Wave transformation in the nearshore zone: comparison between a Boussinesq model and field data

Coastal Engineering 39 Ž2000. 149–171 www.elsevier.comrlocatercoastaleng

Wave transformation in the nearshore zone: comparison between a Boussinesq model and field data Atilla Bayram ) , Magnus Larson Department of Water Resources Engineering, UniÕersity of Lund, Box 118, S-221 00 Lund, Sweden Received 23 February 1999; received in revised form 4 August 1999; accepted 28 September 1999

Abstract A numerical model based on the Boussinesq equations was developed to simulate wave transformation in the nearshore zone with emphasis on describing the effects of wave breaking. Two different formulations for estimating additional momentum due to depth-limited wave breaking, namely one proposed by Watanabe and Dibajnia wWatanabe, A., Dibajnia, M., 1988. A numerical model of wave deformation in the surf zone. Proceedings of the 21st Coastal Engineering Conference, ASCE, pp. 578–587x and another one by Schaffer et al. wSchaffer, H.A., ¨ ¨ Madsen, P.A., Deigaard, R., 1993. A Boussinesq model for waves breaking in shallow water. Coastal Engineering 20, 185–202x, were incorporated and tested in the numerical wave model. Detailed, high-quality data from Duck, NC, collected during the DUCK85 and SUPERDUCK field experiments using photographic means were employed to evaluate the applicability of the Boussinesq model to surf-zone conditions. Also, comparison with the field data allowed for an assessment of how well the two formulations could describe the effects of wave breaking on the evolution of the waveform. Model simulations were compared with measured time series of water surface elevation as well as with derived statistical quantities, such as the skewness, kurtosis and root-mean-square wave height. Overall, the model results were satisfactory, although the Boussinesq model failed to accurately reproduce the strong nonlinear shoaling that occurred prior to breaking. The roller model developed by Schaffer et al. produced somewhat better agreement with ¨ the data than the model proposed by Watanabe and Dibajnia. However, the difference was small and both formulations are judged to yield acceptable results, although improvements are needed

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Corresponding author. Fax: q46-46-222-44-35; e-mail: [email protected]

0378-3839r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 3 9 Ž 9 9 . 0 0 0 5 6 - 3

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before the predictions can be used for calculating the sediment transport rate in intra-wave models. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Boussinesq equations; Surf zone; Breaking waves; Energy dissipation; Shelf profile; Barred profile; Field data

1. Introduction Modelling sediment transport and beach topography change in the nearshore zone requires accurate predictions of waves and currents. The requirements on the hydrodynamic calculations are directly related to the scales of beach response that the sediment transport model aims at resolving. For example, many of the existing numerical models of beach profile change employ time-averaged statistics, such as root-mean-square Žrms. wave height and velocity moments, to calculate the net sediment transport representative over many wave periods. However, as attempts are made to describe intra-wave sediment transport and bed evolution, the need for robust and reliable models that will predict hydrodynamic quantities at these scales is evident. Since a significant part of the sediment transport takes place in the surf zone, it is necessary to develop models that describe shallow-water wave transformation including the effects of wave breaking. In recent years many studies have focused on modelling wave transformation in shallow water and especially approaches involving the Boussinesq equations have shown promise in producing reliable predictions of water surface elevations and velocities, although mainly for non-breaking waves ŽNwogu, 1993.. A few models have been developed to take into account wave breaking in the Boussinesq equations, but the calculation results have primarily been compared to small-scale laboratory ŽSchaffer et al., 1993.. However, Ozanne et al. Ž1997. used a one-dimen¨ sional time domain Boussinesq-type model and applied it to large-scale laboratory data. Water surface elevation and depth-averaged velocity computed by the model were compared with measurements from the shoaling and surf zone obtained in the SUPERTANK experiment ŽKraus and Smith, 1994.. They found that the computations agreed reasonably well with the measurements for regular waves, whereas noticeable discrepancies occurred for irregular waves. Before wave transformation models based on the Boussinesq equations are employed for predictions of hydrodynamic quantities in the surf zone, they should be validated against field data, especially evaluating the models developed to describe the effects of wave breaking. Also, the generality of parameter values in the models describing wave breaking should be confirmed since these values were determined from laboratory studies. In summary, for engineering applications, it is desirable to have the validity of these models investigated using field data in order to establish confidence in their capability to simulate conditions on natural beaches. The main objective of this study was to investigate the possibility of simulating wave transformation in shallow water using the Boussinesq equations with special emphasis on depth-limited wave breaking. Predictions made with a numerical model based on the Boussinesq equations were compared with field measurements from the U.S. Army

