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Edge Waves and Beach Cusps

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Edge Waves and Beach Cusps VOL. 80, NO. 21 JOURNALOF GEOPHYSICALRESEARCH JULY 20, 1975 EdgeWaves and BeachCusps ROBERTT. GUZA AND DOUGLASL. INMAN ScrippsInstitution ofOceanography, Unioersity ofCalifornia, LaJolla, California 92037 Genetically,beach cus. psare of at least two types: those linked with incident waves which are surging andmostly reflected (reflective systems) andthose generated onbeaches where wave breaking and nearshorecirculation cells are important (dissipative systems). The spacings ofsome cusps formed under reflectivewave conditions both in the laboratory and in certain selected natural situations areshown to be consistentwithmodels hypothesizing formation byeither (1) subharmonic edgewaves (period twice that ofthe incident waves) ofzero mode number or(2) synchronous (period equal to'that of incident waves) edgewaves of lowmode. Experiments showthat visible subharmonic edgewave generation occurs on nonerodableplanelaboratory beaches only when the incident waves are strongly reflected atthe beach, andthis observation isquantified. Edge wave resonance theory and experiments suggest that synchronous potentialedge wave generation canalso occur on reflective beaches andis a higher-order,weaker resonancethan the subharmonic type. In dissipativesystems, modes of longshoreperiodic motion other thanpotential edge waves may be important incontrolling thelongshore scaleof circulation cellsand beachmorphologies. Onreflective plane laboratory beaches, initially large subharmonic edgewaves rear- ragesand tracers into shapes which resemble natural beach cusps, butthe edge wave amplitudes decrease asthe cusps grow. Cusp growth isthus limited by negative feedback from the cusps tothe edge wave ex- citationprocess. Small edge waves can form longshore periodic morphologies byproviding alestabilizing perturbationsona berm properly located inthe swash zone. In this case the retreating incident wave surge ischannelized intobreeches inthe berm caused by the edge waves, and there is an initially positive feed- backfrom the topography to longshoreperiodic perturbations. INTRODUCTION modelsare feasibleon erodablebeaches, they cannotexplain later laboratoryexperiments [Gah)in, 1965; Harris, 1967; Rhythmicallongshore patterns have often been observed in Bowen,1967; Bowen and Inntan, 1969, 1971] on nonerodable shorelineand nearshøre morphologies, both in thefield and in planebeaches which showed the existence of longshore wavetanks [e.g., Køntar, 1973; Bowen and Inntan, 1971; Dolan, periodicwave motions (edge waves), some of whichdevelop 1971'Harris, 1967; Johnson, 1919]. Cuspate patterns, which circulationcells. Either the edgewaves or the associatedcir- will be consideredhere, are concaveseaward, are usually culationcells can imposetheir longshoreperiodicity on an formedat theshoreline, and have longshore wavelengths vary- erodablebed. The present experiments indicate that although ingfrom less than 1 m onlakeshores [Komar, 1973] to thescale the inducedtopographic changes eventually have a feedback of capes(105 m) or larger[Dolan and Fernt, 1968]. Dolan and to the waves and currentsand thus alter the further rearrange- Ferm enumerate some traditional names given cuspate mentof sediment,the primary longshore periodic generative featureson ocean ShorelineS: (1) 'typical beach cusps,' 8-25 m; mechanismof beach cusps issometimes present in theewaves (2)'storm cusps,' 70'120 m; and (3) 'giant cusps,' also known andcurrents occurring on nonerodablebeds, Inclusion of the as,shOreline rhythms,' 700-1500 m.A Classificationaccording feedback of changingtopography is fOUndto be necessaryto to sizeis, hOweVer ' genetically unconvincing. Kuenen [1948] determinethe equilibriumamplitude and the permanenceof statedthat 'tYpical' and 'storm ' cusps are 'practically the same' morphologicchanges. exceptfor size.The presentstudy centers on a single It hasalso been hypothesized thatrhythmic topograPhie son generatingmechanism, edge waves, which have wavelengths an unboundedcoast are 'sand wave' trains which result from rangingfrom centimeters onlakeshores to hundreds of meters an instabilityof thesurf zone bed to perturbationsby (or more)near oceanic coastlines. Edge waves capable of longshorecurrents [e.g., Hom-nta and Sonu, 1963; Sonu., I968, producingcuspate morphologies are generatedin the 1973;Schwartz, 1972]. The analogy here is to thedunes and laboratorywith Wavelengths varying from tens of centimetersantidUnes of fluvialsystems; initially small bottom ir- to l0 m, thewave basin size precluding longer waves. regularitiesgrow because of a reinforcingcoupling to the culationsCuspsareaSsociated noted for withtheir them.regularity JOhnson and [1919]the distinctive summarizes cir- longshore current.Their proponents