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Model Predictions of Radiation Stress Profiles for Nonlinear Shoaling Waves

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MODELPREDICTIONSOFRADIATIONSTRESSPROFILESFORNONLINEARSHOALINGWAVESByBRETM.WEBBATHESISPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFMASTEROFSCIENCEUNIVERSITYOFFLORIDA2004

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Thisworkisdedicatedtomywife,Shannon.

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ACKNOWLEDGMENTSFirst,Iwishtothankmywifeforhersupportandencouragementoverthepastfewyears.Myparents,therestofmyimmediatefamily,myextendedfamilyandcircleoffriendsalsodeservemuchcreditforthepersonIhavebecome,bothsociallyandacademically.Iwouldliketothankmyadvisor,DonSlinn,forhisencouragement,ideas,andsupport.Drs.RobertG.DeanandAndrewKennedy,oftheUniversityofFlorida,deserverecognitionfortheirserviceasmembersofmysupervisorycommittee.Thanksgo,also,totheremainderofthefacultyandstaffintheDepartmentofCivilandCoastalEngineering.Myofce-mates,allofwhomprovidedencouragementandguidanceoverthepastfewyears,wereinstrumentalinmysuccessasagraduatestudentandtheyshouldberecognizedfortheircontributionsaswell.ToddHollandandJackPuleodeserveacknowledgmentfortheirsupportandencouragement,andfornancialcontributionsthatpermittedmetoattendvariouslaboratoryandeldexperimentsoverthepastfewyears.ThanksshouldalsobeexpressedtotheOfceofNavalResearchandtheUniversityofFloridaforprovidingnancialsupport.Aportionofthiswork,aswellasmyeducation,wasmadepossiblebytheUniversityofFloridaGraduateFellowshipInitiative. iii

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................ iii LISTOFTABLES ................................... vi LISTOFFIGURES ................................... vii ABSTRACT ....................................... ix 1INTRODUCTION ................................ 1 1.1Background ................................ 1 1.2LiteratureSurvey ............................. 3 1.3Organization ................................ 5 2METHODOLOGY ................................ 7 2.1ModelCharacteristics ........................... 7 2.2GoverningEquations ........................... 8 2.3ImprovedBoundaryConditions ..................... 12 2.3.1WaveForcing ........................... 12 2.3.2FreeSurfaceVelocityBoundaryConditions ........... 14 2.3.3RigidBottomVelocityBoundaryConditions .......... 16 3EXPERIMENTS ................................. 18 3.1VisserExperiment ............................. 18 3.2ModelFormulation ............................ 19 3.3ModelSimulations ............................ 21 3.3.1NumericalConvergence ...................... 21 3.3.2ComputationalCost ........................ 22 3.3.3CompensatingforMassFlux ................... 22 3.3.4NonlinearityofModelPredictions ................ 24 4RESULTS ..................................... 26 4.1Model-DataComparison ......................... 26 4.1.1WaveTransformation ....................... 26 4.1.2LongshoreCurrentFormation .................. 30 4.2Three-DimensionalFlowFeatures .................... 32 4.2.1FreeSurfaceVisualizations .................... 32 4.2.2Depth-DependentWave-CurrentInteraction ........... 33 iv

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4.2.3LongshoreCurrentVariability .................. 37 4.3VerticalDistributionsofVelocityandMomentum ............ 40 4.3.1Time-AveragedVelocityProles ................. 40 4.3.2VerticalDistributionofRadiationStress ............. 44 4.3.3StressGradientsandNearshoreForcing ............. 55 5SUMMARY .................................... 61 5.1Applications ................................ 61 5.2Sensitivity ................................. 61 5.3Discoveries ................................ 62 APPENDIX ATHETADIFFERENCING ............................ 65 BCROSS-SHOREMASSBALANCE ....................... 68 REFERENCES ..................................... 72 BIOGRAPHICALSKETCH .............................. 76 v

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LISTOFTABLES Table page 2Breakingwavetypesclassiedbytheinshoresurfsimilarityparameter 12 3BeachandwaveparametersusedinVisser'sExperiment4 ........ 19 3Summaryofsimulationsperformedtoinvestigatenumericalconvergenceandcomputationalcost .......................... 22 vi

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LISTOFFIGURES Figure page 2Atypicalcomputationalcellusedinastaggeredgridandtheassociatedcoordinateaxissystem .......................... 8 2Contourplotsofthevelocityeldsattheforcingboundary ....... 14 2Specialcasesforsettingthevelocitycomponentsonthefreesurface .. 16 2Prescriptionsforsettingthetangentialvelocitycomponentsaroundastep .................................... 17 3PhysicaldomainusedinthesimulationofVisser'sExperiment4 .... 20 3Predictedroot-mean-squarewaveheightsforvedifferentgridresolutions 21 3Computationaltimerequiredforvariousgridresolutions ......... 23 3Theresponseoftheuidsurface,H,tomassuxneartheforcingboundary ................................. 24 3Contrastingvelocitytime-seriesplotstakenatoffshoreandinshorelocations ................................. 25 4Comparisonofmeasuredandpredictedwaveheights .......... 27 4Theinstantaneousfreesurfaceandwavesteepness ............ 28 4Statisticalpropertiesofthewaveeldatvarioustimelevels ....... 29 4Thepredictedandmeasuredlongshorecurrentvelocities ......... 31 4Timeevolutionoftheaveragelongshorecurrent ............. 33 4Averagelongshorecurrentvelocitiesoverdifferentsilldepths ...... 34 4Three-dimensionalvisualizationsoftheinstantaneousfreesurface ... 35 4Depth-dependentwave-currentinteractionsinthecross-shore ...... 36 4Depth-averaged u; wvelocityelds ................... 38 4Colorcontourplotsofthedepth-averagedlongshorevelocity ...... 39 4Time-averagedvelocityproles ...................... 42 vii

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4Thedepth-averagedcross-shorevelocity ................. 43 4Acomparisonoftheshapeandmagnitudeofvariouscomponentsofsxx 46 4Predictedprolesoftime-averagedradiationstresssxx .......... 50 4Predictedprolesoftime-averagedradiationstresssxz .......... 52 4Thedepth-integratedmagnitudesofthepredictedradiationstresses ... 53 4Theratioofmomentumuxoverthevertical ............... 55 4Theverticaldistributionofradiationandshearstressgradients ..... 57 4Depth-integratedvaluesofthenearshoreforcingcomponents ...... 60 AEffectonnumericaldiffusiononmodelpredictions ........... 67 BThetime-meanvelocityeldtakenatz=2:5m ............. 69 BSpatialfeaturesoftheaveragevelocityandfreesurfaceelds ...... 70 viii

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AbstractofThesisPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofMasterofScienceMODELPREDICTIONSOFRADIATIONSTRESSPROFILESFORNONLINEARSHOALINGWAVESByBretM.WebbDecember2004Chair:DonaldN.SlinnMajorDepartment:CivilandCoastalEngineeringTheuxofmomentumdirectedshorewardbyanincidentwaveeld,commonlyreferredtoastheradiationstress,playsasignicantroleinnearshorecirculationand,therefore,hasaprofoundimpactonthetransportofpollutants,biota,andsedimentinnearshoresystems.HavingreceivedmuchattentionsincetheinitialworkofLonguet-HigginsandStewartintheearly1960's,useoftheradiationstressconceptcontinuestoberenedandevidenceofitsutilityiswidespreadinliteraturepertainingtocoastalandoceanscience.Anumberofinvestigations,bothnumericalandanalyticalinnature,haveusedtheconceptoftheradiationstresstoderiveappropriateforcingmechanismsthatinitiatecross-shoreandlongshorecirculation,buttypicallyinadepth-averagedsenseduetoalackofinformationconcerningtheverticaldistributionofthewavestresses.Whiledepth-averagednearshorecirculationmodelsarestillwidelyusedtoday,advancementsintechnologyhavepermittedtheadaptationofthree-dimensional-Dmodelingtechniquestostudyowpropertiesofcomplexnearshorecirculationsystems.Ithasbeenshownthattheresultingcirculationinthese3-Dmodelsisverysensitive ix

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totheverticaldistributionofthenearshoreforcing,whichhavebeenimplementedaseitherdepth-uniformordepth-lineardistributions.Recently,analyticalexpressionsdescribingtheverticalstructureofradiationstresscomponentshaveappearedintheliterature,typicallybasedonlineartheory,butdonotfullydescribethemagnitudeandstructureintheregionboundbythetroughandcrestofnonlinear,propagatingwaves.Utilizingathree-dimensional,nonlinear,numericalmodelthatresolvesthetime-dependentfreesurface,wepresentmeanowpropertiesresultingfromasimulationofalaboratoryexperimentonuniformlongshorecurrents.Morespecically,weprovideinformationregardingtheverticaldistributionofradiationstresscomponentsresultingfromobliquelyincident,nonlinearshoalingwaves.Verticalprolesoftheradiationstresscomponentspredictedbythenumericalmodelarecomparedwithpublishedanalyticalsolutions,expressionsgivenbylineartheory,andobservationsfromaninvestigationemployingsecond-ordercnoidalwavetheory. x

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CHAPTER1INTRODUCTION1.1BackgroundThenearshorecoastalregion,typicallytakenastheareabetweentheinstantaneousshorelineandapointjustseawardofwavebreaking,isadynamicandcomplexsystem.Aswavespropagateintothisareatheyreleasetheirenergyintothewater,formingcurrentsthatcanpersistinboththecross-shoreandlongshoredirections.Thesecurrents,alongwiththecouplingthattakesplacebetweentheincidentwaveeldandthecurrents,makeupthenearshorecirculation.Nearshorecirculationcontinuestobestudiedandinvestigatedasitsimportanceismadeevidentthroughthetransportofmaterials,organicandnon-organic,inthecoastalzone.Ofparamountconcerntothecoastalscientististhetransportofsedimentinthecross-shoreandlongshoredirections,whichcontinuallyactstoshapeandreshapethenearshoreseabed.Thestudyofnearshorecirculationhasbeenapproachedmanydifferentways.Numerouslaboratoryandeldexperimentshavebeen,andcontinuetobe,performedinordertoincreaseunderstandingofnearshorecurrentdynamics.Analyticalstudieshavepavedthewayfornumericalsimulationsofnearshoreprocessesandadvancesincomputationalsciencepromotethedevelopmentofcomprehensivemodelingtools.Whiletwo-dimensional-Dnumericalmodelscontinuetobeusedforsimulatingnearshorecirculations,severallimitationsareinherentintheirapplication.Longshore-averaged2-Dmodelsignorethelongshorevariabilityofthewaveclimateandcurrentsystems,whiledepth-integrated2-Dmodelsneglecttheverticalstructureofnearshorecurrents.Neitherapproachcanprovidecompleteinformationaboutsedimenttransportinthenearshoresystem:therstexcludesthelongshoredegreeoffreedomwhilethelatterignoresthevertical-dependenceoftheowandsedimentsuspension 1

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2 cannotbeaccuratelydescribedbydepth-integratedquantities.Therequiredcouplingbetweenseparateshoalingwavemodelsandphase-averagedcirculationmodelsbecomesmorecomplexinthreedimensions.Thiscouplingisguidedbylineartheoryandempiricallybasedapproximationsdevelopedunderidealizedconditions.Astandardapproachistosolveadepth-integratedwaveenergyequationincorporatingempiricalformulationsforwaveenergydissipation.Thetransferofmomentumbetweenthebreakingwavesandthemeancurrentsisrepresentedbyradiationstressgradientterms.Distributingtheseforcesappropriatelyoverthewatercolumnrepresentedinaphase-averaged,3-Dcirculationmodelrequiresknowledgeoftheirverticaldistribution.Theintroductionofthree-dimensional-Dwave-phase-averagedcirculationmodels,however,hasnotprovidedthemodelingpanaceahopedfor.Theadditionaldimensiondemandsenormouscomputationalpowerandmorecomprehensiveforcingtechniquesthatrequire,inmanycases,aprioriknowledgeoftheoweldinalldimensions.Moreover,thecirculationpredictedbythesemodelshasbeenshowntobehighlysensitivetotheverticaldistributionofforcing.Thetermradiationstressdescribestheowofmomentuminpropagatingwavesandisdiscussedatlengthin Longuet-HigginsandStewart 1964 .Incidentwavesprovidethemajorityofmomentumtothenearshorecirculation.Gradientsintheradiationstresseldsrepresenttheforcingappliedtothesurfzone.Theverticaldistributionofradiationstressoverthewatercolumnaffectsthecross-shoreandlongshorecirculation,aswellasverticalmixing.Thus,choosingappropriateverticaldistributionsfortheradiationstresstermsisessentialforaccuratemodelingofnearshorecirculation.Usingathree-dimensional,nite-difference,Navier-Stokesmodelthatresolvesthetime-dependentfreesurface,weperformsimulationsofalaboratoryexperimentoflongshorecurrents.Meanowpropertiesofthesimulationarethenusedtoplotverticalprolesoftheshoreward-andlongshore-directedcomponentsofthe

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3 cross-shoreradiationstressresultingfromobliquelyincident,nonlinearshoalingwaves.Prolesofradiationstressarecomputedusingamodiedformoftheequationssuggestedby Longuet-HigginsandStewart 1964 andcomparedagainstprolesplottedusinglinearsolutionspresentedintheliterature.Whileweunderstandthatthisapproachisneitherthebestnortheonlywaytomodelwave-currentinteractions,wehopetoconveyourndingsinamannerthatishelpfultoothercoastalscientistsinterestedintheseprocesses,especiallythoseinvolvedwiththeresearchanddevelopmentofmodelingtools.1.2LiteratureSurveyApparentstressterms,whichappearasvelocityproductsintheconvectiveaccelerationtermsfoundintheNavier-Stokesequations,andtheradiationstresstermsexplicitinthewaveenergyequation,playsignicantrolesintheresultingnearshorecirculationinnumericalmodels.Numeroustheoretical,numerical,laboratory,andeldstudieshaveattemptedtoquantifyorjudgetherelativeimportanceofthenormalandshearstressesresultingfromshoalingandbreakingwaves.Recentattentionhasbeengiventotheverticaldistributionoftheseapparentstresses[see DeigaardandFredse 1989 ; SobeyandThieke 1989 ;and RiveroandArcilla 1995 ]and,morespecically,theradiationstressesthatresultfromtheincidentwaveeld[ Mellor 2003 and Xiaetal. 2004 ].Thepioneeringworkof Longuet-HigginsandStewart 1964 isevidentinmostnearshoreinvestigationsandtheirconceptscontinuetobeusedanddevelopedfurther.Unfortunately,thistheoreticalinvestigationreliedheavilyonthelinearapproximationofapropagatingwaterwaveandexcludedcontributionsfromtheincidentwavesabovethemeanwaterlevel,citingtheinsignicanceofthethird-orderintegrandsthatresultfromextendingthesolutionabovethesurface.Disregardingthecontributionoftheshoreward-directedmomentumuxabovethemeanwaterlevel,accordingto Svendsen 1984 and SobeyandThieke 1989 ,signicantlyunderestimatesthe

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4 magnitudeoftheradiationstressduetoreal,nonlinearpropagatingwaterwaves.Also,theresultinganalyticalexpressionspresentedin Longuet-HigginsandStewart 1964 aredepth-integratedquantitiesand,therefore,ignorepotentiallyimportantverticalinformation.Contrarytothemethodologyfollowedby Xieetal. 2001 ,thesedepth-integratedquantitiesarenotsuitableforuseasforcingtermsinverticallydependentmomentumequations.Traditionalradiationstressexpressions,however,havebeenusedinbothnumericalandtheoreticalcalculations,indepth-integratedform,todescribethewave-inducedsetupandsetdown,aswellasnearshorecurrents,andtheresultsagreequalitativelywithobservationsmadeby BadieiandKamphuis 1995 Bowenetal. 1968 ,and LarsonandKraus 1991 .Recentattemptstoprovidethree-dimensionalradiationstresssolutionshavebeenmadeby DolataandRosenthal 1984 Mellor 2003 ,and Xiaetal. 2004 ,buteachhasitslimitations. DolataandRosenthal 1984 neglectedpressureeffectsintheiranalyticalsolution,which,aswewillshowlater,representasignicantcomponentoftheshoreward-directedcomponentofthecross-shoreradiationstress.Theanalyticalsolutionsgivenby Mellor 2003 alsoemploylinearwavetheoryandspecicallyaddressdeepwaterpropagatingwaves.Inthecontextofnearshorecirculation,wherewavesareoftencharacterizedasshallowwaterwaves,thesesolutionsdonotappeartobeappropriate. Xiaetal. 2004 beginwiththedepth-integratedequationsforradiationstresspresentedby Longuet-HigginsandStewart 1964 ,disregardtheverticalintegrationoftheterms,andthensubstitutelinearexpressionsforthevelocityandpressuretermsintothemodiedequations. 1 Theresultinganalyticalsolutionsgivenby Xiaetal. 2004 areultimatelylimitedbytheirsmall-amplitudeassumptionthat 1Whilethesolutionsof Xiaetal. 2004 wereconsideredinthisinvestigation,theresultingprolescomputedwiththeirexpressionsarenotpresentedinourresultsbecausetheyweresimilartotheprolescomputedwiththeexpressionsof Mellor 2003

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5 is,theratioofwaveamplitudetowavelengthismuchlessthanunity.Intheshallowwatersofthenearshoresystem,itisnotuncommonforthisassumptiontobeviolatedasthewaveamplitudeincreasesthroughshoalingandthewavelengthdecreasestosatisfythelineardispersionrelationship.Whensmall-amplitudeassumptionsareemployedinthismanner,analyticalsolutionsbasedonthembegintodifferfromobservationsanderrorsareinherentinsubsequentcalculations StiveandWind 1982 .Finite-amplitudeandcnoidalwavetheoriesprovidebetterestimatesofwavevelocitiesinshallowwaters Xiaetal. 2004 .Theverticaldistributionsofapparentstresstermsvelocityproducts,explicitintheclassicalradiationstressexpressionsgivenby Longuet-HigginsandStewart 1964 ,resultingfromaninvestigationemployingsecond-ordercnoidalwavetheorywerepresentedin SobeyandThieke 1989 .Althoughtheydidnotprovidethree-dimensionalradiationstresssolutions,commentsandobservationsregardingtheirinvestigationproveusefultoourstudyoftheEulerianmeanowandtheresultingverticalproles.1.3OrganizationInthechaptersthatfollow,weprovideinformationabouttheverticaldistributionsofvelocityandmomentum,andtheneteffectsfromobliquelyincident,monochromaticwavesonaEulerianmeanow.Characteristics,abilities,andlimitationsofthenumericalmodelusedinthisstudyarepresentedinChapter 2 ;thegoverningequationsandimprovedboundaryconditionsforforcingandfreesurfacevelocitiesarediscussedhereaswell.InChapter 3 ,weprovideabriefsummaryofthelaboratoryexperiments[ Visser 1984 Visser 1991 ]usedtovalidatemodelpredictions,explainhowthemodelwasadaptedtosimulatetheexperiment,anddiscussphysicalandnumericalresultsofperformingsuchasimulationwiththemodel.Theresultsofthemodel-datacomparisonandsubsequentsimulationsareprovidedinChapter 4 ,alongwithanalysisofthemodelpredictions.Thisanalysisultimatelyleadstodiscussionoftheverticaldistributionofradiationstress,alsopresentedinChapter 4 ,wherewecompare

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6 predictedprolesfromthenonlinearmodeltothedistributionssuggestedbyanalyticalsolutionsfoundintheliterature.ConcludingremarksonthecapabilityofthenumericalmodeltoreproducenearshoreprocessesandasummaryofndingsontheverticaldistributionofradiationstressfornonlinearshoalingwavesaregiveninChapter 5 .Detailsconcerningthetheta-weighted,nite-differencingschemesandthesensitivityofmodelpredictionstothedifferencingschemesusedarepresentedinAppendix A .Adiscussiononthecross-shoremassbalanceisfoundinAppendix B whereweprovideadditionalinformationtosupportourclaimthatthenumericalmodelisconservative.

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CHAPTER2METHODOLOGY2.1ModelCharacteristicsThenumericalmodelconsidered,SOLA-SURF,isathree-dimensionalmodelthatemployscomputationaluiddynamicsCFDtodescribeunsteadyuidows.SOLA-SURFisanextensionofatwo-dimensionalCFDmodel,SOLA,thatwascreatedtostudytime-dependentuidowsinconnedregions.AlternateextensionsoftheoriginalSOLAcodehavebeenusedtostudyprocessessuchasbuoyancy-drivenows,owsofstratieduids,andowsinporousmedia Hirtetal. 1975 .Incontrasttotheoriginalmodel,SOLA-SURFhastheabilitytomodeluidowsboundbyfreeorcurvedrigidsurfaces.Theadditionofthesesurfaceboundaryconditionspermitstheusertosimulatewaterwavepropagationovervariablebathymetry.Variouslateralboundaryconditionsmaybeusedinthemodel:possiblecongurationsincludefree-slipandno-slipwalls,continuativeoroutowboundaries,andperiodicboundaryconditions.ThediscretisedequationsofmotionmaybesolvedoneitheraxisymmetricorCartesiancoordinatesystems.Therearenophysical,empiricalcoefcientstotuneandfewnumericalcoefcientstodeneinthismodel,makingthecodeadaptabletoabroadrangeofapplications.SOLA-SURFisbasedontheMarker-and-CellMACmethodwheretheprimarydependentvariables,pressureandvelocity,aresolvedinaEulerianreferenceframe.CharacteristicofMACschemes,astaggeredgridisusedtodenethelocationsofthepressureandthethreecomponentsofvelocity.Onthisgrid,thepressureisdenedatthecenterofeachcontrolvolumewhilethecomponentsofvelocityarelocatedoncellfaces.AtypicalgridcellusedinMACschemesisshowninFig. 2 ,wheretheorientationofthecoordinateaxesisrepresentativeofthesystemusedinSOLA-SURF. 7

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8 Figure2:Atypicalcomputationalcellusedinastaggeredgridandtheassociatedcoordinateaxissystem. ACartesiancoordinatesystemx;y;zrepresentsthecross-shore,vertical,andlongshoredirections,respectively,andsimulationsareperformedonarectangulargrid.2.2GoverningEquationsSOLA-SURFsolvestheNavier-StokesequationsEqs. 2 2 ,and 2 indiscretisedformandsatisesthemasscontinuityequationEq. 2 throughtheincorporationofaPoissonequationforthepressureeld.Inthecontextofthiswork,theequationsofmotioncharacterizeanunsteady,incompressible,homogeneousuid.OurCartesiancoordinatesystemassociatesthex,y,andzdirectionswiththeu,v,andwvelocitycomponents,respectively.@u @x+@v @y+@w @z=0@u @t+@u2 @x+@uv @y+@uw @z=)]TJ/F25 11.955 Tf 10.883 8.088 Td[(@p @x+"@2u @x2+@2u @y2+@2u @z2#

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9 @v @t+@uv @x+@v2 @y+@vw @z=)]TJ/F25 11.955 Tf 10.625 8.088 Td[(@p @y+"@2v @x2+@2v @y2+@2v @z2#+g@w @t+@uw @x+@vw @y+@w2 @z=)]TJ/F25 11.955 Tf 10.542 8.087 Td[(@p @z+"@2w @x2+@2w @y2+@2w @z2# whereg=gravitationalacceleration=coefcientofkinematicviscosityFollowingfromtheassumptionthattheuidisincompressibleandhomogeneous,pressureinthemodelisdenedastheratioofpressuretoconstantdensity.Simulationsareperformedwitharigidbottomboundarythatbestrepresentsthebathymetryofourproblem.Theinitialpressureeldforauidatrestishydrostatic,whichweusetoinitializesimulationsstartingfromanequilibriumstatewherenomotionexists.Oncethesimulationsbegin,themodelusesthetwo-stepprojectionmethodof Chorin 1968 tosolveforthepressureeld.Theintermediatestepvelocityprojectioneldsarefoundbysubstitutingtheinitialpressureandvelocityeldsintothediscretisedequationsofmotionthatutilizeathetaschemetocontroltheamountofdonorcelldifferencingfortheconvectiveuxterms.Theviscousuxtermsarediscretisedusingathree-pointstencilthatyieldssecond-ordercentraldifferences.BoundaryconditionsarethenappliedtothevelocityeldandcellpressuresareadjustediterativelyinordertosatisfythemasscontinuityequationEq. 2 .IfthedivergenceofthevelocityeldinacellthelefthandsideofEq. 2 islessthanzero,thepressureofthatcellisincreasedtodecreasetheowofmassintothecell;theconverseofthisstatementisalsotrueandthecellpressureisdecreasedtoencouragetheowofmassbackintoacellwhenthedivergenceofthecellisgreaterthanzero.SincetheMACschemeusesonlyonepointtoapproximatethepressureofeachcell,thedivergenceofthevelocityeldmaybedriventozero,ortoadesiredlevelofaccuracy,inaniterativemanner.Typicalvaluesoftheconvergencecriteria,,areontheorderof10)]TJ/F24 7.97 Tf 6.587 0 Td[(3or

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10 smaller Hirtetal. 1975 .Inordertodecreasethenumberofiterationsrequiredtosatisfy,anover-relaxationfactor,,isappliedtothepredictedpressuredifferential.Findinganoptimalvalueoftheover-relaxationfactorrequires,inpart,performingaratherexhaustiveeigenvalueexpansionofthesystemofequationsataskthatwedonotseektoundertake Fletcher 2000 .Therefore,followingguidanceprovidedbytheSOLAmanualwetaketheover-relaxationfactortobe1.8,whichisstillwellbelowthestabilitythresholdof2.0.Whilesuccessiveover-relaxationSORmethodstypicallydecreasethenumberofiterationsrequiredtoreachsomedesiredlevelofconvergence,theycanbecomecomputationallyexpensiveforincreasingnumbersofgridcells.AdditionalinformationregardingtheeffectofSORoncomputationaltimeispresentedinChapter 3 .ContrarytosomeMACformulations,SOLA-SURFdoesnotemploymarkingparticlestotrackthefreesurface.Instead,thelocationofthefreesurfaceispredictedbysubstitutingvelocitypredictionsprovidedbythemomentumequationsintothekinematicfreesurfaceboundarycondition.Oncethecontinuityequationhasbeensatised,theresultingvelocityeldisthenusedinconjunctionwiththekinematicfreesurfaceboundaryconditionKFSBCtodeterminethetime-dependentfreesurface,providinguniquethree-dimensionalrepresentationsoftheuidsurfacethroughoutasimulation.TheKFSBCisgivenbyEq. 2 ,wherethefreesurfaceisdenedastheheight,,ofthesurfaceabovetherigidbottomboundary.@ @t+u@ @x+w@ @z=vThediscretisedformoftheKFSBCemploysanalternativethetaschemeindependentofthatusedintheconvectiveaccelerationtermstocontroltheamountofdonorcelldifferencing,andusesaone-stepprojectionmethodtopredictthefreesurfacelocationatthenewesttime-level.Inthecontextofoursimulations,wehavefounditadvisabletousesecond-ordercentraldifferencesinthespatialdiscretisationof

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11 Eq. 2 .Additionalinformationregardingthepredictiveskillofvariousdiscretisationschemes,aswellasexamplesofthediscretisedformsofaconvectiveaccelerationtermandtheKFSBCarepresentedinAppendixA.ThefreesurfaceinSOLA-SURFmustbedenablebyasinglepointinbothhorizontaldirections.Thisconstraintarisesfromthesimpliedapproachusedtosolvethediscretisedequationsofmotion,theKFSBC,andalsothecellpressures.Waterwavesexhibitingthistypeofsurfacefeatureareoftenclassiedasspillingbreakersatthelimitofbreaking DeanandDalrymple 1991 astheslopeoftheuidsurfaceneverreachesinnityvertical.SinceSOLA-SURFdoesnotallowtheslopeofthefreesurfacetoexceedthecellaspectratio,eithery=xory=z,themodelisunabletoresolveplungingbreakersorothercomplex,multi-valuedfreesurfaces.Thebreakertypeisoftenclassiedbythesurfsimilarityparameter,rstusedby IribarrenandNogales 1949 ,andisdenedinEq. 2 .=tan q H Lo where=slopeofbathymetryH=waveheightLo=deep-waterwavelengthForwavestravelingatobliqueangles,thesurfsimilarityparameterischangedslightly,yieldingtheinshoresurfsimilarityparameterbrof Battjes 1974 showninEq. 2 .br=tan q Hbr Locosbr wherebr=waveangleatbreakingHbr=waveheightatbreakingTheclassicationofbreakingwavetypes,alsopresentedby Battjes 1974 ,isgiveninTable 2

