abruptx

366NonlinearFourWaveInteractionsandFreakWavesPeterA.
E.
M.
JanssenResearchDepartment1May2002ForadditionalcopiespleasecontactTheLibraryECMWFShineldParkReadingRG29AXlibrary@ecmwf.
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NonlinearFourWaveInteractionsandFreakWavesAbstractFour-waveinteractionsareshowntoplayanimportantroleintheevolutionofthespectrumofsurfacegravitywaves.
ThisfollowsfromdirectsimulationsofanensembleofoceanwavesusingtheZakharovequation.
Thetheoryofhomogeneousfour-waveinteractions,extendedtoincludeeffectsofnonresonanttransfer,comparesfavourablywiththeensembleaveragedresultsoftheMonteCarlosimulations.
Inparticular,thereisgoodagreementregardingspectralshape.
Also,thekurtosisofthesurfaceelevationprobabilitydistributioniswell-determinedbytheoryevenforwaveswithanarrowspectrumandlargesteepness.
Theseextremeconditionsarefavourablefortheoccurenceoffreakwaves.
TechnicalMemorandumNo.
3661NonlinearFourWaveInteractionsandFreakWaves1IntroductionPresently,thereisaconsiderableinterestinunderstandingtheoccurrenceoffreakwaves.
ThenotionoffreakwaveswasrstintroducedbyDraper(1965),andthistermisappliedforsinglewavesthatareextremelyunlikelyasjudgedbytheRayleighdistributionofwaveheights(Dean,1990).
Inpractice,awavewithwaveheightH(denedasthedistancefromcresttotrough)exceedingthesignicantwaveheightHSbyafactor2.
2isconsideredtobeafreakwave.
Itisdifculttocollecthardevidenceonsuchextremewavephenomenabecausetheyoccursorarely.
Nevertheless,observationalevidencefromtimeseriescollectedoverthepastdecadedoessuggestthatforlargesurfaceelevationstheprobabilitydistributionforthesurfaceelevationdeviatessubstantiallyfromtheonethatfollowsfromlineartheorywithrandomphase,namelytheGaussiandistribution(cf.
e.
g.
WolframandLinfoot,2000).
Thereareanumberofreasonswhyfreakwavephenomenamayoccur.
Often,extremewaveeventscanbeexplainedbythepresenceofoceancurrentsorbottomtopographythatmaycausewaveenergytofocusinasmallareaduetorefraction,reectionandwavetrapping.
Thesemechanismsarewellunderstoodandmaybeexplainedbylinearwavetheory(cf.
e.
g.
Lavrenov,1998).
TrulsenandDysthe(1997)argue,however,thatitisnotwellunderstoodwhyexceptionallylargewavesmayoccurintheopenoceanawayfromnon-uniformcurrentsorbathymetry.
AsanexampletheydiscussthecaseofanextremewaveeventthathappenedonJanuary1,1995intheNorwegiansectoroftheNorthSea.
Theirbasicpremiseisthatthesewavescanbeproducedbynonlinearselfmodulationofaslowlyvaryingwavetrain.
Anexampleofnonlinearmodulationorfocussingistheinstabilityofauniformnarrow-bandwavetraintoside-bandperturbations.
Thisinstability,knownastheside-band,modulationalorBenjamin-Feir(1967)instability,willresultinfocusingofwaveenergyinspaceand/ortimeasisillustratedbytheexperimentsofLakeetal(1977).
Toarstapproximationtheevolutionintimeoftheenvelopeofanarrow-bandwavetrainisdescribedbythenonlinearSchr¨odingerequation.
Thisequation,whichoccursinmanybranchesofphysics,wasrstdiscussedinthegeneralcontextofnonlineardispersivewavesbyBenneyandNewell(1967).
ForwaterwavesitwasrstderivedbyZakharov(1968)usingaspectralmethodandbyHasimotoandOno(1972)andDavey(1972)usingmultiple-scalemethods.
ThenonlinearSchr¨odingerequationinone-spacedimensionmaybesolvedbymeansoftheinversescatteringtransform.
ForvanishingboundaryconditionsZakharovandShabat(1972)foundthatforlargetimesthesolutionconsistsofacombinationofenvelopesolitonsandradiationmodes,inanalogywiththesolutionoftheKorteweg-deVriesequation.
However,fortwo-dimensionalpropagation,ZakharovandRubenchik(1974)discoveredthatenvelopesolitonsareunstabletotransverseperturbations,whileCohen,WatsonandWest(1976)foundthatarandomwaveeldwouldbreakupenvelopesolitons.
Thismeantthatsolitonscouldnotbeusedasbuildingblocksofthenonlinearevolutionofgravitywaves.
ForperiodicboundaryconditionsthesolutionofthenonlinearSchr¨odingerequationismorecomplex.
Linearstabilityanalysisofauniformwavetrainshowsthatcloseside-bandsgrowexponentiallyintimeingoodqualitativeagreementwiththeexperimentalresultsofBenjaminandFeir(1967)andLakeetal(1977).
Forlargetimesthereisaconsiderableenergytransferfromthecarrierwavetotheside-bands.
Inone-spacedimension,ifthereisonlyoneunstableside-band,Fermi-Pasta-Ulamrecurrenceoccurs(YuenandFerguson,1978)inqualitativeagreementwiththeexperimentsofLakeetal(1977).
Inthepresenceofmanyunstableside-bands,theevolutionofanarrowbandwavetrainbecomesmuchmorecomplex.
Norecurrenceisthenfound(Caponietal,1982)andtheseauthorshavetermedthisconnedchaosinanonlinearwavesystembecausemostoftheenergyresidesintheunstablemodes.
Also,intwo-spacedimensions(2D)thephenomenonofrecurrenceistheexceptionratherthantherule.
Inaddition,in2Dtheinstabilityregionisunboundedintheperturbationwavevectorspace,resultinginenergyleakagetohighwavenumbermodes,hencethereisnoconnedchaosin2D(MartinandYuen,1980).
Thissuggeststhatthe2DnonlinearSchr¨odingerequationisinadequatetodescribethe2TechnicalMemorandumNo.
366NonlinearFourWaveInteractionsandFreakWavesevolutionofweaklynonlinearwaves.
ThiswaspointedoutalreadybyLonguet-Higgins(1978)whoperformedastabilityanalysisontheexactequationsandfoundthattheinstabilityregionisniteinextent.
Morerealisticevolutionequationssuchasthefourth-orderevolutionequationofDysthe(1979)ortheZakharovequation(1968)areneededtogiveanappropriatedescriptionofnonlineargravitywavesintwo-spacedimensions.
Nevertheless,studiesofthepropertiesofthenonlinearSchr¨odingerequationhavebeenvitalinunderstandingtheconditionsunderwhichfreakwavesmayoccur.
ThiswasdiscussedindetailbyOsborneetal(2000).
Forperiodicboundaryconditionstheone-dimensionalnonlinearSchr¨odingerequationmaybesolvedbytheinversescatteringmethodaswell.
Theroleofthesolitonsisthenreplacedbyunstablemodes.
Inthelinearregime,thesemodesjustdescribetheevolutionintimeaccordingtotheBenjamin-Feirinstability,whilebymeansoftheinversescatteringtransformthefateoftheunstablemodemaybefollowedrightintothenonlinearregime.
Usingtheinversescatteringtransformthesolutionofthe1DnonlinearSchr¨odingerequationmaybewrittenasa"linear"superpositionofstablemodes,unstablemodesandtheirmutualnonlinearinteractions.
Here,thestablemodesformaGaussianbackgroundwaveeldfromwhichtheunstablemodesoccasionallyriseupandsubsequentlydisappearagain,repeatingtheprocessquasi-periodicallyintime.
Makinguseoftheinversescatteringtransformtheseauthorsreadilyconstructafewexamplesofgiantwavesfromtheone-dimensionalnonlinearSchr¨odingerequation.
Thequestionnowiswhathappensinthecaseoftwo-dimensionalpropagation.
Thenotionofsolitonsisnolongeruseful,becausesolitonsareunstableintwo-dimensions.
Osborneetal(2000)showthatunstablemodesdoindeedstillexistandthatinthenonlinearregimetheycantaketheformoflargeamplitudefreakwaves.
Furthermore,thenotionofunstablemodesseemstobeagenericpropertyofdeepwaterwavetrains,astheauthorsndnonlinearunstablemodesinboththeoneandtwo-dimensionalversionsofDysthe'sfourthorderevolutionequation.
Tosummarizethisdiscussion,itseemsthatfreakwavesarelikelytooccuraslongasthewavetrainissubjecttononlinearfocussing.
Inaddition,weonlyneedtostudythecaseofone-dimensionalpropagation,becauseitcapturestheessentialsofthegenerationoffreakwaves.
Therefore,inthecontextofthedeterministicapproachtowaveevolutionthereseemstobeareasonablethe-oreticalunderstandingofwhyintheopenoceanfreakwavesoccur.
Inoceanwaveforecastingpracticeonefollows,however,astochasticapproach,i.
e.
oneattemptstopredicttheensembleaverageofaspectrumofrandomwaves,becauseknowledgeonthephasesisnotavailable.
Themainproblemthenistowhatextentonecanmakestatementsregardingtheoccurrenceoffreakwavesinarandomwaveeld.
Ofcourse,inthecontextofwaveforecastingonlystatementsofaprobablisticnaturecanbemade.
AsfreakwavesimplyconsiderabledeviationsfromtheNormal,Gaussianprobabilitydistributionfunction(pdf)ofthesurfaceelevation,themainquestionthereforeiswhetherwecandetermineinareliablemannerthepdfofthesurfaceelevation.
Sincethewavespectrumplaysacentralroleinthestochasticapproachthequestionthereforeiswhetherforgivenwavespectrumtheprobabilityofextremeeventsmaybedetermined.
Presentdaywaveforecastingsystemsarebasedontheenergybalanceequation(Komenetal,1994),includingaparametrisedversionofHasselmann'sfour-wavenonlineartransfer(Hasselmann,1962).
Resonantfour-waveinteractionsforarandom,homogeneousseaplayanimportantroleintheevolutionofthespectrumofwindwaves,becauseontheonehandtheydeterminethehigh-frequencypartofthespectrum,givingrisetoanω4tail(Zakharov&Filonenko,1968),whileontheotherhandthepeakofthespectrumisshiftedtowardslowerfrequencies.
ThehomogeneousnonlinearinteractionsgiverisetodeviationsfromtheGaussianpdfforthesurfaceelevation,becausethethirdordernonlinearitygeneratesfourthcumulantsofthepdf,whilethenitefourthcumulantresultsinspectralchange.
Animportantissueis,however,whetherthestandardhomogeneoustheorycanproperlydescribethegenerationoffreakwaves,simplybecauseitdoesnotseemtoincorporatetheBenjamin-Feirinstabilitymechanism(Alber,1978,AlberandSaffman,1978,Crawfordetal,1980,Janssen,1983b).
ThisfollowsfromsimplescalingconsiderationsappliedtotheHasselmannevolutionequationforfour-waveinteractions.
SincetherateofchangeoftheactiondensityNisproportionaltoN3,thenonlineartransferoccursonthetimescaleTNL=O(1/ε4ω0).
Here,εisatypicalwavesteepness,whichisassumedtobeTechnicalMemorandumNo.
3663NonlinearFourWaveInteractionsandFreakWavessmall,andω0isatypicalangularfrequencyofthewaveeld.
Incontrast,theBenjamin-FeirinstabilityoccursonthemuchfastertimescaleofO(1/ε2ω0).
TheBenjamin-Feirinstabilityisanexampleofanonresonantfour-waveinteractionwherethecarrierwaveisphase-lockedwiththesidebands.
ThisprocesscannotbedescribedbyatheorythatassumesthattheFourieramplitudesarenotcorrelated(i.
e.
ahomogeneouswaveeld),andinwhichonlyresonantfour-waveinterac-tionsareconsidered.
Foraninhomogeneous,Gaussiannarrow-bandwavetrain,AlberandSaffman(1978),andAlber(1978)derivedanevolutionequationfortheWignerdistributionoftheseastate.
Inhomogeneitiesgaverisetoamuchfasterenergytransfer,comparablewiththetypicaltimescaleofthemodulationalinstability.
Infact,theseauthorsdiscoveredtherandomversionoftheBenjamin-Feirinstability:arandomnarrowbandwavetrainisunstabletoside-bandperturbationsprovidedthewidthofthespectrumissufcientlynarrow.
Therefore,onewouldexpecttheAlberandSaffmanapproachtobeanidealstartingpointfortreatingfreakwavesinarandomwavecontext.
However,itisemphasizedthatthisapproachhasitlimitationsbecausedeviationsfromNormalityhavenotyetbeentakenintoaccount.
Inthispaperitwillbeshown,usingnumericalsimulationsofanensembleofoceanwaves,thatnon-Gaussianeffectsarequiteimportantwhileinhomogeneitiesplayonlyaminorroleintheevolutionoftheensemble-averagedwavespectrum.
Ontheotherhand,nonresonantinteractionsappeartoberelevant.
