Apr 15, 1997 - by an angular bottom which necessitates a matching condition at a corner in the ...... spatial scales of the oscillating breaker line match the edge.
JOURNAL OF GEOPHYSICAL
RESEARCH, VOL. 102, NO. C4, PAGES 8663-8679, APRIL 15, 1997
Generation of edge waves in shallow water T. C. Lippmann Centerfor CoastalStudies,ScrippsInstitutionof Oceanography, La Jolla,Califomia R. A. Holman
Collegeof OceanandAtmospheric Sciences,OregonStateUniversity,Corvallis A. J. Bowen
Departmentof Oceanography, DalhousieUniversity,Halifax, Nova Scotia,Canada
Abstract. Theoreticalgrowthratesfor resonantlydrivenedgewavesin the nearshoreare estimmedfrom the forced,shallowwaterequationsof motionfor the caseof a plane slopingbed. The forcingmechanism arisesfrom spatialandtemporalvariationsin radiationstressgradients inducedby a modulatingincidentwavefield. Only thecaseof exactresonance is considered, where thedifference frequencies ,andwavenumbers satisfytheedgewavedispersion relation(thespecific carrierfrequencies arenotimportant,onlytheforceddifferencevalues).The forcingis ex,'unined in theregionseawardof thebreakpoint andalsowithinthefluctuatingregionof surfzonewidth. In eachregion,theforcingis dominatedby the cross-shore gradientof onshoredirectedmomentum flux, exceptfor largeanglesof incidenceandthe lowestedgewavemodes.Outsidethe surfzone, thespatialandtemporalvariationof theforcingis determined by considering theinteractionof two progressive shallowwaterwavesapproaching thebeachobliquely.In the surfzone,incident waveamplitudesareassumed to be proportionalto the waterdepth. Thusinsidethebreakpoint, radiationstressgradientsareconstantandno forcingoccurs.However,at the breakpoint,gradients arisingfrombreakingandnonbreaking wavesareturnedon andoff (like a wavemaker)with ti•nescales and lengthscalesdeterminedby the modulationof the breakerposition. The tbrcingin thisregionis stronger,with inviscidgrowthratesresultingin edgewavesgrowingto the sizeof the incidentwavesof the orderof about10 edgewaveperiods,a factorof 2-10 timeslargerthanin the offshoreregion. Usinga si•nplepar,-uneterization for frictionaldamping,edgewave
equilibrium mnplitudes arelbundtodepend linearlyontheratiotan]•/Ca , wherefl isthebeach slopeandCais a bouomdragcoefficient. For tan•/Ca about3-10,equilibrium amplitudes can be as •nuchas 75% of the incidentwavesover most of the infragravityportionof the spectrum.
Whentheforcingis turnedoff, thesedissipation ratesresultin a half-lifedecaytimescaleof the orderof 10-30 edgewave periods. Introduction
1985; Oltman-Shay and Guza, 1987; Howd et al., 1991;
Since the initial observations of Munk [19491 and Tucker
[1950] much effort has been aimed at understandingthe origin and importance of low-frequency (relative to wind waves) surfacegravity wavesin shallow water. Field data obtainedon
natural beaches haveshown thatlong-period (O(102-10 3 s)) infragravity motions often dominate power spectra in the inner surf zone and swash, particularly during storms when incident wave heightsin shallow water are severelylimited by breaking [Huntley, 1976; Huntley et al., 1981; Holman, 1981; Thornton and Guza, 1982; Holman and Sallenger, 1985; Sullenget and Holman, 1987]. The data also indicate that infragravity surface gravity waves are predominantly composed of edge waves, longshore progressive waves trappedto the shorelineby refraction,together with a smaller componentof leaky waves, normally reflected waves which escapefrom the nearshoreinto deepwater [Guza and Thornton,
Copyright1997 by the AmericanGeophysicalUnion. Papernumber96JC03722. 0148-0227/97/96JC-03722509.00 8663
Herbets et al., 1995b].
