wave energy transformation on natural profiles 1996

8/7/2019 Wave energy transformation on natural profiles 1996

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ELSEVIER Coastal Engineering 27 ( 1996) I-20COASTALENGINEERING

Wave energy transformation on natural profilesT.C. Lippmann, A.H. Brookins , E.B. Thornton

Departmeni of Oceanography, Naval Postgraduate School, Mont erey. CA, USAReceived 17 June 1994; accepted 2 November 1995

AbstractA wave energy transformation model, which includes wave breaking within the surf zone

described by surface rollers, is developed for randomly varying waves over arbitrary bathymetry.The model includes roller energy gradients in the energy flux balance, and further specifies thedissipation function based on roller theory following Svendsen. Root-mean-square wave heights,H rms, are found across the surf zone by numerical iteration, and compare well with field dataacquired on both barred and near-planar beaches. The model has two free parameters, CT, he meanangle to vertical of the wave/roller interface roughly representing the type of breaker, and y, ameasure of breaking wave saturation equal to the ratio of Hm s to local depth. Optimal values ofboth parameters are chosen by model fitting, and show the model to be insensitive to o. Althoughthe model is sensitive to the choice of y, values are consistent with field data. The surface rollermodel behaves similarly to the bore dissipation model of Thornton and Guza for a particularparameter choice, and is used to decrease model dependence to one input parameter, y.

1. Introduction

As surface gravity waves approaching a coast propagate into intermediate depths ofthe shelf, wave energy flux is approximately conserved. However, as waves move intovery shallow water they become unstable and break, and organized wave energy isconverted to turbulent water motion and dissipated. Turbulence generated at the surfaceboundary layer by wave breaking is the primary dissipative mechanism in the surf zone(dissipation at the bottom boundary layer is small). Thus, to model wave transformationacross the surf zone, the wave breaking dissipative mechanism must be specified.Currently, there is a (growing) wide body of literature aimed at parameterizing thebreaking process in order to predict the transformation of wave heights all the way to theshoreline (Battjes and Janssen, 1978; Mase and Iwagaki, 1982; Thornton and Guza,

Present address: Naval Atlantic Meteorology and Oceanography Center, Virginia Beach, VA, USA.037%3839/96/$15.00 0 1996 Elsevier Science B.V. All rights reservedSSDI 0378.3839(95)00036-4


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1983; Dally et al., 1985; and many others). The goal of these models is to make accuratesurf zone predictions knowing the bottom profile and a limited number of offshore waveparameters, such as wave period, height, and direction. Models in the literature generallyrequire extensive empirical formulation and/or the specification of free unconstrainedparameters, thus limiting their general application. These free parameters are neededsince so little is understood about the detailed breaking processes.

Quantitative approaches have relied on the relationship between wave height, isI, andwater depth, h, inside the surf zone (where the vast majority of incident wave energy isexpended). The earliest monochromatic models describe H inside the surf zone as alinear function of h

H= Kh (1)where K is an empirical coefficient of O( 1). Galvin and Eagleson (1965) foundK = 0.8-1.2 for monochromatic laboratory waves. These early models worked well inthe lab, but are limited in their application to natural situations where waves are nevertruly monochromatic. In stochastic models, wave heights are described statistically (e.g.,Battjes and Janssen, 1978; Thornton and Guza, 1983). and are often described by

Hm1s= yh (2)Thornton and Guza (1982) show that y _ 0.4 for field measurements on near-planarbeaches. Using field data acquired from five beaches (including barred profiles),Sallenger and Holman (1985) found

y = 3.2tan p f 0.30 (3)where ,!j is the beach slope and y is independent of wave steepness. Dally (1990), basedon Weggels (1972) criterion for incipient breaking, employs a more complicatedempirical relationship for the decay of individual waves which depend on beach slopebut not necessarily wave saturation. The dependence on wave saturation is produced bya wave-by-wave analysis across arbitrary topography. For an unsaturated wave field(i.e., not all waves are depth limited), (2) and (3) do not fully define wave heightseverywhere in the surf zone, and more sophisticated models must be used to describewave transformation.

The most common approach has been to govern wave transformation by the energyflux balance and a physically based dissipation function,

;( Ec,cosa) = --6where E is linear wave energy, c, is the group velocity, CY s the incident wave angle.and E is the dissipation function commonly modeled with simple periodic bores (afterLeMehaute. 1962). In Thornton and Guza ( 19831, E is found following Stoker (1957)and Battjes and Janssen (1978) by applying conservation of mass and momentum atregions of flow upstream and downstream of a fully developed bore-like wave approxi-mated by an hydraulic jump

where p is density of water, g is gravity, and f is the frequency of the wave. B is an


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T.C. Lippm ann et al./Coast aI Engineering 27 (1996) I -20 3

empirical coefficient of O(l), representing the fraction of foam on the face of the wave,and loosely accounts for various breaker types. The model is integrated through theRayleigh distribution, weighted by an empirical function describing the distribution ofbroken waves. Although the Thornton and Guza (1983) model results agreed with fieldobservations within 9% error, average best fits gave B > 1, suggesting that the simplebore function (5) underestimated the actual dissipation.