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Field Research Facility in Duck, NC. The data employed encompassed the time history of the water surface elevation recorded at selected locations across the surf zone and for several cases using photographic techniques. Two different formulations Ži.e., models. proposed in the literature for including wave breaking in the Boussinesq equations ŽWatanabe and Dibajnia, 1988; Schaffer et al., 1993. were evaluated and compared ¨ using a number of these cases, which included different profile shapes and incident wave conditions. In the following, a brief theoretical description is first given of the Boussinesq equations and the two models employed for describing the effects of wave breaking. Then, the field data is discussed and the cases selected for the model testing summarised. Detailed comparisons are made between calculated and measured water surface elevations at selected locations through the surf zone and for several different cases using the two models of wave breaking. Furthermore, comparisons are made for statistical wave quantities such as rms wave height Ž Hrms ., significant wave height Ž H1r3 ., the mean of 1r10 highest waves Ž H1r10 ., and skewness Ž zs . and kurtosis Ž z k . of water surface elevation. The paper ends with some overall conclusions regarding the applicability of Boussinesq models to simulate wave transformation in the surf zone and how well the two different models of wave breaking reproduced the field data.

2. Wave transformation model 2.1. Boussinesq equations The numerical wave transformation model developed in this study is based on the Boussinesq equations, with improved linear dispersion characteristics as documented in Madsen et al. Ž1991. and Madsen and Sørensen Ž1992.. The governing equations Žmass and momentum conservation, respectively. for the one-dimensional case may be written,

ht q Px s 0 Pt q

Ž 1.

P2

ž / D

q gDhx y Bgh3hx x x y B q x

q M D ,b s 0

ž

1 3

/

h 2 Px x t y h x 2 Bgh2hx x q

ž

1 3

hPx t

/ Ž 2.

where h is the instantaneous water surface elevation, P the depth-integrated velocity component Žflux. in the wave propagation direction x, D the instantaneous water depth, h the still water depth, g the acceleration of gravity, M D,b a momentum correction term due to wave breaking Žto be discussed in detail later., and the subscripts x and t denote differentiation with respect to space and time. The parameter B, called dispersion parameter, is a coefficient with a value that is selected to improve the linear dispersion characteristics of the model. For example, a value of B s 1r15 corresponds to the

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frequency dispersion obtained from a Pade´ w2,2x expansion of the Stokes linear dispersion relationship ŽMadsen and Sørensen, 1992.. The most important element in time domain models of surf zone hydrodynamics is the formulation of energy dissipation Žimplying an additional term in the momentum equation. due to wave breaking. This formulation should be made so that wave motion on scales smaller than the wavelength and period is resolved. Several corrections to the momentum equation have been proposed to simulate the effects of wave breaking in the surf zone at these scales during the last decade Že.g., Watanabe and Dibajnia, 1988; Karambas and Koutitas, 1992; Schaffer et al., 1993.. In the following sections, two ¨ different dissipation models proposed in the literature that are theoretically well-founded and have demonstrated predictive capabilities are briefly reviewed. These models will be employed later to simulate wave transformation in the surf zone. 2.2. Watanabe and Dibajnia model for breaking waÕes Watanabe and Dibajnia Ž1988. presented a numerical model of wave transformation based on linear wave theory. The model employed a set of time-dependent mass and momentum conservation equations equivalent to the mild-slope equations. In their model, energy dissipation due to wave breaking was described through a force obtained from the flow resistance relative to the wave orbital velocity. Calculating the energy dissipation with this model involves two steps, where the first step is to determine the location of the break point. Watanabe et al. Ž1984. reanalysed the data collected by Goda Ž1970. in terms of the horizontal water particle velocity amplitude and the wave celerity Žinstead of wave height and water depth at breaking, as was done by Goda.. They defined the criterion for incipient breaking from the ratio between water particle velocity Ž u p ., based on small-amplitude wave theory, and wave celerity Ž c .. The criterion, which was originally given in graphic form, may be approximated by the following formula ŽIsobe, 1987.,

gbs

up

ž / c

ž ( /

s 0.53 y 0.3exp y3 b

db

Lo

q 5 tan

5r2

½ ž(

b y45

db Lo

2

y 0.1

/5 Ž 3.

where d b is the water depth at incipient breaking, Lo the deep-water wavelength, and b the beach slope Žtypically determined as an average over some distance seaward of the break point.. Eq. Ž3. was derived based on linear theory and should mainly be applicable in such wave models. However, this equation was still employed in the calculations given in the paper because of the weakly nonlinear characteristics of the Boussinesq formulation used. Computations were also made where the critical value for incipient breaking was increased by a factor of 1.5–1.7 in order to take into account wave nonlinearity, as suggested by Watanabe Žpersonal communication.. Further discussion on the effect of this modification of Eq. Ž3. is presented in Section 4. The second step involves determining how the momentum loss develops as the breaking wave propagates onshore. The functional form of M D,b used in the wave

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transformation computations carried out by Watanabe and Dibajnia Ž1988., which produced results in good agreement with laboratory data, is,

)

M D ,b s yf D Q s ya D tan b

g Qˆ y Q r d Qs y Q r

Q

Ž 4.

where f D is an energy dissipation coefficient, Q the flow per unit width, a D a coefficient Ž2.5 inside the surf zone and zero elsewhere., tan b the bottom slope, d the mean total depth, Qˆ the flow amplitude, Qs the wave-induced flow inside the surf zone, and Q r the flow amplitude of reformed waves. Based on experimental data from Maruyama and Shimuzu Ž1986. and Isobe Ž1987., Qs and Q r can be estimated as:

(

Qs s 0.4 Ž 0.57 q 5.3 tan b . gd 3

(

Qr s 0.135 gd 3 .

