Harmonic generation by nonlinear self-interaction of a single internal wave mode

Abstract

Weakly nonlinear theory is used to explore the dynamics of a single-mode internal tide in variable stratification with rotation. Nonlinear self-interaction in variable stratification generates a perturbation which is forced with double the original frequency and wavenumber. The dynamics of the perturbation is analogous to a forced harmonic oscillator, with the steady-state solution corresponding to a bound harmonic matching the forcing frequency and wavenumber. When the forcing frequency is near a natural frequency of the system, even a small-amplitude (nearly linear) internal tide may induce a significant harmonic response. Idealized stratification profiles are utilized to explore the relevance of this effect for oceanic$M_{2}$baroclinic internal tides, and the results indicate that a rapidly growing harmonic may occur in some environments near the Equator, but is unlikely at higher latitudes. The results are relevant to recent observations of$M_{4}$(harmonic) internal tides in the South China Sea and elsewhere. More generally, nonlinear self-interaction may contribute to the transfer of energy to smaller scales and the dissipation of baroclinic internal tides, especially in equatorial waters.