Noticehowah ighfrequency signalisaliasedintoalowerfrequencyifthehigherfreque ncyisabovethe criticalfrequency.Fortunately,wecancaneasilyavoidth eproblem:(i)use instrumentsthatdonotrespondtoveryshort,highfrequenc ywavesifweare

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282 CHAPTER16.OCEANWAVES t=0.2 s f=4Hz f=1Hz one second Figure16.4Samplinga4Hzsinewave(heavyline)every0.2sa liasesthefrequencyto1Hz (lightline).Thecriticalfrequencyis1/(2 0.2s)=2.5Hz,whichislessthan4Hz. interestedinthebiggerwaves;and(ii)chose t smallenoughthatweloselittle usefulinformation.Intheexampleshowningure16.3,ther earenowavesin thesignaltobedigitizedwithfrequencieshigherthan Ny =1 : 5625Hz. Let'ssummarize.Digitizedsignalsfromawavestacannotb eusedtostudy waveswithfrequenciesabovetheNyquistcriticalfrequenc y.Norcanthesignal beusedtostudywaveswithfrequencieslessthanthefundame ntalfrequency determinedbytheduration T ofthewaverecord.Thedigitizedwaverecord containsinformationaboutwavesinthefrequencyrange: 1 T
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16.3.WAVESANDTHECONCEPTOFAWAVESPECTRUM 283 Power Spectral Density (m2/H z)10410610210010-210-410-310-1101Frequency (Hz) 10-2100108 Figure16.5Theperiodogramcalculatedfromtherst164sof data fromgure16.2.TheNyquistfrequencyis1.5625Hz. where S N isnormalizedsuchthat: N 1 X j =0 j j j 2 = N= 2 X n =0 S n (16.27) Thevarianceof j isthesumofthe( N= 2+1)termsintheperiodogram.Note, thetermsof S n abovethefrequency( N= 2)aresymmetricaboutthatfrequency. Figure16.5showstheperiodogramofthetimeseriesshownin gure16.2. Theperiodogramisaverynoisyfunction.Thevarianceofeac hpointisequal totheexpectedvalueatthepoint.Byaveragingtogether10{ 30periodograms wecanreducetheuncertaintyinthevalueateachfrequency. Theaveraged periodogramiscalledthespectrumofthewaveheight(gure 16.6).Itgivesthe distributionofthevarianceofsea-surfaceheightatthewa vestaasafunction offrequency.Becausewaveenergyisproportionaltothevar iance(16.12)the spectrumiscalledthe energyspectrum orthe wave-heightspectrum .Typically threehoursofwavestadataareusedtocomputeaspectrumof waveheight. Summary Wecansummarizethecalculationofaspectrumintothefollo wing steps: 1.Digitizeasegmentofwave-heightdatatoobtainusefulli mitsaccordingto (16.26).Forexample,use1024samplesfrom8.53minutesofd atasampled attherateof2samples/second.

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284 CHAPTER16.OCEANWAVES Power Spectral Density (m2/H z)10410510310210110010-110-210-210-1100 Frequency (Hz) Figure16.6Thespectrumofwavescalculatedfrom11minutes ofdatashowningure7.2 byaveragingfourperiodogramstoreduceuncertaintyinthe spectralvalues.Spectralvalues below0.04Hzareinerrorduetonoise. 2.Calculatethedigital,fastFouriertransform Z n ofthetimeseries. 3.Calculatetheperiodogram S n fromthesumofthesquaresoftherealand imaginarypartsoftheFouriertransform. 4.Repeattoproduce M =20periodograms. 5.Averagethe20periodogramstoproduceanaveragedspectr um S M 6. S M hasvaluesthatare 2 distributedwith2 M degreesoffreedom. Thisoutlineofthecalculationofaspectrumignoresmanyde tails.Formore completeinformationsee,forexample,PercivalandWalden (1993),Presset al.(1992: x 12),OppenheimandSchafer(1975),orothertextsondigital signal processing.16.4Ocean-WaveSpectra Oceanwavesareproducedbythewind.Thefasterthewind,the longerthe windblows,andthebiggertheareaoverwhichthewindblows, thebiggerthe waves.Indesigningshipsoroshorestructureswewishtokn owthebiggest wavesproducedbyagivenwindspeed.Supposethewindblowsa t20m/sfor manydaysoveralargeareaoftheNorthAtlantic.Whatwillbe thespectrum ofoceanwavesatthedownwindsideofthearea?

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16.4.OCEAN-WAVESPECTRA 285 0 10 30 50 60 80 90 110100 704020 00.050.100.150.200.250.300.35 Frequency (Hz)Wave Spectral Density (m 2 / Hz)20.6 m/s 18 m/s 15.4 m/s 12.9 m/s 10.3 m/s Figure16.7Wavespectraofafullydevelopedseafordieren t windspeedsaccordingtoMoskowitz(1964). Pierson-MoskowitzSpectrum Variousidealizedspectraareusedtoanswer thequestioninoceanographyandoceanengineering.Perhap sthesimplestis thatproposedbyPiersonandMoskowitz(1964).Theyassumed thatifthe windblewsteadilyforalongtimeoveralargearea,thewaves wouldcomeinto equilibriumwiththewind.Thisistheconceptofa fullydevelopedsea .Here,a \longtime"isroughlyten-thousandwaveperiods,anda\lar gearea"isroughly ve-thousandwavelengthsonaside. Toobtainaspectrumofafullydevelopedsea,theyusedmeasu rementsof wavesmadebyaccelerometersonBritishweathershipsinthe northAtlantic. First,theyselectedwavedatafortimeswhenthewindhadblo wnsteadilyfor longtimesoverlargeareasofthenorthAtlantic.Thentheyc alculatedthewave spectraforvariouswindspeeds(gure16.7),andtheyfound thatthefunction S ( )= g 2 5 exp 0 4 (16.28) wasagoodttotheobservedspectra,where =2 f f isthewavefrequency inHertz, =8 : 1 10 3 =0 : 74, 0 = g=U 19 : 5 and U 19 : 5 isthewindspeed ataheightof19.5mabovetheseasurface,theheightofthean emometerson theweathershipsusedbyPiersonandMoskowitz(1964). Formostairrowovertheseatheatmosphericboundarylayerh asnearly

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286 CHAPTER16.OCEANWAVES neutralstability,and U 19 : 5 1 : 026 U 10 (16.29) assumingadragcoecientof1 : 3 10 3 ThefrequencyofthepeakofthePierson-Moskowitzspectrum iscalculated bysolving dS=d! =0for p ,toobtain p =0 : 877 g=U 19 : 5 : (16.30) Thespeedofwavesatthepeakiscalculatedfrom(16.7),whic hgives: c p = g p =1 : 14 U 19 : 5 1 : 17 U 10 (16.31) Hencewaveswithfrequency p travel14%fasterthanthewindataheightof 19.5mor17%fasterthanthewindataheightof10m.Thisposes adicult problem:Howcanthewindproducewavestravelingfastertha nthewind?Iwill returntotheproblemafterIdiscussthe jonswap spectrumandtheinruence ofnonlinearinteractionsamongwind-generatedwaves. Byintegrating S ( )overall wegetthevarianceofsurfaceelevation: n 2 = Z 1 0 S ( ) d! =2 : 74 10 3 ( U 19 : 5 ) 4 g 2 (16.32) Rememberingthat H 1 = 3 =4 n 2 1 = 2 ,thesignicantwaveheightis: H 1 = 3 =0 : 21 ( U 19 : 5 ) 2 g 0 : 22 ( U 10 ) 2 g (16.33) Figure16.8givessignicantwaveheightsandperiodsforth ePierson-Moskowitz spectrum. P er i odHeight 0 5 10 15 20 0510 15 20 25Significant Wave Height (m) Period (s)Wind Speed U10 (m/s) 0 5 10 15 20 Figure16.8Signicantwaveheightandperiodatthepeakoft hespectrumofafully developedseacalculatedfromthePierson-Moskowitzspect rumusing(16.35and16.32).

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16.4.OCEAN-WAVESPECTRA 287 80 km 52 km 37 km 20 km 9.5 km 0.10.20.30.40.50.6 0.7 0.1 0.2 0.3 0.4 0.5 0.6 0.7Wave Spectral Density (m2/ Hz)Frequency (Hz) 0 0 Figure16.9Wavespectraofadevelopingseafordierentfet ches measuredat jonswap .AfterHasselmannetal.(1973). JONSWAPSpectrum Hasselmannetal.(1973),afteranalyzingdatacollectedduringtheJointNorthSeaWaveObservationProject jonswap ,found thatthewavespectrumisneverfullydeveloped(gure16.9) .Itcontinuesto developthroughnon-linear,wave-waveinteractionsevenf orverylongtimesand distances.Theythereforeproposedthespectrum: S ( )= g 2 5 exp 5 4 p 4 r r (16.34a) r =exp ( p ) 2 2 2 2 p (16.34b) Wavedatacollectedduringthe jonswap experimentwereusedtodetermine thevaluesfortheconstantsin(16.36): =0 : 076 U 2 10 Fg 0 : 22 (16.35a) p =22 g 2 U 10 F 1 = 3 (16.35b) r =3 : 3(16.35c) = 0 : 07 p 0 : 09 !>! p (16.35d) where F isthedistancefromaleeshore,calledthe fetch ,orthedistanceover whichthewindblowswithconstantvelocity.

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288 CHAPTER16.OCEANWAVES Theenergyofthewavesincreaseswithfetch: n 2 =1 : 67 10 7 ( U 10 ) 2 g F (16.36) where F isfetch. The jonswap spectrumissimilartothePierson-Moskowitzspectrumexce pt thatwavescontinuestogrowwithdistance(ortime)asspeci edbythe term, andthepeakinthespectrumismorepronounced,asspeciedb ythe r term. Thelatterturnsouttobeparticularlyimportantbecauseit leadstoenhanced non-linearinteractionsandaspectrumthatchangesintime accordingtothe theoryofHasselmann(1966).GenerationofWavesbyWind Wehaveseeninthelastfewparagraphs thatwavesarerelatedtothewind.Wehave,however,putoun tilnowjust howtheyaregeneratedbythewind.Supposewebeginwithamir ror-smooth sea(BeaufortNumber0).Whathappensifthewindsuddenlybe ginstoblow steadilyatsay8m/s?Threedierentphysicalprocessesbeg in: 1.Theturbulenceinthewindproducesrandompressureructu ationsatthe seasurface,whichproducessmallwaveswithwavelengthsof afewcentimeters(Phillips,1957). 2.Next,thewindactsonthesmallwaves,causingthemtobeco melarger. Windblowingoverthewaveproducespressuredierencesalo ngthewave prolecausingthewavetogrow.Theprocessisunstablebeca use,asthe wavegetsbigger,thepressuredierencesgetbigger,andth ewavegrows faster.Theinstabilitycausesthewavetogrowexponential ly(Miles,1957). 3.Finally,thewavesbegintointeractamongthemselvestop roducelonger waves(Hasselmannetal.1973).Theinteractiontransfersw aveenergy fromshortwavesgeneratedbyMiles'mechanismtowaveswith frequencies slightlylowerthanthefrequencyofwavesatthepeakofthes pectrum. Eventually,thisleadstowavesgoingfasterthanthewind,a snotedby PiersonandMoskowitz. 16.5WaveForecasting Ourunderstandingofoceanwaves,theirspectra,theirgene rationbythe wind,andtheirinteractionsarenowsucientlywellunders toodthatthewave spectrumcanbeforecastusingwindscalculatedfromnumeri calweathermodels. Ifweobservesomesmalloceanarea,orsomeareajustoshore ,wecanseewaves generatedbythelocalwind,the windsea ,pluswavesthatweregenerated inotherareasatothertimesandthathavepropagatedintoth eareaweare observing,the swell .Forecastsoflocalwaveconditionsmustincludebothsea andswell,hencewaveforecastingisnotalocalproblem.Wes aw,forexample, thatwavesoCaliforniacanbegeneratedbystormsmorethan 10,000kmaway. Varioustechniqueshavebeenusedtoforecastwaves.Theear liestattempts werebasedonempiricalrelationshipsbetweenwaveheighta ndwavelength andwindspeed,duration,andfetch.Thedevelopmentofthew avespectrum

