ObservationsofsurfbeatforcinganddissipationStephenMHendersonandAJBowen - PDF document

1 Observationsofsurfbeatforcinganddissipat
ObservationsofsurfbeatforcinganddissipationStephenM.HendersonandA.J.BowenDepartmentofOceanography,DalhousieUniversity,Halifax,NovaScotia,CanadaReceived15June2000;revised14August2001;accepted6March2002;published13November2002.2002.1]Weusedasimpleenergybalanceequation,andestimatesofthecross-shoreenergyfluxcarriedbyprogressivesurfbeat,tocalculatetherateofnetsurfbeatforcing(ordissipation)onabeachnearDuck,NorthCarolina.Farinsidethesurfzone,surfbeatdissipationexceededforcing.Outsidethesurfzone,surfbeatforcingexceededdissipation. TotalenergyofmodeEnergyofmodedissipatedinasinglewaveperiodNotethatisapproximatelythetimescaleforwavedissipationdividedbythewaveperiod,soalowrapiddissipationandahighindicatesslowdissipation.Onlyif BryanandBowen,1996;¨ffer,1993,1994]orassumethatdissipationisasymptoticallyweak[,1971;FodaandMei,1981;Lippmannetal.,1997].Thesehighmodelssuccessfullypredictmanyfeaturesoftheobservedcross-shore[,1976;Hollandetal.,1995],longshoreeHuntleyetal.,1981;Oltman-ShayandGuza,1987;etal.,1998],andfrequencydomain[,1981;andThornton,1985]structureofsurfbeat.ThesuccessofmodelssuggeststhattheofsurfbeatisoftenhighighHolman,1981].However,strongcorrelationsareoftenobservedbetweensurfbeatandlocalnonlinearforcingngMunk,1949;Tucker,1950;HuntleyandKim,1984;etal.,1984;,1986,1992;,1995;1998a],suggestingthatasignificantamountofenergyiscarriedbymodesthatarenotresonantlyforced(‘forcedmodes’).ObservationsofstrongcorrelationsbetweensurfbeatandlocalnonlinearforcingleadHuntleyandKim[1984]toconcludethatthelow-frequency()surfbeattheyobservedwasnotdominatedbyfree(i.e.,highresonantlyforced)waves.waves.7]Longuet-HigginsandStewart[1962]showedthatthestrengthoftheresponseofsurfbeattoforcingincreaseswithdecreasingwaterdepth.Symondsetal.[1982]sug-gestedthatasignificantamountofsurfbeatforcingoccursattheedgeofthesurfzone,wherewavebreakingisintermittent,andnoforcingoccursinsidethesaturatedsurfHuntleyandKimKimList[1986],Masselink[1995],and[1998a,1998b]foundthatcorrelationsbetweensurfbeatandnonlinearforcingincreasewithincreasingwaveheightanddecreasingwaterdepthuntilintensewavebreakingoccurs,atwhichstagecorrelationssuddenlydecline.TheseobservationsareinqualitativeagreementwiththetheoriesofLonguet-HigginsandStew-[1962]andSymondsetal.[1982];however,noquanti-tativerelationshipbetweentheobservedcorrelationstrengthandthestrengthofsurfbeatforcingneartheshorehasbeenderived.Fartheroffshore(8mwaterdepth)onlyasmallproportionoflow-frequencyenergycanbeexplainedbylocalnonlinearforcing,althoughthisproportiondoesincreaseduringstorms[Okihiroetal.,1992;Elgaretal.Herbersetal.,1994,1995b].Munketal.al.foundthatstronglyresonant(high)edgewavesdomi-natedtheinfragravityenergytheyobservedin7mwaterdepthontheCalifornianContinentalShelf.Shelf.8]GuzaandBowenBowenBowen[1977],BowenandandHuntleyetal.[1981],andBryanandBowen[1996]suggestedthatwavebreakingmightleadtopartic-ularlystrongsurfbeatdissipationinsidethesurfzone.Feddersenetal.[1998]showedthatthebottomdragcoefficientforthemeanlongshorecurrentisaboutthreetimeslargerinsidethesurfzonethanoutsidethesurfzone.Observationsofprogressiveinfragravitywaves[Elgaret,1994]andinfragravityenergyfluxes[Herbersetal.1995a]in8mwaterdepthsuggestthatinfragravitywavescanbesignificantlydissipatedonthecontinentalshelfduringstorms.torms.9]Thepurposeofthispaperistodeterminetheapprox-imatestrengthandcross-shorestructureofsurfbeatforcinganddissipation.Insection2wepresentasimpleenergybalanceequationforsurfbeat.Weshowthatthisenergybalanceequationcanbecombinedwithmeasurementsofwaterpressure,velocity,anddepthtoyields