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12 Table2:Breakingwavetypesclassiedbytheinshoresurfsimilarityparameter. surgingorcollapsingifbr>2:0plungingif0:4
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13 enforcestheLaplacianofthevelocitypotentialtobeequaltozero.ThisconstraintisfoundbysubstitutingEq. 2 intoEq. 2 ,whichgivesEq. 2 .r2=0ThethreecomponentsofvelocityusedtoforcethewavesignalarederivedfromthevelocitypotentialusingEq. 2 andaregivenbyu=@ @x;v=@ @y;andw=@ @zandtheresultingforcingequationsforthevelocitycomponentsaregivenbyEqs. 2 2 ,and 2 .ux;y;z;t=H 2gk!)]TJ/F24 7.97 Tf 6.587 0 Td[(1coshkh+y coshkhcoskzsin+kxcos)]TJ/F25 11.955 Tf 11.955 0 Td[(!tcosvx;y;z;t=H 2gk!)]TJ/F24 7.97 Tf 6.587 0 Td[(1sinhkh+y coshkhsinkzsin+kxcos)]TJ/F25 11.955 Tf 11.955 0 Td[(!twx;y;z;t=H 2gk!)]TJ/F24 7.97 Tf 6.587 0 Td[(1coshkh+y coshkhcoskzsin+kxcos)]TJ/F25 11.955 Tf 11.955 0 Td[(!tsinGraphicalrepresentationsofthevelocityeldsarepresentedinFig. 2 .SOLA-SURFrequiresonlybasicwaveparameterstoforcethemonochromaticwavesignal.Thewaveamplitude,period,anddirectionmustbespeciedalongwiththewaterdepth.Usingthisinformationthemodelcomputesthevaluesof,u,v,andwusingtheprecedingexpressions.Thesevaluesarespeciedintherstgridcell,foreachtimestep,andtheequationsofmotionthengovernthepropagationofthewavethroughoutthedomain.Thelengthofthecross-shoredomainandtheboundaryconditionusedneartheonshoreboundaryresultinminimalreectedwavestravelingbackoffshore.Thisallowstheincidentwavestoretainaconsistentshapeand

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14 Figure2:Contourplotsofthevelocityeldsattheforcingboundaryderivedfromthethree-dimensionalvelocitypotentialequationforanincidentangleof=15:4.ThevelocitycontoursofAu,Bv,andCwdemonstratethedepth-dependencyofthelinearequations. magnitudethroughoutasimulation,therebyeliminatingtheneedtotuneoradjustthetime-dependentboundaryconditionstoallowforoutgoingwavecharacteristics.2.3.2FreeSurfaceVelocityBoundaryConditionsThevelocityboundaryconditionsforthefreesurfacemustbetreateddifferentlythanthelateralboundariesinthemodelandrequirespecialattention.Amodiedfree-slipconditionisspeciedonthefreesurface,@u @y=0@w @y=0@v @y=)]TJ/F31 11.955 Tf 9.299 13.27 Td[(@u @x+@w @zwhichprohibitsshearinthevelocityeldacrosstheuidinterfaceandsolvesfortheverticalcomponentofvelocityvinamannerthatexplicitlysatisesthemasscontinuityequationEq. 2 .Theboundaryconditionsappliedtothehorizontal

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15 velocitycomponentsnearthefreesurface,usingghostpoints,arediscretisedasui;jt;k=ui;jt)]TJ/F24 7.97 Tf 6.586 0 Td[(1;kandwi;jt;k=wi;jt)]TJ/F24 7.97 Tf 6.587 0 Td[(1;kwheretheindexnotationjtisusedtorepresenttheuppermostgridcellcontainingthefreesurface.Whileappropriateforamildlyslopingfreesurface,theseboundaryconditionsmustbealteredforthelimitingcaseofmaximumsteepness:whentheslopeofthefreesurfaceapproachesthecellaspectratio.Whentheslopeofthefreesurfaceapproachesthislimit,thehorizontalvelocitycomponentsinthecellcontainingthefreesurfacearesetusingamethodthatproducesverticalmomentumtransfer[see Chenetal. 1995 ].DemonstratedinpanelBofFig. 2 ,theimprovedmethodsforprescribingthehorizontalvelocityboundaryconditionsaregivenbyui;jt;k=ui)]TJ/F24 7.97 Tf 6.586 0 Td[(1;jt;kandwi;jt;k=wi;jt;k)]TJ/F24 7.97 Tf 6.587 0 Td[(1:Thisimprovedmethod,suggestedby Chenetal. 1995 ,betterrepresentsafree-slipconditionwhentheslopeofthefreesurfacenearsthelimitofmaximumsteepness.WhileonlytheuvelocityisdepictedinFig. 2 ,asimilarprescriptionisappliedtotheorthogonalhorizontalvelocitycomponentw.Wheretheoriginalvelocityboundaryconditionssatisfythefree-slipconditionforamildlyslopingsurface,itisobservedthatbyneglectingverticalgradientsofthehorizontalvelocitycomponents@u @y=0and@w @y=0inthecaseofmaximumsteepness,thereislittlemomentumtransferfromthespillingwaveintothewatercolumn.SinceSOLA-SURFdoesnotuseasubgridmodeltosimulatewavebreaking,andintheabsenceofaparameterizationforenergydissipationduetowavebreaking,theimprovedvelocityboundaryconditionsprovideasufcientmechanismtopromotemomentumtransferfromthewavetothemeanow.The

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16 Figure2:Specialcasesforsettingthevelocitycomponentsonthefreesurface,representedherebythedashedline.AistheoriginalmethodemployedbythemodelandBistheimprovedmethod. effectsofthesevelocityboundaryconditionsonthegenerationoflongshorecurrentsandwavetransformationarediscussedinChapter 4 .2.3.3RigidBottomVelocityBoundaryConditionsAno-slipboundaryconditionisappliedtotherigidbottomboundary,requiringthatthehorizontalvelocitycomponentsequalzeroatthebed.Thevelocitycomponentnormaltothebedisalsosetequaltozerosinceowisnotpermittedtocrosstherigidbottomboundary.Theseconditions,prescribedusingghostpointsoutsideofthecomputationaldomain,aregivenbyui;jb)]TJ/F24 7.97 Tf 6.586 0 Td[(1;k=)]TJ/F25 11.955 Tf 9.298 0 Td[(ui;jb;k;wi;jb)]TJ/F24 7.97 Tf 6.587 0 Td[(1;k=)]TJ/F25 11.955 Tf 9.299 0 Td[(wi;jb;k;andvi;jb)]TJ/F24 7.97 Tf 6.586 0 Td[(1;k=0wherejbistheindexoftheverticalgridcellcontainingthebottomboundary.Usingtheseprescriptionsforthehorizontalvelocitycomponentsforcestheirvaluetobezeroatthebed.Forsimulationsperformedonarectangulargrid,aslopingboundaryisrepresentedbyaseriesofsmallsteps.Additionalinformationaboutthebehaviorofhorizontal

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17 velocitycomponentsaroundthesestepsmustbesuppliedtothemodel.Fig. 2 isaschematicofasinglestepintherigidbottomboundary.Thevelocityboundaryconditionsusedaroundastepareui;jb;k=0andui;jb)]TJ/F24 7.97 Tf 6.586 0 Td[(1;k=0whichstatesthatowisnotpermittedtocrossthebottomboundary.Similarprescriptionsareusedforthewvelocitycomponent.Forbathymetrythatdoesnotvaryinthelongshorezdirection,onlytheno-slipconditionontheboundarymustbespecied. Figure2:Prescriptionsforsettingthetangentialvelocitycomponentsaroundastep.Thedarkblacklinerepresentstherigidbottomboundary.

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CHAPTER3EXPERIMENTSInordertodeterminetheabilityofSOLA-SURFtopredictnearshoreprocessessuchaswavetransformationandthegenerationoflongshorecurrents,simulationsofalaboratoryexperimentwereconducted.Aseriesoflaboratoryexperimentsconductedby Visser 1984 wereperformedinarangeofwaveandbasinparameterssuitableforsimulatingwithourmodel.Inparticular,wechosetosimulatehisExperiment4withSOLA-SURFforitsuniquesetofparametersandfortheaccompanyingcomprehensivedatasetpresentedin Visser 1991 .3.1VisserExperimentThepurposeofVisser'slaboratoryexperimentwastwofold:rst,hesoughttodevelopamethodforgeneratinguniformlongshorecurrentsinalaboratorysettingandsecond,toprovidescientistswithalargesetofdatacharacterizinglongshorecurrentsforapproximatelylongshore-uniformconditions.Visserevaluatedanumberofwavebasincongurationsbeforeelectingtouseabasinwithapumpedrecirculationsystem.Thisrecirculationsystemprovidedastabilizingmechanismforthelongshoreuniformityofthecurrentsandcarewastakentodetermineappropriatepumpingrates.Detailedinformationregardingthewavebasinandtherecirculationsystemmaybefoundin Visser 1984 and Visser 1991 .Numerousexperimentswereconductedusingavarietyofwaveparameters,twodifferentbeachslopes,andtwodistinctbeachsurfaces.OneexperimentinparticularExperiment4wasperformedwithasetofparametersconducivetoperformingsimulationswithSOLA-SURF.Thereisalsoasignicantamountofdatapresentedin Visser 1991 thatcorrespondstothisexperiment.TheparametersusedinExperiment4arepresentedinTable 3 ,whereisthebeachslopeangle,Tisthe 18

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19 Table3:BeachandwaveparametersusedinVisser'sExperiment4. tanTdHscmdegrcm 0.0501.0235.015.47.8 waveperiod,disthestillwaterdepthatforcing,isthewaveangleatforcing,andHistheforcedwaveheight.3.2ModelFormulationThetotaldimensionsofVisser'swavebasinmlongshoreby16.6mcross-shorewerequitelargeincomparisontothewavelengthandwaveheightassociatedwiththeexperiment.InordertoreducethesizeofthecomputationaldomainusedinthesimulationsofVisser'sexperiment,onlytheslopingpartofthebasinisincludedinthemodelbathymetryandthelongshoreextentofthedomainisequivalenttothelongshorewavelengthoftheforcedwaves.AsdiscussedinChapter 2 ,wavesareforcedbyapplyingtime-dependentboundaryconditionsonthefreesurfaceEq. 2 andthethreevelocitycomponentsEqs. 2 2 .Thesewaveforcingboundaryconditionsareappliedatthelocationx=0minFig. 3 .Inordertosimulateobliquelyincidentwavessimilartothosecreatedinthelab,periodicboundaryconditionsareusedinthelongshorezdirectionofthecomputationaldomain.Theseperiodicconditions,ascomparedtofreeorno-slipwalls,preventwavereectionfromthelateralboundariesandresultinaconsistentwaveeld.EnforcingperiodiclongshoreboundaryconditionsinSOLA-SURFrequiresthelongshoreextentofthedomaintoequalthelongshorecomponentofthewavelength.Thewavelengthmaybefoundbysolvingthelineardispersionrelationshipforthewavenumberkinaniterativefashion.ThelineardispersionrelationshipisgiveninEq. 3 .!2=gktanhkh

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20 Inthiscase,krepresentsthewavenumbermagnitude.Forobliquelyincidentwaves,thewavenumbermagnitudeistheresultantofthecross-shoreandlongshorewavenumbercomponents,whicharegivenbythefollowingexpressionskx=kcoskz=ksin;whereisthelocalangleofwaveincidencemeasuredfromtheshore-normaldirection.Theparametersoftheexperiment,combinedwiththeperiodicityrequirement,resultintypicalphysicaldomainlengthsofabout10minthecross-shore,approximately5.6minthelongshoreand0.5mintheverticaldirection.ArepresentativedomainisshowninFig. 3 .Inordertoaccuratelyresolvewaveparameters,gridcellswerechosentobe0.01mintheverticaland0.04minthehorizontaldirections.Theselengthscalesresultinacomputationaldomaincontainingnearly1.75milliongridcells;however,sincethemodeldoesnotcomputevaluesbelowthebottomboundary,aboutone-halfofthosecellsremainunused. Figure3:PhysicaldomainusedinthesimulationofVisser'sExperiment4showingthebathymetry,stillwaterlevel,anddomainlengths.

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21 3.3ModelSimulations3.3.1NumericalConvergenceAvarietyofgridresolutionsweretestedduringsimulationsofVisser'sexperimenttoensurethatnumericalconvergencehadbeenreached.ThevedifferentgridresolutionsTable 3 werealsousedtodeterminetheeffectofthecellaspectratioonthepredictionofcross-shorewaveheights,whichisdemonstratedinFig. 3 .Nearlyallofthepredictedroot-mean-squareRMSwaveheightsfromvariousgridresolutionsfallwithinonestandarddeviationofthemeanvalue.Wechosetoperformsimulationswithacellaspectratioof1:4,inordertoenhancethepredictiveabilityofSOLA-SURFandtominimizethecomputationalcostofrunningasimulation. Figure3:Predictedroot-mean-squarewaveheightsforvedifferentgridresolutions.Thedottedlinerepresentsthemeanofthepredictionsateachcross-shorelocationandtheerrorbarssignifyonestandarddeviationfromthemean.

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22 3.3.2ComputationalCostAsnotedinChapter 2 ,theSORmethodappliedtothepressuresolverreducesthenumberofiterationsnecessarytosatisfytheconvergencecriterion,butincreasesthetotalcomputationaltimeaseachiterationtakeslongertoperform.ThesimulationsperformedtoinvestigatetheeffectofgridresolutionarepresentedinTable 3 ,alongwiththenumberofhoursrequiredtocompleteonefull,200ssimulationofVisser'sexperiment.AgraphicalrepresentationofthedatapresentedinTable 3 isshowninFig. 3 ,wherethedatahasbeenplottedonalog-logplottodemonstratetherelationshipbetweenthenumberofcomputationalgridcellsandthetimerequiredtocompleteasimulation. Table3:Summaryofsimulationsperformedtoinvestigatenumericalconvergenceandcomputationalcost. CellAspectRatioGridCellDistributionElapsedTimey:xnx;ny;nzhrs 1:8125,50,70671:5200,50,1001331:4250,50,1402331:3265,50,1854001:2400,50,2802000 3.3.3CompensatingforMassFluxFiniteamplitudewaterwavesproduceameantransportofmass,ormassux,inthedirectionofpropagation.Thismasstransport,inaEuleriansense,stemsfromthedifferenceinuidvolumecontainedunderthewavecrestandwavetrough.Inthecontextofourmodel,ifthevelocitiesattheforcingboundarywereleftintheiroriginalformtherewouldbeanincreaseofmassinthecomputationaldomain.Inordertocounteractthismassuxattheforcingboundary,theforcingvelocitiesmustbealteredinsomecogentmanner.Weallowedthemodeltorunforanumberofwaveperiodsusuallytenwithoutapplyingcorrectionstotheforcingvelocitiesandcalculatedtheirdepthand

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23 Figure3:Computationaltimerequiredforvariousgridresolutions.Thediagramshowsthelogarithmicincreaseincomputationalcostasafunctionofthenumberofcomputationalgridcells.Thepower-tlineisoftheformy=eAlogx+B. longshore-averagedvaluesatafrequencyoftenhertz.Bythentakingatime-averageofthevelocitycomponentsoverthetenwaveperiods,wendtheexcessvelocityduetomasstransport.Theseexcessvelocityvaluesarethensubtractedfromtheirrespectiveforcingvelocitycomponents,uniformlyoverdepthandtime,andthesimulationisstartedfromrestandallowedtoruntosteady-state.TheeffectsofmassuxontheuidsurfaceasafunctionoftimeisshowninFig. 3 .Thedottedredlineinthisgurerepresentstheuidsurfaceasafunctionoftimeforasimulationwithoutmassuxvelocitycorrections.ComparingtheaveragetrendofthisseriesHavgtotheanalyticalsolutionprovidedbylineartheory DeanandDalrymple 1991 ,whichisgivenbythethin,dashedblackline,yieldsanagreeableresult.Theresultingaverageuidsurfaceafterthevelocitycorrectionshavebeenappliedisgivenbythedashedand

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24 Figure3:Theresponseoftheuidsurface,H,tomassuxneartheforcingboundary,x;z=:01;2:0m.AtimeseriesofHforasimulationwithoutvelocitycorrectionsisrepresentedbythedottedredline.ThesolidredlineistheaveragetrendofthesurfaceHt,andthevelocitycorrectionscalculatedpredictauid-surfacerisegivenbythesolidblacklinewithcircles.Thedashed,thinblacklineistheanalyticalsolutiongivenbylineartheoryandthedash-dot,darkblacklinerepresentsthecorrectedsurfaceHt. dottedblackline,whichremainsataconstantelevationof0.35mabovethebottomboundary.3.3.4NonlinearityofModelPredictionsThetime-dependentboundaryconditionsappliedonthevelocityeldtoforcetheoblique,monochromaticwavesignalwerederivedfromalinearvelocitypotentialequation.Watersurfacedisplacementsneartheforcingboundary,therefore,aresinusoidalinformandsoarethevelocityelds.ThemomentumequationsEqs. 2 2 thatgovernthepropagationofthewavesignalthroughoutthedomainarenonlinearequationsandprovide,accordingly,nonlineardevelopmentstothe

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25 velocityeld.TheselinearandnonlinearcharacteristicsaredemonstratedinpanelsBandCofFig. 3 .The30stime-seriesofthevelocitycomponentswastakenfromthefull200stime-seriesrecordedduringthesimulationataninshorepointinshallowwaterandisdenotedbythedarkblackboxinpanelA. Figure3:Contrastingvelocitytime-seriesplotstakenatoffshore.2m,0.28m,2.0mandinshore.6m,0.3m,2.0mlocations.Ashowsthecomplete,200stime-seriesofu;v;andwtakenattheinshorepoint.Bisa30stime-seriesofthethreevelocitycomponentsneartheforcingboundary.Cshowsthenonlinearityandasymmetryofthevelocitycomponentsovera30stime-seriestakenin3cmofwater.

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CHAPTER4RESULTS4.1Model-DataComparisonPerformingsimulationsofVisser'sExperiment4allowsustoevaluatethepredictiveskillofSOLA-SURFbycomparingpredictednearshoreprocessestothoseobservedandmeasuredinthelaboratory.Asignicantamountofdatarepresentingwavetransformationandtheformationoflongshorecurrentsisprovidedby Visser 1991 andserveasbenchmarkstoassessthecapabilitiesofourmodel.4.1.1WaveTransformationThepredictedfreesurfaceelevations,takenfromatransectthroughthemiddleofthelongshoredomain,arerecordedatafrequencyof10Hzthroughoutasimulation.Atime-seriescorrespondingtoapproximatelytenwave-periodsisthentakenfromthetotalrecordandanalyzedtocomputewaveheightstatistics.Themeanofthetime-seriesiscalculatedandsubsequentlyremovedfromthedata,resultinginpositiveandnegativeoscillationsaboutzero.Azerodown-crossingtechniqueisthenusedtoextractindividualwaveeventsfromtherecord,therebyallowingustocalculatestatisticalpropertiesassociatedwiththewaverecord.ThesestatisticalpropertiesarepresentedinFig. 4 ,wherewecomparethepredictedsignicantwaveheightsHs,maximumwaveheightsHmax,andRMSwaveheightsHrms,tothewaveheightsmeasuredduringthelabexperiment.Performingsimulationswiththeoriginalfree-slipsurfacevelocityboundaryconditionsproducedwaveheightsthatweresimilarinmagnitudetothosemeasuredintheexperiment.Simulationsperformedwiththeoriginalfree-slipboundaryconditions,however,producedlongshorecurrentvelocitiesthatwereonlyabout10%oftheexpectedvalues.AssuggestedinChapter 2 ,theoriginalvelocityboundary 26

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27 Figure4:Comparisonofmeasuredredandpredictedwaveheightsasafunctionofthecross-shoredepth.Theerrorbarsrepresentpossiblemeasurementerrorsof10%,suggestedin Visser 1991 .ThedatapointscorrespondingtothelabelOldHmaxwerepredictedbySOLA-SURFusingtheoriginalfreesurfacevelocityboundaryconditions.Hmax,Hs,andHrmsrepresentthepredictedmaximum,signicant,androotmeansquarewaveheights,respectively. conditionsneglectedverticalgradientsinthevelocityeldacrossthefreesurface,therebyprohibitingmomentumtransferfromthesteepeningwavetothemeanow.Simulationsimplementingtheimprovedfreesurfacevelocityboundaryconditions Chenetal. 1995 providereasonableestimatesofthelongshorecurrentvelocities,butunder-predicttheshoalingwaveheights.DemonstratedinFig. 4 ,theRMSwaveheightspredictedbythemodelaresmallerthantheaveragevaluescollectedduringtheexperiment.Thecomparisonshowsreasonableagreementfortherstfewdatapoints,thoseindeeperwater,andalsoforthelastfewdatapoints,butdemonstratestheinabilitytoaccuratelyreproducetheshoalingwaveheightsobservedintheexperiment.Apossibleexplanationforthelargedifferencesbetweenthemeasuredandpredicted

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28 Figure4:Theinstantaneousfreesurfaceandwavesteepness.Atransecttakennearthemidpointofthelongshoredomainshowstheinstantaneousfreesurfaceh+,bathymetrysolidblackline,andtheapproximatewavesteepnessH/Linthecross-shoredirection. waveheightsinvolvesthesteepeningoftheforcedwavesandthecellaspectratio:4selectedforthesimulation.Whenthesinusoidalwaveisforcedattheoffshoreboundaryithasanapproximatesteepnessof1:6,whichiswellbelowthecellaspectratioof1:4.Asthesewavesshoaltheybecomenonlinearandthefaceofthewavesteepensquicklyanddramaticallywithintherstfewmetersofthecross-shoredomain,asseeninFig. 4 .Thissteepeningpresentsaproblemwhentheslopeofthefreesurfacereachesornearsthecellaspectratio,asthemodiedfreesurfacevelocityboundaryconditionsbegintotranslatemomentumdownthefaceofthewaveabitsoonerthannecessarytomatchthelabdata.Oneresultofthismomentumtransferappearstobeareductioninthepredictedwaveheights.Wesaythatthesteepnessratios,showninFig. 4 ,are

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29 approximatebecauseweassumethatthewaveformissinusoidalwhenwecalculatetheslopeofthewaveface.Themodelednonlinearwavesexhibitsurfaceslopesthatexceedthosecalculatedbyoursinusoidalapproximation.OneparticularadvantageofemployingSOLA-SURFtosimulatenearshoreprocessesliesinitstime-dependency.Whilesteady-statewavemodelsassumethatwaveshoalingisastationaryprocess,time-dependentprocessessuchasthegenerationofalongshorecurrentandundertowcanaffectwavetransformationovertime.ShowninFig. 4 ,theRMSwaveheightspredictedbySOLA-SURFdonotremainconstantthroughoutthesimulation.Thissuggeststhatthewaveeldrespondscontinuouslytothedevelopingundertowandlongshorecurrent.TheseRMSvalueswerecalculatedfromaten-waveaverage,centeredaboutthesimulationtimeshowninthelegend. Figure4:Statisticalpropertiesofthewaveeldatvarioustimelevelssuggestthatthewaveshoalinginthisexperimentisnotastationaryprocess.

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30 4.1.2LongshoreCurrentFormationTheformationofthelongshorecurrentandthemagnitudeofthecurrentvelocitiesishighlysensitivetotheimplementationofthefreesurfacevelocityboundaryconditions,asdiscussedabove.Usingtheimprovedvelocityboundaryconditionsprovidesreasonableestimatesofthelongshorecurrentvelocities,butfailstoaccuratelypredictthecross-shorewaveheights.Ourpredictedcurrentiscalculatedbyrsttakingthedepthandlongshoreaveragesofthelongshorevelocitycomponentwandthenbyaveragingthesevaluesover30wave-periods. 1 AsdemonstratedbyFig. 4 ,themodelapproximatelypredictsthecorrectmagnitudeofthemaximumlongshorecurrentvelocity,butthepeakisshiftedshorewardofthelocationobservedinthelaboratory.Thisisperhapsduetotheabsenceofsetupattheshorelineowingtoourchoiceofashallowsillasanonshoreboundaryinsteadofanintersectingprole.Anotherfeatureofthepredictedlongshorecurrentthatwasnotobservedinthelaboratoryexperimentistheadditionallongshore-directedowseawardofthepeakvelocity.ThismaybeexplainedbyreferringtothepredictedwaveheightsshowninFig. 4 .UnderestimatingtheRMSwaveheightsuggestsanoverestimationoftheenergydissipation Chenetal. 2003 ,whichmanifestsitselfintheoffshoreregionasadditionallongshorecurrentvelocity.Thelaboratorywaveheights,onthecontrary,remainsomewhatconstantuntiltheynearthepointofbreakingresultinginlesslongshorecurrentvelocityintheoffshoreregion.Apparently,otherhorizontalmixingprocessesinthelabexperimentarenotsufcientlystrongtospreadthemeanlongshorecurrentsignicantlyoffshoreofthebreakpoint.Infact,theopportunityforlongshoreowtopersistintheoffshore 1Henceforth,wedenotedepth-averagedvalueswithasingleoverbar ,depthandlongshore-averagedvalueswithtwooverbars ,andtime-averagedvalueswiththeanglebracketshi.

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31 Figure4:Thepredictedandmeasuredlongshorecurrentvelocities.Acomparisonofthepredictedtime-mean,depthandlongshore-averaged,longshorecurrentvelocitytothemeasuredcurrentvelocity.Thepredictedcurrentiscalculatedateachcross-shoregridlocationandisrepresentedherebytheblackline.TheredlineisasplinettoVisser'sdatacollectionpoints,whichareshownbytheredsquares. regionmaybesuppressedbythewave-guidingwallslocatedatbothendsofVisser'swavetank.Thesewallsextendfromabovethemeanwaterleveltothebed,thusrestrictinganymean,longshore-directedowbetweenthem.Inaseriesofexperimentsoflongshorecurrentsonplanarslopes, GalvinandEagleson 1965 and MizuguchiandHorikawa 1978 measuredsignicantlongshorevelocitiesto40%ofthemaximumseawardofthesurfzone.Theseexperimentswereperformedinsimilarwavetanksthatprovidedclearancebetweenthebedandthebottomofthewaveguideandtheresultssuggestthatthiscongurationpermitsrecirculationintheoffshoreregion,whereitwassuppressedinVisser'sexperiment.Weallowedsimulationstorunfor200secondsofmodeltime,bywhichpointtheoweldhadreachedsteady-state.InFig. 4 weseetheformationandgrowth

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32 ofthepredictedaveragelongshorecurrent.Formationofthelongshorecurrentbeginswellseawardofthelocationofthepredictedpeakvelocityandspreadslaterallyacrossthedomainovertime.Afterapproximately150secondsofmodeltime,thelongshorecurrenthasreachedsteady-state:themagnitudeandshapeofthecurrentvarylittleduringtheremainderofthesimulation.Simulationswerealsorunwithtwodifferentsilldepthstoensurethatitslocationdidnotdeterminewherethepeaklongshorevelocityoccurred.Silldepthsof1cmand2cmwereusedinthesesimulationsandtheresultingaveragelongshorecurrentsareshowninFig. 4 .Regardlessofthedepthused,themaximumlongshorecurrentvelocityremainsinthesamelocation,seawardofthesill,wherethemeanwaterdepthisapproximately3cm.Thelongshorecurrentformsmoreslowlyforthesilldepthof2cmascomparedtothedepthof1cm,butthisoutcomewasexpectedastwiceasmuchvolumeispermittedtocrossthesill.4.2Three-DimensionalFlowFeatures4.2.1FreeSurfaceVisualizationsNormallyincidentwavespropagatingoverlongshore-uniformbathymetryproducecirculationandowfeaturesthatarecharacteristicallytwo-dimensional.Longshoregradientsinvelocityeldsandwaveheightsresultingfromobliquelyincidentwaves,however,producethree-dimensionalowfeaturesinthenearshoreregion.Onesuchfeatureisdemonstratedbythelongshorenon-uniformitiesinthewavesastheytraveloverashearingcurrent.Figure 4 contraststheincidentwaveeldfromoursimulationatanearlytime,whentheundertowandlongshorecurrentsareundeveloped,andatalatertimewhenbothcurrentshavereachedsteady-state.TheincidentwaveeldinFig. 4 Ademonstratestheshoalingandrefractionofthewavesovertheslopingbathymetryusedinthelaboratoryexperiment.ThecolorcontoursinFigs. 4 Aand 4 Bcorrespondtothelocationofthefreesurfaceintheverticaldomain,referencedtothestillwaterlevelaty=0:35m.Atthisearlytime,whenthecurrentshaveyettodevelop,weseethatthewavecrestsaremoreorlessparallelto

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33 Figure4:Timeevolutionoftheaveragelongshorecurrent.Shownherearethedepthandlongshore-averaged,longshorecurrentvelocitiestakenatvarioustimesthroughoutthesimulation.Theseproleswereaveragedovertenwave-periods,centeredaroundthetimelabelsshowninthelegend,toremovethewavecomponentofthelongshorevelocity. oneanotherastheytravelacrossthedomainandthattheperiodicboundaryconditionsperformproperly.ThecontrastingwaveeldshowninFig. 4 B,takenwhenthemodelhasreachedsteady-state,showstheresponseofthewaveeldtothedevelopedundertowandlongshorecurrents.Therelativelyundisturbed,straightwavecrestsseeninFig. 4 Aarenowonlyfoundneartheoffshoreboundaryandthosepropagatingovertheshearingcurrentshavesignicantlongshorenon-uniformities.Clearly,theincidentwavesbegintofeelthemeancurrentsforx>3mandproducespatiallydependentwavebreakingpatterns.4.2.2Depth-DependentWave-CurrentInteractionThethree-dimensionalnatureofSOLA-SURFisperhapsevenmoreusefulforanalyzingthedepth-dependentcirculationsthatdevelopinacomplexoweld.