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16.6.MEASUREMENTOFWAVES 289 allowedevolutionofindividualwavecomponentswithfrequ ency f travellingin direction ofthedirectionalwavespectrum ( f; )using @ 0 @t + c g r 0 = S i + S nl + S d (16.37) where 0 = 0 ( f; ; x ;t )variesinspace( x )andtime t S i isinputfromthewind givenbythePhillips(1957)andMiles(1957)mechanisms, S nl isthetransfer amongwavecomponentsduetononlinearinteractions,and S d isdissipation. Thethird-generationwave-forecastingmodelsnowusedbym eteorological agenciesthroughouttheworldarebasedonintegrationsof( 16.39)usingmany individualwavecomponents(The swamp Group1985;The wamdi Group,1988; Komenetal,1996).Theforecastsfollowindividualcompone ntsofthewave spectruminspaceandtime,allowingeachcomponenttogrowo rdecaydependingonlocalwinds,andallowingwavecomponentstointe ractaccordingto Hasselmann'stheory.Typicallytheseaisrepresentedby30 0components:25 wavelengthsgoingin12directions(30 ).Toreducecomputingtime,themodels useanestedgridofpoints:thegridhasahighdensityofpoin tsinstormsand nearcoastsandalowdensityinotherregions.Typically,gr idpointsinthe openoceanare3 apart. Somerecentexperimentalmodelstakethewave-forecasting processonestep furtherbyassimilatingaltimeterandscatterometerobser vationsofwindspeed andwaveheightintothemodel.Forecastsofwavesusingassi milatedsatellitedataareavailablefromtheEuropeanCentreforMediumRangeWeather Forecasts. Noaa 'sOceanModelingBranchattheNationalCentersforEnviron mental Predictionsalsoproducesregionalandglobalforecastsof waves.TheBranchuse athird-generationmodelbasedontheCycle-4 wam model.Itaccommodates ever-changingiceedge,anditassimilatesbuoyandsatelli tealtimeterwave data.Themodelcalculatesdirectionalfrequencyspectrai n12directionsand25 frequenciesat3-hourintervalsupto72hoursinadvance.Th elowestfrequency is0.04177Hzandhigherfrequenciesarespacedlogarithmic allyatincrements of0.1timesthelowestfrequency.Wavespectraldataareava ilableona2.5 2.5 gridfordeep-waterpointsbetween67.5 Sand77.5 N.Themodelisdriven using10-meterwindscalculatedfromthelowestwindsfromt heCentersweather modeladjustedtoaheightof10meterbyusingalogarithmicp role(8.20).The Branchistestinganimprovedforecastwith1 1.25 resolution(gure16.10). 16.6MeasurementofWaves Becausewavesinruencesomanyprocessesandoperationsats ea,many techniqueshavebeeninventedformeasuringwaves.Hereare afewofthe morecommonlyused.Stewart(1980)givesamorecompletedes criptionof wavemeasurementtechniques,includingmethodsformeasur ingthedirectional distributionofwaves.SeaStateEstimatedbyObserversatSea Thisisperhapsthemostcommonobservationincludedinearlytabulationsofwaveheigh ts.Thesearethe

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290 CHAPTER16.OCEANWAVES Global 1 x 1,25 grid 20 August 1998 00:00 UTC Wind Speed in Knots 12322122123325444111GM Figure16.10Outputofathird-generationwaveforecastmod elproducedby Noaa 'sOcean ModelingBranchfor20August1998.Contoursaresignicant waveheightinmeters,arrows givedirectionofwavesatthepeakofthewavespectrum,andb arbsgivewindspeedinm/s anddirection.From noaa OceanModelingBranch. signicantwaveheightssummarizedintheU.S.Navy's MarineClimatological Atlas andothersuchreportsprintedbeforetheageofsatellites. SatelliteAltimeters Thesatellitealtimetersusedtomeasuresurfacegeostrophiccurrentsalsomeasurewaveheight.Altimeterswer erownonSeasatin 1978,Geosatfrom1985to1988, ers{1&2 from1991,Topex/Poseidonfrom 1992,andJasonfrom2001.Altimeterdatahavebeenusedtopr oducemonthly meanmapsofwaveheightsandthevariabilityofwaveenergyd ensityintime andspace.Thenextstep,justbegun,istousealtimeterobse rvationwithwave forecastingprograms,toincreasetheaccuracyofwavefore casts. Thealtimetertechniqueworksasfollows.Radiopulsefroma satellitealtimeterrerectrstfromthewavecrests,laterfromthewave troughs.The rerectionstretchesthealtimeterpulseintime,andthestr etchingismeasured andusedtocalculatewaveheight(gure16.11).Accuracyis 10%. AccelerometerMountedonMeteorologicalorOtherBuoy Thisisa lesscommonmeasurement,althoughitisoftenusedformeasu ringwavesduringshortexperimentsatsea.Forexample,accelerometerso nweatherships measuredwaveheightusedbyPierson&Moskowitzandthewave sshownin gure16.2.Themostaccuratemeasurementsaremadeusingan accelerometer stabilizedbyagyrosotheaxisoftheaccelerometerisalway svertical.

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16.6.MEASUREMENTOFWAVES 291 0.56m SHW 1.94m SWH -15-10-50510 15 20 Time (ns) Received power Figure16.11ShapeofradiopulsereceivedbytheSeasatalti meter,showingtheinruence ofoceanwaves.Theshapeofthepulseisusedtocalculatesig nicantwaveheight.After Stewart(1985:264). Doubleintegrationofverticalaccelerationgivesdisplac ement.Thedouble integration,however,amplieslow-frequencynoise,lead ingtothelowfrequency signalsseeningures16.5and16.6.Inaddition,thebuoy's heaveisnotsensitive towavelengthslessthanthebuoy'sdiameter,andbuoysmeas ureonlywaves havingwavelengthsgreaterthanthediameterofthebuoy.Ov erall,careful measurementsareaccurateto 10%orbetter. WaveGages Gaugesmaybemountedonplatformsoronthesearoorin shallowwater.Manydierenttypesofsensorsareusedtomea suretheheight ofthewaveorsubsurfacepressurewhichisrelatedtowavehe ight.Sound, infraredbeams,andradiowavescanbeusedtodeterminethed istancefrom thesensortotheseasurfaceprovidedthesensorcanbemount edonastable platformthatdoesnotinterferewiththewaves.Pressurega ugesdescribedin x 6.8canbeusedtomeasurethedepthfromtheseasurfacetothe gauge.Arrays ofbottom-mountedpressuregaugesareusefulfordetermini ngwavedirections. Thusarraysarewidelyusedjustoshoreofthesurfzonetode termineoshore wavedirections. Pressuregaugemustbelocatedwithinaquarterofawaveleng thofthe surfacebecausewave-inducedpressureructuationsdecrea seexponentiallywith depth.Thus,bothgaugesandpressuresensorsarerestricte dtoshallowwateror tolargeplatformsonthecontinentalshelf.Again,accurac yis 10%orbetter. SyntheticApertureRadarsonSatellites Theseradarsmaptheradarrerectivityoftheseasurfacewithspatialresolutionof6{25 m.Mapsofrerectivity oftenshowwave-likefeaturesrelatedtotherealwavesonth eseasurface.Isay `wave-like'becausethereisnotanexactone-to-onerelati onshipbetweenwave heightandimagedensity.Somewavesareclearlymapped,oth erslessso.The maps,however,canbeusedtoobtainadditionalinformation aboutwaves,especiallythespatialdistributionofwavedirectionsinshall owwater(Veseckyand

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292 CHAPTER16.OCEANWAVES Stewart,1982).Becausethedirectionalinformationcanbe calculateddirectly fromtheradardatawithouttheneedtocalculateanimage(Ha sselmann,1991), datafromradarsandaltimeterson ers {1&2arebeingusedtodetermineifthe radarandaltimeterobservationscanbeuseddirectlyinwav eforecastprograms. 16.7ImportantConcepts 1.Wavelengthandfrequencyofwavesarerelatedthroughthe dispersion relation. 2.Thevelocityofawavephasecandierfromthevelocityatw hichwave energypropagates. 3.Wavesindeepwateraredispersive,longerwavelengthstr avelfasterthan shorterwavelengths.Wavesinshallowwaterarenotdispers ive. 4.Thedispersionofoceanwaveshasbeenaccuratelymeasure d,andobservationsofdispersedwavescanbeusedtotrackdistantstorm s. 5.Theshapeoftheseasurfaceresultsfromalinearsuperpos itionofwavesof allpossiblewavelengthsorfrequenciestravellinginallp ossibledirections. 6.Thespectrumgivesthecontributionsbywavelengthorfre quencytothe varianceofsurfacedisplacement. 7.Waveenergyisproportionaltovarianceofsurfacedispla cement. 8.Digitalspectraarebandlimited,andtheycontainnoinfo rmationabout waveswithfrequencieshigherthantheNyquistfrequency. 9.Wavesaregeneratedbywind.Strongwindsoflongduration generatethe largestwaves. 10.Variousidealizedformsofthewavespectrumgeneratedb ysteady,homogeneouswindshavebeenproposed.Twoimportantonesarethe PiersonMoskowitzand jonswap spectra. 11.Observationsbymarinersonshipsandbysatellitealtim etershavebeen usedtomakeglobalmapsofwaveheight.Wavegaugesareusedo nplatformsinshallowwaterandonthecontinentalshelftomeasur ewaves. Bottom-mountedpressuregaugesareusedtomeasurewavesju stoshore ofbeaches.Andsynthetic-apertureradarsareusedtoobtai ninformation aboutwavedirections.

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Chapter17CoastalProcessesandTidesInthelastchapterIdescribedwavesontheseasurface.Noww ecanconsider severalspecialandimportantcases:thetransformationof wavesastheycome ashoreandbreak;thecurrentsandedgewavesgeneratedbyth einteraction ofwaveswithcoasts;tsunamis;stormsurges;andtides,esp eciallytidesalong coastsandinthedeepocean.17.1ShoalingWavesandCoastalProcesses Wavephaseandgroupvelocitiesareafunctionofdepthwhent hedepth islessthanaboutone-quarterwavelengthindeepwater.Wav eperiodand frequencyareinvariant(don'tchangeasthewavecomesasho re);andthisis usedtocomputethepropertiesofshoalingwaves.Waveheigh tincreasesas wavegroupvelocityslows.Wavelengthdecreases.Wavescha ngedirectiondue torefraction.Finally,wavesbreakifthewaterissucient lyshallow;andbroken wavespourwaterintothesurfzone,creatinglong-shoreand ripcurrents. ShoalingWaves Thedispersionrelation(16.3)isusedtocalculatewavepro pertiesasthewavespropagatefromdeepwateroshoretoshal lowwaterjust outsidethesurfzone.Because isconstant,(16.3)leadsto: L L 0 = c c 0 = sin sin 0 =tanh 2 d L (17.1) where L 0 = gT 2 2 ;c 0 = gT 2 (17.2) and L iswavelength, c isphasevelocity, istheangleofthecrestrelative tocontoursofconstantdepth,and d iswaterdepth.Thesubscript0indicates valuesindeepwater. Thequantity d=L iscalculatedfromthesolutionof d L 0 = d L tanh 2 d L (17.3) 293

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294 CHAPTER17.COASTALPROCESSESANDTIDES 1.0 2.00.2 0.50.1 0.010.05 0.020.10.5 0.21.0 ()CGC0dL tanh 2pd L C C0, L L0 sin asin a0 H H0 ,,d / L) ( tanh 2pd L C C0, L L0 sin asin a0 CG / C0d / L0CG /CH / H0deep shallow Figure17.1Propertiesofwavesinintermediatedepthsbetw eendeepandshallowwater. d = depth, L =wavelength, C =phasevelocity, C g =groupvelocity, =angleofcrestsrelative tocontoursofconstantdepth, H =waveheight.Subscript0referstopropertiesindeep water.FromvaluesinWiegel(1964:A1).usinganiterativetechnique,orfromgure17.1,orfromWie gel(1964:A1). Becausewavevelocityisafunctionofdepthinshallowwater ,variations inoshorewaterdepthcanfocusordefocuswaveenergyreach ingtheshore. Considerthesimplecaseofwaveswithdeep-watercrestsalm ostparalleltoa Beach deposit Headland Headland Wave fronts Water depth (m) Orthogonal lines Bay 80 60 40 20 Figure17.2sub-seafeatures,suchassubmarinecanyonsand ridges,oshoreofcoastscan greatlyinruencetheheightofbreakersinshoreofthefeatu res.AfterThurman(1985:229).