2 patiallyaver-agedratesofnetsurfbeatforci
patiallyaver-agedratesofnetsurfbeatforcing(ordissipation).Thismethodprovidesestimatesofthedifferencebetweensurfbeatforcinganddissipation,butdoesnotallowustoevaluateforcinganddissipationseparately.Wewillusethemethoddevelopedinsection2toanalyzedatacollectedfromabeachnearDuck,NorthCarolina.Wedescribethefieldsiteandinstrumentationinsection3andpresentresultsinsection4.Wefindthatnetsurfbeatforcing(ordis-sipation)wasstrongwhenincidentwaveswerelargeandweakwhenincidentwavesweresmall.Forcingexceededdissipationoutsidethesurfzoneanddissipationexceededforcinginsidethesurfzone.Duringstorms,shorewardsurfbeatpropagationmaintainedsurfbeatenergyinsidethesaturatedsurfzone.Wediscusstheimplicationsofourresultsandpresentourconclusionsinsection5.2.EnergyEquationEquation10]Scha¨ffer[1993]derivedanenergybalanceequationforsurfbeatbyassumingthatbeatfrequenciesaremuchlowerthanincidentwavefrequencies.InthissectionweshowthataslightlymodifiedformofScha¨ffer’senergyequationappliesevenwhenbeatfrequenciesarenotmuchlowerthanincidentwavefrequencies.Thisresultisusefulbecausebeatfrequenciesareoftennotmuchlowerthanincident-wavefrequencies(surfbeatfrequenciesareashighas0.05Hzandthepeakfrequencyofincidentwavesisoftenonly0.1Hz).Wealsoderiveasimplerenergyequationforastatisticallysteadywavefieldonalong,straightbeach.beach.11]Let=time,’thhorizontalcoordinate(=1or2),=horizontalwatervelocity,’thcomponentof=stillwaterdepth,=seasurfaceelevation(abovestillwaterlevel),=gravitationalacceleration.Also,foranyvariable,letbethetime-varyingcomplexamplitudeofafrequencyFouriercomponentof(seeAppendixA1fordefinition).definition).12]InAppendixAwederivetheequationforthetime-varyingenergyoffrequencysurfbeat   


)=anddepth-integratedrateofdissipationatfrequency=realpart, 2HENDERSONANDBOWEN:SURFBEATFORCINGANDDISSIPATION Tjk
hujukjk0 Thesummationconventionhasbeenused,sorepeatedindicesaresummedoverallallowablevalues.values.13]Equation(2)appliestostronglynonlinearwaves.However,inastronglynonlinearwavefield,someenergyissharedbetweenwaveswithdifferentfrequencies,andthewaveenergyatasinglefrequencyisnotwelldefined.Consequently,interpretationofequation(2)isdifficultunlessnonlinearinteractionsareweak.weak.14]u˜isthedepth-integratedmasstransportdividedbythestillwaterdepth.Themomentumfluxtensorcontributionsfromboththemeanflowandthewavefield.Variationsinradiationstress[Longuet-HigginsandStewart1964]areincludedinistheStokesdrift.drift.15]()is,exceptforasmallStokesdriftcontribution,thedepth-integratedenergyofawavewithfrequency)isthethewaveenergyflux,andrepresentsthedepth-integratedrateofworkingbythewaterpressureonthewatermotion.)isthedepth-integratedrateofworkingbythemomentumfluxgradientonthewatermotion.motion.16]NonlinearinteractionsbetweenwavesareassociatedwiththemomentumfluxandStokesdriftofequation(2).Forexample,considerthenonlinearcom-ponentofthewaveenergyflux,,intheshallowwaterlimitwhere.FromageneralizationofParseval’stheorem[e.g.,,1960],formatriadofnonlinearlyinteractingwavesforeveryvalueofof17]Equation(2)wasderivedbyassumingthatbeatfrequencypressurefluctuationsarehydrostatic(equation(A13)),andbyneglectingthedepthdependenceofbeatfrequencyfluctuationsinhorizontalvelocity(equation(A14)).TheseassumptionsarecorrecttoleadingorderforBoussinesqsurfbeatandexactintheshallowwaterlimit.Neglectedeffectsincludereductionsinwaterpres-sureassociatedwiththeverticalfluxofverticalmomentum(which,asweshowinAppendixA,probablyleadstoerrorsoflessthan5%),andnonpotentialflowinthebottomboundarylayer(whichisnegligibleifboundarylayerthicknessiss