Ž 5. Ž 6.

Wave breaking, however, involves strong turbulence and momentum mixing, particularly at the front face of the wave. The energy dissipation is a direct result of the diffusion of momentum in the surf zone and it is therefore necessary to use a modified form of Eq. Ž4. to describe M D,b in order to better describe these phenomena. Sato et al. Ž1992. presented a model where the energy dissipation was proportional to the diffusion of momentum. The following formulation for M D,b proposed by Sato et al. Ž1992. employs an eddy viscosity n b to describe the momentum exchange due to turbulence: M D ,b s n b

E2 Q

Ž 7.

Ex2

An expression for n b may be deduced from Eq. Ž4. by assuming a sinusoidally varying Q, yielding the following expression for n b Žvalid for long waves.,

nbs

a D gdtan b s2

)

g Qˆ y Qr d Qs y Q r

Ž 8.

where s is angular wave frequency Žs 2prT, where T is the wave period.. The form of the n b in Eq. Ž8. is consistent with Eq. Ž4., and Sato et al. Ž1992. confirmed that the momentum-mixing model given by Eqs. Ž7. and Ž8. is capable of simulating the tilting of the wave profile that the waves experience when they propagate into shallow water. The extension of the model to irregular wave conditions was done by evaluating the eddy viscosity on a wave-by-wave basis ŽSato et al., 1992.. In the numerical calculations the instantaneous water surface profile was divided into individual waves using a zero-up crossing method Ždone at every time step.. Wave breaking was checked for each individual wave using Eq. Ž3., and for each identified breaking wave the dissipation factor f D,ir was estimated as,

)

f D ,ir s a r a D tan b

g Qˆ y Q r d Qs y Q r

Ž 9.

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where a r is constant determined through comparison with data. Sato et al. Ž1992. suggested a r s 0.5 based on laboratory experiments and for non-breaking waves, f D,ir is set to zero. The spatial distribution of the energy dissipation coefficient f D was estimated through linear interpolation of f D,ir . In the following, when the above-described model of wave breaking is employed here it will be referred to as the Watanabe–Dibajnia ŽW–D. model, although Sato et al. Ž1992. developed the final form. 2.3. Roller model (Schaffer ¨ et al., 1993) for breaking waÕes Svendsen Ž1984. was a pioneer in introducing the surface roller concept and employed it in time-averaged wave transformation modeling. A simplified velocity field was hypothesized for the breaking wave resulting in additional terms in the depth-integrated momentum equation. Deigaard Ž1989. developed a breaking criterion based on the roller concept and a technique for geometrically determining the shape of the roller in the surf zone. The initial version of the roller model introduced the effect of the surface roller as an additional force in the momentum equation assuming a hydrostatic pressure distribution in the roller. Thus, the impact from the rollers modified the wave motion and extracted energy from the waves. The first comprehensive implementation and testing of the roller model, including the definition and calibration of the main empirical parameters, were made by Schaffer et al. Ž1993.. ¨ In the model by Schaffer et al. Ž1993., wave breaking was incorporated in the ¨ depth-integrated momentum equation through an additional convective term resulting from the non-uniform velocity profile. The roller, considered to be a passive mass of water carried by the wave, is defined as the water above a line given by tan f and breaking is initiated if tan f exceeds tan f B , where f B is the maximum breaking angle. Locally, breaking is assumed to cease when the maximum angle becomes less than tan f B and exponentially change to a smaller terminal angle fo . The time variation of f is described by,

ž

tan f Ž t . s tan fo q Ž tan f B y tan fo . exp yln2

t y tB t 1r2

/

Ž 10 .

where t 1r2 defines the time scale for the development of the roller, thus, controlling the rate of decay of f , and the parameter t B is the time when breaking starts. Once a roller has been identified at a certain time step, the roller thickness Ž d . is multiplied by a shape factor Ž fd . in order to compensate for the simple approach of separating the roller from the rest of the flow. The contribution to the momentum flux from the rollers is defined by Schaffer et al. ¨ Ž1993. as, E M D ,b s

Ex

ž

d cy

P D

2

/ž

d 1y

D

y1

/

Ž 11 .

where c is the roller celerity. As a first approximation, Schaffer et al. Ž1993. suggested ¨ wave celerity equal to 1.3 times the linear shallow-water wave celerity Žusing the local water depth.. This is in agreement with empirical observations of the breaking wave

