N. Booij

A third‐generation wave model for coastal regions: 1. Model description and validation

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 wave model for coastal regions: 2. Verification

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 prediction model for stationary, short-crested waves in shallow water with ambient currents

Abstract A numerical model for the hindcasting of waves in shallow-water ( hiswa ) is described and comparisons are made between observations and model results in a realistic field situation. The model is based on a Eulerian presentation of the spectral action balance of the waves rather than on the more conventional (at least in coastal engineering) Lagrangian presentation. Wave propagation is correspondingly computed on a grid rather than along rays. The model accounts for refractive propagation of short-crested waves over arbitrary bottom topography and current fields. The effects of wave growth and dissipation due to wind generation, bottom dissipation and wave breaking (in deep and shallow water) are represented as source terms in the action balance equation. The computational efficiency of the model is enhanced by two simplifications of the basic balance equation. The first one is the removal of time as an independent variable to obtain a stationary model. This is justified by the relatively short travel time of waves in coastal regions. The second simplification is the parameterization of the basic balance equation in terms of a mean frequency and a frequency-integrated action density, both as function of the spectral wave direction. The discrete spectral representation of wave directionality is thus retained. An untuned version of hiswa has been tested in a closed branch of the Rhine estuary where measurements with buoys and a wave gauge are available. In this situation, where wave breaking and short-crestedness dominate, rms-errors in the significant wave height and mean wave period are about 10 and 13% respectively of the observed values.

A note on the accuracy of the mild-slope equation

Abstract The mild-slope equation is a vertically integrated refraction-diffraction equation, used to predict wave propagation in a region with uneven bottom. As its name indicates, it is based on the assumption of a mild bottom slope. The purpose of this paper is to examine the accuracy of this equation as a function of the bottom slope. To this end a number of numerical experiments is carried out comparing solutions of the three-dimensional wave equation with solutions of the mild-slope equation. For waves propagating parallel to the depth contours it turns out that the mild-slope equation produces accurate results even if the bottom slope is of order 1. For waves propagating normal to the depth contours the mild-slope equation is less accurate. The equation can be used for a bottom inclination up to 1:3.

Phase-decoupled refraction–diffraction for spectral wave models

Abstract Conventional spectral wave models, which are used to determine wave conditions in coastal regions, can account for all relevant processes of generation, dissipation and propagation, except diffraction. To accommodate diffraction in such models, a phase-decoupled refraction–diffraction approximation is suggested. It is expressed in terms of the directional turning rate of the individual wave components in the two-dimensional wave spectrum. The approximation is based on the mild-slope equation for refraction–diffraction, omitting phase information. It does therefore not permit coherent wave fields in the computational domain (harbours with standing-wave patterns are excluded). The third-generation wave model SWAN (Simulating WAves Nearshore) was used for the numerical implementation based on a straightforward finite-difference scheme. Computational results in extreme diffraction-prone cases agree reasonably well with observations, analytical solutions and solutions of conventional refraction–diffraction models. It is shown that the agreement would improve further if singularities in the wave field (e.g., at the tips of breakwaters) could be properly accounted for. The implementation of this phase-decoupled refraction–diffraction approximation in SWAN shows that diffraction of random, short-crested waves, based on the mild-slope equation can be combined with the processes of refraction, shoaling, generation, dissipation and wave–wave interactions in spectral wave models.

THE "SWAN" WAVE MODEL FOR SHALLOW WATER

A study of alternatives including a shoreline evolution numerical modelization has been carried out in order to both diagnose the erosion problem at the beaches located between Cambrils Harbour and Pixerota delta (Tarragona, Spain) and select nourishment alternatives.

