Effects of the bottom friction formulation on the energy balance for gravity waves in shallow water

The energy transfer equation for wind‐generated ocean waves is solved numerically for fetch‐limited conditions in waters of limited depth. The resulting growth curves for the total energy and the peak frequency are obtained for five different bottom friction dissipation formulations: an empirical Joint North Sea Wave Project (JONSWAP) expression ( Hasselmann et al. , 1973), three expressions based on the drag law friction model ( Hasselmann and Collins , 1968; Collins , 1972; Madsen et al. , 1988a) and a form based on the eddy viscosity friction model ( Weber , 1991a). The effects of different bottom friction dissipation formulations on the energy balance are intercompared and evaluated. The results are also compared with the Coastal Engineering Research Center growth curves. Numerical simulation and mathematical analysis show that only the drag law formulations with a fixed drag coefficient scale in terms of the air friction velocity. The empirical JONSWAP, the drag law formulation with dynamically changing drag coefficient, and the eddy viscosity model do not scale. It is possible to tune the dissipation coefficients for one particular wind friction velocity but different water depths so that all five formulations give almost the same growth curves for the total energy and the peak frequency. This can be explained by the fact that the five tuned bottom friction dissipation models have the same effect on the energy balance when integrated over the frequency and angle space.