JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 92, NO. C9, PAGES 9487-9493, AUGUST 15, 1987 EdgeWaves on the SydneyCoast JASONH. •[IDDI.ETON AND MADELEINE L. CAmLL Schoolof Mathematics,UniversityoJ'NewSouthWales,Kensington,Australia WILLIAM W. HSIEH Departmentof Oceanography,Universityof BritishColumbia,Vancouver,Canada Pressureand currentoscillationsat periodsof 40 s to 17 min observedduring stormconditionsat two locationsseparated by 560 m in thealongshore directionin thecoastaloceannearSydney,Australia,indicatethe existence of infragravity waves having amplitudes of -20 cmandvelocities of -10 cms-1. Theobserved infragravitywavesappearto be locally forcedby the wind wave envelopethroughradiationstress,yet the observedalongshore phasedifferences of the infragravitywavesare consistent with thosepredictedfrom free edgewave theoryfor low-modeedgewavestravellingnorthwardandthe relationshipof pressureto velocityat eachlocationis alsoconsistent with freeedgewavetheory. As a functionof time,the infragravitywave spectral energygrowsanddecaysin stepwith thelonger-period windwaves,suggesting a continuous transferof energy. The infragravitywavesappearto containenergyin both directlyforcedand freely propagating(edgewave) oscillations.The edgewavesmay be generatedeitherby radiationstressas outlinedabove,by a resonanttriad mechanism,or by a combinationof the two. 1. INTRODUCTION having periodsperhapsa few minutesor less. For somewhat Edge waves aresurface gravity waves which aretrapped alonglonger period waves, Greenspan [1956], Buchwald anddeSzoeke shore linesby theshoaling topography. Theygenerally have[1973], VieraandBuchwald [1982] andWorthy [1984] found periods somewhat longer thanthose of windwaves andare theoretical solutions for edgewavesdrivenby travelling believed toplayanimportant roleincoastal sedimentation and atmospheric events. surfbeat[Bowen andInman, 1971; Huntley andBowen, 1975; Theworkreported herewasprompted by suggestions by Guza andThornton, 1985; Holman andSallenger, 1986].The Buchwald and de Szoeke[1973]thatthe 3-min-period earliest theory ofedge waves appears tobedue toStokes [1846],oscillations ofsea level, observed inPort Kembla harbor (70km whofound wavelike solutions to thebarotropic equations of south of Sydney) during storms, wereduetostorm-generated motion appropriate forshallow water ona beach ofconstant edge waves travelling northward along thecoast. Ourexperiment slope. Subsequently, these theories were extended byEckartwasdesigned asa pilotstudy toobserve themagnitude of [1951], Ursell [1952], andReid[1958], whoidentified the oscillations with periods of0.5-17mininthecoastal ocean south discrete modal structure. Thedynamics ofedge wave propagation ofSydney and todetermine if theobserved propagation rates are arenow wellunderstood inprinciple [LeBlond and Mysak, 1978],consistent withedge wave theory. Since edge waves propagate asisthegeneral relationship ofedge waves tothewider class of because of refraction byshoaling topography, a section of coastal trapped waves. relatively straight coastline withsteep cliffsandfewenergy Observations of2-to30-min-period oscillations insea level off absorbing inlets was chosen, this occurring offtheRoyal National theCalifornia coast ledMunk etal. [1956] topropose thata Park-25kmsouth ofSydney and-45 kmnorth ofPort Kembla "wake" ofedge waves of10-to30-min period was produced by (Figure 1). theimpulsive forcing ofamoving atmospheric front, while higher Thispaper isorganized asfollows. Section 2 outlines edge frequency waves of2-to5-min period were due tosurf beat. An wave theory andexperiment design while section 3 describes the extensive alongshore array ofpressure gauges was later used on experimental method andtheobservations. A comparison of thesame beach byMunk etal.[1964] toobtain wave numberobservation withtheory isundertaken insection 4,while section 5 frequency relations ofoscillations withperiods of1 minto10 concludes with adiscussion. hours, andwavelengthof 1.5 to 30 km. They concluded thatthe 2. EDGE WAVE TttEORY AND DESIGN OF EXPERIMENT observedoscillations werepredominantly causedby discreteedge We choose a Cartesian coordinatesystem with the x axis wave modes,and that little energywas associatedwith leaky pointing offshore and the y axis alongshore.The sea level waves.More recently,Huntleyet al. [1981]usedalongshore and displacement rl is assumedto havea travellingwaveform, with offshorearraysof currentmetersto showthat 1- to 4-min-period oscillationsoff Scrippsbeachin Californiahave both structures rI = F (x)ei(•y-•) (1) anddispersion curvesconsistent with thosepredictedfromedge From the unforcedshallow water equations,the coastaltrapped wave theory. While thepropagation ofedge waves iswell understood, theirwave equation forlinear waves inabarotropic ocean isderived by generation isnot.Gallagher [1971], Guza andDavis [1974], LeBlond andMysak [1978, equation (25.7)]. Neglecting the Bowen andGuza [1978] and Symonds etal.