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celerity presented in Stive Ž1984.. Schaffer et al. Ž1993. evaluated the roller model for ¨ predicting the evolution of the wave height during shoaling and breaking, and compared the calculated low-order wave statistics Žwave height, mean water level, and standard deviation of the water surface. with data from small-scale laboratory experiments. The agreements were good in general but the wave height was underestimated in the region around the break point. In the first of series of papers on modelling various types of surf zone phenomena, Madsen et al. Ž1997a. extended the time domain formulation by Schaffer et al. Ž1993. to ¨ two horizontal dimensions. Their model included improved accuracy of linear dispersion and nonlinear energy transfer and a moving boundary at the shoreline. In a companion paper, the same model was used for investigating high-order statistics and surf beat due to bichromatic and irregular waves ŽMadsen et al., 1997b.. Wave-induced circulation was studied in the third paper by Sørensen et al. Ž1998b.. However, it was clear that these models underestimated the nonlinear shoaling and higher order statistics near the break point. In order to overcome these shortcomings of Boussinesq models, Sørensen et al. Ž1998a. included higher-order nonlinearity in the dispersive terms as well as a more accurate linear dispersion relationship. Their modelling results showed that the description of the nonlinear shoaling and higher-order statistics improved significantly. 2.4. Numerical solution In the numerical computations, the continuity equation was discretized based on the variables h and P Žsee Eq. Ž1... The solution technique basically followed Peregrine Ž1967., however, in the treatment of the continuity equation the conservation form was used rather than expanding the volume flux term. Initially, an explicit formulation of the continuity equation is used to obtain a first estimate of the water surface elevations at the next time level. Then, this estimate of the elevations is used in an implicit discretization of the momentum equation to compute the depth-integrated velocity component Ž P . in the wave propagation direction at the next time level. Central difference formulations are used both for the time and space derivatives, which are computed at Ž i, j q 1r2., where the indices i and j denote the discretization in space and time, respectively. Finally, using the estimated values for P at the next time level the initial water surface elevations are improved through a set of new elevation estimates from the continuity equation. The numerical solution scheme employs a double-sweep algorithm, which is the most efficient way of solving a tri-diagonal matrix system as long as the diagonal elements are dominant. The initial condition used in all cases presented here was the undisturbed state; that is, at the beginning of the calculations the surface elevation and the depth-integrated velocity were set to zero throughout the computational domain. At the seaward boundary the measured surface elevation was specified as input at the first node and the depth-integrated velocity was calculated from linear wave theory. The onshore boundary condition implied that all waves were absorbed at the still water shoreline and no reflection occurred. The time and spatial steps Ž D t and D x, respectively. used in all computations were approximately D t s Tpr100 ŽTp is the incident peak period. and

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D x s Lor75 Ž L o is the deep-water wavelength at the offshore boundary.. At the seaward boundary, the Courant number c Ž D trD x . was approximately equal to one. In the entire test cases default parameter values were employed for the two models of wave breaking ŽSato et al., 1992; Schaffer et al., 1993.. Thus, in the roller model the four ¨ parameters were set to f B s 208, fo s 108, t 1r2 s Tr5, and fd s 1.5. In the W–D model the parameters a D and a r were given the values 2.5 and 0.5, respectively.

3. Field data Data on water surface elevation collected during two extensive field experiments on surf-zone hydrodynamics and sediment transport carried out at the U.S. Army Field Research Facility ŽFRF., Duck, NC, were employed to evaluate the wave transformation model. The wave measurements during the two experiments known as DUCK85 and SUPERDUCK were discussed in Ebersole and Hughes Ž1987. and Ebersole and Hughes Ž1988., respectively. These two unique data sets on water surface elevation were chosen for two main reasons: Ž1. The technique employed to record the water surface elevation Žphotopole technique; discussed below. is highly accurate and well-suited for wave measurements in the surf zone compared to conventional techniques such as pressure transducers or capacitance gages; and Ž2. the experiments were highly successful and conducted during conditions that were ideal for investigating wave transformation in the surf zone encompassing a range of beach profile shapes and incident wave conditions. In the following, a brief description of the experiments, data collection methods, and analysis techniques are given. Ebersole and Hughes Ž1987; 1988. should be consulted for a more detailed discussion of these items. Extensive field measurements of waves in the nearshore zone were conducted on September 4–6, 1985 ŽDUCK85., and on September 7 to October 6, 1986 ŽSUPERDUCK., at the FRF. In the experiments, synchronized 16-mm movie cameras were used to photograph the propagation of the waves past brightly painted poles Žknown as ‘‘photopoles’’ and used as geometric references. placed across the beach profile, from the swash zone out to a depth where wave breaking typically did not occur during the experiment. This photographic technique, which had previously been successfully used in Japan ŽHotta and Mizuguchi, 1980., allows the free-surface elevation to be digitized directly from the photographic images. The photopoles were placed at intervals between 5 and 8 m in the cross-shore direction. Each experimental run encompassed almost 13 min of photography with pictures taken at a frequency of 5 Hz yielding in total 3800 frames per run. In the DUCK85 experiment, six cameras were employed to photograph the water surface elevation at 14 poles. The wave conditions during the experiment were characterized as long-crested swell Žwave periods of 10 to 12 s. with a majority of the waves breaking as plunging breakers. The energy-based significant wave height Ž Hmo . at the most seaward pole varied between 0.4 and 1.0 m during the experiment. A steady offshore wind Ž6 to 7 mrs. was typically present during the measurements. In total nine runs were carried out during DUCK85, and the runs selected in this study for comparison with the numerical simulations are given in Table 1 together with some

