Asymptotes of the nonlinear transfer and wave spectrum in the frame of the kinetic equation solution

Abstract. The kinetic equation for a gravity wave spectrum is solved numerically to study the high frequencies asymptotes for the one-dimensional nonlinear energy transfer and the variability of spectrum parameters that accompany the long-term evolution of nonlinear waves. The cases of initial two-dimensional spectra S(ω,θ) of modified JONSWAP type with the frequency decay-law S(ω) ~ ω⁻ⁿ (for n = 6, 5, 4 and 3.5) and various initial functions of the angular distribution are considered. It is shown that at the first step of the kinetic equation solution, the nonlinear energy transfer asymptote has the power-like decay-law, Nl(ω) ~ ω^−p, with values p ≤ n − 1, valid in cases when n ≥ 5, and the difference, n-p, changes significantly when n approaches 4. On time scales of evolution greater than several thousands of initial wave periods, in every case, a self-similar spectrum S_sf(ω,θ) is established with the frequency decay-law of form S(ω) ~ ω⁻⁴. Herein, the asymptote of nonlinear energy transfer becomes negative in value and decreases according to the same law (i.e., Nl(ω) ~ −ω⁻⁴). The peak frequency of the spectrum, ω_p(t), migrates in time t to the low-frequency region such that the angular and frequency characteristics of the two-dimensional spectrum S_sf(ω,θ) remain constant. However, these characteristics depend on the degree of angular anisotropy of the initial spectrum. The solutions obtained are interpreted, and their connection with the analytical solutions of the kinetic equation by Zakharov and co-authors for gravity waves in water is discussed.