The existenceof edge waves in nature has been known for some time. Bowen and (;uza [1978] and Holman [1981] suggest that the growth of edge wave amplitudes, an, is
determinedby the coupling between the nonlinear forcing,
F(a1, a2, x), arisingfrom an interacting incidentwavefidld andthecross-shore edgewavewaveform,On(x), (1) rYe f F(al,a2,x)On(x)dx g
o
where al and a2 are incidentwave amplitudes,rYeis the edge wave radian frequency, g is gravity, t is time, and x is the cross-shorecoordinate. Growth occurs when the forcing pattern is not orthogonalto the edge wave waveform. The details of the forcing are describedby the spatially varying form of F. In this work we derivean analyticexpressionfor F and numerically integrate the coupling integral to obtain an estimatefor the growth rate. The principalforcingis believedto be derivedfrom pairsof incident wind waves, which produce spatial and temporal
8664
LIPPMANN
ET AL.: GENERATION
OF EDGE WAVES IN SHALLOW
variations in radiation stress gradients of the same timescale
and lengthscaleas the edge waves. Outsidethe surf zone in intermediate and shallow depths, forcing occurs through a typical second-ordernonlinearinteraction[Gallagher, 1971;
WATER
and Mei [1981], they assumeconstantbreakingcriteria so that modulations in incident wave heights are manifested in fluctuations in the width of the surf zone.
However, their
in specifying the forcing functions all the way to the shoreline. Here we assume that amplitude variations are eliminated by saturationand no oscillatory forcing occurs. However,thereis a time dependentmomentumflux inducedby the modulation in breakpoint position associatedwith the variation in incident wave height [Foda and Mei, 1981; Symondset al., 1982; Symondsand Bowen, 1984; Schaffer,
model is limited to only two dimensions (no inclusion of longshore variability), precluding the possibility of edge wave forcing. Nevertheless, the results of Symondset al. suggestthat edge wave generationby temporal variations in surf zone width is possible. Schaffer and Svendsen[1988] retain the ideasof Foda and Mei [1981] and Symonds et al. [1982], by allowing fluctuations in both breakpoint positions and incident wave energy inside the breakpoint, although they limit their
1993].
discussion to the two-dimensional case. List [1992] extends
Bowen and Guza, 1978]. Inside the surf zone, difficulties arise
Outsidethe surf zone (referredto herein as offshoreforcing)
this mechanism in a numerical model to include arbitrary
the generationmechanismevolves from the second-order topography and further includes incoming bound waves, forced (or bound) wave generated by wave groups in crudely•nodeledempiricallyusingfield data,as do Schafferand intermediatewater depths[Biesel, 1952; Longuet-Higginsand Svendsen [1988]. They forego a more theoreticalapproach Stewart, 1962, 1964; Hasselmann, 1962]. In two dimensions,
the bound wave is, in principal, released at breaking and reflected at the shoreline; thus the incoming and outgoing waves form a complex pattern in the cross-shore,however, without the longshorecomponentnecessaryfor edge wave
since it is not clear what the correct boundaryconditionsare for boundwave dynamicsin the shoalingand breakingregion, although it is expected the forced motions become nearly resonant in shallow water [Longuet-Higgins and Stewart, 1964; Okihiro et al., 1992].
Schaffer [1994] usesa WKB (optics) approachto describe generation. This idea was extendedto three dimensions by Gallagher [1971], who showed that nonlinearinteractions the incident waves and drives the edge waves with radiation between incident wave pairs could producelow-frequency stressmodulationsdeterminedby fluctuating amplitudes(i.e.,
oscillationswith a nonzero alongshorewavenumber. Bowen and Guza [1978] generalized Gallagher's model to include forcingof all modesand showedwith laboratoryexperiments that resonantresponsewas greaterthan forcedresponse.This same conclusionwas reachedby Foda and Mei [1981] in a fourth-orderWKB expansionof the momentumequations. In the surf zone, forcing occurs because an interacting incidentwave field producesspatial and temporalvariationsin locations of the position at which a wave breaks (herein referred to as surf zone forcing). Momentum is transferred from incidentto lower frequenciesthroughwave breaking,and the interactionsagain lead to fluctuations in the flow field with timescalesand spacescalesof the order of wave groups. This idea was first explored as a generatingmechanismfor long waves by Foda and Mei [1981] and Symondset al. [1982]. As will be shown later, there is also a contribution from nonbreakingwaves within the region of fluctuatingsurf zone width.