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Table 1 Beach and wave characteristics for experimental runs selected from the DUCK85 and SUPERDUCK experiments to be compared with model simulations Ž Hmo senergy-based significant wave height at the most offshore pole; Tp s peak spectral period; So and S br sslope parameter at offshore pole and break point, respectively; ´ 0 and ´ br s nonlinearity parameter at offshore pole and break point, respectively; m 0 s dispersion parameter at offshore pole. Date

Profile type

Hmo Žm.

Tp Žs.

S0

S br

´0

´ br

m 0 Ž10y3 .

Sept. 5, 1985, 09:55 Sept. 5, 1985, 10:15 Sept. 5, 1985, 13:52 Sept. 5, 1985, 15:25 Sept. 15, 1986, 14:45 Sept. 19, 1986, 11:00

Shelf Shelf Shelf Shelf Shelf Bar

0.61 0.42 0.64 0.53 1.07 0.70

11.4 13.1 10.9 11.1 10.1 11.2

0.81 0.93 0.77 0.83 0.62 0.79

1.12 1.29 1.00 1.13 0.72 1.00

0.16 0.11 0.17 0.12 0.15 0.10

0.70 0.66 0.68 0.68 0.48 0.47

1.68 1.27 1.88 1.61 2.84 2.5

characteristics of each run Žthe notation is explained in the table caption.. The measured water surface elevation at the most offshore pole ŽP14. was used as input in the numerical model at the seaward boundary ŽFig. 1.. In the SUPERDUCK experiment, eight cameras were used to photograph the water surface elevation at 22 poles, thus providing a higher resolution of wave transformation in the nearshore than during the DUCK85 experiment. The wave conditions during SUPERDUCK involved both oblique and normal sea and swell waves; however, during the experiment runs mostly swell prevailed with periods between 10 and 12 s. The Hmo wave height recorded at the most seaward pole varied between 0.6 and 1.3 m during the experiment, and the recorded water surface elevation at this pole was used as input in the numerical model. During the experiment, the seabed elevation at the each of the photopoles was surveyed once a day and Fig. 2 shows, as an example, the surveyed bottom profile on September 19, 1986. In contrast to the DUCK85 experiment, barred profiles occurred several times during the SUPERDUCK experiment. Table 1 summa-

Fig. 1. Bottom topography measured along the photopole transects on September 6, 1985 ŽDUCK85 experiment..

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Fig. 2. Bottom topography measured along the photopole transects on September 19, 1986 ŽSUPERDUCK experiment..

rizes the experimental runs selected from the SUPERDUCK experiment Žin total 10 runs were made. for comparison with the numerical simulations.

Fig. 3. Comparison between measured and computed time series of water surface elevation using the roller model at poles P12, P9 and P8 Ž09:55 EDT, September 5, 1985..

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4. Model comparisons with field data Although simulations were carried out for all experimental runs given in Table 1, only two of the runs were selected for detailed discussion here. However, the general conclusions given are based on the results from all the simulations. Two different types of beach profile shapes were chosen to be discussed, namely a shelf and a barred profile Žsee Figs. 1 and 2, respectively.. The beach profiles displayed fairly mild slopes, typically in the range 1r30–1r25, also for the cases when a bar was present. Characteristic parameters to quantify the wave conditions, such as the dispersion parameter m Žs hrgT 2 . and the non-linearity parameter ´ Žs Hr2 h., yielded values that are typical of nonlinear, long waves Ž ´ was as high as 0.7 near the break point.. A slope parameter S s h x Lrh, where h x is the local beach slope, was also calculated to indicate the reflective properties of the beach. Table 1 gives the values of these parameters for the experimental cases studied. Default parameter values ŽSato et al., 1992; Schaffer et al., 1993. were employed in ¨ all simulations and no special calibration was performed. The measured mean water depths at the poles during a run were used to derive the input beach profile, implying

Fig. 4. Comparison between measured and computed time series of water surface elevation using the roller model at Poles P7, P6 and P5 Ž09:55 EDT, September 5, 1985..