[1982] argue thatEarth's rotation, wecan rewrite their equation as nonlinear resonant interactions between incoming surface gravity waves are the main mechanismsfor the generationof waves Copyright 1987 bytheAmerican Geophysical Union. F"-k2F+H-I(H'F' + to2g-iF) =0 (2) where prime denotesdifferentiationwith respectto x, H = H (x) denotesthe depth,and g is the gravitation constant.The Paper number 7C0393. eigenvalues of (2), whichdetermine the dispersion relation 0148-0227/87/007C-0393505.00 to= to(k), and the eigenfunctionsF (x) were found numerically 9487 9488 MIDDLETON ET AL.: EDGE WAVES IN 0'15T--mode I) iii/ ROYAL if' / NATIONAL/•/'. // w (s-•) 34ø06'S o1• J•.•'• /// PACIFIC OCEAN • / / •-• _•o• / 0 0.010 k (m -'• ) // Fig. 2b. The dispersion curvesfor the first threeedgewavemodes calculated fromthetheoryusingtheactualdepthprofile. J•Oø/?'i lkmI •/ / I 0.00,5 / / I / phase speed c is confined to Cm•< c <Cm,•. 151 ø10'E where on the Cm•, = [gH(0)]v' and Cm•= [gH(L)]V'. Depending Fig.1. Thecoastal region off theRoyalNational Park,Sydney,alongshore propagation speed C of theatmospheric front,three Australia, including the locations oftheSeadata 635-12S pressure-current situations mayoccur: (1)C >Cmax, (2)Cmin <C < cnm, and (3) recorders used inthis study. C<cx•.Incase 1the waves generated are not coastally trapped, [Holman andBowen, 1979] using F'/F=-{k2-co2/[gH (L)]]• while incase 3,except forsome transients, nowavelike motions where H'(L)=0 inthedeep ocean x =L. Since thecoastline at aregenerated. Onlyin case 2 areedgewaves withc = C x =0 is composed of cliffs,wealsoused H(0)=10m andresonantly excited. Fordepth profiles withH = 0 atthecoast, F'(0)= 0. Cmi• = 0, andcase 3 does notarise.WithH(0)= 10mandH (L) Thedepth profile H(x)inourexperimental area together with=5000 m,wehave Cmi• = 10ms-•and Cm• =220ms-x. thepredicted offshore dependence ofthemodal structure for1.3- From ourdispersion diagram (Figure 2b),thezeroth mode edge and3.3-minwavesare shownin Figure2a while themode0, 1, wave with c matchingthe fastestfrontalspeedC of about23 m and 2edge wave dispersion curves are shown inFigure 2b. s-•have aperiod ofaround 8min and awavelength of12km.On From LeBlond andMysak [1978, equation (25.3) with f = 0], theother hand, thezeroth mode wave witha 1-min period hasa one can easily show that theedge wave velocity components obeywavelength of1kmand aphase speed ofc = 17ms-•.These spatial and time scales basically determinedthe instrument u=gCO-•F' exp [i(ky - cot - •)] v = gkco-lFexp[i (ky- cot )] (3) locations andthesampling rates. (4) 3. Tim EXPERIMENT AND Tim OBSERVATIONS For this experimentwe were able to obtain two Seadata635- andthepressure perturbation p obeys 12S instruments capableof measuring bothwavepressures and currents at wind wave periods and longer. These were deployed p = pgF exp [i (ky - tot)] (5) on bottom-mounted tripodsso that the sensorswere 1.5 m off the wherep is the waterdensityand (5) followsfromp = pgTI. bottom. To resolve wavelengthsof-1 km or longer the Hence v andp arein phase,eachleadingu by 90ø. instrumentswere deployed560 m apart and 100 m from the Theatmospherically forced problem wasstudied byBuchwaldcoastal cliffsin 21 m of water.Thesampling strategy wasto anddeSzoeke [1973]andVieraandBuchwald [1982].Astheir simultaneously burstsample 512samples at 2-sintervals (fora depthprofiles havea vertical wallat thecoast, theedgewave duration of 17minand4 s)withbursts occurring each30min. 0 50 •_.