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Model based Volcanic Plume Propagation with Parametric Uncertainty

Permanent Link: http://ufdc.ufl.edu/UFE0045558/00001

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Title: Model based Volcanic Plume Propagation with Parametric Uncertainty
Physical Description: 1 online resource (37 p.)
Language: english
Creator: Lin, Hongnan
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: 1volcanicplumesystem -- 2deterinisticparemeters -- 3parametricunvertainty
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This thesis considered model based plume propagation for volcanic eruptions. The interaction between the volcanic plume system and wind is considered, accounting for enhanced entrainment of air and horizontal momentum, distortion of the plume and a decrease of maximum plume rise height at the same mass eruption rate. To obtain the system response in the presence of uncertainties in system parameters, a generalized polynomial chaos approach is used. The resulting residual is minimized using a stochastic Galerkin method, leading to a set of integration differential equations that must be solved numerically. The final column height is computed and compared for the deterministic and uncertain cases.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Hongnan Lin.
Thesis: Thesis (M.S.)--University of Florida, 2013.
Local: Adviser: Kumar, Mrinal.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045558:00001

Permanent Link: http://ufdc.ufl.edu/UFE0045558/00001

Material Information

Title: Model based Volcanic Plume Propagation with Parametric Uncertainty
Physical Description: 1 online resource (37 p.)
Language: english
Creator: Lin, Hongnan
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: 1volcanicplumesystem -- 2deterinisticparemeters -- 3parametricunvertainty
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This thesis considered model based plume propagation for volcanic eruptions. The interaction between the volcanic plume system and wind is considered, accounting for enhanced entrainment of air and horizontal momentum, distortion of the plume and a decrease of maximum plume rise height at the same mass eruption rate. To obtain the system response in the presence of uncertainties in system parameters, a generalized polynomial chaos approach is used. The resulting residual is minimized using a stochastic Galerkin method, leading to a set of integration differential equations that must be solved numerically. The final column height is computed and compared for the deterministic and uncertain cases.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Hongnan Lin.
Thesis: Thesis (M.S.)--University of Florida, 2013.
Local: Adviser: Kumar, Mrinal.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045558:00001


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MODELBASEDVOLCANICPLUMEPROPAGATIONWITHPARAMETRICUNCERTAINTYByHONGNANLINATHESISPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFMASTEROFSCIENCEUNIVERSITYOFFLORIDA2013

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c2013HongnanLin 2

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Idedicatethistoeveryonethathelpedmeduringthisprocess. 3

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ACKNOWLEDGMENTS Firstofall,IwouldliketoexpressmygratitudetoDr.MrinalKumaratMechanicalandAerospaceEngineeringofUniversityofFlorida,whogavemealotofhelpofmyMasterofScienceThesisProject.Thanksgotohimnotonlyforhissupervisionandvaluablecontributionsconcerningmywork,butalsofortheimprovementofsolvingproblems.Moreover,Ialsowanttothankformyparentwhogavememuchencouragementandcondenceduringtheprocess.Besides,specialthanksgotoZinanZhaoandYifeiSunbecauseinthelab,wecontributetogoodatmospheretodoresearchtogetherandtheygavememanyhelpfuladvice.Finally,Iwouldliketothankallthepeoplewhohelpme. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 6 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 8 CHAPTER 1INTRODUCTION ................................... 9 2MODELOFPLUMEMOTIONINWIND ...................... 14 2.1Equationsofdynamicsofavolcanicplume ................. 14 2.2DiscussionaboutVolcanicPlumewithDeterministicParameters ..... 17 3VOLCANICPLUMEEVOLUTIONUNDERPARAMETERUNCERTAINTY ... 22 3.1StochasticSystems .............................. 22 3.2IntroductionofgPCexpansionandorthogonalproperty .......... 23 3.2.1ThegPCexpansion .......................... 23 3.2.2StochasticGalerkinMethod ...................... 25 3.2.3GeneralProcedure ........................... 26 3.3gPCGalerkinMethodtoStochasticVolcanicPlumes ............ 26 3.4DiscussionaboutVolcanicPlumewithUncertainParameters ....... 28 4CONCLUSION .................................... 34 REFERENCES ....................................... 36 BIOGRAPHICALSKETCH ................................ 37 5

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LISTOFTABLES Table page 3-1CorrespondencebetweentheTypeofGeneralizedPolynomialChaosandTheirUnderlyingRandomVariables. ........................ 25 6

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LISTOFFIGURES Figure page 2-1diagramillustratingavolcanicplumebehaviorintwotypesofcoordinates. ... 15 2-2variationofmassrate. ................................ 18 2-3variationofmomentumrate. ............................. 18 2-4variationofangle. .................................. 18 2-5variationofenthalpyrate. .............................. 18 2-6Relationshipbetweenriseheightandaxialvelocity. ................ 18 2-7RelationshipbetweenriseheightandTemperature. ................ 18 2-8Relationshipbetweenriseheightandbulkdensity. ................ 19 2-9EruptionColumnHeight. ............................... 20 3-1variationofmassrate. ................................ 29 3-2variationofmomentumrate. ............................. 29 3-3variationof. ..................................... 29 3-4variationofspecicenthalpy. ............................ 29 3-5massratewithd=2)]TJ /F7 7.97 Tf 6.58 0 Td[(8mm. ............................. 31 3-6massratewithd=2)]TJ /F7 7.97 Tf 6.58 0 Td[(3mm. ............................. 31 3-7massratewithd=2)]TJ /F7 7.97 Tf 6.58 0 Td[(2mm. ............................. 31 3-8massratewithd=2)]TJ /F7 7.97 Tf 6.58 0 Td[(1mm. ............................. 31 3-9massratewithd=25mm. .............................. 31 3-10massratewithd=210mm. ............................. 31 3-11AComparisonofEruptionColumnHeightsintwocases. ............. 32 7

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AbstractofThesisPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofMasterofScienceMODELBASEDVOLCANICPLUMEPROPAGATIONWITHPARAMETRICUNCERTAINTYByHongnanLinMay2013Chair:MrinalKumarMajor:MechanicalEngineeringThisthesisconsideredmodelbasedplumepropagationforvolcaniceruptions.Theinteractionbetweenthevolcanicplumesystemandwindisconsidered,accountingforenhancedentrainmentofairandhorizontalmomentum,distortionoftheplumeandadecreaseofmaximumplumeriseheightatthesamemasseruptionrate.Toobtainthesystemresponseinthepresenceofuncertaintiesinsystemparameters,ageneralizedpolynomialchaosapproachisused.TheresultingresidualisminimizedusingastochasticGalerkinmethod,leadingtoasetofintegrationdifferentialequationsthatmustbesolvednumerically.Thenalcolumnheightiscomputedandcomparedforthedeterministicanduncertaincases. 8

