Asymptotes of the nonlinear transfer and the swell spectrum in the frame of the kinetic equation

TL;DR

This paper numerically investigates the asymptotic behavior of nonlinear energy transfer in gravity wave spectra, revealing self-similar spectra with specific decay laws and the influence of initial anisotropy.

Contribution

It provides new numerical insights into the high-frequency asymptotes and long-term evolution of gravity wave spectra based on the kinetic equation, including the development of self-similar solutions.

Findings

Nonlinear energy transfer asymptote follows a power-law decay with p less than n-1 for n > 5.

A self-similar spectrum with a -4 power-law decay emerges after several hundred wave periods.

Spectrum peak shifts to lower frequencies while angular and frequency characteristics stabilize, depending on initial anisotropy.

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 swell. The cases of initial two-dimensional spectra of the different frequency decay-law with the power n 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 with values p less n-1, valid for cases in which n greater 5, and the difference, n-p, changes significantly when n approaches 4. On time scales of evolution greater than several hundred initial wave-periods, in every case, a self-similar spectrum Ssf is established with the frequency decay-law of power -4. Herein, the asymptote of nonlinear energy transfer becomes negative in value and decreases according to the same law The peak frequency of the spectrum migrates to the low-frequency region such that the angular and frequency characteristics of the two-dimensional spectrum Ssf 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.