WeextendHasselmann'streatmentoffour-waveinteractionsbyincludingtheeffectsofnonresonantinteractions.
Asaconsequence,theresonancefunctionisforshorttimesbroaderthantheusualδ-functionanddependsontheangularfrequencyresonanceconditions`andontime.
Thestandardnonlineartransferisbasedontheassumptionthattheactiondensityspectrumisaslowlyvaryingfunctionoftime.
Itisthenarguedthattheresonancefunctionmaybereplacedbyitslargetimelimit,givingtheusualdeltafunction.
However,thetimespanrequiredfortheresonancefunctiontoevolvetowardsadeltafunctionissolargethatconsiderablechangesintheactiondensityfunctionmayhaveoccurredinthemeantime.
Thiswillbeshownforthespecialcaseofonedimensionalpropagationofsurfacegravitywaves.
Inthosecircumstancesthestandardapproachtononlinearwave-waveinteractionswouldnotgiverisetononlineartransfer,whereasconsiderablechangesofthewavespectrumoccurinthenewapproach.
Infact,thereiscloseagreementbetweenresultsontheensembleaveragedspectrumandthekurtosisofthepdfofthesurfaceelevation,asobtainedfromnumericalsimulationsofanensembleofoceanwaves.
Sincetimeseriesfromthenumericalsimulationsindicatetheoccurenceoffreakwaveswhenthewavesaresufcientlysteep(seealsoTrulsenandDysthe(1997)orOsborneetal(2000)),theimplicationisthatanapproachtononlineartransfer,thatincludesnonresonantinteractionsseemstocapturefreakwaveevents.
However,itisstronglyemphasizedthatsuchanapproachcanonlygivestatementsofaprobablisticnatureontheoccurrenceofextremewaveevents.
Thestructureofthispaperisasfollows.
InSection2wereviewdevelopmentsregardingtheevolutionofarandomwaveeld,butwediscussonlytheideasneededforunderstandingresultsintheremainderofthispaper.
Inparticular,weextendthestandardtheoryoffourwaveinteractionsbyincludingeffectsofnonresonantinteractionsandderiveanexplicitexpressionforthekurtosisintermsoftheactiondensityspectrum.
WealsodiscussAlberandSaffman'skeyresult,thataccordingtolowestorderinhomogeneoustheorythereisonlyBenjamin-Feirinstabilitywhenthewavespectrumissufcientlynarrow.
InSection3wepresentresultsfromMonteCarlosimulationsofthenonlinearSchr¨odingerequationfollowingsimilarworkbyOnoratoetal(2000).
Onlyone-dimensionalwavepropagationisdiscussed.
Apartfromreasonsofeconomy(wetypicallydorunswith500memberensembles),themainreasonforthischoiceisthatforonedimensionthenonlineartransferaccordingtothestandardhomogeneoustheoryoffourwaveinteractionsvanishesidentically.
Theensembleaveragedevolutionofthewavespectrumclearlyshowsthatthereisanirreversibleenergytransferresultinginabroadeningofthespectrum,whilethepdfofthesurfaceelevationhasconsiderabledeviationsfromtheGaussiandistribution.
ThesedeviationsfromNormalitymaybedescribed,asexpectedfromfour-waveinteractions,bymeansofthefourthcumulant.
Incaseofnonlinearfocussing,thecorrectiontothepdfissuchthatthere4TechnicalMemorandumNo.
366NonlinearFourWaveInteractionsandFreakWavesisanenhancedprobabilityofextremeevents,whileinthecaseofnonlineardefocusing(thiswasachievedbychangingthesignofthenonlinearterm)theoppositeoccurs,namelytheprobabilityofextremeeventsisreduced.
ThisisinagreementwithresultsbyTanaka(1991)whofoundanincreaseingroupinessincaseofnonlinearfocussingwhileintheoppositecaseofastablewavetraingroupinessreduces.
Boththespectralbroadeningandthefourthcumulant(orkurtosis)arefoundtodependonasingleparametercharacterisingthenarrow-bandwavetrain,namelytheratioofmeansquareslopetothenormalisedwidthofthe(frequency)spectrum.
ItissuggestedtocallthisratiotheBenjamin-FeirIndex(BFI).
IftheBFIislargerthan1thenaccordingtoAlberandSaffman(1978)therandomwaveeldismodulationallyunstable.
ThisresultwouldsuggestthatiftheBFIislessthan1nochangesinthespectrumoccur,whileintheoppositecasetheunstableside-bandswouldgiverisetoabroadeningofthewavespectrum.
Hence,BFI=1isabifurcationpoint.
OurnumericalsimulationsprovidenoconvincingevidenceofabifurcationatBFI=1.
Rather,thereisalreadyaconsiderablebroadeningofthewavespectrumaroundBFI=1,whilethedependenceofthebroadeningontheBFIappearstobesmoothratherthenabrupt(cf.
Tanaka(1991)).
WecontinueinSection3bypresentingresultsfromMonteCarlosimulationsoftheZakharovequation(Za-kharov,1968).
ResultsaresimilarinspirittothosesobtainedwiththeNonlinearSchr¨odingerequation,exceptthatthemodulationalinstabilityseemstooccurforlargerBFI.
ForthenonlinearSchr¨odingerequationthespectralchangeowingtononlineartransferissymmetricalwithrespecttothespectralmaximum,butthisisnotthecaseforZakharovequation.
Inthelattercasethenonlineartransfercoefcientsandtheangularfrequencyareasymetricalwithrespecttothespectralpeakandasaconsequencethereisadown-shiftofthepeakofthespectrum.
Itisemphasizedthatthisdown-shiftoccursintheabsenceofdissipation,whilequantitiessuchasaction,wavemomentumandtotalwaveenergyareconserved.
InSection4aninterpretationofthenumericalresultsofSection3isgiven.
Firstly,itisshownthatinhomo-geneitiesonlyplayaminorroleintheevolutionofthewavespectrum,whiledeviationsfromNormalityaremorerelevant.
Secondly,resultsfromthenumericalsolutionoftheextendedversionofHasselmann'swave-waveinteractionapproacharepresentedandcomparedwiththeresultsfromMonte-Carlosimulations.
Agoodagreementisobtained.
ApartfromthefactthatwehavegivenadirectvalidationofHasselmann'sfour-wavetheory,italsoshowsthateveninextremeconditionssuchasoccurduringthegenerationoffreakwaves,reliableestimatesofdeviationsfromNormalitycanbemade.
InSection5asummaryofconclusionsisgiven.
Muchtooursurprise,effectsofinhomogeneityonlyplayaminorroleinunderstandingtheensembleaveragedevolutionofsurfacegravitywaves.
Homogeneousfour-waveinteractions,albeitextendedbyallowingforatimedependentresonancefunction,seemtocapturemostessentialfeaturesoftheaveragednonlinearwaveevolution.
Itseemsnowpossibletoestimatetheenhancedoccurrenceofextremewavesandfreakwavesontheopenoceansincethekurtosismaybeestimateddirectlyfromthewavespectrum.
2ReviewofthetheoryofarandomwaveeldOurstartingpointistheZakharovequation,whichisadeterministicevolutionequationforsurfacegravitywavesindeepwater.
ItisobtainedfromtheHamiltonianforwaterwaves,rstfoundbyZakharov(1968).
Considerthepotentialowofanidealuidofinnitedepth.
Coordinatesarechoseninsuchawaythattheundisturbedsurfaceoftheuidcoincideswiththex-yplane.
Thez-axisispointedupward,andtheaccelerationofgravitygispointedinthenegativez-direction.
Letηbetheshapeofthesurfaceoftheuid,andletφbethepotentialoftheow.
Hence,thevelocityoftheowfollowsfromu=φ.
TechnicalMemorandumNo.
3665NonlinearFourWaveInteractionsandFreakWavesBychoosingascanonicalvariablesη,and,ψ(x,t)=φ(x,z=η,t),(1)Zakharov(1968)showedthatthetotalenergyEoftheuidmaybeusedasaHamiltonian.
Here,E=12η∞dzdx(φ)2+(φz)2+g2dxη2.
(2)Thex-integralsextendoverthetotalbasinconsidered.
Ifaninnitebasinisconsideredtheresultingtotalenergyisinnite,unlessthewavemotionislocalizedwithinaniteregion.
Thisproblemmaybeavoidedbyintroducingtheenergyperunitareabydividing(2)bythetotalsurfaceL*L,whereListhelengthofthebasin,andtakingthelimitofL→∞afterwards.
Asaconsequence,integralsoverwavenumberkarereplacedbysummationswhileδ-functionsarereplacedbyKroneckerδ's.
Foramorecompletediscussioncf.
Komenetal(1994).
Wewilladoptthisapproachimplicitelyintheremainderofthispaper.
Theboundaryconditionsatthesurface,namelythekinematicboundaryconditionandBernoulli'sequation,arethenequivalenttoHamilton'sequations,ηt=δEδψ,ψt=δEδη,(3)whereδE/δψisthefunctionalderivativeofEwithrespecttoψ,etc.
InsidetheuidthepotentialφsatiesLaplace'sequation,2φ+2φz2=0(4)withboundaryconditionsφ(x,z=η)=ψ(5)andφ(x,z)z=0,z→∞.
(6)Ifoneisabletosolvethepotentialproblem,thenφmaybeexpressedintermofthecanonicalvariablesηandψ.
ThentheenergyEmaybeevaluatedintermsofthecanonicalvariables,andtheevolutionintimeofηandψfollowsatoncefromHamilton'sequations(Eq.
(3)).
ThiswasdonebyZakharov(1968),whoobtainedthedeterministicevolutionequationsfordeepwaterwavesbysolvingthepotentialproblem(4-6)inaniterativefashionforsmallsteepnessε.
Inaddition,theFouriertransformsofηandφwereintroduced,whileresultscouldbeexpressedinaconcisewaybyuseoftheactionvariableA(k,t).
Forexample,intermsofAthesurfaceelevationηbecomesη=∞∞dkk2ω1/2[A(k)+A(k)]eik.
x.
(7)Here,kisthewavenumbervector,kitsabsolutevalue,andω=√gkdenotesthedispersionrelationofdeep-water,gravitywaves.
SubstitutionoftheseriessolutionforφintotheHamiltonian(2)givesanexpansionofthetotalenergyEoftheuidintermsofwavesteepness,E=ε2E2+ε3E3+ε4E4+O(ε5).
(8)Retainingonlythesecond-ordertermofEcorrespondstothelineartheoryofsurfacegravitywaves,thethird-ordertermcorrespondstothree-waveinteractions,andthefourth-ordertermcorrespondstofour-waveinterac-tions.
Sinceresonantthree-waveinteractionsareabsentfordeep-watergravitywaves,ameaningfuldescription6TechnicalMemorandumNo.
366NonlinearFourWaveInteractionsandFreakWavesofthewaveeldisonlyobtainedbygoingtofourthorderinε.
Infact,Krasitskii(1990)hasshownthatintheabsenceofresonantthreewaveinteractionsthereisanonsingular,canonicaltransformationfromtheac-tionvariableAtothenewvariableathatallowseliminationofthethirdordercontributiontothewaveenergy.
Looselyspeaking,thenewvariableadescibesthefreewavepartofthewaveeld.
Apartfromaconstantfactor,theenergyofthefreewavesbecomes,E=dk1ω1a1a1+12dk1,2,3,4T1,2,3,4a1a2a3a4δ1+234,(9)wherea1=a(k1),etc.
,δistheDiracdeltafunctionandtheinteractionmatrixTisgivenbyKrasitskii(1990).
Theinteractionmatrixenjoysanumberofsymmetryconditions,ofwhichthemostimportantoneisT1,2,3,4=T3,4,1,2asthisconditionimpliesthatEisconserved.
Hamilton'sequationsnowbecomethesingleequationiat=δEδa,(10)and,evaluatingthefunctionalderivativeofEwithrespecttoa,theevolutionequationforabecomesa1t+iω1a1=idk2,3,4T1,2,3,4a2a3a4δ1+234,(11)knownastheZakharovequation.
Apartfromthefreewaveenergy(9)theZakharovequationadmitsconserva-tionofactionandofwavemomentumasa)ddtdk1a1a1=0,b)ddtdk1k1a1a1=0.
(12)2.
1CommentsontheZakharovEquationThepropertiesoftheZakharovequationhavebeenstudiedingreatdetailby,forexample,Crawfordetal(1981)(foranoverviewseeYuenandLake,1982).
Thusthenonlineardispersionrelation,rstobtainedbyStokes(1947),followsfromEq.
(11),whilealsotheinstabilityofaweaklynonlinear,uniformwavetrain(theso-calledBenjamin-Feirinstability)iswelldescribedbytheZakharovequation;theresultsongrowthrates,forexample,arequalitativelyingoodagreementwiththeresultsofLonguet-Higgins(1978).
However,theseresultswereobtainedwithaformoftheinteractionmatrixTthatdidnotresultinaHamiltonianformofEq.
(11).