In general, the random wave models have been successful at describing Hmstransformation in both lab and field situations. However, they do not adequately describethe spatial characteristics of the dissipation in order to predict, for example, thelongshore current profile, particularly on barred profiles (Roelvink and Stive, 1989;Church and Thornton, 1993; Smith et al., 1994). This deficiency has promoted muchinterest in mixing mechanisms within the surf zone (e.g., Church and Thornton, 1993,and references therein), but also a reexamination of the validity of the bore dissipationmodel. One mechanism which has received considerable attention in recent years issurface roller theory, initially investigated in the lab by Duncan (1981) and first appliedto the surf zone by Svendsen (1984a,b).

The objective of this paper is to develop an improved model framework for waveenergy transformation in the surf zone by reducing and constraining with data thenumber of free parameters required. The principal difference from previous work is theinclusion of surface roller energy gradients in the energy flux balance, and furtherspecifying the dissipation through the shear stress at the wave/roller interface. Thewave field is considered random with waves having irregular amplitude, and approach-ing from angles over arbitrary bottom profiles with straight and parallel contours. Forthe purpose of this paper, we will only be concerned with prediction of cross-shoreenergy transformation, and not on the behavior of rollers advected out of theirimmediate generation region (i.e., the dissipation and advection terms will be describedby the same probability weightin, 0 function). In doing so we will have assumed, thatonce formed, the roller has no affect on wave height distributions. Thus all energy lossesfrom the wave/roller system will determine only the wave height transformation, andnot necessarily accurately describe the spatial distribution of surface generated turbu-lence. The spatial characteristics of rollers and associated shear stresses is investigated inLippmamr and Thornton (1996).

The surface roller model is presented in the next section. Field measurements from anaturally barred beach during the Delilah nearshore processes experiment (Duck, NC,19901, and on near-planar beaches from the National Sediment Transport Study (NSTS;Torrey Pines Beach, San Diego, 1980 and Leadbetter Beach, Santa Barbara, 1978) arethen described, and roller model generated H,, compared with observations. Resultsare discussed in terms of the sensitivity of the model to y and (T, the angle of thewave/roller interface, and comparisons are made to the bore dissipation model ofThornton and Guza (1983; henceforth TG83).

2. ModelAs waves break, turbulence is generated in a volume of water which rushes down the

front face of the wave. This volume of water has been modeled by Duncan (1981) and


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T.C. Lippmann et ~l./Coastal Engineering 27 (1996) I-20

x7/////////////////////,,,,, *

Fig. 1. Roller definition diagram taken with slight modification from Fredsoe and Deigaard (1992). The normalforce acting on the roller at the wave/roller interface is denoted f,. Other variables are described in the text.

Svendsen (1984a,b) as a detached body of fluid separated from the wave form andperched on the wave face. This body of detached turbulent, aerated water is defined asthe surface wave roller. The surface roller is believed to play an important role in surfzone dynamics including the generation of setup and wave driven currents (Svendsen,1984a,b; Roelvink and Stive, 1989; Nairn et al., 1990; Fredsoe and Deigaard, 1992;Smith et al., 1994). Svendsen envisioned a mass of turbulent water pushed forward bythe wave front with horizontal velocity equal to the wave phase speed (Fig. 1). Theshear layer at the interface between the turbulent roller and the organized wave motionresults in the local dissipation of turbulent kinetic energy (TKE) out of the roller.Because the surface roller propagates with the phase speed of the wave, roller turbulencehas been thought to be a potential source of TKE in the inner surf zone region (Basco,1985; Roelvink and Stive, 1989).

As with bore dissipation models, the roller model is approached through the energyflux balance (4). We assume stationary wave conditions, straight and parallel bottomcontours, and random waves, which are narrow-banded in both frequency and direction(i.e., the wave field can be described by a single frequency, 7, and mean deep waterwave direction, z). Wave energy is considered to be contained in two terms representingcontributions from the wave, E,, and the roller, E,,

E=E, +E,Thus (4) becomes

(6)

aE, c,cosE) + z ( E,ccosz) = - EBecause the roller is associated with depth-limited breaking, the energy of the rollertravels at the phase speed, c, of the waves.

The first term on the 1.h.s. of (7) is the energy flux gradient used by TG83, and isformulated using linear theory

1E, = ~~,vgff~where p, is the density of sea water, and the group velocity, cg, is described by

1 khcg = c5+----sinh2 kh (9)

where k is the wavenumber associated with 3 Wave propagation direction for narrow-


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banded linear waves over straight and parallel contours is approximated by Snells lawfor wave refractionc

o= sin- ( 1-since,c, (10)where cr, and c, are the initial (offshore) wave angle and phase speed.The second term on the 1.h.s. of (7) is the gradient in energy flux due to tbe presenceof wave rollers. Following the work of Svendsen (1984a,b), surface roller energy densityis derived from the kinetic energy density of the roller volume with unit crest length,KE = mu2/2 L where u is a velocity. In the roller it is assumed that u = c, so that