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CHAPTER1INTRODUCTIONVolcaniceruptionscanaffectourlivesinavarietyofwaysfromcausingcatastrophicdamagetobeingahindranceinourdailylives.Particlefalloutfromeruptioncolumnsanddispersingplumesisamajorhazardtohumanslivingclosetovolcanos.Thehazardsofvolcanicplumessuchasfalloutofparticles,mudandacidrain,thepresenceofnoxiousgases,lightningstrikesandshockwavesarealsoassociatedwitheruptions.Manyofthesedangerscouldhappensimultaneouslyduringasingleeruption.Particlefalloutthreatenshumansafetybecauseitcouldgeneraterespiratoryandeyeproblemsinhumans.Themostseriousdangerforanimalsiscontaminationofthefoodsupplyresultingfromthedepositsofashbearingtoxichalogenprecipitatesonthevegetationtheyfeedon.Forexample,thepoisoningofapproximately7500farmanimalsfollowedthe1970eruptionofHeklawhenuorinecontentsofupto4000ppmwererecordedonash-contaminatedvegetation.Besides,theheavyfalloutfromvolcanicplumescancauseenormousdamagetoproperty.TakingtheeruptionofEyjafjallajokullin2010inIcelandasanexample,thevolcanicplumeroseto7000metersaltitudetoformavolcanicashcloud,andproceededtoexpandtoalargeareaduetothewind.ThisledtoacompleteshuttingdownofairtransportacrossEurope.Over5days,Europeanairlinescanceledatotalofmorethan70,000ightsandsufferedalossof200millioneurosperday.Consideringsuchseriousdamage,thereisaneedtocomeupwithvariousmethodstobothavoiddestructionandaccuratelypredicteruptionsofvolcanoessoastosafelyevacuatepeoplelivingclosetovolcanoes.Sofarsignicantprogresshasbeenmadetopredictvolcaniceruption.Inparticular,3methodshavehadsomesuccess.First,detectingseismicwavesiskeytopredictvolcaniceruptions.Somevolcanoesnormallyhavecontinuallow-levelseismicactivity,butanincreasemightsignalanimpendingeruption.Second,becausethesimilaritiesbetweenvolcanicandicebergtremors 9

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includelongdurationsandamplitudes,aswellascommonshiftsinfrequencies,abettermethodforpredictioneruptionsthroughobservingicebergtremorshasbeendeveloped.Finally,remotesensingcanalsopredictvolcaniceruption.Asatellite'ssensorsdetecttheelectromagneticenergywhichisabsorbed,reected,radiatedorscatteredfromvolcano'ssurfaceoreruptedmaterialinaneruptioncloud.However,thereisacommondrawbackinthattheabovemethodsonlypredictwhenandwhereavolcanowilleruptinsteadofaccuratelypredictinghowmuchareawillbeseriouslyaffectedinordertopreventorreduceasmuchdamageaspossible.Asforthepredictionofthedistributionofvolcanicplumes,althoughtheresearchinthisareahasn'tbeenperfected,someprogresshasbeenmade.Modelingthedynamicsofavolcaniceruptionplumeisagoodwaytosolvethisproblem.Beforedevelopingaphysicalmodelofthedynamicsforvolcanicplumes,itisessentialtoclarifyplumerisetheory.Inastillenvironment,therearethreebasiczonesofaplumebasedonthedominantforcesthatcontrolthemotionofplume.Thesezonesaregasthrustregion,convectiveregionandumbrellaregion.Forgasthrustregion,themomentumofowdominatestheplumemotion.Inthisregion,amixtureofhotgasandpyroclastsisejectedfromthevolcanicventatahighspeedandatmosphericcoolaircouldbeentrained,heatedbytheeruptedthermalenergyandthenisexpanded.Ifsufcientairisabletobeentrainedandheated,bulkdensityoftheplume,whichisdensityofmixturewhichconsistofpyroclastsandgascouldbecomesmallerthanthatofsurroundingatmosphere.Thisphenomenoncausestheplumemotioncouldenterintonextstageconvectiveregion.Inconvectionregion,buoyantforceswhichresultsfromtheentrainmentandheatingofatmosphereairmainlycontrolsthebehaviorofthevolcanicplume.Therstinstanceofparticlefalloutoccursintheconvectionregion,startingwithlargestsizedparticles.However,themajorityofpyroclastsaretransferredtoagreaterheightduetotheforcesofbuoyancy.Intheumbrellaregion,becauseofvariationofthedensityofatmosphere,thevolcanicplumewillrisetoaheightwhere 10

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itsbulkdensityisequaltothatofsurroundingatmosphere.Thenthevolcanicplumecontinuestorisebecauseofitsinertiaaswellasexpandlaterally.Finally,itattainsamaximumheightuntiltheupwardspeeddecreasestozero.Thisprocessisalsoimportantinthatmostofsedimentationtakesplaceinthisphase.Theabovedescribedprocedureisconrmedbyrecenteruptionsandlaboratoryexperiments.Themotionofvolcanicplumesisaverycomplexprocessthatinvolvesnumerousfactors.Onesuchfactoriswindentrainmentthatleadstoplumebending.Plumebendingisacomplexissuewhichisofgreatimportanceforhumanbeings.Thisissuehasbeenaddressedbyseveralresearcherswhoareinterestedinthemovementofplume.Monron[ 1 ],whoseworkissolidfoundationinthisarea,modeledthebulkmotionofabuoyantplumeinmoistatmosphereintermsofconservationofmass,momentumandenergy(intermsofspecicenthalpy).Briggs[ 4 ],tookaderivationofplumeriseforbuoyantplumes,underbothstillandwindyconditions,inlaboratoryandnaturebyemployingdimensionalarguments.SlawsonandCsandy[ 10 ]derivedaplumeriserelationforbent-overplumesincross-windbyusingauidmechanicalentrainmentmodelandcomparedtheirresultswithobservations.Hewett[ 6 ]developedatheoreticalmodelofmovementofbuoyantsmokestackplumesinastableatmospherebasedonpreviousmodelstopredictplumerise.Wright[ 11 ]consideredthemovementofbuoyantjetsindensity-stratiedcrossowtopredictthemaximumriseheights.ErnstG.G.J[ 5 ],developedatheoreticalmodelwhichleadstoapredictionforsedimentationfromturbulentjetsandplumesinastillenvironment;M.Bursik[ 8 ],consideringwindentrainmenttoplumebending,modeleddynamicsofavolcanicplumeinaplume-centeredcoordinatesystemsoastopredictmaximumplumeriseheightandexplainsedimentationofpyroclasts.Allthesestudiesinvolveadeterministicanalysisoftheplumedynamics.Themainissuesarethatthedynamicalsystemishighlynonlinearanddependingonthedegreeofsophistication,relativelyhighdimensional.Noneofthesestudiesconsidertheproblem 11