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that setuprsetdown was included in the depths. There is no easy method to obtain the still-water depths and the effect of this disturbance on the calculation results should have been minor. The time series of water surface elevation measured at the most offshore pole was filtered before being used as input to the model Ža band-pass filter with period cutoffs at 3 and 12 s in accordance with Ebersole and Hughes, 1987.. This was primarily done to eliminate high-frequency oscillations resulting from complex transformations of the water surface in connection with breaking and associated difficulties in digitising this surface. For the same reason, measured time series at other poles used for comparison with the simulations were also filtered Žsame filter and cutoff frequencies.. In the following, comparisons are first provided between measured and simulated time series of water surface elevation. These comparisons were made for selected poles representing different aspects of the wave transformation Že.g., rapid shoaling, incipient breaking, outer and inner surf zone.. Then, the agreement between calculated and measured statistical wave properties are presented for properties such as skewness, kurtosis and selected wave height measures. The W–D and roller model were employed in all simulation cases to describe the effects of wave breaking in order to compare the performance of the two formulations.

Fig. 5. Comparison between measured and computed time series of water surface elevation using the W–D model at Poles P12, P9 and P8 Ž09:55 EDT, September 5, 1985..

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Computations with the W–D model were performed both with Eq. Ž3. in its original form and multiplied with a factor of 1.5, as recommended by Watanabe Žpersonal communication.. Multiplication by this factor somewhat improved the agreement with the data compared to the original formulation, which was used in all calculations presented here. However, the improvements were not significant enough to warrant a change in the applied criterion considering that Eq. Ž3. in its original form has been used in several applications. As pointed out previously, the weakly nonlinear properties of the Boussinesq equations employed in this study should well conform with the assumptions behind Eq. Ž3.. 4.1. Time series of water surface eleÕation This section focuses on the agreement between measured and simulated water surface elevation. Fig. 3 shows a comparison between the measured time series of water surface elevation at 09:55 on September 5, 1985, and the corresponding calculation with the roller model Žshelf profile.. The subplots in Fig. 3 show the water surface elevation for

Fig. 6. Comparison between measured and computed time series of water surface elevation using the W–D model at Poles P7, P6 and P5 Ž09:55 EDT, September 5, 1985..

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the time period between 5.83 and 12 min at poles P12, P9 and P8 Žsee Fig. 1 for the pole locations.. The period of the individual waves was quite constant during the run with a typical value of T s 11.4 s. As expected, we observe that there is greater asymmetry in the higher waves compared to the lower waves, and the asymmetry increases for all the waves as they propagate onshore. Near pole P12 the three largest waves break Ž d s 1.58 m., and they may be observed as the three marked peaks in the beginning of the time series at pole P9. Most of the remaining waves break in the vicinity of poles P8 and P7 Ž d s 0.93 and 0.81 m, respectively.. The water surface elevation is underestimated by the calculations in the area of incipient breaking, which is not unexpected considering the type of Boussinesq formulation employed in this study. Madsen et al. Ž1997a; b. pointed out that this type of formulation tends to underestimate the nonlinearities. However, other Boussinesq formulations, for example, the model proposed by Nwogu Ž1993., instead overestimates the nonlinearities and associated wave shoaling. In order to improve the nonlinear shoaling it is necessary to include nonlinear dispersive terms as demonstrated by Sørensen et al. Ž1998a.. The latter type of Boussinesq formulation was outside the scope of the present work and calculation results from this type of model are not discussed here.

Fig. 7. Comparison between measured and computed time series of water surface elevation using the roller model at Poles P18, P16 and P14 Ž11:00 EDT, September 19, 1986..

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Fig. 4 shows the water surface elevation as a function of time at poles P7, P6 and P5 Žmeasured and calculated with the roller model.. Since the waves are nearly non-dispersive at these pole locations, implying that all waves travel with nearly the same speed, waveforms observed in the time domain are indicative of those that occur in the spatial domain. After incipient breaking, the waveforms resemble those of periodic bores; that is, they are saw-toothed with very steep forward faces and gradual slopes on the rear face of the crest. The roller model was fairly successful in reproducing this steepening of the waveform, at least for the higher waves. It is noted that both in the model simulations and measurements there are creation of secondary peaks within some wave crests and the formation of small oscillations within the waveform. However, there is a tendency for the roller model to generate too many of these small oscillations, sometimes almost resulting in complete wave disintegration. At poles P6 and P5 all wave heights in the group are similar to each other, which is characteristic for the inner surf zone ŽSvendsen, 1984.. Fig. 5 shows the results of calculations with the W–D model for the same case and poles as presented in Fig. 3. The calculation results agree fairly well with the

Fig. 8. Comparison between measured and computed time series of water surface elevation using the roller model at Poles P12, P10 and P8 Ž11:00 EDT, September 19, 1986..