In the work by Foda and Mei [1981], low-frequencywaves are generatedby wave-wave interaction assuminga fixed breakpoint position for all waves. This allows incident modulationsto progressto the shoreline,and edge waves are forced everywhere. In nature,however,the initial breakpoint of individual waves is not constant through time, nor is the breaker line uniform along the beach. Under most conditions, temporal and spatial variations in the width of the surf zone occur on large timescalesand space scales,of the order of infragravity scaling [Symondset al., 1982]. Although the basic physicsinshore from the breakpointis doubtful, Foda and Mei do consider resonant edge wave growth, also consideredhere, and thus a brief comparisonwith their results is made later.
Symondset al. [1982] considerthe problemof long wave forcing by time modulationsin surf zone width for the caseof a planeslopingbed and later includedinteractionswith barred topography[Symondsand Bowen, 1984]. In contrastto Foda
the two incidentwaves are assumedto approachfrom the same direction). Numerical solutionsare considerablycomplicated by an angularbottomwhich necessitates a matchingcondition at a corner in the profile. Furthermore, Schaffer's offshore profile is horizontal; thus waves propagatingoffshore beyond this "shelf" are no longer refracted, thus limiting the possible edgewaveswhich can exist. Low-frequencyresponseis found through numerical calculation, with resonant modes determinedcritically by the single-incidentwave angle and the offshoreprofile configuration. This resultsin a very specific set of results only valid for theserestrictedset of conditions. In this paper we present a mechanism for the resonant forcingof edgewavesin shallowwater throughmodulationsin radiation stress gradients. We consider two interacting shallow
water
incident
waves whose difference
wavenumbers
and frequenciessatisfythe edge wave dispersionrelation. The total forcing is found by integratingthe growth rate equation froin the shoreline to deep water (relative to wind waves). Contributionsto the forcing from inside and outside the surf zone are examined by separatingthe forcing integral at the breakpoint. We considerthe simplestcaseof a planar beach profile, where forcing in the surf zone is determinedby a time and spacedependentfluctuationin the width of the surf zone. In the next sectionwe review the theoryfor edgewaveson a plane sloping beach beginning with the forced shallow water (depth integrated),linearized equationsof motion, leading to an analytic expressionfor the initial (undamped) edge wave growthrate. Model resultsare then presented,comparingthe relative contributionsto the forcing from the componentsof the radiation stressand also the strengthof inviscid growth rates in the surf zone and offshore regions. Resultsare then discussed in terms of the validity of model assumptions, parameter sensitivity, and implications in field situations (spectralforcing). The effect of a linearizeddissipationterm on edge wave growth is discussed. In Appendix B the extensionof the calculationsto an incident wave spectrumis briefly discussed.
LIPPMANN ET AL.: GENERA•ON
OF EDGE WAVES IN SHALLOW WATER
where L n is the Laguerre polynomial of order n. The approximate shallow water edge wave dispersionrelation is given by [Eckart, 1951]
Model
Edge Wave Theory
The forced, shallow water (depth integrated), linearized equationsof motionincludinga linearizeddissipationterm are [Phillips, 1977]
2
cye- gke(2n+1)tanfi Growth
m +
3t X-b--fxph
-
Sxx+
•+
Sxy+
-
a, gTfy ph
- •,u
3y xy
Tyy s"
-•v
(2)
3t
(h.)+
(8)
Rates
We seek an expressionfor the time rate of change of edge
wavemodalamplitudes, or growthrate, 3an/3t. Substituting
(3)
(6) into (5) and allowing an to be a slowly varying function of time give
2a n 3a n -anZ(qJn)]e -ire -F
and the continuityequation
--+
8665
(9)
3--'•qn +i2tYe 3y
hu)-0
(4)
where
g-3xlh3•-•l+(CYe2-ghke