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ofuncertaintyintheparametersofthesystem.Thedynamicsystemofvolcanicplumemotioncontainsnumerousparameters,severalofwhicharebasedonheurisicandorempiricalmodels.Asaresult,thereisroomforsignicantuncertaintyinthemthatmayinuencethepropagationoftheplume[ 7 ]Asweknow,infact,thereisuncertaintyforsomeinputparametersinthemodelofdynamicsofvolcanicplumes.Inordertosolvetherandomnessofthesystem,generalizedPolynomialChaos(gPC)[ 3 ]canbewellemployed.ThedevelopmentofgPCapproachisbasedonpolynomialchaos(PC).Wiener[ 9 ]rstlyintroducedPCexpansionandusedHermitepolynomialstomodelstochasticprocesseswithGaussianrandomvariables.Animportantpaper,CameronandMartin's[ 2 ]showedthatPCexpansionconvergestoL2senseforanyrandomprocesswithnitesecondmoment.However,PCwasdevelopedonlyforinstandardGaussianstochasticprocesses.SoastogeneralizetheresultofCameron-MartintononGaussiananddiscretedistributions,XiuandKarniadakis[ 3 ]developedgeneralizedPCmethodonthebasisofthecorrespondencebetweenpolynomialfunctionsderivedfromtheAskey-scheme[14]andtheweightfunctionwhichistheprobabilitydensityfunctionoftherandomvariable.Inthiswork,wepresentanalysisoftheeruptioncolumnheightandotherstateslikemassrate,momentumux,theanglebetweentheplumecenterlineandthehorizon,underdifferentwind-speeds.Oneisthatcompletedynamicsisdeterministicallyknown(DeterministicCase)inChapter2,theotheristhatoneofparametersisassumedtobearandomvariable(StochasticCase)inChapter3.Amodelofdynamicsofvolcanicplumesinastandardstate-spaceisrederivedbasedonM.Bursik[ 8 ]includingtheinteractionofwind.Thenarelationshipbetweenmaximumriseheightsofplumesandvariousmasseruptionratesisshownunderdifferentwindspeedsindeterministiccase.Moreover,inrandomcase,generalizedPolynomialChaosandStochasticGalerkinmethod,GaussQuadratureandstandard4th-orderRunge-Kuttaschemeareincorporatedtosolvethestochasticsystem.Weusetheaveragesofstatesin 12

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randomcasetocomparethoseindeterministiccaseanduseaveragewithin3standarddeviationtoshowthedifferenceofapproximationsinbothtwocases.Thispaperisorganizedasfollows:inChapter2,themodelofdynamicsofvolcanicplumesconsideringwindeffectwithdeterministicparameterswillbeintroducedandreviewed.InChapter3,generalizedPolynomialChaosGalerkintechniquewillbeemployedtohandletheparametricuncertaintyofthesystem.InChapter4,conclusionsaredrawn. 13

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CHAPTER2MODELOFPLUMEMOTIONINWINDInthischapter,weconsiderthedeterministicanalysisofvolcanicplumepropagation.Aspreviouslymentioned,thedynamicsofplumepropagationishighlycomplexandmustcapturenumerousphysicalprocesses.Forexample,windcanaffectthemotionofvolcanicplumesandpyroclastsfallingfromavolcanicplume.Smallvolcanicplumescanbeeasilybentoverbymoderatewinds.Inthefollowingsection,amodelforvolcanicplumesthatentrainatmosphericairhasbeendevelopedbasedonthepaperbyBursik[ 8 ]totrackbehaviorsofvolcanicplumesinaplume-centeredcoordinatesystem.ThersttwoEquations2-1and2-2belowexpressthecalculationofplumetrajectoryz=Zs0sinds(2.1)x=Zs0cosds(2.2)wherezverticaldirectionandxrepresentshorizondownwinddirection.Thevariablesmeasuresdistancesalongtheplume(i.e.tangentialtoit)andtheanglerepresentstheinclinationbetweentheplumecenterlineandthehorizonaxis.ThesevariablesareillustratedinFigure2-1: 2.1EquationsofdynamicsofavolcanicplumeThebasicequationsofplumepropagationarebasedontheprinciplesofconservationofmass,momentumandenergy.Accordingtomassconservation,wehavethefollowingequation:dY(1) ds=2aY1Y2 p U+23Xi=5dY(i) ds(2.3)whereY(1)=b2U(2.4) 14

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Figure2-1. diagramillustratingavolcanicplumebehaviorintwotypesofcoordinates. 1 =1)]TJ /F4 11.955 Tf 11.95 0 Td[(n +n a(2.5)Y1Y2=s Y(1)2 Y(2)(2.6)U=jU)]TJ /F4 11.955 Tf 11.96 0 Td[(VcosY(3)j+jVsinY(3)j(2.7)Y(1)representsmassrateofthematerialintheplumeandY(2)isthemomentumrateoftheplume(describedingreaterdetailbelow),Y(i)ismassrateofpyroclastsofithsizewithintheplume,isbulkplumedensitywhichincludesentrainedairandvolcanicgases.ItsexpressionisgivenasEquation2-4.InEquation2-4,isthedensityofpyroclastsandnrepresentsthemassfractionofgas.ThevariableinEquation2-5,brepresentsplumecolumnradius,Uistheaxialplumespeed,arepresentsambientatmospheredensity.UisentrainmentvelocitywhichisafunctionofwindspeedVand 15