E,= ;P,cl; (11)where A is area of the roller, L is the wavelength of the wave, and pr is the density ofthe roller. The density of the roller will generally be less than that of the undisturbed seawater owing to entrainment of air by turbulent vortices generated at the wave surface atthe time of breaking. Longuet-Higgins and Turner (1974) estimated the maximumpercent air entrained to be at most 15-20%, in qualitative agreement with laboratoryresults of Hwung et al. (1993). Thus, the density difference between the aerated regionof the breaker and the undis~rbed water will not always be negligible. However, havingP, = pW= p is equivalent to keeping the mass of the roller constant by balancing theincrease in p by a corresponding decrease in A, a good approximation since it is themass of the roller which is the important quantity to accurately describe.The mass of the roller can be estimated using the analogy of a hydraulic jump.Engelund (1981) shows that under the assumption of fully developed rollers (i.e., theroller covers the entire face of the wave), A is approximated by

HbA=:- 4htan u (12)where CT s the angle of the wave/roller stress vector, rs (Fig. I), and H, is the heightof the wave at breaking. The value of o is a function of breaker type and is believed tovary from a maximum value when breaking is initiated to a minimum value within theinner surf zone or at the cessation of breaking @chaffer et al., 1993). In the developmentof (12), Engelund has assumed that the roller is a stationary volume of water, and so inour application the roller must be instantaneously generated at breaking and thetransition distance and time from unbroken wave to newly formed roller is ignored. Forrapidly forming rollers of a given size ~dete~ined initially by the largest vortexproduced at breaking) relative to the phase speed of the wave form, this is probably agood approximation. Using (12) and c = $!,, (11) becomes

1 HbET= -p$- htan c (13)The largest region of energy loss from the wave/roller system occurs at thewave/roller interface (Fredsoe and Deigaard, 19921, so that E = E,. Dissipation due to


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6 T.C. Lippmunn et al./Coustul Engineering 27 (1996) I-20

bottom friction is assumed negligible (a good approximation in the field shown byTG83). E, is found by consideration of the work done by the roller through the shearstress between the roller bottom boundary and the wave surface boundary

7sc,y=-r L (14)If we assume the wave/roller interface is linear, then

rs = pgAsinaas shown in Fig. 1. Using (15) and (121, (14) becomes

(15)

1 MbE, = -pgf-cosa4 h ( 16)2.1. Ensemble averaging

For random waves, TG83 and Whitford (1988) show that wave heights both insideand outside the surf zone can be reasonably described by the Rayleigh distribution,p(H),

2Hp(H) = _..&/d (17)Other distributions (e.g., Goda, 1975; Battjes and Janssen, 1978; Klopman and Stive,1989; Roelvink, 1993) or a wave-by-wave transformation (Dally, 1992) could also beused as adequate descriptors of the wave field. Since the Rayleigh distribution hasworked well for field data (TG83; Whitford, 1988; Church and Thornton, 1993; andothers), it will be the only distribution considered. In a random wave field, the ensembleaveraged wave energy, (E,), is found by integrating (8) through the Rayleigh distribu-tion

(18)The generation of turbulence and energy dissipation only apply to breaking waves,

identified by integrating through a breaking wave distribution, as in TG83. The averagerate of energy dissipation, ( F~), (as well as the average roller energy flux, (E,>> iscalculated by multiplying the dissipation for a single broken wave of height H by theprobability of wave breaking at each height, ( pb( H)) and integrating over all H. Theprobability that a wave is breaking is found by weighting the Rayleigh function for allwaves, breaking and non-breaking, by an empirical weighting function, W(H), thatdetermines statistically how many waves in the ensemble are breaking,

Pb(H) = W(H)P(H) (19)The area under the distribution p,,(H) is equal to the percent of breaking waves. Thechoice of W(H) is based on field data acquired by TG83 and Whitford (1988). The


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weighting functions were developed by visually identifying breaking waves in timeseries of sea surface elevation, and have the form

W( H) = M( 1 - e-(H/yh)*) (20)where

in TG83or

M= 1 + tanh[8( 3 - l)] inWhitford(1988)

(21)

(22)For a true weighting function in nature, W(H) I 1. However, W(H) in TG83 andWhitford (1988) both slightly exceed one at saturation, indicating that the model predictsmore than 100% of the waves to be breaking, a consequence of (5) underestimating thedissipation. Nevertheless, the wave height probability density functions observed arewell approximated by either (21) and (22) for the respective data sets (see TG83 andWhitford, 1988, for details).

To find the total roller energy flux gradient and dissipation, the roller terms in (7)must be integrated through the breaking wave distribution. All variables in the rollerterms (13 and 16) are assumed independent of H, thus the integration is simply

(23)

Assuming v is independent of x, the ensemble averaged energy flux balance is

(24)There are two free parameters in (24): y, the (constant) ratio of H,.,,,$ to local depth

defined by (21, and (T, the angle of the wave/roller interface (Fig. 1). For simplicity,and for lack of either theoretical or experimental guidance, u is assumed a constant forall breaking waves (i.e., independent of H and x>. The effect of the two forms for theweighting parameter M, (21) and (22), will be evaluated in section 4.