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17.1.SHOALINGWAVESANDCOASTALPROCESSES 295 1 2 3 4(c) Surging breaker (b) Plunging breaker (a) Spilling breaker 1 2 3 4 1 2 3 4 Spillingbreaker Plungingbreaker Surging 0.06 0.070.050.04 0.03 0.020.01 0 00.020.040.060.080.100.12 Beach slope tanbDeepwater wave steepness Ho/Lo Figure17.3 Left :Classicationofbreakingwavesasafunctionofbeachstee pnessandwave steepnessoshore. Right :Sketchoftypesofbreakingwaves.AfterHorikawa(1988:79 ,81). straightbeachwithtworidgeseachextendingseawardfroma headland(gure 17.2).Wavegroupvelocityisfasterinthedeeperwaterbetw eentheridges,and thewavecrestsbecomeprogressivelydeformedasthewavepr opagatestoward thebeach.Waveenergy,whichpropagatesperpendiculartow avecrests,is refractedoutoftheregionbetweentheheadland.Asaresult ,waveenergyis focusedintotheheadlands,andbreakerstherearemuchlarg erthanbreakersin thebay.Thedierenceinwaveheightcanbesurprisinglylar ge.Onacalmday, breakerscanbekneehighshorewardofasubmarinecanyonatL aJollaShores, California,justsouthoftheScrippsInstitutionofOceano graphy.Atthesame time,wavesjustnorthofthecanyoncanbehighenoughtoattr actsurfers. BreakingWaves Aswavesmoveintoshallowwater,thegroupvelocitybecomessmall,waveenergypersquaremeterofseasurfaceincr eases,andnonlineartermsinthewaveequationsbecomeimportant.Thesep rocessescause wavestosteepen,withshortsteepcrestsandbroadshallowt roughs.When waveslopeatthecrestbecomessucientlysteep,thewavebr eaks(gure17.3 Right).Theshapeofthebreakingwavedependsontheslopeof thebottom, andthesteepnessofwavesoshore(gure17.3Left). 1.Steepwavestendtoloseenergyslowlyasthewavesmovesin toshallower waterthroughwaterspillingdownthefrontofthewave.Thes earespilling breakers. 2.Lesssteepwavesonsteepbeachestendtosteepensoquickl ythatthe crestofthewavemovesmuchfasterthanthetrough,andthecr est,racing aheadofthetrough,plungesintothetrough(gure17.4). 3.Ifthebeachissucientlysteep,thewavecansurgeupthef aceofthe beachwithoutbreakinginthesensethatwhitewaterisforme d.Orifit isformed,itisattheleadingedgeofthewaterasitsurgesup thebeach. Anextremeexamplewouldbeawaveincidentonaverticalbrea kwater.

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296 CHAPTER17.COASTALPROCESSESANDTIDES Figure17.4Steep,plungingbreakersarethearchetypicalb reaker.Theedge ofsuchbreakersareidealforsurng.FromphotobyJeDevin e. Wave-DrivenCurrents Wavesbreakinanarrowbandofshallowwateralong thebeach,the surfzone .Afterbreaking,wavescontinuesasanear-verticalwall ofturbulentwatercalleda bore whichcarrieswatertothebeach.Atrst,the boresurgesupthebeach,thenretreats.Thewatercarriedby theboreisleft intheshallowwatersinsidethebreakerzone. Waterdumpedinsidethebreakerzonemustreturnoshore.It doesthisby rstmovingparalleltothebeachasan along-shorecurrent .Thenitturnsand rowsoshoreperpendiculartothebeachinanarrow,swift ripcurrent .The ripsareusuallyspacedhundredsofmetersapart(gure17.5 ).Usuallythereis abandofdeeperwaterbetweenthebreakerzoneandthebeach, andthelongshorecurrentrunsinthischannel.Thestrengthofaripcurr entdependsonthe Breaker Zone Beach Longshore CurrentRip CurrentRip CurrentCoastal Current Figure17.5Sketchofripcurrentsgeneratedbywatercarrie dtothebeachbybreakingwaves.

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17.2.TSUNAMIS 297 heightandfrequencyofbreakingwavesandthestrengthofth eonshorewind. Ripsareadangertounwaryswimmers,especiallypoorswimme rsbobbingalong inthewavesinsidethebreakerzone.Theyarecarriedalongb ythealong-shore currentuntiltheyaresuddenlycarriedouttoseabytherip. Swimmingagainst theripisfutile,butswimmerscanescapebyswimmingparall eltothebeach. Edgewaves areproducedbythevariabilityofwaveenergyreachingshor e. Wavestendtocomeingroups,especiallywhenwavescomefrom distantstorms. Forseveralminutesbreakersmaybesmallerthanaverage,th enafewverylarge waveswillbreak.Theminute-to-minutevariationinthehei ghtofbreakers produceslow-frequencyvariabilityinthealong-shorecur rent.This,inturn, drivesalow-frequencywaveattachedtothebeach,anedgewa ve.Thewaves haveperiodsofafewminutes,along-shorewavelengthofaro undakilometer, andanamplitudethatdecaysexponentiallyoshore(gure1 7.6). z x y (north) Figure17.6Computer-assistedsketchofanedgewave.Suchw avesexistinthebreaker zonenearthebeachandonthecontinentalshelf.AfterCutch inandSmith(1973). 17.2Tsunamis Tsunamisarelow-frequencyoceanwavesgeneratedbysubmar ineearthquakes.Thesuddenmotionofsearooroverdistancesofahund redormore kilometersgenerateswaveswithperiodsof15{40minutes( gure17.7).Aquick calculationshowsthatsuchwavesmustbeshallow-waterwav es,propagatingat aspeedof180m/sandhavingawavelengthof130kminwater3.6 kmdeep (gure17.8).Thewavesarenotnoticeableatsea,butafters lowingonapproach tothecoast,andafterrefractionbysub-seafeatures,they cancomeashoreand surgetoheightstenormoremetersabovesealevel.Inanextr emeexample, thegreat2004IndianOceantsunamidestroyedhundredsofvi llages,killingat least200,000people. Shepard(1963,Chapter4)summarizedtheinruenceoftsunam isbasedon hisstudiesinthePacic. 1.Tsunamisappeartobeproducedbymovement(anearthquake )alonga linearfault. 2.Tsunamiscantravelthousandsofkilometersandstilldos eriousdamage.

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298 CHAPTER17.COASTALPROCESSESANDTIDES 3:304:004:305:005:306:006:307:00 3:002:30 4000 2:00AM KEY OrthogonalsWave Fronts 2:30A.M.Submarine Contours: m 5500 3700 1800 ALEUTIANISLANDS ALASKA HAWAIIANISLANDS 20o 40o45o50o55o35o30o25o-160o-170o-165o-155o-150o-175o 20 22o0 10 50 302010 40 50 -160o0 910 6:26 6:24 6:22 6:20 6:28 6:28 3700 2700 1800 910 550 1800 2700 180 2700 1800 1800 2700WAIMEA 14 12m 5.2 2.4 4.6 2.4 7.6 4.6 8.5 3.7 7.3 3.7 6.1 14 12 2.1 05:00 Approximate Time (hrs) 08:00 06:0007:00 1.0 0 0.80.4 Height (m) Waimea gauge Figure17.7 Left Hourlypositionsofleadingedgeoftsunamigeneratedbythe large earthquakeintheAleutianTrenchonApril1,1946at1:59AMH awaiiantime(12:59GMT). Right:top SealevelrecordedbyarivergaugeintheestuaryoftheWaime aRiver. Right: lower MapofKauaishowingtheheightsreachedbythewater(inmete rsabovelowerlow water)duringthetsunami,wavefronts,orthogonals,andsu bmarinecontours.Timesreferto thecomputedarrivaltimeoftherstwave.AfterShepard,Ma cDonald,andCox(1950). 3.Therstwaveofatsunamiisnotlikelytobethebiggest.4.Waveamplitudesarerelativelylargeshorewardofsubmar ineridges.They arerelativelylowshorewardofsubmarinevalleys,provide dthefeatures extendintodeepwater. 5.Waveamplitudesaredecreasedbythepresenceofcoralree fsbordering thecoast. 6.Somebayshaveafunnelingeect,butlongestuariesatten uatewaves. 7.Wavescanbendaroundcircularislandswithoutgreatloss ofenergy,but theyareconsiderablysmalleronthebacksidesofelongated ,angularislands. Numericalmodelsareusedtoforecasttsunamiheightsthrou ghoutocean basinsandtheinundationofcoasts.Forexample, noaa 'sCenterforTsunami ResearchusestheMethodofSplittingTsunami most model(TitovandGonzalez,1997).Themodelusesnestedgridstoresolvethetsun amiwavelength,it

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17.3.STORMSURGES 299 Figure17.8TsunamiwavesfourhoursafterthegreatM9Casca diaearthquakeothecoastof Washingtonon26January1700calculatedbyanite-element ,numericalmodel.Maximum open-oceanwaveheight,aboutonemeter,isnorthofHawaii. AfterSatakeetal.(1996). propagatesthewaveacrossoceanbasins,andthencalculate srun-upwhenthe wavecomesashore.Itisinitializedfromagrounddeformati onmodelthatuses measuredearthquakemagnitudeandlocationtocalculateve rticaldisplacement ofthesearoor.Theforcingismodiedoncewavesaremeasure dnearthe earthquakebysearoorobservingstations.17.3StormSurges Stormwindsblowingovershallow,continentalshelvespile wateragainstthe coast.Theincreaseinsealevelisknownasastormsurge.Sev eralprocesses areimportant: 1.Ekmantransportbywindsparalleltothecoasttransports watertoward thecoastcausingariseinsealevel. 2.Windsblowingtowardthecoastpushwaterdirectlytoward thecoast. 3.Waverun-upandotherwaveinteractionstransportwatert owardthecoast addingtothersttwoprocesses. 4.Edgewavesgeneratedbythewindtravelalongthecoast.5.Thelowpressureinsidethestormraisessealevelbyonece ntimeterfor eachmillibardecreaseinpressurethroughtheinverted-ba rometereect. 6.Finally,thestormsurgeaddstothetides,andhightidesc anchangea relativeweaksurgeintoamuchmoredangerousone. SeeGraberetal(2006)and x 15.5foradescriptionofAdvancedCirculation ModelusedbytheNationalHurricaneCenterforpredictings torm-surges. Toacruderstapproximation,windblowingovershallowwat ercausesa slopeintheseasurfaceproportionaltowindstress. @ @x = 0 gH (17.4)

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300 CHAPTER17.COASTALPROCESSESANDTIDES 1953 1894 1916 1906 1928 1904 Mean High Water 10310210110-110-210-310-40 1 2 3 4 5Storm Surge (m)Frequency per year Figure17.9Probability(peryear)densitydistributionof verticalheightofstormsurgesin theHookofHollandintheNetherlands.Thedistributionfun ctionisRayleigh,andthe probabilityoflargesurgesisestimatedbyextrapolatingt heobservedfrequencyofsmaller, morecommonsurges.AfterWiegel(1964:113).where issealevel, x ishorizontaldistance, H iswaterdepth, T 0 iswindstress attheseasurface, iswaterdensity;and g isgravitationalacceleration. If x =100km, U =40m/s,and H =20m,valuestypicalofahurricane oshoreoftheTexasGulfCoast,then T =2 : 7Pa,and =1 : 3mattheshore. Figure17.9showsthefrequencyofsurgesattheNetherlands andagraphical methodforestimatingtheprobabilityofextremeeventsusi ngtheprobability ofweakerevents.17.4TheoryofOceanTides Tideshavebeensoimportantforcommerceandscienceforsom anythousandsofyearsthattideshaveenteredoureverydaylanguage : timeandtidewait fornoone theebbandrowofevents ahigh-watermark ,and turnthetideof battle 1.Tidesproducestrongcurrentsinmanypartsoftheocean.T idalcurrents canhavespeedsofupto5m/sincoastalwaters,impedingnavi gationand mixingcoastalwaters. 2.Tidalcurrentsgenerateinternalwavesoverseamounts,c ontinentalslopes, andmid-oceanridges.Thewavesdissipatetidalenergy.Bre akinginternal wavesandtidalcurrentsareamajorforcedrivingoceanicmi xing. 3.Tidalmixinghelpsdrivethedeepcirculation,anditinru encesclimate andabruptclimatechange. 4.Tidalcurrentscansuspendbottomsediments,eveninthed eepocean.