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measurements, as was the case for the roller model, with the same underestimation of the nonlinear shoaling. In Fig. 6 is the calculated water surface elevation with the W–D model at poles P7, P6 and P5 shown. The cross-shore variations in the shape of the broken waves are well predicted by the model, that is, the development of a saw-tooth shape in the surf zone. However, the maximum water surface elevations are somewhat underestimated prior to breaking. The arrival of the individual bores is also predicted reasonably well. Overall, a smoother water surface elevation is obtained with the W–D model, implying that wave disintegration is less pronounced. An example of the calculation results for the barred beach profile is illustrated in Fig. 7, where the water surface elevation as predicted by the roller model are compared with the measurements. The water surface elevation recorded at poles P18, P16 and P14 Žsee Fig. 2 for the pole locations. at 11:00 on September 19, 1986, are shown in respective subplot together with the calculation results. The larger waves started to break on the offshore bar just seaward of P17 Ž d s 1.97 m.. Fig. 8 shows measured and calculated Žroller model. time series of water surface elevation at poles P12, P10 and P8. It is clear that the water surface elevation is somewhat underestimated in the calculations with the

Fig. 9. Comparison between measured and computed time series of water surface elevation using the W–D model at Poles P18, P16 and P14 Ž11:00 EDT, September 19, 1986..

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roller model. The observed fine-scale variations in the water surface elevation are fairly well described by the roller model. Fig. 9 compares the measured water surface elevation with computations using the W–D model at poles P18, P16 and P14, whereas Fig. 10 shows the results of comparisons at poles P12, P10 and P8 Žcompare results from the roller model in Figs. 7 and 8, respectively.. To a large extent, similar comments regarding the simulation results as were made for the roller model can be made for the W–D model. However, as for the shelf-profile case, the water surface elevation simulated with the W–D model is much smoother than the simulations with the roller model. This sometimes causes a lower resolution of the fine-scale variations, especially in the inner surf zone, implying that only the major features of the waveshape are described. In summary, the agreement between the simulation results and the measurements is acceptable, both for the roller model and W–D model, considering that the model parameters were not adjusted in order to objectively assess the predictive capability of the models. Nonlinear shoaling was underestimated in the calculations due to the inherent limitations of the Boussinesq equations. The steepening of the wave profile was

Fig. 10. Comparison between measured and computed time series of water surface elevation using the W–D model at Poles P12, P10 and P8 Ž11:00 EDT, September 19, 1986..

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Fig. 11. Comparison between measured and computed skewness and kurtosis for shelf profile case Ž09:55 EDT, September 5, 1985..

reasonably well simulated by both models. Considering that the two models of wave breaking were developed for spilling breakers and that plunging breakers prevailed in the experiments, the simulation results must be regarded as satisfactory. 4.2. Statistical waÕe properties This section presents the results of comparisons for statistical quantities derived from the time series of water surface elevation. Fig. 11 shows the measured and calculated

Fig. 12. Comparison between measured and computed skewness and kurtosis for barred profile case Ž11:00 EDT, September 19, 1986..

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Table 2 Comparison between calculated and measured statistical wave quantities expressed in terms of rms error Date

j H rms

j H 1r3

j H 1r10

Roller model W–D model Roller model W–D model Roller model W–D model Sept. 5, 1985, 09:55 Sept. 5, 1985, 10:15 Sept. 5, 1985, 13:52 Sept. 5, 1985, 15:25 Sept. 15, 1986, 14:45 Sept. 19, 1986, 11:00

19.0 16.8 26.0 17.0 12.5 8.8

25.0 27.4 28.0 23.0 15.0 13.7

18.0 27.0 27.0 18.0 13.0 6.3

23.0 21.0 28.0 19.0 16.0 15.1

18.0 30.0 24.0 12.0 16.7 13.8

24.0 23.0 28.0 13.0 22.3 19.4

Žboth with the roller and W–D model. cross-shore variation in the skewness Ž zs . and kurtosis Ž z k . of the water surface elevation for the experimental run carried out at 09:55 on September 5, 1985 Žshelf profile.. The figure shows that both zs and z k are seriously underestimated in the shoaling region. Table 1 gives ´ o s 0.16 at pole P14 based on Hrms , however, ´ br Žbased on Hmax . is already as high as 0.7 around the average break point located close to P9. Thus, for the highest waves the employed Boussinesq formulation clearly underestimates zs and z k by not including higher-order nonlinear dispersive terms ŽSørensen et al., 1998a.. Although both zs and z k are underestimated

Fig. 13. Comparison between measured and calculated statistical wave height parameters for shelf profile case Ž09:55 EDT, September 5, 1985..

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in the shoaling region, seaward of the average breakpoint at about 152 m ŽFRF’s local coordinate system. the models largely overestimate these quantities. Similarly to Fig. 11, Fig. 12 compares measured and calculated cross-shore variations in zs and z k but for the experimental case at 11:00 on September 19, 1986 Žbarred profile.. As for the shelf profile, in the region of wave shoaling the calculations significantly underestimates zs and z k . Again, one major cause for the discrepancies is the limitations of the present Boussinesq formulation to describe wave propagation in regions where non-linear dispersive effects are pronounced. Another possible cause for the disagreement could be difficulties in unambiguously determining the location of the water surface during the digitisation when strong turbulence occur on the surface due to wave breaking. An rms error was defined to objectively determine the agreement between model calculations and measurements according to,

js

ž

Ý Ž hmeas y hpred . Ý Ž hmeas .