A simple forward stepping numerical scheme is sufficiently accurate (TG83) to solve(24)

(J5,CpJ2 +(&c,), = (dAx+ (KvCg.), + (Qr), (25)where the subscripts 1 and 2 indicate grid points in the cross-shore direction, with 1


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Table 1Wave and beach conditions

H,, Cm) fr (Hz) CY,, ) P (fore) i,, p (off) tide breaker typeSanta Barbara 4 Feb 80 0.52 0.070 19Santa Barbara 5 Feb 80 0.41 0.078 19Torrey Pines 4 Nov 78 0.35 0.063 0Torrey Pines 10 Nov 78 0.56 0.055 0Delilah 0723 lOOct90 0.77 0.093 38Delilah 0954 100ct90 0.83 0.103 40Delilah 1305 11 Ott 90 1.07 0.113 IODelilah 1750 12 act 90 1.65 0.074 16Delilah 1330 14Oct90 0.80 0.103 22

0.040 1 o 0.011 high0.052 1.3 0.011 high0.037 1.2 0.016 high0.029 1.1 0.013 high0.080 1.2 0.054 low0.080 1.1 0.054 high0.113 1.2 0.054 mid0.113 1.5 0.054 mid0.107 1.4 0.054 mid

plungeplunge/spillspillspillplunge/spillplunge/spillplunge/spillplunge/spillplunge/spill

&, defined by (261, is calculated using the foreshore slope, p (fore). Ho is the deep water rms wave height.LYEdetermined as the angle of the peak frequency, &, in 9 m depth for Santa Barbara and 8 m depth forDelilah.

being further offshore than 2, and Ax is the spacing between grid points. Due to thenonlinear nature of (25), (Hrm,12 is solved iteratively.

3. Field dataThe surface roller model is evaluated by comparing with data acquired from three

field experiments conducted on two near-planar beaches of different slope (NSTS) and abarred beach (Delilah). The experiments used extensive cross-shore arrays of bi-direc-tional current meters and wave sensors to estimate H,,,,, = @, where s* is thevariance of sea surface elevation (estimated from currents and pressures using lineardepth corrections) for the incident wave portion of the spectra, 0.05 Hz


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T.C. Lippmann et cd./ Coastal Engineering 27 (1996) I-20 9