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17.4.THEORYOFOCEANTIDES 301 5.Earth'scrustiselastic.Itbendsundertheinruenceofth etidalpotential. Italsobendsundertheweightofoceanictides.Asaresult,t hesearoor, andthecontinentsmoveupanddownbyabout10cminresponset o thetides.Thedeformationofthesolidearthinruencealmos tallprecise geodeticmeasurements. 6.Oceanictideslagbehindthetide-generatingpotential. Thisproduces forcesthattransferangularmomentumbetweenearthandthe tideproducingbody,especiallythemoon.Asaresultoftidalforces,ea rth'srotation aboutit'saxisslows,increasingthelengthofday;therota tionofthemoon aboutearthslows,causingthemoontomoveslowlyawayfrome arth;and moon'srotationaboutit'saxisslows,causingthemoontoke epthesame sidefacingearthasthemoonrotatesaboutearth. 7.Tidesinruencetheorbitsofsatellites.Accurateknowle dgeoftidesis neededforcomputingtheorbitofaltimetricsatellitesand forcorrecting altimetermeasurementsofoceanictopography. 8.Tidalforcesonotherplanetsandstarsareimportantforu nderstanding manyaspectsofsolar-systemdynamicsandevengalacticdyn amics.For example,therotationrateofMercury,Venus,andIoresultf romtidal forces. Marinershaveknownforatleastfourthousandyearsthattid esarerelatedto thephaseofthemoon.Theexactrelationship,however,ishi ddenbehindmany complicatingfactors,andsomeofthegreatestscienticmi ndsofthelastfour centuriesworkedtounderstand,calculate,andpredicttid es.Galileo,Descartes, Kepler,Newton,Euler,Bernoulli,Kant,Laplace,Airy,Lor dKelvin,Jereys, Munkandmanyotherscontributed.Someoftherstcomputers weredeveloped tocomputeandpredicttides.Ferrelbuiltatide-predictin gmachinein1880that wasusedbytheU.S.CoastSurveytopredictnineteentidalco nstituents.In 1901,Harrisextendedthecapacityto37constituents. Despiteallthisworkimportantquestionsremained:Whatis theamplitude andphaseofthetidesatanyplaceontheoceanoralongthecoa st?Whatis thespeedanddirectionoftidalcurrents?Whatistheshapeo fthetideson theocean?Whereistidalenergydissipated?Findinganswer stothesesimple questionsisdicult,andtherst,accurate,globalmapsof deep-seatideswere onlypublishedin1994(LeProvostetal.1994).Theproblemi shardbecause thetidesareaself-gravitating,near-resonant,sloshing ofwaterinarotating, elastic,oceanbasinwithridges,mountains,andsubmarine basins. Predictingtidesalongcoastsandatportsismucheasier.Da tafromatide gaugeplusthetheoryoftidalforcinggivesanaccuratedesc riptionoftidesnear thetidegauge.TidalPotential Tidesarecalculatedfromthehydrodynamicequationsfor aself-gravitatingoceanonarotating,elasticearth.Thed rivingforceisthe gradientofthegravityeldofthemoonandsun.Iftheearthw ereanocean planetwithnoland,andifweignoretheinruenceofinertiaa ndcurrents,the gravitygradientproducesapairofbulgesofwateronearth, oneontheside

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302 CHAPTER17.COASTALPROCESSESANDTIDES facingthemoonorsun,oneonthesideawayfromthemoonorsun .Aclear derivationoftheforcesisgivenbyPugh(1987)andbyDietri ch,Kalle,Krauss, andSiedler(1980).HereIfollowthediscussioninPugh(198 7: x 3.2). Notethatmanyoceanographicbooksstatethatthetideispro ducedby twoprocesses:i)thecentripetalaccelerationatearth'ss urfaceastheearthand mooncirclearoundacommoncenterofmass,andii)thegravit ationalattraction ofmassonearthandthemoon.However,thederivationofthet idalpotential doesnotinvolvecentripetalacceleration,andtheconcept isnotusedbythe astronomicalorgeodeticcommunities. Celestial body A P R Earth O r r1j Figure17.10Sketchofcoordinatesfordeterminingthetide -generatingpotential. Tocalculatetheamplitudeandphaseofthetideonanoceanpl anet,we beginbycalculatingthetide-generatingpotential.Thisi smucheasierthan calculatingtheforces.Ignoringfornowearth'srotation, therotationofmoon aboutearthproducesapotential V M atanypointonearth'ssurface V M = rM r 1 (17.5) wherethegeometryissketchedingure17.10, r isthegravitationalconstant, and M ismoon'smass.Fromthetriangle OPA inthegure, r 2 1 = r 2 + R 2 2 rR cos (17.6) Usingthisin(17.5)gives V M = rM R 1 2 r R cos + r R 2 1 = 2 (17.7) r=R 1 = 60,and(17.7)maybeexpandedinpowersof r=R usingLegendre polynomials(WhittakerandWatson,1963: x 15.1): V M = rM R 1+ r R cos + r R 2 1 2 (3cos 2 1)+ (17.8) Thetidalforcesarecalculatedfromthespatialgradientof thepotential.The rsttermin(17.8)producesnoforce.Thesecondterm,whend ierentiated withrespectto( r cos )producesaconstantforce rM=R 2 paralleltoOAthat keepsearthinorbitaroundthecenterofmassoftheearth-mo onsystem.The thirdtermproducesthetides,assumingthehigher-orderte rmscanbeignored. Thetide-generatingpotentialistherefore: V = rMr 2 2 R 3 (3cos 2 1)(17.9)

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17.4.THEORYOFOCEANTIDES 303 Z 30o-30o0o60o Figure17.11Thehorizontalcomponentofthetidalforceone arthwhenthe tide-generatingbodyisabovetheEquatorat Z .AfterDietrichetal.(1980:413). Thetide-generatingforcecanbedecomposedintocomponent sperpendicular P andparallel H totheseasurface.Tidesareproducedbythehorizontal component.\Theverticalcomponentisbalancedbypressure ontheseabed, buttheratioofthehorizontalforceperunitmasstovertica lgravityhastobe balancedbyanopposingslopeoftheseasurface,aswellasby possiblechanges incurrentmomentum"(Cartwright,1999:39,45).Thehorizo ntalcomponent, showningure17.11,is: H = 1 r @V @' = 2 G r sin2 (17.10) where G = 3 4 rM r 2 R 3 (17.11) Thetidalpotentialissymmetricabouttheearth-moonline, anditproduces symmetricbulges. Ifweallowourocean-coveredearthtorotate,anobserverin spacesees thetwobulgesxedrelativetotheearth-moonlineasearthr otates.Toan observeronearth,thetwotidalbulgesseemstorotatearoun dearthbecause moonappearstomovearoundtheskyatnearlyonecycleperday .Moon produceshightidesevery12hoursand25.23minutesontheeq uatorifthe moonisabovetheequator.Noticethathightidesarenotexac tlytwiceperday becausethemoonisalsorotatingaroundearth.Ofcourse,th emoonisabove theequatoronlytwiceperlunarmonth,andthiscomplicates oursimplepicture ofthetidesonanidealocean-coveredearth.Furthermore,m oon'sdistancefrom earth R variesbecausemoon'sorbitisellipticalandbecausetheel lipticalorbit isnotxed. Clearly,thecalculationoftidesisgettingmorecomplicat edthanwemight havethought.Beforecontinuingon,wenotethatthesolarti dalforcesare derivedinasimilarway.Therelativeimportanceofthesuna ndmoonare

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304 CHAPTER17.COASTALPROCESSESANDTIDES nearlythesame.Althoughthesunismuchmoremassivethanmo on,itismuch furtheraway. G sun = G S = 3 4 rS r 2 R 3 sun (17.12) G moon = G M = 3 4 rM r 2 R 3 moon (17.13) G S G M =0 : 46051(17.14) where R sun isthedistancetothesun, S isthemassofthesun, R moon isthe distancetothemoon,and M isthemassofthemoon. CoordinatesofSunandMoon Beforewecanproceedfurtherweneedto knowthepositionofmoonandsunrelativetoearth.Anaccura tedescriptionof thepositionsinthreedimensionsisverydicult,anditinv olveslearningarcane termsandconceptsfromcelestialmechanics.Here,Iparaph raseasimplied descriptionfromPugh(1987).Seealsogure4.1. Anaturalreferencesystemforanobserveronearthistheequ atorialsystem describedatthestartofChapter3.Inthissystem, declinations ofacelestial bodyaremeasurednorthandsouthofaplanewhichcutstheear th'sequator. Angulardistancesaroundtheplanearemeasuredrelativeto apoint onthiscelestialequatorwhichisxedwithrespecttothest ars.Thepoint chosenforthissystemisthe vernalequinox ,alsocalledthe`FirstPointof Aries'...Theanglemeasuredeastward,betweenAriesandth eequatorial intersectionofthemeridianthroughacelestialobjectisc alledthe right ascension oftheobject.Thedeclinationandtherightascensiontoget her denethepositionoftheobjectonacelestialbackground.. [Anothernaturalreference]systemusestheplaneoftheear th'srevolutionaroundthesunasareference.Thecelestialextensi onofthis plane,whichistracedbythesun'sannualapparentmovement ,iscalled the ecliptic .Conveniently,thepointonthisplanewhichischosenfor azeroreferenceisalsothevernalequinox,atwhichthesunc rossesthe equatorialplanefromsouthtonortharound21Marcheachyea r.Celestialobjectsarelocatedbytheireclipticlatitudeandecli pticlongitude. Theanglebetweenthetwoplanes,of23.45 ,iscalledtheobliquityofthe ecliptic...|Pugh(1987:72). TidalFrequencies Now,let'sallowearthtospinaboutitspolaraxis.The changingpotentialataxedgeographiccoordinateonearth is: cos =sin p sin +cos p cos cos( 1 180 )(17.15) where p islatitudeatwhichthetidalpotentialiscalculated, isdeclination ofmoonorsunnorthoftheequator,and 1 isthehourangleofmoonorsun. The hourangle isthelongitudewheretheimaginaryplanecontainingthesu n ormoonandearth'srotationaxiscrossestheEquator.

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17.4.THEORYOFOCEANTIDES 305 Theperiodofthesolarhourangleisasolardayof24hr0m.The period ofthelunarhourangleisalunardayof24hr50.47m. Earth'saxisofrotationisinclined23.45 withrespecttotheplaneofearth's orbitaboutthesun.Thisdenestheecliptic,andthesun'sd eclinationvaries between = 23 : 45 withaperiodofonesolaryear.Theorientationofearth's rotationaxisprecesseswithrespecttothestarswithaperi odof26000years. Therotationoftheeclipticplanecauses andthevernalequinoxtochange slowly,andthemovementcalledthe precessionoftheequinoxes Earth'sorbitaboutthesuniselliptical,withthesuninone focus.Thatpoint intheorbitwherethedistancebetweenthesunandearthisam inimumiscalled perigee .Theorientationoftheellipseintheeclipticplanechange sslowly withtime,causingperigeetorotatewithaperiodof20942ye ars.Therefore R sun varieswiththisperiod. Moon'sorbitisalsoelliptical,butadescriptionofmoon's orbitismuch morecomplicatedthanadescriptionofearth'sorbit.Herea rethebasics.The moon'sorbitliesinaplaneinclinedatameanangleof5.15 relativetothe planeoftheecliptic.Andlunardeclinationvariesbetween =23 : 45 5 : 15 withaperiodofonetropicalmonthof27.32solardays.Theac tualinclination ofmoon'sorbitvariesbetween4.97 ,and5.32 Theshapeofmoon'sorbitalsovaries.First,perigeerotate swithaperiod of8.85years.Theeccentricityoftheorbithasameanvalueo f0.0549,and itvariesbetween0.044and0.067.Second,theplaneofmoon' sorbitrotates aroundearth'saxisofrotationwithaperiodof18.613years .Bothprocesses causevariationsin R moon NotethatIamalittleimpreciseindeningthepositionofth esunand moon.Lang(1980: x 5.1.2)givesmuchmoreprecisedenitions. Substituting(17.15)into(17.9)gives: V = rMr 2 R 3 1 4 3sin 2 p 1 3sin 2 1 +3sin2 p sin2 cos 1 +3cos 2 p cos 2 cos2 1 (17.16) Equation(17.16)separatestheperiodofthelunartidalpot entialintothree termswithperiodsnear14days,24hours,and12hours.Simil arlythesolar potentialhasperiodsnear180days,24hours,and12hours.T husthereare threedistinctgroupsoftidalfrequencies:twice-daily,d aily,andlongperiod, havingdierentlatitudinalfactorssin 2 ,sin2 ,and(1 3cos 2 ) = 2,where is theco-latitude(90 ). Doodson(1922)expanded(17.16)inaFourierseriesusingth ecleverlychosen frequenciesintable17.1.Otherchoicesoffundamentalfre quenciesarepossible, forexamplethelocal,mean,solartimecanbeusedinsteadof thelocal,mean, lunartime.Doodson'sexpansion,however,leadstoanelega ntdecomposition oftidalconstituentsintogroupswithsimilarfrequencies andspatialvariability. UsingDoodson'sexpansion,eachconstituentofthetidehas afrequency f = n 1 f 1 + n 2 f 2 + n 3 f 3 + n 4 f 4 + n 5 f 5 + n 6 f 6 (17.17)

