The numerical model SWAN (Simulating WAves Nearshore) for the computation of wave conditions in shallow water with ambient currents is briefly described. The model is based on a fully spectral representation of the action balance equation with all physical processes modelled explicitly. No a priori limitations are imposed on the spectral evolution. This makes the model a third-generation model. In Holthuijsen et al. (1993) and Ris et al. (1994) test cases for propagation, generation and dissipation have been shown without currents. Current effects have now been added and academic cases are shown here. The model is also applied in a fairly academic case of a shallow lake (Lake George, Australia) and in a complex, realistic case of an inter-tidal area with currents (Friesche Zeegat, the Netherlands). The results are compared with observations. A new development to formulate the model on a curvi-linear grid to accommodate linkage to hydro-dynamic circulation models is presented and a first test is shown. INTRODUCTION Over the last decade, the traditional wave ray models in coastal engineering to compute waves in nearshore conditions are being replaced by models that formulate the wave evolution in terms of a spectral energy balance on a regular grid (or the action balance in the presence of ambient currents). In third-generation versions of such models the wave spectrum is allowed to evolve free of any a priori limitations and all relevant physical processes are represented explicitly in a discrete spectral formulation. Such a wave model (the SWAN model), with the inclusion of ambient currents is described here. Conceptually it is an extension of deep water thirdgeneration wave models but the physical processes and the numerical techniques involved are more complicated. The SWAN wave model has been conceived to be a computationally feasible third-generation spectral wave model for waves in shallow water (including the surf zone) with ambient currents in a consulting environment with return times of less than 30 min on a desk top computer. Delft University of Technology, Department of Civil Engineering, P.O. Box 5048, 2600 GA Delft, Netherlands. "SWAN" WAVE MODEL 669 THE SWAN WAVE MODEL The SWAN wave model (Ris et al., 1994) is a fully discrete spectral model based on the action balance equation which implicitly takes into account the interaction between waves and currents through radiation stresses (e.g., Phillips, 1977): |NW)+VIy.[ciyNW)]4KNW)]4[c,NM]=M ot do do o The first term in the left-hand side is the rate of change of action density in time, the second term is the rectilinear propagation of action in geographical x-,yspace. The third term describes the shifting of the relative frequency due to currents and time-varying depths with propagation velocity c„ in cr-space. The fourth term represents the propagation in 0-space (depthand current-induced refraction) with propagation velocity ce. The term S(<x,6) at the right hand side of the action balance equation is the source term representing the growth by wind, the wave-wave interactions and the decay by bottom friction, whitecapping and depth-induced wave breaking. To reduce computer time, we remove time from the action balance equation (i.e., d/dt = 0). This is acceptable for most coastal conditions since the residence time of the waves is usually far less than the time scale of variations of the wave boundary conditions, the ambient current, wind or the tide. For cases in which the time scale of these variations becomes important, i.e., variable incoming waves at the boundary, or variable winds or currents, a quasi-stationary approach can be taken by repeating the computations for predefined time intervals. The formulations for the generation, the dissipation and the quadruplet wavewave interactions are taken from the WAM model (WAM Cycle 3, WAMDI group, 1988 and optionally WAM Cycle 4, Komen et al., 1994 as presently operational at the European Centre for Medium Range Weather Forecasting). For the present study the formulations from WAM Cycle 3 are used. These are supplemented with a spectral version of the dissipation model for depth-induced breaking of Battjes and Janssen (1978) (with the maximum wave height to depth ratio from Nelson, 1987) and a recently formulated discrete interaction approximation for the triad wave-wave interactions (Eldeberky and Battjes, 1995). Fully implicit numerical schemes are used in the SWAN model for propagation in both geographic space and spectral space (an iterative, forward-marching, foursweep technique, Ris et al., 1994). This scheme is unconditionally stable in contrast with the explicit schemes of conventional spectral wave models which are only conditionally stable and which require therefore very small time steps in shallow water (typically 10 s for 100 m resolution in water depth of 10 m where in the SWAN model the time increment may be as large as 15 min). The formulation is basically in terms of finite differences on a regular, rectangular grid. This is inconvenient in regions with highly variable scales such as tidal inlets, tidal flats and 670 COASTAL ENGINEERING 1996 estuaries. Nesting of grids with decreasing resolution is the conventional approach in such cases but it requires extra computations. A variable resolution grid would avoid such extra computations, in particular if the grid would conform to the topography of the region. It would also accommodate the linkage with hydrodynamic circulation models which are often formulated on such grids. Such a curvilinear approach is implemented in the SWAN model by considering in the numerical scheme of each spatial grid point, two separate, non-equidistant finite up-wind differences in each of two orthogonal directions. For the x -direction this is for grid point i ,j (the grid points are ordered in x , y-space):