where x andy are the cross-shoreand alongshorehorizontal Cartesiancoordinamswith x positiveseaward(for a right-hand systemwith z positiveupward),u andv are the corresponding life- key- Getß horizontalcomponentsof velocity, • is sea surfaceelevation, • is a friction coefficient (discussedlater), h is the still water In (9), and in subsequentequationsfor the growth rate, we depth,andp is the densi• of water. The equationshavebeen have droppedthe complex conjugatefor brevity, thus later we averaged over an incident wave period so that the time will take the real part to obtain the magnitudeand phase. The dependence is on infragravi W andlongerscales.Spqare the function Z(cpn) is the homogeneousequation for shallow radiation stresses of short (incident) waves introduced by
water waves and vanishes for the case of resonance considered
Longuet-Higginsand Stewart [1962, 1964] and describethe flux of pth directedmomentumin the qth direction. Combining (2)-(4) and temporarily dropping the damping term yield a single second-order(inviscid) equation in sea
here. If we assumethat the edge wave growth rate is slow relative to its period, then the first term describing the accelerationin growth can be neglected. Thus an equationfor the initial edge wave growth rate is
surface elevation [Mei and Benmoussa,1984],
i2o'•,c-•--c),•e-iV•' -F (10)
c-)271 c-)t 2
p•
ax+ ay +•yy(ax+ ay
(5)
(describedlater), (ai)o, the normalizedgrowthrate becomes
= Fxx+ Fyx+ Fxy+ Fyy= F whereFxx,Fyx,Fxy,andFyyarethecomponents of thetotal forcing, F, correspondingto the four secondderivativeson the right-handside. The forcingand frictionaldissipationare assumedto be of secondorder, and we ignore their effects on both the wave solutions and the dispersionrelation (which
couldbe significantif the forcingand friction are large). For a planebeach,h= xtanfl, where/8 is the beachslope, the homogenous case(free waves)is satisfiedby edgewavesof the form [Eckart, 1951]
rln(x,y,t) - •-anc)n (x)e -i(l•y-cr•t) +(*)
1 e3a• =(ai)ogtanfl(2n+l) -2rri I F0,•dx (ai)of •gt e-iv/• (11) 0
in which thenormalizing factor [•cpn2dx - 1/2k ehas been used.Thecoupling integralexpress4s therateat whichenergy is transferred
from
(6)
complex conjugate of theprevious term,andi-4Z-f. The cross-shorestructureof the edge wave waveform, qS,•(x), is given by
(7)
the
incident
waves
to resonant
lower-
frequencymotions. The inverseof the magnitudeof (11) is the number of edge wave periods necessaryfor the edge wave to grow to the size of the primary incidentwave.
Coupling Integral
where CYe=2nfeand ke=27r/Le are the edge wave radian frequencyand alongshorewavenumberOreandL e are the edge wave frequency and wavelength), n is the edge wave mode number,an are the complexmodal amplitudes(which can be resolved into a magnitude and phase), (*) indicates the
c),• (x)- e-k•XL,• (2kex)
The cross-shoredependenceis eliminated by multiplying by qS,• and integratingfrom the shorelineto infinity in the crossshore direction, as in (1). Incorporating the dispersion relation and dividing by the primary incident wave amplitude
and Incident
Wave Amplitudes
In deep water, resonant interactions are quartic and only forced waves are produced by triad interactions. In shallower water, resonant triad interactions can occur, producing variation in radiation stresswhich can match the edge wave dispersion relation. This spatial and temporal modulation gives rise to the possibility of edge wave growth. Although the modulationscalesremain the same, the forcing function F, which dependson wave amplitude, varies as waves shoal in intermediatewater and eventuallybreak in the surf zone. As a consequence,there are two distinct forcing regions separated by the contourof the breakerline, xb(y,t) (shown graphically in Figure 1), and the coupling integral can be written
8666
LIPPMANN ET AL.: GENERATION OF EDGE WAVES IN SHALLOW WATER
OffshoreForcing
Breakpoint •b '4'6•b
I
No Forcing Shoreline
!•Ie ---keY-- (Je t Figure 1. Graphical representationof the edge wave forcing regionsrelative to an oscillatingbreaker line.
Thevertical axisis Z= cre2x/gtan,B andthehorizontal axisis gre=key-Get.Theoffshore forcing region is defined as seawardof the •nostseawardbreakpoint. Inside the minimum breakpoint,no forcing occurs. The
rangeof forcingby a modulatingbreakpointpositionis a functiono.__f incidentwave modulation,8. The regionswhere waves are alternatelyshoalingand breaking(between Zb and the oscillatingbreakerline) are labeled I and II, respectively.