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34 Figure4:Averagelongshorecurrentvelocitiesoverdifferentsilldepths.Theseproleswereaveragedovertenwave-periods,centeredaroundthetimelabelsshowninthelegend,toremovethewavecomponentofthelongshorevelocity. Verticalvariationsofthecombinedcross-shoreandlongshorecurrentvelocitieswereinvestigatedby SvendsenandLorenz 1989 andweresaidtohavesignicanteffectsonthedirectionofsedimenttransportinthenearshoreregion.Whilemanynumericalinvestigationsregardingundertowandlongshorecurrentvelocityproleshavemadeuseofdepth-integratedordepth-averagedequations,themomentumequationsusedinSOLA-SURFexplicitlyincludedepth-dependentterms.Thedepth-dependentnatureofthevelocityeldandcross-shorecurrentisdemonstratedbyFig. 4 ,whereplotsattwodifferenttimescomparethedevelopingvelocityeldatanearlytimewiththefullydevelopedeldthatoccurslaterinthesimulation.ThelargerframesinFigs. 4 Aand 4 Bshowthevelocityeldtakenateverysecondcross-shoregridpoint,wherethevelocityvectoristheresultantoftheuandvvelocitycomponents.Theareaenclosedbythedark,blacklineineachframecorrespondstotheareaplottedinthesmaller

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35 Figure4:Three-dimensionalvisualizationsoftheinstantaneousfreesurface.Thecontrastingwaveeldsshowtheeffectoftheshearingcurrentontheincidentwaveeld,takenatAt=10sandatBt=150s. insetframes,wherethevelocityvectorsareplottedateverycomputationalgridcell.Theseinstantaneousguresofthex;yplanearetransectstakenfromthemiddleofthelongshoredomain.Figure 4 Ashowsthedepth-dependentvelocityeldresultingfromtheincidentwaveeldwhere,demonstratedbythecolorcontourplotofuinthepanelinset,theundertowhasyettodevelop.InFig. 4 B,however,theundertowisfullydevelopedandtheresultingvelocityeldisnowmuchmorecomplicated,as

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36 Figure4:Depth-dependentwave-currentinteractionsinthecross-shore.VelocityeldstakenatAt=10sandatBt=150sshowthedepth-dependentvelocitypredictionsandthesuperpositionofapropagatingwaveonthedevelopedundertow. shownbyboththeinsetpanelandthelargerframe.Atthislatertime,theincidentwavesarepropagatingacrossalower-frequency,seaward-directedownearthebedthatisofsimilarmagnitudetotheshoreward-directedvelocitiesoftheobliquelyincidentwaveeld.NotetheincreasedstrengthofthevelocityatthewavecrestintheinsetofFig. 4 Bandtheresultingshearinthevelocityeldunderthewavecrest.

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37 4.2.3LongshoreCurrentVariabilityWhilenotdiscussedin Visser 1991 ,thetime-dependentlongshorecurrentpredictedbySOLA-SURFisalsospatiallyvariableinbothhorizontaldirections.Thistime-dependentcurrentstructureisshowninpanelsA,B,andCofFig. 4 ,wherewehaveplottedtheinstantaneous,depth-averaged u; wvelocityeldinthex;zplaneatthreetimesduringthesimulation.ThevelocityeldshowninFig. 4 Drepresentsathirtysecondaverage,centeredaboutt=165s,ofthedepth-averaged u; wvelocitiestakenoncethemodelhadreachedsteady-state.Forclarity,everyfthvelocityvectorinthelongshoredirectionisplottedinthisgurebutnonehavebeenomittedfromthecross-shoredirection.InFig. 4 A,att=20s,weseethatthepredominantfeatureofthevelocityeldistheincidentwavesignalandnotethat,relativetotheonshore-directedvelocities,thevelocitymagnitudesofreectedwavesonthesillareverysmall.Atroughlyhalfofthetimeittakesthelongshorecurrenttoreachsteady-state,thereappearstobealongshore-periodicstructureassociatedwiththelongshorecurrent,whichisshowninFig. 4 B.Figure 4 Cdemonstratesthepersistenceofthismeanderingperiodicstructure,althoughmuchweaker,evenasthemeanlongshorecurrentnearssteady-state.Thethirtysecondaverageofthedepth-averaged,steady-statevelocityeldshowninFig. 4 Dremovesthevelocitysignatureoftheincidentwaveeld,resultinginamostlylongshoreuniformcurrent.FurtherinvestigationoftheperiodicstructureassociatedwiththedevelopinglongshorecurrentFig. 4 Bsuggeststhatoscillationsofthecurrentoccurinboththelongshoreandcross-shoredirections.Overathirtysecondperiodspanningfromt=90stot=120s,whenthemeancurrentspeedandproleareevolving,thedepth-averagedlongshorevelocityisnotuniforminthelongshoredirection.TheperiodicstructureshowninFig. 4 Arepresentsthetime-mean,depth-averagedlongshorecurrentfor90
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38 Figure4:Depth-averaged u; wvelocityeldstakenatAt=20s,Bt=80s,Ct=150s,andDovera30secondaveragetoremovetheeffectsoftheincidentwaveeld.NotethelongshorevariabilityofthecurrentstructureinpanelsBandC. Nearshoremotionswithfrequenciesofthismagnitudeareoftenclassiedasshearwaves,butthepresenceofthesemotionswerenotdiscussedin Visser 1984 or Visser 1991 .PanelsBandCofFig. 4 arecolorcontourplotsoftheinstantaneous,depth-averagedresiduallongshorevelocityatt=100sandt=120srespectively.TheresidualvelocitycomponentisfoundusingaReynoldsdecomposition Pope 2000 forthedepth-averagedvelocityandisgivenbyEq. 4 .Here,theresiduallongshorevelocityconsideredaccountsforthewave-inducedvelocityaswellastheturbulence,butnodistinctionismadebetweenthetwocomponents. w0= w)-222(h wiAlternatingvelocitydecitblueandvelocityexcessredsignaturesseeninFigs. 4 Band 4 Cdemonstratethelongshorepropagationofthisinstability,

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39 aswellasitscross-shoredisplacement.AscomparedtotheresidualvelocitysignaturesshowninFig. 4 B,noticethattheexcessanddecitsignatureshavereversedtheirpositionsinthelongshoredirectionandhavemigratedseawardinFig. 4 C.Shearwavesareenergetic,low-frequencyvorticalstructuresthatpropagateinthelongshoredirectionandexhibitsignicantexcursionsinthehorizontalplane BowenandHolman 1989 .Thus,theseshearwavemotionscontributegreatlytothenearshore Figure4:Colorcontourplotsofthedepth-averagedlongshorevelocityforAa30stime-periodpriortosteady-state,Btheinstantaneouslongshorevelocityresidualatt=100sandCtheinstantaneouslongshorevelocityresidualatt=120s.

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40 circulationandultimatelyaffectthetransportofsediment,pollutants,andbiologicalmaterial Doddetal. 2000 .Whilebarredbeacheshavebeenfoundtofacilitatetheonsetofshearinstabilities[see Oltman-Shayetal. 1989 ; BowenandHolman 1989 ],observationsofthesemotionsonplanarbeachesarenotwidespreadbuthavebeenknowntooccur Doddetal. 2000 Oltman-ShayandHowd 1993 discoveredevidenceofshearwavemotionsontwoplanarbeachesinCaliforniaLeadbetterBeachandTorryPinesafterreanalyzingdatafromtheNearshoreSedimentTransportStudyNSTSconductedin1980.Shearwaveshavenotbeenreadilyidentiedinlaboratoryexperimentsconductedwithplanarslopes Reniersetal. 1997 eitherbecauseoflimitationsindatacollectingorduetosuppressionbyconnementofthewavebasin BowenandHolman 1989 .Intheirnumericalmodel, Allenetal. 1996 weresuccessfulingeneratingshearwavesoveraplanarbeachand PutrevuandSvendsen 1992 suggestedthattypicallengthandtime-scalesofshearwavemotionsfromtheexperimentsof Visser 1984 ,iftheyhadbeenidentied,wouldbeofOmandOs,respectively.Thesescalesagreewellwiththosepredictedbythesimulation20sand5.6m,resultinginaphasespeedofroughly70%ofthemaximumlongshorecurrentvelocitysimilartothevaluesuggestedby Doddetal. 2000 .4.3VerticalDistributionsofVelocityandMomentum4.3.1Time-AveragedVelocityProlesAlthoughthecross-shorecirculationwasnotdiscussedin Visser 1991 ,time-averagedvelocityprolesofthepredictedoweldshowastrongseaward-directedownearthebed,commonlyreferredtoasundertow.ThevelocityprolesplottedinFig. 4 demonstratethisbehaviorofthereturnow,specicallyinpanelsB,C,andD,wheretheundertowdominatesthecirculation.VelocityprolesshowninFig. 4 wereaveragedoverthirtysecondsandweretakenfromatransectlocatednearthemid-pointofthelongshoredomain.TheslightlynegativeuandwvelocityprolesinFig. 4 Aresultfromthevelocitycorrectionsappliedattheforcingboundary

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41 thataccountformassux.Figure 4 Aalsodemonstratesthat,asexpected,thereisashoreward-directeduxofmomentumduetothewavesthatroughlyoccupiestheregionboundbythewavetroughandwavecrest.Whilethismomentumuxduetothewavespersiststhroughouttheremainingpanels,theverticalstructureofthevelocitieschangesignicantlyasthelocationsprogressfromtheforcingboundaryshoreward.Thetime-averagedlongshorevelocityprolesinFigs. 4 BandCdemonstrateauniquebehaviornearthetwogridpointsclosesttothebed.Here,thetime-averagedlongshorecurrentreceivesanasymmetricimpulsefromthestaircaserepresentationofthebottomslope.Astheshorewardvelocityofthephaseofashoalingwaveencountersasteprise,thereisacorrespondingpulseofnear-bottomwaterthatowsinthepositivez-directionsincetheuidislesslikelytoowupwardduetothemassaboveit.Similarly,fortheseaward-directedvelocityoftheopposingphasethereisapulseofwaterinthenegativez-direction.Themagnitudeoftheundertowdiminishessignicantlyinthelocationswherethelongshorecurrentvelocitiesarehigh,asdemonstratedinpanelsEthroughHofFig. 4 .Thepredicteddistributionoflongshorevelocityissimilar,inmostproles,tothenearlydepth-uniformstructurereportedby Visser 1991 anddiscussedby SvendsenandLorenz 1989 ,whosuggestedthatadepth-uniformvelocityprolewouldbeabetterassumptionthanthelogarithmicprolesmorecommonlyusedtodescribesteady,open-channelows.Thisdepth-uniformstructureismostevidentinthestrongesthighervelocitypartofthelongshorecurrentandisdemonstratedbyFigs. 4 Gand 4 H.Figure 4 alsoshowsthelargedifferencebetweenthemagnitudesofthehorizontalvelocities,uandw,andtheverticalvelocityv,whichissosmallthatyoucanbarelydistinguishitinthegure.Thisimpliesthattheaverageverticalvelocitycontributesverylittletothemeanowandbecomesevenlesssignicantifconsideredinadepth-averagedordepth-integratedsense.Productsofthe

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42 Figure4:Time-averagedvelocityprolesatAtheforcingboundary,Bh=0:246m,Ch=0:226m,Dh=0:206m,Eh=0:145m,Fh=0:124m,Gh=0:085m,andHh=0:069m.Thedashedlineineachpanelrepresentsthemeanwaterlevelwhilethedottedlineshowstheapproximatelocationofthebottomboundary.ThesepanelsrepresentaprogressionfromtheoffshoreforcingboundaryAtotheshallowdepthsoftheinshoreHwherethelongshorecurrentdominatesthenearshorecirculation. verticalvelocitywithhorizontalvelocitycomponents,however,maynotnecessarilybeinsignicantinthepresenceofanonlinear,propagatingwaveeld.Time-meanvelocityprolesofuinpanelsB,C,andDofFig. 4 suggestthattheconservationofmassisbeingsatisedasdepth-averagedvaluesoftheprolesareveryclosetozero.ThisisnotthecaseinpanelsEthroughH,wherethereappearstobeanetowofmassintheshorewarddirection.However,theshoreward-directedmomentumcarriedbytheincidentwaveeldisbalancedbytheundertowonlyin

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43 Figure4:Thedepth-averagedcross-shorevelocity,u,averagedoverthirtysecondsandplottedinthex;zplane. astrictlytwo-dimensionalsense Svendsen 1984 ;previousguresanddiscussionshaveemphasizedthethree-dimensionalityoftheoweld.Acolorcontourplotofthetime-mean,depth-averagedcross-shorevelocityisshowninFig. 4 wherehottercolorscorrespondtoshoreward-directedowandcoolercolorsdesignateseaward-directedow.Theconcentrationofshoreward-directedowbetweenx=5mandx=6:5mcorrespondstotheareawherethevelocityprolesinpanelsEHofFig. 4 arelocated.Thedepth-averagedcross-shorevelocitywasaveragedoverathirtysecondperiodjustpriortoreachingsteady-state,duringwhichtimethereweremanycomplex,spatialandtime-dependentowfeaturesincludingseichingandtheperiodicoscillationsofthelongshorecurrent.Thesedifferentowfeaturesalloccurondifferentlengthandtimescalesmakingitdifculttoextractanensemblethatexplicitlyprovesthatthedepth-averagedvelocityprolesobeytheconservationofmass.We

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44 know,however,thatthemodelconservesmassoverthecourseofa200secondsimulation:theinitialvolumeofuidcontainedwithinthephysicaldomainisthesameasthevolumecontainedattheendofthesimulation.Anothersignicantfeatureofthisregionisthesetup,andthelocationofthetime-meanfreesurfacedisplacementseemstocorrespondwellwiththelocationoftheexcessshoreward-directedvelocity.AdditionalinformationaboutthespatialrelationshipbetweenthesetwofeaturesispresentedinAppendix B .4.3.2VerticalDistributionofRadiationStressTheverticaldistributionsoftheshoreward-directedcomponentsoftheradiationstresshavereceivedparticularattentionrecently[ Mellor 2003 Xiaetal. 2004 ].Aspresentedin Longuet-HigginsandStewart 1964 ,thesetwohorizontalcomponentsarethecross-shorecomponentoftheshoreward-directedradiationstressandthelongshorecomponentoftheshoreward-directedradiationstress,denotedhereasSxxandSxz, 2 respectively,andaregivenbyEqs. 4 and 4 .Thelongshorecomponentoftheshoreward-directedradiationstressisnon-zeroonlyforathree-dimensionalwaveclimateproducedbyeitherobliquelyincidentwavesorlongshorevariablebathymetry.Wehaveneglectedathirdhorizontalradiationstresscomponent,Szzthelongshore-directedcomponentfortworeasons:rst,theverticalstructureofthetime-averagedlongshoreowisessentiallydepth-uniformand,assuch,verticaldistributionswouldnotbeascomplexasthoseinthecross-shoreand,second,sincelongshoregradientsofthetime-averagedquantitiesthatcontributetothisstress 2AlthoughthenotationofSxy,presentedbyLonguet-HigginsandStewart,ismorecommonlyused,weshallusethisalternativenotationsinceitisconsistentwithourcoordinatesystem.

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45 componentwouldbezeroowingtoourperiodiclongshoredomain.Following Longuet-HigginsandStewart 1964 ,wedene:Sxx=Z)]TJ/F26 7.97 Tf 6.587 0 Td[(hp+u2dy)]TJ/F31 11.955 Tf 11.955 16.273 Td[(Z0)]TJ/F26 7.97 Tf 6.586 0 Td[(hp0dySxz=Z)]TJ/F26 7.97 Tf 6.586 0 Td[(huwdy whereh=bedelevation=verticallocationofthefreesurface=densityoftheuidp=totalpressurep0=hydrostaticpressureintheabsenceofwaveshi=time-averagingoperatorInordertoobtaininformationregardingtheverticalstructureoftheradiationstresscomponents,wesimplyneglecttheverticalintegrationoftheexpressionspresentedinEqs. 4 and 4 anddenotethedepth-dependentvaluesoftheradiationstresscomponentsbysxxandsxz.Theresultingequationsfortheshorewardcomponentoftheshoreward-directedradiationstressandthelongshorecomponentoftheshoreward-directedradiationstressaregivenbyEqs. 4 and 4 ,respectively,sxxy=hu2i+hpi)]TJ/F25 11.955 Tf 19.261 0 Td[(p0sxzy=huwiTheformulationofEq. 4 ismucheasiertounderstandwhenevaluatingtheverticalstructureofeachcomponentandtheirrelativemagnitudes.Figure 4 showsthedistributionandmagnitudeofeachcomponentofsxx,aswellasthetotal,wherevalueshavebeenaveragedoverthirtyseconds.Insteadofplottingbothpressurecomponents,thedifferencebetweenthetwoisshowninordertoreducethescaleoftheabscissaandincreasetheresolutionofeachcomponent.Notethattheunitsof

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46 Figure4:Acomparisonoftheshapeandmagnitudeofvariouscomponentsofsxxtakenfromalocationinthemiddleofthelongshoredomainwherethelocaldepthish=0:246m. sxxshownontheabscissacharacterizeastress,whiledepth-integratedvaluesoftheradiationstresstermsSxxandSxzhavetheunitsofstresstimeslength.ThedashedlineinFig. 4 representsthemeanwaterleveloverthethirtysecondaverageatalocationwherethemeanwaterdepthis0.246mandthelocationofthebedisaty=0:104m.ThetotalpressureterminEqs. 4 and 4 isthetime-meanpressureintheprogressivewaveeldand,thus,representsboththepressureduetothewaterwavesaswellasthehydrostaticpressureoverthewatercolumn.Plottingthedifferencebetweenthetime-meantotalpressureandthehydrostaticpressureintheabsenceofawaveeldFig. 4 resultsinthetime-meandynamicpressureduetotheincidentwaves.Betweenthetroughandcrestlevels,Fig. 4 showsthattheapparentstresstermhu2iandthegravitationaltermhpi)]TJ/F25 11.955 Tf 20.095 0 Td[(p0contributeapproximatelyequalamounts

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47 ofmomentumux.Thisisincontrasttothendingsof SobeyandThieke 1989 whostatedthatthegravitationaltermwasdominantabovetroughlevelandthattheapparentstresswaslesssignicantinthisregion;however,themagnitudeofthewaveapparentstressabovethetroughisroughlyve-timesgreaterthanthepredictedvalueintheundertow,whichissimilartotheirndings.Whilenonlinearcnoidaltheorywasusedintheirinvestigation,thewavespredictedinthismodelarestronglynonlinear,whichisevidentinthetranslationoftheapparentstresspeakabovethemeanwaterlevel.Theexpressionhv2i,whilenotincludedinourformulationofsxxEq. 4 ,isalsoincludedinFig. 4 todemonstrateitsrelativesignicancetotheotherterms.Morespecically,itshowsthatthereisnotanexactbalancebetweentheverticaluxofmomentumhv2iandthetime-meandynamicpressurehpi)]TJ/F25 11.955 Tf 20.089 0 Td[(p0belowthemeanwaterlevelthedifferenceinsign,however,iscorrect.Thiscontradictsthemethodologyfollowedby Longuet-HigginsandStewart 1964 intheirformulationoftheradiationstressequationforSxx,wherethetermswereconsideredtoexplicitlybalanceoneanotherbelowthemeanwaterlevel.Aformulationforsxxsuggestedby SobeyandThieke 1989 accountedforthisinequity:sxxy=hu2i)]TJ/F25 11.955 Tf 19.261 0 Td[(hv2i+p;wherewehaveadaptedournotationforthetime-averagingandaccountedforthedifferenceinthecoordinatesystemsbyreplacingw2withv2.Herethepressuretermpaccountsforthetime-averagedynamicpressureintheregionboundbythewavecrestandwavetrough;therefore,theverticalmomentumuxtermhv2iisassumedtoexactlybalancethetime-averagedynamicpressurebelowthetroughlevel.AsshowninFig. 4 ,theshapeandmagnitudeoftheprolescorrespondingtothetermsinquestionarenotsimilar,especiallynearthebedwherethemomentumtendstowardzeromuchfasterthanthedynamicpressure.

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48 UsingtheformulationforsxxgivenbyEq. 4 ,wehaveplottedthepredictedprolesoftheshoreward-directedradiationstress,alongwiththeanalyticalsolutionprovidedby Longuet-HigginsandStewart 1964 Eq. 4 forthelineardistributionofthetotalenergyE,atriangulardistributionofE 2Eq. 4 abovethemeantroughlevel Dean 1995 ,andtheverticaldistributionofsxxsuggestedby Mellor 2003 Eq. 4 .Sxx=E2hkx sinh2hkx+1 2sxxy=gHcos 2"r 1)]TJ/F31 11.955 Tf 11.955 13.271 Td[(2y H2)]TJ/F15 11.955 Tf 13.15 8.088 Td[(2y Hcos)]TJ/F24 7.97 Tf 6.586 0 Td[(12y H#+8><>:0ify>MWLgyify
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49 andapproximatelylinear.Themagnitudesofthetriangularanddepth-uniformsegmentswouldbelargerhadwenotappliedcorrectionstotheforcingvelocitiestoaccountformassux.Althoughthewavesareforcedwithalinear,sinusoidal-typesignal,theysteepenandbecomeasymmetricastheyshoal,dramaticallyaffectingthedistributionofmomentumux.ThepredictedprolesofsxxFig. 4 betweenthetroughandcrestlevelshaveatriangular-shapeddistributionofmomentumuxthataccountsforthemajorityoftheshoreward-directedux,similartoobservationsmadeby Svendsen 1984 and SobeyandThieke 1989 .Whilethepeaksofthepredictedradiationstressproleshavemagnitudessimilartothosepredictedbylineartheory,thenonlinearityofthesimulatedwavesredistributesthemajorityofthemomentumabovetroughlevelandshiftsthepeakoftheproleabovethemeanwaterlevel.Thepredictedtime-averagecirculationgivestheradiationstressproleaverydistinctshapebelowthetroughlevel,especiallyinFigs. 4 BDwheretheundertowisthedominantowfeature.Progressinguptheslopeofthebathymetry,inFigs. 4 EH,weseethatthemeanowisrelativelydepth-uniform,belowthetroughlevel,inthelocationswherethelongshorecurrentvelocitiesarehighest;however,theaveragemagnitudeoftheprolesbelowthetroughlevelvarylittlethroughoutthecross-shore.Mellor'sequationfortheverticallydependentshoreward-directedradiationstressEq. 4 providesanalternateapproximationforthedistributionofmomentum.InsomeinstancesFigs. 4 EH,themagnitudeoftheradiationstressnearthebedisactuallygreaterthanthevalueatthecrestlevel.Onepossiblebenetoftheanalyticalsolutionprovidedby Mellor 2003 isthatitprovidessomelevelofapproximationconcerningtheterminationoftheverticaldistributionatthecrestlevelwhilethelineardistributionofmomentumuxsuggestedby Longuet-HigginsandStewart 1964 simplystopsatthemeanwaterlevel.Perhapsasuitablecompromisebetweentheanalyticalsolutionprovidedbylineartheoryandthatprovidedby Mellor 2003 would

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50 Figure4:Predictedprolesofradiationstresssxx.AcomparisonofthepredictedshapeandmagnitudeofsxxtolineartheoryandMellor'sanalyticalsolutionatAtheforcingboundary,Bh=0:246m,Ch=0:226m,Dh=0:206m,Eh=0:145m,Fh=0:124m,Gh=0:085m,andHh=0:069m.ThetriangulardistributionofE=2hasbeenshiftedtotherightforpurposesofcomparison.Thedashedlinerepresentsthemeanwaterlevelandthesloping,dottedlinerepresentstheapproximatelocationofthebathymetry. betoaddthetriangulardistributionofE 2toMellor'sprole,startingatthecrestlevel.Neglectingthecharacteristicsandverticalstructureofthemeanoweld,itispossiblethatadepth-uniformdistributionoftheradiationstressbelowtroughlevelwouldprovideasufcientlyrealisticrepresentationofthemomentumuxcausedbythetime-averagedwaveforcing.Verticaldistributionsofthelongshorecomponentoftheshoreward-directedradiationstress,sxz,areshowninFig. 4 wheretheplottingconventionis

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51 similartothatusedinFig. 4 .Thepredictedproleofsxzwasplottedusingthevertically-dependentformulationgivenbyEq. 4 ,whichwasderivedfromthedepth-integratedequationEq. 4 suggestedby Longuet-HigginsandStewart 1964 .Figure 4 showsthecomparisonofthepredictedradiationstressproletothedepth-uniformdistributionofmomentumuxEpredictedbylineartheoryandsuggestedin DeanandDalrymple 1991 Eq. 4 ,andtheverticaldistributionproposedby Mellor 2003 Eq. 4 .Sxz=E 42hk sinh2hk+1sin2sxzy=kDEF12F11kxkz k2ThemodeledverticalprolesofsxzabovethetroughlevelexhibitthetriangulardistributionseeninthesxxprolesofFig. 4 ,butthepeakoftheproleisshiftedfurtherabovethemeanwaterlevel.AsshownintheradiationstresscomponentprolesofFig. 4 ,thegravitationaltermreducesthesuper-elevationofthepeakofthesxxprole.Sincetherearenoadditionaltermsintheformulationofthevertically-dependentproleofsxz,thewaveapparentstresstermcontrolstheshapeoftheproleandthedistributionofmomentumux.Thissuggeststhatassumingsimilardistributionsofsxxandsxzabovetroughlevel,andatdifferentcross-shorepositions,isinaccurateinthepresenceofpropagatingnonlinearwaterwaves.Belowtroughlevel,however,thedistributionofmomentumisrelativelydepth-uniformwiththeexceptionofFigs. 4 Eand 4 Fwheretheseaward-directedundertowproducesaninectionpointnearmid-depth.Comparisonsofthepredictedproleofsxztothelinearsolutionandtheanalyticalsolutiongivenby Mellor 2003 givemixedresults.SimilartothecomparisonofsxxprolesinFig. 4 ,Mellor'sformulationEq. 4 dramaticallyunder-predictsthemagnitudeofmomentumdistributedabovethetroughlevelbutcorrectlyestimates

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52 Figure4:Predictedprolesoftime-averagedradiationstresssxz.AcomparisonofthepredictedshapeandmagnitudeofsxztolineartheoryandMellor'sanalyticalsolutionatAtheforcingboundary,Bh=0:246m,Ch=0:226m,Dh=0:206m,Eh=0:145m,Fh=0:124m,Gh=0:085m,andHh=0:069m.Thedashedlinerepresentsthemeanwaterlevelandthesloping,dottedlinerepresentstheapproximatelocationofthebathymetry. theupper-mostextentofthedistributionprole.AtthetwomostoffshorelocationspanelsAandBofFig. 4 ,however,thedistributionofmomentumbelowthetroughlevelgivenbyMellor'sequationlooksverysimilarinshapeandmagnitudetoboththepredictedproleandthedepth-uniformdistributionsuggestedbylineartheory.Moreover,Fig. 4 AshowsaverycloseagreementbetweenMellor'ssolutionandthepredictedproleattheforcingboundary.ThepreviousguresdemonstratingthedistributionofradiationstressFigs. 4 and 4 showthatthemajorityofthemomentumuxislocatedabovethetrough

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53 Figure4:Thedepth-integratedmagnitudesofthepredictedradiationstressesSxxandSxzshowgoodagreementtothemagnitudesfoundfromlineartheoryandMellor'sequations. level.Althoughthevariousmethodsforplottingverticalprolesofsxxandsxzgivevastlydifferentdistributionsoverthevertical,theirdepth-integratedvaluesaresimilar,whichisanencouragingresult.Thedepth-integratedradiationstressmagnitudesareshowninFig. 4 anddemonstratethesimilarityofthepredictedmagnitudewiththatgivenbylineartheoryE+E 2andMellor'sequationsgivenpreviously.Depth-integratedvaluesoftheradiationstresscomponentSxxarerepresentedbythehollowsymbolswhereeachshaperepresentsthetechniqueortheorygiveninthesymbollegend.ThelongshorecomponentofradiationstressSxzisdenotedbythelledsymbols,whichagaincorrespondtotheapproximationlistedinthelegend.Amultiple-applicationtrapezoidalscheme ChapraandCanale 1998 wasusedtointegratethevertically-dependentprolesplottedinFigs. 4 and 4 ,andatveadditionalcross-shorelocationsaswell.Withtheexceptionofthevaluespredictedat

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54 theforcingboundary,thepredictedmagnitudesofSxxandSxzlookquitesimilartotheanalyticalsolutionsprovidedbytheothertwotheoriesconsideredhere.Theforcingboundarypresentsaspecialcasesincecorrectionshavebeenappliedtothevelocitycomponentstocompensateformassux.Asdiscussedearlier,andshowninFig. 4 ,thevelocitycorrectionsappliedattheforcingboundaryresultinamean-oweldbelowthetroughlevelthatisjustslightlylargerthanzero;therefore,thelargestcontributiontothedepth-integratedvalueofSxxcomesfromthemomentumuxassociatedwiththeforced,monochromaticwavesignal.Thisisnotthecaseatothercross-shorelocationswheretheundertowandlongshorecurrentcontributesignicantlytothestructureandmagnitudeoftheradiationstressprolebelowthetroughlevelofthewaves.Sincewedonothaveaformulationforthetriangulardistributionoftheradiationstresscomponentsxzbetweenthewavecrestandtrough,thedepth-integratedvalueofSxzonlyincludestheportionoftheprolebelowthemeanwaterlevel.Thedistributionoftheradiationstressproducedbythenonlinearwavesinoursimulationssuggeststhatthemajorityofthemomentumuxiscontainedabovethetroughlevel.Similarobservationsweremadeby Svendsen 1984 and SobeyandThieke 1989 regardingtheshoreward-directedcomponentoftheradiationstress,Sxx.FromthepredictedprolesofthelongshorecomponentpresentedinFig. 4 ,itappearsthatthisobservationholdstrueforSxzaswell.Thisdistributionofenergyoverthewatercolumnisnotatrivialmatterasphase-averagedmodelsmaybeforcedwithvariousapproximationstostressorforcedistributions,acommonchoicebeingdepth-uniformordepth-linearproles.Lineartheorysuggeststhatroughlyone-thirdofthetotalmomentumuxisadvectedshorewardbythewavesintheregionbetweenthetroughandthecrest.Byintegratingourradiationstressprolesofsxxandsxzandcomparingtheareacontainedintheregionboundbythecrestandthetroughtothetotalareaoftheprole,wendratherthatroughlytwicethisamountiscarriedbythenonlinearwavespresentinoursimulations.ThisratioisplottedinFig. 4 forboth