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306 CHAPTER17.COASTALPROCESSESANDTIDES Table17.1FundamentalTidalFrequencies FrequencyPeriodSource /hour f 1 14.492052111lunardayLocalmeanlunartime f 2 0.549016531monthMoon'smeanlongitude f 3 0.041068641yearSun'smeanlongitude f 4 0.004641848.847yearsLongitudeofmoon'sperigee f 5 -0.0022064118.613yearsLongitudeofmoon'sascendingnod e f 6 0.0000019620,940yearsLongitudeofsun'sperigee wheretheintegers n i arethe Doodsonnumbers n 1 =1 ; 2 ; 3and n 2 { n 6 are between 5and+5.Toavoidnegativenumbers,Doodsonaddedveto n 2 6 EachtidalwavehavingaparticularfrequencygivenbyitsDo odsonnumberis calleda tidalconstituent ,sometimescalleda partialtides .Forexample,the principal,twice-per-day,lunartidehasthenumber255.55 5.Becausethevery long-termmodulationofthetidesbythechangeinsun'speri geeissosmall,the lastDoodsonnumber n 6 isusuallyignored. Iftheoceansurfaceisinequilibriumwiththetidalpotenti al,whichmeans weignoreinertiaandcurrentsandassumenoland(Cartwrigh t1999:274),the largesttidalconstituentswouldhaveamplitudesgivenint able17.2.Notice thattideswithfrequenciesnearoneortwocyclesperdayare splitintoclosely spacedlineswithspacingseparatedbyacyclepermonth.Eac hoftheselinesis furthersplitintolinesseparatedbyacycleperyear(gure 17.12).Furthermore, eachoftheselinesissplitintolinesseparatedbyacyclepe r8.8yr,andsoon. Table17.2PrincipalTidalConstituents Equilibrium Tidal Amplitude y Period SpeciesName n 1 n 2 n 3 n 4 n 5 ( m )(hr) Semidiurnal n 1 =2 Principallunar M 2 200000.24233412.4206 Principalsolar S 2 22-2000.11284112.0000 Lunarelliptic N 2 2-10100.04639812.6584 Lunisolar K 2 220000.03070411.9673 Diurnal n 1 =1 Lunisolar K 1 110000.14156523.9344 Principallunar O 1 1-10000.10051425.8194 Principalsolar P 1 11-2000.04684324.0659 Ellipticlunar Q 1 1-20100.01925626.8684 LongPeriod n 1 =0 Fortnightly Mf 020000.041742327.85 Monthly Mm 010-100.022026661.31 Semiannual Ssa 002000.0194464383.05 y AmplitudesfromApel(1987)

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17.4.THEORYOFOCEANTIDES 307 Frequency (deg/hr) 3N2e22N2m2N2n2g2a2M2b2d2l2L22T2T2S2K2R2z2h2 2T2T2S2K2R2 Amplitude (cm) 10-210-110010110226 27 28 29303132Amplitude (cm) 29.8029.8529.9029.9530.0030.0530.1030.1530.20 Frequency (deg/hr) 10-210-1100101102 Figure17.12 Upper: Spectrumofequilibriumtideswithfrequenciesneartwicep erday. Thespectrumissplitintogroupsseparatedbyacyclepermon th(0.55deg/hr). Lower: Expandedspectrumofthe S 2 group,showingsplittingatacycleperyear(0.04deg/hr). Thenestsplittinginthisgureisatacycleper8.847years (0.0046deg/hr).FromRichard Eanes,CenterforSpaceResearch,UniversityofTexas.Clearly,thereareverymanypossibletidalconstituents. Whyisthetidesplitintothemanyconstituentsshowningur e17.12?To answerthequestion,supposemoon'sellipticalorbitwasin theequatorialplane ofearth.Then =0.From(17.16),thetidalpotentialontheequator,where p =0,is: V = rMr 2 R 3 1 4 cos(4 f 1 )(17.18) Iftheellipticityoftheorbitissmall, R = R 0 (1+ ),and(17.18)isapproximately V = a (1 3 )cos(4 f 1 )(17.19) where a = rMr 2 = 4 R 3 isaconstant. varieswithaperiodof27.32days, andwecanwrite = b cos(2 f 2 )where b isasmallconstant.Withthese simplications,(17.19)canbewritten: V = a cos(4 f 1 ) 3 ab cos(2 f 2 )cos(4 f 1 )(17.20a) V = a cos(4 f 1 ) 3 ab [cos2 (2 f 1 f 2 )+cos2 (2 f 1 + f 2 )](17.20b)

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308 CHAPTER17.COASTALPROCESSESANDTIDES whichhasaspectrumwiththreelinesat2 f 1 and2 f 1 f 2 .Therefore,theslow modulationoftheamplitudeofthetidalpotentialattwocyc lesperlunarday causesthepotentialtobesplitintothreefrequencies.Thi sisthewayamplitude modulatedAMradioworks.Ifweaddintheslowchangesinthes hapeofthe orbit,wegetstillmoretermseveninthisveryidealizedcas eofamooninan equatorialorbit. Ifyouareveryobservant,youwillhavenoticedthatthetida lspectrum ingure17.12doesnotlookliketheocean-wavespectrumofo ceanwavesin gure16.6.Oceanwaveshaveallpossiblefrequencies,andt heirspectrumis continuous.Tideshaveprecisefrequenciesdeterminedbyt heorbitofearthand moon,andtheirspectrumisnotcontinuous.Itconsistsofdi scretelines. Doodson'sexpansionincluded399constituents,ofwhich10 0arelongperiod, 160aredaily,115aretwiceperday,and14arethriceperday. Mosthavevery smallamplitudes,andonlythelargestareincludedintable 17.2.Thelargest tideswerenamedbySirGeorgeDarwin(1911)andthenamesare includedin thetable.Thus,forexample,theprincipal,twice-per-day ,lunartide,whichhas Doodsonnumber255.555,isthe M 2 tide,calledthe M-two tide. 17.5TidalPrediction Iftidesintheoceanwereinequilibriumwiththetidalpoten tial,tidalpredictionwouldbemucheasier.Unfortunately,tidesarefarf romequilibrium. Theshallow-waterwavewhichisthetidecannotmovefasteno ughtokeepup withsunandmoon.Ontheequator,thetidewouldneedtopropa gatearound theworldinoneday.Thisrequiresawavespeedofaround460m /s,whichis onlypossibleinanocean22kmdeep.Inaddition,thecontine ntsinterruptthe propagationofthewave.Howtoproceed? Wecanseparatetheproblemoftidalpredictionintotwopart s.Therst dealswiththepredictionoftidesinportsandshallowwater wheretidescanbe measuredbytidegauges.Theseconddealswiththepredictio noftidesinthe deepoceanwheretidesaremeasuredbysatellitealtimeters TidalPredictionforPortsandShallowWater Twomethodsareusedto predictfuturetidesatatide-gaugestationusingpastobse rvationsofsealevel measuredatthegauge. TheHarmonicMethod Thisisthetraditionalmethod,anditisstillwidely used.Themethodtypicallyuses19yearsofdatafromacoasta ltidegaugefrom whichtheamplitudeandphaseofeachtidalconstituent(the tidalharmonics) inthetide-gagerecordarecalculated.Thefrequenciesuse dintheanalysisare speciedinadvancefromthebasicfrequenciesgivenintabl e17.1. Despiteitssimplicity,thetechniquehaddisadvantagesco mparedwiththe responsemethoddescribedbelow. 1.Morethan18.6yearsofdataareneededtoresolvethemodul ationofthe lunartides. 2.Amplitudeaccuracyof10 3 ofthelargesttermrequirethatatleast39 frequenciesbedetermined.Doodsonfound400frequenciesw ereneeded foranamplitudeaccuracyof10 4 ofthelargestterm.

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17.5.TIDALPREDICTION 309 3.Non-tidalvariabilityintroduceslargeerrorsintothec alculatedamplitudes andphasesoftheweakertidalconstituents.Theweakertide shaveamplitudessmallerthanvariabilityatthesamefrequencyduetoo therprocesses suchaswindsetupandcurrentsnearthetidegauge. 4.Atmanyports,thetideisnon-linear,andmanymoretidalc onstituentsare important.Forsomeports,thenumberoffrequenciesisunma nageable. Whentidespropagateintoveryshallowwater,especiallyri verestuaries, theysteepenandbecomenon-linear.Thisgeneratesharmoni csoftheoriginalfrequencies.Inextremecases,theincomingwavesstee penssomuch theleadingedgeisnearlyvertical,andthewavepropagates assolitary wave.Thisisa tidalbore TheResponseMethod Thismethod,developedbyMunkandCartwright (1966),calculatestherelationshipbetweentheobservedt ideatsomepointand thetidalpotential.Therelationshipisthespectraladmit tancebetweenthe majortidalconstituentsandthetidalpotentialateachsta tion.Theadmittance isassumedtobeaslowlyvaryingfunctionoffrequencysotha ttheadmittance ofthemajorconstituentscanbeusedfordeterminingtheres ponseatnearby frequencies.Futuretidesarecalculatedbymultiplyingth etidalpotentialby theadmittancefunction. 1.Thetechniquerequiresonlyafewmonthsofdata.2.Thetidalpotentialiseasilycalculated,andaknowledge ofthetidalfrequenciesisnotneeded. 3.Theadmittanceis Z ( f )= G ( f ) =H ( f ). G ( f )and H ( f )aretheFourier transformsofthepotentialandthetidegagedata,and f isfrequency. 4.Theadmittanceisinversetransformedtoobtaintheadmit tanceasafunctionoftime. 5.Thetechniqueworksonlyifthewavespropagateaslinearw aves. TidalPredictionforDeep-Water Predictionofdeep-oceantideshasbeen muchmoredicultthanpredictionofshallow-watertidesbe causetidegauges wereseldomdeployedindeepwater.Allthischangedwiththe launchofTopex/ Poseidon.Thesatellitewasplacedintoanorbitespecially designedforobservingoceantides(Parkeetal.1987),andthealtimetricsyste mwassuciently accuratetomeasuremanytidalconstituents.Datafromthes atellitehavenow beenusedtodeterminedeep-oceantideswithanaccuracyof 2cm.Formost practicalpurposes,thetidesarenowknownaccuratelyform ostoftheocean. Twoavenuesledtothenewknowledgeofdeep-watertidesusin galtimetry. PredictionUsingHydrodynamicTheory Purelytheoreticalcalculationsof tidesarenotveryaccurate,especiallybecausethedissipa tionoftidalenergy isnotwellknown.Nevertheless,theoreticalcalculations providedinsightinto processesinruencingoceantides.Severalprocessesmustb econsidered: 1.Thetidesinoneoceanbasinperturbearth'sgravitationa leld,andthe massinthetidalbulgeattractswaterinotheroceanbasins. Theselfgravitationalattractionofthetidesmustbeincluded.