2

1r2

2

/

Ž 12 .

where hmeas and hpred represent the measured and predicted statistical quantity to be compared. The j-values for the quantities Hrms , H1r3 , and H1r10 and for all test cases are given in Table 2. Figs. 13 and 14 show the measured and calculated cross-shore variations of the statistical wave height parameters for the aforementioned shelf and barred profiles,

Fig. 14. Comparison between measured and calculated statistical wave height parameters for barred profile case Ž11:00 EDT, September 19, 1986..

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respectively. The rms wave height Ž Hrms ., the significant wave height Ž H1r3 ., the mean of the 1r10 highest waves Ž H1r10 . were determined using the zero down-crossing method. The rms wave height was reasonably well simulated for both types of profiles and the rms error was typically less than 15% ŽTable 2.. However, the results for H1r10 showed slight underestimation in breaking region with rms errors around 25%. Overall, the roller model predicted the statistical parameters slightly better than the W–D model.

5. Conclusions A numerical model was developed to simulate wave transformation in the surf zone based on the Boussinesq equations. The effects of depth-limited wave breaking were described in the momentum equation with an additional momentum term. Two different formulations from the literature were employed to model this loss term, namely the formulations by Watanabe and Dibajnia Ž1988; W–D model. and Schaffer et al. Ž1993; ¨ roller model.. Detailed data on water level elevations collected with photographic techniques during two major field experiments ŽDUCK85 and SUPERDUCK. at the U.S. Army Field Research Facility, Duck, NC, were used to evaluate the wave transformation model and the two formulations for wave breaking effects. Thus, the overall objectives of the study were to test how well a Boussinesq-type model could simulate surf-zone conditions and to assess the applicability of two state-of-the-art formulations of the effects of wave breaking at the intra-wave scale. Overall, the Boussinesq equations in combination with either of the models of wave breaking reasonably well described the time history of water level elevation throughout the surf zone for a range of wave conditions and profile shapes. The comparison between measurements and simulations included the region before breaking Žgradual and rapid shoaling., the region of rapid transformation after breaking Žouter surf zone., and the region where bore-type waves prevailed Žinner surf zone.. However, detailed comparison between measurements and model simulations indicated that the Boussinesq equations employed here failed to accurately capture the rapid, strongly nonlinear shoaling just before breaking, and the maximum water surface elevation was typically underestimated in this region. Both the W–D and the roller model simulated the effects of breaking on the development of the waveshape through the surf zone satisfactorily, considering that the models were applied with default parameter values determined mainly from small-scale laboratory data. Furthermore, both models were developed for spilling breakers whereas plunging breakers were predominant during the field experiments. Wave steepening and bore formations were reproduced by the models as well as wave disintegration, although the latter was more pronounced in the roller model. The W–D model had a tendency to produce time series of water level elevation that were smoother than the roller model, which reduced wave disintegration compared to the roller model but also prevented fine-scale resolution of the wave motion in some cases. The prediction of the third- and fourth-order moments Žskewness and kurtosis, respectively. of the water surface elevations were not well predicted, which was probably due to the inability of the employed Boussinesq equations to describe the

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nonlinear dispersive effects the waves exhibited in the region of rapid shoaling. Simulation of statistical wave quantities, such as rms and significant wave height, displayed acceptable agreement with the field measurements with an error of typically less than 25%. The roller model produced somewhat better results than the W–D model for the statistical wave quantities, especially for the higher wave measures Že.g., H1r10 .. On the other hand, from a practical point of view, the W–D model is easier to implement into numerical models for engineering applications and gives more robust calculation results. The roller model needs a more refined computational grid in the numerical model. In summary, the numerical wave transformation model produced satisfactory results considering the assumptions underlying the modelling and that no special calibration was employed to tune the parameter values in the breaking wave models. However, for some applications, which rely on accurate predictions of the waveshape Žand velocity., such as sediment transport calculations, a Boussinesq-type model as the one presented here would not yield accurate enough predictions without additional improvements. A natural next step in the model development would be to include nonlinear dispersive terms in accordance with Sørensen et al. Ž1998a..

Acknowledgements The authors would like to thank Mr. Bruce Ebersole, Coastal and Hydraulics Laboratory ŽCHL., U.S. Army Engineer Waterways Experiment Station, USA, for providing the data from the DUCK85 and SUPERDUCK field experiments. Professor Shintaro Hotta, Department of Construction Engineering, Nihon University, and Dr. Nicholas C. Kraus, CHL, were instrumental in launching and carrying out the photopole experiments and their contributions are gratefully acknowledged. The Swedish Natural Science Research Council financially supported this study. ML also acknowledges the support of the European Commission through the project INTAS-96-2077. The comments by Professor Akira Watanabe, University of Tokyo, and one anonymous reviewer considerably improved the paper and are greatly appreciated.

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