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0

~~~~ _,

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0Cross-shore Distance (m)

Fig. 2. Beach profiles from three experiments examined: (a) Torrey Pines, NSTS, 1978, (b) Santa Barbara,NSTS, 1980, and ( c ) Duck, NC, Delilah, 1990. Cross-shore locations of wave measuring instrumentation areshown by the crosses.

The second NSTS experiment was conducted at Leadbetter Beach in Santa Barbara,California over a one month period in February of 1980. Bottom contours inside 6 mdepth were nearly straight and parallel. Beach profiles from the days analyzed are shownin Fig. 2b. The beach face was concave, with mean nearshore slope varying between1 : 33 at low tide to 1 : 16 at high tide. A cross-shore array of 16 current meters and 6pressure sensors measured the cross-shore wave transformation properties. Two waveslope arrays offshore measured incident wave spectra. To reach Santa Barbara, theopen-ocean North Pacific swell must pass between Point Conception and the ChannelIslands, a narrow window of f9, making the wave field approximately unidirectional.Data are examined on two narrow-banded, stationary wave days (4-5 February).

The third data set was acquired on a barred beach during the Delilah nearshoreprocesses experiment held in October 1990 at the US Army Corps of Engineers FieldResearch Facility at Duck, North Carolina. Beach profiles are shown in Fig. 2c. Both barlocation and foreshore slope, which varied between 1 : 10 and 1 : 15, responded to a widevariety of wave conditions including two storms which had substantial effects on thebathymetry. The mean offshore slope of 1 : 150 was not substantially modified by thewave climate. Directional wave spectra were obtained from an alongshore array ofbottom mounted pressure sensors in 8 m depth. A cross-shore array of 9 pressure sensorswas located from 4.5 m depth to the shoreline. Four days (Oct. 10, 11, 12, and 14) arechosen for model comparison when waves were relatively narrow-banded, 7= 0.113-


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0.074 Hz, Ho = 0.77-1.65 m, and arriving at angles between 16 and 40 from normal tothe beach.

4. ResultsExample model results for each experiment are presented in Figs. 3-5. As in

Roelvink (1993), isolines of percent error between model and data are shown for theexpected parameter range. Results are shown for the roller model (Eq. 24) and TG83 forW(H) given by both (21) and (22). Free parameters plotted against y are B for TG83and CT for the roller model. Results show similar behavior for all data runs, although theexact parameter range may vary. The roller model is found to be insensitive to V, withisolines of constant error approximately flat across a range of a> 5-10. The modelalso is insensitive to the choice of M. The model is, however, sensitive to y, allowingonly a narrow range of values which give errors less than about 10%. Best fitparameters, discussed below, are given in Table 2.

On the other hand, although B and y can be chosen to give excellent fit to the data(with errors comparable to the roller model), the TG83 model is sensitive to the choiceof both y and B, in which a wide range of y-B pairs yield similar results. In order to

Torrev Pines 10 Nov 78Roller Model * TG83 W(H):

Q. (21)

BFig. 3. lsolines of percent errors between H,, model predictions and observations for 10 November duringthe Torrey Pines experiment. The left hand panels are from the roller model (Eq. 24) and the right hand panelsare from the bore model of TG83. The panels on the top are model runs with W(H) given by M from Eq.(211, and on the bottom for M given by (22). The ordinate is y in all plots. The abscissa is B for the TG83model, and (T (in ) for the roller model.


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Santa Barbara 4 Feb 80

11

Roller Model TGS3 W(H):

Eq. (21)

Eq. (22)

Fig. 4. Same as Fig. 3 for 4 February during the Santa Barbara experiment.

choose a reasonable value of y for the TG83 model which is comparable with the rollermodel, we force B to unity so that the entire face of the wave is conceived to be coveredwith foam, consistent with Engelunds (1981) derivation of the roller area in which theroller covers the entire face of the wave. With B = 1, y is then chosen uniquely to yieldminimum percent error. In this way we forcibly reduce the sensitivity of the TG83model to one parameter.

H rms predictions as a function of cross-shore distance from the TG83 and rollermodels are shown for best fits to the data in Figs. 6-8. Results are shown for W(H)with M given by (21) and (22). The roller model well predicts the transformation ofH rms for both the near planar NSTS data and the barred Delilah profile. The TG83 boremodel also fits the data well. Since we have chosen y in TG83 by fixing B, and sincethe roller model is insensitive to CT, the behavior of the transformation profile dependson y and W( H >. The results show that (21) and (22) do about an equally good job in thesurf zone, suggesting that both models are generally insensitive to the choice of M (it isnoted that Whitford, 1988, found that (22) has better properties than (21) in intermediatewater depth). Thus the transformation of Hms in the surf zone can be adequatelyapproximated using either (21) or (221, as long as y is known.

The models are sensitive to y, with percent errors ranging 30% over the expectedrange predicted by (3): y = 0.3-0.6 for tan p = 0.01-0.10. A plot of H,.,,,, versus h is


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Roller Model DEL.ILAH 0723 10 Ott 90 TG83 W(H):

Eq. (21)

Eq. (22)

Fig. 5. Same as Fig. 3 for 10 October, at 0723 hrs, during the Delilah experiment.