3 mallcomparedtothewaterdepth).Equation(2)
mallcomparedtothewaterdepth).Equation(2)isverysimilartotheenergyequationof¨ffer[1993],butdoesnotrelyontheassumptionthatthesurfbeatperiodismuchlongerthantheincidentwavewave18]Letbethecross-shoreandlongshorecoor-dinatesrespectively(positiveonshore).Letbethecomponentsofthevelocity.Applyingtheexpect-ationoperator[.]toequation(2),assumingstationarityrityt=0)andlongshorehomogeneity((y=0),andgatheringnonlineartermstogetherintoasingleterm)gives 
gMxuj 19]Fromequation(12)

isthedensityofthecospectrumbetweenisthefrequencyresolutiondefinedbyequation(A2)).A2)).20]EldeberkyandBattjesBattjesElgaretal.al.ChenandGuza[1997],andHerbersetal.[2000]usedequation(11)tostudywaveshoaling,buttheyassumedshorewardpropagation(althoughHerbersetal.[2000]allowedforsmalldeparturesfromthisassumption),sotoleadingorder
where)=groupvelocity.Theassumptionofshorewardpropagationisprobablyagoodapproximationfortheincidentwavefrequenciestowhichequation(16)hasprimarilybeenapplied,butitsapplicationtosurfbeatisnotjustified.Reflectionofsurfbeatfromtheshore,broaddirectionalspread,andrefractivetrappingofedgewavesensurethatequation(16)doesnotapplytosurfbeat.Incontrast,equation(14)allowsforreflectionanddirectionalspreadandisfreefromtheseproblems.roblems.21]Integratingequation(11)fromapplyingequation(14)gives)isthedissipationperunitfrequency(similarlyforfor22]Givenmeasurementsofwaterpressure,waterveloc-ity,andmeanwaterdepthattwopointsthecross-shore,equation(17)canbesolvedfortheexcessofdissipationoverforcing(negativeifforcingexceedsdissipation)between.Alternatively,wecanchoosetobetheshoreline,acrosswhichthereisnofluxofsurfbeatenergy.ThentheshorewardenergyfluxatHENDERSONANDBOWEN:SURFBEATFORCINGANDDISSIPATION theexcessofdissipationoverforcingonshoreof.Unfortu-nately,itisdifficulttoseparatetheforcinganddissipationtermsofequation(17),butthedifferencebetweenforcinganddissipationstillprovidesusefulinformation.Equation(17)appliesateverysurfbeatfrequency,butforsimplicitywewillintegrateequation(17)overallsurfbeatfrequenciestoobtainatotalsurfbeatenergybalance.3.FieldSiteandInstrumentationInstrumentation23]WewillanalyzedatacollectedonanoceanbeachnearDuck,NorthCarolinabytheDalhousienearshoreresearchgroupduringtheSandyduckbeachexperimentof1997.Figure1showsthecross-shorearrayoffourinstru-mentedframesfromwhichthedatawerecollected,togetherwithmeasuredbeachprofiles.Waterpressuresandveloc-itiesweremeasuredat2Hzateveryframe.DuringtheSandyduckexperiment,theU.S.ArmyCorpsofEngineersregularlymeasuredseabedelevationprofilesusingtheCoastalResearchAmphibiousBuggy[LeeandBirkemeier1993].Seabedelevationsbeneaththeinstrumentedframeswerealsomeasuredcontinuouslyusingsonaraltimeters.4.ResultsResults24]Cross-periodogramsbetweenwaterpressureandvelocitywerecalculatedfromquadraticallydetrendedhalf-hourtimeseries.Spectraofthewaveenergyfluxwereestimatedbyaveragingthesecross-periodogramsandapplyingequations(14)and(A13).Figure2showsaspectrumoftheshorewardenergyfluxmeasuredovertwohoursduringastormonday293.TheshadedregionofFigure2isthesurfbeatband(0.005–0.05Hz).Foreveryhalfhouroftheexperimentweestimatedthetotalsurfbeatenergyfluxbyintegratingrawcross-periodogramsbetweenpressureandvelocityovertheentiresurfbeatbandandmultiplyingbythewaterdepth.Otherstatistics,suchassignificantwaveheightandthesurfbeatseasurfaceelevationvariance,werealsocalculatedeveryhalfhour.Weestimatedseasurfaceelevationsfromwaterpressuremeas-urementsusinglinearwavetheory.Thesignificantwaveheightwascalculatedas4 ,where istheseasurfaceelevationvarianceduetowaveswithf

4 requenciesgreaterthan0.05Hz.Hz.25]Equati
requenciesgreaterthan0.05Hz.Hz.25]Equation(11)neglectsthelongshoregradientofthelongshoreenergyflux.Theratioofthelongshorefluxgradienttothecross-shorefluxgradientis qyqx  qyqx =longshorecomponentofsurfbeatenergyflux,=cross-shorelengthscaleoverwhich=longshorelengthscaleoverwhichSincethebeachatDuckislongandstraight,weassume1.Whenincidentwaveswerelarge,theobservedmeansquarelongshoresurfbeatenergyfluxwasusuallyanorderofmagnitudesmallerthanthemeansquarecross-shoresurfbeatenergyflux.Whenincidentwavesweresmall,thelongshoreandcross-shoreenergyfluxeswereofthesamemagnitude.Therefore,whenincidentwaveswerelarge1,sothelongshoreuniformapproximationmadeinthederivationofequation(11)wasreasonable.Wecannotbesurehowaccuratethelongshoreuniformapproximationwaswhenincidentwavesweresmall.small.26]Figure3showsthetimeseriesofsignificantwaveheight,surfbeatseasurfaceelevationvariance,andshore-wardsurfbeatenergyflux,measuredatframe4.Similarresultswereobtainedfromtheotherthreeframes.Surfbeatenergyincreasedwithincreasingsignificantwaveheight,consistentwiththefindingsofTucker[1950],HolmanlmanGuzaandThornton[1985],andothers.Thesurfbeatenergyfluxwasdirectedonshorein86%ofcases,suggestingthatthenearshorezone(mdepth)wasusuallyaregionofnetsurfbeatdissipationduringtheSandyduckexperiment(equation(17)).Shorewardenergyfluxesweremostpronouncedwhenincidentwaveswerelarge:whenthesignificantwaveheightwasgreaterthan1mthesurfbeatenergyfluxwasalwaysdirectedonshore.Theobservedshorewardenergyfluxdoesnotimplythatsurfbeatforcingwasweakneartheshore,butitdoesimplythatdissipationwasusuallystrongerthanforcing.DuringtheSandyduckexperiment,incidentwave(0.05–0.33Hz)energyfluxesrangedfrom0.08mto12mwerealwaysdirectedonshore. Figure1.Measuredbeachprofilesandlocationofinstrumentedframes1–4. Figure2.Spectrumoftheenergyfluxdensityatframe1foratwohourperiodfollowing10:30pmonday293,estimatedusingequation(14).Theshadedregionisthesurfbeatband.Thecospectralestimateshave10dof.4HENDERSONANDBOWEN:SURFBEATFORCINGANDDISSIPATION 27]Figure4showsthesignificantwaveheightasafunctionofwaterdepthatframes1and4.Waveheightswerelimitedbybreaking,asdescribedbytherelationwherethebreakerratioequals0.66atframe1and0.43atframe4.SallengerandHolman[1985]andRaubenheimeretal.[1996]discusstheapplicationofequation(19)tothesurfzoneatDuck.Inordertodetermineapproximatelywhetherframe1wasinsideoroutsidethesaturatedsurfzone,wedefinetheshoaledsignificantwaveheight(4)and(4)arethesignificantwaveheightandwaterdepthatframe4,andisthelocalwaterdepth.thewaveheightthatwouldbeobservedgivenlinear,nondissipativeshoalingofshorenormalshallowwaterwaves.Roughly,frame1isinside(outside)thesaturatedsurfzonewhenisgreaterthan(lessthan)atframe1.Atframe4,4,28]TheshorewardsurfbeatenergyfluxshowninFigure3impliessomeshorewardpropagationofsurfbeat.Wedefinetheonshoreprogressivenessastheratiobetweentheactualsurfbeatenergyflux,calculatedfromequation(14),andthesurfbeatenergyfluxexpectedforapurelyshorenormal,shorewardpropagatingwave,calculatedfromequation(16).Ifallsurfbeatpropagatesdirectlyshoreward=1,ifallsurfbeatpropagatesdirectlyseawardandifallsurfbeatisstandinginthecross-shore=0.Directionalspreadingofwaves(awayfromshorenormal)reducesthemagnitudeof.Weestimatedthetotalsurfbeatenergydensityastwicethepotentialenergydensity,or 2sb istheseasurfaceelevationvarianceduetosurfbeat.Theuseofequation(20)introducedsmallerrorsintoestimates,butallowedustoseparatesurfbeat(gravitywave)energyfromtheshearwaveenergythatoftencontributesalargeproportionofthetotallow-frequencykineticenergy[,1999].Figure5show

5 sthe Figure4.Significantwaveheight,,vers
sthe Figure4.Significantwaveheight,,versuswaterdepth,:(a)frame1and(b)frame4.Dashedlineisis0.66atframe1and0.43atframe4.Eachdatapointisestimatedfromahalf-hourtimeseriessegment. Figure5.Onshoreprogressivenessofsurfbeat,),versusshoaledsignificantwaveheight(equation(20))dividedbywaterdepth,:(a)frame1and(b)frame4.TheverticaldashedlinesindicateEachdatapointisestimatedfromahalf-hourtimeseries Figure3.Timeseriesofhalf-hourlysignificantwave,surfbeatseasurfaceelevationvariance, ,andshorewardsurfbeatenergyflux,,duringSandyduckatframe43.5mdepth).HENDERSONANDBOWEN:SURFBEATFORCINGANDDISSIPATION observeddependenceofonthenondimensionalshoaledwaveheight.Whenthenondimensionalwaveheightwassmall,0andsurfbeatwasapproximatelystandinginthecrossshore.Whenthenondimensionalwaveheightwaslarge,�0andtherewasasignificantshorewardpropagatingcomponentofsurfbeat.beat.29]Wewillnowshowthattheobservednetsurfbeatdissipationcanbepredictedusingastandardbottomstressparameterization.Thedissipationoftheenergyofafre-wavebybottomfrictionis
isadimensionlessenergydissipationfactor.Valuesobservedinthefieldforincidentfrequencywavesareusuallyintherange0.01–1[,1984].1984].30]Feddersenetal.[1998]showedthatthedragforce(dividedbythewaterdensity),,retardingthemeanlong-shorecurrent,,atDuckiswherethebottomdragcoefficient,,isroughly0.001outsidethesurfzoneand0.003insidethesurfzone.Applyingthesameparameterizationtosurfbeat

Equation(22)differsfromequation(21)onlyinthemagnitudeofthenondimensionalcoefficient:dissipationfactorsforwavesareoneortwoordersofmagnitudelargerthandragcoefficientsforthemeancurrent[,1992].].31]Fromequations(21)and(22)thetotaldissipationofsurfbeatenergybybottomdragscaleswith u212 u2sb isthecontributiontovelocityvariancefromsurfbeatfrequencies.Toseparategravitywavedissipationfromshearwavedissipationwerewriteequation(23)intermsoftheseasurfaceelevationvariance.Forgravitywavesinshallowwater[Lippmannetal.,1999] u2g 2h  212 Fromequations(17)and(24) 212 32]Figure6showsthattheshorewardsurfbeatenergyfluxatframe1scaleswith 212 =0.92),aspredictedbyequation(25)ifthenonlinearenergyexchangetothesurfbeatband,,isneglected.Anorderofmagnitudeestimate isthedistancefromframe1totheshore,istheslopeofthedashedlineinFigure6,andisatypicalwaterdepthshorewardofframe1(takentobehalfthedepthatframe1).Applyingequation(26)gives0.08,whichisanormalvalueforawavedissipationfactorandis27–80timeslargerthanthedragcoefficientsFeddersenetal.[1998]foundwereappropriateforthemeanlongshorecurrent.current.33]Weneglectedinthederivationofequation(26).