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310 CHAPTER17.COASTALPROCESSESANDTIDES 2.Theweightofthewaterinthetidalbulgeissucientlygre atthatit deformsthesearoor.Theearthdeformsasanelasticsolid,a ndthe deformationextendsthousandsofkilometers. 3.Theoceanbasinshaveanaturalresonanceclosetothetida lfrequencies. Thetidalbulgeisashallow-waterwaveonarotatingocean,a nditpropagatesasahightiderotatingaroundtheedgeofthebasin.Thu sthetides areanearlyresonantsloshingofwaterintheoceanbasin.Th eactualtide heightsindeepwatercanbehigherthantheequilibriumvalu esnotedin table17.2. 4.Tidesaredissipatedbybottomfrictionespeciallyinsha llowseas,bythe rowoverseamountsandmid-oceanridges,andbythegenerati onofinternalwavesoverseamountsandattheedgesofcontinentals helves.If thetidalforcingstopped,thetideswouldcontinuesloshin gintheocean basinsforseveraldays. 5.Becausethetideisashallow-waterwaveeverywhere,itsv elocitydepends ondepth.Tidespropagatemoreslowlyovermid-oceanridges andshallow seas.Hence,thedistancebetweengridpointsinnumericalm odelsmust beproportionaltodepthwithveryclosespacingoncontinen talshelves (LeProvostetal.1994). 6.Internalwavesgeneratedbythetidesproduceasmallsign alatthesea surfacenearthetidalfrequencies,butnotphase-lockedto thepotential. Thenoisenearthefrequencyofthetidescausesthespectral cuspsin thespectrumofsea-surfaceelevationrstseenbyMunkandC artwright (1966).Thenoiseisduetodeep-water,tidallygenerated,i nternalwaves. AltimetryPlusResponseMethod SeveralyearsofaltimeterdatafromTopex/ Poseidonhavebeenusedwiththeresponsemethodtocalculat edeep-seatides almosteverywhereequatorwardof66 (Maetal.1994).Thealtimetermeasuredsea-surfaceheightsingeocentriccoordinatesateac hpointalongthesubsatellitetrackevery9.97days.Thetemporalsamplingalia sedthetidesintolong frequencies,butthealiasedperiodsarepreciselyknownan dthetidescanbe recovered(Parkeetal.1987).Becausethetidalrecordissh orterthan8years, thealtimeterdataareusedwiththeresponsemethodtoobtai npredictionsfor amuchlongertime. Recentsolutionsbytendierentgroups,haveaccuracyof 2.8cmindeep water(Andersen,Woodworth,andFlather,1995).Workhasbe guntoimprove knowledgeoftidesinshallowwater. AltimetryPlusNumericalModels Altimeterdatacanbeuseddirectlywith numericalmodelsofthetidestocalculatetidesinallareas oftheoceanfrom deepwaterallthewaytothecoast.Thusthetechniqueisespe ciallyuseful fordeterminingtidesnearcoastsandoversea-roorfeature swherethealtimeter groundtrackistoowidelyspacedtosamplethetideswellins pace.Tidemodels usenite-elementgridssimilartotheoneshowningure15. 3.Recentnumerical calculationsby(LeProvostetal.1994;LeProvost,Bennett ,andCartwright, 1995)giveglobaltideswith 2{3cmaccuracyandfullspatialresolution.

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17.5.TIDALPREDICTION 311 60o120o180o-120o-60o0o -60o-30o0o30o60o 10101010101010101020202020202020202020202020202020202020202020303030303030303030303030303030303030404040404040404040404040405050505050505050505060606060707080 00000 0000060606060606060 120120120120120120120180180180180180180 240240240240300300300300300300 M2 Figure17.13Globalmapof M 2 tidecalculatedfromTopex/Poseidonobservationsofthe heightoftheseasurfacecombinedwiththeresponsemethodf orextractingtidalinformation. Fulllinesarecontoursofconstanttidalphase,contourint ervalis30 .Dashedlinesarelines ofconstantamplitude,contourintervalis10cm.FromRicha rdRay, nasa GoddardSpace FlightCenter. Mapsproducedbythismethodshowtheessentialfeaturesoft hedeep-ocean tides(gure17.13).Thetideconsistsofacrestthatrotate scounterclockwise aroundtheoceanbasinsinthenorthernhemisphere,andinth eoppositedirectioninthesouthernhemisphere.Pointsofminimumampli tudearecalled amphidromes .Highesttidestendtobealongthecoast. Themapsalsoshowtheimportanceofthesizeoftheoceanbasi ns.The semi-diurnal(12hrperiod)tidesarerelativelylargeinal loceanbasins.But thediurnal(24hrperiod)tidesaresmallintheAtlanticand relativelylarge inthePacicandIndianocean.TheAtlanticistoosmalltoha vearesonant sloshingwithaperiodnear24hr.TidalDissipation Tidesdissipate3 : 75 0 : 08TWofpower(Kantha,1998), ofwhich3.5TWaredissipatedintheocean,andmuchsmallera mountsin theatmosphereandsolidearth.Thedissipationincreasest helengthofday byabout2.07millisecondspercentury,itcausesthesemima joraxisofmoon's orbittoincreaseby3.86cm/yr,anditmixeswatermassesint heocean. ThecalculationsofdissipationfromTopex/Poseidonobser vationsoftides areremarkablyclosetoestimatesfromlunar-laserranging ,astronomicalobservations,andancienteclipserecords.Thecalculationssho wthatroughlytwo thirdsoftheM2tidalenergyisdissipatedonshelvesandins hallowseas,and onethirdistransferredtointernalwavesanddissipatedin thedeepocean(Egbertandray,2000).85to90%oftheenergyoftheK1tideisdis sipatedin shallowwater,andonlyabout10{15%istransferredtointer nalwavesinthe deepocean(LeProvost2003,personalcommunication). Overall,ourknowledgeofthetidesisnowsucientlygoodth atwecan begintousetheinformationtostudymixingintheocean.Rec entresultsshow

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312 CHAPTER17.COASTALPROCESSESANDTIDES that\tidesareperhapsresponsibleforalargeportionofth everticalmixing intheocean"(Jayneetal.2004).Remember,mixinghelpsdri vetheabyssal circulationintheoceanasdiscussedin x 13.2(MunkandWunsch,1998).Who wouldhavethoughtthatanunderstandingoftheinruenceoft heoceanon climatewouldrequireaccurateknowledgeoftides?17.6ImportantConcepts 1.Wavespropagatingintoshallowwaterarerefractedbyfea turesofthe searoor,andtheyeventuallybreakonthebeach.Breakingwa vesdrive near-shorecurrentsincludinglong-shorecurrents,ripcu rrents,andedge waves. 2.Stormsurgesaredrivenbystrongwindsinstormsclosetos hore.The amplitudeofthesurgeisafunctionofwindspeed,theslopeo fthesearoor, andthepropagationofthestorm. 3.Tidesareimportantfornavigation;theyinruenceaccura tegeodeticmeasurements;andtheychangetheorbitsandrotationofplanet s,moons,and starsingalaxies. 4.Tidesareproducedbyacombinationoftime-varyinggravi tationalpotentialofthemoonandsunandthecentrifugalforcesgenerated asearth rotatesaboutthecommoncenterofmassoftheearth-moon-su nsystem. 5.Tideshavesixfundamentalfrequencies.Thetideisthesu perpositionof hundredsoftidalconstituents,eachhavingafrequencytha tisthesum anddierenceofvefundamentalfrequencies. 6.Shallowwatertidesarepredictedusingtidemeasurement smadeinports andotherlocationsalongthecoast.Tidalrecordsofjustaf ewmonths durationcanbeusedtopredicttidesmanyyearsintothefutu re. 7.Tidesindeepwaterarecalculatedfromaltimetricmeasur ements,especiallyTopex/Poseidonmeasurements.Asaresult,deepwate rtidesare knownalmosteverywherewithanaccuracyapproaching 2cm. 8.Thedissipationoftidalenergyintheoceantransfersang ularmomentum frommoontoearth,causingthedaytobecomelonger. 9.Tidaldissipationmixeswatermasses,anditisamajordri verofthedeep, meridional-overturningcirculation.Tides,abyssalcirc ulation,andclimate arecloselylinked.

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Indexabsolutevorticity, 200 200 210 abyss, 211 227 abyssalcirculation, 105 211 211 219 222 312 accelerationequation, 109 accuracy, 17 19 19 88 altimeter, 157 170 Argos, 174 AVHRRtemperature, 90 maps, 92 Boussinesqapproximation, 154 CTD, 97 currentmeter, 181 density, 76 84 101 163 depthfromXBT, 96 dragcoecient, 48 echosounders, 30 ElNi~noforecasts, 254 equation momentum, 153 ofstate, 88 soundspeed, 35 equations geostrophic, 169 ruxes frommodels, 64 ICOADS, 62 radiative, 58 geoid, 156 158 heatruxes, 59 numericalmodels, 255 257 coastal, 264 coupled, 270 pressure, 95 160 rainfall, 64 251 salinity, 73 76 93 94 fromtitration, 93 satellitetrackingsystems, 158 short-waveradiation, 60 stormsurge, 265 temperature, 77 bucket, 90 sea-surfacemaps, 92 ship-injection, 90 thermistor, 90 XBT, 96 thermometers mercury, 88 platinum, 88 tides, 308 { 310 312 topography, 181 waveheight, 290 291 winds Beaufort, 44 calculated, 47 scatterometer, 45 ship, 46 SSM/I, 46 acoustic-dopplercurrentproler, 180 adiabatically, 85 AdvancedVeryHighResolutionRadiometer(AVHRR), 58 60 90 92 102 177 178 267 268 advection, 51 Aguasatellite, 100 altimeters, see satellitealtimetry amphidromes, 311 anomalies atmosphericpressure, 249 density, 163 sea-surfacetemperature, 77 92 159 243 245 253 sealevel, 159 332

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INDEX 333 specicvolume, 163 wind, 253 windstress, 253 AntarcticCircumpolarCurrent, 83 222 229 231 { 233 271 calculationsof, 271 272 AntarcticIntermediateWater, 225 226 AntarcticPolarFront, 232 anti-cyclonic, 135 Argossystem, 97 135 173 { 175 251 atmosphericboundarylayer, 43 atmosphericcirculation causes, 235 drivenbyocean, 39 atmosphericconditions ndinghistorical, 215 atmospherictransmittance, 56 -plane, 105 207 239 262 EkmanPumping, 206 ruiddynamicson, 206 baroclinicrow, 163 barotropicrow, 163 basins, 27 bathymetriccharts ETOPO-2, 34 GEBCO, 33 bathythermograph(BT), 70 95 expendable(XBT), 95 268 bore, 296 bottomwater, 215 Antarctic, 225 { 227 NorthAtlantic, 211 213 233 boundarycurrents, 105 boundarylayer, 115 Boussinesqapproximation, 112 114 154 256 257 259 262 263 boxmodel, 107 168 breakers andedgewaves, 297 andlong-shorecurrents, 296 heightof, 294 295 plunging, 295 296 spilling, 295 surging, 295 typesof, 295 Brunt-Vaisalafrequency, 128 see stability,frequency bulkformulas, 58 buoyancy, 87 103 114 123 127 frequency, 128 buoyancyrux, 51 canyon, 27 carbondioxide, 212 chlorinity, 74 chlorophyll calculatingconcentration, 101 measurementfromspace, 100 circulation abyssal, 211 211 219 222 312 deep AntarcticCircumpolarCurrent, 229 fundamentalideas, 219 importanceof, 212 observationsof, 222 theoryfor, 219 { 222 GulfStreamrecirculationregion, 194 meridionaloverturning, 14 212 215 { 217 258 269 271 312 NorthAtlantic, 192 Sverdrup'sTheory, 183 CircumpolarDeepWater composition, 230 closureproblem, 122 conductivity, 75 measurementof, 93 conservationlaws, 103 conservationofmass, 106 111 conservative, 202 conservativerow, 202 conservativeproperties, 224 continentalshelves, 27 continentalslopes, 27 continuityequation, 111 112 113 coordinatesystems, 104 -plane, 105 Cartesian, 104 f-plane, 104