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axialplumespeedUandequationisobtainedfromHewettetal.,1971.InEquation2-3,thersttermontherightsideisintermsofmassratebyentrainmentofairwhilethesecondtermisthelossofmassratebyfalloutofpyroclasts.Foraxialmomentumconservation,wehavethefollowingequation:dY(2) ds=Y1Y22 ()]TJ /F3 11.955 Tf 11.96 0 Td[(a)gsinY(3)+VcosY(3)dY(1) ds+Y(2) Y(1)23Xi=5dY(i) ds(2.8)whereY(2)=b2U2(2.9)U=Y(2) Y(1)(2.10)Y(2)representstheaxialmomentumoftheplume.ThersttermofEquation2-8ontherightsideisthechangeinmomentumresultingfromthecomponentofgravitationalacceleration,g,intheaxialdirectionandthesecondtermrepresentsentrainmentofmomentumfromthewindwhilethethirdtermisthemomentumofallkindsofpyroclasts.Theequationforconservationofradialmomentumisgivenby:dY(3) ds=1 Y(2)")]TJ /F3 11.955 Tf 9.3 0 Td[((Y1Y2)2 ()]TJ /F3 11.955 Tf 11.96 0 Td[(a)gcosY(3))]TJ /F4 11.955 Tf 11.96 0 Td[(VsinY(3)dY(1) ds#(2.11)Y(3)isintermsoftheanglebetweentheplumecenterlineandhorizonaxial.Thechangeinthetaresultedfrombothgravitywhichisthersttermontheright-handsideandwind,thesecondtermontheright-handside.Forspecicenthalpyconservation,wehavethefollowingequation:dY(4) ds=2Y1Y2 p UaCaTa)]TJ /F3 11.955 Tf 11.95 0 Td[(Y1 gsinY(3)+CpY(4) CY(1)23Xi=5dY(i) ds(2.12)whereY(4)=b2UCvT(2.13)Y(4)isspecicenthalpyrateandthersttermontheright-handsiderepresentsenergyofentrainmentofair,thesecondoneisthechangeinthermalenergycausedby 16

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conversiontogravitationalpotentialenergy,andthelasttermislossofheatresultingfromsedimentationofpyroclasts.Carepresentsheatcapacityoftheair,Taisthetemperatureoftheair,andCpisintermsoftheheatcapacityofthepyroclastsintheplume.InEquation2-13,CvisheatcapacityofmaterialintheplumeandTistemperatureofthematerial.Forconservationofmassrateofdifferentkindsofparticlesisgivenas:dY(i) ds=)]TJ /F5 11.955 Tf 9.3 0 Td[(2PS(i)]TJ /F5 11.955 Tf 11.96 0 Td[(4)Y(i) Y1Y2Y2Y1+6fY(2) Y(1) 5(Y1Y2)2 (2.14)wherePS=0.226(sinY(3))2+0.5(cosY(3))2(2.15)f=0.43 1+)]TJ /F7 7.97 Tf 6.67 -4.97 Td[(0.78 6(2.16)=)]TJ /F9 7.97 Tf 11.44 -4.8 Td[(F0 LMF1 3 (2.17)Y(i)representsmassrateformultipleparticlesizes,fisre-entrainmentparameter,F0isspecicthermaluxatthevent,isgivenassettlingspeedofaparticleinthegivensizeclassandLMFisintermsofmomentumuxatthevent.fisre-entrainmentparameterandbecausethediametersofparticlesthataresufcientsmallerthancouldbesweptbackatlowheightbythebackowintotheplumeafterfallingoutfromlargeheight. 2.2DiscussionaboutVolcanicPlumewithDeterministicParametersFiguresfrom2-1to2-4displayvariationsoffourstatesinthemodelwithwind-speed25m/snamelythemassrate,momentumrate,andenthalpyrate.Theinitialmasseruptionrateisassumedtobe8.2243107kg=swithinitialplumeradius,b0=500mandventspeedU0=100m=s.Massrateincreasesalongthetrajectorybecausesurroundingairiscontinuouslyentrainedintotheplume.Themomentumrateincreasesuntiltheplumereachesneutralbuoyancyheightandthendecreases.Theplumerisestomaximumheightviaitsinertia.Consideringwindeffect,theplume 17

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Figure2-2. variationofmassrate. Figure2-3. variationofmomentumrate. Figure2-4. variationofangle. Figure2-5. variationofenthalpyrate. isdistortedtosomedegreeandthereforethereisacontinuousdecreaseof.Asforenthalpyrate,itincreasesduringthewholeprocess. Figure2-6. Relationshipbetweenriseheightandaxialvelocity. Figure2-7. RelationshipbetweenriseheightandTemperature. 18

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Figure2-8. Relationshipbetweenriseheightandbulkdensity. InFigure2-5,thebluecurvedisplaystherelationshipbetweenriseheightandaxialvelocityunderwind-speedbeing25m/s.Astheheightrises,axialvelocityrstlydecreases,thenincreasesandnallydecreasesto0.Thereasonforthischangeisthatinthegasthrustregion,thebulkdensityislargerthenthatofsurroundingatmosphere,thereforegravityishigherthanbuoyancyandtheaxialvelocitydecreasesrstly.Astheriseheightincreases,theairiskeptbeingentrainedintothevolcanicplumeandthecolumnbecomesbuoyantsothatmaterialintheplumecouldaccelerate.Figure2-6andFigure2-7showvariationsoftemperatureandbulkdensityunderthesamewind-speedoftheplumerelativetoheightrises.Bothtemperatureandbulkdensitydecreaseduringthewholeprocessandnallyfallbelowofthoseinenvironment.Becauseduringthewholeprocess,thevolcanicplumekeepsentrainingambientcoolairandsolidpyroclastsintheplumeheattheentrainedairwhichwillbeexpanded,thereisadecreaseoftemperatureandbulkdensity.InFigure2-8,thereare7curvesunder7differentwind-speeds.Whenthewind-speediszero,theeruptioncolumnheightishighestcomparedwiththatofanyotherwind-speedatthesamemasseruptionrate.Moreover,whenmaximumwindspeedinjetcaseisthesameasthatinconstantwindspeedcase,theeruptioncolumnheightisgenerallyhigherthanthatofconstantwindspeed.Besides,injetcase,thelargerthemaximumspeedis,thelowertheeruptioncolumnheightwillbe.Similarly, 19