• • 2 x(km) 4 6 I I I I I•- H(m)J 1 Wave periods(<4 s) unresolvedby the measurements were not expectedto seriouslyalias the data as a resultof the depth attenuationof shorter-period waves. During the experimental observation period,only one significantstormoccurred,andthis storm developedgraduallyrather than arriving as an easily identifiable A typicalburstof stormdatacomprisingpressure p, offshore velocity u , and alongshore velocityv components is shownin Figure3. Superimposed on the raw time seriesare low-passfilteredtime seriescalculatedusinga discreteFouriertransform filter with frequency domaintaperinganda half-powerperiodof 40 s. Wind waveamplitudes areclearlyconsiderable (-1 m) with significant orbitalcurrents for waveswith periodsof-8-20 s. The low-passeddata show the existenceof persistentlonger-period (-2-20 min) oscillations of muchsmalleramplitude(-20 cm) and T = 1 28 m•n P 4 front. 6 current(--10cms-x). Suchlow-frequency oscillations areoften referredto asinfragravity waves.Similarplotswereobtained for many otherburstsduringthe storm,and in all casestheselower- frequency oscillations existed.Beforethestorm,amplitudes were Fig.2a. The offshore depth profile together with theoffshore structure of negligible, andasthestorm gradually developed, theamplitudes thepressure (sealevel)fieldforthezeroth andfirstmode edge waves of of thelower-frequency oscillations increased at aboutthesame period1.28min and3.33 minfoundfromthetheory. rate as did the wind waves. MIDDLETON ET AL.: EDGE WAVES 9489 - f: 0 073 s -1 _ '1'1 / _ - .5- /1'x'J// ............. ...... .' •0.0 -.--T/X•/'• '',___- ......... 0'02 .... _-- _/ .... - (] '__ 12 24 12 24 /• 12 - 5- - 00 24 12 /15x•x 10 uJ Z 12 DAY 2 24 12 DAY 3 24 12 DAY 4 24 12 24 DAY 5 Fig.5. Spectral estimates (b(/') of windwave,edgewaveandenvelope series plottedin theformf •(f ) asa function of timefor selected bands centeredaroundthe indicatedcyclicfrequencies f Wind wavespectraare plotted on the upper axes, while infragravity (solid line) and envelope TIME (MINS) (dashedline) spectraare shownon the lower axes. Eachestimatehas 120 Fig. 3. Time seriesof pressure p, offshorevelocityu andalongshoredegrees of freedomandisplottedusingsixconsecutive bursts of data. velocityv for a singleburstof 512 samples at 2 secondintervals.Overlaid on therawdata arelow-passed time series obtained from therawdatainfragravity current andpressure fluctuations in thetimeseries, usinga frequencydomainfilter with a cosineshapedtapercenteredat a frequency of0.025 Hz. thespectral estimates riseoncemore.Although thetimeseries used were not strictly stationary,the spectraldistributionsof To examinetheenergycontributions in thefrequency domain, Figure4 aretypicalof thosefoundfor individual bursts, although spectral estimates forp, u andv werecalculated for eachburst.To theburstspectra wereof coursesubstantially noisier. maximise resolution, whilemaintaining goodstatistical reliability, To investigate thegrowthof thespectral energies, calculations thespectral estimates at eachfrequency wereensemble averagedof spectral estimates asa function of timeweremadeforselected with thosefrom all otherburstson the sameday, giving96 frequency bands.In eachcase,spectral estimates wereobtained degreesof freedom.Figure4 showsspectralestimates from the by ensemble averaging over6 bursts(covering3 hours),andband southern locationfor thedayof maximumwindwaveenergy.It averaging over 10 adjacentestimates in the frequency domain is clearthatin eachcase,thereissignificant energy atperiods of givinganeffective cyclicbandwidth of -0.01s-a. Theestimates 5-15 s (windwaves)andthata spectral gapexistsat periods of foreachconsecutive 3-hourperiodareplottedfortheentirestorm 2040 s. At periodslongerthan-40 s wherewe observethe for the northernSeadatain Figure5. For the wind wavebands centered on cyclicfrequencies of 0.073s-a, 0.083s-• and0.102 s-a,thespectrum for thehighest frequency risesfirst,followed by t000 99 % p :too, the spectrumfor the next highestfrequency.The spectrafor the longer-period wind wavesrise later,but to largervalues. Energy is thus fed to longer and longer period waves as the storm develops, withthehighfrequency waves(0.102s-a) approaching 1oi some "steadystate" value quite early in day 2. The spectral estimates for the infragravity wavebandcentredat 0.005 s-a include contributionsfrom -0.001 s-a to 0.010 s-a and thus are t u .01 0.0 .05 .t0 .t5 .20 essentiallyestimatesof the entire observedinfragravity wave spectral energy. Of interest here is the similar shape of the lower-frequency wind wavebandand the infragravitywaveband; both bandsrise and fall in proportion.It thereforeseemslikely that it is the wind wavesof longerperiodthat are feedingenergy into the infragravitywaves. This linear dependence