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334 INDEX forsunandmoon, 304 sphericalcoordinates, 105 Copenhagenseawater, 75 core, 226 coremethod, 225 227 tracers, 227 Coriolisforce, 104 110 136 139 153 Coriolisparameter, 120 122 134 137 152 153 155 164 198 199 207 nearequator, 238 CTD, 14 94 96 97 102 164 166 168 170 180 182 264 currentshear, 129 currents alongshore, 296 rip, 296 surface, 237 tidal, 300 wave-driven, 296 Dansgaard/Oeschgerevent, 216 data validated, 17 dataassimilation, 46 68 266 { 268 datasets, 17 whatmakesgooddata?, 17 declinations, 304 deepcirculation, 105 219 AntarcticCircumpolarCurrent, 229 fundamentalideas, 219 importanceof, 212 observationsof, 222 theoryfor, 219 { 222 densication, 224 density, 84 absolute, 84 accuracyof, 88 anomalyorsigma, 84 equation ofstate, 87 neutralsurfaces, 87 potential, 85 diapycnalmixing, 123 125 127 diusiveconvection, 131 dispersionrelation, 274 doldrums, 237 Doodsonnumbers, 306 doublediusion, 127 131 saltngers, 130 drag coecient, 48 48 49 121 138 139 192 264 265 286 form, 202 232 drifters, 2 149 173 182 accuracyofcurrentmeasurements, 173 174 andnumericalmodels, 255 holey-sock, 174 inKuroshio, 170 inPacic, 140 141 measurementofEkmancurrents, 142 rubberduckie, 179 dynamicinstability, 129 dynamicmeter, 160 dynamictopography, 155 earth equinox, 305 inspace, 39 perigeeof, 305 radiiof, 21 rotationrate, 108 earth-systemscience, 8 echosounders, 29 { 30 errorsinmeasurement, 30 eddydiusivity, 120 eddyviscosity, 120 Ekmanlayer, 115 135 { 143 151 164 202 bottom, 136 140 characteristics, 148 coastalupwelling, 145 dened, 135 depth, 139 Ekman'sassumptions, 137 141 inruenceofstability, 142 observationsof, 142 seasurface, 135 { 139 surface-layerconstants, 138

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INDEX 335 theoryof, 136 EkmanNumber, 139 Ekmanpumping, 2 145 { 147 149 205 207 { 210 219 239 dened, 147 example, 208 Ekmantransport, 143 { 145 186 232 236 299 masstransportdened, 143 uses, 145 volumetransportdened, 144 ElNi~no, 240 { 246 dened, 243 forecasting, 251 253 atmosphericmodels, 252 coupledmodels, 253 oceanicmodels, 252 observing, 250 teleconnections, 248 theoryof, 244 ElNi~no{SouthernOscillation(ENSO), see SouthernOscillation Envisat, 33 equationofstate, 87 equatorialprocesses, 236 ElNi~no, 240 Ni~na, 240 undercurrent, 237 theory, 238 equatoriallytrappedwaves, 247 equinox, 305 precessionof, 305 ERSsatellites, 11 33 48 63 156 251 269 292 Eulerequation, 109 Eulerianmeasurements, 179 acoustic-dopplercurrentproler, 180 f -plane, 104 ruiddynamicson, 205 Taylor-ProudmanTheorem, 205 fetch, 287 roats, 175 ALACE, 175 Argo, 97 175 269 inNorthAtlantic, 195 row conservative, 202 typesof, 105 rux buoyancy, 51 directcalculationof gustprobemeasurement, 57 radiometermeasurements, 58 globaldatasetsfor, 61 indirectcalculationof bulkformulas, 58 latentheatrux calculationof, 60 netlong-waveradiation, 60 sensibleheatrux calculationof, 61 waterrux calculationof, 60 ruxadjustments, 271 ruxcorrections, 271 formdrag, 202 232 Fourierseries, 279 Fouriertransform, 279 friction, 104 frictionvelocity, 121 andwindstress, 121 fullydevelopedsea, 285 generalcirculation, 105 geoid, 32 32 33 155 156 158 160 161 181 errors, 158 undulations, 32 33 155 geophysicalruiddynamics, 8 geophysics, 8 geopotential, 155 160 anomaly, 162 meter, 160 surface, 160 161 Geosat, 11 33 156 290 Follow-Onmission, 269 geostrophicapproximation, 3 152 169 geostrophicbalance, 151,152 181 andRossbywaves, 248 ignoresfriction, 170

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336 INDEX limitationsof, 169 notnearequator, 170 239 geostrophiccurrents, 3 14 15 149 163 altimeterobservationsof, 126 andEkmanpumping, 145 208 andEkmantransports, 209 andlevelofnomotion, 170 assimilatedintonumericalmodels, 267 calculatedbynumericalmodel, 260 cannotchange, 169 commentson, 164 deepinterior, 219 eddies, 125 126 equationsfor, 153 { 155 160 fromaltimetry, 155 { 158 fromhydrographicdata, 158 { 172 fromslopeofdensitysurfaces, 171 inGulfStream, 170 inocean'sinterior, 207 inPacic, 210 measuredbyaltimetry, 170 181 Munk'stheoryfor, 198 notnearequator, 239 relative, 164 relativetolevelofnomotion, 181 relativetotheearth, 164 surface, 155 Sverdrup'stheoryfor, 197 velocityof, 147 verticalandEkmanpumping, 207 vorticity, 201 vorticityconstraints, 205 206 geostrophicequations, 152,153 limitationsof, 169 geostrophictransport, 189 GlobalConveyerBelt, 213 GlobalOceanDataAssimilationExperiment roats, 175 products, 268 globalprecipitation mapof, 61 GlobalPrecipitationClimatologyProject, 64 GRACE, 143 156 158 gravity, 103 greenhouseeect, 55 groupvelocity, 275 GulfStream, 5 172 194 198 anddeepmixing, 222 andmixing, 125 asawesternboundarycurrent, 105 193 calculatedbyMICOMnumerical model, 261 calculationof, 266 267 269 crosssectionof, 168 223 densitysurfaces, 172 eddies, 172 forecasts, 267 268 forecastsof, 272 Franklin-Folgermapof, 13 isbaroclinic, 163 mappedbyBenjaminFranklin, 7 9 13 mappedbyroats, 193 mappedbyTopex/Poseidon, 156 northeastofCapeHatteras, 170 observationsof, 192 positionof, 266 267 recirculationregion, 194 sketchof, 197 southofCapeCod, 164 Stommel'stheoryfor, 189 T-Splots, 224 transport, 192 214 transportofheatby, 213 velocityof, 172 vorticity, 200 wiggles, 266 gyres, 105 heatbudget, 51 geographicaldistributionofterms, 65 importanceof, 52 termsof, 51 53 throughthetopoftheatmosphere, 69 zonalaverage, 65

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INDEX 337 heatrux, 2 47 51 65 237 252 259 261 267 270 272 from icoads 72 fromnumericalmodels, 64 65 globalaverage, 67 infrared, see infraredrux latent, see latentheatrux meanannual, 68 measurementof, 72 measurementsof, 57 net, 67 70 72 netthroughthetopofatmosphere, 68 70 71 Oberhuberatlas, 253 poleward, 215 sensible, see sensibleheatrux solar, see insolation unitsof, 52 zonalaverage, 66 70 heatstorage, 69 seasonal, 53 heattransport calculationof, 70 directmethod, 70 residualmethod, 70 surfaceruxmethod, 70 GlobalConveyerBelt, 213 meridional, 68 68 oceanic, 213 Heinrichevents, 216 hydrographicdata, 158 162 acrossGulfStream, 172 196 223 andaltimetry, 181 andgeostrophiccurrents, 163 164 166 168 { 170 181 182 andnorthAtlanticcirculation, 192 andpotentialvorticity, 202 andSverdruptransport, 186 andwatermasses, 225 disadvantageof, 168 fromCarnegie, 185 fromEndeavor, 164 168 hydrographicsections, 171 176 hydrographicstations, 96 161 { 163 167 acrossAntarcticCircumpolarCurrent, 230 andacousticDopplercurrentproler, 180 usedforsalinity, 93 hydrography, 8 hydrostaticequilibrium, 151 ice-age, 215 ICOADS(internationalcomprehensive ocean-atmospheredataset), 43 60 61 62 65 92 93 243 insitu, 11 84 85 87 inertial current, 134 135 147 motion, 133 oscillation, 3 134 period, 135 142 148 infraredrux, 68 annualaverage, 56 factorsinruencing, 54 net, 55 insolation, 51 53 70 absorptionof, 65 annualaverage, 54 67 atsurface, 65 attopofatmosphere, 54 65 70 balancedbyevaporation, 65 calculationof, 59 60 64 factorsinruencing, 53 mapsof, 59 maximum, 40 zonalaverage, 62 65 66 instability dynamic, 129 InternationalHydrographicBureau, 22 27 InternationalHydrographicOrganization, 34 internationalnauticalmile, 21 InternationalSatelliteCloudClimatologyProject, 64 inverseproblem, 266 irradiance, 99 isobaricsurface, 153 isopycnalmodel, 260 isopycnalsurfaces, 163 isotropicturbulence, 122

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338 INDEX Jason, 33 63 156 157 181 269 290 accuracyof, 33 157 Jason-2, 156 jets, 105 JPOTS(ProcessingofOceanographic StationData), 88 Kelvinwave, see waves,Kelvin Kuroshio, 43 99 asawesternboundarycurrent, 105 geostrophicbalancein, 170 observedbydrifters, 141 thermocline, 129 transportof, 192 widthof, 192 LaNi~na, 240 { 246 forecasting, 251 253 atmosphericmodels, 252 coupledmodels, 253 oceanicmodels, 252 observing, 250 teleconnections, 248 theoryof, 244 Lagrangianmeasurements, 172 holey-sockdrifters, 174 satellitetrackedsurfacedrifters, 173 tracers, 175 Langmuircirculation, 147 latentheatrux, 51 56 calculationof, 60 latitude, 21 levelsurface, 30 32 127 153 155 162 light, 97 absorptionof, 97 linearity, 19 longitude, 21 meridionaltransport, 68 mesoscaleeddies, 105 151 156 172 258 261 270 272 mixedlayer, 81 84 101 120 223 228 andEkmanlayer, 141 142 andEkmanpumping, 207 andinertialoscillations, 148 andphytoplankton, 145 currents, 237 currentswithin, 148 deepenedbyKelvinwaves, 244 equatorial, 236 externalforcingof, 2 82 highlatitude, 83 ineasternbasins, 146 innumericalmodels, 253 measuredbybathythermograph, 95 mid-latitude, 82 mixingin, 125 mixingthroughbaseof, 126 seasonalgrowthanddecay, 82 solarheatingandphytoplankton, 99 T-Splot, 223 theory, 122 tropicalPacic, 83 upwellingthrough, 146 velocitywithin, 142 watermassformationwithin, 223 mixing, 123 aboveseamounts, 222 alongconstant-densitysurfaces, 227 amongwatermasses, 224 225 andcoremethod, 225 anddeepcirculation, 225 andrushingtime, 107 andpolewardheatrux, 215 averagehorizontal, 125 { 126 averagevertical, 123 betweenwatermasses, 223 226 bywinds, 215 diapycnal, 123 125 127 energyfor, 215 equatorial, 236 horizontal, 126 inCircumpolarCurrent, 230 271 innumericalmodels, 259 260 262 { 264 coastal, 264 Gent-McWilliamsscheme, 271

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INDEX 339 MellorandYamadascheme, 262 263 PacanowskiandPhilanderscheme, 270 quasi-geostrophic, 267 Smagorinskischeme, 264 inthermocline, 211 219 increasesdensity, 224 measuredvertical, 125 meridional, 122 oceanic, 123 ofdeepwaters, 125 211 215 219 221 233 ofheatdownward, 219 ofmomentum, 120 121 oftracers, 227 oftritium, 228 onsurfacesofconstantdensity, 259 tidal, 39 103 125 215 219 300 311 vertical, 19 28 123 227 measured, 125 zonal, 122 mixing-lengththeory, 15 121 122 MODIS, 100 molecularviscosity, 116 momentumequation, 108 Cartesiancoordinates, 109 Coriolisterm, 110 gravityterm, 110 moon, 7 103 273 301 { 304 308 312 coordinates, 304 Munk'ssolution, 190 Nansenbottles, 96 nauticalmile, 21 Navier-Stokesequation, 109 netinfraredradiation, 51 netlong-waveradiation, 60 neutralpath, 87 neutralsurfaceelement, 87 neutralsurfaces, 87 NorthAtlanticDeepWater, 94 216 225 { 227 numericalmodels, 4 assimilation, 266 HarvardOpen-OceanModel, 267 Mercator, 269 NLOM, 269 atmospheric, 252 coastal, 262 DartmouthGulfofMaineModel, 262 PrincetonOceanModel, 262 coupled, 253 269 accuracyof, 270 ClimateSystemModel, 270 ruxadjustmentsin, 270 { 272 HadleyCenterModel, 270 PrincetonCoupledModel, 270 deepcirculation, 222 isopycnal, 260 limitationsof, 255 mechanisticmodels, 257 numericalweathermodels, 46 reanalysisfrom, 47 reanalyzeddatafrom, 64 sourcesofreanalyzeddata, 47 oceanic, 252 primitive-equation, 259 climatemodels, 261 GeophysicalFluidDynamicsLaboratoryModularOceanModel(MOM), 259 HybridCoordinateOceanModel, 259 ParallelOceanProgramModel, 259 simulationmodels, 257 257 spin-up, 258 storm-surge, 264 AdvancedCirculationModel, 265 Sea,Lake,andOverlandSurges Model, 265 tidalprediction, 310 Nyquistcriticalfrequency, 281 observations, 4 16 ocean, 1 AtlanticOcean, 22 dened, 22