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Figure2-9. EruptionColumnHeight. whenthewind-speedholdsaconstantvalue,thereisalsoaninverselyproportionalrelationshipbetweeneruptioncolumnheightandmassrate.Asthewind-speedsincreases,theeruptioncolumnheightdecreases.Besides,wemakealongitudinalcomparison,underthesamewind-speed,theeruptioncolumnheightincreasesasmassrateincreases.Actually,allvolcanicplumesarenallydistortedbywindtosomedegree.Aswind-speedbecomeshigher,therewillbemoreairentrainedtoavolcanicplumeandtheentrainmentcoefcientalsowillbelarger.Therefore,thebulkdensityofavolcanicplumewillreducefasterandresultsinadecreaseoferuptioncolumnheightatlast.It'sveryobviousforsmallweakvolcanicplumessufferinghighwind-speeds,aplentyofairwithhorizontalmomentumisentrainedintotheplumeandtheplumeiseasilybentover.Asaresult,theplumewillfailtoformumbrellaclouds.Whileforlarge,strongvolcanic 20

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plumeandlowwind-speed,theplumecouldnotonlyrisetomaximumheightbutalsocouldn'tbebentoverbywindaswellasthatinstillenvironment.Formostplumeswithatotalriseheightgreaterthan20km[ 12 ],there'shardlynowind-speedaffectingplumeriseandallvolcanicplumescoulddevelopumbrellaclouds.However,plumeswitheruptioncolumnheightlessthan10kmwillbeseriouslyaffectedbywind.Insummary,fromY(1)toY(23),allthevariationsof23statesareemployedtowelldescribetheevolutionofvolcanicplumemodel.Inaddition,severalinputparameterswhicharenotverypreciselymodelledcanmakesomeuncertainties.Iftheplumesystemissensitivetouncertaintiesoftheinputparameters,thesensitivitycancausevariationsinnalresults.Asaresult,weneedaframeworkthatcanhandlesuchparametricvariationstohelpusdrawsomeconclusionsaboutnatureofplumepropagation. 21

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CHAPTER3VOLCANICPLUMEEVOLUTIONUNDERPARAMETERUNCERTAINTYAsweknow,inmostengineeringapplications,weusuallyconvertaphysicalproblemintoamathematicalmodelinwhichalltheinputparametersaredeterministic.Inpreviouschapter,weapproximatethevariationsofstatesinacaseinwhichalltheparametersaredetermined.However,inreality,theseinputparameterslikere-entrainmentcoefcient,waterfractioncoefcient,windspeedmayshowrandomnesswhichcouldaffectthenalsolution.Thisrandomnessfailstobeconsideredinthedeterministiccase.Fortheuncertaintyinthemathematicalmodel,it'snecessarytousesomeprobabilisticmethodstosolvethisproblem.MonteCarloSimulation,astatisticalmethod,needsalargenumberofsamples.Thequalityoftheresultgreatlydependsonthesamplesweselectandgeneralizedstatisticsaredifculttoobtain.However,generalizedPolynomialChaos(gPC)iscommonlyappliedinframe-workforproblemsinvolvingparametricuncertainty.Inpracticalapplication,gPCmethodwhichwasdevelopedbyXiuandKarniadakis[ 3 ]hasbeenwidelyusedinuidmechanics,optimalcontrolareastohandleparametricuncertaintyproblems.That'swhatI'mgoingtodoaswellinthischapter. 3.1StochasticSystemsInthissection,threepartswillbeintroduced.Atrst,stochasticsystemswillbeintroduced.Moreover,howgeneralizedpolynomialchaosexpansionandorthogonalpropertywillbepresented.Finally,specicstepstohandletheparametricuncertaintyofthevolcanicplumesystemwillbedisplayed.Stochasticmathematicalmodelsareonthebasisofaprobabilityspace(,A,P)whereistheeventspace,A2its-algebraandPitsprobabilitymeasure[ 3 ].Here 22

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apartialdifferentialequations(PDEs)ischosenasgeneralandbasicmodel.8>>>>>><>>>>>>:ut(x,t,Z)=L(u),D(0,T]Rd,B(u)=0,@D[0,T]Rd,u=u0,Dft=0gRd.(3.1)whereListhedifferentialoperatorwhichcanbenonlinear,Bistheboundaryconditionoperator,u0istheinitialconditionandZ=(Z1,Z2,...Zd)2Rd,d1,beasetofindependentrandomvariablesdescribingtherandominputs. 3.2IntroductionofgPCexpansionandorthogonalproperty 3.2.1ThegPCexpansionGeneralizedPolynomialChaosisbasedonPolynomialChaos(PC)whichisintroducedbyWienerin1938[ 9 ].HermitePolynomialswasusedbyhimtorepresentastochasticprocesswitharandomvariableofstandardGaussiandistributionanditsPCexpansionisgivenas:^X=NX0ckHk(Z)(3.2)whereZisstandardGaussianrandomvariable,Hk(Z)isHermitePolynomialandckistime-evolutioncoefcientofeachHermitepolynomial.MoreoverPCexpansionisonthebasisofatheoryofCameronandMartinwhichisgivenas:E(jX)]TJ /F5 11.955 Tf 13.64 2.65 Td[(^Xj2)=ZjX)]TJ /F5 11.955 Tf 13.65 2.65 Td[(^Xj2dFZ(z)!0(3.3)whereFZ(z)=P(Zz)isprobabilitydensityfunction.Whereas,PCexpansioncanonlysatisfythetheoryofCameronandMartinwhenHermitePolynomialscorrespondstoStandardGaussianrandomvariables.InordertomakethetheoryofCameronandMartinbeappliedinawiderarea,gPCwasdevelopedandtheexpressionofgPC 23

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expansionisshownas:Q(x,t,Z)=1Xj=0qj(x,t)j(Z)(3.3)wherej(Z)isasetoforthogonalpolynomialsandZ=fZj(w)gNj=0representsavectorofrandomvariablesandqj(x,t)istime-evolutioncoefcientsoforthogonalpolynomialbasisandiswhatweneedtodetermine.Consideringcomputationalrelevance,stochasticprocessesarerepresentedasgPCexpansionwithaniteorder.Duetotheordertruncation,thelastsolutionbecomesanapproximation.BeforeemployinggPC,aprerequisiteneedstobesatisedwhichisthereisaspecialcorrespondencebetweendistributionofrandomvariablesandorthogonalpolynomials.Intable3-1[ 13 ],someofthewell-knowncorrespondencesbetweentheprobabilitydistributionofZanditsgPCbasispolynomialsarelisted.OrthogonalpolynomialfunctionsneedtosatisfythefollowingconditionE[i(Z)j(Z)]=jij,i,j2M,(3.4)wherej=E[2j(Z)],j2M,(3.5)ij=8>><>>:0fori6=j;1otherwise.(3.6)jarethenormalizationfactors,M=f0,1,2,...,NgisaniteindexsetandijisKroneckerdeltafunction.Asweknow,generalizedpolynomialschaosworkwellinbothdiscreteandcontinuouscases.IfrandomvariableZiscontinuous,itsprobabilitydensityfunctionexistsasdFZ(z)=p(z)dzandorthogonalpolynomialfunctionscouldbewrittenasE[i(Z)j(Z)]=Zi(z)(j)(z)p(z)dz=jij,i,j2M,(3.7) 24

