Resonant Interactions of Poincare Waves in the Shallow Water Approximation

TL;DR

This paper develops a weakly nonlinear theory for Poincare waves in shallow water, revealing four-wave resonant interactions, deriving their equations, and predicting a saturation effect significant for geophysical hydrodynamics.

Contribution

It introduces a novel four-wave interaction framework for Poincare waves and derives equations describing their nonlinear dynamics and energy transfer.

Findings

Four-wave interactions are present in Poincare waves.

An analytical solution describes wave saturation over time.

The study predicts a saturation effect relevant to geophysical applications.

Abstract

The paper develops a weakly nonlinear theory of Poincare waves. The nondegeneracy of the Poincare wave dispersion law leads to the presence of resonant interactions in perturbation theory. A study of the dispersion relation of Poincare waves showed that three-wave interactions are absent in the quadratic nonlinear approximation. In this paper, a linear equation of the envelope is derived. A qualitative study of the dispersion law showed the existence of four-wave interactions of Poincare waves. Equations of nonlinear interactions of four waves for the amplitudes of Poincare waves are derived. The Manley-Rowe equations are obtained, which determine the distribution of energy and its transfer between interacting waves. The nonlinear dynamics of interacting waves is investigated. The saturation effect of Poincare waves, which is important for geophysical hydrodynamics, has been predicted. An analytical solution is obtained that describes the saturation effect of Poincare waves in time.