Maximumsteepnessofoceanicwaves:FieldandlaboratoryA.Toffoli,A.Babanin,M - PDF document

Herefor A rticle Maximumsteepnessofoceanicwaves:FieldandlaboratoryA.Toffoli,A.Babanin,M.Onorato,andT.WasedaReceived15November2009;revised22January2010;accepted28January2010;published9March2010.2010.1]Thebreakingofwavesisanimportantmechanismforanumberofphysical,chemicalandbiologicalprocessesintheocean.Intuitively,wavesbreakwhentheybecometoosteep.Unfortunately,ageneralconsensusontheultimateshapeofwaveshasnotbeenachievedyetduetothecomplexityofthebreakingmechanismwhichstillremainstheleastunderstoodofallprocessesaffectingwaves.Toestimatethelimitingshapeofoceanwaves,herewepresentastatisticalanalysisofalargesampleofindividualwavesteepness.Datawerecollectedfrommeasurementsofthesurfaceelevationinlaboratoryfacilitiesandtheopenseaunderavarietyofseastateconditions.ObservationsrevealthatwavesareabletoreachsteeperprofilesthantheStokeslimitforstationarywaves.Duetothelargenumberofrecordsthisfindingisstatisticallyrobust.Citation:Toffoli,A.,A.Babanin,M.Onorato,andT.Waseda(2010),Maximumsteepnessofoceanicwaves:Fieldandlaboratoryexperiments,Geophys.Res.Lett.,L05603,doi:10.1029/2009GL041771.1.IntroductionIntroduction2]Thebreakingofdeepwatersurfacewavesisanin-trinsicfeatureoftheoceanandappearsintheformofspo-radicwhitecaps.Beneathandabovethesurfaceofbreakingwaves,amixtureofairandwatergeneratesaturbulentflowwhichisresponsiblefortheexchangeofgasses,watervapor,energyandmomentumbetweentheatmosphereandtheocean[,1996;Jessupetal.,1997].Theseprocessesplayaveryimportantroleinmanyphysical,chemicalandbiologicalphenomenaintheupperoceanlayerandloweratmosphere.Apartfrombeingdirectlyresponsibleforthedissipationofenergyinthewavefield[Komenetal.,1994],thebreakinggeneratesmarineaerosols[Jessupetal.,1997]whichinfluencecloudphysics,atmosphericradiationbal-anceandhurricanedynamics[MelvilleandMatusov,2002],changestheseasurfaceroughnesswhichmoderatestheairseamomentumandenergyexchange[Babaninetal.,2007b]andfacilitatestheupperoceanmixing[Babaninetal.2009].Hence,appropriateaccountforthewavebreakingphysicsandstatisticsisafundamentalpartofapplicationsrangingfromforecastingthewaves[e.g.,,2009]totheestimationoftheglobalweatherandclimate[Csanady1990].Furthermore,owingtotheviolentnatureofsteepbreakingwaves,shipsandoffshorestructuresmaysufferseriousdamagesespeciallyinharshseaconditions.conditions.3]Becausethewavebreaking(whitecapping)playssuchavitalroleattheairseainterface,thereisaneedforaccurateandquantitativeestimatesofitsproperties.Amongthem,theultimatesteepnessthatbreakingwavescanreachisofparticularinterest.Unfortunately,breakingisaverycomplicatedprocessandsuchpropertieshavebeenelusivefordecades.Afullunderstandingofthismechanismandtheabilitytoquantifyithavebeenhinderedbythestrongnonlinearityoftheprocess,togetherwithitsirregularandintermittentnature(anextendedreviewofthebreakingmechanismcanbefoundinworkbyby)[4]Intuitively,itisreasonabletoassumethatanindi-vidualwavemaynolongersustainitsshapeandhencebreakwhenitsheightbecomestoolargewithrespecttoitslength,i.e.,thewavebecomestoosteep.Overacenturyago,[1880]predictedtheoreticallythataregular,sta-tionaryprogressiveonedimensionalwavewouldbecomeunstableandbreakonlyiftheparticlevelocityatthecrestexceededthephasevelocity.Intermsofwaveprofile,thiscorrespondstoawavehavingananglebetweentwolinestangenttothesurfaceprofileatthewavecrestof120°(i.e.,60°oneachside).Indeepwater,[