PAGE 348

340 INDEX dimensionsof, 23 dominantforcesin, 103 featuresof, 27 { 28 IndianOcean, 23 mapsof, 33 meansalinity, 79 meantemperature, 79 milestonesinunderstanding, 13 { 15 PacicOcean, 22 processesin, 4 oceaniccirculation abyssal, 211 211 219 222 312 deep AntarcticCircumpolarCurrent, 229 fundamentalideas, 219 importanceof, 212 observationsof, 222 theoryfor, 219 { 222 GulfStreamrecirculationregion, 194 Sverdrup'sTheory, 183 oceanicexperiments, 17 oceaniccirculation MeridionalOverturning, 212 oceanography, 8 erasofexploration, 8 { 12 newmethodsof, 4 pathline, 188 perigee, 305 periodogram, 282 phasevelocity, 274 physicaloceanography, 8 bigpicture, 3 goalsof, 2 plains, 27 planetaryvorticity, 199 potential density, 85 temperature, 84 85 potentialvorticity, 202 conservation consequencesof, 204 conservationof, 203 precision, 19 pressure measurementof, 95 quartzbourdongage, 95 quartzcrystal, 95 straingage, 95 vibratron, 95 standardatmospheric, 75 unitsof, 153 pressuregradient horizontal, 104 pseudo-forces, 104 pycnocline, 83 quasi-geostrophic, 267 QuikScat, 45 63 radiance, 98 rainfall calculationof, 60 cumulative, 60 equatorial, 235 global, 80 map, 61 overcoldocean, 244 patterns, 243 rates, 251 Texas, 250 tropical, 251 rainfall andENSO, 249 reducedgravity, 247 ReferenceSeawater, 76 referencesurface, 164 relativevorticity, 200 ReynoldsStress, 117 119 calculationof, 119 RichardsonNumber, 129 ridges, 27 ripcurrents, 293 296 312 Rossbywave, see waves,Rossby RubberDuckieSpill, 178 salinity, 73,74 accuracyof, 88 93 basedonchlorinity, 74

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INDEX 341 basedonconductivity, 75 conservationof, 224 geographicaldistributionof, 77 measurementof, 93 measurementwithdepth, 95 practical, 76 PracticalSalinityScale, 75 Reference, 76 simplevs.complete, 74 saltngering, 131 samplingerror, 17 18 44 46 60 158 satellitealtimetry, 156 290 310 errorsin, 157 mapsofthesea-roortopography, 33 systems, 31 useinmeasuringdepth, 30 satellitetrackedsurfacedrifters, 173 scatterometer Quikscat, 44 scatterometers, 44 45 47 49 59 252 289 accuracyof, 45 seamounts, 28 seas marginal, 23 Mediterranean, 23 Seasat, 33 sensibleheatrux, 51 57 annualaverage, 69 calculationof, 61 globalaverage, 65 mapsof, 68 uncertainty, 63 zonalaverage, 65 66 sequentialestimationtechniques, 46 267 sills, 28 28 solarconstant andinsolation, 60 69 value, 60 variabilityof, 70 71 sound absorptionof, 36 channel, 35 36 36 inocean, 34 rays, 36 speed, 35 andBoussinesqapproximation, 112 asfunctionofdepth, 35 inincompressibleruid, 112 typical, 37 variationof, 35 useof, 34 37 usedtomeasuredepth, 29 30 37 SouthernOscillation, 241 242 242 243 ElNi~noSouthernOscillation(ENSO), 236 243 248 253 Index, 236 242 243 250 specichumidity, 57 specicvolume, 161 anomaly, 162 squirts, 105 stability, 128 dynamic, 127 equation, 128 frequency, 127 128 128 132 inPacic, 129 sketchof, 127 static, 127 standardgeopotentialdistance, 162 Stommel'sTheory, 189 Stommel,Arons,Fallertheory, 219 { 221 stormsurges, 299 straingage, 95 streamfunction, 188 streamlines, 188 sun, 7 39 40 103 273 coordinates, 304 equinox, 39 heightabovehorizon, 53 54 perigeeof, 305 warmsequatorialwatewrs, 235 surfzone, 296 surfaceanalysis, 46 47 65 surfacecurrents, 237 surfacetemperature, 236 Sverdrup, 107 144 Sverdrup'sassumptions, 186

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342 INDEX Sverdrup'sTheory, 183 Taylor-ProudmanTheorem, 206 teleconnections, 248 temperature, 77 77 absolute, 77 accuracyof, 88 conservationof, 224 geographicaldistributionof, 77 globalmapsof, 92 InternationalTemperatureScale, 77 measurementatsurface, 88 AdvancedVeryHighResolution Radiometer(AVHRR), 269 byAdvancedVeryHighResolutionRadiometer(AVHRR), 90 bybucketthermometers, 90 bymercurythermometers, 88 byplatinumresistancethermometers, 90 bythermistors, 90 errorsin, 91 { 92 fromshipinjectiontemperatures, 90 measurementwithdepth, 95 bybathythermograph(BT), 95 byCTD, 97 byexpendablebathythermograph (XBT), 95 byreversingthermometers, 96 potential, 84 85 practicalscale, 77 surface, 236 Terrasatellite, 100 thermistor, 90 thermocline, 82 82 83 123 { 125 129 147 202 221 andcurrentshear, 142 andKelvinwaves, 244 belowGulfStream, 196 deep, 240 eddydiusivityin, 124 222 equatorial, 240 244 246 { 248 254 inNorthAtlantic, 192 innumericalmodels, 253 mixingin, 126 211 219 permanent, 83 seasonal, 82 83 147 shallow, 236 stabilityof, 129 upper, 237 238 ventilated, 126 verticalvelocityin, 219 220 thermometer mercury, 88 reversing, 88 89 96 97 102 onNansenbottles, 96 tidal currents, 105 amphidromes, 311 bore, 309 constituents, 306 309 principal, 306 currents, 300 dissipation, 311 Doodsonnumbers, 306 frequencies, 304 { 308 hourangle, 304 potential, 301 302 prediction, 308 altimetryplusnumericalmodels, 310 altimetryplusresponsemethod, 310 deepwater, 309 fromhydrodynamictheory, 309 harmonicmethod, 308 responsemethod, 309 shallowwater, 308 309 tides, 15 273 312 andperigee, 305 andstormsurges, 265 andtheequinox, 305 diurnal, 311 equilibrium, 306 307 hourangle, 304 partial, 306 semi-diurnal, 311 theoryof, 300 { 308

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INDEX 343 Topex/Poseidon, 11 15 33 63 156 157 181 251 290 310 312 accuracyof, 33 157 groundtracks, 12 observationsofdissipation, 311 observationsofElNi~no, 159 observationsofGulfStream, 156 observationsoftides, 309 observationsoftopography, 157 tidemap, 311 topographicblocking, 204 topographicsteering, 204 topography, 32 dynamic, 155 measuredbyaltimetry, 181 oceanic, 155 totalderivative, 107 tracers, 175 227 transport acrossequator, 220 andEkmanpumping, 145 147 andupwelling, 145 atmospheric, 81 byAntarcticCircumpolarCurrent, 222 229 { 231 233 272 byequatorialundercurrent, 238 byGulfStream, 192 194 198 214 bywaves, 278 calculatedbyStommelandArons, 220 carbondioxide, 213 convergenceof, 210 eastward, 144 Ekman, 142 143 145 206 { 209 Ekmanmass, 143 144 147 Ekmanvolume, 144 Ekman,observationsof, 144 equatorial, 236 geostrophicmass, 189 globalSverdrup, 187 heat, 2 51 67 70 { 72 122 212 { 215 heatinNorthAtlantic, 233 heatupward, 123 inNorthAtlantic, 193 194 inPacic, 185 186 210 inSouthernOcean, 229 inwesternboundarycurrents, 192 220 mass, 199 212 222 massandstormsurges, 299 meridional, 68 70 momentum, 119 northward, 184 198 northwardheat, 216 northwardintrades, 208 southwardinwesterlies, 208 streamfunction, 189 { 191 surfacemass, 189 Sverdrup, 183 { 185 throughDrakePassage, 229 volume, 107 188 volume,indeepocean, 212 wind-driven, 136 trenches, 28 tropicalconvergencezone, 237 tsunami, 3 258 293 Cascadia1700, 299 characteristics, 297 Hawaiian, 298 IndianOcean, 297 tsunamis, 105 297 turbulence, 25 103 193 256 atmospheric, 288 calculationof, 256 262 closureproblem, 122 263 indeepocean, 28 inGulfStream, 198 inmixedlayer, 82 innumericalmodels, 255 isotropic, 122 laboratory, 194 measurementof, 48 117 125 oceanic, 194 Reynoldsnumber, 132 subgrid, 262 272 theoryof, 15 16 121 123 transitionto, 117 twodimensional, 25 turbulent boundarylayer, 121

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344 INDEX nestructure, 125 mixing, 122 123 125 126 130 131 stress, 117 119 183 upwelling, 145 215 andcarbonstorage, 213 anddeepcirculation, 219 220 andsheries, 145 andwatertemperature, 145 coastal, 145 146 149 duetoEkmanpumping, 2 145 equatorial, 236 244 246 importanceof, 145 inNorthPacic, 218 ofCircumpolardeepWater, 232 radiance, 100 vibratron, 95 viscosity, 115 eddy, 120 molecular, 116 turbulent, 116 vorticity, 199 absolute, 200 210 conservationof, 202 water clarityof, 99 compressibilitycoecient, 113 type, 3 224 typemixing, 224 225 waterrux calculationof, 60 watermass, 223 226 AntarcticBottomWater, 226 AntarcticIntermediateWater, 226 CircumpolarDeepWater, 226 deepcirculation, 224 NorthAtlanticDeepWater, 226 waves breaking, 295 currents, 296 dispersionrelation, 274 edge, 297 edgewaves, 105 energy, 276 equatorial, 105 fetch, 287 forecasting, 288 Fourierseries, 279 Fouriertransform, 279 generationbywind, 288 groupvelocity, 275 internalwaves, 105 Kelvin, 105 244 246 { 248 254 273 275 length, 274 lineartheory, 273 measurementof, 289 { 292 gages, 291 satellitealtimeters, 290 syntheticapertureradars, 291 momentum, 278 nonlinear, 278 Nyquistcriticalfrequency, 281 period, 274 periodogram, 282 phasevelocity, 274 planetarywaves, 105 Rossby, 105 244 246 { 248 275 shoaling, 293 signicantheight, 277 solitary, 309 spectra, 284 calculating, 282 concept, 278 energy, 283 JONSWAP, 287 Pierson-Moskowitz, 285 wave-height, 283 surfacewaves, 105 tidalcurrents, 105 tsunamis, 105 Yanai, 105 westerlywindbursts, 244 westernboundarycurrents StommelsTheory, 189 wind, 104 Beaufortscale, 43 44 49 fromnumericalweathermodels, 46 47 49

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INDEX 345 fromscatterometers, 44 47 49 59 252 289 generationofwaves, 288 globalmap, 40 globalmean, 42 measurementof, 43 { 46 speed, 59 windsea, 288 windstress, 48 49 184 191 192 anddragcoecient, 48 andEkmanlayer, 138 andEkmanpumping, 147 andmasstransportinocean, 144 andnumericalmodels, 259 261 262 andstormsurges, 299 andsurfacecurrents, 147 andSverdruptransport, 187 annualaverage, 185 191 anomalies, 253 calculationof, 59 components, 137 184 curlof, 183 184 192 198 199 207 { 210 dailyaveragesof, 64 equatorial, 240 244 fromnumericalmodels, 65 isavector, 59 meanannual, 259 zonalaverage, 185 wind-drivencirculation, 105 Windsat, 45 WorldOceanCirculationExperiment, 11 12 16 230 zonal, 77