shown in Fig. 9 for the NSTS near-planar beaches examined earlier (the Delilah data arenot included since the beach profiles are not monotonic). Also plotted are the range of yvalues (Table 2) found by fitting model H,, to the data. Each model yields y values

Table 2Model parameter resultsModel: TG83 RollerWeighting function: Eq. 21 Eq. 22 Eq. 21 Eq. 22

B Y %err y %err (T (1 y %err y %errSanta Barbara 4 Feb 80 1.0 0.35 6.2 0.35 6.9 12.5 0.32 6.5 0.32 7.8Santa Barbara 5 Feb 80 1.0 0.32 5.6 0.34 4.4 10.5 0.30 6.2 0.3 1 5.2Torrey Pines 4 Nov 78 I.0 0.24 6.5 0.25 6.3 26.5 0.22 8.1 0.24 7.7Torrey Pines 10 Nov 78 1.0 0.28 6.0 0.29 5.6 17.5 0.26 6.5 0.26 5.6Delilah 0723 lOOct90 1.0 0.39 4.3 0.38 5.4 10.5 0.33 4.9 0.35 3.2Delilah 0954 lOOct90 1.0 0.41 2.7 0.37 5.8 5.5 0.37 2.5 0.34 2.2Delilah 1305 11 act 90 1.0 0.42 10.1 0.36 9.0 7.5 0.38 10.0 0.32 8.2Delilah 1750 12Oct90 1.0 0.43 8.1 0.39 9.7 10.5 0.38 8.8 0.37 6.7Delilah 1330 14Oct90 1.0 0.42 4.3 0.38 5.4 9.5 0.38 4.9 0.34 3.4

Mean 6.0 6.5 6.5 5.6y values in the TG83 model are best tits for B = 1.0 in all cases


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T.C. Lippmann et al./ Coastal Engineering 27 (1996) I-20 13Tomy Pines IO Nov 78

0.75 - EA = =. _.x --_ - _&&Lm eq. 22* - --__.5-

2X */xf, .

60 50 100 150 200 250 300

0.75- *,,a-= - ----x_ _.

s *_ _ _~~H))?rcm eq. 21

g 0.5----____

PY

0.25- ,fI

I0 0 50 100 150 2w 250 300

-0 50 100 150 200 250 300Cross-shore Distance (m)

Fig. 6. Model predictions of Hms as a function of cross-shore distance for best fit values to observations (*)from 10 November during the Torrey Pines expeximent. Results are shown for the roller model (dashed line)and TG83 (dash-dot line). The middle and upper panels are results using W(H) from Eqs. (21) and (22),respectively. The beach profile is shown in the lower panel.

which are not inconsistent with the data; however, with the scatter in the data it is notpossible to resolve the differences.

5. Discussion

Dissipation rate and turbulent penetration depth are dictated by breaker characteris-tics, observed generally as spilling or plunging for the data examined (Table 1). Whileplunging wave turbulence can penetrate to the bottom, spilling breaker turbulence isprimarily confined to the surface layer between the crest and trough. One commonparameter which grossly distinguishes between plunging, spilling, and other generic


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14 T.C. Lippmmn et al./ Coastal Engineering 27 (1996) l-20Santa Barbara 04 Feb 80

* fl--*-;__X/ W(H) from eq. 22~----__

+x0.25 /I

o-0 25 50 75

-

-

100 125 150 175 200

-- ---__ W(H) from eq. 21

100 125 150 175 200

-1010 25 50 75 100 125 150 175 200Cross-shore Distance (m)

Fig. 7. Same as Fig. 6 for 4 February during the Santa Barbara experiment.

classes of breaker types, and which is widely believed to be relevantprocesses is the Iribarren number,

tan B5o Ho/Lo) 2to nearshore

(26)where Ho is the deep water wave height, L, is the deep water wave length, and tan /3 isthe beach slope (Bowen et al., 1968; Battjes, 1974). Early conceptual models relatedwave breaking to the Iribarren number, which is in general agreement with observationsof wave breaking on natural beaches (Galvin, 1968), and is qualitatively related to thesteepness of the front face of the breaker or bore.

In nature, the breaker steepness would be expected to evolve across the surf zone aswaves propagate shoreward. However, in the derivation of the roller model, we haveassumed that the angle of the wave/roller interface, U, is constant. Thus, we have notallowed the shape of the breaking wave to evolve across the surf zone. This is probablynot a realistic assumption, particularly for barred beaches where waves are substantially


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T.C. Lippm ann et al./Coast al Engineering 27 (1996) I-20 15

EBX

DELILAH IO Ott 90 0723 Estl-

X7 - - --._ W(H) fro m eq. 220.75 - r *

- -~---- - --- - _-__

3y 0.5.,-s__ary

0.25 - ;I

l-* W(H) from eq. 21

0.75 - , _-_- _* -_-_- = 1 g_ -_-_- z = -_---i > _ _ -__3

X,I,I

j o5&,c--*-__-1 /

0.25- ,I

0%0 50 100 150 200 250 300

100 150 200 250 300Cross-shore Distance (m)

Fig. 8. Same as Fig. 6 for 10 October, at 0723 hrs, during the Delilah experiment.

A. Santa Barbara 4 Feb 80 B. Torrey Pines 10 Nov 78l-

0.90.6 y= 0.350.7. 0

y=o.30 o

2.5 3 3.5 4Depth Cm)

10.90.60.70.60.50.40.30.20.1

00 1 2 3 4Depth

00

k-d6 7 6

Fig. 9. Plot of Hm s VS. h for (A) 4 February, Santa Barbara, and (B) IO November, Torrey Pines. Also shownare the range of y found from model fitting (Table 2).


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modified in the trough of the bar. The behavior of CT is not currently constrained byfield measurements. The model is insensitive to the choice of IT greater than about5-lo, suggesting that although breaker characteristics are not necessarily modeledcorrectly, H,, transformation is adequately represented empirically.In Engelunds (1981) derivation of the roller area, it is assumed that the horizontallength of the roller covers the entire front face of the wave. This is equivalent tochoosing B = 1 (as we have done in the previous section) in the bore dissipationfunction of LeMehaute (1962). With B constant across the surf zone, the Engelundmodel does not consider the possibility of roller length evolution after breaking has beeninitiated. The dissipation of roller energy results in a decrease in thickness of the rollerwithout changing the horizontal length of the roller along the face of the wave, and isdependent on the breaker height to depth ratio and the choice of W(H). The modifica-tion to the roller thickness compensates by keeping the length of the roller constant.Thus, the approximation to the dissipation function of Svendsen (1984a, Svendsen(1984b), which depends on only the roller area, does not effect the H,, profiles.