betweenthe incident longer-period wind wave energy level and the infragravity energy level has been observedby others [e.g., Tucker, 1950; Guza et al., 1984], althoughthe data are usually presentedby meansof significantwave height estimateswhich mask any frequency dependence. The present spectral calculationsshow clearly that the infragravity(---200-speriod) waves grow linearly only with the longer-period(11-14 s) wind waves,but not with the shorterperiod(---10s) waves. FRF. OUENC¾ (CPS) Tofurther investigate therelationship between wind waves and Fig. 4. Spectral estimates ofpressure p0cP_a2/.cps), Offshore velocity u infragravity waves, thepressure time series were band passed to (m 2s-2/cps) and alongshore velocity v(m 2s-2/cps) from the southern retain cyclic frequencies between 0.065 and 0.095 s-z' Envelope Seadata.Eachestimate wascalculated by ensemble averaging overthe spectral andcross-spectral estimates for48consecutive bursts, giving 96 timeseries fortheband-passed datawerethenfound using the degrees offreedom. methodoutlinedby Bracewell[1978] andMelville [1983]. Figure 9490 Mn)DLETON ET ^L.: EDOE WAV• ./envelope N 0.50 0 25 (M) -1 10' • bond-pos$ filtered IV V v- • -•J 1800 •ow-pass filtered vV .... .... .... ' 2000 2200 19 JUN 2400 1984 Fig. 8. Time seriesof sealevelfoundfrom thepressures recorded by two Aanderaapressure gaugesseparated by 10 km in the alongshore direction. The characteristic oscillationsbeginningat -1900 LT in the southem(S) , TINE (NINS) data and -1920 LT in the northem (N) data are associatedwith the H•. 6. • •ad-p•s•ad (0.•5 <f < 0.•5 s-;) (• safia• of•i•um• passageof a strongfrontonJune19. as a •ar f•uency of f = 0.08 s-; m•uhtad wi• • p•am• en•al•a f•ctioa • • •a•es •bow• u•r axes,with • low-passed i•ragra•ky t•a below. range, while there appearsto be a phase lag of--re. LonguetHiggins and Stewart [1962, 1964] coined the term "radiation 6 shows theband-passed timeseries fromFigure 3 together with stress" for the nonisotropic flow of momentum dueto the theenvelope function plotted ontheupper setof axes.Forvisual presence ,ofwaterwaves.Theyshowed theoretically thatgroups comparison thelow-passed infragravity wavepressure timeseriesof waveshavinglargeamplitudes havea highradiation stress for the sameburstis shownon the lower set of axes. The wind which causesan expulsionof fluid below, thus leading to a waveenvelope hassignificant low-frequency energy, andthereis reduced meansealevel.Highwaveenvelopes aretherefore rcout some visualcorrelation between theenvelope andtheinfragravity of phase withvariations in themeansealevel.Thisfeature is wavetimeseries.Similarcalculations of thespectral energy of clearlyevidentin thepresent data,indicating thata substantial theenvelope timeseries weremadeasa function of timefor a amount of energyin theobserved low-frequency oscillations is frequency bandcentered onf = 0.005s-] andthesearepresentedlocallyforced bythewindwaveenvelope. in thelowersection of Figure5 for directcomparison withthe Asexplained earlier,fortheparticular stormobserved withthe infragravity waveresults.The envelope spectra(dashed line) Seadams, the wind stressgrewgradually, thuspreventing any growsimultaneously with boththe longer-period windwave studyof the impulsive generation mechanism. Someweeks spectra andtheinfragravity wavespectra (solidline),indicatingearlier, however, twoAanderaa pressure gauges weredeployed in thatwindwaveenergyis transferred to theinfragravity wave thesame region witha 10-kmseparation during thepassage of a frequencies through thewindwaveenvelopes. Thisconclusion is strongfront.Thisfrontarrivedin theSydney regionat about reinforced by calculations summarized byFigure7, whichshows2000LT onJune19. Calculations of thespeed of thefrontfrom spectral energies, coherences, gainandphaselag between time Bureau of Meteorology meansealevelatmospheric pressure data series of windwaveenvelopes andtheobserved timeseries at suggest anaverage frontspeed of-8 m s-]. Timeseries ofsea infragravity wavefrequencies. Herespectral andcross-spectral leveldataareshown in Figure8. Oscillations looking remarkably estimates havebeenensemble averaged overthe48 bursts of day like the Cauchy-Poisson oscillations depicted by Lamb[1932, 4. Spectral energies fortheenvelope series arelargerthanthose article238],andobserved by Munket al. [1956]areevidentat for theinfragravity wavetimeseries by a factorof 20;however,-1900 