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55 SxxandSxz,wherewehaveonceagainusedthemultiple-applicationruletointegratetheproles. Figure4:Theratioofmomentumuxinthewavetrough-to-crestregiontothetotaldepth-integratedvalueforSxxandSxzasafunctionofcross-shoredepth.Forprogressivenonlinearwaterwaves,between40and80%ofthetotalmomentumuxiscarriedintheregionboundbythewavecrestandwavetrough. 4.3.3StressGradientsandNearshoreForcingThepreviousguresshowingverticalprolesofradiationstresscomponentsprovidevaluableinformationaboutthedistributionofmomentumuxproducedbynonlinearshoalingwaves.Althoughtheseradiationstressprolespresentusefulinformationconcerningthenonlineardistributionofmomentum,itisthegradientsofradiationandshearstressesthatdrivenearshorecirculation.Thesignicanceofcross-shoregradientsofSxxonsurfzoneforcinginproducingsetupandundertowhasbeennotedby Longuet-HigginsandStewart 1964 Svendsen 1984 ,andmorerecentlyby Christensenetal. 2002 .Similarly,forcinginthelongshoredirection

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56 isproportionaltothecross-shoregradientofthelongshorecomponentofradiationstress,@ @xSxz DeanandDalrymple 1991 .Shearstressdistributionsi.e.huviplaysignicantrolesincirculationaswell,butarecommonlyimplementedasdepth-uniformordepth-linearforcesactingoverthewatercolumn DeigaardandFredse 1989 iftheyareconsideredatall. RiveroandArcilla 1995 discussedtheimportanceofshearstressdistributionsinthecontextofmorecomplexconditionsforwavesencounteringambientcurrentsandslopingbottoms. DeigaardandFredse 1989 alsonotedtheimportanceofthetermhuviasanecessarymechanismformomentumtransferinnon-uniformorunsteadywaveconditionsanddiscusseditsrelativeimportanceintheirsimpliedmomentumequationgivenbyEq. 4 =)]TJ/F15 11.955 Tf 9.299 0 Td[(2U@U @xh)]TJ/F25 11.955 Tf 11.955 0 Td[(y)]TJ/F25 11.955 Tf 11.955 0 Td[(g@ @x)]TJ/F25 11.955 Tf 11.955 0 Td[(Sgh)]TJ/F25 11.955 Tf 11.955 0 Td[(y+UV where=shearstressU=horizontalwave-inducedvelocityV=verticalwave-inducedvelocityh=averagewaterdepthy=verticallocation=verticallocationofthefreesurfaceS=meanslopeofthewatersurfacesetupWehaveplottedtheverticaldistributionofradiationandshearstressgradientsinFig. 4 toemphasizethethree-dimensionalnatureofthesenearshoreforcingmechanisms.Cross-shoregradientsoftheradiationstresscomponentswerecomputedovertwoadjacentgridlocationsandthenminimallysmoothedusingadiffusive-typelter.Thegradientsoftheshearstresstermhuviwerecalculatedoveradjacentgridlocationsintheverticaldirectionandwerealsosmoothedaslittleasnecessarytoensurethatthestructureandmagnitudeoftheprolewouldberetained,whilesimultaneouslyprovidingsufcientlysmoothproles.GradientsofsxxshowninFig. 4 representthedistributionoftheforcingthatisappliedinthecross-shoredirectionandhasthedimensionalunitsofaforceperunit

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57 Figure4:TheverticaldistributionofradiationandshearstressgradientsatAtheforcingboundary,Bh=0:246m,Ch=0:226m,Dh=0:206m,Eh=0:145m,Fh=0:124m,Gh=0:085m,andHh=0:069m.Thedashedlinerepresentsthemeanwaterlevelandthesloping,dottedlinerepresentstheapproximatelocationofthebathymetry. area,perunitdepth. Longuet-HigginsandStewart 1964 statedthatthisforcingonthesurfzonewouldbebalancedbyagradientinthemeanwaterlevelEq. 4 .@ @xhi=)]TJ/F15 11.955 Tf 16.973 8.088 Td[(1 gh@ @xSxxThisforcebalanceisusuallyconsideredinadepth-integratedsense,resultinginthelossofinformationregardingtheverticaldistributionoftheforcebalance.Positivevaluesof@xsxxinFig. 4 ,whendepth-integrated,indicatethepresenceofashoreward-directedforcebalancingthewave-inducedsetdown,whilenegativevalues

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58 representaseaward-directedforcebalancingthesetup.Figure 4 demonstratesthatthereisauniqueverticaldistributionoftheforcingateachcross-shorelocation,especiallyinpanelsFandGwheretherearesignicantsigndifferencesaboveandbelowthewavetroughlevel.Asnotedby Dyhr-NielsenandSorensen 1970 ,thebalancingforceswillnothavesimilarverticaldistributionsandtheinclusionofshearandturbulentstressesandlocalaccelerationsforunsteadyowsmustalsobeconsideredtoobtaintheappropriateforcebalanceoverthevertical Christensenetal. 2002 .Thegradientsofthelongshoreradiationstresscomponent@xsxzplottedinFig. 4 aremanifestedaslongshore-directedforcesperunitareainthenearshoreregion DeanandDalrymple 1991 .Theselongshore-directedforcesarebalancedbybottomshearstressesaswellastheresistancetolateralshearoverthewatercolumn Longuet-Higgins 1970 .Thenetmagnitudeoftheresistingforcesisfoundbydepth-averagingtheprolesof@xsxzinFig. 4 wherethesignofthedepth-averagedvalueindicatesthedirectionofaction.Depth-averagedvaluesofthe@xsxzprolesinFigs. 4 ADaresmallrelativetotheprolesinpanelsEHduetothelowerlongshorecurrentvelocitiesatthecorrespondingcross-shorelocations.Thissuggeststhatthebalancingforceswouldberelativelysmallattheselocations,aswell.InpanelsEandF,however,decreasingmagnitudesofsxzresultinstrongerlongshoreforcinginthepositivez-direction. RiveroandArcilla 1995 correctlydrawattentiontothesignicanceofthetermhuvithatappearsinthedepth-averagedmomentumequations.Previously,theargumenthadbeenmadethatthisvalueisessentiallyzerosincethehorizontaluandverticalvvelocitycomponentsare90outofphase;thisassumptiononlyholdstrueforsteady,periodic,linearwavesthatarenottypicallyseeninnearshoreenvironments.Recentattemptsweremadeby RiveroandArcilla 1995 and DeigaardandFredse 1989 toemphasizetheimportanceofincludingappropriatevaluesand

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59 distributionsofhuviinmomentumequationsasthistermplaysimportantrolesintheverticaldistributionofshearstressaswellaswaveenergydissipation.TheverticalgradientofhuviisshowninFig. 4 atvariouscross-shorelocationswherethedensityconstanthasbeenincludedtocomparetherelativemagnitudeofthistermtotheradiationstressgradients.Astheypredicted,thegradientsofhuviarenotnegligiblysmallandinmostcaseshavemagnitudessimilartothoseoftheradiationstressgradientsseeninFig. 4 .Theresponsetothenearshoreforcingproducedbygradientsintheradiationstresseldsisfoundbyintegratingtheprolesof@xsxxand@xsxzplottedinFig. 4 .Relationshipsforthecross-shoreandlongshoreforcingcomponentsaregivenbyEqs. 4 and 4 ,respectively.ThetermFxEq. 4 ,therefore,istheforcingappliedonthesurfzoneinthecross-shoredirection,perunitarea.Forcingappliedinthelongshoredirection,perunitarea,isgivenbythecomponentFzEq. 4 .ThesetwonearshoreforcingcomponentsareplottedinFig. 4 andprovideinformationconcerningthedistributionofforcinginthecross-shoredomain,themagnitudeofeachforcingcomponent,andtheircombinedeffect.Figure 4 AdemonstratesthemagnitudeandlocationoftheappliedforcingcomponentsFxandFz,whichappeartohavebi-modaldistributions,withtheirpeaksoccurringatsimilarcross-shorelocations.ThevectorsshowninFig. 4 Brepresentthecombinednearshoreforcingfoundbycomputingtheresultantofthetwoforcingcomponentsateachcross-shorelocation;however,forclarityonlyone-fthofthetotalnumberofpointsinthecross-shoredomainarerepresentedbythisgure.Fx/)]TJ/F25 11.955 Tf 23.114 8.088 Td[(@Sxx @xFz/)]TJ/F25 11.955 Tf 23.114 8.088 Td[(@Sxz @x

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60 Figure4:Depth-integratedvaluesofthenearshoreforcingcomponentsFxandFzplottedasafunctionofcross-shorelocationAindependentlyandBasaschematicofthecombinednearshoreforcing.

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CHAPTER5SUMMARY5.1ApplicationsSOLA-SURFisaportableCFDmodelthatcanbeusedtostudyawidevarietyofuidows.Thesimplisticnatureofthecode,evidentinbothitsorganizationandnumerics,enablesuserswithminimalnumericalmodelingexperiencetosimulatetime-dependentows.Indeed,verylittleinformationmustbesuppliedtothemodel:bathymetrydata,waveheight,waveperiod,andthedepthattheoffshoreforcingboundary.Initsmostbasicstatethismodelcouldbeutilizedasateachingortrainingtool,providingstudentsorotherinterestedindividualsachancetouseandmanipulateaverysimple,yetrobustlystable,three-dimensionalCFDmodel.Forthemoreadvancedmodeler,however,thiscodeservesasagoodstartingpointforbuildinganevenmoreadvancedcomputationaltoolthatcanbeappliedtostudymorecomplexproblems.PerhapsthegreatestattributeofthisNavier-Stokescodeistheabilitytoresolvethetime-dependentfreesurface.Thisfeaturecouldbeutilizedtostudywavediffractionpatternsinharborsandbasinsoraroundbreakwatersandjetties.Providingthemodelwithathree-dimensionalbathymetryeld,onecouldstudytheresultingwaverefractionaround,forinstance,acuspatebeachforbothshore-normalandobliquelyincidentwaveelds.Cross-shoreandlongshorecurrentstructuresovervariablebathymetrycouldalsobeinvestigatedforavarietyofwaveclimates.5.2SensitivityThethree-dimensionalNavier-Stokesmodel,SOLA-SURF,providesreasonableestimatesofnearshoreprocesses,butreliesonboththeimplementationofthefreesurfacevelocityboundaryconditionsaswellasthecellaspectratio.Caremustalsobetakenwhenselectingappropriatevaluesofthetheta-differencingcoefcientsinorder 61

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62 topreventamplitudeerrorsinthewaveeldandexcessivediffusioninthevelocityelds.Theseconditionslimittheparameter-spacetoarangeofspecicvalues,buttheconditionsusedinVisser'slaboratoryexperimentwereconducivetoperformingsimulationswiththismodel.WaveswithlowerIribarrennumbers,thoseinthespillingbreakerrange,couldbemoreappropriatelymodeledwithSOLA-SURFandtheresultingwavetransformationwouldyieldbetterresults.Thesensitivitytothefreesurfacevelocityboundaryconditionsisunfortunate,butperhapsacombinationbetweenthetwomethodsdiscussedinChapter 2 exists,suchthatimprovedresultscouldbeobtainedforboththewaveheightandvelocityelds.Extendingtheseboundaryconditionsusinghigher-orderdifferencingschemesisperhapsanotherviablesolution,butultimatelytheslopeofthefreesurfacewillbelimitedbythecellaspectratioarst-orderlimitationatbest.WhenelectingtouseSOLA-SURF,itisperhapsbesttounderstandthephysicallimitationsofthemodelandtakethemintoaccountwhenanalyzingtheresults.Inthiscase,webelievethattheabilityofthemodeltoreproducetheexperimentalobservationspreciselyisofminorsignicanceinthecontextofourdiscoveries.Theabilitytoreproduceapproximatelyaccurateorrealisticresultswasamoreimmediateconcernandthecomparisonofmodelpredictionstolaboratoryobservations,aswellasvariousanalyticaltheories,bolsteredourcondenceintheapplicabilityofSOLA-SURFtothisinvestigation.5.3DiscoveriesPerformingsimulationsofVisser'slaboratoryexperimentonuniformlongshorecurrents[ Visser 1984 ; Visser 1991 ]withSOLA-SURFhasprovidednewinformationregardingtheverticaldistributionofmomentumand,morespecically,thehorizontalcomponentsofradiationstressinthepresenceofnonlinearshoalingwater-waves.Whiletheverticalstructureofradiationstresscomponentshasbeendiscussedintheliterature,particularlyinthepresenceofmorerealisticwavespropagatingovervariablebathymetry,verylittleinformationexiststhatquanties

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63 itsimportance.Ourmodelsimulationshaveleadtotheconclusionthatinthepresenceofnonlinearshoalingwaves,themajorityofmomentumuxdirectedbothshorewardSxxandalongshoreSxziscontainedwithintheregionboundbythewavetroughandcrest.Lineartheorysuggeststhatroughlyone-thirdofthetotalmomentumuxadvectedshorewardisfoundinthisregion,butouranalysissuggestsanaveragevaluethatisroughlytwicethisamountinthesurfzone.Earlyestimatesofradiationstressmagnitudesneglectedcontributionsfromthewave-inducedvelocityabovethemeanwaterleveland,althoughincorrectinthecontextofrealprogressivewaves,thismethodologycontinuestobefollowed.Aninteresting,andpertinent,resultfromthisinvestigationisthatthedepth-integratedmagnitudesoftheradiationstressprolespredictedbySOLA-SURFaresimilartothemagnitudescalculatedusinganalyticalsolutionsprovidedby Mellor 2003 Longuet-HigginsandStewart 1964 DeanandDalrymple 1991 ,and Dean 1995 .Thisresultispertinentbecauseitprovidescondencethatthemodelpredictions,whilenotprecise,areaccurate.Therefore,nonlinearshoalingwaveshaveradiationstressmagnitudesthatarerelativelysimilartothevaluesgivenbylineartheory,butthedistributionofmomentumuxovertheverticalisdifferent.Inthecontextofthree-dimensionalmodeling,thisresultisimportantsincemanycirculationmodels,specicallywave-phase-averagedmodels,areforcedwithradiationstressgradientproles.Ithasbeensuggestedintheliteraturethattheresultingcirculationinthesemodelsishighlysensitivetothedistributionofforcingoverthevertical,butuntilrecentlyscientistsandnumericalmodelershavehadlittleinformationregardingmoreplausibledistributions.Theanalyticalexpressionsgivenby Mellor 2003 ,however,tendtounderestimatetheuxofmomentumnear,andabove,thefreesurfaceandoverestimatethecontributionnearthebedinshallowwater.Thislatterresultwascertainlyunexpectedandthepredictedprolesofradiationstressgivenbythemodelshowthatthisfeatureisnotevident.Althoughtheundertowprovidessome

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64 verticalstructuretotheradiationstressprolebelowthetroughlevel,itscontributionistypicallymanytimessmallerthanthatfoundabovethemeanwaterlevel.Althoughradiationstressprolesdocontributesignicantlytotheoveralloweld,gradientsintheradiationstresseldsrepresentthetrueaveragecross-shoreandlongshoreforcinginanearshoresystem.Analysisoftheverticaldistributionofradiationstressgradients,however,providedlessconclusiveinformation.Gradientsofsxxseemedtohaveuniqueverticalprolesateachcross-shorelocationconsideredinoursimulation.Surely,theverticalstructureofthemean-owhasmuchtodowiththisvariability,especiallynearthebedwheretheundertowisadominantowfeature.Someinshorelocationsyieldedprolesof@xsxxthathadsignicantsigndifferencesaboveandbelowthemeanwaterlevel;theselocationscorrespondtoareaswheretheobliquelyincidentwavesarenonlinear,asymmetric,andverysteep.Ontheotherhand,theverticalproleoftheradiationstressgradient,@xsxx,attheforcingboundarysuggeststhatthemajorityofthiscross-shoredirectedforcingiscontainedinadepth-uniformdistributionbelowthemeanwaterlevel,withlittlecontributionaboveit.AsdiscussedinChapter 4 ,thislocationrepresentstheforcinginducedbythemonochromatic,sinusoidalwavesignalandmoreorlessexcludestheEulerianmeanow.Insharpcontrasttotheproleof@xsxx,theproleof@xsxzattheforcingboundaryshowsthatthemajorityoftheforcingisfoundinthewavetrough-to-crestregionwithalmostnocontributionevidentbelowthewavetroughlevel.Thisdistributionisessentiallyreversedinthelocationofthepeaklongshorecurrent,wherethelargestforcingcontributionisfoundbelowthemeanwaterlevel.

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APPENDIXATHETADIFFERENCINGPredictionsofferedbynumericalmodelscanbesensitivetothenite-differencingschemeusedtodiscretisethegoverningequations.SOLA-SURFemploystheta-differencinginthediscretisationofconvectiveuxtermsfoundinthemomentumequationsEqs. 2 2 andtocontroltheamountofdonorcelldifferencinginthekinematicfreesurfaceboundaryconditionEq. 2 .ThediscretisedformsoftheconvectiveuxtermsseeEq. A andtheKFSBCEq. A areweightedwithuniquethetacoefcients,therebyallowingtheusertocontrolthedifferencingschemesindependently.@u2 @x=1 4x"ui;j;k+ui+1;j;k2+jui;j;k+ui+1;j;kjui;j;k)]TJ/F25 11.955 Tf 11.955 0 Td[(ui+1;j;k)-222()]TJ/F15 11.955 Tf 11.291 0 Td[(ui)]TJ/F24 7.97 Tf 6.586 0 Td[(1;j;k+ui;j;k2)]TJ/F25 11.955 Tf 11.955 0 Td[(jui)]TJ/F24 7.97 Tf 6.586 0 Td[(1;j;k+ui;j;kjui)]TJ/F24 7.97 Tf 6.586 0 Td[(1;j;k)]TJ/F25 11.955 Tf 11.955 0 Td[(ui;j;k# A Hn+1i;k=Hni;k+t)]TJ/F15 11.955 Tf 9.298 0 Td[(1 4xui;jt;k+ui)]TJ/F24 7.97 Tf 6.587 0 Td[(1;jt;kHi+1;k)]TJ/F25 11.955 Tf 11.955 0 Td[(Hi)]TJ/F24 7.97 Tf 6.586 0 Td[(1;k)-222()]TJ/F25 11.955 Tf 11.291 0 Td[(jui;jt;k+ui)]TJ/F24 7.97 Tf 6.586 0 Td[(1;jt;kjHi+1;k)]TJ/F15 11.955 Tf 11.956 0 Td[(2Hi;k+Hi)]TJ/F24 7.97 Tf 6.586 0 Td[(1;k)-222()]TJ/F15 11.955 Tf 18.288 8.088 Td[(1 4zwi;jt;k+wi;jt;k)]TJ/F24 7.97 Tf 6.587 0 Td[(1Hi;k+1)]TJ/F25 11.955 Tf 11.955 0 Td[(Hi;k)]TJ/F24 7.97 Tf 6.587 0 Td[(1)-222()]TJ/F25 11.955 Tf 11.291 0 Td[(jwi;jt;k+wi;jt;k)]TJ/F24 7.97 Tf 6.586 0 Td[(1jHi;k+1)]TJ/F15 11.955 Tf 11.955 0 Td[(2Hi;k+Hi;k++hvvi;jt;k+)]TJ/F25 11.955 Tf 11.955 0 Td[(hvvi;jt)]TJ/F24 7.97 Tf 6.587 0 Td[(1;k A AfterdiscoveringthesensitivityofthefreesurfacevelocityboundaryconditionsseeChapter 2 ,werananumberofsimulationsusingdifferentvaluesforthe 65

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66 theta-schemecoefcients,and,inordertondthebestcombinationofnite-differenceschemestouseinournalsimulation.Appropriatevaluesof,assuggestedby Hirtetal. 1975 forthetwo-dimensionalSOLAmodel,arefoundusingthefollowinginequality:1maxut x;vt y;wt z:In Hirtetal. 1975 ,areasonablevalueofissaidtobe1.2to1.5timesgreaterthantheright-handsideofthisinequality,butwefoundthisrangeofvaluestobetoolowtoproduceastablesimulation,givenoursetofparameters.Thelowestvalueofthatproducedastablesimulationwas0.5,whichisroughlyfourtimesgreaterthanthesuggestedvalueusingtherulestatedabove.Valuesofbelow0.5resultedinhighlydispersivevelocityeldsthatcausethesimulationtobecomeunstable,whilevaluesofgreaterthan0.5diffused,orsmoothed,thevelocityeldsdampingtheunsteadinessarticially.While Hirtetal. 1975 suggestsettingthevalueofequalto,wefoundthatthestabilityofthesimulationswasmuchlesssensitivetothevalueofused.Aftertryingvariouscombinationsofand,wefoundthemostagreeablevaluestobe0.5and0.0,respectively.Equations A and A representthediscretisedformsofEqs. A and A ,respectively,wherewehavesubstituted=0:5and=0:0intotheappropriateequations.Thesensitivityofthepredictedcross-shorewaveheightsRMStothecombinationoftheta-schemecoefcientsisshowninFig. A forsevenofthecasestested.@u2 @x=1 4x"u2i+1;j;k+2ui;j;kui+1;j;k)]TJ/F25 11.955 Tf 11.955 0 Td[(ui;j;kui)]TJ/F24 7.97 Tf 6.587 0 Td[(1;j;k)]TJ/F25 11.955 Tf 11.955 0 Td[(u2i)]TJ/F24 7.97 Tf 6.586 0 Td[(1;j;k++1 2jui;j;kjui;j;k)]TJ/F25 11.955 Tf 11.956 0 Td[(ui+1;j;k)]TJ/F25 11.955 Tf 11.956 0 Td[(ui)]TJ/F24 7.97 Tf 6.587 0 Td[(1;j;k+jui+1;j;kjui;j;k)]TJ/F25 11.955 Tf 11.955 0 Td[(ui+1;j;k++jui)]TJ/F24 7.97 Tf 6.586 0 Td[(1;j;kjui;j;k)]TJ/F25 11.955 Tf 11.955 0 Td[(ui)]TJ/F24 7.97 Tf 6.587 0 Td[(1;j;k# A

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67 Hn+1i;k=Hni;k+t)]TJ/F15 11.955 Tf 9.299 0 Td[(1 4xui;jt;k+ui)]TJ/F24 7.97 Tf 6.587 0 Td[(1;jt;kHi+1;k)]TJ/F25 11.955 Tf 11.955 0 Td[(Hi)]TJ/F24 7.97 Tf 6.586 0 Td[(1;k)-222()]TJ/F15 11.955 Tf 18.288 8.087 Td[(1 4zwi;jt;k+wi;jt;k)]TJ/F24 7.97 Tf 6.586 0 Td[(1Hi;k+1)]TJ/F25 11.955 Tf 11.955 0 Td[(Hi;k)]TJ/F24 7.97 Tf 6.587 0 Td[(1+hvvi;jt;k++)]TJ/F25 11.955 Tf 11.955 0 Td[(hvvi;jt)]TJ/F24 7.97 Tf 6.587 0 Td[(1;k A FigureA:Effectonnumericaldiffusiononmodelpredictions.Shownarepredictedcross-shorewaveheightsfordifferentcombinationsofthetheta-schemediscretisationcoefcients.

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APPENDIXBCROSS-SHOREMASSBALANCEInChapter 4 ,webrieytouchedonthesubjectofconservationwhenpointingoutfeaturesofthevelocityprolestakenatvariouscross-shorelocationsFig. 4 .SomeoftheprolesFigs. 4 E 4 Happearedtobenon-conservative:thatis,therewasanabsenceofreturnownearthebedthatwouldacttobalancetoshoreward-directedownearthesurface.Whileithasbeennotedintheliteraturethatconservationintime-meanvelocityprolesshouldbeexplicitforpurelytwo-dimensionalproblems,weexpectedatleastaminoramountofundertowtoappearinalloftheproles.TheindividualvelocityprolesplottedinFig. 4 ,however,onlyprovideinformationaboutthemeanoweldatonediscretecross-shorelocation.Figure B showsthetime-meanvelocityeldatatransecttakennearthemidpointofthelongshoredomain.Fromthisgure,weseethatthereissignicantlymorestructuretothecross-shorecirculationthanisdescribedbythevelocityprolestakenatdiscretelocations.JustaswediscoveredinFig. 4 byplottingthetime-mean,depth-averagedcross-shorevelocity,themeanvelocityeldbetweenx=5mandx=6:5m,showninFig. B ,appearstobedirectedshorewardwithverylittlereturnownearthebed.Ontheotherhand,seawardofx=4mthereisalarge,conservativecirculationcellthatisalsoevidentinthevelocityprolesplottedinFigs. 4 B 4 D.Itispossiblethatconservationinthisareamx6.5missatised,however,iftheexcesscross-shorevelocityisbalancedbysomephysicalstoragemechanism.ThismechanismcanbedescribedthroughamodiedformofthecontinuityequationEq. B thatrelatesthetime-dependentfreesurfacetogradientsinthevelocityelds DeanandDalrymple 2002 .Here,wearemoreconcernedwith 68

PAGE 79

69 FigureB:Thetime-meanvelocityeldtakenatz=2:5mandplottedasAvelocityvectorsandBasstreamlines.Thered,dashedlinerepresentsthestillwaterlevelatinitialization. thecross-shoregradientsintheuvelocityeldsincetheremainingvelocitygradienttermsarequitesmallincomparison.@ @t+@uh @x+@wh @z=0BThetime-averagedfreesurfaceplottedinFig. B Bshowsthatthereisapositivedisplacementofthefreesurfaceneartheareaoftheexcesscross-shorevelocity,whichisshownagaininFig. B A.Inordertoseeifthesetwophysicalprocesseswerespatiallyrelated,theywereeachscaledbytheirmaximumorminimumdeparture,dependingonthesignofthevalue,therebynon-dimensionalisingthetwoelds.

PAGE 80

70 FigureB:Spatialfeaturesoftheaveragevelocityandfreesurfaceelds.Acontourplotofthetime-meanAdepth-averagedcross-shorevelocity,Bfreesurfaceeld,CscaledvelocityoodandfreesurfacelineseldsandDspatialcorrelationbetweenthedepth-averagedcross-shorevelocityandthewave-inducedsetupandsetdown. Scalingforthevelocityandfreesurfaceeldswasaccomplishedusingthefollowingmethodology: umax=max[h ux;zi] umin=min[h ux;zi]ux;z=h ux;zi uswhereus=8><>: umaxforh ux;zi>0 uminforh ux;zi<0

PAGE 81

71 max=max[h x;zi] min=min[h x;zi]x;z=h x;zi swheres=8><>: maxforh x;zi>0 minforh x;zi<0whereux;zandx;zarethescaledvelocityandfreesurfaceeldsrespectively.Thisscalingprocessremovesthedimensionalityofeachvariableandresultsinvelocityandfreesurfaceeldsthatvarybetween)]TJ/F15 11.955 Tf 9.298 0 Td[(1and1:)]TJ/F15 11.955 Tf 9.298 0 Td[(1ux;z1and)]TJ/F15 11.955 Tf 11.955 0 Td[(1x;z1:ThescaledvelocityandfreesurfaceeldsareplottedinFig. B C.Weseethatmaximumpositivedisplacementsofthetime-averagedfreesurfaceeldcontourlinescorrespondtolocationswherethereisalsoamaximum,shoreward-directedvelocitycontourood.ThecorrelationbetweenthesescaledeldswascalculatedandthecorrelationcoefcientisplottedinFig. B D.Whileweknowthatcorrelationdoesnotnecessarilyimplycausation,Fig. B Dsuggeststhatthereisaspatialcorrelationbetweenthesetwophysicalprocesses.Therefore,itisquitepossiblethattheexcessdepth-averagedcross-shorevelocityevidentinFig. B Aisbeingstoredaspotentialenergyintheformofsetup,particularlyovertheshallowsill.AgreatdealoftimewasspentattemptingtouseEq. B toexplainthecause-and-effectrelationshipofthiscross-shoreforcebalance,buttheprocessprovedratherdifcult;thehighdegreeofspatialandtemporalvariabilitymadeitverydifculttoshowthatthisequationissatisedateachtime-step,andoverallgridlocations,tosufcientprecisioninthenitedifferencemodel.