As pointed out by Roelvink and Stive (1989) and Nairn et al. (19901, there is a slightlandward transition in dissipation due to the roller gradient term in the energy fluxbalance. This distance was referred to by Nairn, et al. as the transition zone width, whosuggest that the transition zone substantially modifies the distribution of longshorecurrents and set-up profile. Although we assume instantaneous genesis of the rollervolume at breaking, a transition zone arises in the model because the roller and thedissipation each depend on the same distribution function, making the roller gradientnon zero where there is dissipation, and forcing the transition zone width to bedependent on the choice of W( H ). Our results (Table 2) show that roller y values areslightly less than for the bore model. In effect, smaller y compensates for the transitionregion by causing breaking to be initiated further offshore. The transition zone isnecessarily small in the roller model because dissipation occurs very rapidly near thepoint of breaking; thus, we have not allowed the rollers to be advected away from theirgeneration region. This is also the case for Nairn et al. (1990).

Since the distance of the transition zone is small, the distribution of H,, is notsignificantly effected. Thus, because radiation stress forcing functions are describedthrough gradients in wave energy flux, the spatial modification to this forcing mecha-nism should also be small. However, the spatial distribution of the surface shear stressesinduced by rollers, not considered in this work, can be important in the momentumbalance. The role of rollers depends on the horizontal distances that rollers are advectedoutside their generation region (i.e., Smith et al., 1994; Southgate and Wallace, 1994;Lippmann and Thornton, 1996; and others).

In TG83, for the approximately planar Torrey Pines beach, y = 0.42 was determinedfrom the data by plotting observations of H,, against local depth (as in Fig. 9>, andthen fitting the transformation model to the cross-shore distribution of the data byadjusting the B parameter. However, the a priori determination of y is not alwaysgenerally possible for all beaches, particularly those with substantial longshore bars thatcause breaking to be interrupted in the greater depths of the trough. In the present work,the roller model is not sensitive to (T, thus y becomes the unknown fitting parameter.For the near-planar NSTS beaches the (free) model fit for y is consistent with the


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T.C. Lippm ann et al./ Coasta l Engineering 27 (1996) I-20 17

OB-0.7.0.6.

?- 0.5.aB 0.4.E 0.3

0.2j A:0 0.5 1 1.5 2

j B.0 0.05 0.1 0.15

it c. ,0 0.002 o.oc 4 0.006 0.006 0.01

Iribarren Number Foreshore Slope Wave SteepnessFig. 10. Plots of model y as a function of (A) [,,, Eq. 26, (B) foreshore beach slope, and (C) deep water wavesteepness, Ho /L,. Data points in the plots are 0 for Santa Barbara, * for Torrey Pines, and x for Delilah.definition given by Eq. 2 (Fig. 9). Although for planar beaches y can be approximatedby the data in this manner, slight errors in its approximation can lead to substantialerrors in H,,,,, prediction, thus it is important to constrain y as much as possible. The apriori specification of y requires further investigation of the incipient wave breakingprocess.

TG83 conclude that the periodic bore model of Stoker (1957) underestimatesdissipation, for which the B parameter partially compensates. They also conclude that Band y could be combined into one coefficient, but are left separate for greater physicalinsight. Cacina (1989) combined B and y as

B3B= 2Y (27)

and iterated TG83 over the B parameter to improve agreement with NSTS observa-tions. Cacina (1989) concluded that B is correlated with the &, described by (26). Sincewe have forced B to unity, it should follow that the roller models single sensitive freeparameter, y. should also be a function of I&,. Results of varying y with 6, foreshoreslope, and deep water wave steepness (HO/L,) are shown in Fig. 10.

A plot of y vs. l, does not reveal any significant trends, except to show that yvalues (best fits to the models) are slightly larger for the barred Delilah data. The data(Fig. 10A) are clustered around 5, between 1 O and 1.3. The limited and narrow rangeof 5, are inadequate to examine its relation to y, a dependence predicted by Battjes(1974). Furthermore, in computing 5, we have used the foreshore slope because it ismeasurable and well approximated by a linear fit. The validity of using this p torepresent the barred beach profiles (Fig. 2) is not known.

A weaker relationship is found between y and p (Fig. IOB) than found by Sallengerand Holman (1985; Eq. 3). Sallenger and Holman used y obtained by fitting to plots ofH vs. h, as in Fig. 9. The values from model predictions (Table 2) all are less thanthzpredicted by (31, and have an approximate dependence of p.4. We have used theforeshore slope to represent the profile, whereas Sallenger and Holman used a slope fit


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18 T.C. Lippmann et al./Coustal Engineering 27 (1996) I-20

to the entire width of the surf zone, generally much less than the steeper beach face.Additionally, there is a weak dependence of y on deep water wave steepness (Fig. lOC),the denominator in & (Eq. 1). The similar dependencies of beach slope and wavesteepness combine to limit the range of &, for these data.

6. ConclusionsA wave energy transformation model, with wave breaking described by surface