Propagation of ocean waves in discrete spectral wave models

Abstract In many numerical models for hindcasting or forecasting ocean waves, wave energy is propagated over large distances. In the class of discrete spectral models such propagation suffers from a disintegration of the initial wave field into many individual wave fields. This “garden sprinkler” effect is due to the treatment of finite spectral bands as individual wave components. It is shown in the present study that this effect can be avoided by including two correction terms in the commonly used energy balance equation of the waves. One of these terms accounts for longitudinal (frequency) dispersion, the other term accounts for lateral (directional) dispersion. These terms are derived from the energy balance of finite spectral bands and they are expressed in terms of the spectral band characteristics. Since their nature is that of diffusion terms, they are local operators, which is computationally convenient. However, the coefficients of these terms are not locally determined. To illustrate the effect of the proposed correction terms, the propagation of swell from a distant storm (oceanic scale) is computed with and without the proposed correction terms.

Diffusion reduction in an arbitrary scale third generation wind wave model

Abstract The numerical schemes for the geographic propagation of random, short-crested, wind-generated waves in third-generation wave models are either unconditionally stable or only conditionally stable. Having an unconditionally stable scheme gives greater freedom in choosing the time step (for given space steps). The third-generation wave model SWAN (“Simulated WAves Nearshore”, Booij et al., 1999 ) has been implemented with this type of scheme. This model uses a first order, upwind, implicit numerical scheme for geographic propagation. The scheme can be employed for both stationary (typically small scale) and nonstationary (i.e. time-stepping) computations. Though robust, this first order scheme is very diffusive. This degrades the accuracy of the model in a number of situations, including most model applications at larger scales. The authors reduce the diffusiveness of the model by replacing the existing numerical scheme with two alternative higher order schemes, a scheme that is intended for stationary, small-scale computations, and a scheme that is most appropriate for nonstationary computations. Examples representative of both large-scale and small-scale applications are presented. The alternative schemes are shown to be much less diffusive than the original scheme while retaining the implicit character of the particular SWAN set-up. The additional computational burden of the stationary alternative scheme is negligible, and the expense of the nonstationary alternative scheme is comparable to those used by other third generation wave models. To further accommodate large-scale applications of SWAN, the model is reformulated in terms of spherical coordinates rather than the original Cartesian coordinates. Thus the modified model can calculate wave energy propagation accurately and efficiently at any scale varying from laboratory dimensions (spatial scale O(10 m) with resolution O(0.1 m)), to near-shore coastal dimension (spatial scale O(10 km) with resolution O(100 m)) to oceanic dimensions (spatial scale O(10 000 km) with resolution O(100 km).

A SPECTRAL MODEL FOR WAVES IN THE NEAR SHORE ZONE

A special reflecting wall 12 m long and 2.1 m high was built off the beach at Reggio Calabria, and 30 wave gauges were assembled before the wall and were connected to an electronic station on land. It was possible to observe the reflection of wind waves generated by a very stable wind over a fetch of 10 Km. The experiment aimed to verify the general closed solution for the wave group mechanics (Boccotti, 1988, 1989), for the special case of the wave reflection.Significant features on Wadden Sea wave climate are evaluated in respect of the state of the art. Main emphasis was laid on an analysis of the governing boundary conditions of local wave climate in island sheltered Wadden Sea areas with extensions being sufficient for local wind wave growth. Explanatory for significant wave heights a reliable parametrization of local wave climate has been evaluated by using generally available data of water level and wind measurements.

A GRID MODEL FOR SHALLOW WATER WAVES

More than 150 tests have been analyzed in order to describe the dynamically stable profiles of rock slopes and gravel beaches under wave attack. Relationships between profile parameters and boundary conditions have been established. These relationships have been used to develop a computer program. This program is able to predict the profiles of slopes with an arbitrary shape under varying wave conditions, such as those found in storm surges and during the tidal period.This paper investigates the utility of winds obtainable from a numerical weather prediction model for driving a spectral ocean-wave model in an operational mode. Wind inputs for two operational spectral wave models were analyzed with respect to observed winds at three locations in the Canadian east coast offshore. Also, significant wave heights obtainable from the two spectral models were evaluated against measured wave data at these locations. Based on this analysis, the importance of appropriate wind specification for operational wave analysis and forecasting is demonstrated.