PAGE 82

REFERENCES ALLEN,J.S.,NEWBERGER,P.A.&HOLMAN,R.A.1996Nonlinearshearinstabilitiesofalongshorecurrentsonplanebeaches.JournalofFluidMechanics310,181. 4.2.3 BADIEI,P.&KAMPHUIS,J.W.1995Physicalandnumericalstudyofnearshorecurrents.InProc.CoastalDynamcis,ASCE,Gdansk,Poland,377. 1.2 BATTJES,J.A.1974Computationofset-up,longshorecurrents,run-upandovertoppingduetowind-generatedwaves.Comm.onHydraulics,Rep.74-2,Dep.CivilEng.,DelftUniv.ofTechnology,Delft,TheNetherlands. 2.2 BOCCOTTI,P.2000WaveMechanicsforOceanEngineering.OceanographySeries64.Elsevier,Amsterdam,TheNetherlands. 2.3.1 BOWEN,A.J.&HOLMAN,R.A.1989Shearinstabilitiesofthemeanlongshorecurrent,1.Theory.JournalofGeophysicalResearch94C12,18023. 4.2.3 BOWEN,A.J.,INMAN,D.L.&SIMMONS,V.P.1968Waveset-downandset-up.JournalofGeophysicalResearch73,2569. 1.2 CHAPRA,S.C.&CANALE,R.P.1998NumericalMethodsforEngineers.WCB/McGraw-Hill,Boston,Massachusetts. 4.3.2 CHEN,Q.,KIRBY,J.T.,DALRYMPLE,R.A.,SHI,F.ÞTON,E.B.2003Boussinesqmodelingoflongshorecurrents.JournalofGeophysicalResearch118C11,3362. 4.1.2 CHEN,S.,JOHNSON,D.B.&RAAD,P.E.1995Velocityboundaryconditionsforthesimulationoffreesurfaceuidow.JournalofComputationalPhysics116,262. 2.3.2 4.1.1 CHORIN,A.J.1968Numericalsolutionofthenavier-stokesequations.MathematicsofComputation22,745. 2.2 CHRISTENSEN,E.D.,WALSTRA,D.J.&EMERAT,N.2002Verticalvariationoftheowacrossthesurfzone.CoastalEngineering45,169. 4.3.3 DEAN,R.G.1995ClassnotesforOCP6167,OceanWavesII:Non-linearTheory. 4.3.2 5.3 72

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73 DEAN,R.G.&DALRYMPLE,R.A.1991WaterWaveMechanicsforEngineersandScientists.WorldScienticPress,RiverEdge,NewJersey. 2.2 3.3.3 4.3.2 4.3.3 5.3 DEAN,R.G.&DALRYMPLE,R.A.2002CoastalProcesseswithEngineeringApplications.CambridgeUniversityPress,NewYork,NewYork. B DEIGAARD,R.&FREDSE,J.1989Shearstressdistributionindissipativewaterwaves.CoastalEngineering13,357. 1.2 4.3.3 DODD,N.,IRANZO,V.&RENIERS,A.2000Shearinstabilitiesofwave-drivenalongshorecurrents.ReviewofGeophysics38,437. 4.2.3 DOLATA,L.F.&ROSENTHAL,W.1984Wavesetupandwave-inducedcurrentsincoastalzones.JournalofGeophysicalResearch89,1973. 1.2 DYHR-NIELSEN,M.&SORENSEN,T.1970Sandtransportphenomenaoncoastswithbars.InProc.ofthe12thInt.Conf.onCoastalEng.2,ASCE,Washington,DC,855. 4.3.3 FLETCHER,C.A.J.2000ComputationalTechniquesforFluidDynamics1.SpringerSeriesinComputationalPhysics.Springer-Verlag,NewYork,NewYork. 2.2 GALVIN,C.J.&EAGLESON,P.S.1965Experimentalstudyoflongshorecurrentsonaplanebeach.U.S.ArmyCoastalEng.Res.Cent.,Vicksburg,Mississippi,Tech.Mem.10. 4.1.2 HIRT,C.W.,NICHOLS,B.D.&ROMERO,N.C.1975SOLAANumericalSolutionAlgorithmforTransientFluidFlows,LA-5852edn.LosAlamosScienticLaboratoryoftheUniversityofCalifornia,LosAlamos,NewMexico. 2.1 2.2 A IRIBARREN,C.R.&NOGALES,C.1949ProtectiondesPorts,SectionII,Comm.4,XVIIthInt.Nav.Congress,Lisbon,31. 2.2 LARSON,M.&KRAUS,N.C.1991Numericalmodeloflongshorecurrentforbarandtroughbeaches.JournalofWaterway,Port,CoastalandOceanEng.117,ASCE,326. 1.2 LONGUET-HIGGINS,M.S.1970Longshorecurrentsgeneratedbyobliquelyincidentseawaves,1.JournalofGeophysicalResearch75. 4.3.3 LONGUET-HIGGINS,M.S.&STEWART,R.W.1964Radiationstressesinwaterwaves;aphysicaldiscussion,withapplications.Deep-SeaResearch11,529. 1.1 1.2 4.3.2 4.3.3 5.3 MELLOR,G.2003Thethree-dimensionalcurrentandsurfacewaveequations.JournalofPhysicalOceanography33,1978. 1.2 4.3.2 5.3

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74 MIZUGUCHI,M.&HORIKAWA,K.1978Experimentalstudyonlongshorecurrentvelocitydistribution.Bull.Fac.Sci.Eng.21,ChuoUniv.,Tokyo,Japan. 4.1.2 OLTMAN-SHAY,J.&HOWD,P.A.1993Edgewavesonnonplanarbathymetryandalongshorecurrents:Amodelanddatacomparison.JournalofGeophysicalResearch98,2495. 4.2.3 OLTMAN-SHAY,J.,HOWD,P.A.&BIRKEMEIER,W.A.1989Shearinstabilitiesofthemeanlongshorecurrent,2.Fieldobservations.JournalofGeophysicalResearch94C12,18031. 4.2.3 POPE,S.B.2000TurbulentFlows.CambridgeUniversityPress,NewYork,NewYork. 4.2.3 PUTREVU,U.&SVENDSEN,I.A.1992Shearinstabilityoflongshorecurrents:Anumericalstudy.JournalofGeophysicalResearch97,7283. 4.2.3 RENIERS,A.J.H.M.,BATTJES,J.A.,FALQUES,A.&HUNTLEY,D.A.1997Alaboratorystudyontheshearinstabilityoflongshorecurrents.JournalofGeophysicalResearch102C4,8597. 4.2.3 RIVERO,F.J.&ARCILLA,A.S.1995Ontheverticaldistributionofhuwi.CoastalEngineering25,137. 1.2 4.3.3 SOBEY,R.J.&THIEKE,R.J.1989Meanowcirculationequationsforshoalingandbreakingwaves.JournalofEngineeringMechanics115,285. 1.2 4.3.2 STIVE,M.J.F.&WIND,H.G.1982Astudyofradiationstressandset-upinthenearshoreregion.CoastalEngineering6,1. 1.2 SVENDSEN,I.A.1984Massuxandundertowinasurfzone.CoastalEngineering8,347. 1.2 4.3.1 4.3.2 4.3.3 SVENDSEN,I.A.&LORENZ,R.S.1989Velocitiesincombinedundertowandlongshorecurrents.CoastalEngineering13,55. 4.2.2 4.3.1 VISSER,P.J.1984Amathematicalmodelofuniformlongshorecurrentsandthecomparisonwithlaboratorydata.Comm.onHydraulics,Rep.84-2,Dep.CivilEng.,DelftUniv.ofTechnology,Delft,TheNetherlands. 1.3 3 3.1 4.2.3 5.3 VISSER,P.J.1991Laboratorymeasurementsofuniformlongshorecurrents.CoastalEngineering15,563. 1.3 3 3.1 4.1 4 4.2.3 4.3.1 5.3 XIA,H.,XIA,Z.&ZHU,L.2004Verticalvariationinradiationstressandwave-inducedcurrent.CoastalEngineering51,309. 1.2 1 4.3.2

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75 XIE,L.,WU,K.,PIETRAFESA,L.&ZHANG,C.2001Anumericalstudyofwave-currentinteractionthroughsurfaceandbottomstresses:WinddrivencirculationintheSouthAtlanticBightunderuniformwinds.JournalofGeophysicalResearch106C8,16841. 1.2

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BIOGRAPHICALSKETCHIwasbornandraisedinFortMyers,Florida.AsachildmostweekendsconsistedofsailingaroundSanCarlosBayandPineIslandSoundonmyfather'sboat,oftentimesstoppingoffatlocalislandsandbeachestoplayandrelax.Occasionallymyparentswouldtakemysisterandmeonlongertrips:weekendjourneystoNaplesorMarcoIslandandevenweek-longtripstotheDryTortugas.IloggedmyrstlongdistancecruisewhenIwasjustafewmonthsold,thusbeginningmyfascinationwithoceans,islands,water,andbeaches.ObtainingmySCUBAcerticationattheageof13allowedmetolearnaboutandexperiencewhatlifeislikebelowthesurfaceofthewater.Upuntilthispoint,mostofmytimewasspentsailingonthesurfaceofthewaterwithverylittleknowledgeofwhatwashappeningbelowme.SailingandscubadivingaffordedmemanyopportunitiestolearnaboutweatherandoceanprocessesandevenfromanearlyageIknewthatIwantedtospendmylifelearningmoreaboutboth.Inthesummerof1996,havingjustturned17,Iwasgiventheopportunitytodosomepart-timeworkforaconsultingrmthatspecializedincoastalandoceanographicengineering.Iwasexposedtoanumberofdifferentprojectsthatsummerandwasimmediatelyfascinatedbythework.Attheendofthesummer,IwasquitesurethatIhadfoundmypassion.ItransferedtoBishopVerotCatholicHighSchoolinAugust1994afterleavingthepublicschoolsystemattheendofninth-grade.Itwasdifcultchangingschools,butitwascertainlyoneofthebestdecisionsImadeasateenager.Smallclasssizesandattentive,capableteachersmadethelearningexperienceintheclassroomthatmuchmoreenjoyable.Whilethepublicschoolsystemprovidedmewithastrongeducationalfoundationinscience,mathematics,andlanguagearts,theteachersat 76

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77 BishopVerotencouragedbothcriticalandcreativethinking...somethingIfoundmuchmorestimulating.AftergraduatingfromBishopVerotin1997,IwentontostudycivilengineeringattheUniversityofFlorida.Ifoundthefacultytobesupportiveandmanyencouragedmetocontinuemystudiesafterobtainingabachelor'sdegree.Sincecoastalengineeringwasmytrueinterest,itseemedappropriatetoobtainaspecializationinthisareabycontinuingontograduateschool.AlthoughIappliedtomanydifferentprograms,aftergraduatingwithaBachelorofScienceinCivilEngineeringinDecember,2001,IreceivedawonderfuloffertocontinuestudyingattheUniversityofFloridaunderDonSlinn.ThedecisiontoremaininGainesvillewasmadeeveneasierbythefactthatmygirlfriendwasworkingtowardobtainingamaster'sdegreeineducation.Soonafterbeginninggraduateschool,IproposedtoShannonandweweremarriedthenextyear.Mywifemadegraduateschoolmuchmoretolerableandwasalwaysthevoiceofencouragementattheendofafrustratingday.Graduateschoolhasbeen,forthemostpart,awonderfulexperience.InAprilof2002IwasawardedastipendfromtheAssociationofWesternUniversitiestoperformresearchattheNavalResearchLaboratoryatStennisSpaceCenterforatwelve-weekperiod.Duringthattime,IhadthechancetoassistinalaboratoryexperimentattheU.S.ArmyCorp'sWaterwayExperimentStationlocatedinVicksburg,Mississippi,whereIlearnedaboutparticleimagevelocimetryPIVmeasurementtechniquesanddatacollecting.Ayearlater,inOctober2003,IassistedscientistsfromtheNavalResearchLabwiththeNCEXNearshoreCanyonExperimenteldexperimentinLaJolla,California.Theopportunitytoassistinlaboratoryandeldexperiments,combinedwithtraditionallearningintheclassroom,hasenrichedmyeducationandhasallowedmetoapplytheoreticalsciencetoexplainrealandobservedprocesses.


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MODEL PREDICTIONS OF RADIATION STRESS PROFILES FOR NONLINEAR
SHOALING WAVES















By

BRET M. WEBB


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA


2004















This work is dedicated to my wife, Shannon.















ACKNOWLEDGMENTS

First, I wish to thank my wife for her support and encouragement over the past

few years. My parents, the rest of my immediate family, my extended family and circle

of friends also deserve much credit for the person I have become, both socially and

academically.

I would like to thank my advisor, Don Slinn, for his encouragement, ideas, and

support. Drs. Robert G. Dean and Andrew Kennedy, of the University of Florida,

deserve recognition for their service as members of my supervisory committee. Thanks

go, also, to the remainder of the faculty and staff in the Department of Civil and

Coastal Engineering. My office-mates, all of whom provided encouragement and

guidance over the past few years, were instrumental in my success as a graduate

student and they should be recognized for their contributions as well.

Todd Holland and Jack Puleo deserve acknowledgment for their support and

encouragement, and for financial contributions that permitted me to attend various

laboratory and field experiments over the past few years. Thanks should also be

expressed to the Office of Naval Research and the University of Florida for providing

financial support. A portion of this work, as well as my education, was made possible

by the University of Florida Graduate Fellowship Initiative.
















TABLE OF CONTENTS


page


ACKNOWLEDGMENTS ....

LIST OF TABLES ...

LIST OF FIGURES ....


A B STR A C T . . . . . . . . . .


1 INTRODUCTION ....


Background.....
Literature Survey .
Organization .....


2 METHODOLOGY ....


2.1 M odel Characteristics . . . . . . .
2.2 Governing Equations .. .......................
2.3 Improved Boundary Conditions .. .................
2.3.1 W ave Forcing . . . . . . .
2.3.2 Free Surface Velocity Boundary Conditions . . .
2.3.3 Rigid Bottom Velocity Boundary Conditions .. ........

3 EXPERIM EN TS .. . ... .. ... ... ... ... .. ... ... ..


Visser Experiment ..............
Model Formulation .............
M odel Simulations .............
3.3.1 Numerical Convergence .......
3.3.2 Computational Cost .........
3.3.3 Compensating for Mass Flux ....
3.3.4 Nonlinearity of Model Predictions .


. .. 1 8
. .. 19
. .. 2 1
. .. 2 1
. .. 2 2
. ... . 2 2
... . 24


4 RESULTS


4.1 Model-Data Comparison . . . . . . 26
4.1.1 Wave Transformation . . . . . 26
4.1.2 Longshore Current Formation . . . . 30
4.2 Three-Dimensional Flow Features . . . . . 32
4.2.1 Free Surface Visualizations . . . . . 32
4.2.2 Depth-Dependent Wave-Current Interaction . ..... 33











4.2.3 Longshore Current Variability . .
4.3 Vertical Distributions of Velocity and Momentum .
4.3.1 Time-Averaged Velocity Profiles . ...
4.3.2 Vertical Distribution of Radiation Stress .
4.3.3 Stress Gradients and Nearshore Forcing .


. . 37
. . 40
. . 40
. . 44
. . 55


5 SU M M ARY . . . . . . . . .


A applications . . . . .
Sensitivity . . . . . .
D discoveries . . . . .


. .. 6 1
. .. 6 1
. .. 62


APPENDIX


A THETA DIFFERENCING .......

B CROSS-SHORE MASS BALANCE . . .

REFERENCES ............

BIOGRAPHICAL SKETCH .. ...............


. .. 6 8















LIST OF TABLES
Table page

2-1 Breaking wave types classified by the inshore surf similarity parameter 12

3-1 Beach and wave parameters used in Visser's Experiment 4 . . 19

3-2 Summary of simulations performed to investigate numerical convergence
and computational cost . . . . . . 22















LIST OF FIGURES
Figure page

2-1 A typical computational cell used in a staggered grid and the associated
coordinate axis system . . . . . . 8

2-2 Contour plots of the velocity fields at the forcing boundary . 14

2-3 Special cases for setting the velocity components on the free surface 16

2-4 Prescriptions for setting the tangential velocity components around a
step . . . . . . . . . 17

3-1 Physical domain used in the simulation of Visser's Experiment 4 . 20

3-2 Predicted root-mean-square wave heights for five different grid resolutions 21

3-3 Computational time required for various grid resolutions . . 23

3-4 The response of the fluid surface, H, to mass flux near the forcing
boundary . . . . . . . . 24

3-5 Contrasting velocity time-series plots taken at offshore and inshore
locations . . . . . . . . 25

4-1 Comparison of measured and predicted wave heights . . 27

4-2 The instantaneous free surface and wave steepness . ..... 28

4-3 Statistical properties of the wave field at various time levels . 29

4-4 The predicted and measured longshore current velocities . . 31

4-5 Time evolution of the average longshore current . . . 33

4-6 Average longshore current velocities over different sill depths . 34

4-7 Three-dimensional visualizations of the instantaneous free surface 35

4-8 Depth-dependent wave-current interactions in the cross-shore . 36

4-9 Depth-averaged (u, ) velocity fields . . . . 38

4-10 Color contour plots of the depth-averaged longshore velocity . 39

4-11 Time-averaged velocity profiles . . . . . 42









4-12 The depth-averaged cross-shore velocity . . . . 43

4-13 A comparison of the shape and magnitude of various components of sx 46

4-14 Predicted profiles of time-averaged radiation stress sx . . 50

4-15 Predicted profiles of time-averaged radiation stress s. . . 52

4-16 The depth-integrated magnitudes of the predicted radiation stresses 53

4-17 The ratio of momentum flux over the vertical . . . 55

4-18 The vertical distribution of radiation and shear stress gradients . 57

4-19 Depth-integrated values of the nearshore forcing components . 60

A-1 Effect on numerical diffusion on model predictions . ..... 67

B-1 The time-mean velocity field taken at z = 2.5 m ... . ...... 69

B-2 Spatial features of the average velocity and free surface fields . 70















Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

MODEL PREDICTIONS OF RADIATION STRESS PROFILES FOR NONLINEAR
SHOALING WAVES

By

Bret M. Webb

December 2004

Chair: Donald N. Slinn
Major Department: Civil and Coastal Engineering

The flux of momentum directed shoreward by an incident wave field, commonly

referred to as the radiation stress, plays a significant role in nearshore circulation and,

therefore, has a profound impact on the transport of pollutants, biota, and sediment

in nearshore systems. Having received much attention since the initial work of

Longuet-Higgins and Stewart in the early 1960's, use of the radiation stress concept

continues to be refined and evidence of its utility is widespread in literature pertaining

to coastal and ocean science. A number of investigations, both numerical and analytical

in nature, have used the concept of the radiation stress to derive appropriate forcing

mechanisms that initiate cross-shore and longshore circulation, but typically in a

depth-averaged sense due to a lack of information concerning the vertical distribution

of the wave stresses.

While depth-averaged nearshore circulation models are still widely used today,

advancements in technology have permitted the adaptation of three-dimensional (3-D)

modeling techniques to study flow properties of complex nearshore circulation systems.

It has been shown that the resulting circulation in these 3-D models is very sensitive









to the vertical distribution of the nearshore forcing, which have been implemented as

either depth-uniform or depth-linear distributions. Recently, analytical expressions

describing the vertical structure of radiation stress components have appeared in the

literature, typically based on linear theory, but do not fully describe the magnitude and

structure in the region bound by the trough and crest of nonlinear, propagating waves.

Utilizing a three-dimensional, nonlinear, numerical model that resolves the

time-dependent free surface, we present mean flow properties resulting from

a simulation of a laboratory experiment on uniform longshore currents. More

specifically, we provide information regarding the vertical distribution of radiation

stress components resulting from obliquely incident, nonlinear shoaling waves. Vertical

profiles of the radiation stress components predicted by the numerical model are

compared with published analytical solutions, expressions given by linear theory, and

observations from an investigation employing second-order cnoidal wave theory.















CHAPTER 1
INTRODUCTION

1.1 Background

The nearshore coastal region, typically taken as the area between the instantaneous

shoreline and a point just seaward of wave breaking, is a dynamic and complex system.

As waves propagate into this area they release their energy into the water, forming

currents that can persist in both the cross-shore and longshore directions. These

currents, along with the coupling that takes place between the incident wave field and

the currents, make up the nearshore circulation. Nearshore circulation continues to

be studied and investigated as its importance is made evident through the transport of

materials, organic and non-organic, in the coastal zone. Of paramount concern to the

coastal scientist is the transport of sediment in the cross-shore and longshore directions,

which continually acts to shape and reshape the nearshore seabed.

The study of nearshore circulation has been approached many different ways.

Numerous laboratory and field experiments have been, and continue to be, performed

in order to increase understanding of nearshore current dynamics. Analytical studies

have paved the way for numerical simulations of nearshore processes and advances

in computational science promote the development of comprehensive modeling

tools. While two-dimensional (2-D) numerical models continue to be used for

simulating nearshore circulations, several limitations are inherent in their application.

Longshore-averaged 2-D models ignore the longshore variability of the wave climate

and current systems, while depth-integrated 2-D models neglect the vertical structure of

nearshore currents. Neither approach can provide complete information about sediment

transport in the nearshore system: the first excludes the longshore degree of freedom

while the latter ignores the vertical-dependence of the flow and sediment suspension









cannot be accurately described by depth-integrated quantities. The required coupling

between separate shoaling wave models and phase-averaged circulation models

becomes more complex in three dimensions. This coupling is guided by linear theory

and empirically based approximations developed under idealized conditions. A standard

approach is to solve a depth-integrated wave energy equation incorporating empirical

formulations for wave energy dissipation. The transfer of momentum between the

breaking waves and the mean currents is represented by radiation stress gradient

terms. Distributing these forces appropriately over the water column represented in a

phase-averaged, 3-D circulation model requires knowledge of their vertical distribution.

The introduction of three-dimensional (3-D) wave-phase-averaged circulation models,

however, has not provided the modeling panacea hoped for. The additional dimension

demands enormous computational power and more comprehensive forcing techniques

that require, in many cases, a priori knowledge of the flow field in all dimensions.

Moreover, the circulation predicted by these models has been shown to be highly

sensitive to the vertical distribution of forcing.

The term "radiation stress" describes the flow of momentum in propagating

waves and is discussed at length in Longuet-Higgins and Stewart (1964). Incident

waves provide the majority of momentum to the nearshore circulation. Gradients in

the radiation stress fields represent the forcing applied to the surfzone. The vertical

distribution of radiation stress over the water column affects the cross-shore and

longshore circulation, as well as vertical mixing. Thus, choosing appropriate vertical

distributions for the radiation stress terms is essential for accurate modeling of

nearshore circulation.

Using a three-dimensional, finite-difference, Navier-Stokes model that resolves

the time-dependent free surface, we perform simulations of a laboratory experiment

of longshore currents. Mean flow properties of the simulation are then used to

plot vertical profiles of the shoreward- and longshore-directed components of the









cross-shore radiation stress resulting from obliquely incident, nonlinear shoaling

waves. Profiles of radiation stress are computed using a modified form of the equations

suggested by Longuet-Higgins and Stewart (1964) and compared against profiles

plotted using linear solutions presented in the literature. While we understand that this

approach is neither the best nor the only way to model wave-current interactions, we

hope to convey our findings in a manner that is helpful to other coastal scientists

interested in these processes, especially those involved with the research and

development of modeling tools.

1.2 Literature Survey

Apparent stress terms, which appear as velocity products in the convective

acceleration terms found in the Navier-Stokes equations, and the radiation stress terms

explicit in the wave energy equation, play significant roles in the resulting nearshore

circulation in numerical models. Numerous theoretical, numerical, laboratory, and

field studies have attempted to quantify or judge the relative importance of the normal

and shear stresses resulting from shoaling and breaking waves. Recent attention has

been given to the vertical distribution of these apparent stresses [see Deigaard and

Fredsoe (1989); Sobey and Thieke (1989); and Rivero and Arcilla (1995)] and, more

specifically, the radiation stresses that result from the incident wave field [Mellor

(2003) and Xia et al. (2004)].

The pioneering work of Longuet-Higgins and Stewart (1964) is evident in most

nearshore investigations and their concepts continue to be used and developed further.

Unfortunately, this theoretical investigation relied heavily on the linear approximation

of a propagating water wave and excluded contributions from the incident waves

above the mean water level, citing the insignificance of the third-order integrands that

result from extending the solution above the surface. Disregarding the contribution

of the shoreward-directed momentum flux above the mean water level, according

to Svendsen (1984) and Sobey and Thieke (1989), significantly underestimates the









magnitude of the radiation stress due to real, nonlinear propagating water waves.

Also, the resulting analytical expressions presented in Longuet-Higgins and Stewart

(1964) are depth-integrated quantities and, therefore, ignore potentially important

vertical information. Contrary to the methodology followed by Xie et al. (2001),

these depth-integrated quantities are not suitable for use as forcing terms in vertically

dependent momentum equations. Traditional radiation stress expressions, however, have

been used in both numerical and theoretical calculations, in depth-integrated form, to

describe the wave-induced setup and setdown, as well as nearshore currents, and the

results agree qualitatively with observations made by Badiei and Kamphuis (1995),

Bowen et al. (1968), and Larson and Kraus (1991).

Recent attempts to provide three-dimensional radiation stress solutions have been

made by Dolata and Rosenthal (1984), Mellor (2003), and Xia et al. (2004), but each

has its limitations. Dolata and Rosenthal (1984) neglected pressure effects in their

analytical solution, which, as we will show later, represent a significant component of

the shoreward-directed component of the cross-shore radiation stress. The analytical

solutions given by Mellor (2003) also employ linear wave theory and specifically

address deep water propagating waves. In the context of nearshore circulation, where

waves are often characterized as shallow water waves, these solutions do not appear

to be appropriate. Xia et al. (2004) begin with the depth-integrated equations for

radiation stress presented by Longuet-Higgins and Stewart (1964), disregard the vertical

integration of the terms, and then substitute linear expressions for the velocity and

pressure terms into the modified equations.1 The resulting analytical solutions given

by Xia et al. (2004) are ultimately limited by their small-amplitude assumption-that



1 While the solutions of Xia et al. (2004) were considered in this investigation, the
resulting profiles computed with their expressions are not presented in our results
because they were similar to the profiles computed with the expressions of Mellor
(2003).









is, the ratio of wave amplitude to wavelength is much less than unity. In the shallow

waters of the nearshore system, it is not uncommon for this assumption to be violated

as the wave amplitude increases through shoaling and the wavelength decreases to

satisfy the linear dispersion relationship. When small-amplitude assumptions are

employed in this manner, analytical solutions based on them begin to differ from

observations and errors are inherent in subsequent calculations (Stive and Wind, 1982).

Finite-amplitude and cnoidal wave theories provide better estimates of wave velocities

in shallow waters (Xia et al., 2004). The vertical distributions of apparent stress

terms (velocity products), explicit in the classical radiation stress expressions given

by Longuet-Higgins and Stewart (1964), resulting from an investigation employing

second-order cnoidal wave theory were presented in Sobey and Thieke (1989).

Although they did not provide three-dimensional radiation stress solutions, comments

and observations regarding their investigation prove useful to our study of the Eulerian

mean flow and the resulting vertical profiles.

1.3 Organization

In the chapters that follow, we provide information about the vertical distributions

of velocity and momentum, and the net effects from obliquely incident, monochromatic

waves on a Eulerian mean flow. Characteristics, abilities, and limitations of the

numerical model used in this study are presented in Chapter 2; the governing equations

and improved boundary conditions for forcing and free surface velocities are discussed

here as well. In Chapter 3, we provide a brief summary of the laboratory experiments

[Visser (1984), Visser (1991)] used to validate model predictions, explain how the

model was adapted to simulate the experiment, and discuss physical and numerical

results of performing such a simulation with the model. The results of the model-data

comparison and subsequent simulations are provided in Chapter 4, along with analysis

of the model predictions. This analysis ultimately leads to discussion of the vertical

distribution of radiation stress, also presented in Chapter 4, where we compare







6

predicted profiles from the nonlinear model to the distributions suggested by analytical

solutions found in the literature. Concluding remarks on the capability of the numerical

model to reproduce nearshore processes and a summary of findings on the vertical

distribution of radiation stress for nonlinear shoaling waves are given in Chapter 5.

Details concerning the theta-weighted, finite-differencing schemes and the sensitivity

of model predictions to the differencing schemes used are presented in Appendix A. A

discussion on the cross-shore mass balance is found in Appendix B where we provide

additional information to support our claim that the numerical model is conservative.















CHAPTER 2
METHODOLOGY

2.1 Model Characteristics

The numerical model considered, SOLA-SURF, is a three-dimensional model

that employs computational fluid dynamics (CFD) to describe unsteady fluid flows.

SOLA-SURF is an extension of a two-dimensional CFD model, SOLA, that was

created to study time-dependent fluid flows in confined regions. Alternate extensions

of the original SOLA code have been used to study processes such as buoyancy-driven

flows, flows of stratified fluids, and flows in porous media (Hirt et al., 1975). In

contrast to the original model, SOLA-SURF has the ability to model fluid flows bound

by free or curved rigid surfaces. The addition of these surface boundary conditions

permits the user to simulate water wave propagation over variable bathymetry. Various

lateral boundary conditions may be used in the model: possible configurations include

free-slip and no-slip walls, continuative or outflow boundaries, and periodic boundary

conditions. The discretised equations of motion may be solved on either axisymmetric

or Cartesian coordinate systems. There are no physical, empirical coefficients to tune

and few numerical coefficients to define in this model, making the code adaptable to a

broad range of applications.

SOLA-SURF is based on the Marker-and-Cell (MAC) method where the primary

dependent variables, pressure and velocity, are solved in a Eulerian reference frame.

Characteristic of MAC schemes, a staggered grid is used to define the locations of the

pressure and the three components of velocity. On this grid, the pressure is defined

at the center of each control volume while the components of velocity are located on

cell faces. A typical grid cell used in MAC schemes is shown in Fig. 2-1, where the

orientation of the coordinate axes is representative of the system used in SOLA-SURF.











y
Vj,k






Pi.j, k''








Figure 2-1: A typical computational cell used in a staggered grid and the associated
coordinate axis system.