Krasitskii(1990)foundthecorrectcanonicaltransformationtoeliminatethecubicinteractions,whichresultedinaTthatsatisedtheappropriatesymmetryconditionsforEq.
(11)tobeHamiltonian.
KrasitskiiandKalmykov(1993)studiedthedifferencesbetweentheHamiltonianandthenon-HamiltonianformsoftheZakharovequationbutonlyforlargeamplitudedifferencesinthesolutionwerefound.
Inthispaperweinitiallyuseanarrow-bandapproximationtotheZakharovequation,becausethemainimpactoftheBenjamin-Feirinstabilityisfoundnearthespectralpeak.
ThisapproximateevolutionequationisobtainedbymeansofaTaylorexpansionofangularfrequencyωandtheinteractionmatrixTaroundthecarrierwavenumberk0.
ThenonlinearSchr¨odingerequationisthenobtainedbyusingonlythelowestorderapproximationtoTgivenbyk30,whileangularfrequencyωisexpandedtosecondorderinthemodulationwavenumberp=kk0.
ThemainadvantageoftheuseofthenonlinearSchr¨odingerequationisthatmanypropertiesofthisequationareknownandthatitcanbesolvednumericallyinanefcientway.
Thedrawbackis,however,thatitoverestimatesthegrowthratesoftheBenjamin-FeirinstabilityandthatthenonlinearenergytransferTechnicalMemorandumNo.
3667NonlinearFourWaveInteractionsandFreakWavesissymmetricalwithrespecttothecarrierwavenumber.
Forthisreason,westudysolutionsofthecompleteZakharovequationaswell,usingtheKrasitskii(1990)expressionfortheinteractionmatrixT.
Similarly,onecouldstudyhigher-orderevolutionequationssuchastheonebyDysthe(1979),butwefoundthatspectramaybecomesobroadthatthenarrow-bandapproximationbecomesinvalid.
AnotherreasonforstudyingthenonlinearSchr¨odingerequationisthatitallowsustointroduceanimportantparameterwhichwillbeusedtostratifythenumericalandtheoreticalresults.
Fromthephysicalpointofviewwearebasicallystudyingaproblemthatconcernsthebalancebetweendispersionofthewavesanditsnonlinearity.
ForthefullZakharovequationitwillbedifculttointroduceauniquemeasureof,forexample,nonlinearitybecausethenonlineartransfermatrixTisacomplicatedfunctionofwavenumber.
However,inthenarrow-bandapproximation,givingthenonlinearSchr¨odingerequation,thisismorestraight-forwardtodo.
Balancingthenonlineartermandthedispersiveterminthenarrow-bandversionofEq.
(11)thereforegivesthedimensionlessnumbergT0ω01k40ω0s2σ2ω.
(13)Sinceourinterestisinthedynamicsofacontinuousspectrumofwavestheslopeparametersandtherelativewidthσωofthefrequencyspectrumrelatetospectralproperties,hences=(k20)12,withtheaveragesurfaceelevationvariance,andσω=σω/ω0.
Forpositivesignofthedimensionlessparameter(13)thereisfocussing(modulationalinstability)whileintheoppositecasethereisdefocussingoftheweaklynon-linearwavetrain.
BasedonthisweintroducetheBenjamin-Feir(BF)Index,which,apartfromaconstant,isthesquarerootofthedimensionlessnumber(13).
Usingthedispersionrelationfordeep-watergravitywavesandtheexpressionforthenonlinearinteractioncoefcient,T0=k30,theBFIndexbecomes,BFI=s√2/σω.
(14)TheBFIndexturnsouttobeveryusefulinorderingthetheoreticalandnumericalresultspresentedinthefollowingSections.
Forsimpleinitialwavespectrathatonlydependonthevarianceandonthespectralwidth,itcanbeshownthatforthenonlinearSchr¨odingerequationthesolutioniscompletelycharacterizedbytheBFIndex.
FortheZakharovequationthisisnotthecase,buttheBFIndexisstillexpectedtobeausefulparameterfornarrow-bandwavetrains.
2.
2StochasticapproachTheZakharovequation(11)predictsamplitudeandphaseofthewaves.
Forpracticalapplicationssuchaswaveprediction,thedetailedinformationregardingthephaseofthewavesisnotavailable.
Therefore,atbestonecanhopetopredictaveragequantitiessuchasthesecondmomentB1,2=,(15)wheretheanglebracketsdenoteanensembleaverage.
Here,webrieysketchthederivationoftheevolutionequationforthesecondmomentfromtheZakharovequation,assumingazeromeanvalue,=0.
Itisknown,however,thatbecauseofnonlinearity,theevolutionofthesecondmomentisdeterminedbythefourthmoment,andsoon,resultinginaninnitehierarchyofequations(Davidson,1972).
Toobtainameaningfultruncationofthishierachy,itiscustomarytoassumethattheprobabilitydistributionfora1isclosetoaGaussiandistribution,anassumptionwhichisareasonableoneforsmallwavesteepnessε.
Inthatevent,higher-ordermomentscanbeexpressedinlower-ordermoments.
Ingeneral,forazero-meanstochasticvariablea1,onendsthatthefourthmomentbecomes=Bj,lBk,m+Bj,mBk,l+Dj,k,l,m,(16)8TechnicalMemorandumNo.
366NonlinearFourWaveInteractionsandFreakWaveswhereDistheso-calledfourthcumulant,whichvanishesforaGaussianseastate.
Resonantnonlinearinterac-tions,however,willtendtocreatecorrelationsinsuchawaythatanitefourthcumulantresults.
ButforsmallsteepnesDisexpectedtobesmall,sothatanapproximateclosureoftheinnitehierarchyofequationsmaybeachieved.
LetusnowsketchthederivationoftheevolutionequationforthesecondmomentfromtheZa-kharovequation(11).
Tothatend,wemultiplyEq.
(11)foraibyaj,addthecomplexconjugatewithiandjinterchanged,andtaketheensembleaverage:t+i(ωiωj)Bi,j=idk2,3,4[Ti,2,3,4δi+234c.
c.
(ij)],(17)wherec.
c.
denotescomplexconjugate,andijdenotestheoperationofinterchangingindicesiandjinthepreviousterm.
Becauseofnonlinearitytheequationforthesecondmomentinvolvesthefourthmoment.
Similarly,theequationforthefourthmomentinvolvesthesixthmoment.
Itbecomest+i(ωi+ωjωkωl)=idk2,3,4[Ti,2,3,4δi+234+(ij)]+idk2,3,4[Tk,2,3,4δk+234+(kl)].
(18)Sofar,noapproximationshavebeenmade.
InthenextSection,wediscusstheimplicationsoftheassumptionsofahomogeneousweaklynonlinearwaveeld.
Homogeneityofthewaveeld,however,doesnotallowadescriptionoftheBenjamin-Feirinstability,andthereforeinthefollowingSectionwediscusstheconsequencesforspectralevolutionwhenthewaveeldisallowedtobeinhomogeneous.
2.
3EvolutionofahomogeneousrandomwaveeldAwaveeldisconsideredtobehomogeneousifthetwopointcorrelationfunctionx1)η(x2)>dependsonlyonthedistancex1x2.
Usingtheexpressionforthesurfaceelevation,Eq.
(7),itisthenstraightforwardtoverifythatawaveeldishomogeneousprovidedthatthesecondmomentBi,jsatisesBi,j=Niδ(kikj),(19)whereNiisthespectralactiondensity,whichisequivalenttoanumberdensitybecauseωiNiisthespectralenergydensity,whilekiNiisthespectralmomentumdensity(apartfromafactorρw).
ForweaklynonlinearwavesthefourthcumulantDissmallcomparedtotheproductofsecond-ordercumu-lants(thismaybeveriedafterwards,itfollowsimmediatelyfromEq.
(18).
Now,invokingtherandom-phaseapproximation(i.
e.
Eq.
(16))withD=0)onEq.
(17),combinedwiththeassumptionofahomogeneouswaveeldresultsinconstancyofthesecondmomentBi,j.
Hence,theneedtogotohigherorder;thatisthefourthmomenthastobedeterminedthroughEq.
(18).
ApplicationoftherandomphaseapproximationtothesixthmomentandsolvingEq.
(18)forthefourthcumu-lantD,subjecttotheinitialvalueD(t=0)=0,givesDi,j,k,l=2Ti,j,k,lδi+jklG(ω,t)[NiNj(Nk+Nl)(Ni+Nj)NkNl](20)TechnicalMemorandumNo.
3669NonlinearFourWaveInteractionsandFreakWaveswhereωisshorthandforωi+ωjωkωl,andwehavemadeextensiveuseofthesymmetrypropertiesofthenonlineartransfermatrixT,inparticulartheHamiltoniansymmetry.
Inaddition,weusedthepropertythat,accordingtoEq.
(17)theactiondensityNonlyevolvesontheslowtimescale.
ThefunctionGisdenedasG(ω,t)=it0dτeiω(τt)=Rr(ω,t)+iRi(ω,t),(21)whereRr(ω,t)=1cos(ωt)ω,(22)whileRi(ω,t)=sin(ωt)ω.
(23)ThefunctionGdevelopsforlargetimetintotheusualgeneralisedfunctionsP/ω,andδ(ω),since,limt→∞G(ω,t)=Pω+πiδ(ω),(24)arelationwhichis,strictlyspeaking,onlymeaningfulinsideintegralsoverwavenumberwhenmultipliedbyasmoothfunction.
SubstitutionofEq.
(20)intoEq.
(17)eventuallyresultsinthefollowingevolutionequationforfour-waveinter-actions,tN4=4dk1,2,3T21,2,3,4δ(k1+k2k3k4)Ri(ω,t)*[N1N2(N3+N4)N3N4(N1+N2)],(25)wherenowω=ω1+ω2ω3ω4.
ThisevolutionequationisusuallycalledtheBoltzmannequation.
TwolimitsoftheresonancefunctionRi(ω,t)areofinteresttomention.
Forsmalltimeswehavelimt→0Ri(ω,t)=t(26)whileforlargetimeswehavelimt→∞Ri(ω,t)=πδ(ω).
(27)Hence,accordingtoEq.
(25),forshorttimestheevolutionoftheactiondensityNiscausedbybothresonantandnonresonantfour-waveinteractions,whileforlargetimes,whentheresonancefunctionsevolvestowardsaδ-function,onlyresonantinteractionscontributetospectralchange.
Inthestandardtreatmentofresonantwavewaveinteractions(cf.
,forexampleHasselmann(1962)andDavid-son(1972))itisarguedthattheresonancefunctionRi(ω,t)maybereplacedbyitstime-asymptoticvalue(Eq.
(27)),becausetheactiondensityspectrumisaslowlyvaryingfunctionoftime.
However,thetimerequiredfortheresonancefunctiontoevolvetowardsadeltafunctionmaybesolargethatinthemeantimeconsider-ablechangesintheactiondensitymayhaveoccurred.
Forthisreasonwewillkeepthefullexpressionfortheresonancefunction.
AnimportantconsequenceofthischoiceconcernstheestimationofatypicaltimescaleTNLforthenonlinearwave-waveinteractionsinahomogeneouswaveeld.
Withεatypicalwavesteepnessandω0atypicalangular10TechnicalMemorandumNo.
366NonlinearFourWaveInteractionsandFreakWavesfrequencyofthewaveeld,onendsfromtheBoltzmannequation(25)thatforshorttimesTNL=O(1/ε2ω0),whileforlargetimesTNL=O(1/ε4ω0).
Hence,althoughthestandardnonlineartransfer,whichusesasreso-nancefunctionEq.
(27),doesnotcapturethephysicsofthemodulationalinstability(whichoperatesonthefasttimescale1/ε2ω0),thefullresonancefunctiondoesnotsufferfromthisdefect.
Itisalsoimportanttonotethataccordingtothestandardtheorythereisonlynonlineartransferfortwo-dimensionalwavepropagation.
Intheone-dimensionalcasethereisnononlineartransferinahomogeneouswaveeld.
Thereasonforthisisthatonlythosewavesinteractnonlinearlythatsatisfytheresonanceconditionsk1+k2=k3+k4andω1+ω2=ω3+ω4.
Inonedimensiontheseresonanceconditionscanonlybemetforthecombinationsk1=k3,k2=k4ork1=k4,k2=k3.
Then,therateofchangeoftheactiondensity,asgivenbyEqns.
(25and27),vanishesidenticallybecauseofthesymmetrypropertiesoftheterminvolvingtheactiondensities.
ThiscontrastswiththeBenjamin-Feirinstabilitywhichhasitslargestgrowthratesforwavesinonedimension.
Ontheotherhand,usingthecompleteexpressionfortheresonancefunction,thereisalwaysanirreversiblenonlineartransfereveninthecaseofone-dimensionalpropagation.
TheBoltzmannequation,Eq.
(25),admitsjustasthedeterministicZakharovequation,conservationoftotalaction,wavemomentum,whiletheensembleaverageoftheHamiltonian(Eq.
(9))isconservedaswell(ThelastconservationlawfollowsfromEqs.
(25)byconsistentlyutilizingtheassumptionofaslowlyvaryingactiondensity).