LT (southern gauge)and-1920LT (northern gauge). The asis shownby thegain(wherein onlythecoherent contributions timelag between theoscillations suggests an apparent speedof arecompared), thereis approximately 10%asmuchcoherent propagation of-8 m s-•, consistent withthespeed of thefront, energyin the infragravity wavespectrum as in the envelopeand it thereforeseemslikely that the oscillations are indeed specmun.Coherences are significant overmuchof theplotted Cauchy-Poisson oscillations caused by the impulsedueto the •000 / 100• .50 • 5oo• o• • movingfront. Calculationof the phase speedsof lower-modeedge waves showstheirspeed(>17 m s-•) to be muchlargerthanthefront speed. Thus it is unlikely thatthe resonantmechanisms proposed by Buchwaldand de Szoeke[1973] and Viera and Buchwald [1982] would be effectivein thiscase. "' o/ , o.o .5• , • .• (u 1. i tj j z -.r- o o t,_) o o o o øo Do Doo ......... 0.0 .5• , 360 .& _ **+, .d• .d• I I •oe,, o ..•95 • oo .... O0 o.o , o.o .01 .02 FI IEflUENCY (CPS) 0 0 o FRE•UENCY (CPS) *• 4. COMPARISON OF OBSERVATIONS WITH EDGE-WAVE THEORY One methodof comparingthe infragravitydatawith theoriesof edge wave propagationis to measure coherenceand phase betweentime seriesof observations separatedalongthe direction of propagation. Values of coherencesquaredand phasewere calculatedfor day 4 of the storm (June30), this particularday beingchosenbecausethe stormandthe infragravitywave spectra were largest. The spectral and cross-spectralestimatesfor pressureandvelocitytime serieswere ensembleaveragedover all bursts taken on the day, and values of coherencesquaredand Fig.7. Spectral and cross-spectral quantities ofthepressure envelope phase areshown inFigure 9. Allvalues ofcoherence squared are timeseries (E)andtheobserved timeseries (R)atinfragravity frequencies, plotted,whileplottingof phases wasrestricted to thosepoints ensemble averaged over48 bursts. where the coherencesquaredexceededthe 95% confidencelimit MIDDLETON ETA!..: EDop. WAVF.S 9491 t.0 P 0 0 .0! 0.0 .02 .0! 270, ,,, 90 .. • 9o -90 0.0 0.0 .02 .0! 27ø/ , ø•1x2/ -90 .01 .02 0.0 .0! FFIEOUENCY FFIE9UENCY (CPS) .02 90 •• ,l' ,• -90 .0! 0.0 .02 .02 FRE9UENCY (CPS) Fig. 9. Coherence squared andcoherence phaseplottedasa functionof frequencyfor the pressure time series,the offshorevelocitytime series,and the alongshore veloctytime series.The point estimatesare foundfrom the data, while the solid lines indicatephaselags predictedfor northwardpropagation of edgewavesby thetheory. for the null hypothesis. The nonstationarynature of the time of the current signals being more easily affected by local seriesis less importantfor calculationsof phasewhich are not topographicirregularities.Phasesare also somewhatnoisy but expectedto be dependent on theenergylevels. arereasonably consistent with thetheoretical predictions. There are severalnoteworthyfeatures. Observedcoherences There appearto be no phaseestimatessuggesting a southward betweenthepressure timeseriesaresignificant overmostof the propagating wave, so we concludethat the majorityof the rangeof periods(1-17 min). The observed phaselags (point observedlow frequencyenergylies in northwardpropagating estimates) showa generalincreasewith increasing frequency,infragravity waves,withpropagation ratesbeingconsistent with indicating a northward propagation. Predicted phaselags(solid edgewavetheory. lines)fromthetheory forthefirstthreeedgewavemodes arealso Another wayof examining theproperties of theoscillations is plottedfor comparison.The observations are in generaltocompare therelativeamplitudes andphases of thepressure data agreement with the theoryovermostof the frequency range andtheonshore-offshore velocity fieldasa •anction of frequency. plotted,althoughthe scatterin the observations precludes a Thesecalculations weremadefromthedatausinga frequency precise determination of whichmodes areresponsible. response typeof analysis, wherein onlythecoherent contributions Theoffshore andalongshore velocity coherence squared and atanyfrequency areconsidered. Theresults predicted bytheory phase calculations aresomewhat different. Coherences arevery for thegainandphase(solidlines)arecompared against the much lowerthanthose forthepressure series, probably asaresultestimates obtained fromthedata(point estimates) inFigure 10for both oa t.0 S ,.,% • .5 ,.,'" t'0 4N '" .5 z {D• ILl n DO ø oø'"!3 ool 3 ø o oø direction. on 0 