1893]foundthatthisparticularshapeimpliesthatthewaveheight()is0.14timesthewavelength(),whichcorrespondstoawave/2=0.44,whereisthewavenumber.wavenumber.5]However,finiteamplitudeStokeslikewavestendtobeunstabletomodulationalperturbations[,1966;BenjaminandFeir,1967].Thus,aninitiallyregularwavetraindevelopsintoaseriesofwavepackets.Withinsuchgroups,individualwavescanthengrowandeventuallybreak[HigginsandCokelet,1978;,1982].Interestinglyenough,recentnumericalandlaboratoryexperiments[DyachenkoandZakharov,2005;Babaninet,2007a]revisitedtheprocessofmodulationalinstabilityandconsequentbreakingforinitialquasidimensionalwavetrainswithmeansteepnesswellbelowthevalueforthelimitingStokeswave.Thesestudiesshowedthatthewavesteepnessofunstableindividualwavesdoesgrowuptothethresholdvalueof/2=0.44,afterwhichtheirreversibleprocessofbreakingbegins.begins.6]Anotherpossiblemechanismforthedeepwaterwavebreakingislineardispersivefocusingofwaves[Rappand,1990;Piersonetal.,1992].Suchfocusingwillleadtoabreakingonsetalsoatasteepnessof/2=0.440.44BrownandJensen,2001].Mostofthewaveresearchhasbeenconductedinquasienvironments[RappandMelville,1990;Piersonetal.BrownandJensen,2001],butthedirectionalfocus-inghasalsobeenhighlightedasapossiblebreakingcauseofsteepcoherentwavetrains[Fochesatoetal.,2007].2007].7]Onedimensionalstudiesdealwithsimplificationofrealoceanwavesastheyexcludeeffectsrelatedtothe FacultyofEngineeringandIndustrialSciences,SwinburneUniversityofTechnology,Hawthorn,Victoria,Australia.DipartimentodiFisicaGenerale,UniversitádiTorino,Turin,Italy.DepartmentofOceanTechnologyPolicyandEnvironment,UniversityofTokyo,Tokyo,Japan.Copyright2010bytheAmericanGeophysicalUnion.GEOPHYSICALRESEARCHLETTERS,VOL.37,L05603,doi:10.1029/2009GL041771,20101of directionalpropertiesofthewavefields.Inthisrespect,laboratoryexperimentsontheevolutionofshortregularwavesandwavegroups[Sheetal.,1994;Nepfetal.1998]suggestedthatthebreakingonsetissensitivetowavedirectionality.Inparticular,breakingwaveswereobservedtobecomebiggerandwithasteeperfrontasthedirectionalspreadingwasincreased;incontrast,therearsteepnesswasobservedtobeindependentfromwavedirectionality.Ageneralquantitativeconsensusonthewaveshapeatthetimeofbreaking,however,hasnotbeenachievedyet.Thus,amajorquestionwhichstillremainsunanswered(andisthesubjectofthepresentLetter)pertainstothemaximum(ulti-mate)shapethatrealisticoceanwavescanexhibit.Inordertoprovideananswertotheaforementionedquestion,herewepresentastatisticalanalysisoflargesamplesofindividualwavesteepnesswhichwerecollectedfrommeasurementsofthesurfaceelevationinlaboratoryfacilitiesandopensealocationswithinavarietyofseastateconditions.2.DataSetsSets8]Theadvantageofusinglaboratoryexperimentsisduetothefactthatthewaveconditionsareundercontrol.Twodatasetsfromtwoindependentdirectionalwavebasinswereemployed.OneoftheexperimentstookplaceattheUniver-sityofTokyo,Japan(KinoshitaLaboratory/RheemLabora-tory)[Wasedaetal.,2009].ThesecondonewasconductedattheMarinteksoceanbasininTrondheim,Norway,whichisoneofthelargestwavetanksintheworld[Onoratoetal.2009].Theexperimentaltestswerecarriedoutinaverysimpleway.Anumberofrandomwavefieldsweremechan-icallygeneratedatthewavemakerbyimposinganinput(initial)spectralenergydensityandrandomizingthewaveamplitudesandphases.AJONSWAPformulationwasusedtomodeltheenergyinthefrequencydomainandacosdirectionalfunctionwasusedforthedirectionaldomaindomainKomenetal.,1994].Differentcombinationsofsignificantwaveheight,peakperiodanddirectionalspreading(fromunidirectionaltodirectionalseastates)weretested[etal.,2009;Wasedaetal.,2009].However,thepeakperiod1s)waschosentohavedeepwaterwavesonly.Wementionthattherandomtestsweremainlyperformedtostudythestatisticalpropertiesofextremewaves.Therefore,theinitialconditionswereselectedsuchthattheoccurrenceofwavebreakingwasminimized;spectralconditionswithsteepness0.16,whereisthespectralpeakwave-numberandisthesignificantwaveheight,wereusedtothisend.Anumberoftestswerealsoperformedwithhighersteepness(/2�0.2)sothatwaveswereforcedtoreachtheirbreakinglimit.Inaddition,aseriesofexperimentsspecificallydesignedtostudythewavebreakingwereper-formedbygeneratingindividualtwodimensionalwavegroups(onlyattheUniversityofTokyo).Asthewavefieldpropagatedalongthetank,thesurfaceelevationsweremonitoredbymeasuringtimeseriesatdifferentlocationswithwireresistancewavegauges.gauges.9]Theuseofmechanicallygeneratedwavesprovidesaclearoverviewofeffectsrelatedtothedynamicsofthewavefieldnotinfluencedbythewindforcing.Additionally,wealsoinvestigatedfieldobservations,i.e.,timeseriesofthesurfaceelevationsobtainedinrealdirectionalwindwavesunderabroadvarietyofconditions.Fieldmeasure-mentswerecollectedattwodistinctlydifferentlocations:oneinthenorthwesternpartoftheBlackSea[Babaninand,1998]andasecondoneintheIndianOceanofftheNorthWestcoastofAustralia[,2006].Thelatterdataset,whichwerecollectedbyWoodsideEnergyLtd.attheNorthRankinAGasPlatform,containsobservationsofharshseaconditionsincludingseveraltropicalcyclonesbe-tween1995and1999.UnliketheBlackSeadatasetwhichwasrecordedwithwireresistancewavegauges,dataatNorthRankinwerecollectedwithdirectionalwavebuoys.Adis-cussiononthedifferencesbetweenLagrangianandEuleriansensorscanbefoundinworkbyby[10]Fromtherecordedsurfaceelevations,weextractedindividualwavesbyusingzerodowncrossingandupcross-ingdetectionwhichassumethatanindividualwaveistheportionofarecordbetweentwoconsecutivezerocrossingorupcrossingpointsrespectively.Thewaveheightisthendefinedastheverticaldistancebetweenthelowestandthehighestelevation,whilethewaveperiodisthetimeintervalbetweentwoconsecutivezerodowncrossing(orupcrossing)points.Asaforementioned,thewavesteepnessisdefinedasthewavenumbertimeshalfthewaveheight.Becauseofthenonlinearnatureofbreakingwaves,thewavenumberofindividualwavesiscalculatedfromthewaveperiodusinganonlineardispersionrelation[Yuenand,1982].Wementionthatthedowncrossingdefinitionprovidesameasureofthesteepnessatthewavefront,whiletheupcrossingdefinitionprovidesameasureofthesteep-nessatthewaverear.Onthewhole,about5×10vidualwaveswereextractedfromeachsetofobservations;wavesshorterthan0.5timesthepeakperiodwereexcludedfromtheanalysisthough.3.LimitingSteepnessofIndividualWavesWaves11]Anoverviewoftheindividualwaveshapeisprovidedbythejointcumulativedistributionfunctionofthelocal Figure1.Jointcumulativedistributionfunctionofwaveheightandperiod.Wavefieldswith(top)(a)downcrossingwavesand(b)upcrossingwaves.