Swell enhances wind-induced wave growth in the third-generation wave models of the WAM family (e.g., WAM, SWAN, TOMAWAC). This is contrary to observations in laboratory conditions where swell reduces wind-induced wave growth. To improve the performance of these models, it is hypothesized here that high-frequency whitecapping is increased by surface-straining near the crests of low-frequency waves. The pulse-based parameterization of whitecapping in these models is accordingly scaled at every frequency with the ratio of high-frequency steepness over total wave steepness (in an experimental version of SWAN). A generic laboratory experiment which illustrates the phenomenon can now be reproduced. The growth of young wind sea in idealized conditions is not affected. Moreover, the spectral shape (both in frequency and direction) improves. However, in the one field case that is considered here, the effects are marginal.

Abstract. A third-generation numerical wave model to compute random, short-crested waves in coastal regions with shallow water and ambient,currents (Simulating Waves Nearshore (SWAN)) has been developed, implemented, and validated. The model is based on a Eulerian formulation,of the discrete spectral balance of action density that accounts,for refractive propagation,over arbitrary bathymetry,and current fields. It is driven by boundary conditions and local winds. As in other third-generation wave models, the processes of wind generation, whitecapping, quadruplet wave-wave interactions, and bottom dissipation are represented explicitly. In SWAN, triad wave-wave interactions and depth-induced,wave,breaking are added. In contrast to other third-generation wave models, the numerical propagation scheme is implicit, which implies that the computations are more,economic,in shallow water. The model,results agree well with analytical solutions, laboratory observations, and (generalized) field observations.

A third‐generation spectral wave model (Simulating Waves Nearshore (SWAN)) for small‐scale, coastal regions with shallow water, (barrier) islands, tidal flats, local wind, and ambient currents is verified in stationary mode with measurements in five real field cases. These verification cases represent an increasing complexity in two‐dimensional bathymetry and added presence of currents. In the most complex of these cases, the waves propagate through a tidal gap between two barrier islands into a bathymetry of channels and shoals with tidal currents where the waves are regenerated by a local wind. The wave fields were highly variable with up to 3 orders of magnitude difference in energy scale in individual cases. The model accounts for shoaling, refraction, generation by wind, whitecapping, triad and quadruplet wave‐wave interactions, and bottom and depth‐induced wave breaking. The effect of alternative formulations of these processes is shown. In all cases a relatively large number of wave observations is available, including observations of wave directions. The average rms error in the computed significant wave height and mean wave period is 0.30 m and 0.7 s, respectively, which is 10% of the incident values for both.

A third-generation numerical wave model to compute random, short-crested waves in coastal regions with shallow water and ambient currents (Simulating Waves Nearshore (SWAN)) has been developed, implemented, and validated. The model is based on a Eulerian formulation of the discrete spectral balance of action density that accounts for refractive propagation over arbitrary bathymetry and current fields. It is driven by boundary conditions and local winds. As in other third-generation wave models, the processes of wind generation, whitecapping, quadruplet wave-wave interactions, and bottom dissipation are represented explicitly. In SWAN, triad wave-wave interactions and depth-induced wave breaking are added. In contrast to other third-generation wave models, the numerical propagation scheme is implicit, which implies that the computations are more economic in shallow water. The model results agree well with analytical solutions, laboratory observations, and (generalized) field observations.