o f_• øo on _p o -; t o.o ko -oo.., 95, 0.0[--,,,., '".o2,Yøø oo..6• , .6•,-,,., o.o .o• 0.0 ß08 v .06 mode •_.03 .03 z ,-, 0 0.0 o.o .6• .6a 270 -90 o.o o.o .... .o• .oa 270 ua90, -Jt 0.0 90 , .6• , .6• -90 +' ' o.o , .6• the northem and southem Seadatas. In each case the observedphase lags are a little larger than the value of 90% predicted by edge wave theory and this may be due to the orientation chosen for the alongshore(and hence offshore) The orientation of the coastline was chosen to be 035øT, however as was outlined earlier, local small-scale topographicvariationsmight play an importantrole in "steering" the currents. Estimatesof the gain also show good general quantitativeagreementwith that predictedby edge wave theory, althoughagainthereappearsto be substantialnoise. In both the plots for the nonhem and southemdata, thereare frequency bands where both the gain and the phase appear anomalous.These might be due to the relatedclassof "leaky" edge-waves or to reflected infragravity waves whose energy propagatesbut is not confined to the modal structure,nor to propagatingalongthecoastalwaveguide. In general,however,thesepressure-velocity resultsshow that the observed infragravity oscillationshave propertiesfully consistentwith unforcededgewave theory. 5. DISCUSSION Observationsof pressureand velocity made in the coastal , .62 ocean offtheRoyalNational Parksouth of Sydney, Australia, show that infragravitywaveswith heightof-20 cm andvelocity of-10 cm s4 existat periodsof 40 s to 17 min duringstorm Fig. 10. Estimatesof coherence squared, coherence phase,andgainfor events. Theseinfragravitywavesare apparentlylocally forcedby the pressureand offshorevelocitycomponents for eachof the Seadata the radiation stress associated with the incident wind wave records.Solidlinesin the phaseandgainplotsarefoundfrom the theory envelope, yet the alongshorepropagation speeds and the whilethepointestimates arefoundfromthedata. 9492 MIDDLETONET •a..: EDOmWAVES P u v 0.0 .0• 0.0 .0! .02 ', .02 FREI3UENCY(CPS) Fig. 11. AsforFig.9, except thatcalculations aremadefortheenvelopes ratherthanthetimeseries. pressure-velocity relationships areconsistent withthose predictedwheretoois thecentral windwavefrequency andtx0is theangle from unforcedlinearedgewavetheory.Theseideasseem of incidence indeepwater.Knowing theangleofincidence atthe somewhat contradictory; however, fromearliercalculations it was pointof observation, tx0maybefoundby assuming conservation foundthatvaluesof coherence squared for the envelope and of alongshore wavenumberas the windwavesshoal.For the infragravity timeseries (Figure 7) aregenerally significant but present casewheretoo-- 0.44-0.56 s-] (corresponding to wave lessthan0.5, as are valuesfor the southern andnorthern periods of 11-14s4),or0.=10ø-30 ø,andtherange ofspeed ofthe infragravity timeseries (Figure 10). Thusanexplanation forthe forcing is= 17-64m s-], againquiteconsistent withtherange of apparent contradiction is that the observed infragravity waves speeds forthelower-mode edgewaves. most probablycompriseof both locallyforcedand freely Theedgewavespectral energies areobserved to develop and propagating (edgewave)oscillations. Theseparation of forceddecaywithtimethrough a stormin stepwiththelonger-period andfreeinfragravity oscillations andtheidentification of the (-14 s) windwaves,suggesting a continuing energytransfer, forcingmechanisms haveprovedelusivein otherstudies even through possibly oneorbothof theresonant mechanisms outlined wheretheinstrument arraysarelarge,andwedonotbelievethat above.Themajorityof energyappears tobe transferred viathese theseissues canbeproperly addressed withthepresent dataset. mechanisms rather than throughan impulsivegeneration Somediscussion of possible forcingmechanisms is, however,mechanism, although thisexperiment wasnota goodtestforthe appropriate. impulsive generation theoryor for Worthy's[1984]theoryof Onepossible mechanism is a resonant matching of alongshore wind-generated edgewaveswhichis practically applicable to propagation speeds of envelopes andedgewaves.To investigatewavesofperiodlongerthan15min. thismechanism, coherence andphaseestimates werecalculated Thereareseveralimportantfactorswhichneedconsideration, for envelope timeseriesfromthesouthern andnorthern Seadatas,giventhata relativelyenergetic infragravity wavesignalexistsin andtheseareplottedaspointsin Figure11.Superimposed onthe