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Similarly,ifZisdiscrete,theorthogonalitycanbeexpressedas:E[i(Z)j(Z)]=Xii(zi)(j)(zi)p(zi)=jij,i,j2M,(3.8) Table3-1. CorrespondencebetweentheTypeofGeneralizedPolynomialChaosandTheirUnderlyingRandomVariables. DistributionofZgPCbasispolynomialsSupport ContinuousGaussianHermite(,1)GammaLaguerre[0,1)BetaJacobbi[a,b]UniformLegendre[a,b]DiscretePoissonCharlierf0,1,2,...gBinomialKrawtchoukf0,1,...,NgNegativebinoialMeixnerf0,1,2,...gHypergeometricHahnf0,1,...,Ng GeneralizedPolynomialChaoscouldbeappliedtostochasticprocessesrepresentedbyrandomvariablesofcommonlyuseddistributionsinsteadofthatofstandarddistribution.InordertoeffectivelyusegPC,thecorrespondencerelationshipbetweenprobabilitydensityfunctionofcertainrandomvariablesandtheweightfunctionsoforthogonalpolynomialsofAskey-scheme. 3.2.2StochasticGalerkinMethodAsmentionedintheprevioussection,gPCexpansionservesasacompletebasisofsolvingstochasticdifferentialequations.Inthispart,gPCexpansionandGalerkinmethodwillbecombinedtogetherforsolvingstochasticsystems.WeintroduceStochasticGalerkinMethodviaageneralstochasticpartialdifferentialequation(PDE).ThestochasticGalerkinprocedureisemployedtotransformthestochasticequationstoasetofdeterministicequations.Theseequationscanbediscretizedfromcontinuousonesbystandardnumericaltechniques.Afterusingthismethod,wecanhaveasetofcoupleddiffusionequationswhichcanbewritteninavectorormatrixform. 25

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3.2.3GeneralProcedureLetZ=(Z1,Z2,...Zd)2Rd,d1,beasetofindependentrandomvariablescharacterizingtherandominputs.Thestochasticsystemisgivenas8>>>>>><>>>>>>:ut(x,t,Z)=L(u),D(0,T]Rd,B(u)=0,@D[0,T]Rd,u=u0,Dft=0gRd.(3.9)whereListhedifferentialoperatorwhichcanbenonlinear,Bistheboundaryconditionoperator,u0istheinitialcondition.TherststepforsolvingstochasticsystemsistoapplygPCexpansiontoeachcomponentofuindividually.Forthesecondstep,accordingtothegPCbasisfunctionsEquation3-3,foranyarbitraryxandt,wecouldgetuN(x,t,Z)=NXj=0^uj(x,t)j(Z)(3.10)sothatforallKwhichislessandequalthanN,wewillobtain[Xiu]8>>>>>><>>>>>>:E[@tuN(x,t,Z)k(Z)]=E[L(uN)k],D(0,T],E[B(uNk)]=0,@D[0,T],^uk=^u0,k,Dft=0g.(3.11)where^u0,k=E[u0k] karetheinitialgPCprojectioncoefcients.Thestochasticsystembecomeoneofcoupleddeterministicequations.ThedimensionofthesystemisdimdN=0B@N+dd1CA(3.12) 3.3gPCGalerkinMethodtoStochasticVolcanicPlumesInthemodelofthevolcanicplume,f,re-entrainmentcoefcient,isregardedasarandomparameterwhichsatisesGaussiandistributionfN(,2)with0.43 26

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meanand0.0001variance.Onthebasisofprevioustwosections,weemploygPCGalerkinmethodtosolvestochasticsystemofvolcanicplumes.Atrst,23statesareexpandedviagPCexpansion.Accordingtocorrespondencebetweenthedistributionoffandorthogonalpolynomialsbasis,weapplyHermitePolynomialsfHm(Z)gtoworkasorthogonalpolynomialbasis.TherearetwodifferentwaysofstandarizingtheHermitePolynomials:probabilist'sHermitepolynomialsandphysicists'HermitePolynomials.Hereweapplyprobabilist'sHermitepolynomialsandtheexpressionsaregivenas:H0(Z)=1;H1(Z)=Z;(3.13)Hn+1(Z)=ZHn(Z))]TJ /F4 11.955 Tf 11.95 0 Td[(nHn)]TJ /F7 7.97 Tf 6.58 0 Td[(1(Z)(3.14)whereZN(0,1)beastandardGaussianrandomvariablewithzeromeanandunitvariance.ItsPDFisp(z)=1 p 2e)]TJ /F9 7.97 Tf 6.59 0 Td[(z2=2(3.15)whichissimilartoweightfunctionw(Z)w(Z)=e)]TJ /F9 7.97 Tf 6.59 0 Td[(Z2=2(3.16)Therefore,basedontheHermitepolynomialsandweightfunction,wecanobtainorthogonalpropertiesE[Hi(Z)Hj(Z)]=Z1Hi(Z)Hj(Z)w(Z)dZ=8>><>>:0fori=jp 2n!mnotherwise,(3.17)Asweknow,inHermitepolynomials,randomvariableisofstandardGaussiandistributionbutfdoesn'tsatisfynormaldistribution.Asaresult,werepresentsfintermsofZandtheexpressionisobtainedasf=+Z(3.18) 27