ItisemphasizedthattheHamiltonianconsistsoftwoparts,theenergyaccordingtolinearwavetheoryandanonlinearinteractionterm.
Therefore,unlikethestandardtheoryoffour-waveinteractions,thelinearexpressionforthewaveenergyisnotconserved.
TheexceptionoccursforlargetimeswhentheresonancefunctionRihasevolvedtowardsaδ-function,andthenjustasinthestandardtheorythelinearwaveenergyisconserved.
ThisfollowsalsofromthenumericalsimulationspresentedinSection3whichshowthattheensembleaverageoftheHamilonianisconservedbut,inparticularforshorttimes,notthelinearwaveenergy.
Furthermore,itshouldbementionedthattheBoltzmannequation(25)hasthetimereversalsymmetryoftheoriginalZakharovequation,sincetheresonancefunctionchangessignwhentimetchangessign.
Also,asRivanishesfort=0,thetimederivativeoftheactiondensityspectrumiscontinuousaroundt=0anddoesnotshowacusp.
(cf.
Komenetal,1994).
Nevertheless,despitethefactthatthereistimereversal,Eq.
(25)hastheirreversibilityproperty:thememoryoftheinitalconditionsgetslostinthecourseoftimeowingtophasemixing.
Thestandardnonlineartransferinahomogeneouswaveeldhasbeenstudiedextensivelyinthepastfourdecades.
TheJONSWAPstudy(Hasselmannetal,1973)hasshowntheprominentroleplayedbyfour-waveinteractionsinshapingthewavespectrum,andinshiftingthepeakofthespectrumtowardslowerfrequencies.
Modernwaveforecastingsystemsthereforeuseaparametrizationofthenonlineartransfer(Komenetal,1994).
Ourmaininterestinthispaperisinthestatisticalaspectsofrandom,weaklynonlinearwavesinthecontextoftheZakharovequation.
InparticularweareinterestedintherelationbetweenthedeviationsfromtheGaussiandistributionandfour-waveinteractions.
BecauseofthesymmetriesoftheZakharovequation,therstmomentofinterestisthenthefourthmomentandtherelatedkurtosis.
Thethirdmomentanditsrelatedskewnessvanishes:informationontheoddmomentscanonlybeobtainedbymakingexplicituseofKrasitskii's(1990)canonicaltransformation.
Now,thefourthmomentmaybeobtainedinastraightforwardmannerfromEq.
(16)andtheexpressionforthefourthcumulantEq.
(20)as=34g2dk1,2,3,4(ω1ω2ω3ω4)12+c.
c(28)Denotingthesecondmomentbym0,deviationsfromNormalityarethenmostconvenientlyestablishedbycalculatingthekurtosisC4=/3m201,TechnicalMemorandumNo.
36611NonlinearFourWaveInteractionsandFreakWavessinceforaGaussianpdfC4vanishes.
TheresultforC4isC4=4g2m20dk1,2,3,4T1,2,3,4δ1+234(ω1ω2ω3ω4)12*Rr(ω,t)N1N2N3,(29)whereRrisdenedbyEq.
(22).
Forlargetimes,unliketheevolutionoftheactiondensity,thekurtosisdoesnotinvolveaDiracδ-functionbutratherdependsonP/ω.
Therefore,thekurtosisisdeterminedbytheresonantandnonresonantinteractions.
ItisinstructivetoapplyEq.
(29)tothecaseofanarrowbandwavespectruminonedimension.
Hence,performingtheusualTaylorexpansionsaroundthecarrierwavenumberk0tolowestsignicantorder,onendsforlargetimesC4=8ω20g2m20T0ω0dp1,2,3,4δ1+234p21+p22p23p24N1N2N3,(30)wherep=kk0isthewavenumberwithrespecttothecarrier.
ItisseenthatthesignofthekurtosisisdeterminedbytheratioT0/ω0,whichisthesameparameterthatdetermineswhetherawavetrainisstableornottoside-bandperturbations.
Remarkthatnumericallytheintegralisfoundtobenegative,atleastforbell-shapedspectra.
Hence,fromEq.
(30)itisimmediatelyplausiblethatforanunstablewavesystemwhichhasnegativeT0/ω0thekurtosiswillbepositiveandthuswillresultinanincreasedprobabilityofextremeevents.
Ontheotherhandforastablewavesystemtherewillareductionintheprobabilityofextremeevents.
Finally,afurthersimplicationoftheexpressionforthekurtosismaybeachievedifitisassumedthatthewavenumberspectrumF(p)=ω0N(p)/gonlydependsontwoparametersnamely,thevariancem0andthespectralwidthσk.
Introducethescaledwavenumberx=p/σkandthecorrespondinglyscaledspectrumm0H(x)dx=F(p)dp.
Then,usingthedeep-waterdispersionrelationandT0=k30,Eq.
(30)becomesC4=8sσω2J,(31)wheresisthesignicantsteepnessk0m120whileσωistherelativewidthinangularfrequencyspaceσω/ω0=0.
5σk/k0.
TheparameterJisgivenbytheexpressionJ=dx1,2,3,4δ1+234x21+x22x23x24H1H2H3,andisindependentofthespectralparametersm0andσk.
Therefore,Eq.
(31)suggestsasimpledependenceofthekurtosisonspectralparameters.
Infact,thekurtosisdependsonthesquareoftheBFindexintroducedinEq.
(13).
2.
4EvolutionofaninhomogeneousrandomwaveeldTheBenjamin-Feirinstabilityistheresultofanonlinearinteractionofwavesthatarephase-locked,asthecarrierwaveisphase-lockedwiththesidebandsandthereforethisprocesscannotbedescribedbyatheorythatassumesthattheFourieramplitudesarenotcorrelated,asexpressedbytheassumptionofhomogeneityofthewaveeld(cf.
Eq.
(19)).
Therefore,thissuggeststhatlocalnonlineareventssuchasfreakwavescouldbebeyondthescopeofthestandarddescriptionofoceanwaves.
TheinvestigationoftheeffectofinhomogeneitiesonthenonlinearenergytransferstartedwiththeworkofAlber(1978),andAlberandSaffman(1978),whileCrawfordetal(1980)combinedtheeffectsofinhomo-geneityandnon-Normalityontheevolutionofweaklynonlinearwaterwaves.
Areviewofthismaybefound12TechnicalMemorandumNo.
366NonlinearFourWaveInteractionsandFreakWavesinYuenandLake(1982).
Wewillonlydiscussthelowestordereffectsofinhomogeneity,disregardinganyeffectsresultingfromdeviationsfromNormality,andweonlydiscussone-dimensionalwavepropagation.
Hence,wedonotimposetheconditionofahomogeneouswaveeld(cf.
Eq.
(19)).
NowinvokingtheGaussianapproximationonthefourthmoment(16withD=0)andsubstitutingtheresultintheevolutionequationforthesecondmoment,Eq.
(17),givest+i(ωiωj)Bi,j=2idk2,3,4[Ti,2,3,4δi+234B3,jB4,2Tj,2,3,4δj+234Bi,3B2,4](32)Here,weusedthepropertythatthesecondmomentBishermitian,Bi,j=Bj,i,andwemadeuseofthesymmetrypropertiesofT.
Inprinciple,Eq.
(32)couldbeusedtostudythe(in)stabilityofahomogeneouswavespectra,buttoourknowl-edgethishasnotbeendonesofar.
Insteadofthis,Alber(1978)andAlberandSaffman(1978)studiedthestabilityofanarrow-band,homogeneouswavespectrum.
FollowingCrawfordetal(1980)andYuenandLake(1982),aconsiderablesimplicationoftheevolutionequationforBi,jmaybeachievedbyexpandingangularfrequencyωandinteractioncoefcientTaroundthecarrierwavenumberk0.
Atthesametimeoneintroducesthesumanddifferencewavenumbersn=12(ki+kj),m=kikj(33)whileweintroducetherelativewavenumberp=nk0.
ThecorrelationfunctionBisfromnowonregardedasafunctionmandn.
Realisingthatinthenarrowbandapproximationnisclosetok0whilemissmall,oneobtainsfromEq.
(32)thefollowingapproximateevolutionequationforB,t+im(ω0+pω0)Bn,m=2iT0dl[Bn12l,mlBn+12l,ml]dkBk,l.
(34)Here,aprimedenotesdifferentationwithrespecttothecarrierwavenumberk0,whileT0=k30.
AkeyroleintheworkofAlberandSaffmanisplayedbytheenvelopespectralfunctionW(p,x,t),whichisinfactaWignerdistribution(Wigner,1932).
ItisrelatedtotheFouriertransformofB(n,m,t)withrespecttom,W(p,x,t)=2ω0gdmeimxB(n,m,t).
(35)andahomogeneousseastatesimplyhasaWignerdistributionwhichisindependentofthespatialcoordinatex,inagreementwiththedenitionofhomogeneousseagiveninEq.
(19).
IntermsoftheWignerdistributionEq.
(34)becomesatransportequationinx,pandt,whichbearsasimilaritywiththeVlasovEquationfromplasmaphysics.
ThistransportequationisobtainedbymeansofaTaylorexpansionofthedifferencetermintheright-handsideof(34)withrespecttol,givinganinnitesum.
Theresultist+(ω0+pω0)xW=gT0ω0ρxWp+.
.
.
,(36)whereρ(x,t)=2isthemeansquareenvelopevariance,givenbyρ(x,t)=dpW(p,x,t),(37)whilethedotsontheright-handsideofEq.
(36)representtheremainingtermsoftheTaylorseriesexpansion.
Notethatalltermsoftheseriesarerequiredtoproperlyrecovertherandom-versionoftheBenjamin-Feirinstability.
TechnicalMemorandumNo.
36613NonlinearFourWaveInteractionsandFreakWavesAlberandSaffman(1978)andAlber(1978)studiedthestabilityofahomogeneousspectrumandfoundthatitisunstabletolongwavelengthperturbationsifthewidthofthespectrumissufcientlysmall.
Inotherwords,incaseofinstabilityinhomogeneitieswouldbegeneratedbywhatwetermtherandomversionoftheBenjamin-Feirinstability,thereforeviolatingtheassumptionofhomogeneitymadeinthestandardtheoryofwave-waveinteractions.
ToseewhetherahomogeneousspectrumW0(p)isstableornot,oneproceedsintheusualfashionbyperturbingW0(p)slightlyaccordingtoW=W0(p)+W1(p,x,t),W1W0.
(38)LinearizingtheevolutionequationforWaroundtheequilibriumW0andconsideringnormalmodeperturbationsoneobtainsadispersionrelationbetweentheangularfrequencyωandthewavenumberkoftheperturbation.
InstabilityisfoundforIm(ω)>0.
Alber(1978)consideredasspecialcasetheGaussianspectrumW0(p)=σk√2πexp(p22σ2k),(39)whereisaconstantenvelopevarianceandσkisthewidthofthespectruminwavenumberspace.
Stabilityoftherandomwavetrainwasfoundwhentherelativewidthofthespectrum,σk/k0,exceedsameasureofmeansquareslope.
Intermsoftherelativewidthσω/ω0ofthefrequencyspectrum,whichisjusthalftherelativewidthforthewavenumberspectrum,onendsstabilityifσωω0>(k20)12,(40)whileintheoppositecasethereisinstabilityoftherandomwavetrain.
NotethatintermsoftheBFIndexthestabilityconditionEq.
(40)simplybecomesBFIxpecttondinnaturespectrawithawidthlargerthantherighthandsideofEq.
(40),becauseforsmallerwidththerandomversionoftheBenjamin-Feirinstabilitywouldoccur,resultinginarapidbroadeningofthespectralshape.
Forarandomnarrow-bandwavetrainthisbroadeningisanirreversibleprocessbecauseofphasemixing(Janssen,1983b).
Thebroadeningofthespectrumisassociatedwiththegenerationofinhomogeneitiesinthewaveeld.
Toappreciatethispoint,wementionthattheevolutionequation(36)satisesanumberofconservationlaws.
Usingthealreadyintroducedenvelopesurfaceelevationvarianceρ(x)therstfewconservationlawsaregivenbya)ddtdxρ(x)=0,b)ddtdxdppW=0,(41)c)ddtω0dxdpp2W+gT0ω0dxρ2(x)=0,assumingperiodicboundaryconditionsinx-spaceandthevanishingofWforlargep.
Therstequationexpressesconservationofwavevariance,thesecondoneimpliesconservationofwavemomentumwhilethelastoneisthemostinterestingoneinourpresentdiscussionbecauseitrelatestherateofchangeofspectralwidthtotheinhomogeneityofthewaveeld.
Ifthewaveeldishomogeneousthenρ(x)isindependentofxandthesecondintegralinEq.
(41c)isthen,becauseoftherstconservationlaw,independentoftime.
Therefore,14TechnicalMemorandumNo.
366NonlinearFourWaveInteractionsandFreakWavesforahomogeneouswaveeldthereis,asexpected,nochangeinspectralwidthwithtime;onlyinhomogeneitieswillgiverisetospectralchangeaccordingtolowestorderinhomogeneoustheoryofwavewaveinteractions.