A Cartesian coordinate system (x, y, z) represents the cross-shore, vertical, and

longshore directions, respectively, and simulations are performed on a rectangular grid.



2.2 Governing Equations

SOLA-SURF solves the Navier-Stokes equations (Eqs. 2-2, 2-3, and 2-4) in

discretised form and satisfies the mass continuity equation (Eq. 2-1) through the

incorporation of a Poisson equation for the pressure field. In the context of this work,

the equations of motion characterize an unsteady, incompressible, homogeneous fluid.

Our Cartesian coordinate system associates the x, y, and z directions with the u, v, and

w velocity components, respectively.

au av aw
= + 0 (2-1)
ax y + az


au au2 auv auw a9 [2U 02U 02U
+ a+ a + + v [-+ + (2-2)
at ax ay z r x 9x2 9y2 9z2











+ + + + + + +g (2-3)
at ax ay az ay ox2 ay2 az2


aw auw avw aw2 ap a2W 2 W 2w
at + a + a + + v i (2-4)
at ax ay 6z 6z 6x2 6y2 6z2
where
g = gravitational acceleration
v = coefficient of kinematic viscosity

Following from the assumption that the fluid is incompressible and homogeneous,

pressure in the model is defined as the ratio of pressure to constant density. Simulations

are performed with a rigid bottom boundary that best represents the bathymetry of our

problem. The initial pressure field for a fluid at rest is hydrostatic, which we use

to initialize simulations starting from an equilibrium state where no motion exists.

Once the simulations begin, the model uses the two-step projection method of Chorin

(1968) to solve for the pressure field. The intermediate step velocity projection fields

are found by substituting the initial pressure and velocity fields into the discretised

equations of motion that utilize a theta scheme to control the amount of donor cell

differencing for the convective flux terms. The viscous flux terms are discretised using

a three-point stencil that yields second-order central differences. Boundary conditions

are then applied to the velocity field and cell pressures are adjusted iteratively in order

to satisfy the mass continuity equation (Eq. 2-1). If the divergence of the velocity field

in a cell (the left hand side of Eq. 2-1) is less than zero, the pressure of that cell is

increased to decrease the flow of mass into the cell; the converse of this statement is

also true and the cell pressure is decreased to encourage the flow of mass back into

a cell when the divergence of the cell is greater than zero. Since the MAC scheme

uses only one point to approximate the pressure of each cell, the divergence of the

velocity field may be driven to zero, or to a desired level of accuracy, in an iterative

manner. Typical values of the convergence criteria, c, are on the order of 10-3 or









smaller (Hirt et al., 1975). In order to decrease the number of iterations required to

satisfy c, an over-relaxation factor, Q, is applied to the predicted pressure differential.

Finding an optimal value of the over-relaxation factor requires, in part, performing a

rather exhaustive eigenvalue expansion of the system of equations-a task that we do

not seek to undertake (Fletcher, 2000). Therefore, following guidance provided by the

SOLA manual we take the over-relaxation factor to be 1.8, which is still well below the

stability threshold of 2.0. While successive over-relaxation (SOR) methods typically

decrease the number of iterations required to reach some desired level of convergence,

they can become computationally expensive for increasing numbers of grid cells.

Additional information regarding the effect of SOR on computational time is presented

in Chapter 3.

Contrary to some MAC formulations, SOLA-SURF does not employ marking

particles to track the free surface. Instead, the location of the free surface is predicted

by substituting velocity predictions provided by the momentum equations into the

kinematic free surface boundary condition. Once the continuity equation has been

satisfied, the resulting velocity field is then used in conjunction with the kinematic free

surface boundary condition (KFSBC) to determine the time-dependent free surface,

providing unique three-dimensional representations of the fluid surface throughout a

simulation. The KFSBC is given by Eq. 2-5, where the free surface is defined as the

height, (, of the surface above the rigid bottom boundary.


at ax az

The discretised form of the KFSBC employs an alternative theta scheme

(independent of that used in the convective acceleration terms) to control the amount

of donor cell differencing, and uses a one-step projection method to predict the free

surface location at the newest time-level. In the context of our simulations, we have

found it advisable to use second-order central differences in the spatial discretisation of









Eq. 2-5. Additional information regarding the predictive skill of various discretisation

schemes, as well as examples of the discretised forms of a convective acceleration term

and the KFSBC are presented in Appendix A.

The free surface in SOLA-SURF must be definable by a single point in both

horizontal directions. This constraint arises from the simplified approach used to solve

the discretised equations of motion, the KFSBC, and also the cell pressures. Water

waves exhibiting this type of surface feature are often classified as spilling breakers

at the limit of breaking (Dean and Dalrymple, 1991) as the slope of the fluid surface

never reaches infinity (vertical). Since SOLA-SURF does not allow the slope of the

free surface to exceed the cell aspect ratio, either 6y/6x or 6y/6z, the model is unable

to resolve plunging breakers or other complex, multi-valued free surfaces. The breaker

type is often classified by the surf similarity parameter (i), first used by Iribarren and

Nogales (1949), and is defined in Eq. 2-6.

tan c
(2-6)


where
a = slope of bathymetry
H = wave height
L, = deep-water wavelength

For waves traveling at oblique angles, the surf similarity parameter is changed slightly,

yielding the inshore surf similarity parameter--br-of Battjes (1974) shown in

Eq. 2-7.
tan c
,br = n COS Obr (2-7)


where
Obr = wave angle at breaking
Hbr = wave height at breaking

The classification of breaking wave types, also presented by Battjes (1974), is given in

Table 2-1.







12

Table 2-1: Breaking wave types classified by the inshore surf similarity parameter.

surging or collapsing if Ibr > 2.0
plunging if 0.4 < br < 2.0
spilling if


2.3 Improved Boundary Conditions

2.3.1 Wave Forcing

A new forcing method was added to SOLA-SURF and is applied as a time-dependent

offshore boundary condition. This method specifies both the potential and kinetic

energy of a monochromatic wave at the offshore boundary. This implies that both the

free surface and velocity are forced at the boundary. The equations for the free surface

(Eq. 2-8) and the three-dimensional velocity potential (Eq. 2-9) were suggested by

Boccotti (2000).

H
l(x, z, t) = cos (kz sin 0 + kx cos 0 ut) (2-8)
2



((x, yz,, t) = _- oh k(h + )sin (kz sin 0 + kx cos 0 wt) (2-9)
2 cosh kh

where
H = wave height
k = wavenumber
h = water depth
0 = wave angle with respect to shore normal
a = angular frequency
t = time

Assuming that the model describes irrotational incident waves, a velocity potential

for the fluid exists and the total fluid velocity is described by Eq. 2-10.


u= VO (2-10)


Combining this with the assumption that the fluid is incompressible (Eq. 2-11)


V7'= 0 0


(2-11)









enforces the Laplacian of the velocity potential to be equal to zero. This constraint is

found by substituting Eq. 2-10 into Eq. 2-11, which gives Eq. 2-12.


V2 = 0 (2-12)


The three components of velocity used to force the wave signal are derived from

the velocity potential using Eq. 2-10 and are given by


u v -, v and w -
Ox' Oy' dOz

and the resulting forcing equations for the velocity components are given by Eqs. 2-13,

2-14, and 2-15.


u(x, y, z,t) = gkuwl-co (h+ cos (kzsin 0 + kxcos 0 t) cos (2-13)
2 cosh kh



H _sinhk(h + y)
v(x,y,z,t) g kw sin (ks sin O + kxcos O t) (2-14)
2 cosh kh




w(x,y, z,t) = gku-w c+ k os 0 ut) sin (2-15)
2 cosh kh

Graphical representations of the velocity fields are presented in Fig. 2-2.

SOLA-SURF requires only basic wave parameters to force the monochromatic

wave signal. The wave amplitude, period, and direction must be specified along with

the water depth. Using this information the model computes the values of Tr, u, v,

and w using the preceding expressions. These values are specified in the first grid

cell, for each time step, and the equations of motion then govern the propagation

of the wave throughout the domain. The length of the cross-shore domain and the

boundary condition used near the onshore boundary result in minimal reflected waves

traveling back offshore. This allows the incident waves to retain a consistent shape and








14


A) 05 u0(m.2
04 0.16

0.04
"0
+ -0.04
-0.08
u -0.12
-0.16
1 2 3 4 5 -0.2
Longshore (m)
B) o05 v(m/s)
5 0.1
0.08
0.06
01 _-0.04
0.02
2: -0.
-0.02

-0.08
2 3 4 5 -0.1
Longshore (m)
C o (ms)
0.05
0.03
E 002
S0.01
"0
S -0.01
-0.02
-0.04
-0.05
Longshore (m)


Figure 2-2: Contour plots of the velocity fields at the forcing boundary derived from
the three-dimensional velocity potential equation for an incident angle of 0 = 15.40.
The velocity contours of A) u, B) v, and C) w demonstrate the depth-dependency of
the linear equations.



magnitude throughout a simulation, thereby eliminating the need to tune or adjust the

time-dependent boundary conditions to allow for outgoing wave characteristics.

2.3.2 Free Surface Velocity Boundary Conditions

The velocity boundary conditions for the free surface must be treated differently

than the lateral boundaries in the model and require special attention. A modified

free-slip condition is specified on the free surface,


9Bu 9w 9v fau aw\

0y 0 y (y (\x z/


which prohibits shear in the velocity field across the fluid interface and solves for

the vertical component of velocity (v) in a manner that explicitly satisfies the mass

continuity equation (Eq. 2-1). The boundary conditions applied to the horizontal









velocity components near the free surface, using ghost points, are discretised as


Ui,jt,k Uijt-I,k and "' it,k = "' it-,k

where the index notation jt is used to represent the uppermost grid cell containing

the free surface. While appropriate for a mildly sloping free surface, these boundary

conditions must be altered for the limiting case of maximum steepness: when the

slope of the free surface approaches the cell aspect ratio. When the slope of the free

surface approaches this limit, the horizontal velocity components in the cell containing

the free surface are set using a method that produces vertical momentum transfer [see

Chen et al. (1995)]. Demonstrated in panel B of Fig. 2-3, the improved methods for

prescribing the horizontal velocity boundary conditions are given by


i,jt,k Ui-l,jt,k and i' it,k = "' it,k-1.

This improved method, suggested by Chen et al. (1995), better represents a free-slip

condition when the slope of the free surface nears the limit of maximum steepness.

While only the u velocity is depicted in Fig. 2-3, a similar prescription is applied to

the orthogonal horizontal velocity component w.

Where the original velocity boundary conditions satisfy the free-slip condition

for a mildly sloping surface, it is observed that by neglecting vertical gradients of the

horizontal velocity components

9u aw
= 0 and = 0
6y 6y

in the case of maximum steepness, there is little momentum transfer from the spilling

wave into the water column. Since SOLA-SURF does not use a subgrid model to

simulate wave breaking, and in the absence of a parameterization for energy dissipation

due to wave breaking, the improved velocity boundary conditions provide a sufficient

mechanism to promote momentum transfer from the wave to the mean flow. The












A) B)

S Vi,jt,k Vi,jt,k


u i,jt-,k
Uijt -,k


\

iU l.. i,jt,k
Uj ~ U


Figure 2-3: Special cases for setting the velocity components on the free surface,
represented here by the dashed line. A) is the original method employed by the model
and B) is the improved method.


effects of these velocity boundary conditions on the generation of longshore currents

and wave transformation are discussed in Chapter 4.

2.3.3 Rigid Bottom Velocity Boundary Conditions

A no-slip boundary condition is applied to the rigid bottom boundary, requiring

that the horizontal velocity components equal zero at the bed. The velocity component

normal to the bed is also set equal to zero since flow is not permitted to cross the

rigid bottom boundary. These conditions, prescribed using ghost points outside of the

computational domain, are given by


Ui,jb-l,k U : .1 1 it' ,,-1,k = -"' .',k, and I' .-1,k = 0


where jb is the index of the vertical grid cell containing the bottom boundary. Using

these prescriptions for the horizontal velocity components forces their value to be zero

at the bed.

For simulations performed on a rectangular grid, a sloping boundary is represented

by a series of small steps. Additional information about the behavior of horizontal










velocity components around these steps must be supplied to the model. Fig. 2-4 is

a schematic of a single step in the rigid bottom boundary. The velocity boundary

conditions used around a step are


u ,, 0 and Uii,jbl,k =


which states that flow is not permitted to cross the bottom boundary. Similar

prescriptions are used for the w velocity component. For bathymetry that does not

vary in the longshore (z) direction, only the no-slip condition on the boundary must be

specified.


Vi,jb-l,k
Uijb,k


Ui,jb-1,k


Figure 2-4: Prescriptions for setting the tangential velocity components around a step.
The dark black line represents the rigid bottom boundary.















CHAPTER 3
EXPERIMENTS

In order to determine the ability of SOLA-SURF to predict nearshore processes

such as wave transformation and the generation of longshore currents, simulations of a

laboratory experiment were conducted. A series of laboratory experiments conducted

by Visser (1984) were performed in a range of wave and basin parameters suitable for

simulating with our model. In particular, we chose to simulate his Experiment 4 with

SOLA-SURF for its unique set of parameters and for the accompanying comprehensive

data set presented in Visser (1991).

3.1 Visser Experiment

The purpose of Visser's laboratory experiment was twofold: first, he sought to

develop a method for generating uniform longshore currents in a laboratory setting

and second, to provide scientists with a large set of data characterizing longshore

currents for approximately longshore-uniform conditions. Visser evaluated a number of

wave basin configurations before electing to use a basin with a pumped recirculation

system. This recirculation system provided a stabilizing mechanism for the longshore

uniformity of the currents and care was taken to determine appropriate pumping rates.

Detailed information regarding the wave basin and the recirculation system may be

found in Visser (1984) and Visser (1991).

Numerous experiments were conducted using a variety of wave parameters,

two different beach slopes, and two distinct beach surfaces. One experiment in

particular-Experiment 4-was performed with a set of parameters conducive to

performing simulations with SOLA-SURF. There is also a significant amount of data

presented in Visser (1991) that corresponds to this experiment. The parameters used

in Experiment 4 are presented in Table 3-1, where a is the beach slope angle, T is the







19

Table 3-1: Beach and wave parameters used in Visser's Experiment 4.

tan a T d 0 H
(s) (cm) (degr) (cm)
0.050 1.02 35.0 15.4 7.8


wave period, d is the still water depth at forcing, 0 is the wave angle at forcing, and H

is the forced wave height.

3.2 Model Formulation

The total dimensions of Visser's wave basin-34 m longshore by 16.6 m

cross-shore-were quite large in comparison to the wavelength and wave height

associated with the experiment. In order to reduce the size of the computational

domain used in the simulations of Visser's experiment, only the sloping part of the

basin is included in the model bathymetry and the longshore extent of the domain is

equivalent to the longshore wavelength of the forced waves. As discussed in Chapter 2,

waves are forced by applying time-dependent boundary conditions on the free surface

(Eq. 2-8) and the three velocity components (Eqs. 2-13-2-15). These wave forcing

boundary conditions are applied at the location x = 0 m in Fig. 3-1. In order to

simulate obliquely incident waves similar to those created in the lab, periodic boundary

conditions are used in the longshore (z) direction of the computational domain. These

periodic conditions, as compared to free or no-slip walls, prevent wave reflection

from the lateral boundaries and result in a consistent wave field. Enforcing periodic

longshore boundary conditions in SOLA-SURF requires the longshore extent of the

domain to equal the longshore component of the wavelength. The wavelength may

be found by solving the linear dispersion relationship for the wavenumber-k-in an

iterative fashion. The linear dispersion relationship is given in Eq. 3-1.


w2 gk tanh kh


(3-1)










In this case, k represents the wavenumber magnitude. For obliquely incident waves, the

wavenumber magnitude is the resultant of the cross-shore and longshore wavenumber

components, which are given by the following expressions


k, = k cos 0


kz = k sin ,


where 0 is the local angle of wave incidence measured from the shore-normal direction.

The parameters of the experiment, combined with the periodicity requirement,

result in typical physical domain lengths of about 10 m in the cross-shore, approximately

5.6 m in the longshore and 0.5 m in the vertical direction. A representative domain

is shown in Fig. 3-1. In order to accurately resolve wave parameters, grid cells were

chosen to be 0.01 m in the vertical and 0.04 m in the horizontal directions. These

length scales result in a computational domain containing nearly 1.75 million grid cells;

however, since the model does not compute values below the bottom boundary, about

one-half of those cells remain unused.







0.4
0.3
0.2
0.1


10 5
010




Figure 3-1: Physical domain used in the simulation of Visser's Experiment 4 showing
the bathymetry, still water level, and domain lengths.










3.3 Model Simulations

3.3.1 Numerical Convergence

A variety of grid resolutions were tested during simulations of Visser's experiment

to ensure that numerical convergence had been reached. The five different grid

resolutions (Table 3-2) were also used to determine the effect of the cell aspect ratio

on the prediction of cross-shore wave heights, which is demonstrated in Fig. 3-2.

Nearly all of the predicted root-mean-square (RMS) wave heights from various grid

resolutions fall within one standard deviation of the mean value. We chose to perform

simulations with a cell aspect ratio of 1:4, in order to enhance the predictive ability of

SOLA-SURF and to minimize the computational cost of running a simulation.


0.3 0.25 0.2 0.15 0.1
Cross-shore Depth (m)


0.05 0


Figure 3-2: Predicted root-mean-square wave heights for five different grid resolutions.
The dotted line represents the mean of the predictions at each cross-shore location and
the error bars signify one standard deviation from the mean.


E 1:2
A 1:3
1:4
F 1:5
S 1:8










l i
17









3.3.2 Computational Cost

As noted in Chapter 2, the SOR method applied to the pressure solver reduces

the number of iterations necessary to satisfy the convergence criterion, but increases

the total computational time as each iteration takes longer to perform. The simulations

performed to investigate the effect of grid resolution are presented in Table 3-2, along

with the number of hours required to complete one full, 200 s simulation of Visser's

experiment. A graphical representation of the data presented in Table 3-2 is shown

in Fig. 3-3, where the data has been plotted on a log-log plot to demonstrate the

relationship between the number of computational grid cells and the time required to

complete a simulation.

Table 3-2: Summary of simulations performed to investigate numerical convergence
and computational cost.

Cell Aspect Ratio Grid Cell Distribution Elapsed Time
(6y : 6x) (nx, ny, nz) (hrs)
1:8 125, 50, 70 67
1:5 200, 50, 100 133
1:4 250, 50, 140 233
1:3 265, 50, 185 400
1:2 400, 50, 280 2000


3.3.3 Compensating for Mass Flux

Finite amplitude water waves produce a mean transport of mass, or mass flux,

in the direction of propagation. This mass transport, in a Eulerian sense, stems from

the difference in fluid volume contained under the wave crest and wave trough. In the

context of our model, if the velocities at the forcing boundary were left in their original

form there would be an increase of mass in the computational domain. In order to

counteract this mass flux at the forcing boundary, the forcing velocities must be altered

in some cogent manner.

We allowed the model to run for a number of wave periods (usually ten)

without applying corrections to the forcing velocities and calculated their depth and













104

-I




) 101


Q.
U)


0 1:4




10
%I--


I-




2E+06 4E+06 6E+06
Number of Grid Cells


Figure 3-3: Computational time required for various grid resolutions. The diagram
shows the logarithmic increase in computational cost as a function of the number of
computational grid cells. The power-fit line is of the form y e(Alog(x)+B)


longshore-averaged values at a frequency of ten hertz. By then taking a time-average

of the velocity components over the ten wave periods, we find the excess velocity

due to mass transport. These excess velocity values are then subtracted from their

respective forcing velocity components, uniformly over depth and time, and the

simulation is started from rest and allowed to run to steady-state. The effects of mass

flux on the fluid surface as a function of time is shown in Fig. 3-4. The dotted red line

in this figure represents the fluid surface as a function of time for a simulation without

mass flux velocity corrections. Comparing the average trend of this series (Havg) to

the analytical solution provided by linear theory (Dean and Dalrymple, 1991), which is

given by the thin, dashed black line, yields an agreeable result. The resulting average

fluid surface after the velocity corrections have been applied is given by the dashed and













0.75
Linear Theory
0.7 H
Ha
Correction
0.65 -.- Corrected Hav

0.6

0.55
E
0.5
+
0.45

0.4

0.35 -------------

0.3

0.25
50 100 150 200
t (s)


Figure 3-4: The response of the fluid surface, H, to mass flux near the forcing
boundary, (x, z)=(0.01, 2.0) m. A time series of H for a simulation without velocity
corrections is represented by the dotted red line. The solid red line is the average
trend of the surface H(t), and the velocity corrections calculated predict a fluid-surface
rise given by the solid black line with circles. The dashed, thin black line is the
analytical solution given by linear theory and the dash-dot, dark black line represents
the corrected surface H(t).


dotted black line, which remains at a constant elevation of 0.35 m above the bottom

boundary.

3.3.4 Nonlinearity of Model Predictions

The time-dependent boundary conditions applied on the velocity field to

force the oblique, monochromatic wave signal were derived from a linear velocity

potential equation. Water surface displacements near the forcing boundary, therefore,

are sinusoidal in form and so are the velocity fields. The momentum equations

(Eqs. 2-2-2-4) that govern the propagation of the wave signal throughout the domain

are nonlinear equations and provide, accordingly, nonlinear developments to the









velocity field. These linear and nonlinear characteristics are demonstrated in panels B

and C of Fig. 3-5. The 30 s time-series of the velocity components was taken from

the full 200 s time-series recorded during the simulation at an inshore point in shallow

water and is denoted by the dark black box in panel A.


En
0
0


C)


Figure 3-5: Contrasting velocity time-series plots taken at offshore (0.2 m, 0.28 m,
2.0 m) and inshore (5.6 m, 0.3 m, 2.0 m) locations. A) shows the complete, 200 s
time-series of u, v, and w taken at the inshore point. B) is a 30 s time-series of the
three velocity components near the forcing boundary. C) shows the nonlinearity and
asymmetry of the velocity components over a 30 s time-series taken in 3 cm of water.


Inshore


A' j lI 1


A)


~iil:















CHAPTER 4
RESULTS

4.1 Model-Data Comparison

Performing simulations of Visser's Experiment 4 allows us to evaluate the

predictive skill of SOLA-SURF by comparing predicted nearshore processes to those

observed and measured in the laboratory. A significant amount of data representing

wave transformation and the formation of longshore currents is provided by Visser

(1991) and serve as benchmarks to assess the capabilities of our model.

4.1.1 Wave Transformation

The predicted free surface elevations, taken from a transect through the middle of

the longshore domain, are recorded at a frequency of 10 Hz throughout a simulation.

A time-series corresponding to approximately ten wave-periods is then taken from

the total record and analyzed to compute wave height statistics. The mean of the

time-series is calculated and subsequently removed from the data, resulting in positive

and negative oscillations about zero. A zero down-crossing technique is then used

to extract individual wave events from the record, thereby allowing us to calculate

statistical properties associated with the wave record. These statistical properties are

presented in Fig. 4-1, where we compare the predicted significant wave heights (Hs),

maximum wave heights (Hmax), and RMS wave heights (Hrms), to the wave heights

measured during the lab experiment.

Performing simulations with the original free-slip surface velocity boundary

conditions produced wave heights that were similar in magnitude to those measured

in the experiment. Simulations performed with the original free-slip boundary

conditions, however, produced longshore current velocities that were only about

10% of the expected values. As suggested in Chapter 2, the original velocity boundary



















E
0.06
.)
'-
I
4)
> 0.04
Q


0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
Cross-shore Depth (m)


Figure 4-1: Comparison of measured (red) and predicted wave heights as a function of
the cross-shore depth. The error bars represent possible measurement errors of 10%,
suggested in Visser (1991). The data points corresponding to the label Old Hax were
predicted by SOLA-SURF using the original free surface velocity boundary conditions.
Hmax, Hs, and H,,m represent the predicted maximum, significant, and root mean
square wave heights, respectively.


conditions neglected vertical gradients in the velocity field across the free surface,

thereby prohibiting momentum transfer from the steepening wave to the mean flow.

Simulations implementing the improved free surface velocity boundary conditions

(Chen et al., 1995) provide reasonable estimates of the longshore current velocities,

but under-predict the shoaling wave heights. Demonstrated in Fig. 4-1, the RMS wave

heights predicted by the model are smaller than the average values collected during

the experiment. The comparison shows reasonable agreement for the first few data

points, those in deeper water, and also for the last few data points, but demonstrates the

inability to accurately reproduce the shoaling wave heights observed in the experiment.

A possible explanation for the large differences between the measured and predicted












0.5



0.4



0.3
E

+
S0.2



0.1
[] h+i|
HU/L

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
Cross-shore Depth (m)


Figure 4-2: The instantaneous free surface and wave steepness. A transect taken near
the midpoint of the longshore domain shows the instantaneous free surface (h+Tr),
bathymetry (solid black line), and the approximate wave steepness (H7r/L) in the
cross-shore direction.


wave heights involves the steepening of the forced waves and the cell aspect ratio (1:4)

selected for the simulation.

When the sinusoidal wave is forced at the offshore boundary it has an approximate

steepness of 1:6, which is well below the cell aspect ratio of 1:4. As these waves shoal

they become nonlinear and the face of the wave steepens quickly and dramatically

within the first few meters of the cross-shore domain, as seen in Fig. 4-2. This

steepening presents a problem when the slope of the free surface reaches or nears

the cell aspect ratio, as the modified free surface velocity boundary conditions begin

to translate momentum down the face of the wave a bit sooner than necessary to

match the lab data. One result of this momentum transfer appears to be a reduction in

the predicted wave heights. We say that the steepness ratios, shown in Fig. 4-2, are










approximate because we assume that the waveform is sinusoidal when we calculate

the slope of the wave face. The modeled nonlinear waves exhibit surface slopes that

exceed those calculated by our sinusoidal approximation.

One particular advantage of employing SOLA-SURF to simulate nearshore

processes lies in its time-dependency. While steady-state wave models assume that

wave shoaling is a stationary process, time-dependent processes such as the generation

of a longshore current and undertow can affect wave transformation over time. Shown

in Fig. 4-3, the RMS wave heights predicted by SOLA-SURF do not remain constant

throughout the simulation. This suggests that the wave field responds continuously to

the developing undertow and longshore current. These RMS values were calculated

from a ten-wave average, centered about the simulation time shown in the legend.




0.1



0.08



.O 0.06 .-
I 9

S 0.04- o t = 10s
V) t =60S
2 t = 90s
v t = 120s
0.02 ~ t = 150s o
> t = 180s 6


0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
Cross-shore Depth (m)


Figure 4-3: Statistical properties of the wave field at various time levels suggest that
the wave shoaling in this experiment is not a stationary process.









4.1.2 Longshore Current Formation

The formation of the longshore current and the magnitude of the current velocities

is highly sensitive to the implementation of the free surface velocity boundary

conditions, as discussed above. Using the improved velocity boundary conditions

provides reasonable estimates of the longshore current velocities, but fails to accurately

predict the cross-shore wave heights. Our predicted current is calculated by first taking

the depth and longshore averages of the longshore velocity component (w) and then by

averaging these values over 30 wave-periods.1

As demonstrated by Fig. 4-4, the model approximately predicts the correct

magnitude of the maximum longshore current velocity, but the peak is shifted

shoreward of the location observed in the laboratory. This is perhaps due to

the absence of setup at the shoreline owing to our choice of a shallow sill as an

onshore boundary instead of an intersecting profile. Another feature of the predicted

longshore current that was not observed in the laboratory experiment is the additional

longshore-directed flow seaward of the peak velocity. This may be explained by

referring to the predicted wave heights shown in Fig. 4-1. Underestimating the RMS

wave height suggests an overestimation of the energy dissipation (Chen et al., 2003),

which manifests itself in the offshore region as additional longshore current velocity.

The laboratory wave heights, on the contrary, remain somewhat constant until they near

the point of breaking resulting in less longshore current velocity in the offshore region.

Apparently, other horizontal mixing processes in the lab experiment are not

sufficiently strong to spread the mean longshore current significantly offshore of

the breakpoint. In fact, the opportunity for longshore flow to persist in the offshore



1 Henceforth, we denote depth-averaged values with a single overbar (-), depth and
longshore-averaged values with two overbars (=), and time-averaged values with the
angle brackets (( )).












0.6
a Measured
Predicted
0.5


0.4


E, 0.3


0.2


0.1



2 4 6 8 10
Cross-shore (m)


Figure 4-4: The predicted and measured longshore current velocities. A comparison of
the predicted time-mean, depth and longshore-averaged, longshore current velocity to
the measured current velocity. The predicted current is calculated at each cross-shore
grid location and is represented here by the black line. The red line is a spline fit to
Visser's data collection points, which are shown by the red squares.


region may be suppressed by the wave-guiding walls located at both ends of Visser's

wave tank. These walls extend from above the mean water level to the bed, thus

restricting any mean, longshore-directed flow between them. In a series of experiments

of longshore currents on planar slopes, Galvin and Eagleson (1965) and Mizuguchi

and Horikawa (1978) measured significant longshore velocities (20 to 40% of the

maximum) seaward of the surfzone. These experiments were performed in similar

wave tanks that provided clearance between the bed and the bottom of the wave guide

and the results suggest that this configuration permits recirculation in the offshore

region, where it was suppressed in Visser's experiment.

