Energy balance in a wind-driven sea

In this paper, we offer the answers to certain questions extremely important for the development of a self-consistent analytical theory for the wind-driven sea. (i) We discuss the separation into 'resonant' and 'slave' harmonics in an ensemble of weakly nonlinear gravity waves on the surface of deep water, and we construct an explicit form of the generation function for canonical transformation that eliminates the slave harmonics. (ii) When two waves compiling a quadruple are short in comparison with two others, we find an asymptotic form for the four-wave coupling coefficient. This result makes it possible to reduce the Hasselmann equation to the nonlinear diffusion equation, whose solution describes the well-known effect of angular spreading of wave spectra on its rear face. (iii) Studying the isotropic Kolmogorov-Zakharov solution of the Hasselmann equation, we find numerically the values of Kolmogorov constants. (iv) We calculate the nonlinear damping of surface waves appearing due to four-wave interaction and compare the damping with the growth rate of the instability of the wave surface induced by the wind. It is found that for all known models of wind input, the nonlinear damping surpasses the instability at least in order of magnitude. This result, supported by numerical simulation of the Hasselmann equation, leads to the conclusion: in a real sea, except for the case of very young waves, four-wave interaction is the dominant process. This statement opens the way for the development of a well-justified analytical theory for the wind-driven sea.