(bottom)Wavefieldswith/2�0.2:(c)downcrossingwavesand(d)upcrossingwaves.Thecurvesrepresentthenonexceedanceprobabilitylevels:lowestprobability(innercurve);highestprobability(outercurve).Curveofequalsteepnessarepresentedforcomparison:/2=0.55(solid/2=0.44(dashedline).TOFFOLIETAL.:MAXIMUMSTEEPNESSOFOCEANICWAVES2of4 (individual)waveheightandperiod.ThisispresentedinFigure1fromdatacollectedinthelaboratoryfacilitiesonly.Fromvisualobservations,weknowthatindividualwavesrecordedunderinitialspectralconditionswith(hereaftertestA,Figures1aand1b)seldomreachedthebreakingpoint,whileforinitial/2�0.20(hereaftertestB,Figures1cand1d)wavebreakingwasadistinctivedistinctive12]Thedistributionindicatesthatthereexistsanupperboundforthewaveshape.Thislimitcanbeconvenientlydescribedbycurveswithconstantsteepness.FortestA,wherewavesweregenerallyfarfrombreaking,theprofilewasrathersymmetric.Inthisrespect,thejointdistributionsofdowncrossingandupcrossingwavesshowasimilarupperlimitslightlybelow0.44,whichcorrespondstothebreakingonsetforunidirectionalwaves[seeBabaninetal.,2007a].ForthesteeperseastatesintestB,ontheotherhand,wavesweremorepronetobreaking.Atthepointofbreakingthewavesaresymmetric,butwhilealreadybreakingtheybecomeasymmetricandwithasteeperfront[Babaninetal.2007a].Ingeneral,thechangeofwaveshapeismorerelatedtothereductionofthedowncrossingwaveperiod(short-eningofthewavefront)ratherthantotheincreaseofwaveheight.Inthejointdistribution,theincreaseofthefrontfaceordowncrossingwavesteepnessisreflectedbytheenhancementoftheupperbound,whichrisesuptothevalueof0.55(Figure1c).Avisualanalysisofwavesapproachingthiscriticalsteepness,i.e.,waveswith/2�0.44,suggeststhatthesewavesarealreadybreakingratherthanimminentbreakers[Babaninetal.,2007a].Inthisrespect,althoughthefinalcollapseofthewavestructurecanoccuranytimeafterthebreakingonset,wavesdonotappeartoovercomeadowncrossingsteepnessof0.55.Thisisthemaximalsteepnessthatwatersurfacewavesseemtobeabletoto13]Itisinterestingtonote,however,thattheupperlimitofthejointdistributionisreducedforperiodclosetoorgreaterthantheinitialpeakwaveperiod(1s).Becausewavesaresubjectedtoashorteningofthedowncrossingperiodastheyareabouttobreak,itisnottotallyunexpectedtoobserveaconcentrationofverysteepwavesatperiodslowerthanthedominant.Asimilarresultwasalsorecoveredfromtheindependentsetofwavegroupexperiments.experiments.14]Onthecontrary,weobservedthatthewaverearsdidnotmodifysubstantiallytheirshape,inagreementwithpre-viousthreedimensionalobservationsbyNepfetal.al.Asaresult,despitethemorefrequentoccurrenceofbreak-ing,theupperboundforupcrossingwavesdoesnotdeviatefromthelimitingvalueof0.44(Figure1d).Nonetheless,unliketherandomtests,thewavegroupexperimentsshowthatupcrossingwavescanactuallyexceedthisthresholdlimit,atleastwithinashortrangeofperiods(seeFigure1d).Again,thesemustbethewavesalreadybreakingasthelimitingrearfacesteepnessatthebreakingonsetis0.440.44Babaninetal.,2007a].Thedistributionremainsnotablybelowthelimitof0.55though.Itisalsoimportanttomentionthatbothlimits(downcrossingandupcrossing)werenotparticularlysensitivetothevariationofthedirec-tionalspreading.spreading.15]Itisnowinstructivetoanalyzetheprobabilitydensityfunctionofthewavesteepness.InFigure2,theexceedanceprobabilityofthesteepnessispresentedforthedown-crossingandupcrossingdefinitionrespectively;alldatasets,i.e.,thelaboratoryandfieldobservations,aredisplayed.Becausethejointdistributionofwaveheightandperiodisupperboundedbyalimitingsteepness,itisreasonabletoexpectthatthetailoftheprobabilitydensityfunctionwouldnotextendfartherthantheaforementionedlimits.Inthisrespect,wesawthatthedistributionofthefrontsteepness(Figure2a)dropsatamaximumvalueofabout0.55andaprobabilitylevelof10.Consideringthatthetotalnumberofobservationsinoursampleis=5×10theminimumdetectableprobabilitylevelcorrespondsto=2×10(see,e.g.,Figure2a).Thislevelisaboutoneorderofmagnitudelowerthantheoneactuallydetected.Thusthemaximumsteepnesscanberegardedasacutlimit.Interestinglyenough,thisthresholdalsoappearstobeindependentfromthenatureoftheobservationsasitisinfactobtainedfromallthesetsofmeasurements.Likewise,thedistributionoftherearfacesteepness(Figure2b)dropsatalimitingvaluecloserto0.44.Nonetheless,thefieldobservationsshowaslightlyhigherlimitthanintherandomlaboratoryexperiments,inagreementwiththefindinginthewavegrouptests.Itisimportanttostressthat,whilethelaboratoryandfieldprobabilitydensityfunctionsareessentiallydifferent,theircutoffsareclose.Thishighlightsthenotionthatthemaximalpossiblesteepnessofdeepwaterbreakingwavesisnotafeatureofwavedevelopmentcon-ditionsorenvironmentalcircumstances,butisratherapropertyofwatersurfaceinthegravityfield.field.16]Theprobabilitydistributionmaysufferofstatisticaluncertainty,especiallytowardslowprobabilitylevels(tailofthedistribution).Inthisrespect,anestimateofthe95%confidenceintervalswascalculatedbymeansofbootstrapmethods,whicharebasedonthereproductionofrandomcopiesoftheoriginaldataset[see,e.g.,EmeryandThomson2001].Becauseofthelargenumberofobservations,the95%confidenceintervalsremainrathersmall.Atprobabilitylevelsaslowas10(i.e.,exceedanceprobabilityforthemaximumdetectedsteepness),thedegreeofuncertaintyisoneorderofmagnitudesmallerthantheexpectedvalueofsteepness.Thus,wecanregardourestimatefortheexceedanceprobabilityasstatisticallysignificant.Itishoweverimportanttomentionthatmaximumsteepnesscanalsobesubjecttouncertaintywhichderivesfromthefluc-tuationofthezerocrossingpointduetoshortwavesriding Figure2.Wavesteepnessdistributionfor(a)downcross-ingand(b)upcrossingwaves.Thestarindicatesthemini-mumpossiblelevelthatcouldbedetectedwithasampleof5×10TOFFOLIETAL.:MAXIMUMSTEEPNESSOFOCEANICWAVES3of4 ontopofthelongwavemainlyandhenceperturbsthewave4.ConclusionsConclusions17]Wepresentedananalysisofthesteepnessofindi-vidualwaves.Observationswerecollectedfromindepen-dentlaboratoryandfieldmeasurementcampaignsunderabroadvarietyofseastateconditionsandmechanicallygenerateddirectionalwavefields.Despitethediversityoftheobservations,alldatasetsshowedconsistentresults.Precisely,thefindingsindicatethatthereexistsawelldefinedvalueforthewavesteepnessabovewhichwavescannolongersustaintheirshape.Intermsoffrontsteepness,thisultimatethresholdisequivalenttoasteepnessof0.55,whichisnotablyhigherthantheStokeslimitforstationarywaves.Intermsoftherearfacesteepness,how-ever,thethresholdvaluesisslightlyabove0.44,confirmingacertainasymmetryoftheultimateshape.Theselimitswerenotsignificantlyaffectedbythedirectionalspreading.Moreover,duetothelargenumberofobservationsinvolved,thisfindingisstatisticallyrobust.robust.18]Itisimportanttoclarifythattheaforementionedlimitsonlyrepresentthemaximumsteepnessthatwatersurfacewavescanreach.Thisimpliesthatthestructureofabreakingwavecancollapseanytimeaftertheonsetoftheprocess.Inthecourseofthebreaking,however,wecanexpectwithhighconfidencethatthesteepnessbecomeshigherthantheonsetthresholdof0.44andlikelyreachesavaluearound0.55.Nevertheless,theupperboundissubjecttosomeuncertaintywhichoriginatesfromthefluctuationofthezerocrossingpointsduetoshortwavesridingontopofthelongwaveandhenceperturbsthewaveperiodandnotsomuchthewaveheight.Thepreciseupperboundshouldbedeterminedfromhydrodynamicconsiderationinamoredeterministicmanner.manner.19]Acknowledgments.A.ToffoliandA.BabaningratefullyacknowledgefinancialsupportoftheAustralianResearchCouncilandWoodsideEnergyLtd.throughthegrantLP0883888.TheexperimentalworkinMarintekwassupportedbytheEuropeanCommunitysSixthFrameworkProgramme,IntegratedInfrastructureInitiativeHYDROLABIII,contract022441(RII3).TheexperimentattheUniversityofTokyowassupportedbyGrantAidforScientificResearchoftheJSPS,Japan.DatafromNorthWestAustralia(NorthRankinPlatform)werekindlymadeavailablebyJasonMcConochie(WoodsideEnergyLtd.).Babanin,A.V.