Numerical verification of the weak turbulent model for swell evolution

The purpose of this article is to numerically verify the theory of weak turbulence. We have performed numerical simulations of an ensemble of nonlinearly interacting free gravity waves (a swell) by two different methods: by solving the primordial dynamical equations describing the potential flow of an ideal fluid with a free surface, and by solving the kinetic Hasselmann equation, describing the wave ensemble in the framework of the theory of weak turbulence. In both cases we have observed effects predicted by this theory: frequency downshift, angular spreading and formation of a Zakharov-Filonenko spectrum I_ω ∼ ω^-4. To achieve quantitative coincidence of the results obtained by different methods, we have to augment the Hasselmann kinetic equation by an empirical dissipation term S_diss modeling the coherent effects of white-capping. Using the standard dissipation terms from the operational wave predicting model (WAM) leads to a significant improvement on short times, but does not resolve the discrepancy completely, leaving the question about the optimal choice of S_diss open. In the long run, WAM dissipative terms essentially overestimate dissipation.