The kinetic Hasselmann equation for wind waves

∂Nk/∂t + C~g∇N = Snl + Sin + Sds

is a theoretical basis for wave-prediction models. It prescribes changes of wave action spectral density to the effects of resonant nonlinear interactions, the wind generation and the wave dissipation (mostly by white-capping). While the nonlinear term is known from the ‘first principles’, our knowledge of generation term Sin is based on experimental results, which are not accurate enough to make a dispersion in measurements of the wave growth-rate less than the growth-rate itself. Less is known about the dissipation term Sdiss. This uncertainty hinders the development of well-justified prediction models. We modified the Resio-Tracy code for numerical solution of the Hasselmann equation and made it approximately ten times faster, more accurate and stable. With this new code we performed a massive duration-limited simulation of the Hasselmann equation using different versions of generation and dissipation terms, also choosing a small white noise as initial conditions. In all cases we observed the same effect: soon after beginning of computation the nonlinear term Snl becomes dominating over Sin and Sdiss, while the spectral evolution tends to a self-similar regime. The Hasselmann equation in absence of Sin and Sdiss has a rich family of self-similar solutions. At the fronts of these solutions propagating to low frequencies strong nonlinearity is balanced by non-stationarity or non homogeneity, while the rare faces of the solutions are described by power-like weak-turbulent exact Kolmogorov solutions of the stationary Hasselmann equation Snl = 0. The whole scope of experimental data on wind-driven sea implies the self-similarity of the spectral shape. Also, the self-similarity hypothesis corroborates with the evidence of power-like dependence of main energy and peak frequency on dimensionless fetch. We propose to describe experimental spectra of wind-driven sea by self-similar solution of the ‘conservative’ Hasselmann equation. The parameters of these solutions can be found from averaged by~k-space balance of wave action. The balance equation includes the generation and dissipation terms in an integral form, which can be parameterized by a detailed comparison with experiment. So far the shapes of our self-similar spectra display perfect coincidence with the spectra measured in the JONSWAP and other major fetch-limited studies.