theSydney coastal region.First,harbor resonance isliabletobea phasediagramarethepredicted phases for thefirstthreeedge continuing problemfor harbors suchasPortKemblawherethe wavemodes(solidlines),andthegeneralagreement confirms the firstmodeseiching period(-3 min) occursat periods for which matchingof envelopeand edgewavespeeds.Coherences are infragravitywavesare active. Second,standingedgewaves generallylowerthanfor the infragravitytimeseriescalculationsproduced by reflectionfromheadlands mayprovidea generation (Figure9), addingfurtherweightto thecontention thatsomeof mechanism or a transportmechanism for the largesubmerged theenergyresides in naturaloscillation (edgewaves).Estimatessandbedswhichexistfurthernorthin theSydneyarea[Gordon of envelopespeedsare readilyfoundin the followingway. andHoffman, 1985].Thesesandbedsexistat30- to40-mdepth, Calculationsof the direction of the incident wind waves as corresponding to distancesoffshoreof 1-3 km. As is indicatedin measured by the instruments indicatethatthewavespropagateFigure2, theedgewavestructure is suchthatfor periods of-3 basically fromthesouthto southeast quadrant. Waveenvelopesminor longer,significant edgewavecurrents will existasfar as approaching the coastline,whichhas an orientation of -035ø- 4.5 km offshore.At a 3-minperiodthemode0 wavelength is 215øT,will propagate alongthecoastline at a phase(andgroup) approximately 3.2 km long,a lengthscalecomparable withthe velocityof c/sinct. Here c is the phasespeedof the shallow observed sandbeds. water waves,and ot is the anglebetweenthe incidentwavesand thenormal tothecoastline asmeasured bytheSeadatas. Forthe Acknowledgements. Thiswork wassupported byMarine Sciences and Technologies grant 83/1176.We thankGregNippardand David present case where thecoastal depth is-20m,c--13ms4, ot= Griffin fortheir mooring expertise, Steedman Lidforthe provision ofone 5ø-20 ø,andtheapparent speed ofpropagation of theenvelopes Seadata 635-12s and forreading thedata tapes, and John Biddlecombe and alongthecoast is 35-150m s4, overlapping therangeof speedsDaveAdams for theuseof M.V. Laura-EandM.V. Elizabeth, of freeedgewaves of zerothmode(17-32m s4), firstmode(29- respectively. BobGuza, Graham Symonds, andtheanonymous reviewers 68ms4),andsecond mode (38-192 ms4) atperiods of3.3to17 provided helpful comments onanearlier draft. min. Theconcept of forcing byresonant triads, proposed byBowen REFERENCES andGuza[1978] andsupported by laboratory experiments, Bowen, A.J.,and D.L.Inman, Edge waves and crescentic bars, J. predictsresonantforcingat a phasespeedof g(2co0 sinor0) 4, Geophys. Res.,76,8662-8671, 1971. MIDDLETON ET AL.: EDGE WAVES 9493 Bowen,A. J., andR. T. Guza,Edgewavesandsurfbeat.J. Geophys.Res., Longuet-Higgins,M. S., and R. W. Stewart, Radiationstressand mass 83, !913-1920,1978. transportin gravitywaves,with application to "surf-beats," J. Fluid Bracewell,R. N., The Fourier Transformand its Applications,2nd ed, Mech., 13, 481-504,1962. McGraw-Hill,New York, 1978. Longuet-Higgins, M. S., andR. W. Stewart. Radiationstresses in water Buchwald,V. T., andR. A. de Szoeke,The response of a continental shelf waves:A physicaldiscussion, Deep-SeaRes.,11,529-562, 1964. to travellingpressuredisturbances, Aust.J. Mar. FreshwaterRes.,24, Melville, W. K., Wave modulationandbreakdown, J. Fluid Mech.,128, 143-158, 1973. 489-506, 1983. Eckart,C., Surfacewaveson waterof variabledepth,WaveRep.100,99 Munk, W., F, Snodgrass, andG. Carder,Edgewaveson thecontinental pp.,Scripps Inst.of Oceanogr., Univ.of Calif.,La Jolla,1951. shelf,Science, 123,127-132,1956. Gallagher, B., Generation of surfbeatby non-linear waveinteractions, J. Munk, W., F. Snodgrass, andF. Gilbert,Longwaveson thecontinental Fluid Mech.,49, 1-20, 1971. shelf: An experimentto separatetrappedand leaky modes,J. Fluid Gordon, A.D., andJ. G. Hoffman, Sediment features andprocesses of the Mech.,20,529-554,1964. Sydneycontinental shelf,Tech.Memo 85/2, 37 pp., N.S.W. Public Reid,R. O., Effectof Coriolis forceonedgewaves, I, Investigation of the normalmodes,J. Mar. Res., 16(2), 24-33, 1958. WorksDep.CoastalBranch,Sydney,1985. in hydrodynamics, Rep. 16th Greenspan,H. P., The generationof edge wavesby movingpressure Stokes,G. G., Reporton recentresearches distributions,J. Fluid Mech., 1,574-592, 1956. Meet. Brit. Assoc.Adv. Sci., Southampton, 1846,Murray, London,pp, 1-20, 1846. Guza, R. T., and A. J. Bowen, The