The third-generation wave model SWAN for small-scale, coastal regions with shallow water, (barrier) islands, tidal flats, local wind and ambient currents is verified in real field conditions by applying the model in stationary mode to eight cases at four different sites in the southern North Sea. The sites represent an increasing complexity in bathymetry. The simplest of the cases are quasi 1-dimensional whereas the more complex cases are two-dimensional: tidal gaps between barrier islands with channels, shoals, tidal currents and local wind. The agreement between computed and observed spectra in the quasi-1D cases is excellent. This supports (a) a spectral version of a well established model for depth-induced wave breaking and (b) a relatively new model for triad wave-wave interactions. In the rather more complicated 2-D cases of the tidal inlets, SWAN performs well in terms of significant wave height and mean wave period, the rms-error being about 10% of the incident values.

The numerical model SWAN (Simulating WAves Nearshore) for the computation of wave conditions in shallow water with ambient currents is briefly described. The model is based on a fully spectral representation of the action balance equation with all physical processes modelled explicitly. No a priori limitations are imposed on the spectral evolution. This makes the model a third-generation model. In Holthuijsen et al. (1993) and Ris et al. (1994) test cases for propagation, generation and dissipation have been shown without currents. Current effects have now been added and academic cases are shown here. The model is also applied in a fairly academic case of a shallow lake (Lake George, Australia) and in a complex, realistic case of an inter-tidal area with currents (Friesche Zeegat, the Netherlands). The results are compared with observations. A new development to formulate the model on a curvi-linear grid to accommodate linkage to hydro-dynamic circulation models is presented and a first test is shown.

Waves travelling against an increasing opposing current tend to dissipate energy and part of the energy reflects back (in blocking conditions). The kinematic behaviour of these waves can be approximated with the linear theory for surface gravity waves. This theory has been implemented for random, short-crested waves in the third-generation wave model SWAN with numerical schemes that are fully implicit. Ad hoc assumptions that are made in other, similar models for blocking conditions are therefore not required and the model is consistent with the underlying theory. To represent the dissipation of the breaking waves in these blocking conditions, the pulse-based model of Hasselmann (1974) as adapted by Komen et al. (1984) has been chosen. Computations have been compared with the flume observations of Lai et al. (1989) in which random waves are blocked with violent breaking by an increasing counter current in relatively deep water. The computations underestimate the dissipation considerably but the addition of the bore-based model of Battjes and Janssen (1978) for steep, breaking waves in deep water improves the agreement with the observations significantly although some discrepancy remains.

The present paper describes the second phase in the development of a fully spectral wave model for the near shore zone (SWAN). Third-generation formulations of wave generation by wind, dissipation due to whitecapping and quadruplet wave-wave interactions are added to the processes of (refractive) propagation, bottom friction and depth-induced wave breaking that were implemented in the first phase (Holthuijsen et al., 1993). The performance and the behaviour of the SWAN model are shown in two observed cases in which waves are regenerated by the wind after a considerable decrease due to shallow water effects. In the case of the Haringvliet (a closed branch of the Rhine estuary, the Netherlands) reasonable results in terms of significant wave heights were obtained. In the case of Saginaw Bay (Lake Huron, USA), the SWAN model underestimates the significant wave height deep inside the bay (as did two other models). Due to the absence of triad wave-wave interactions in the model the mean period is not properly shifted to the higher frequencies in shallow water. Adding these triads is planned for the next phase of developing the SWAN model.

Spectral wave models that represent the evolution of the waves on a grid superior in several respects to conventional wave ray models. Spectral models on a grid have been developed for applications in the deep ocean and for shelf seas. However, they are not economically feasible in coastal waters due to numerical limitations. We present the first step in implementation of a version that does not have these limitations. we remove time as an independent variable (reducing the computations to stationery or quasi-stationery computations, which is proper considering the residence time of the waves in the area) and we use an unconditionally stable propagation scheme. The propagation scheme is successfully tested in academic cases, including a case with complete reversal of wave direction. As a preliminary test of propagation in an observed field case, computations are carried out for waves travelling across and around an extended (5 km) shoal. With a limited representation of the bottom induced processes (bottom friction and surf dissipation), realistic results are obtained for the significant wave height. This test also shows the relevance of the planned implementation of the wave-wave interactions (in particular grid interactions) and wind generation. The model is planned to be optionally second-or-third-generation (with or without predefined spectral constraints).