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where=0.43,=0.01Besides,viagPCexpansionwithorderN=2,the23statesinthesystemcouldberepresentedasY(i)=NXj=0cij(s)Hij(Z)(3.19).ThentakeY(i)intotheordinarydifferentialequationsofdynamicsofvolcanicplume,wecanobtaindY(i) ds=_Y(i)=NXj=0_cij(s)Hij(Z)=Li(s,Z,f(Z)).(3.20)whereListhedifferentialoperatorofthesystem.ByusingGalerkinprojectionaccordingtoEquation3-7and3-17,thederivativesofdeterministiccoefcientsofHermitepolynomialscanbederivedZf(NXj=0_cij(s)Hij(Z))Him(Z)w(Z)df=ZfLi(s,Z,f(Z))Him(Z)w(Z)df(3.21)wherem=0,1,...N,ifm=j,wecanobtain_cij(s)=RfLi(s,Z,f(Z))Hij(Z)w(Z)df Rf(Hij(Z))2w(Z)df(3.22)Assincreases,thecoefcientsalsochangeandcouldapplyastandardODEsolvertosolvetheresultingsystem.Inordertorealisethis,standardfourth-orderRunge-Kuttaschemeisemployedinthiswork. 3.4DiscussionaboutVolcanicPlumewithUncertainParametersApproximatedmean,i(s),andvariance,2i(s),canbedeterminedviausingcoefcientsweobtainedfromsolving(3.22).i(s)===NXj=0cij(s)(3.24)2i(s)=<(yi(s))]TJ /F3 11.955 Tf 11.95 0 Td[(i(s))2>=NXj=1(cij(s))2<(ij(Z))]TJ /F3 11.955 Tf 15.46 0 Td[()2>(3.25) 28

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Figure3-1. variationofmassrate. Figure3-2. variationofmomentumrate. Figure3-3. variationof. Figure3-4. variationofspecicenthalpy. FromFigure3-1toFigure3-4,those4guresdisplayevolutionsof4statesinthevolcanicplumesystemwithwind-speedbeing25m/s.Theinitialmasseruptionrateisstill8.2243107kg=swithinitialplumeraidus,b0=500mandoriginalvelocityU0=100m=s.Ineachgure,thereare4curvesandthebluecurverepresentsevolutionofastateinDeterministicCase,blackonedisplaysvariationofaverageofastateinStochasticCase.yup(i)=yaverage(i)+3,ylow(i)=yaverage(i))]TJ /F5 11.955 Tf 12.07 0 Td[(3;respectivelyredandgreenlines,areintermsofoutcomeswithin3standarddeviationofastatetoshowhowfarthenumbersliefromthemean.InFigure3-1,itdisplaysevolutionofmassratewhichincreasesacrossthewholeprocessbecauseofcontinuousentrainmentofambientair.WeobservethatgPCGalerkinsolution,yaverage(1),don'tdeviatefromthat,y1,inDeterministicCase. 29

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Moreover,ylow(1)andyup(1)arealmostthesameandtverywellwithothertwocurvesfromwhichstandarddeviation,isalmostzero.Therefore,thegPCGalerkinapproachpresentsgoodapproximationresults.InFigure3-2,itpresentsvariationofmomentumrate.Atrst,momentumratedecreasessinceinthegasthrustregion,theplumerisesviaitsinertia.Besides,intheconvectiveregion,momentumrateincreasesuntiltheplumereachestoneutralbuoyancyheight(aheightwhichitsdensityequalsthatoftheatmosphere)becauseinthisregion,buoyancywhichislargerthangravityleadstoaccelerationoftheplume.What'smore,themomentumratedecreasesagainsincestratieddensityresultsinbuoyancylessthangravityandtheplumerisestorestbyitsinertia.ForgPCGalerkinsolutions,yaverage(2)convergeswelltoy2inDeterministicCase.Takingylow(2)andyup(2)intoconsideration,thereisalsonodeviationrelativetoy2whichdisplaysthatstandarddeviationiszero.InFigure3-3,theanglebetweenplume-centerlineandhorizonkeepsdecreasingallthetimebecausewithsuchwindeffectofhighspeed,theplumeiseasilybentover.ThefourcurvesarealmostcoincidedwhichshowsthatgPCGalerkinoutcomessatisfyconvergencetoy3.FromFigure3-5toFigure3-10,those5guresdisplaytheevolutionofmassratesofdifferentsizesofpyroclastsintheplumeconsideringwindspeedof25m/s.Whenparticlesizearesmall,thepyroclastsintheplumecanbeheldinsteadofescapingoutofthemarginoftheplume.Whileforpyroclastsoflargesize,becausegravityishigherthanbuoyancy,theyareejectedatlowheights.InFigure3-5andFigure3-6,themassratesofpyroclastswithdiametersof2)]TJ /F7 7.97 Tf 6.58 0 Td[(8mm,2)]TJ /F7 7.97 Tf 6.58 0 Td[(3mm,and2)]TJ /F7 7.97 Tf 6.59 0 Td[(2mmalmostkeepthesameacrossthewholeprocess.However,forpyroclastswithdiameter2)]TJ /F7 7.97 Tf 6.58 0 Td[(1mm,themassratecontinuouslydecreasesasplumeheightrises.Asdiameterofpyroclastscontinuestoincrease,thedegreeofdecreaseofmass 30

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Figure3-5. massratewithd=2)]TJ /F7 7.97 Tf 6.59 0 Td[(8mm. Figure3-6. massratewithd=2)]TJ /F7 7.97 Tf 6.59 0 Td[(3mm. Figure3-7. massratewithd=2)]TJ /F7 7.97 Tf 6.59 0 Td[(2mm. Figure3-8. massratewithd=2)]TJ /F7 7.97 Tf 6.59 0 Td[(1mm. Figure3-9. massratewithd=25mm. Figure3-10. massratewithd=210mm. 31

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Figure3-11. AComparisonofEruptionColumnHeightsintwocases. rateofpyroclastsalsobecomesmoreintense.AsforpyroclastsinFigure3-9andFigure3-10withlargediameters,massratesbothdecreasetoalmostzero.Ineachgure,weobservethatfourcurvesarealmostcoincidedandgPCGalerkinsolutionsconvergewelltothosefromDeterministicCase.Asforylow(i)andyup(i),there'snodeviationrelativetoyaverage(i).Figure3-11presentsacomparisonoferuptioncolumnheightunderdifferentwindspeedsandmasseruptionratesinbothtwocases.Theapproximationresultsarethesame.Atconstantmasseruptionrate,asthewindspeedincreases,eruptioncolumnheightbecomessmaller.While,forthesamewindspeed,thereisapositivelyproportionalrelationshipbetweenmaximumriseheightandmasseruptionrate.Thelargermasseruptionrateis,thehighereruptioncolumnheightis.ForgPCGalerkinapproach,itprovidesaccurateapproximations. 32