Go
x!,(y,t )
the mean wave amplitudeand the modulation(aboutthe mean) is determinedfrom a2 [Symondset al., 1982]. Becauseof sharpdifferencesin the shoalingand breaking regions, the amplitudes of the incident waves are treated differently. In the offshoreregion the amplitudesare described by progressive shallow water waves over a plane sloping bottom; thus an interactingwave field has modulationsall the way to the breakpoint. Wave amplitudes in this region are referenced to a convenient location (e.g., the breakpoint). Inside the surf zone, wave amplitudesare stronglyattenuated shorewardby breakingand are describedas a linear functionof local depth,
IF(al,a2,x)cPnCx)dxIFsz(al,a2,x)cpn(x)dx o
o
+ IFoff(al,a2,x)cpn(x)dx (12) xt,(y,t)
whereFszandFor f aretheforcinginsideandoutside the breakpoint, respectively. The forcing functions are determinedby the temporal fluctuations of radiation stress gradients,which are in turn determinedby •nodulationsin incident wave amplitude.
It is the slowly varying breakerline, xb(y,t ), that creates first-order fluctuations in the surf zone coupling integral. If we assume that the breakpoint varies sinusoidally with gre tilne and lengthscales(describedlater) andfurtherseparatethe couplingintegral at •'b and expandin a Taylor'sseriesabout
1
ai =•-Th
x 3
(equivalent to H>-•(Ttanfi)2/2,so validfor nearlyall Parameterizing
the Forcing
oceaniccases),the besselfunctionsinay be approximated by
Following Phillips [1977], the general forin of the radiation stressis given by
,q'pq --[3__jh C_)W 3X q (S pq Z--
[Stoker, 1947]
Cos(X/--•)(23a) Jø (Xi)= '•i 2/¾2
•
sin(X i-•)
(18)
where x is the horizontal spacevector in which subscriptsp and q denote either horizontal coordinate, (I) is the velocity potential of the incident waves, fi is the mean pressure,and
Introducingcomplexnotationyieldsthe followingform for (I)i
(hi--• aig(--2)l/2ei(X'-%+ cr i •,•rx i +(*) (24)
5pqis theKronecker delta.Themomentum balance hasbeen spatially averagedin the directionof the wave crests(allowing separation of the turbulent and wave terms) and temporally averaged over the incident wave period (allowing mean or slowly varying properties to be evaluated), denoted with the overbarsin (18). In shallow water •--pgz and assuming 7]= 0 at secondorder,
(23b)
The form of our velocity potentialis only strictly valid for waves approachingthe beachshorenormally. However, Guza and Bowen [1975] show that nonnormalanglesof incidence have only small effectson the forin of the velocity potential due to refraction decreasingthe angle of incidence and show that where shallow water solutions are valid, the solutions for
Spq •ph•x• •x•+ •pq • • We consider
the bichromatic
case where
(19) the incident
wave
field is composedof two discrete wave trains with slightly differentwavenumberand frequency,
• - •
+ •2
shore-normal and obliqueincidenceangleare nearlythe same. Substitutionof (24) into (21) producesterms with sum and differencewavenumbers andfrequencies.The sumterms(highfrequencyforcedwaves) are unimportantto the generationof infragravity motions [Bowen and Guza, 1978] and are not consideredfurther. The difference terms describethe long timescalesand space scalesof the incident wave modulation and allow for the possibilityfor edge waves at infragravity
(20)
where the subscripts1 and 2 refer to the two incident waves.
frequencies
Substituting• into (19) produces
Spq -Spq(*l*l)+2Spq(*l*2)+Spq(*2*2)(21) The first and last terms represent the self-self interactions which generate•nean flows and harmonics. The harmonicsdo not contribute to low-frequency forcing in either region and are therefore neglected. Letting the amplitude of a primary wave •i be much larger than any secondary waves •2 (equation (14)), then •1•1 terms are of O(1), while cross-
interaction terms•1•2 areO(8) and•2•2 termsareO(82). The stressarising from •i•i interactionswill generate a setdown outsidethe breakpointand a setupinside the breakpoint
Of = CY1-CY 2
(25a)
[kfl-Ikl -k21
(25b)
and
where k! and k2 are the longshorewavenumbersof the incident wavesand the subscriptf indicatesthe forced differencevalues associated with the incident wave inodulation.