�0(i.e.,ifnonlinearinteractionsforce,ratherthandissipate,surfbeat)thenactualratesofdissipationandtruevalueswerehigherthanweestimated.Alternatively,nonlineardamping(,[e.g.,¨ffer,1993;VanDongerenetal.,1996])couldaccountforsomeoftheobservedenergyloss,inwhichcasetruevalueswouldbelowerthanestimated.However,theslopewaslargelydeterminedbythestrongenergyfluxesthatweremeasuredduringstorms,whenframe1waswellinsidethesaturatedsurfzone(Figure4)andnonlinearforcingwasprobablyweak.Wealsoassumedthatwavevelocityvarianceswereconstantonshoreofframe1.Thisassumptionmightintro-duceasignificanterrorintoourestimate.Becauseofthesecrudeassumptions,ourestimatecanonlyberegardedasanorderofmagnitudeapproximation.Furthermore,wecannotbesurethatbottomstresswasthemechanismresponsiblefordissipatingsurfbeat,inspiteofthegoodagreementbetweenobserveddissipationandthestandardbottomstressparameterization.Nevertheless,itisusefultonotethatnetsurfbeatdissipationinthesurfzonecanbeparameterizedinasimplemanner..3

6 4]Insection1wediscussedasameasureofthest
4]Insection1wediscussedasameasureofthestrengthofsurfbeatdissipation.Sincewecanmeasureonlyspatiallyaveragednetforcingordissipation,wecannotvaluesforindividualsurfbeatmodes.Instead,wedefinethenetforcingstrength NetsurfbeatenergygeneratedinonebeatperiodTotalsurfbeatenergy  Figure6.Onshoresurfbeatenergyflux,,versus 212 atframe1.Dashedlineis 212 ,withchosentogiveleastsquaresfit.Eachdatapointisestimatedfromahalf-hourtimeseriessegment.6HENDERSONANDBOWEN:SURFBEATFORCINGANDDISSIPATION ,andareintegratedoverthesurfbeatband.Fromequation(17) where0.025hasbeenchosenasatypicalsurfbeatfrequency.Sincethis‘‘typical’’beatfrequencywaschosensomewhatarbitrarily,onlythesignandorderofmagnitudearesignificant.Wherepossibleweestimatedtheintegralinequation(28)usingthetrapezoidalrule.Toestimatetheintegralbetweenbetweenframe1andtheshore,wemultipliedtheenergydensityatframe1bythedistancetotheshore.Theenergydensitywasestimatedusingequation(20).20).35]Sispositive(negative)iftotalforcingexceedstotaldissipation(dissipationexceedsforcing)between.Ifisorderone,thenthesurfbeatenergyforced(ordissipated)betweeninasinglebeatperiodisofthesameorderasthetotalamountofsurfbeatenergystoredbetween.If1,thennetforcing(ordissipation)isweak,butthisdoesnotimplythatactualforcinganddissipationareweak.Ifstrongforcingandstrongdissipationhappentocancel,then1.Therefore,largevaluesindicatestrongforcingordissipation,whereassmallvaluesareambiguous.ambiguous.36]Figure7ashowstheestimatedstrengthofnetsurfbeatforcing,,fortheregionbetweenframes1and4.Figure7bshowsfortheregionbetweenframe1andtheshore.Theverticaldashedlinesindicatethenondimensionalwaveheightatwhichframe1isapproximatelyontheedgeofthesaturatedsurfzone.Whenincidentwavesweresmall,valueswerescatteredaroundzero.Whenincidentwaveswerelarge,wasorderone,soforcinganddissipationwerestrong.Betweenframes1and4(betweenabout2and3.5mdepth),netforcingincreasedwithincreasingwaveheightuntilitreachedamaximumvaluewhenframe1wasslightlyoutsidethesaturatedsurfzone.Whenframe1wasinsidethesaturatedsurfzone,dissipationsometimesexceededforcingbetweenframes1and4.Onshoreofframe1,dissipationusuallyexceededforcingandnetdissipationgrewstrongerasincidentthewaveheightincreased.Wesuggestthat,whenincidentwaveswerelarge,dissipationwasstronginsidethesurfzoneandforcingwasstrongjustoutsidethesurfzone.zone.37]Fromequations(1)and(28)itisclearthatrelatedto2,butthereareimportantdifferencesbetweenmeasuresthedissipationofasinglewave(modeofmotion),whereasmeasuresthenetforcingordissipa-tionofallmodesinsomelimitedregion.Consequently,wecannotusethemeasuredvaluestoestimatevaluesforindividualsurfbeatmodes.Nevertheless,thenegative,orderonevaluesobservedduringstormsdoindicatethatthesurfbeatenergylostwithinthesurfzoneinasinglebeatperiodwasofthesameorderasthetotalsurfbeatenergystoredwithinthesurfzone.5.DiscussionandConclusionsConclusions38]Asimpleenergybalanceequationrelatestheenergyfluxcarriedbyprogressivesurfbeattonetsurfbeatforcingordissipation.Weusedwaterpressures,velocities,anddepthsmeasuredonabeachnearDuck,NorthCarolina,toestimatecross-shoresurfbeatenergyfluxes,andappliedtheenergybalanceequationtoestimatenetsurfbeatforcingordissipation.DuringtheSandyduckexperiment,thenear-shorezone(mdepth)wasusuallyaregionofnetsurfbeatdissipation.Shorewardpropagatingsurfbeatcarriedenergyintothenearshorezonetobalancethisnetdissipa-tion.Thisshorewardpropagatingcomponentofsurfbeatwaslargeduringstorms,whenthenetshorewardenergyfluxwasabouthalfaslargeastheenergyfluxthatcouldbecarriedifallsurfbeatpropagateddirectlyonshore.onshore.39]Thestrongs

7 horewardenergyfluxesthatweobservedarecon
horewardenergyfluxesthatweobservedareconsistentwiththeincompletereflectionofsurfbeatobservedinthethesurfzonebyNelsonandGonsalvesGonsalvesRaubenheimeretal.al.Saulteretal.l.Hendersonetal.[2001],andSheremetetal.al.[40]Mostexistingsurfbeatmodelsdonotpredictthestrongshorewardpropagationthatweobservedbecausetheydonotsimulatestrongsurfzonedissipation.Unforced,undampedsurfbeatmodels[,1951;,1952;,1970;HolmanandBowen,1979,1982;Howdet,1992;BryanandBowen,1996]predictthatsurfbeathasacross-shorestandingstructure.Nondissipativeandweaklydissipativemodelsthatallowforthepossibilityofedgewaveresonance[Gallagher,1971;BowenandGuza,1978;¨ffer,1994;Lippmannetal.,1997]alsopredictthatsurfbeatiscross-shorestanding.Thebreakpoint-forcingmodel Figure7.Estimatednetforcingstrength,,definedbyequation(28)versusshoaledsignificantwaveheight(equation(20))dividedbywaterdepth,,atframe1:(a)netforcingstrengthbetweenframes1and4and(b)netforcingstrengthonshoreofframe1.Theverticaldashedlinesindicate(1)=.Eachdatapointisestimatedfromahalf-hourtimeseriessegment.HENDERSONANDBOWEN:SURFBEATFORCINGANDDISSIPATION Symondsetal.[1982]predictscross-shorestandingwavesonshoreofthebreakpointandseawardpropagatingwavesoffshoreofthebreakpoint.Themodelsof¨ffer[1993]andVanDongerenetal.[1996]predictthatnon-linearforcingduetointermittentwavebreakingopposesincidentboundwavemotions,leadingtonetnonlineardampingofsurfbeatatthebreakpointandshorewardpropagationjustoutsidethebreakpoint.However,theselastthreemodelsexcludethepossibilityofedgewaveresonancethroughthearbitraryassumptionthatforcingisentirelyshorenormal.normal.41]DuringtheSandyduckexperiment,dissipationwasstrongestwellinsidethesaturatedsurfzone,exactlywhereincidentwaveswerelimitedbybreakingandsurfbeatmadeanimportantcontributiontothetotalflowfield.Wesuggestthatmodelsofsurfbeatdynamicsshouldincorporaterapidsurfzonedissipation.ion.42]Astandardbottomdissipationparameterizationpre-dictedtheobservednetsurfbeatdissipationwell.Thewavedissipationfactorforsurfbeatwas),withintherangeofdissipationfactorsusuallyobservedforhigher-frequencyincidentwaves.waves.43]Theregionbetween2mand3.5mdepthwasaregionofnetsurfbeatforcing,exceptwhenincidentwaveswereverylargeandthesaturatedsurfzoneextendedbeyond2mdepth.Wesuggestthatsurfbeatforcingusuallyexceededdissipationoutsidethesurfzone,whereasdissi-pationexceededforcinginsidethesurfzone.Thisisconsistentwiththecross-shorestructureofsurfbeatforcingpredictedbyLonguet-HigginsandStewart[1962]andSymondsetal.