Weremarkthatthersttwoconservationlawsof(41)mayalsobeobtainedimmediatelyfromtheensembleaverageofEqns.
(12),whilethelastconservationlawfollowsfromtheexpressionofthefreewaveenergygiveninEq.
(9)byperformingensembleaveragingandbyinvokingthenarrow-bandapproximation.
Letusgivesomeofthedetailsofthislastderivation.
Thus,inthersttermtheangularfrequencyisexpandedaroundthecarrierwavenumberk0uptosecondorder,whileinthesecondtermtheinteractionmatrixisreplacedbyitsvalueatk0.
Forone-dimensionalpropagationwethereforeget,E=dp1(ω0+p1ω0+12p21ω0)a1a1+T02dp1,2,3,4a1a2a3a4δ1+234.
Now,thersttwotermsarealreadyconservedbecauseofconservationofactionandmomentum,sowewillomitthem.
PerformingensembleaveragingwhileinvokingtheassumptionofaGaussianstate,i.
e.
Eq.
(16)withD=0,andrenamingoftheintegrationvariablesgives=ω02dp1p21+T0dp1,2,3,4δ1+234.
UsingthedenitionfortheWignerdistribution,Eq.
(35),onethennallyarrivesattheconservationlaw(41).
InordertosummarizethepresentdiscussionweremarkthatthecentralroleoftheBFIndexisimmediatelyevidentinthecontextofthelowest-orderinhomogeneoustheoryofwave-waveinteractions.
Accordingtothestabilitycriterion(40)thereischangeofstabilityforBFI=1.
Inotherwords,BFIisabifurcationparameter:ontheshorttimescalespectrawillbestableandthereforedonotchangeifBFIximatetheorythatneglectsdeviationsfromNormality.
Ingeneral,considerabledeviationsfromNormalityaretobeexpected,inparticularincaseofBenjamin-FeirInstability.
Itisthereforeofinteresttoexploretheconsequencesofnon-Normality.
ThiswillbedoneinthenextSectionbymeansofanumericalsimulationofanensembleofsurfacegravitywaves.
3NumericalSimulationofanEnsembleofWaves.
Itisimportanttodeterminetherangeofvalidityofboththehomogeneousandinhomogeneoustheoriesoffourwaveinteractions.
Boththeoriesassumethatthewavesteepnessissufcientlysmall,whilethehomogeneoustheoryignoreseffectsofinhomogeneity,andtheinhomogeneoustheoryassumesthatdeviationsfromNormalityaresmall.
Inordertoaddressthesequestionswesimulatetheevolutionofanensembleofwavesbyrunningadeterministicmodelwithrandominitialconditions.
Onlywavepropagationinonedimensionwillbeconsideredfromnowon.
ForgivenwavenumberspectrumF(k),whichisrelatedtotheactiondensityspectrumthroughF=ωN/g,initialconditionsfortheamplitudeandphaseofthewavesaredrawnfromaGaussianprobabilitydistributionofthesurfaceelevation.
Thephaseofthewavecomponentsisthenrandombetween0and2πwhiletheamplitudeshouldbedrawnfromaprobabilitydistributionaswell(cf.
Komenetal,1994).
Regardingeachwavecomponentasindependent,narrow-bandwavetrains,aRayleighdistributionseemstobeappropriatefortheamplitude.
Weremarkthatbothphaseandamplitudeofthewavesshouldberegardedasrandomvariables.
Choosingonlythephaseasrandomvariablewouldimplythatthewavespectrumisknownprecisely,whichisincontrastwithobservationalexperience.
Thereisaconsiderableuncertaintyinthewavespectrumaswell,whichcanonlybereducedbyobtainingfrequencyspectrafromverylongtimeseriesorwavenumberTechnicalMemorandumNo.
36615NonlinearFourWaveInteractionsandFreakWavesspectrafromsufcientlylargeareas.
Itisstraightforwardtoimplementsuchanapproach.
However,sincethesurfaceelevationisonlydeterminedbyanitenumberofwaves,extremestatesarenotwell-represented.
Asaconsequencethekurtosisofthepdfisunderestimated.
Forexample,forlinearwavesitwascheckedthatevenwith51wavecomponentsandawavenumberresolutionof0.
2σkthekurtosiswasunderestimatedbymorethan5%.
Thesizeoftheensemblewasvariedbetween500and5000.
Ontheotherhand,drawingrandomphasesbutchoosingtheamplitudesofthewavesinadeterministicfashion,asiscommonpractice,gaveonlyanunderestimationofkurtosisby0.
1%.
Sinceourmaininterestisintheproperrepresentationofextremeevents,andsincecomputerresourcesarelimited,itwasthereforedecidedtoonlytaketheinitialphaseasrandomvariable,hence,a(k)=N(k)keiθ(k),(42)whereθ(k)isarandomphase=2πxr,xrisarandomnumberbetween0and1,andktheresolutioninwavenumberspace.
Eachmemberoftheensembleisintegratedforalongenoughtimetoreachequilibriumconditions,typicallyoftheorderof60dominantwaveperiods.
AteverytimestepofinteresttheensembleaverageofquantitiessuchasthecorrelationfunctionB,thepdfofthesurfaceelevationandintegralparameterssuchaswaveheight,spectralwidthandkurtosisistaken.
Typically,thesizeoftheensembleNensis500members.
Thischoicewasmadetoensurethatquantitiessuchasthewavespectrumweresufcientlysmoothandthatthestatisticalscatterinthespectra,whichisinverselyproportionalto√Nens,issmallenoughtogivestatisticallysignicantresults.
WenowapplythisMonteCarloapproachtothenonlinearSchr¨odingerequationandtotheZakharovequation.
3.
1NonlineartransferaccordingtothenonlinearSchr¨odingerequationAsastartingpointwechoosetheZakharovequation(11)withtransfercoefcientsanddispersionrelationappropriateforthenonlinearSchr¨odingerequation.
Theactionvariableiswrittenasasumofδ-functions,a(k)=i=N∑i=Naiδ(kik),(43)wherekistheresolutioninwavenumberspaceand2N+1isthetotalnumberofmodes.
SubstitutionofEq.
(43)intoEq.
(11)givesthefollowingsetofordinarydifferentialequationsfortheamplitudea1,ddta1+iω1a1=i∑1+234=0T1,2,3,4a2a3a4(44)WehavesolvedthissetofdifferentialequationswithaRunge-Kutta34methodwithvariabletimestep.
Relativeandabsoluteerrorofthesolutionhavebeenchoseninsuchawaythatconservedquantitiessuchasaction,wavemomentumandwaveenergyareconservedtoatleastvesignicantdigits.
IncaseofthenonlinearSchr¨odingerequationweexpandtheangularfrequencyaroundthecarrierwavenumberk0uptosecondorder.
Againusingthedifferencewavenumberp=kk0,wendω=ω0+pω0+12p2ω0andweeliminatethecontributionofthersttwotermsbytransformingtoaframemovingwiththegroupvelocity.
Furthermore,theinteractionmatrixTisreplacedbyitsvalueatk0.
Asaresultweobtainddta1+i2p21ω0a1=iT0∑1+234=0a2a3a4(45)16TechnicalMemorandumNo.
366NonlinearFourWaveInteractionsandFreakWavesFigure1:Detailofsurfaceelevationηasfunctionofdimensionlesstimet.
Thetoppanela)showsthetimeseriesforaxedchoiceofinitialphase,θ=0,whilethebottompanelb)showsthetimeseriesforarandomchoiceofinitialphase.
whereT0=k30.
AmplitudeandphaseneededfortheinitialconditionforEq.
(45)aregeneratedbyEq.
(42)wherethewavenumberspectrumisgivenbyaGaussianshape,F(p)=σk√2πexp(p22σ2k).
(46)Beforeresultsontheevolutionofthespectralpropertiesofthesystem(45-46)arepresented,wementionthatthenonlinearSchr¨odingerequationadmitsastraightforwardscalingrelation.
Inordertoseethis,letusremovethedependenceoftheinitialconditiononthevarianceandthewidthσkbyintroducingdimensionlessvariablesp=p/σk,t=(σk/k0)2ω0t,(47)a=k0a/(s√c0),wheresisthewavesteepnessdenedbelow(13)andc0isthephasespeedcorrespondingtothecarrierwavenumber,c0=ω0/k0.
WritingthenonlinearSchr¨odingerequationintermsofthesenewvariablesitisimme-diatelyevidentthatforlargetimesitssolutioncanonlydependonasingleparameter,namelyk0s/σk,whichapartfromaconstantisjusttheBFindexasdenedinEq.
(13).
InitialresultsobtainedfromtheensembleaverageofMonteCarloForecastingdidnotshowthesimplescalingbehaviourwithrespecttotheBFIndex,untilitwasrealisedthatonlyresultsshouldbecomparedforthesamedimensionlesstimet,whichdependsinasensitivemanneronthespectralwidthσk.
WethereforeintegratedtheTechnicalMemorandumNo.
36617NonlinearFourWaveInteractionsandFreakWavesFigure2:Thesurfaceelevationprobabilitydistributionasfunctionofnormalizedheight,η/√m0(withm0thevariance),correspondingtothecasesofFig.
1.
ForreferencetheGaussiandistributionisshownaswell.
systemofequations(45)untilaxeddimensionlesstimet=15.
Aspectralwidthσk=0.
2k0waschosenandwithoutlossofgeneralitythecarrierwavenumberk0=1wastaken.
Theintegrationintervalthencorrespondstoabout60wavepeakperiods.
Furthermore,theresolutioninwavenumberspacewastakenask=σk/3whilethetotalnumberofwavecomponentswas41,thereforecoveringawiderangeinwavenumberspace.
Asalreadynotedthischoicegaveforlinearwavesareasonablesimulationofthepdfofthesurfaceelevation.
Weremarkthatthespecicationofarandominitialphasehasimportantconsequencesfortheevolutionofanarrow-bandwavetrain.
ThisisimmediatelyevidentwhenwecompareinFig.
1timeseriesforthesurfaceelevationfromarunwithaxedinitialphaseθ(k)=0withresultsfromarunwitharandomchoiceoftheinitialphase.
WhilewithadeterministicchoiceofinitialphasethenonlinearSchr¨odingerequationgeneratesinanalmostperiodicfashionextremeevents(Fig.
1a),satisfyingthecriteriaforfreakwaves,witharandomchoiceofinitialphase(Fig.
1b)thisismuchlessevident.
Comparingthetimeseriesfromthetwocasesindetailitisclearthatforxedphasesmallwavesandlargewavesoccurrmorefrequentlythanintherandomphasecase.
ThisimpressionisconrmedbythepdfofthesurfaceelevationshowninFig.
2.
ForreferencewehavealsoshowntheGaussianprobabilitydistribution.
InbothcasesthereareconsiderabledeviationsfromNormality,butinparticularfordeterministicphasethedeviationsarelarge.
SimilardeviationsfromtheNormaldistributionwerefoundbyJanssenandKomen(1982).
TheirapproachwasentirelyanalyticalandtheystartedfromtheassumptionthatforlargetimethesolutionofthenonlinearSchr¨odingerequationwouldevolvetowardsaseriesofenvelopesolitons,describedbyanellipticfunction.
Althoughtheyonlyconsideredthepdfoftheenvelope(whichundernormalconditionsisgivenbytheRayleighDistribution),onemayobtainthepdfofthesurfaceelevationaswell.
Theresultinganalyticalpdfhassimilarcharacteristicsasthepdfforthecaseofdeterministicphase.
TheMonteCarloapproachwasadoptedbecauseitisnotevidentthatforthesystemunderdiscussiontheergodichypothesisapplies.
Thishypothesisimpliesreplacementoftheensembleaveragebyatimeaverage.
However,ifonehappenstochooseinitialphasesinawaythatisfavourableforthegenerationofenvelopesolitons,thenthereisahighprobabilitythatthesolutionstaysclosetotheenvelopesolitonbranchandwillhardlyevervisitotherpartsofphasespace.
InordertoguaranteearepresentativepicturewethereforedecidedtoperformNensrunswhereforeachrunamplitudeandphasearedrawninanindependentmanner.
Intheremainder,onlyensembleaveragedresultswillbediscussed.
18TechnicalMemorandumNo.
366NonlinearFourWaveInteractionsandFreakWavesFigure3:TimeevolutionofspectralwidthforseveralvaluesoftheBFIndex.
Correspondingresultsfromhomogeneoustheoryareshownaswell.
InFig.
3weshowtheevolutionofthespectralwidthσkwithdimensionlesstimetforseveralvaluesofBFI.
Here,σkisdenedusingintegralsofthewavespectrumFoverwavenumberp:σ2k=dpp2F(p)dpF(p).
(48)WeremarkthatforsimulationswiththenonlinearSchr¨odingerequationthisturnedouttogivearemarkablestableestimateofthewidthofthespectralpeak,becausethespectravanishsufcientlyrapidlyforlargep.
Accordingtothissimulationthereisaconsiderablebroadeningofthespectrum,whichoccursonafairlyshorttimescaleofabout10peakwaveperiods.
Inthiscaseofone-dimensionalpropagationthestandardtheoryofnonlineartransferwouldgivenospectralchange.
Notethatσkshowsintheearlystagesoftimeevolutionanovershootfollowedbyarapidtransitiontowardsanequilibriumvalue.
Thenumberofoscilationsaroundthisequilibriumvaluedependontheprecisedetailsofthediscretisationscheme.
Inparticular,more,largeramplitudeoscillationsarefoundforcoarserspectralresolution.
TheovershootisinagreementwithresultsofJanssen(1983b),whostudiedtheevolutionofasingleunstablemodeinthecontextofinhomogeneoustheoryofwave-waveinteractions.
Forsufcientlynarrowspectra,overshootintheamplitudeoftheunstablemodewasfoundfollowedbyadampedoscillationtowardsanequilibriumvalue.
Thedampingtimescalewasfoundtodependonthewidthofthespectrum,vanishingforsmallwidth.
Physically,thedampingiscausedbyphasemixing(ordestructiveinterference)anditseffectdependsonwavenumberresolution.
Asanexampleofspectralevolution,weshowforBFI=1.
40inFig.
4initialandnaltimewavenumberspectrum.
Inordertogiveanideaabouttherepresentativenessoftheresults,95%condencelimits,basedonNens1degreesoffreedom,areshownaswell.
Thebroadeningofthespectrumascausedbythenonlinearinteractionsseemstobestatisticallysignicant.
Althoughthespectralchangeshouldbesymmetricalwithrespecttothecarrierwavenumber,i.
ep=0,itisclearthatthereareasymmetriespresentintheensembleaverageofthenumericalresults.
However,thesedeviationsarewithinthestatisticaluncertainty.
TomakesureofthisweredidthecaseforBFI=1.
40butnowwithanensemblesizeof2000.
Asexpected,statisticaluncertaintywasreducedbyafactoroftwowhileasymmetrieswerereducedaswell.
InordertoexaminewhethertheMonteCarloresultsshowevidenceofabifurcationatBFI=1,weplotinFig.
5therelativeincreaseinspectralwidth,denedas(σk(t∞)σk(0))/σk(0),asfunctionoftheBFindexevaluatedwiththeinitalvalueforspectralwidth.
TheresultssuggestthatthereisonlyevolutionofthespectrumTechnicalMemorandumNo.
36619NonlinearFourWaveInteractionsandFreakWavesFigure4:InitialandnaltimewavenumberspectrumaccordingtotheMonteCarloForecastingofWaves(MCFW)usingthenonlinearSchr¨odingerEquation.
Errorbarsgive95%condencelimits.
Resultsfromtheoryareshownaswell.
Figure5:Relativespectralbroadening(σk(t∞)σk(0))/σk(0)asfunctionoftheBFIndex.
Shownareresultsforfocussing(BF)anddefocussing(noBF)fromthesimulationsandfromtheory,butresultsfromtheoryareidenticalforthesetwocases.
20TechnicalMemorandumNo.
366NonlinearFourWaveInteractionsandFreakWavesFigure6:FinaltimeversusinitialtimevalueoftheBFIndexforthesamecasesasdisplayedinFig.
5.
forsufcientlylargeBFindex,but,incontrasttoinhomogeneoustheoryofwave-waveinteractions,BFI=1doesnotappeartobeabifurcationpoint,asconsiderablechangesinthewavespectrumalreadystarttooccurforBFI=1/2.
Althoughfrominhomogeneoustheoryonewouldexpectasuddentransitionfromnospectralchangetospectralchange,Fig.
5seemstosuggestthatthetransitionisgradual.
WeattributethisdiscrepancytotheassumptionininhomogeneoustheorythatdeviationsfromNormalityaresmall,asthesemaygiverisetoirreversiblechangesofthespectrumaswell.
ThiswillbediscussedmoreextensivelyinthenextSection.
Itisilluminatingtoplottheinformationonspectralwidthinasligthlydifferentmanner,namelybyrelatingthenaltimevalueofBFIwithitsinitialvalue.
ThisisdoneinFig.
6anditclearlyshowsthatforlargetimesBFIhardlyexceedsthevalueof1.
ThisseemstoagreewiththeconjecturegiveninSection2.
2thataccordingtoinhomogeneoustheory(c.
f.
Eq.
(40))oneshouldnotexpectspectratohaveaBFImuchlargerthan1.
AccordingtotheMonteCarloresults(cf.
Fig.
3)thetimescaleofchangeforlargeBFIis,onaverage,onlyafewwaveperiods.
NonlineareffectsgiverisetoconsiderablechangesintheprobabilitydistributionofthesurfaceelevationfromtheGaussiandistribution(cf.
alsoOnoratoetal,2000),althoughthedeviationsareofcoursemuchlesstheninthecasesdiscussedinFig.
2.
ThisisshowninFig.
7forBFI=1.
40.
Toemphasizetheoccurrenceofextremeeventswehaveplottedthelogarithmofthepdfasfunctionofthesurfaceelevationnormalisedwiththewavevariance.
TheGaussiandistributionthencorrespondstoaparabola.
Thesimulatedpdf,intherangeof2-4,showsanalmostlinearbehavioursuggestinganexponentialdecayofthepdf.
Finally,inFig.
8wesummarizeourresultsonthedeviationsfromNormalitybyplottingthenaltimevalueofthekurtosisC4=/3m201asfunctionofthethenaltimeBFIndex.
Here,thefourthmomentwasdeterminedfromthepdfofthesurfaceelevationwhichwasobtainedbysamplingthesecondhalfofthetimeseriesforthesurfaceelevationatanarbitrarilychosenlocation.
Alternatively,thefourthmomentmaybeobtainedfromEq.
(28)givingverysimilaranswers.
Forsmallnonlinearityonewouldexpectavanishingkurtosis,butthesimulationstillunderestimates,asalreadymentioned,thekurtosisby2%.
ThekurtosisdependsalmostquadraticallyontheBenjamin-FeirIndexuptoavaluecloseto1.
ThisquadraticdependencewillbeexplainedinthenextSection,whenaninterpretationofresultsisprovided.
NearBFI=1,ontheotherhand,thekurtosisbehavesinamoresingularfashion,because,inagreementwiththediscussionofFig.
6,theBenjamin-FeirIndexcannotpassthebarriernear1.
ThenonlinearSchr¨odingerequation(45)fordeepwaterwavesisanexamplewherenonlinearityleadstofo-TechnicalMemorandumNo.
36621NonlinearFourWaveInteractionsandFreakWavesFigure7:Probabilitydistributionfunctionforsurfaceelevationasfunctionofnormalizedheightη/√m0.
ResultsfromnumericalsimulationswiththenonlinearSchr¨odingerequationandhomogeneoustheoryincaseoffocussing(BFIndexof1.
4).
ForreferencetheGaussiandistributionisshownaswell.
Freakwavescorrespondtoanormalizedheightof4.
4orlarger.
cussingofwaveenergyandthereforecounteractsthedispersionbythelineartermwhichisproportionaltoω0.
Theresultsfromthenumericalsimulationdoindeedsuggestthatwhennonlinearityissufcientlystrongfocussingofenergyoccursgivingconsiderableenhancementstotheprobabilityofextremeevents,atleastcom-paredtothenormaldistribution.
Intheoppositecase,whenthenonlineartermhasoppositesigndefocussingofwaveenergyoccursandonewouldexpectareductioninthenumberofextremeevents.
InordertoshowthisweperformedsimulationswiththenonlinearSchr¨odingerequation(45)butnowwithnegativenonlineartransfercoefcient(T0=k30).
ResultsofthiscaseareshownintheFigs.
5,6and8whilethelogarithmofthepdfofthesurfaceelevationisshowninFig.
9.
TheseplotsshowthatinthecaseofdefocussingthebroadeningFigure8:NormalizedKurtosisasfunctionoftheBFIndex.
Shownareresultsforfocussing(BF)anddefocussing(noBF)fromthesimulationsandfromtheory.
Thetheoreticalresultfordefocussingcanbeobtainedfromtheresultsoffocussingbyareectionwithrespecttothex-axis.
22TechnicalMemorandumNo.
366NonlinearFourWaveInteractionsandFreakWavesFigure9:Probabilitydistributionfunctionforsurfaceelevationasfunctionofnormalizedheightη/√m0.
ResultsfromnumericalsimulationswiththenonlinearSchr¨odingerequationandhomogeneoustheoryincaseofdefocussing(BFIndexof1.
4).
ForreferencetheGaussiandistributionisshownaswell.
ofthespectrumislessdramatic.
Furthermore,thenaltimeBenjamin-FeirIndexdoesnothavealimitingvalueofabout1.
Ontheotherhand,thekurtosisforthiscaseisnegative,resulting,ascanbeseenfromFig.
9,inalargereductionoftheprobabilityofextremeevents.
ThedependenceofthekurtosisonBFIisdifferentfromthecaseoffocussing,becausethereareclearsignsofsaturationbeyondBFI=1,whileonlyintherangeBFIxtorderinbandwidthandhefoundasurprisinglylargeimpactontheresultsforthegrowthratesofthemodulationalinstability.
Similarly,Crawfordetal(1981)studiedthestabilityofauniformwavetrainusingthecompleteZakharovequationwhichretainsallthehigh-orderdispersioneffects.
Ingeneral,growthratesarereducedcomparedtoresultsfromthenonlinearSchr¨odingerequation,thereforeaccordingtotheZakharovandtheDystheequationauniformwavetrainislessunstable.
Infact,growthratesandthresholdsforinstabilitywereinbetteragreementwithexperimentalresultsofBenjamin&Feir(1967)andLakeetal(1977)(cf.
alsoJanssen,1983a).
TheZakharovandtheDystheequationhave,inaddition,theinterestingpropertythatthenonlineartransfercoefcientandtheangularfrequenciesarenotsymmetricalwithrespecttothecarrierwavenumber.
Itwillbeseenthatthishasimportantconsequencesforthespectralshape.
TheDystheequationfollowsfromtheZakharovequationbyexpandingangularfrequencytothirdorderinthemodulationwavenumberpwhiletheinteractionmatrixTisexpandeduptorstorderinp.
Fornarrow-bandwavetrainsitgivesanaccuratedescriptionoftheseastate.
However,wavespectramaybecomesobroadthatthenarrow-bandapproximationbecomesinvalid,andthereforewehavechosentostudynumericalresultsfromtheZakharovequation.
TheZakharovequationwesolvedwasgivenbyEq.
(44),wherethenonlineartransfercoefcientwasfromKrasitskii(1990),whiletheexactdispersionrelationfordeepwatergravitywaveswastaken.
Theinitialcon-ditionwasprovidedbyEq.
(46).
ThediscretisationdetailswereidenticaltothoseofthenumericalsimulationsTechnicalMemorandumNo.
36623NonlinearFourWaveInteractionsandFreakWavesFigure10:InitialandnaltimewavenumberspectrumaccordingtoMonteCarloForecastingofWaves(MCFW)usingtheZakharovequation.
Errorbarsgive95%condencelimits.
Resultsfromtheoryareshownaswell.
Figure11:ComparisonofcurvesofnaltimeversusinitialvalueoftheBFIndexfromsimulationswiththenonlinearSchr¨odingerequationandfromtheZakharovequation.
Thecorrespondingtheoreticalresultsareshownaswell.
24TechnicalMemorandumNo.
366NonlinearFourWaveInteractionsandFreakWaveswiththenonlinearSchr¨odingerequation.
BecausetheZakharovequationcontainsallhigher-ordertermsinthemodulationwavenumberpitisnotpossibletoprovethatthelargetimesolutionoftheinitialvalueproblemisdeterminedcompletelybytheBFIndex,butingoodapproximationtheBFIndexcanstillbeusedforthispurposeaslongasthespectraarenarrow-banded.
InFig.
10wehaveplottedtheensembleaveragedwavenumberspectrumforBFI=1.
4,anditshowsacleardown-shiftofthepeakofthespectrumwhilealsoconsiderableamountsofenergyhavebeenpumpedintothehigh-wavenumberpartofthespectrum.
Thewavenumberdown-shiftiscausedbytheasymmetriesinthenonlineartransfercoefcientandtothesameextendbytheasymmetriesintheangularfrequencywithrespecttothecarrierwavenumber.
ThiswascheckedbyrunningEq.
(44)withconstantnonlineartransfercoefcient,andsimilarlylookingensemblemeanspectra,butwithhalfthewavenumberdown-shift,wereobtained.
Thereisalsoanoticablebroadeningofthespectrum.
However,becauseoftheincreasedspectrallevelsathighwavenumbersuseofthesecondmomentofthewavenumberspectrum,asdoneforthenonlinearSchr¨odingerequation(cf.
(48)),tomeasurethewidthofthespectralpeakisnotappropriate.
InsteadweuseherethewidthasobtainedfromttingthepeakofthespectrumwithaGaussianshape-function.
TherelationbetweenthenaltimeBFIversustheinitialvalueofBFIisshowninFig.