For resonance
to occur,(25a) and (25b) inustsatisfythe edgewave dispersion relation (8). Incident wave angles are asstunedto follow Snell's law for
wave refraction and are chosenat the breakpointwhere the
8668
LIPPMANN ET AL.: GENERATION OF EDGE WAVES IN SHALLOW WATER
linear shallow water phasevelocity is simply a function of the
local depth, c-•,
_
sothat
ki •yi2 ' g =
sin O•o =
tYisin(o•i)o (26)
X4o
X• 5 e +sina 3(2n+ 1)X• 222sin 3-Xe -i3X l+sina 12(2n+ 1)2Xe o• 1sin a2+1)(29)
ß
where ooindicates thedeep water condition and(o• i)oarethe wave angles of the individual waves taken at some reference depth, ho. If we further referenceai to the breakpoint,where
ho = hi,, thenproductsala2 are givenfrom (24) by
=7r
(Sei(Xe-U/e) p (gtanfi
o
The three terms are derived from the componentsof the
interaction radiation stress fromFxx, 2Fxy,andFyy. The three componentsof the interactionforcing are compared later. Contributions to edge wave forcingby nonbreaking wavesseawardof the breakpointbut within the fluctuating
]1/2
where the subscripto refers to values at the referenceposition
region of surf zone width are consideredin the next section.
Inserting(29) into the couplingintegralin (13) resultsin a complexexpression for the undmnped response of edgewaves andevaluating the appropriate derivatives give an analytic by oscillatingforcingseawardof the •neanbreakpoint. After
(e.g., the breakp.oint).
Now substituting (24) into (19), incorporating(26)-(27),
form of the radiation
taking the real part, expressionsfor the normalized initial
stress
growth ratemagnitude, Goff,andphase, Ooff, aregivenby (28)
Spq =•
(7off= 1 •an= 3•'•-• 1/2(30) fe(al)o at 4(2n +1)•(•bb)3(pn2+Qn2)
where
3 2a• I
lpq= 2a• 2al2+1
where
Pn- I (32cos Xe- B1sin Xe)q),•XedXe
JPq+a2 2ala 2+1 XeA -•(Xl 0•1X20•2)
Kpq= •(Xlal _X2a2)
Qn = I(BicosXe +B2sinXe)q)nXedXe Xb
0
1
B1 =
3-Xe 2+sino•1q-sina 2 - 2sina• sina 2 + 1 X•5
3(2n+l)Xe 2 12(2n +1)2 X• 3
XiX2
O.e2 x/¾2
Xe=2 gtan•
B2- Xe 4 Surf Zone Response From Breaking and Nonbreaking
Waves
The cross-shoreintegral parameterizingthe surf zone contributionto the total edge wave forcing rangesfrom the ß 1• interactions,O(•5) termsare from •2 interactions, shorelineto the breakpoint,0_'" 3.0, increasingthe growthrate abouta factor of 2 from 1ø to 45 ø. Interestingly, in the offshore region, to the breakpoint, the spectral problem seems inherently increases in wave angle tend to reduce the growth rate (a linear where the superposition of many wave components consequenceof the rr phase relationships in the longshore leads to the possibility of resonant triad interactions
forcing in the offshoreregion over a larger length of the edge wave profile. In the surf zone, the length over which the forcing occurs is very much smaller, determined by the incident modulation (•=0.1), and thus nodal points are strongly reflected in the growth rates. The effect of o• on growth rates is shown in Figure 7 for
components of Fpq),whereas theopposite occursin thesurf
occurringfor any numberof different (cyi-cyj, ki-kj)
zone where the growth rate becomes larger as wave angle
incident wave pairs. The forcing of a particular edge wave mode is just a linear sum of all possible interactions satisfyingthe edge wave dispersionrelation. Bowen and Guza [1978] discuss the implications of this resonant restriction
increases (owingto increased contributions fromFxy). Discussion Growth
Rates
The predicted growth rates for the case of phase-locked forcing by deterministicwave trains are not unrealisticif we expect this mechanism to provide reasonable forcing of progressive edge waves under the stochastic forcing conditionsfound in nature [Holman, 1981]. The same type of result arose in the study of energy transfer into internal gravity waves from surfacewave packets. An initial study of Watson et al. [1976] showed a very strong forcing for the phase-lockedcase. Olbers and Hefterich [1979] redid the problem for a stochasticsurface wave field and found the strength of the forcing to be several orders of magnitude weaker thanpredictedby Watsonet al. This suggeststhat our
and show that for
narrow-beam
incident
swells,
some
frequency selection, with a strong modal dependence,may be expected. In the surf zone, the situation is more complicatedbecause amplitudesare functionsof local depth (and hencedistancefor monotonic beaches). However, for the bichromatic case, the forcing is dominated by F xx arising from the self-self interactionof the primary wave. If we consideronly this term, then the growth rate equationtakesthe form
1 c-)a ne-t•e.