hypothesize thatsand earlyattempts tø explain theserhythmic morphologies and wavesinfluence theincident wavefield insuch away asto citesLane[1888] ashYPothesiZing thatrandom irregularities producetheobserved circulation Cells.Like the earlier onthe beach become evenlyspaced through someprocess of hypotheses, thismechanism cannotexplain thelongshore adjustmenttoequilibrium notClearly understood andthat this periodicmore, many motionsauthors observed[e.g., JohnSOn, onnonerodable 1919; Lønguet-Higgins beaches.Further-and equilibriumdistance between Cusps isrelated to theheight of Parkin, 1962;Russell and MClntire, 1965] claim that natural the waves.Johnson basically agreed with this hypothesis,as cuspsform only with normally incident waves, Which generally did laterauthors [e.g., Kuenen, 1948; Otvos, 1964], although do not produceunidirectional, coherent flows like those someauthors stressed the importanceof alepositionalover responsiblefor classicaldune systems. Others have observed erosionalprocesses, and vice versa. This model stresses the im- cuspformation with bothnormal and nonnormal incidence portanceof theinteraction of wave motions and topographic [Worrall, 1969; Otoos, 1964; Mii, 1959].Clearly, any theory changesindetermining cusp spacing. While bottom interactive which requires a longshorecurrent for cuspformation cannot Copyright¸ 1975by theAmerican Geophysical Union. explaina largeportion of observednatural cusps. It should 2997 2998 , GUZAAND INMAN: EDGE WAVES AND BEACH CUSPS notbe assumed, however, that cusps of largephysical dimen- EDGEWAVES ON CONSTANTSLOPE BEACHES sionformed at locationswith large process scales (i.e., in- cidentwave wavelengths) arenecessarily genetically different Coriolisforces being neglected, the dynamics of inviscid from muchsmaller cusps formed elsewhere. edgewaves on a nonerodablebeach of constantslope ex- Edgewaves are the normaltrapped modes of longshoretending into deep water have been discussed byEckart [1951], periodicwave motion that occuralong the edgeof water Ursell[1952], Bowen and Inman [1969, 1971], and others. The bodies,and they may be either standing orprogressive. They edgewave wavelength L in the longshore direction isgiven by existequally well along straight or gentlycurving shorelines Ursell [1952], with no assumptionsabout the shallownessof the water, as andfor uniformlysloping, concavely or convexlycurving offshoredepth profiles. Numerical methods show the structure L = (g/2;r)Te:sin (2n + 1)/•= 2;r/ky (1) of edgewaves on morerealistic topographies [Munk et al., 1964].Edge waves are noteworthy because they are trapped; where x andy arethe offshore and longshore coordinates, theycannot radiate energy out into deep water. Unlike or- respectively;Teis thePeriod; /• is thebeach slope; g is the dinaryswell waves, they cannot propagate offshore and away gravitationalacceleration; n is aninteger known as the mode froma shallowwater energy source. Their energy can only be number,and ky is the longshorewave number. A further dissipatedthrough friction and through interaction with cur- restrictionon n is(2n + 1)/•< ;r/2,where/• is in radians.Thus rentsand other waves. The possible importance of edge waves for a givenedge wave period there are a seriesof possible in thenearshore region was first stressed by lnman'•[Bowen, wavelengths corresponding to the possible values of n. Thex 1967].It hasbeen demonstrated theoretically that edge waves behaviorofthe Ursell solutions isalgebraically quite complex. canbe excited by w•aves incident from deep water [Gallagher, One of us(R.T.G.) has found that Tait [1970] made errors in 1971;Guza and Davis, 1974] or by traveling atmospheric pres- computingthese functions, thereby invalidating his results. It surefronts [Donn and Ewing, 1956]. It is thecontention of the is usefulto considerthe simplershallow water solutions in presentwork that edge waves, both directly and via theirin- termsof thevelocity potential 4•; the velocity is V4•,and the teractionswith other water motions, are responsible formany surfaceelevation • is (-I/g) Ock/Ot.Eckart [1951] gives the casesof cuspatetopography. Related ideas are discussed by shallowwater velocity potential solution for edge waves on a Bowen[1967, 1969, 1972], Bowen and Inman [1969, 1971, beachsloping into deepwater as 1972],Komar [1971, 1973], and Inman [1971]. Thedynamics of water motion on the beach face change ck= (a•g/co)cos k•y exp(-k•x) L,,(2k•x)cos cot (2) drasticallyaccording towhether the incident wave is strongly where ae is the wave amplitude, L,,is the Laguerre polynomial reflectedor breakscleanly and propagates onshore as a dis- of ordern, co= 2;r/Teis theradian frequency, and sipativebore. A distinctionis thereforemade between 'reflec- tive' and 'dissipative'systems. The presentwork deals L = (g/2;r)Te•'(1 - 2s)tan /ff =
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