(2009),Breakingofoceansurfacewaves,ActaPhys.(4),305Babanin,A.V.,andY.P.Soloviev(1998),Fieldinvestigationoftransfor-mationofthewindwavefrequencyspectrumwithfetchandthestageofJ.Phys.Oceanogr.,563Babanin,A.,D.Chalikov,I.Young,andI.Savelyev(2007a),Predictingthebreakingonsetofsurfacewaterwaves,Geophys.Res.Lett.L07605,doi:10.1029/2006GL029135.Babanin,A.V.,M.L.Banner,I.R.Young,andM.A.Donelan(2007b),Wavefollowermeasurementsofthewindinputspectralfunction.Part3.Param-eterizationofthewindinputenhancementduetowavebreaking,J.Phys.,2764Babanin,A.V.,A.Ganopolski,andW.R.C.Phillips(2009),Waveupperoceanmixinginaclimatemodellingofintermediatecomplexity,OceanModell.,189Benjamin,T.B.,andJ.E.Feir(1967),Thedisintegrationofwavetrainsondeepwater.PartI.Theory,J.FluidMech.,417Brown,M.G.,andA.Jensen(2001),Experimentsinfocusingunidirectionalwaterwaves,J.Geophys.Res.,16,917Csanady,G.(1990),Theroleofbreakingwaveletsinairseagastransfer,J.Geophys.Res.,749Dyachenko,A.I.,andV.E.Zakharov(2005),ModulationinstabilityofStokeswavefreakwave,JETPLett.,255Emery,W.,andR.Thomson(2001),DataAnalysisMethodsinPhysicalOceanographyAdv.Ser.OceanEng.,vol.2,638pp.,ElsevierSci.,Fochesato,C.,S.Grilli,andF.Dias(2007),Numericalmodelingofextremeroguewavesgeneratedbydirectionalenergyfocusing,WaveMotionJessup,A.T.,C.J.Zappa,M.R.Loewen,andV.Heasy(1997),Infraredremotesensingofbreakingwaves,,52Komen,G.,L.Cavaleri,M.Donelan,K.Hasselmann,H.Hasselmann,andP.Janssen(1994),DynamicsandModelingofOceanWaves,CambridgeUniv.Press,Cambridge,U.K.Higgins,M.S.(1986),EulerianandlagrangianaspectsofsurfaceJ.FluidMech.,683LonguetHiggins,M.S.,andE.D.Cokelet(1978),Thedeformationofsteepsurfacewavesonwater.II.Growthofnormalmodeinstabilities,Proc.R.Soc.London,Ser.AMelville,E.K.(1982),Instabilityandbreakingofdeepwaterwaves,J.FluidMech.,165Melville,K.W.(1996),Theroleofsurfacewavebreakinginairseainter-Annu.Rev.FluidMech.,279Melville,K.W.,andP.Matusov(2002),Distributionofbreakingwavesattheoceansurface,,58Michell,J.H.(1893),Onthehighestwavesinwater,Philos.Mag.Ser.5,430Nepf,H.M.,C.H.Wu,andE.S.Chan(1998),Acomparisonoftwodimensionalwavebreaking,J.Phys.Oceanogr.,1496Onorato,M.,etal.(2009),Statisticalpropertiesofmechanicallygeneratedsurfacegravitywaves:Alaboratoryexperimentina3Dwavebasin,J.FluidMech.,235Pierson,W.J.,M.A.Donelan,andW.H.Hui(1992),Linearandnonlinearpropagationofwaterwavegroups,J.Geophys.Res.,5607Rapp,R.J.,andW.K.Melville(1990),Laboratorymeasurementsofdeepwaterbreakingwaves,Philos.Trans.R.Soc.London,Ser.A,735She,K.,C.A.Greated,andW.J.Easson(1994),Experimentalstudyofthreedimensionalwavebreaking,J.Water.PortOceanCoastalEng.(1),20Stokes,G.G.(1880),Onthetheoryofoscillatorywaves.AppendixB:Considerationrelativetothegreatestheightofoscillatoryirrotationalwaveswhichcanbepropagatedwithoutchangeofform,Math.Phys.,225Waseda,T.,T.Kinoshita,andH.Tamura(2009),Evolutionofarandomdirectionalwaveandfreakwaveoccurrence,J.Phys.Oceanogr.Young,I.R.(2006),Directionalspectraofhurri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