11andiscomparedwiththecorrespondingonefromthenonlinearSchr¨odingerequation.
Also,Fig.
12showsthenormalizedkurtosisversusthenaltimeBFindex.
ResultsfromtheZakharovequationareinqualitativeagreementwiththeonesfromthenonlinearSchr¨odingerequation.
However,becausegrowthratesaresmaller,thebroadeningofthespectrumisless,thenaltimeBFindexishigherbyabout10%andthenormalizedkurtosisissmalleraswell.
AuniquefeatureoftheZakharovequationisthedown-shiftofthepeakofthewavenumberspectrum.
ThisisshowninFig.
13wherewehaveplottedthenaltimevalueofthepeakwavenumber,normalizedwithitsinitialvalueversustheinitialBFindex.
ForlargevaluesofBFIreductionsinpeakwavenumberofmorethan10%arefoundfromtheresultsofthenumericalsimulations,butthedependenceofthedown-shiftinpeakwavenumberonBFIisnotsmooth.
Thisiscausedbythefactthattheensembleaveragedspectranotalwayshaveawell-denedspectralpeak.
Figure12:NormalizedKurtosisasfunctionoftheBFIndex.
ShownareresultsforfocussingfromsimulationswiththenonlinearSchr¨odingerequationandwiththeZakharovequation.
Thecorrespondingtheoreticalresultsareshownaswell.
TechnicalMemorandumNo.
36625NonlinearFourWaveInteractionsandFreakWavesFigure13:Finaltimepeakwavenumberdown-shiftversusBFIndex.
ShownisacomparisonbetweennumericalsimulationresultsfromtheZakharovequationandtheory.
4InterpretationofnumericalresultsInthepreviousSectionwehavediscussedresultsfromtheMonteCarlosimulationofthenonlinearSchr¨odingerequationandtheZakharovequation.
Theseresultsshowthatonaveragethereisarapidbroadeningofthewavespectrum,whilenonlinearitygivesrisetoconsiderabledeviationsfromGaussianstatistics.
ThequestionnowiswhethertheaverageoftheMonteCarloresultsmaybeobtainedintheframeworkofasimpletheoreticaldescription.
InSection2wehavediscussedtwoattemptstoachievethis.
Therstoneisthestandardtheoryofwave-waveinteractions,extendedwiththeeffectsofnonresonantfourwaveinteractions.
ThisapproachassumesahomogeneouswaveeldbutallowsfordeviationsfromtheGaussianseastate.
Thesecondtheoryistheinhomogeneoustheoryofwave-waveinteractionswhichassumesthateffectsofinhomogeneityinthewaveeldaredominantwhiledeviationsfromNormalityonlyplayaminorrole.
ThisapproachseemstobeanidealstartingpointfortreatinginhomogeneousandnonstationaryphenomenasuchasfreakwavesbecauseitdescribestherandomversionoftheBenjamin-Feirinstability.
LetusthereforerstdiscussthevalidityofinhomogeneoustheoryusingresultsfromtheMonteCarloForecastingofoceanwavesobtainedfromthenonlinearSchr¨odingerequation.
Accordingtoinhomogeneoustheorythebroadeningofthewavespectrumiscausedbytheinhomogeneityofthewaveeld.
Thisisclearlyexpressedbytheconservationlaw(41c)andexplainedinthediscussionthatfollows.
Fromthisconservationlawonemaythereforeobtainameasureofinhomogeneityofthewaveeld,namelyI=I2I21,(49)whereI1=dxρ(x),(50)andI2=dxρ2(x).
(51)26TechnicalMemorandumNo.
366NonlinearFourWaveInteractionsandFreakWavesFigure14:Timeevolutionofameasureforinhomogeneity,I,fortwodifferentensemblesizes,accordingtothenonlinearSchr¨odingerequation.
Here,theintegralsoverspaceareweightedbytheseizeofthedomain,andusingthedenitionsforρ(Eq.
37)andtheWignerdistribution(cf.
Eq.
(35))onemayexpresstheinhomogeneitymeasureIintermsofthecorre-lationfunctionB(n,m,t)asI1=2ω0g∑pB(p,0),(52)whileI2=2ω0g2∑p,p,mB(p,m)B(p,m),(53)ThecorrelationfunctionsB(p,m)maybereadilyobtainedfromthenumericalresultsforthecomplexamplitudea(k,t)afterensembleaveraging.
ForahomogeneouswaveeldB(p,m)=N(p)δ(m),henceI2=I21,orI=1.
TheinitialconditionsusedinthenumericalsimulationofwaveshavebeenchoseninsuchawaythattheseastatecorrespondstoaGaussianone.
Asaconsequence,becausethecomplexamplitudesa(k,t)arenotcorrelated,thisimpliesthatinitiallytheseastateishomogeneousaswell(cf.
Komenetal,1994).
However,thewaveensembleconsistsofanitenumberofmembers,andthismeansthattheinitialprobabilitydistributionisnotaperfectGaussian(thekurtosisisslightlyunderestimated,forexample)butitalsomeansthattheinitialconditionsareslightlyinhomogeneous.
Accordingtotheinhomogeneoustheorytheperturbationsshouldgrowexponentiallyintimeresultinginforexampleabroadeningofthewavespectrum.
ForBFI=1.
4,theevolutionintimeoftheinhomogeneityIisshowninFig.
14.
Initially,inhomogeneityissmallbutgrowsrapidlyinthecourseoftime,whichisthenfollowedbyanoscillationaroundthelevel1.
004.
Thislevelofinhomogeneityandthevariationwithtimeis,however,extremelysmall(notethatforthecaseofFig.
1aIisoftheorderof3)anditcannotexplainthelargechangesinthewavenumberspectrumwehaveseeninthenumericalsimulations.
Inaddition,accordingtotheinhomogeneoustheorytheconservationlaw(41c)shouldbesatised.
InSection3.
2itwasexplainedthatthisconservationlawfollowsfromtheconservationofHamiltonian,assumingthatdeviationsfromNormalitymaybeignored.
Itisofinterest,ofcourse,totestwhetheritisjustiedtoignoreTechnicalMemorandumNo.
36627NonlinearFourWaveInteractionsandFreakWaveseffectsofthefourthcumulant.
TothatendwecompareinFig.
15theevolutionintimeoftheHamiltonianasobtainedfromthenumericalsimulation(thiswillbecalledthe'exact'Hamiltonianfromnowon)withtheHamiltonianaccordingtoinhomogeneoustheory.
Whilethe'exact'Hamiltonianisaconstant(atleastuptovesignicantdigits),itisclearthattheapproximateHamiltonianisnotconservedwhenevaluatedusingthenumericalresults.
Infact,therearelargedeviationsastheapproximateHamiltonianbecomesnegative,whilethe'exact'Hamiltonianispositivedenite.
Thedisagreementbetweentheapproximateandthe'exact'Hamil-tonianiscausedbytheneglectofthehigherordercumulants.
ThisisimmediatelyclearfromFig.
15wherewehavecomparedthenonlinearcontributiontotheHamiltonianaccordingtolowestorderinhomogeneousthe-ory(calledapproximate),withthecorrespondingnonlinearcontributionthatincludeshigherordercumulants(called'exact').
Theapproximatenonlinearcontributionhardlyvarieswithtime,whichisinagreementwiththeresultsfromFig.
14thateffectsofinhomogeneityaresmall.
The'exact'nonlinearcontributionshowsasignicantvariationwithtime.
Thedifferencebetween'exact'andapproximateareconsiderableandthereforeitisnotjustiedtoignoreeffectsofdeviationsfromNormalityinasimpletheoreticaldescriptionoftheevo-lutionoftheseastate.
Asamatteroffact,thedeviationsfromNormalityarethemainreasonforthespectralbroadeningasthetime-varyingnonlinearcontributiontotheHamiltonian,includingeffectsofthefourthordercumulant,justcompensatesforthechangeswithtimeofthelinearpartofthewaveenergy.
Clearly,accordingtotheMonteCarlosimulationsthelinearwaveenergyisnotconserved.
Insummary,ithasbeenshownthatintheinhomogeneoustheoryoffour-waveinteractionseffectsofthegenerationofthefourthcumulantcannotbeignored.
Atthesametimewehaveshownthatthenumericalensembleofoceanwavesmayberegardedtogoodapproximationasahomogeneousensemble.
Hence,thestandardtheoryoffour-waveinteractions(extendedbyincludingnonresonantinteractions),whichassumesahomogeneouswaveeld,maybeagoodcandidatetoexplaintheresultsofthenumericalsimulationsinSection3.
Therefore,weusedtheBoltzmannequation(25)toevolvetheactiondensityN(k)forthesamecasesaspre-sentedinSection3.
ThedifferentialequationwassolvedwithaRunge-Kutta34methodwithvariabletimestep,andthecontinuousproblemwasdiscretizedinthesamewayaswasdoneincaseofthesolutionoftheFigure15:TimeevolutionofHamiltonianEofthenonlinearSchr¨odingerequation.
AlsoshownaretimeevolutionofEaccordingtolowestorderinhomogeneoustheory(Eapprox)andaccordingtohomogeneoustheorythatincludesthefourthcumulant.
Finally,shownarethenonlinearcontributiontoEforastrictlyGaussianstate(NL(C4=0)),andincludingallhigherordercumulants(NL+C4).
28TechnicalMemorandumNo.
366NonlinearFourWaveInteractionsandFreakWavesZakharovequation.
Runtimesusingthehomogeneoustheoryaretypicallytwoordersofmagnitudesfasterthanwhenfollowingtheensembleapproach.
Incontrasttoinhomogeneoustheory,thestandardtheorygivesamuchbetterapproximationtothe'exact'HamiltonianasshowninFig.
15.
Thereare,asshouldbe,smalldifferencesbecausethestandardtheoryisanapproximationaswell,sincebotheffectsofinhomogeneityandthesixthcumulanthavebeenneglected.
FurtherresultsfromthediscretizedversionofthehomogeneoustheoryarecomparedwiththeonesfromthesimulationswiththenonlinearSchr¨odingerequationintheFigs.
3-8.
FromFig.
3whichshowstheevolutionintimeofthespectralwidthforseveralvaluesoftheBenjamin-FeirIndexitisseenthatforlargetimesthereisgoodagreementbetweenhomogeneoustheoryandtheensembleaveragedresultsfromtheMonteCarlosimulations.
ForshorttimesitishoweverevidentthatEq.
(25)doesnotshowtheovershootfoundinthenumericalsimulations.
AlikelyreasonfortheabsenceofovershootinthetheoreticalcalculationsistheassumptionthattheactiondensityvariesslowlycomparedtothetimescaleimpliedbytheresonancefunctionRi(ω,t).
Boththenumericalsimulationsandhomogeneoustheoryshowontheonehandforshorttimesarapidbroadeningofthewavespectrumwhichforlargetimesisfollowedbyatransitiontowardsasteadystate.
Theevolutiontowardsasteadystatecanbeunderstoodasfollows:First,itshouldbenotedthat,accordingtoSection2.
3,forone-dimensionalpropagationthereisnononlineartransferduetoresonantnonlinearinteractions.
Now,initiallytheresonancefunctionRi(ω,t)willbewidesothatnonresonantwave-waveinteractionsareallowedtomodifytheactiondensityspectrum.
Butafterabout5-10waveperiodstheresonancefunctionbecomesprogressivelynarroweruntilitbecomesapproximatelyaδ-function,henceonlyresonantwavesareselected.
Inthateventthereisnochangeoftheactiondensityspectrumpossibleanymoresothatforlargetimesasteadystateisachieved.
AexampleofthecomparisonbetweentheoreticalandsimulatedspectrumisgiveninFig.
4.
Thereisafairagreementbetweenthetwo.
However,itshouldbementionedthattypicallythesimulatedspectrumisslightlymorepeakedthanthetheoreticalonedespitethefactthatthereisacloseagreementinspectralwidth.
ThisagreementinspectralwidthbetweentheoryandsimulationisalsoverymuchevidentintheFigs.
5and6overthefullrangeoftheinitialvalueofBFI.
InparticularnotethatthereiscloseagreementbetweentheupperlimitofthenaltimeBFIfromtheoryandthesimulation.
Hence,homogeneoustheoryalsoprovidesanexplanationofwhythereisalowerboundtospectralwidth.
Wethereforehavethecurioussituationthatbothhomogeneousandinhomogeneoustheoryexplainwhythereisalowerboundtospectralwidthasfoundinthenumericalsimulations.
However,sinceithasbeenshownthatinhomogeneitiesonlyplayaminorroleinthenumericalsimulationsitfollowsthatonlyhomogeneoustheoryprovidesaproperexplanation.
In-situobservationsfromtheNorthSeaseemtoindicatethepresenceofalowerboundtospectralwidthaswell(seeforexampleJanssen,1991).
Althoughitisimpossibletoproveatpresentthatatseainhomogeneitiesdonotplayarole,homogeneoustheoryevenseemstogiveaplausibleexplanationofthelowerboundfoundatsea.