rollers, is developed for randomly varying waves over an arbitrary bottom profile withstraight and parallel contours. The model includes roller energy gradients in thecross-shore energy flux balance, and further specifies the dissipation by the shear at thewave/roller interface following Svendsen (1984a,b). For a narrow-banded wave fielddescribed by a single frequency and wave angle, rms wave height, H,,, is found acrossthe surf zone by numerical iteration. Predictions are compared with field data acquiredon two near-planar and one naturally barred profile. H,, is well modeled on allprofiles, with percent errors for best fit parameters in the range 3-10%.

The cross-shore variation of the average surface shear stress, which depends on theevolution of the roller volume, is determined by the spatial distribution of wavebreaking, specified by an empirical weighting function based on field observations(Thornton and Guza, 1983; Whitford, 1988). The same weighting function is used apriori to describe roller gradients and the dissipation. Thus, we have assumed that onceformed, the roller has no affect on wave height distribution. As a consequence, rollersare not allowed to be advected out of their generation region. The spatial characteristicsof advected rollers and the associated production of turbulence at the wave surface is thesubject of ongoing research (Lippmann and Thornton, 1996).

The model depends on the angle of the wave/roller interface, (T, assumed invariantacross the profile, and a constant y, the limiting ratio of Hm s to local depth a measureof wave saturation. The roller model is found to be insensitive to the choice ofinter-facial angle ((T > 5-lo), but is moderately sensitive to saturation, with a 10%increase in error for y + 0.1. The model is also found to be not strongly dependent onthe choice of the weighted breaking distribution, with similar model performance for thetwo functions examined.

Roller model results are comparable to the bore dissipation model of Thornton andGuza (1983) with their parameter B constrained to unity, where B is an empiricalparameter roughly representing the extent of the breaker along the front face of thewave. The inclusion of roller gradients reduces the requirement of two free parametersin the previous bore model (B and y) to only one (7). Best fit y are consistent with theobserved saturation of energy in the inner surf zone.

AcknowledgementsThis research was sponsored by the Office of Naval Research, Coastal Sciences

Program and Naval Ocean Modeling Program, under contracts NO01 14-95AF-0002 and


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NO01 14-95WR-30021. TCL was supported by a National Academy of Sciences NRCPostdoctoral Fellowship. Chuck Long of the FRF provided directional spectra informa-tion for Delilah. The insightful comments of the anonymous i,eviewers greatly improvedthe paper. We are indebted to the many people who participated in the collection of thedata for the NSTS and Delilah experiments.

References

Basco, D.R., 1985. A qualitative description of wave breaking. J. Waterw. Port Coastal Ocean Eng. AXE,1I l(2): 171-188.

Battjes, J.A., 1974. Surf similarity. In: Proceedings of the 14th International Conference Coastal Engineering,Copenhagen. American Society of Civil Engineers, New York, Vol. 1, pp.466-480.Battjes. J.A. and Janssen, J.P.F.M., 1978. Energy loss and set-up due to breaking of random waves. In:Proceedings of the 16th International Conference Coastal Engineering, Hamburg. American Society ofCivil Engineers, New York, pp. 569-587.

Bowen, A.J., Inman, D.L. and Simmons, V.P., 1968. Wave set-down and setup. J. Geophys. Res., 73:2569-2577.

Cacina, N., 1989. Test and evaluation of surf forecasting model. M.Sc. Thesis, Naval Postgraduate School,Monterey, CA, 41 pp.

Church, C. and Thornton, E.B., 1993. Effects of breaking wave induced turbulence within a longshore currentmodel. Coastal Eng., 20: l-28.

Dally, W.R., Dean, R.G. and Dakymple, R.A., 1985. Wave height variation across beaches of arbitrary profile.J. Geophys. Res., 9OfC6): 11,917- 11,927.

Dally, W.R., 1990. Random breaking waves: a closed-form solution for planar beaches. Coastal Eng., 14:233-263.Dally, W.R., 1992. Random breaking waves: field verification of a wave-by-wave algorithm for engineering

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R. Sot. London, A377: 33 l-348.Engelund, F., 1981. A simple theory of weak hydraulic jumps. Progress Report No. 54, Institute of

Hydrodynamics and Hydraulic Engineering, ISVA, Technical University Denmark, pp. 29-32.Fredsoe, J. and Deigaard, R., 1992. Mechanics of Coastal Sediment Transport. Advanced Series on Ocean

Engineering, Vol. 3. World Scientific, Singapore, 369 pp.GaIvin, C.J. and Eagleson, P.S., 1965. Experimental study of longshore currents on a plane beach. Tech.

Memo. 10, U.S. Army Coastal Eng. Res. Center, Fort Belvoir, VA.GaIvin, C.J., 1968. Breaker type classification in three laboratory beaches. J. Geophys. Res., 73(12):3651-3659.Gcda, Y., 1975. Irregular wave deformation in the surf zone. Coastal Eng. Jpn., 18: 13-26.Guza, R.T. and Thornton, E.B., 1980. Local and shoaled comparisons of sea surface elevation, pressures and

velocities. J. Geophys. Res., 85(C3): 1524-1530.Hwung, H.H., Chyan, J.M. and Chung, Y.C., 1993. Energy dissipation and air bubbles mixing inside surf

zone. In: Proceedings of the 23nd International Conference on Coastal Engineering, Venice. AmericanSociety of Civil Engineers, New York, pp. 308-321.

Klopman, G. and Stive, M.J.F., 1989. Extreme waves and wave loading in shallow water. Paper presented atE&P Forum Workshop Wave and current kinematics and loading, Paris.

LeMehaute, B., 1962. On non-saturated brealcers and the wave mn-up. In: Proceedings of the 8th InternationalConference on Coastal Engineering, Mexico City. American Society of Civil Engineering, New York, pp.77-92.Lippmami, T.C. and Thornton, E.B., 1996. The spatial distribution of wave rollers and turbulent kinetic energyon a barred beach. J. Geophys. Res., in press.


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