[1982]andwiththesuggestionofandBowen[1976a]andothersthatsurfbeatdissipationmightbemostrapidinsidethesurfzone.zone.44]Surfbeatforcinganddissipationwereverystrongduringstorms.Whenincidentwaveswerelarge,thesurfbeatenergydissipatedwithinthesurfzoneduringasinglebeatperiodwasofthesameorderasthetotalsurfbeatenergystoredwithinthesurfzone.Thesurfbeatenergyforcedinasinglebeatperiodneartheedgeofthesurfzonewasalsoofthesameorderasthetotalsurfbeatenergystoredneartheedgeofthesurfzone.AppendixA:DerivationofNonlinearFrequencyDomainEnergyEquationA1.FourierRepresentationofaTime-VaryingWaveave45]Atimeseries)canberepresentedbythepro-gressiveFourierseries
isthefrequencyresolution,and isthecomplexamplitudeofafrequency-sinusoidfittedtoalengthsegmentofcenteredontime.Notethatisafunctionoftimetime46]WenowderiveanidentityforuseinsectionA3.Fromequation(A1), Xt
tjXj  But,bydefinition, Xt
j so,fromequations(A5)and(A6) Xt
iX Xt
 XXt Xt Combiningequations(A7)and(A8),andnotingthatisthecomplexconjugateofforanyreal,gives X2t
2 A2.Massan

8 dMomentumConservationConservation47]Anex
dMomentumConservationConservation47]Anexact,depth-integratedmomentumequationforwavesinwaterofconstantdensityandarbitrarydepthissPhillips,1977,equation(3.6.7)] thuj xkhujuk xjh p p| hxj
00h ujt0h p|xj Tjkxk aredefinedbyequations(7)and(8),waterpressure,and=waterdensity..48]Equation(A11)canberewrittenusingtheFouriercomponentsdefinedbyequations(7)and(8): ujt 1 pxj Tjkxk 8HENDERSONANDBOWEN:SURFBEATFORCINGANDDISSIPATION WenowapproximatethemotionatfrequencyisindependentofEquations(A13)and(A14)arecorrecttoleadingorderiffrequencywavessatisfytheBoussinesqassumptions,andareexactintheshallowwaterlimit.limit.49]Inreality,dependsinpartonfluxesofverticalmomentum.Theeffectsofthesemomentumfluxesareareusuallyincorporatedintotheradiationstress,buthavebeenneglectedinequation(A13).Longuet-HigginsandStewart[1964]dividedthenormalradiationstressintothree,anddLonguet-Higginsand,1964,equation(9)].Wehaveincludedbut,throughequation(A13),haveneglectedthepressuredeficitterm(2).Fromlineartheory,theratiobetweentheneglectedandretainedradiationstresstermsforlongwavegroupsis S2xS1x S3x
sinh2sinh2isthewavenumber.FigureA1showshowwithwaveperiodandwaterdepth.DuringSandyduck,peakperiodswereusuallyabout7–13s,anddepthsattheinstrumentedframesrangedfrom1mto5m.FromFigureA1,assumption(A13)leadstoerrorsofabout2–5%inradiationstressestimates.Usually,sotherelativeerrorinestimatesduetotheneglectedpressuredeficittermwasprobablylessthan5%.5%.50]Equation(A14)neglectsvariationsofinthesurfaceandbottomboundarylayers.Weassumethatthethicknessoftheseboundarylayersismuchlessthanthewaterdepth,sothattheproportionofthedepth-integratedmomen-tumstoredintheboundarylayersisnegligible.negligible.51]Fromequations(6),(A12),(A13),and(A14), 
ujt xj Similarly,theexactdepth-integratedmassconservation t togetherwithequation(A14),leadsto t A3.EnergyEquationEquation52]Addingequation(A16)tothecomplexconjugateofequation(A16)andapplyingequation(A9)gives hu2t 2
uj xjuj Tjkxk0
gh
uj xj

ujxjg sofromequations(A18),(A20),and(A9), xj2 
ujxj Substitutingequation(A21)intoequation(A19)gives t 
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