AsdiscussedinSection2.
3nonlinearitygivesrisetodeviationsfromthenormaldistribution.
WedeterminedthenormalisedkurtosisusingEq.
(29),whichisobtainedfromthefourthcumulantD.
Introducingthenormal-izedheightx=η/√m0,thepdfofthenormalizedsurfaceelevationxisthengivenbyp(x)=1+18C4d4dx4f0,(54)wheref0isgivenbythenormaldistributionf0(x)=1√2πexp(x22).
(55)TechnicalMemorandumNo.
36629NonlinearFourWaveInteractionsandFreakWavesEq.
(54)followsfromanexpansionofthepdfpintermsoforthogonalfunctions(d/dx)nf0.
Here,nisevenbecauseofthesymmetryoftheZakharovequation.
Theexpansioncoefcientsarethenobtainedbydeterminingtherst,secondandfourthmoment.
FortherangeofBFIstudiedinthispaperitwasveriedthathighermomentsonlygaveasmallcontributiontotheshapeofthepdfp(x).
ThepdfaccordingtotheoryiscomparedinFig.
7withthesimulatedone,andagoodagreementisobtained,evenforextremeseastateconditions.
Clearlyinthecaseofnonlinearfocussing,theprobabilityofextremestatesis,asexpected,largerwhencomparedtothenormaldistribution.
Finally,inFig.
8theoreticalandsimulatednaltimekurtosisisplottedasfunctionofthenaltimeBFI.
AgoodagreementbetweenthetworesultsisobtainedevenclosetothelimitingvalueofthenaltimeBFI.
ForBFIxtremevaluesoftheBenjamin-FeirIndex.
Here,itshouldbeemphasizedthatatseaBFIhastypicalvaluesof0.
5orlessandonlyoccasionalyvaluesoftheorder1arereached.
WehaveperformedsimulationstovaluesofBFIofupto3andevenfortheseextremeconditions,havinglargevaluesofkurtosisforexample,areasonableagreementisobtained.
ThisissurprisingbecausethehomogeneoustheorywasderivedundertheassumptionofsmalldeviationsfromNormality.
Inthecaseofnonlineardefocussingtherangeofvalidityofhomogeneoustheoryismuchmorerestricted.
ThisismadeplainlyclearintheFigs.
5,6,8and9,whereresultsfromthesimulationsandhomogeneoustheoryarecomparedforthecaseofnonlineardefocussing.
InordertobeabletointerpretthiscomparisonwenotethathomogeneoustheorydoesnotdistinguishbetweenfocussinganddefocusingbecausethenonlineartransferisindependentofthesignoftheinteractionmatrixT.
OnlythekurtosisdependsonthesignofT.
JudgingfromtheFigs.
5,6and8therangeofvalidityofhomogeneoustheoryisrestrictedtoBFIxthepdfgiveninEq.
(54)maybecomenegative.
Thisisclearlyunrealisticandinordertocorrectthisundesirablepropertyofhomogeneoustheoryoneneeds,forlargeBFI,totakeintoaccounttheeffectsofhigherthenfourthordercumulantsaswell.
Itisbelievedthisisthemainreasonwhyhomogeneoustheoryhassucharestrictedvalidityinthiscase.
Intheoppositecaseofnonlinearfocussingthekurtosisispositive,givingforlargexapositivecorrectiontothenormaldistribution.
Thepdfofthesurfaceelevationisthereforepositive,atleastforthecasesthathavebeenstudiedhere,andasaconsequencehomogeneoustheoryhasamuchlargerrangeofvalidity.
Finally,weappliedhomogeneoustheorytotheZakharovequation.
ResultsarecomparedwiththenumericalsimulationsintheFigs.
10,11,12and13.
Thereisafairagreementbetweensimulatedandtheoreticalspectrum(cf.
Fig.
10),betweensimulatedandtheoreticalnaltimeBFIindex(cf.
Fig.
11)andnormalizedkurtosis(cf.
Fig.
12).
Lessfavourableistheagreementbetweensimulatedandtheoreticalpeakwavenumber,asshowinFig.
13.
Thetheoreticalresultsshowasmoothdependenceofwave-numberdown-shiftonBFI,givingshiftsof20%ormoreforlargevaluesofBFI.
However,thesimulationshowsmorescatterwhilethedown-shiftisatmost15%.
Thereasonforthescatterinthesimulatedresultsisprobablythatthepeakofthewavenumberspectrumisnotalwayswell-dened.
Incontrast,homogeneoustheorygivesasmootherspectrumandawell-denedpeakofthespectrum.
30TechnicalMemorandumNo.
366NonlinearFourWaveInteractionsandFreakWaves5ConclusionsPresentdaywaveforecastingsystemsarebasedonadescriptionoftheensembleaveragedseastate.
Inthisapproachthewavenumberspectrumplaysacentralroleanditsevolutionfollowsfromtheenergybalanceequation.
Thequestiondiscussedhereiswhetheritispossibletomakestatements,necessarilyofastatisticalnature,ontheoccurenceofextremeeventssuchasfreakwaves.
Inordertoshowthatthisispossiblethefollowingapproachhasbeenadopted.
Thestartingpointisadeter-ministicsetofequations,namelytheZakharovequationoritsnarrow-bandlimitthenonlinearSchr¨odingerequation.
Thereisampleevidencethattheseequationsadmitfreakwavetypesolutions.
Thesefreakwavesoc-curwhenthewavesaresufcientlysteepasnonlinearfocussingmaythenovercomethespreadingofenergybylineardispersion.
ForthesamereasontheBenjamin-Feirinstabilityoccurs.
AsshowninFig.
1theoccurrenceoffreakwavesdependsinasensitivemanneronthechoiceoftheinitialphaseofthewaves.
Inaddition,ontheopenoceanwavespropagatefromdifferentlocationstowardsacertainpointofinterest,andmaythereforeberegardedasindependent.
Hence,foropenoceanapplicationstherandomphaseapproximationseemstobeappropriate.
Wethereforesimulatedtheevolutionofanensembleofoceanwavesbyrunningadeterminis-ticmodelwithrandominitialphase.
TheseMonteCarlosimulationsareexpensive(typically,thesizeoftheensembleis500)sothatwerestrictourselvestothecaseofone-dimensionalpropagationonly.
TheensembleaverageoftheresultsfromtheMonteCarlosimulationsshowsthatwhentheBenjamin-FeirIndexissufcientlylarge(asoccursforthecombinationofnarrowspectraandsteepwaves)thewavespectrumbroadenswhileatthesametimeconsiderabledeviationsfromtheGaussianpdfarefound.
IncaseoftheZakharovequationthespectralchangeisevenasymetricalwithrespecttothepeakofasymmetricalspectrumgivingadown-shiftofthepeakwavenumberwhileasaconsequenceoftheconservationofwaveactionandwaveenergyconsiderableamountsofenergyarebeingpumpedintothehigh-wavenumberpartofthespectrum.
Thesespectralchangesoccuronashorttimescale,typicallyoftheorderoftenwaveperiods.
ThistimescaleiscomparablewiththeBenjamin-Feirtimescale.
Thestandardhomogeneoustheoryofresonantnonlineartransferdoesnotgivespectralchangeinthecaseofone-dimensionalpropagation.
Thistheorywasthereforeextendedtoallowfornonresonantinteractionsaswell,becausethenonlinearfocussingrelatedwiththeBenjamin-FeirInstabilityisanexampleofanonresonantfour-waveinteraction.
Thisnonlinearfour-wavetransferisassociatedwiththegenerationofhigherordercumulantssuchasthekurtosis.
Deviationsofthesurfaceelevationpdffromthenormaldistributioncanthereforebeexpressedintermsofasix-dimensionalintegralinvolvingthecubeoftheactionspectrum(cf.
Eq.
(29)).
Incaseofnonlinearfocussingthereis,foralargerangeofvaluesoftheBFI,agoodagreementbetweentheensembleaveragedresultsfromthenumericalsimulationsandhomogeneoustheory.
Thisisinparticulartrueforthebroadeningofthespectrum,thespectralshapeandtheestimationofthekurtosis.
Comparedtothesimulations,theoryoverestimates,however,thepeakwavenumberdown-shift.
Homogeneoustheoryalsoexplainswhyforone-dimensionalpropagationthewavespectrumevolvestowardsasteadystate.
Theresonancefunctionevolvesrapidlytowardsaδ-function,henceforlargetimesonlyresonantwave-waveinteractionsareselectedwhichinonedimensiondonotgiverisetospectralchange.
Thisisinsharpcontrastwiththecaseoftwo-dimensionalpropagation.
Notrendtowardsasteadystateisexpectedinthatcasebecauseresonantfour-waveinteractionsdocontributetoachangeinthespectrum.
Theextendedversionofthehomogeneousfour-wavetheoryhastwotimescales,afastoneonwhichthenonres-onantinteractionstakeplaceandalongtimescaleonwhichtheresonantinteractionsoccur.
Thenonresonantinteractionsplayasimilarroleastransientsinthesolutionofaninitialvalueproblem.
Theyaresimplygen-eratedbecauseinitiallythereisamismatchbetweenthechoiceoftheprobabilitydistributionofthewaves,aGaussian,andbetweentheinitialchoiceofthewavespectrum,representingaseastatewithnarrow-band,steepTechnicalMemorandumNo.
36631NonlinearFourWaveInteractionsandFreakWavesnonlinearwaves.
Forexample,ifonecouldchooseaprobabilitydistributionfunctionwhichisinequilibriumwiththenonlinearseastate(theoreticallyonecan,bytheway),thennonresonantinteractionswouldnotcon-tribute.
Onlyresonantwave-waveinteractionswillthengiverisetononlineartransfer.
Inthegeneralcaseforwhichthereisanitemismatchbetweenpdfandwavespectrum,thenonresonantcontributionwilldieoutveryquicklyowingtophasemixing,butwill,nevertheless,aswehaveseen,resultinconsiderablechangesinthewavespectrum.
Thequestionthereforeiswhetherthereisaneedtoincludeeffectsofnonresonantinteractions.
Thisdependsontheapplication.
Inwavetankexperiments,whereonecanprogramawavemakertoproducetheinitialconditionsusedinthispaper,itseemsthateffectsofnonresonanttransferneedtobetakenaccountfor.
Fortheopenoceancasethisisnotclear.
ThepointisthatinnaturethecombinationofsteepwavesandastrictlyGaussiandistributionmostlikelydoesnotoccur.
Changesinnatureareexpectedtobemoregradualsothatthemismatchbetweenpdfandwavespectrumissmall.
Onlywhenawindstartsblowingsuddenly,henceforshortfetchesandduration,effectsofnonresonantinteractionsareexpectedtoberelevant.
Moreresearchinthisdirectionis,however,required.
TheresultsfromMonteCarloSimulationsdonotprovideevidencethattherearesignicantdeviationsfromhomogeneityoftheensembleofwaves.
DeviationsfromNormalityarefoundtobemuchmoreimportant.
Nevertheless,itcannotbeconcludedfromthepresentstudythattheinhomogeneousapproachofAlberandSaffman(1978)isnotrelevantforrealoceanwaves.
Forexample,effectsofinhomogeneitymightberelevantinfetch-limited,rapidlyvaryingcircumstances.
However,itseffectsareexpectedtobesmallandthereforeonlythelowestorderapproximation,asgivenexplicitelyinEq.
(36),needstoberetained.
Finally,forrealoceanwavesnotonlyfour-waveinteractionsdeterminetheevolutionofthewavespectrum.
Windinputanddissipationduetowhite-cappingarerelevant,andtheseprocessesmayeffectthekurtosisoftheseasurfaceaswell.
However,ithasbeenshowninthispaperthatinparticularthenonresonantnonlineartransferactsonashorttimescalewhichismuchshorterthenthetimescalesassociatedwithwindinputanddissipationsourcefunctionsusedinwaveforecasting(Komenetal,1994).
Hence,onewouldexpectthattheexpressionforkurtosisfoundinthispaper(cfEq.
(29))shouldberelevantinnature.
Nevertheless,nonlinearfocussingmayresultinsteepwaves.
Iftheirsteepnessexceedsacertainthresholdonewouldexpectasignicantamountofwavebreaking,thuslimitingtheheightofthesewaves,andeffectingtheextremestatistics.
Arealistic,deterministicmodelofwavebreakingisneededtoassesstheimportanceofwavebreakinginthesecircumstances.
Itmaybemoreeffective,however,totrytocompareresultsfromthepresentapproachdirectlywithobservationsofextremescollectedoveralongperiod.
AcknowledgementTheauthoracknowledgesusefulandstimulatingdiscussionswithMiguelOnorato.
Hispresentationduringthe2001WisemeetinginCanadagavestimulustohaveafreshlookatwave-wavein-teractionsandfreakwaves.
Furthermore,theauthorthanksTheovanSteynforprovidingtheRunge-Kuttasoftwareforaccurateintegrationofasetofordinarydifferentialequations.
Finally,theauthorthanksAnthonyHollingsworthandMartinMillerforcriticallyreadingthemanuscript.
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