We allowed simulations to run for 200 seconds of model time, by which point

the flow field had reached steady-state. In Fig. 4-5 we see the formation and growth









of the predicted average longshore current. Formation of the longshore current begins

well seaward of the location of the predicted peak velocity and spreads laterally across

the domain over time. After approximately 150 seconds of model time, the longshore

current has reached steady-state: the magnitude and shape of the current vary little

during the remainder of the simulation. Simulations were also run with two different

sill depths to ensure that its location did not determine where the peak longshore

velocity occurred. Sill depths of 1 cm and 2 cm were used in these simulations and the

resulting average longshore currents are shown in Fig. 4-6. Regardless of the depth

used, the maximum longshore current velocity remains in the same location, seaward

of the sill, where the mean water depth is approximately 3 cm. The longshore current

forms more slowly for the sill depth of 2 cm as compared to the depth of 1 cm, but

this outcome was expected as twice as much volume is permitted to cross the sill.

4.2 Three-Dimensional Flow Features

4.2.1 Free Surface Visualizations

Normally incident waves propagating over longshore-uniform bathymetry produce

circulation and flow features that are characteristically two-dimensional. Longshore

gradients in velocity fields and wave heights resulting from obliquely incident waves,

however, produce three-dimensional flow features in the nearshore region. One

such feature is demonstrated by the longshore non-uniformities in the waves as

they travel over a shearing current. Figure 4-7 contrasts the incident wave field

from our simulation at an early time, when the undertow and longshore currents are

undeveloped, and at a later time when both currents have reached steady-state. The

incident wave field in Fig. 4-7A demonstrates the shoaling and refraction of the waves

over the sloping bathymetry used in the laboratory experiment. The color contours

in Figs. 4-7A and 4-7B correspond to the location of the free surface in the vertical

domain, referenced to the still water level at y = 0.35 m. At this early time, when the

currents have yet to develop, we see that the wave crests are more or less parallel to
















0.5-
5 t = 60s
t = 90s
t = 120s
0.4 t = 150s
t = 180s
5- t = 200s
E 0.3


0.2


0.1


0
0 2 4 6 8 10
Cross-shore (m)


Figure 4-5: Time evolution of the average longshore current. Shown here are the depth
and longshore-averaged, longshore current velocities taken at various times throughout
the simulation. These profiles were averaged over ten wave-periods, centered around
the time labels shown in the legend, to remove the wave component of the longshore
velocity.


one another as they travel across the domain and that the periodic boundary conditions

perform properly. The contrasting wave field shown in Fig. 4-7B, taken when the

model has reached steady-state, shows the response of the wave field to the developed

undertow and longshore currents. The relatively undisturbed, straight wave crests seen

in Fig. 4-7A are now only found near the offshore boundary and those propagating

over the shearing currents have significant longshore non-uniformities. Clearly, the

incident waves begin to feel the mean currents for x > 3 m and produce spatially

dependent wave breaking patterns.

4.2.2 Depth-Dependent Wave-Current Interaction

The three-dimensional nature of SOLA-SURF is perhaps even more useful for

analyzing the depth-dependent circulations that develop in a complex flow field.












0.6

1cm: t=90s
0.5 2cm: t=90s
1cm:t=50s
2cm: t=50s
0.4 -


E o0.3


0.2


0.1



0 2 4 6 8 10
Cross-shore (m)


Figure 4-6: Average longshore current velocities over different sill depths. These
profiles were averaged over ten wave-periods, centered around the time labels shown in
the legend, to remove the wave component of the longshore velocity.


Vertical variations of the combined cross-shore and longshore current velocities were

investigated by Svendsen and Lorenz (1989) and were said to have significant effects

on the direction of sediment transport in the nearshore region. While many numerical

investigations regarding undertow and longshore current velocity profiles have made

use of depth-integrated or depth-averaged equations, the momentum equations used in

SOLA-SURF explicitly include depth-dependent terms. The depth-dependent nature of

the velocity field and cross-shore current is demonstrated by Fig. 4-8, where plots at

two different times compare the developing velocity field at an early time with the fully

developed field that occurs later in the simulation. The larger frames in Figs. 4-8A and

4-8B show the velocity field taken at every second cross-shore grid point, where the

velocity vector is the resultant of the u and v velocity components. The area enclosed

by the dark, black line in each frame corresponds to the area plotted in the smaller













A) It= 10.0s


0.4
0.3
0.2
0.1
0


"ss-sh (hrn) 8 10o0--



0.35 : + :
0 31 0315 0 32 0 325 0 33 0 335 0 34 0 345 0 35 0 355 0 36 0 365 0 37 0 375 0 38 0 385 0 39


B) t=150.0


0.4
0.3
0.2
0.1
0


Crosssho(nN) 8 10 0






Figure 4-7: Three-dimensional visualizations of the instantaneous free surface. The
contrasting wave fields show the effect of the shearing current on the incident wave
field, taken at A) t = 10 s and at B) t = 150 s.


inset frames, where the velocity vectors are plotted at every computational grid cell.

These instantaneous figures of the (x, y) plane are transects taken from the middle of

the longshore domain. Figure 4-8A shows the depth-dependent velocity field resulting

from the incident wave field where, demonstrated by the color contour plot of u in

the panel inset, the undertow has yet to develop. In Fig. 4-8B, however, the undertow

is fully developed and the resulting velocity field is now much more complicated, as













A) 2 ---
u 05 2 0 14-008002004 01 016022028034 04










0.5


0.5 t=10.0s

0 2 4 6 8
Cross-shore (m)



B) 2 I_ _
002-0 14-008002004 01 016022028034 04



1.5 -






0.5


S "- t = 150.0 s
0 2 4 6 8
Cross-shore (m)


Figure 4-8: Depth-dependent wave-current interactions in the cross-shore. Velocity
fields taken at A) t = 10 s and at B) t = 150 s show the depth-dependent velocity
predictions and the superposition of a propagating wave on the developed undertow.


shown by both the inset panel and the larger frame. At this later time, the incident

waves are propagating across a lower-frequency, seaward-directed flow near the bed

that is of similar magnitude to the shoreward-directed velocities of the obliquely

incident wave field. Note the increased strength of the velocity at the wave crest in the

inset of Fig. 4-8B and the resulting shear in the velocity field under the wave crest.









4.2.3 Longshore Current Variability

While not discussed in Visser (1991), the time-dependent longshore current

predicted by SOLA-SURF is also spatially variable in both horizontal directions. This

time-dependent current structure is shown in panels A, B, and C of Fig. 4-9, where

we have plotted the instantaneous, depth-averaged (u, w) velocity field in the (x, z)

plane at three times during the simulation. The velocity field shown in Fig. 4-9D

represents a thirty second average, centered about t = 165 s, of the depth-averaged

(u, w) velocities taken once the model had reached steady-state. For clarity, every

fifth velocity vector in the longshore direction is plotted in this figure but none have

been omitted from the cross-shore direction. In Fig. 4-9A, at t = 20 s, we see that

the predominant feature of the velocity field is the incident wave signal and note that,

relative to the onshore-directed velocities, the velocity magnitudes of reflected waves

on the sill are very small. At roughly half of the time it takes the longshore current

to reach steady-state, there appears to be a longshore-periodic structure associated

with the longshore current, which is shown in Fig. 4-9B. Figure 4-9C demonstrates

the persistence of this meandering periodic structure, although much weaker, even

as the mean longshore current nears steady-state. The thirty second average of the

depth-averaged, steady-state velocity field shown in Fig. 4-9D removes the velocity

signature of the incident wave field, resulting in a mostly longshore uniform current.

Further investigation of the periodic structure associated with the developing

longshore current (Fig. 4-9B) suggests that oscillations of the current occur in both

the longshore and cross-shore directions. Over a thirty second period spanning from

t = 90 s to t = 120 s, when the mean current speed and profile are evolving, the

depth-averaged longshore velocity is not uniform in the longshore direction. The

periodic structure shown in Fig. 4-10A represents the time-mean, depth-averaged

longshore current for 90 < t < 120 s and shows that this feature oscillates with a

much lower frequency than the incident wave field: on the order of twenty seconds.













t=20s
t = 20 s













2 4 6 8 10
Cross-shore (m)

t=150s












2 4. ~i6 8 10
Cross-shore (m) -



Cross-shore (m)


t=80s


2, -,
S"I I"-

.,,

^ *;
.- '

' ."`^
-T *,
- T


2 4 6 8 1
Cross-shore (m)

30 s average











Cs '-she
4 6 8 1
Cross-shore (m)


Figure 4-9: Depth-averaged (u, w) velocity fields taken at A) t = 20 s, B) t = 80 s, C)
t = 150 s, and D) over a 30 second average to remove the effects of the incident wave
field. Note the longshore variability of the current structure in panels B and C.



Nearshore motions with frequencies of this magnitude are often classified as shear


waves, but the presence of these motions were not discussed in Visser (1984) or Visser


(1991). Panels B and C of Fig. 4-10 are color contour plots of the instantaneous,


depth-averaged residual longshore velocity at t = 100 s and t = 120 s respectively. The


residual velocity component is found using a Reynolds decomposition (Pope, 2000)


for the depth-averaged velocity and is given by Eq. 4-1. Here, the residual longshore


velocity considered accounts for the wave-induced velocity as well as the turbulence,


but no distinction is made between the two components.


W' = (w)


(4-1)


Alternating velocity deficit (blue) and velocity excess (red) signatures seen in


Figs. 4-10B and 4-10C demonstrate the longshore propagation of this instability,











as well as its cross-shore displacement. As compared to the residual velocity signatures

shown in Fig. 4-10B, notice that the excess and deficit signatures have reversed their

positions in the longshore direction and have migrated seaward in Fig. 4-10C.

Shear waves are energetic, low-frequency vortical structures that propagate in the

longshore direction and exhibit significant excursions in the horizontal plane (Bowen

and Holman, 1989). Thus, these shear wave motions contribute greatly to the nearshore



): 0 004 008 0 01 016 02 024 028 032 036 04


E

0
0

0
-J


2 4 6 8
Cross-shore (m)

W -015 -012 -009 -006 -003 0 003 006 009 012 015










2 4 6 8
Cross-shore (m)







S2 4 6 8




Cross-shore (m)


Figure 4-10: Color contour plots of the depth-averaged longshore velocity for A) a
30 s time-period prior to steady-state, B) the instantaneous longshore velocity residual
at t = 100 s and C) the instantaneous longshore velocity residual at t = 120 s.









circulation and ultimately affect the transport of sediment, pollutants, and biological

material (Dodd et al., 2000). While barred beaches have been found to facilitate

the onset of shear instabilities [see Oltman-Shay et al. (1989); Bowen and Holman

(1989)], observations of these motions on planar beaches are not widespread but have

been known to occur (Dodd et al., 2000). Oltman-Shay and Howd (1993) discovered

evidence of shear wave motions on two planar beaches in California (Leadbetter Beach

and Torry Pines) after reanalyzing data from the Nearshore Sediment Transport Study

(NSTS) conducted in 1980. Shear waves have not been readily identified in laboratory

experiments conducted with planar slopes (Reniers et al., 1997) either because of

limitations in data collecting or due to suppression by confinement of the wave basin

(Bowen and Holman, 1989). In their numerical model, Allen et al. (1996) were

successful in generating shear waves over a planar beach and Putrevu and Svendsen

(1992) suggested that typical length and time-scales of shear wave motions from the

experiments of Visser (1984), if they had been identified, would be of 0(8 m) and

0(20 s), respectively. These scales agree well with those predicted by the simulation

(~20 s and ~5.6 m), resulting in a phase speed of roughly 70% of the maximum

longshore current velocity-similar to the value suggested by Dodd et al. (2000).

4.3 Vertical Distributions of Velocity and Momentum

4.3.1 Time-Averaged Velocity Profiles

Although the cross-shore circulation was not discussed in Visser (1991),

time-averaged velocity profiles of the predicted flow field show a strong seaward-directed

flow near the bed, commonly referred to as undertow. The velocity profiles plotted in

Fig. 4-11 demonstrate this behavior of the return flow, specifically in panels B, C, and

D, where the undertow dominates the circulation. Velocity profiles shown in Fig. 4-11

were averaged over thirty seconds and were taken from a transect located near the

mid-point of the longshore domain. The slightly negative u and w velocity profiles

in Fig. 4-11A result from the velocity corrections applied at the forcing boundary









that account for mass flux. Figure 4-11A also demonstrates that, as expected, there

is a shoreward-directed flux of momentum due to the waves that roughly occupies

the region bound by the wave trough and wave crest. While this momentum flux due

to the waves persists throughout the remaining panels, the vertical structure of the

velocities change significantly as the locations progress from the forcing boundary

shoreward.

The time-averaged longshore velocity profiles in Figs. 4-11B and C demonstrate

a unique behavior near the two grid points closest to the bed. Here, the time-averaged

longshore current receives an asymmetric impulse from the staircase representation

of the bottom slope. As the shoreward velocity of the phase of a shoaling wave

encounters a step rise, there is a corresponding pulse of near-bottom water that flows

in the positive z-direction since the fluid is less likely to flow upward due to the mass

above it. Similarly, for the seaward-directed velocity of the opposing phase there is a

pulse of water in the negative z-direction.

The magnitude of the undertow diminishes significantly in the locations where

the longshore current velocities are high, as demonstrated in panels E through H of

Fig. 4-11. The predicted distribution of longshore velocity is similar, in most profiles,

to the nearly depth-uniform structure reported by Visser (1991) and discussed by

Svendsen and Lorenz (1989), who suggested that a depth-uniform velocity profile

would be a better assumption than the logarithmic profiles more commonly used to

describe steady, open-channel flows. This depth-uniform structure is most evident

in the strongest (higher velocity) part of the longshore current and is demonstrated

by Figs. 4-11G and 4-11H. Figure 4-11 also shows the large difference between

the magnitudes of the horizontal velocities, u and w, and the vertical velocity v,

which is so small that you can barely distinguish it in the figure. This implies that the

average vertical velocity contributes very little to the mean flow and becomes even less

significant if considered in a depth-averaged or depth-integrated sense. Products of the




































Velocity (m/s)


E)
05



04



03-
E

02-



01



-03 -02 -01 0 01 02 03
Velocity (m/s)


B)
05-



04-



03-



02



01 -



-03 -02 -01 0 01 02 03
Velocity (m/s)


F)
05-



04 -



03-



0 2 -



01



-03 -02 -01 0 01 02 03
Velocity (m/s)


C)
05



04 -



03-



02-



01



-03 -02 -01 0 01 02 03
Velocity (m/s)


G)
05



04 -



03-


02
0 2 -



01



-03 -02 -01 0 01 02 03
Velocity (m/s)


D)
05-



04-



03-
E

02 -



01 -



-03 -02 -01 0 01 02 03
Velocity (m/s)


H)
05-



04-



03-
gE

02-



01 -



-03 -02 -01 0 01 02 03
Velocity (m/s)


Figure 4-11: Time-averaged velocity profiles at A) the forcing boundary, B)

h = 0.246 m, C) h = 0.226 m, D) h = 0.206 m, E) h = 0.145 m, F) h = 0.124 m,

G) h = 0.085 m, and H) h = 0.069 m. The dashed line in each panel represents the

mean water level while the dotted line shows the approximate location of the bottom

boundary. These panels represent a progression from the offshore forcing boundary

(A) to the shallow depths of the inshore (H) where the longshore current dominates the

nearshore circulation.



vertical velocity with horizontal velocity components, however, may not necessarily be


insignificant in the presence of a nonlinear, propagating wave field.


Time-mean velocity profiles of u in panels B, C, and D of Fig. 4-11 suggest that


the conservation of mass is being satisfied as depth-averaged values of the profiles are


very close to zero. This is not the case in panels E through H, where there appears


to be a net flow of mass in the shoreward direction. However, the shoreward-directed


momentum carried by the incident wave field is balanced by the undertow only in
















5



4 <( (m/s)
S0.1
0.08
S0.06
S3 0.04
S- 0.02
) 0
a -0.02
0
1 2 --0.04
-0.06
-0.08
-0.1
1 -



2 4 6 8 10
Cross-shore (m)


Figure 4-12: The depth-averaged cross-shore velocity, u, averaged over thirty seconds
and plotted in the (x, z) plane.


a strictly two-dimensional sense (Svendsen, 1984); previous figures and discussions

have emphasized the three-dimensionality of the flow field. A color contour plot

of the time-mean, depth-averaged cross-shore velocity is shown in Fig. 4-12 where

hotter colors correspond to shoreward-directed flow and cooler colors designate

seaward-directed flow. The concentration of shoreward-directed flow between x = 5 m

and x = 6.5 m corresponds to the area where the velocity profiles in panels E-H of

Fig. 4-11 are located. The depth-averaged cross-shore velocity was averaged over a

thirty second period just prior to reaching steady-state, during which time there were

many complex, spatial and time-dependent flow features including seiching and the

periodic oscillations of the longshore current. These different flow features all occur on

different length and time scales making it difficult to extract an ensemble that explicitly

proves that the depth-averaged velocity profiles obey the conservation of mass. We









know, however, that the model conserves mass over the course of a 200 second

simulation: the initial volume of fluid contained within the physical domain is the same

as the volume contained at the end of the simulation. Another significant feature of

this region is the setup, and the location of the time-mean free surface displacement

seems to correspond well with the location of the excess shoreward-directed velocity.

Additional information about the spatial relationship between these two features is

presented in Appendix B.

4.3.2 Vertical Distribution of Radiation Stress

The vertical distributions of the shoreward-directed components of the radiation

stress have received particular attention recently [Mellor (2003), Xia et al. (2004)]. As

presented in Longuet-Higgins and Stewart (1964), these two horizontal components

are the cross-shore component of the shoreward-directed radiation stress and the

longshore component of the shoreward-directed radiation stress, denoted here as S'x

and Sz ,2 respectively, and are given by Eqs. 4-2 and 4-3. The longshore component

of the shoreward-directed radiation stress is non-zero only for a three-dimensional

wave climate produced by either obliquely incident waves or longshore variable

bathymetry. We have neglected a third horizontal radiation stress component, Szz-the

longshore-directed component-for two reasons: first, the vertical structure of the

time-averaged longshore flow is essentially depth-uniform and, as such, vertical

distributions would not be as complex as those in the cross-shore and, second, since

longshore gradients of the time-averaged quantities that contribute to this stress



2 Although the notation of Sy, presented by Longuet-Higgins and Stewart, is more
commonly used, we shall use this alternative notation since it is consistent with our
coordinate system.









component would be zero owing to our periodic longshore domain. Following

Longuet-Higgins and Stewart (1964), we define:


s (p+ pp2)dy} pody (4-2)



S=( K puwdy) (4-3)

where
h = bed elevation
( = vertical location of the free surface
p = density of the fluid
p = total pressure
po = hydrostatic pressure in the absence of waves
( ) time-averaging operator
In order to obtain information regarding the vertical structure of the radiation

stress components, we simply neglect the vertical integration of the expressions

presented in Eqs. 4-2 and 4-3 and denote the depth-dependent values of the radiation

stress components by s,, and sx. The resulting equations for the shoreward

component of the shoreward-directed radiation stress and the longshore component

of the shoreward-directed radiation stress are given by Eqs. 4-4 and 4-5, respectively,


s(xx(Y) p(U2 + (} po (4-4)



szZ(y) = p(uw) (4-5)

The formulation of Eq. 4-4 is much easier to understand when evaluating the

vertical structure of each component and their relative magnitudes. Figure 4-13 shows

the distribution and magnitude of each component of sx, as well as the total, where

values have been averaged over thirty seconds. Instead of plotting both pressure

components, the difference between the two is shown in order to reduce the scale of

the abscissa and increase the resolution of each component. Note that the units of












0.5



0.4

(h+r)

E .3

v(U2)

(P) -P
0.2



0.1 I
0 50 100 150
s, (N/m2)


Figure 4-13: A comparison of the shape and magnitude of various components of sx
taken from a location in the middle of the longshore domain where the local depth is
h = 0.246 m.


s,, shown on the abscissa characterize a stress, while depth-integrated values of the

radiation stress terms (S,, and S,,) have the units of stress times length. The dashed

line in Fig. 4-13 represents the mean water level over the thirty second average at

a location where the mean water depth is 0.246 m and the location of the bed is at

y 0.104 m.

The total pressure term in Eqs. 4-2 and 4-4 is the time-mean pressure in the

progressive wave field and, thus, represents both the pressure due to the water waves as

well as the hydrostatic pressure over the water column. Plotting the difference between

the time-mean total pressure and the hydrostatic pressure in the absence of a wave

field (Fig. 4-13) results in the time-mean dynamic pressure due to the incident waves.

Between the trough and crest levels, Fig. 4-13 shows that the apparent stress term

(p(u2)) and the gravitational term ((p) po) contribute approximately equal amounts









of momentum flux. This is in contrast to the findings of Sobey and Thieke (1989) who

stated that the gravitational term was dominant above trough level and that the apparent

stress was less significant in this region; however, the magnitude of the wave apparent

stress above the trough is roughly five-times greater than the predicted value in the

undertow, which is similar to their findings. While nonlinear cnoidal theory was used

in their investigation, the waves predicted in this model are strongly nonlinear, which is

evident in the translation of the apparent stress peak above the mean water level.

The expression p(v2), while not included in our formulation of sx (Eq. 4-4), is

also included in Fig. 4-13 to demonstrate its relative significance to the other terms.

More specifically, it shows that there is not an exact balance between the vertical

flux of momentum (p(v2)) and the time-mean dynamic pressure ((p) po) below the

mean water level (the difference in sign, however, is correct). This contradicts the

methodology followed by Longuet-Higgins and Stewart (1964) in their formulation

of the radiation stress equation for Sx, where the terms were considered to explicitly

balance one another below the mean water level. A formulation for s,, suggested by

Sobey and Thieke (1989) accounted for this inequity:


s.x(y) = p(U2) p(v2) + Ap, (4-6)

where we have adapted our notation for the time-averaging and accounted for the

difference in the coordinate systems by replacing w2 with v2. Here the pressure term

Ap accounts for the time-average dynamic pressure in the region bound by the wave

crest and wave trough; therefore, the vertical momentum flux term p(v2) is assumed to

exactly balance the time-average dynamic pressure below the trough level. As shown

in Fig. 4-13, the shape and magnitude of the profiles corresponding to the terms in

question are not similar, especially near the bed where the momentum tends toward

zero much faster than the dynamic pressure.









Using the formulation for sx given by Eq. 4-4, we have plotted the predicted
profiles of the shoreward-directed radiation stress, along with the analytical solution

provided by Longuet-Higgins and Stewart (1964) (Eq. 4-7) for the linear distribution
of the total energy (E), a triangular distribution of E (Eq. 4-8) above the mean trough
level (Dean, 1995), and the vertical distribution of sxx suggested by Mellor (2003)
(Eq. 4-9).


sx Es 22h + (4-7)
sinh 2hkx 2



() pgH cos 0 2y2- 2y 2y) ify> (4-8)
27 1 H ,. 0, ify


sx(y) =kDE F12F 11(k2 i) F72 (4-9)

where
cosh kD(1 + v) cosh kD(1 + v) sinh kD(1 + v)
F11 = 12 22
cosh kD sinh kD sinh kD
S-1 if y = -h
v = transformed vertical coordinate such that v = if y (.
O if y = (
and
E = total wave energy = lpgH2
h = local water depth
0 = local wave angle
D = h + ()
MWL = mean water level
The profile of sxx predicted by SOLA-SURF in Fig. 4-14A approximates a
depth-uniform profile since there is essentially no mean-flow below the trough level
at the offshore forcing boundary. This location is of particular interest because the
predicted profile is representative of the radiation stress produced by the time-averaged
wave forcing. At the forcing boundary, the predicted profile agrees well with the
distribution of momentum given by linear theory since the forced waves are sinusoidal









and approximately linear. The magnitudes of the triangular and depth-uniform

segments would be larger had we not applied corrections to the forcing velocities

to account for mass flux. Although the waves are forced with a linear, sinusoidal-type

signal, they steepen and become asymmetric as they shoal, dramatically affecting the

distribution of momentum flux.

The predicted profiles of sx (Fig. 4-14) between the trough and crest levels have

a triangular-shaped distribution of momentum flux that accounts for the majority of the

shoreward-directed flux, similar to observations made by Svendsen (1984) and Sobey

and Thieke (1989). While the peaks of the predicted radiation stress profiles have

magnitudes similar to those predicted by linear theory, the nonlinearity of the simulated

waves redistributes the majority of the momentum above trough level and shifts the

peak of the profile above the mean water level. The predicted time-average circulation

gives the radiation stress profile a very distinct shape below the trough level, especially

in Figs. 4-14B-D where the undertow is the dominant flow feature. Progressing up

the slope of the bathymetry, in Figs. 4-14E-H, we see that the mean flow is relatively

depth-uniform, below the trough level, in the locations where the longshore current

velocities are highest; however, the average magnitude of the profiles below the trough

level vary little throughout the cross-shore.

Mellor's equation for the vertically dependent shoreward-directed radiation stress

(Eq. 4-9) provides an alternate approximation for the distribution of momentum. In

some instances (Figs. 4-14E-H), the magnitude of the radiation stress near the bed is

actually greater than the value at the crest level. One possible benefit of the analytical

solution provided by Mellor (2003) is that it provides some level of approximation

concerning the termination of the vertical distribution at the crest level while the linear

distribution of momentum flux suggested by Longuet-Higgins and Stewart (1964)

simply stops at the mean water level. Perhaps a suitable compromise between the

analytical solution provided by linear theory and that provided by Mellor (2003) would













A) B) C) D)
05- SOLA-SURF 05- 05- 05-
E/2
-E
Mellor
04 04 04 04


03 03- 03 03-


02 02- 02 02-


01 01 01 01


0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150
s, (N/m2) s. (N/m2) s,, (N/m2) s. (N/m2)

E) F) G) H)
05 05- 05- 05-


04 04 04 04


03- 03- 03 03-1
E E. E -

02- 02- 02- 02-


01 01 01 01


0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150
s, (N/m2) s, (N/m2) s,, (N/m2) s, (N/m2)


Figure 4-14: Predicted profiles of radiation stress sx. A comparison of the predicted
shape and magnitude of sx, to linear theory and Mellor's analytical solution at A) the
forcing boundary, B) h = 0.246 m, C) h = 0.226 m, D) h = 0.206 m, E) h = 0.145 m,
F) h = 0.124 m, G) h 0.085 m, and H) h = 0.069 m. The triangular distribution
of E/2 has been shifted to the right for purposes of comparison. The dashed line
represents the mean water level and the sloping, dotted line represents the approximate
location of the bathymetry.



be to add the triangular distribution of to Mellor's profile, starting at the crest level.

Neglecting the characteristics and vertical structure of the mean flow field, it is possible

that a depth-uniform distribution of the radiation stress below trough level would

provide a sufficiently realistic representation of the momentum flux caused by the

time-averaged wave forcing.

Vertical distributions of the longshore component of the shoreward-directed

radiation stress, s, are shown in Fig. 4-15 where the plotting convention is









similar to that used in Fig. 4-14. The predicted profile of s,, was plotted using

the vertically-dependent formulation given by Eq. 4-5, which was derived from

the depth-integrated equation (Eq. 4-3) suggested by Longuet-Higgins and Stewart

(1964). Figure 4-15 shows the comparison of the predicted radiation stress profile

to the depth-uniform distribution of momentum flux (E) predicted by linear theory

and suggested in Dean and Dalrymple (1991) (Eq. 4-10), and the vertical distribution

proposed by Mellor (2003) (Eq. 4-11).


Sxz Es 2hk ,+ 1 sin 20 (4-10)
4 sinh 2hk


sxz(y) = kDE 2FIl l2 k,1 (4-11)

The modeled vertical profiles of sxz above the trough level exhibit the triangular

distribution seen in the sxx profiles of Fig. 4-14, but the peak of the profile is shifted

further above the mean water level. As shown in the radiation stress component

profiles of Fig. 4-13, the gravitational term reduces the super-elevation of the peak

of the sxx profile. Since there are no additional terms in the formulation of the

vertically-dependent profile of s z, the wave apparent stress term controls the shape of

the profile and the distribution of momentum flux. This suggests that assuming similar

distributions of sxx and sxz above trough level, and at different cross-shore positions, is

inaccurate in the presence of propagating nonlinear water waves. Below trough level,

however, the distribution of momentum is relatively depth-uniform with the exception

of Figs. 4-15E and 4-15F where the seaward-directed undertow produces an inflection

point near mid-depth.

Comparisons of the predicted profile of sxz to the linear solution and the analytical

solution given by Mellor (2003) give mixed results. Similar to the comparison of sxx

profiles in Fig. 4-14, Mellor's formulation (Eq. 4-11) dramatically under-predicts the

magnitude of momentum distributed above the trough level but correctly estimates













A) B) C) D)
05- SOLA-SURF 05- 05- 05-
E
-- Mellor

04 04- 04 04

(h+q)
03 / 03- 03- 03-


02 02 02- 02 -


01 01 --L 01 01


0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20
s (N/m2) s (N/m2) sz (N/m2) s (N/m2)

E) F) G) H)
05 05- 05- 05-


04 04 04 04


03- 03- 03 03-


02- 02- 02- 02-


01 01 01 01


0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20
s (N/m2) s (N/m2) sz (N/m2) s (N/m2)


Figure 4-15: Predicted profiles of time-averaged radiation stress sx. A comparison
of the predicted shape and magnitude of sz to linear theory and Mellor's analytical
solution at A) the forcing boundary, B) h = 0.246 m, C) h = 0.226 m, D)
h = 0.206 m, E) h = 0.145 m, F) h = 0.124 m, G) h = 0.085 m, and H) h = 0.069 m.
The dashed line represents the mean water level and the sloping, dotted line represents
the approximate location of the bathymetry.



the upper-most extent of the distribution profile. At the two most offshore locations

(panels A and B of Fig. 4-15), however, the distribution of momentum below the

trough level given by Mellor's equation looks very similar in shape and magnitude to

both the predicted profile and the depth-uniform distribution suggested by linear theory.