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Insummary,ifthevolcanicplumesystemissensitivetouncertaintyofre-entrainmentparameter,f,thereshouldbesameevolutiontrendsbutsomedegreeofdeviationbetweenstatesofthesysteminbothtwocases.While,inreality,thevariationofstatesinbothtwocasesarealmostcompletelythesame.Asaresult,uncertaintyofffailstovarythebehaviorofvolcanicplumealot. 33

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CHAPTER4CONCLUSIONThemodelofdynamicsofavolcanicplumeishighlynonlinear,coupledandofhigh-dimension.Itinvolvesnumerousparameters,manyofwhicharedeterminedinaheuristicorempiricalfashion.Inaddition,therearetime-varyingparameterssuchbulkdensity.Therefore,thenumericalintegralofthissystemisnotatrivialundertaking.Forinstance,thereisscopeforsingularitiesinthesystemofequations(astheaxialvelocity,U,approaches0).Thesedifcultiesmustbeappropriatelyhandled.Wefoundthattheinteractionbetweenavolcanicplumeandwindcannotonlycauseenhancedentrainmentofairandhorizontalmomentumbutalsoleadtoplumebending.Moreover,thereisaninverselyproportionalrelationshipbetweenwindspeedandmaximumplumeriseheight.Thehigherthewindspeedis,thesmallerthemaximumplumeriseheightwillbe.Inaddition,asmasseruptionrateincreases,theeruptioncolumnheightalsobecomeslarger.Inthejetcase,maximumplumeriseheightapproximatelyremainsconstantacrossawiderangedespitethemasseruptionrateincreases.Inaddition,asmaximumwindspeedincreasesinthejetcase,therangeoverwhichmaximumplumeriseheightkeepsapproximatelythesamebecomesbroader.Asforvolcanicplumesunderuncertainparameter,weusedagPCGalerkinapproachtosimulatesystemdynamicsasanalternativetothebruteforceMonteCarlosimulation.TheresultsofMonteCarlosimulationgreatlydependsonqualityoftheselectedsamplesandgeneralizedstatisticsaredifculttoobtain.Whereas,gPCiscommonlyusedinframe-workforproblemsinvolvingparametricuncertainty.Overall,gPCsimplyprovidesamoreelegantsolutiontotheproblemunderconsideration.Duringtheprocessofrealizationofthismethod,it'sdifculttosolveintegraldifferentialequations.TheGaussQuadraturemethodwasusedtosolvenon-standardintegralandapplyitinconjunctionwiththeEulerMethodtosolvedifferentialequationsoforthogonalpolynomialcoefcients.However,theEulermethodwasnotsufcientlyaccurateto 34

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providederivedconvergenceoftheintegrationprocedure,duetothebuildupofroundofferrorsfromtheprevioussteps.Therefore,a4th-orderRunge-Kuttamethodwasemployedtosolvetheintegraldifferentialequationsandtheapproximationresultsweregood.FromFigure3-1toFigure3-11,wecanconcludethatgPCGalerkinmethodprovidesagoodwaytodealwiththesysteminvolvinguncertainparameters.Moreover,randomnessofre-entrainmentcoefcient,f,isn'tsosensitivetothesystemanddoesn'tvaryoutcomesalotinthatthereshouldbesomeerrorbetweentheapproximationsunderthosetwocases.Infact,re-entrainmentprocessissignicant(especiallyinconvectionregion),however,there-entrainmentcoefcientdoesn'taffecttoomuch.Although,gPCdoesnotgiveastarkresult,showinghighsensitivityofthedynamicsystemtof,itistillanimportantresultbecauseitshowsthatfisnotaverysensitiveparameterswhichwasconrmedbyMonteCarlosimulation.It'slikelythatthisresultcouldbeusedtoderivebettermodelsforvolcanicplumepropagation.Infuture,uncertaintyofotherparametersorcombinationofparameterslikewindspeed,V,willbecarriedontochecktheinuencetothedevelopmentoftheplumepropagation. 35

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REFERENCES [1] M.B.R.andT.G.I.,Turbulentgravitationalconvectionfrommaintainedandinstantaneoussources,ProceedingsofTheRoyalSocietyA,vol.234,pp.1,1956. [2] R.CameronandW.Martin,Theorthogonaldevelopmentofnon-linearfunctionalsinseriesoffourier-hermitefunctionals,Ann.Math,vol.48,pp.385,1947. [3] D.XiuandG.Karniadakis,Thewieneraskeypolynomialchaosforstochasticdifferentialequations,SIAM,vol.24,pp.619,2002. [4] B.G.A.,Aplumerisemodelcomparedwithobservations,JournaloftheAirPollutionControlAssociation,vol.15,pp.433,1965. [5] E.G.G.J,C.S.N.,andS.R.S.J.,Sedimentationfromturbulentjetsandplumes,Geo-phys,vol.101,pp.5575,1996. [6] T.HewettandJ.FAY,Laboratoryexperimentsofsmokestackplumesinastableatmosphere,AtmosphericEnvironment,vol.5,pp.767,1971. [7] U.Konda,T.Singh,P.Singla,andP.Scott,Uncertaintypropagationinpuff-baseddispersionmodelsusingpolynomialchaos,EnvironmentalModellingSoftware,vol.25,pp.1608,2010. [8] M.Bursik,Effectofwindontheriseheightofvolcanicplumes,GeophysicalResearchLetters,vol.28,pp.3621,2001. [9] W.N.,Thehomogeneouschaos,AmericanJournalofMathematics,vol.60,pp.897,1938. [10] S.P.R.andC.G.T.,Onthemeanpathofbuoyant,bent-overchimneyplumes,JournalofFluidMechanics,vol.28,pp.311,1967. [11] W.S.,Buoyantjetsindensity-stratiedcrossow,JournalofHydraulicEngineer-ing,vol.110,pp.643,1984. [12] R.Sparks,M.Bursik,andS.Carey,VolcanicPlumes.England:JohnWileyandSons,1997. [13] D.Xiu,NumericalMethodsforStochasticComputations.NewJersey:PrincetonUniversityPress,2010. 36

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BIOGRAPHICALSKETCH HongnanLinwasborninDalian,China.HeattendedNo.12highschoolinDalian.HewenttotheDepartmentofAutomotiveEngineeringin2007,andgothisbachelor'sdegreefromWuhanUniversityofTechnologyin2011.HethenwenttograduateschoolatUniversityofFloridain2011.HecompletedhisMasterofScienceinMechanicalEngineeringin2013. 37