Mitsuhiro Tanaka: Hasselmann nonlinear energy transport in a wave sepctrum, based on direct simulation of primitive equations

The temporal evolution of the energy spectrum of a field of random surface gravity waves in deep water is investigated by means of direct numerical simulations of the deterministic primitive equations. The detected rate of change of the spectrum is shown to be proportional to the cubic power of the energy density and agree quite well with the nonlinear energy transfer $S_{nl}$ as predicted by Hasselmann. In spite of the fact that use of various asymptotic relations which are valid only for $t\to\infty$ or integration with respect to time over a time scale much longer than $O({\rm period}\times (ak)^{-2})$ are necessary in the derivation of Hasselmann's $S_{nl}$, it is clearly demonstrated that the rate of change of the spectrum given by the numerical simulation agrees quite well with Hasselmann's $S_{nl}$ at every instant of ordinary time scale comparable to the period. The result implies that the four-wave resonant interactions control the evolution of the spectrum at every instant of time, while non-resonant interactions do not make any significant contribution even in a short-term evolution. It is also pointed out that the result may call for a reexamination of the process of derivation of the kinetic equation for the spectrum.