Moreover, Fig. 4-15A shows a very close agreement between Mellor's solution and the

predicted profile at the forcing boundary.

The previous figures demonstrating the distribution of radiation stress (Figs. 4-14

and 4-15) show that the majority of the momentum flux is located above the trough












16
E a S: 3E/2
E 14 A S,: Mellor
Z 0 S: SOLA-SURF
a 12 S
SA S,: Mellor
S: SOLA-SURF
10
0 A
.2o

Ci
4 6

2 o o


*00

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
Cross-shore Depth (m)


Figure 4-16: The depth-integrated magnitudes of the predicted radiation stresses
(S,, and S,,) show good agreement to the magnitudes found from linear theory and
Mellor's equations.


level. Although the various methods for plotting vertical profiles of s,, and s,, give

vastly different distributions over the vertical, their depth-integrated values are similar,

which is an encouraging result. The depth-integrated radiation stress magnitudes

are shown in Fig. 4-16 and demonstrate the similarity of the predicted magnitude

with that given by linear theory (E + ') and Mellor's equations given previously.

Depth-integrated values of the radiation stress component S,, are represented by the

hollow symbols where each shape represents the technique or theory given in the

symbol legend. The longshore component of radiation stress S,' is denoted by the

filled symbols, which again correspond to the approximation listed in the legend.

A multiple-application trapezoidal scheme (Chapra and Canale, 1998) was used to

integrate the vertically-dependent profiles plotted in Figs. 4-14 and 4-15, and at five

additional cross-shore locations as well. With the exception of the values predicted at









the forcing boundary, the predicted magnitudes of S,, and S, look quite similar to the

analytical solutions provided by the other two theories considered here. The forcing

boundary presents a special case since corrections have been applied to the velocity

components to compensate for mass flux. As discussed earlier, and shown in Fig. 4-14,

the velocity corrections applied at the forcing boundary result in a mean-flow field

below the trough level that is just slightly larger than zero; therefore, the largest

contribution to the depth-integrated value of S, comes from the momentum flux

associated with the forced, monochromatic wave signal. This is not the case at other

cross-shore locations where the undertow and longshore current contribute significantly

to the structure and magnitude of the radiation stress profile below the trough level of

the waves. Since we do not have a formulation for the triangular distribution of the

radiation stress component s, between the wave crest and trough, the depth-integrated

value of S, only includes the portion of the profile below the mean water level.

The distribution of the radiation stress produced by the nonlinear waves in our

simulations suggests that the majority of the momentum flux is contained above the

trough level. Similar observations were made by Svendsen (1984) and Sobey and

Thieke (1989) regarding the shoreward-directed component of the radiation stress,

Sx. From the predicted profiles of the longshore component presented in Fig. 4-15,

it appears that this observation holds true for S, as well. This distribution of energy

over the water column is not a trivial matter as phase-averaged models may be forced

with various approximations to stress or force distributions, a common choice being

depth-uniform or depth-linear profiles. Linear theory suggests that roughly one-third

of the total momentum flux is advected shoreward by the waves in the region between

the trough and the crest. By integrating our radiation stress profiles of s,, and s, and

comparing the area contained in the region bound by the crest and the trough to the

total area of the profile, we find rather that roughly twice this amount is carried by the

nonlinear waves present in our simulations. This ratio is plotted in Fig. 4-17 for both










Sx and Sxz, where we have once again used the multiple-application rule to integrate

the profiles.






0.9 S
r- S~z
0.8 -
.2
S0.7

.0 0.6

0.5

0.4
>,
( 0.3
LJ
0.2

0.1
l l l l l l l l l l l l l l l l l l l l l l l l l l l l lli i
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
Cross-shore Depth (m)


Figure 4-17: The ratio of momentum flux in the wave trough-to-crest region to the
total depth-integrated value for S, and S, as a function of cross-shore depth. For
progressive nonlinear water waves, between 40 and 80% of the total momentum flux is
carried in the region bound by the wave crest and wave trough.


4.3.3 Stress Gradients and Nearshore Forcing

The previous figures showing vertical profiles of radiation stress components

provide valuable information about the distribution of momentum flux produced by

nonlinear shoaling waves. Although these radiation stress profiles present useful

information concerning the nonlinear distribution of momentum, it is the gradients

of radiation and shear stresses that drive nearshore circulation. The significance of

cross-shore gradients of S,, on surf zone forcing in producing setup and undertow

has been noted by Longuet-Higgins and Stewart (1964), Svendsen (1984), and more

recently by Christensen et al. (2002). Similarly, forcing in the longshore direction









is proportional to the cross-shore gradient of the longshore component of radiation

stress, -S,1 (Dean and Dalrymple, 1991). Shear stress distributions (i.e. p(uv))

play significant roles in circulation as well, but are commonly implemented as

depth-uniform or depth-linear forces acting over the water column (Deigaard and

Fredsoe, 1989) if they are considered at all. Rivero and Arcilla (1995) discussed the

importance of shear stress distributions in the context of more complex conditions

for waves encountering ambient currents and sloping bottoms. Deigaard and Fredsoe

(1989) also noted the importance of the term p(uv) as a necessary mechanism for

momentum transfer in non-uniform or unsteady wave conditions and discussed its

relative importance in their simplified momentum equation given by Eq. 4-12.

=-2(u)(h -y)- g( 9Sg)(h- y)+ UV) (4-12)


where
r = shear stress
U = horizontal wave-induced velocity
V = vertical wave-induced velocity
h = average water depth
y = vertical location
( = vertical location of the free surface
S = mean slope of the water surface setup

We have plotted the vertical distribution of radiation and shear stress gradients

in Fig. 4-18 to emphasize the three-dimensional nature of these nearshore forcing

mechanisms. Cross-shore gradients of the radiation stress components were computed

over two adjacent grid locations and then minimally smoothed using a diffusive-type

filter. The gradients of the shear stress term p(uv) were calculated over adjacent

grid locations in the vertical direction and were also smoothed as little as necessary

to ensure that the structure and magnitude of the profile would be retained, while

simultaneously providing sufficiently smooth profiles.

Gradients of sx shown in Fig. 4-18 represent the distribution of the forcing that

is applied in the cross-shore direction and has the dimensional units of a force per unit














axsx
a~s
aY(P(ILY))


E)
05


04 -


03 -


02-


01 -



-00 -50 0 50 100
N/m2/m


B)
05


04-


03-


02-


01 -


0
50 0 50
N/m2/m

F)
05-


04-


03- '


02-


01 -



0 -50 0 50 100
N/m2/m


C)
05-


04 -


03-


02-


01 -



00 50










0'3 -


02







N/m2/m


D)
05-


04


03-


02


01 -



50 0 50
N/m2/m

H)
05-


04-


03 -


02


01 -



-20 -100 0 100 200
N/m2/m


Figure 4-18: The vertical distribution of radiation and shear stress gradients at A) the

forcing boundary, B) h = 0.246 m, C) h = 0.226 m, D) h = 0.206 m, E) h = 0.145 m,

F) h = 0.124 m, G) h = 0.085 m, and H) h = 0.069 m. The dashed line represents the

mean water level and the sloping, dotted line represents the approximate location of the

bathymetry.



area, per unit depth. Longuet-Higgins and Stewart (1964) stated that this forcing on the


surf zone would be balanced by a gradient in the mean water level (Eq. 4-13).


a
OX)


18s
pgaS
pgh xz


(4-13)


This force balance is usually considered in a depth-integrated sense, resulting in the


loss of information regarding the vertical distribution of the force balance. Positive


values of 9xsxx in Fig. 4-18, when depth-integrated indicate the presence of a


shoreward-directed force balancing the wave-induced setdown, while negative values









represent a seaward-directed force balancing the setup. Figure 4-18 demonstrates

that there is a unique vertical distribution of the forcing at each cross-shore location,

especially in panels F and G where there are significant sign differences above and

below the wave trough level. As noted by Dyhr-Nielsen and Sorensen (1970), the

balancing forces will not have similar vertical distributions and the inclusion of

shear and turbulent stresses and local accelerations for unsteady flows must also be

considered to obtain the appropriate force balance over the vertical (Christensen et al.,

2002).

The gradients of the longshore radiation stress component (dxsxz) plotted in

Fig. 4-18 are manifested as longshore-directed forces (per unit area) in the nearshore

region (Dean and Dalrymple, 1991). These longshore-directed forces are balanced by

bottom shear stresses as well as the resistance to lateral shear over the water column

(Longuet-Higgins, 1970). The net magnitude of the resisting forces is found by

depth-averaging the profiles of Os,, in Fig. 4-18 where the sign of the depth-averaged

value indicates the direction of action. Depth-averaged values of the Os,, profiles

in Figs. 4-18A-D are small relative to the profiles in panels E-H due to the lower

longshore current velocities at the corresponding cross-shore locations. This suggests

that the balancing forces would be relatively small at these locations, as well. In panels

E and F, however, decreasing magnitudes of s, result in stronger longshore forcing in

the positive z-direction.

Rivero and Arcilla (1995) correctly draw attention to the significance of the

term (uv) that appears in the depth-averaged momentum equations. Previously, the

argument had been made that this value is essentially zero since the horizontal (u)

and vertical (v) velocity components are 900 out of phase; this assumption only

holds true for steady, periodic, linear waves that are not typically seen in nearshore

environments. Recent attempts were made by Rivero and Arcilla (1995) and Deigaard

and Fredsoe (1989) to emphasize the importance of including appropriate values and









distributions of (uv) in momentum equations as this term plays important roles in the

vertical distribution of shear stress as well as wave energy dissipation. The vertical

gradient of (uv) is shown in Fig. 4-18 at various cross-shore locations where the

density constant p has been included to compare the relative magnitude of this term

to the radiation stress gradients. As they predicted, the gradients of p(uv) are not

negligibly small and in most cases have magnitudes similar to those of the radiation

stress gradients seen in Fig. 4-18.

The response to the nearshore forcing produced by gradients in the radiation stress

fields is found by integrating the profiles of asxx and xs,, plotted in Fig. 4-18.

Relationships for the cross-shore and longshore forcing components are given by

Eqs. 4-14 and 4-15, respectively. The term F, (Eq. 4-14), therefore, is the forcing

applied on the surf zone in the cross-shore direction, per unit area. Forcing applied in

the longshore direction, per unit area, is given by the component F, (Eq. 4-15). These

two nearshore forcing components are plotted in Fig. 4-19 and provide information

concerning the distribution of forcing in the cross-shore domain, the magnitude of

each forcing component, and their combined effect. Figure 4-19A demonstrates the

magnitude and location of the applied forcing components F, and Fz, which appear to

have bi-modal distributions, with their peaks occurring at similar cross-shore locations.

The vectors shown in Fig. 4-19B represent the combined nearshore forcing found by

computing the resultant of the two forcing components at each cross-shore location;

however, for clarity only one-fifth of the total number of points in the cross-shore

domain are represented by this figure.


as,
Fe c (4-14)
ax (4-15)


F, c s (4-15)
ax








60























6
A) B)

E F


3 +Z
L'.





\ 2 4 6 8 1


0 2 4 6 8 10
Cross-shore (m) Cross-shore (m)


Figure 4-19: Depth-integrated values of the nearshore forcing components F, and F,
plotted as a function of cross-shore location A) independently and B) as a schematic of
the combined nearshore forcing.















CHAPTER 5
SUMMARY

5.1 Applications

SOLA-SURF is a portable CFD model that can be used to study a wide variety

of fluid flows. The simplistic nature of the code, evident in both its organization and

numerics, enables users with minimal numerical modeling experience to simulate

time-dependent flows. Indeed, very little information must be supplied to the model:

bathymetry data, wave height, wave period, and the depth at the offshore forcing

boundary. In its most basic state this model could be utilized as a teaching or training

tool, providing students or other interested individuals a chance to use and manipulate a

very simple, yet robustly stable, three-dimensional CFD model. For the more advanced

modeler, however, this code serves as a good starting point for building an even more

advanced computational tool that can be applied to study more complex problems.

Perhaps the greatest attribute of this Navier-Stokes code is the ability to

resolve the time-dependent free surface. This feature could be utilized to study

wave diffraction patterns in harbors and basins or around breakwaters and jetties.

Providing the model with a three-dimensional bathymetry field, one could study the

resulting wave refraction around, for instance, a cuspate beach for both shore-normal

and obliquely incident wave fields. Cross-shore and longshore current structures over

variable bathymetry could also be investigated for a variety of wave climates.

5.2 Sensitivity

The three-dimensional Navier-Stokes model, SOLA-SURF, provides reasonable

estimates of nearshore processes, but relies on both the implementation of the free

surface velocity boundary conditions as well as the cell aspect ratio. Care must also be

taken when selecting appropriate values of the theta-differencing coefficients in order









to prevent amplitude errors in the wave field and excessive diffusion in the velocity

fields. These conditions limit the parameter-space to a range of specific values, but

the conditions used in Visser's laboratory experiment were conducive to performing

simulations with this model. Waves with lower Iribarren numbers, those in the spilling

breaker range, could be more appropriately modeled with SOLA-SURF and the

resulting wave transformation would yield better results.

The sensitivity to the free surface velocity boundary conditions is unfortunate,

but perhaps a combination between the two methods discussed in Chapter 2 exists,

such that improved results could be obtained for both the wave height and velocity

fields. Extending these boundary conditions using higher-order differencing schemes

is perhaps another viable solution, but ultimately the slope of the free surface will be

limited by the cell aspect ratio-a first-order limitation at best. When electing to use

SOLA-SURF, it is perhaps best to understand the physical limitations of the model and

take them into account when analyzing the results. In this case, we believe that the

ability of the model to reproduce the experimental observations precisely is of minor

significance in the context of our discoveries. The ability to reproduce approximately

accurate or realistic results was a more immediate concern and the comparison of

model predictions to laboratory observations, as well as various analytical theories,

bolstered our confidence in the applicability of SOLA-SURF to this investigation.

5.3 Discoveries

Performing simulations of Visser's laboratory experiment on uniform longshore

currents [Visser (1984); Visser (1991)] with SOLA-SURF has provided new

information regarding the vertical distribution of momentum and, more specifically,

the horizontal components of radiation stress in the presence of nonlinear shoaling

water-waves. While the vertical structure of radiation stress components has been

discussed in the literature, particularly in the presence of more realistic waves

propagating over variable bathymetry, very little information exists that quantifies









its importance. Our model simulations have lead to the conclusion that in the presence

of nonlinear shoaling waves, the majority of momentum flux directed both shoreward

(SA,) and alongshore (S,,) is contained within the region bound by the wave trough

and crest. Linear theory suggests that roughly one-third of the total momentum flux

advected shoreward is found in this region, but our analysis suggests an average

value that is roughly twice this amount in the surfzone. Early estimates of radiation

stress magnitudes neglected contributions from the wave-induced velocity above the

mean water level and, although incorrect in the context of real progressive waves, this

methodology continues to be followed.

An interesting, and pertinent, result from this investigation is that the

depth-integrated magnitudes of the radiation stress profiles predicted by SOLA-SURF

are similar to the magnitudes calculated using analytical solutions provided by Mellor

(2003), Longuet-Higgins and Stewart (1964), Dean and Dalrymple (1991), and

Dean (1995). This result is pertinent because it provides confidence that the model

predictions, while not precise, are accurate. Therefore, nonlinear shoaling waves have

radiation stress magnitudes that are relatively similar to the values given by linear

theory, but the distribution of momentum flux over the vertical is different. In the

context of three-dimensional modeling, this result is important since many circulation

models, specifically wave-phase-averaged models, are forced with radiation stress

gradient profiles. It has been suggested in the literature that the resulting circulation

in these models is highly sensitive to the distribution of forcing over the vertical, but

until recently scientists and numerical modelers have had little information regarding

more plausible distributions. The analytical expressions given by Mellor (2003),

however, tend to underestimate the flux of momentum near, and above, the free surface

and overestimate the contribution near the bed in shallow water. This latter result

was certainly unexpected and the predicted profiles of radiation stress given by the

model show that this feature is not evident. Although the undertow provides some









vertical structure to the radiation stress profile below the trough level, its contribution is

typically many times smaller than that found above the mean water level.

Although radiation stress profiles do contribute significantly to the overall flow

field, gradients in the radiation stress fields represent the true average cross-shore

and longshore forcing in a nearshore system. Analysis of the vertical distribution of

radiation stress gradients, however, provided less conclusive information. Gradients

of sx seemed to have unique vertical profiles at each cross-shore location considered

in our simulation. Surely, the vertical structure of the mean-flow has much to do

with this variability, especially near the bed where the undertow is a dominant flow

feature. Some inshore locations yielded profiles of xs,, that had significant sign

differences above and below the mean water level; these locations correspond to

areas where the obliquely incident waves are nonlinear, asymmetric, and very steep.

On the other hand, the vertical profile of the radiation stress gradient, 80sx, at the

forcing boundary suggests that the majority of this cross-shore directed forcing is

contained in a depth-uniform distribution below the mean water level, with little

contribution above it. As discussed in Chapter 4, this location represents the forcing

induced by the monochromatic, sinusoidal wave signal and more or less excludes

the Eulerian mean flow. In sharp contrast to the profile of 0,s, the profile of 0,sx

at the forcing boundary shows that the majority of the forcing is found in the wave

trough-to-crest region with almost no contribution evident below the wave trough level.

This distribution is essentially reversed in the location of the peak longshore current,

where the largest forcing contribution is found below the mean water level.













APPENDIX A
THETA DIFFERENCING
Predictions offered by numerical models can be sensitive to the finite-differencing
scheme used to discretise the governing equations. SOLA-SURF employs theta-differencing
in the discretisation of convective flux terms found in the momentum equations
(Eqs. 2-2-2-4) and to control the amount of donor cell differencing in the kinematic
free surface boundary condition (Eq. 2-5). The discretised forms of the convective
flux terms (see Eq. A-1) and the KFSBC (Eq. A-2) are weighted with unique
theta coefficients, thereby allowing the user to control the differencing schemes
independently.

U2 4Jx (Ui,j,k + Ui+l,j,k)2 + a Ui,j,k + Ui+lj,k (ij,k Ui+l,j,k) .
cOx 46x

-- (Ui-l,j,k + Ui,j,k)2 -- a I i-l,j,k + Ui,j,kI(Ui-l,j,k -- ij,k) (A-l)



,k+-1 Hk + t (Ui jt,k + i-1,jt,k)(Hi+1,k Hi-1,k)
I-1
H'if = t 46xl-t-

it,k + Ui-,jt,k|(Hi+,k 2Hi,k + Hi-1,k)

46 (' it,k + ,' 1-)(Hi,k+l Hi,k-)

-- ., +" it,k-1(Hi,k+ 2Hi,k + Hi,k) +

+. it,k +(1 h' it-1,k (A-2)

After discovering the sensitivity of the free surface velocity boundary conditions
(see Chapter 2), we ran a number of simulations using different values for the









theta-scheme coefficients, a and 7, in order to find the best combination of finite-difference

schemes to use in our final simulation. Appropriate values of a, as suggested by Hirt

et al. (1975) for the two-dimensional SOLA model, are found using the following

inequality:
( u6t v6t w6t
1 > a > max .
6 6x 6y 6z
In Hirt et al. (1975), a reasonable value of a is said to be 1.2 to 1.5 times greater than

the right-hand side of this inequality, but we found this range of values to be too low

to produce a stable simulation, given our set of parameters. The lowest value of a

that produced a stable simulation was 0.5, which is roughly four times greater than the

suggested value using the rule stated above. Values of a below 0.5 resulted in highly

dispersive velocity fields that cause the simulation to become unstable, while values of

a greater than 0.5 diffused, or smoothed, the velocity fields damping the unsteadiness

artificially. While Hirt et al. (1975) suggest setting the value of 7 equal to a, we found

that the stability of the simulations was much less sensitive to the value of 7 used.

After trying various combinations of a and 7, we found the most agreeable values to

be 0.5 and 0.0, respectively. Equations A-3 and A-4 represent the discretised forms of

Eqs. A-1 and A-2, respectively, where we have substituted a = 0.5 and 7 = 0.0 into

the appropriate equations. The sensitivity of the predicted cross-shore wave heights

(RMS) to the combination of theta-scheme coefficients is shown in Fig. A-1 for seven

of the cases tested.



Ou2 1 2
ax 4x [i+2,j,k + 2(Ui,j,kUi+l,j,k Ui,j,kUi-l,j,k) -- j, +

1 r
+ Ui,j,k(2Ui,j,k Ui+,j,k i-1,j,k) + Ui+l,j,k l(i,j,k- Ui+l,j,k) + "

+ i-1,j,k(i,j,k i-1,j,k) (A-3)












-1
H-,k + 6t ijt,k + Ui- 1,t,k)(Hi+l,k Hi-lk)


-4 (' it,k "' it,k-l)(Hi,k+l Hi,k-1) + it,k +'



+ (1 hv' it-l,k


(A-4)


0.35 0.3 0.25 0.2 0.15 0.1
Cross-shore Depth (m)


0.05 0


Figure A-1: Effect on numerical diffusion on model predictions. Shown are predicted
cross-shore wave heights for different combinations of the theta-scheme discretisation
coefficients.


0.1

0.09

0.08

E 0.07

P. 0.06
I
0 0.05

0.04
C)
S0.03

0.02

0.01















APPENDIX B
CROSS-SHORE MASS BALANCE

In Chapter 4, we briefly touched on the subject of conservation when pointing

out features of the velocity profiles taken at various cross-shore locations (Fig. 4-11).

Some of the profiles (Figs. 4-11E-4-11H) appeared to be non-conservative: that

is, there was an absence of return flow near the bed that would act to balance to

shoreward-directed flow near the surface. While it has been noted in the literature

that conservation in time-mean velocity profiles should be explicit for purely

two-dimensional problems, we expected at least a minor amount of undertow to appear

in all of the profiles. The individual velocity profiles plotted in Fig. 4-11, however,

only provide information about the mean flow field at one discrete cross-shore location.

Figure B-1 shows the time-mean velocity field at a transect taken near the midpoint

of the longshore domain. From this figure, we see that there is significantly more

structure to the cross-shore circulation than is described by the velocity profiles taken

at discrete locations. Just as we discovered in Fig. 4-12 by plotting the time-mean,

depth-averaged cross-shore velocity, the mean velocity field between x = 5 m and

x = 6.5 m, shown in Fig. B-l, appears to be directed shoreward with very little

return flow near the bed. On the other hand, seaward of x = 4 m there is a large,

conservative circulation cell that is also evident in the velocity profiles plotted in

Figs. 4-11B-4-11D.

It is possible that conservation in this area (5 m < x < 6.5 m) is satisfied,

however, if the excess cross-shore velocity is balanced by some physical storage

mechanism. This mechanism can be described through a modified form of the

continuity equation (Eq. B-l) that relates the time-dependent free surface to gradients

in the velocity fields (Dean and Dalrymple, 2002). Here, we are more concerned with












A) 05

04
0 4. 1- --- -,
03 -- --




0 1


00 2 4 6


B) 05

04

03

02

01

0


Cross-shore (m)




- -


4
Cross-shore (m)


Figure B-l: The time-mean velocity field taken at z = 2.5 m and plotted as A)
velocity vectors and B) as streamlines. The red, dashed line represents the still water
level at initialization.


the cross-shore gradients in the u velocity field since the remaining velocity gradient

terms are quite small in comparison.



oa auh awh
aT + x + O = 0 (B-1)
at Ox Oz

The time-averaged free surface plotted in Fig. B-2B shows that there is a positive

displacement of the free surface near the area of the excess cross-shore velocity, which

is shown again in Fig. B-2A. In order to see if these two physical processes were

spatially related, they were each scaled by their maximum or minimum departure,

depending on the sign of the value, thereby non-dimensionalising the two fields.































2 4 6
Cross-shore (m)


2 4 6
Cross-shore (m)


8 1


(UW (m/s)
S10
008
006 E
0 04
0 02
000 M
-0 02
-0 04 r
-0 06 -I
-0 08
-010


8 10


2 4 6
Cross-shore (m)


Co-hAe (n
4 6 8 10
Cross-shore (m)


Figure B-2: Spatial features of the average velocity and free surface fields. A contour
plot of the time-mean A) depth-averaged cross-shore velocity, B) free surface field, C)
scaled velocity (flood) and free surface (lines) fields and D) spatial correlation between
the depth-averaged cross-shore velocity and the wave-induced setup and setdown.



Scaling for the velocity and free surface fields was accomplished using the following


methodology:



Umax = max[(u1, )1


Umin = min[(u1,,)1


(Ux,) where u U max for (u ,) > 0
U Uin for (u,) < 0


- I,







ii L


(1l (m)
1 2E-03
9 6E-04
7 2E-04
4 8E-04
2 4E-04
OE+00
-2 4E-04
-4 8E-04
-7 2E-04
9 6E-04
-1 2E-03


8 1


I I I I I I I 1 1 1 1 1









imnax m ax[(yx )]

7i. = min[(rY,,)]


,z = --- where '7s fo(,
m, rimin for (,z) < 0

where u* and rl*, are the scaled velocity and free surface fields respectively. This

scaling process removes the dimensionality of each variable and results in velocity and

free surface fields that vary between -1 and 1:

-1< <1 and 1< rl <1.


The scaled velocity and free surface fields are plotted in Fig. B-2C. We see that

maximum positive displacements of the time-averaged free surface field (contour lines)

correspond to locations where there is also a maximum, shoreward-directed velocity

(contour flood). The correlation between these scaled fields was calculated and the

correlation coefficient is plotted in Fig. B-2D. While we know that correlation does

not necessarily imply causation, Fig. B-2D suggests that there is a spatial correlation

between these two physical processes. Therefore, it is quite possible that the excess

depth-averaged cross-shore velocity evident in Fig. B-2A is being stored as potential

energy in the form of setup, particularly over the shallow sill. A great deal of time

was spent attempting to use Eq. B-1 to explain the cause-and-effect relationship of

this cross-shore force balance, but the process proved rather difficult; the high degree

of spatial and temporal variability made it very difficult to show that this equation is

satisfied at each time-step, and over all grid locations, to sufficient precision in the

finite difference model.















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BIOGRAPHICAL SKETCH

I was born and raised in Fort Myers, Florida. As a child most weekends consisted

of sailing around San Carlos Bay and Pine Island Sound on my father's boat, often

times stopping off at local islands and beaches to play and relax. Occasionally my

parents would take my sister and me on longer trips: weekend journeys to Naples or

Marco Island and even week-long trips to the Dry Tortugas. I logged my first long

distance cruise when I was just a few months old, thus beginning my fascination with

oceans, islands, water, and beaches.

Obtaining my SCUBA certification at the age of 13 allowed me to learn about and

experience what life is like below the surface of the water. Up until this point, most

of my time was spent sailing on the surface of the water with very little knowledge

of what was happening below me. Sailing and scuba diving afforded me many

opportunities to learn about weather and ocean processes and even from an early age

I knew that I wanted to spend my life learning more about both. In the summer of

1996, having just turned 17, I was given the opportunity to do some part-time work

for a consulting firm that specialized in coastal and oceanographic engineering. I was

exposed to a number of different projects that summer and was immediately fascinated

by the work. At the end of the summer, I was quite sure that I had found my passion.

I transferred to Bishop Verot Catholic High School in August 1994 after leaving

the public school system at the end of ninth-grade. It was difficult changing schools,

but it was certainly one of the best decisions I made as a teenager. Small class sizes

and attentive, capable teachers made the learning experience in the classroom that

much more enjoyable. While the public school system provided me with a strong

educational foundation in science, mathematics, and language arts, the teachers at









Bishop Verot encouraged both critical and creative thinking ... something I found much

more stimulating.

After graduating from Bishop Verot in 1997, I went on to study civil engineering

at the University of Florida. I found the faculty to be supportive and many encouraged

me to continue my studies after obtaining a bachelor's degree. Since coastal

engineering was my true interest, it seemed appropriate to obtain a specialization

in this area by continuing on to graduate school. Although I applied to many

different programs, after graduating with a Bachelor of Science in Civil Engineering

in December, 2001, I received a wonderful offer to continue studying at the University

of Florida under Don Slinn. The decision to remain in Gainesville was made even

easier by the fact that my girlfriend was working toward obtaining a master's degree in

education. Soon after beginning graduate school, I proposed to Shannon and we were

married the next year. My wife made graduate school much more tolerable and was

always the voice of encouragement at the end of a frustrating day.

Graduate school has been, for the most part, a wonderful experience. In April of

2002 I was awarded a stipend from the Association of Western Universities to perform

research at the Naval Research Laboratory at Stennis Space Center for a twelve-week

period. During that time, I had the chance to assist in a laboratory experiment at the

U.S. Army Corp's Waterway Experiment Station located in Vicksburg, Mississippi,

where I learned about particle image velocimetry (PIV) measurement techniques and

data collecting. A year later, in October 2003, I assisted scientists from the Naval

Research Lab with the NCEX (Nearshore Canyon Experiment) field experiment in

La Jolla, California. The opportunity to assist in laboratory and field experiments,

combined with traditional learning in the classroom, has enriched my education and has

allowed me to apply theoretical science to explain real and observed processes.