Numerical Verification of the Weak Turbulent Model for Swell Evolution

The purpose of this article is numerical verification of the theory of weak turbulence. We performed numerical simulation of an ensemble of nonlinearly interacting free gravity waves (swell) by two different methods: solution of primordial dynamical equations describing potential flow of the ideal fluid with a free surface and, solution of the kinetic Hasselmann equation, describing the wave ensemble in the framework of the theory of weak turbulence. Comparison of the results demonstrates applicability of the weak turbulent approach. In both cases we observed effects predicted by this theory: frequency downshift, angular spreading and formation of Zakharov-Filonenko spectrum I ω ∼ ω −4. To achieve quantitative coincidence of the results obtained by different methods, one has to supply the Hasselmann kinetic equation by an empirical dissipation term S diss model-ing the coherent effects of white-capping. Using of the standard dissipation terms from industrial wave predicting model (WAM) leads to significant improvement on short times, but not resolve the discrepancy completely, leaving the question about optimal choice of S diss open. In a long run WAM dissipative terms overestimate dissipation essentially. Numerical modeling of swell evolution in the framework of the dynamical equations is affected by the side effect of resonances sparsity taking place due to finite size of the modeling domain. We mostly overcame this effect using fine integration grid of 512×4096 modes. The initial spectrum peak was located at the wave number k = 300. Similar conditions can be hardly realized in the existing laboratory wave tanks. One of the results of our article consists in the fact that physical processes in finite size laboratory wave tanks and in the ocean are quite different, and the results of such laboratory experiments can be applied to modeling of the ocean phenomena with extra care. We also present the estimate on the minimum size of the laboratory installation, allowing to model open ocean surface wave dynamics.