=
-2rci
xb (y,t)
(ai)of ec-')t (ai)ogtanfi(2n+l) f•(xb)•n(x)dx 0
(40)
8674
LiPPMANN ET AL.: GENERATION OF EDGE WAVES IN SHALLOW WATER
whereF(xt, ) is the meanforcingarisingfrom Fxx(•l•l).
where G is found from (30) and (38) for the two forcing The operandof the couplingintegral is simply a functionof regions. Taking the magnitudeof the real part of (42) results the edge wave waveform with amplitude equal to F. The in a quadratic equationin 3an/3t which has solutionof the possibilityof edge wave forcing arisesif the temporal and fo rill spatial scalesof the oscillatingbreakerline match the edge •a n wave dispersionrelation. For the spectralcase, the breaker •=c, an[-l+f(an)] (43) line is describedby a summationof perturbations abouta mean breakpoint. Edge wave forcing results from a linear where combinationof various incident wave pairs contributingto the alongshoreperturbation,where the contributionfrom any 2•Cre 2 particularpair (with mnplitudesala2) canbe shown(Appendix
c•- •[2+4ere 2
B) to be
Axt, =Ho?,tanfl 4 ala 2cos(•l - •2)
(41)
where H o is the mean breaker height calculated over all incident wave spectralcomponents. Since zXx b is a functionof y and t, the Fourier transformof (41) resultsin a spectralform for the perturbation,now given in terms of longshorewavenumberand frequency. This leads to the attractive result that time and space dependent observations of the width of the surf zone can be related to the
edge wave forcing by the lYequency-wavenumber spectrumof the wave breaking distribution. As indicatedby the growthrate equations(30) and (38), the behaviorof the forcing as a functionof the variousparameters turns out to be quite simple. Growth rate contributionsfrom the surf zone mechanis•nvary with Xl, in a way which largely follows the shapeof On. In the offshoreregion, the influence of mode numberon actualrates is largely throughthe variance distributionof Onin Xb space. That is, the rate of transferring energy from the incident field appearsto be about the samefor all modes,where, for highermodes,the energymust be spread over a larger cross-shoredistance. The only known measurements of growth rates for progressiveedge waves are from the laboratoryinvestigation of Bowen and Guza [1978]. Strict comparisonwith their results
is difficult
since
their
discussion
was
limited
to
Zb = 0.25, and furthermore,they have 6= 1.0, thus violating our assumption of small-amplitude modulation. Their observedgrowthrate is over an order of magnitudeslowerthan predicted by the model. The effect of viscous damping is likely to be important in the laboratory case, and since the scalesof the lab study are much different from thosetypically found in nature, no si•nple comparisonis readily made. In addition, any reasonabledamping mechanismis likely to be different in the two forcing regions and may well have characteristicswhich lead to preferential damping of high
f (an)=•--• e
+ 1+;[2 Ir2an2
The caseof interestis the positiveroot, corresponding to energytransferfrom theincidentwavesto theedgewaves. We havealreadyassumed thatthe frictionaldissipation hasonly a negligible effect on the wave solutionsand dispersion
relation; thus;t2