resonantinstabilitiesof long waves Symonds,G., D. A. Huntley, and A. J. Bowen, Two dimensionalsurf obliquelyincidenton a beach,J. Geophys. Res.,80, 4529-4534,1975. beat: Longwavegeneration by a timevaryingbreakpoint,J. Geophys. Guza,R. T., andR. E. Davis,Excitationof edgewavesby wavesincident Res., 87, 492-498, 1982. on a beach,J. Geophys.Res.,79, 1285-1291,1974. Guza, R. T., and E. B. Thomton,Observations of surfbeat,J. Geophys. Tucker,M. J., Surf beats:Seawavesof 1 to 5 min. period,Proc.R. Soc. Res., 90, 3161-3172, 1985. London,Ser. A, 202, 565-573, 1970. Guza, R. T., E. B. Thorntonand R.A. Holman, Swash on steepand Ursell,F., Edgewaveson a slopingbeach,Proc.R. Soc.London,Ser.A, 214,79-97, 1952. shallowbeaches,in Proceedingsof theNineteenthCoastalEngineering Conference, ASCE, Vol 1, editedby B.L. Edge,pp. 709-723,American Viera, F., and V. T. Buchwald,The responseof the east Australian continentalshelf to a travelling pressuredisturbance,Geophys. Society ofCivilEngineers, 1984. Astrophys.Fluid Dyn., 19, 249-265, 1982. Holman,R. A., andA. J. Bowen, Edgewaveson complexbeachprofiles, Worthy,A. L., Wind-generated, high-frequency edgewaves,Aust.J. Mar. J. Geophys. Res.,84, 6339-6346, 1979. Holman,R. A., andA. H. SallengerJr., High-energynearshore processes, Freshwater Res., 35, 1-7, !984. Eos Trans.AGU, 67(49), 1369-1371, 1986. Huntley,D. A., and A. J. Bowen, Field observations of edgewavesand M. L. Cahill and J. H. Middleton, Schoolof Mathematics,Universityof their effect on beach material, J. Geol. Soc. London, 131, 68-81, 1975. New SouthWales,Kensington,N. S. W., Australia,2033. Huntley,D. A., R. T. Guzaand E. B. Thornton.Field observations of surf W. W. Hsieh, Departmentof Oceanography,University of British beat, 1, Progressiveedge waves, J. Geophys.Res., 86, 6451-6466, Columbia, Vancouver, B.C., Canada V6T 1W5. 1981. Lamb, H. L., Hydrodynamics,6th ed., Dover,New York, 1932. LeBlond, P. H., and L. A. Mysak, Wavesin the Ocean, Elsevier,New York, 1978. (RecievedSeptember16, 1986; revisedApril 12, 1987; accepted April 13, 1987.)
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Edge Waves on the Sydney Coast Middleton, Jason H; Cahill, Madeleine L; Hsieh, William W. Aug 15, 1987
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Title | Edge Waves on the Sydney Coast |
Creator |
Middleton, Jason H Cahill, Madeleine L Hsieh, William W. |
Publisher | American Geophysical Union |
Date Issued | 1987-08-15 |
Description | Pressure and current oscillations at periods of 40 s to 17 min observed during storm conditions at two locations separated by 560 m in the alongshore direction in the coastal ocean near Sydney, Australia, indicate the existence of infragravity waves having amplitudes of ∼20 cm and velocities of ∼10 cm s−1. The observed infragravity waves appear to be locally forced by the wind wave envelope through radiation stress, yet the observed alongshore phase differences of the infragravity waves are consistent with those predicted from free edge wave theory for low-mode edge waves travelling northward and the relationship of pressure to velocity at each location is also consistent with free edge wave theory. As a function of time, the infragravity wave spectral energy grows and decays in step with the longer-period wind waves, suggesting a continuous transfer of energy. The infragravity waves appear to contain energy in both directly forced and freely propagating (edge wave) oscillations. The edge waves may be generated either by radiation stress as outlined above, by a resonant triad mechanism, or by a combination of the two. An edited version of this paper was published by AGU. Copyright 1987 American Geophysical Union. |
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Article |
Type |
Text |
Language | eng |
Date Available | 2016-11-10 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
IsShownAt | 10.14288/1.0041788 |
URI | http://hdl.handle.net/2429/32846 |
Affiliation |
Science, Faculty of Earth and Ocean Sciences, Department of |
Citation | Middleton, Jason H., Cahill, Madeleine L., Hsieh, William W. 1987. Edge Waves on the Sydney Coast. Journal of Geophysical Research 92 C9 9487–9493 |
PublisherDOI | 10.1029/JC092iC09p09487 |
Peer Review Status | Reviewed |
Scholarly Level | Faculty |
Copyright Holder | Hsieh, Wiliam W. |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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