Stability analysis of a resonant triad in a stratified uniform shear flow

The amplitude equations governing the temporal evolution of internal waves forming a resonant triad in a two-dimensional inviscid stably stratified uniform shear flow, bounded between two infinite horizontal parallel plates, are derived in the absence of diffusive effects. The density is considered to be a linear function of the vertical coordinate. The interaction of two vertically confined and horizontally propagating primary internal waves having the same frequency is considered. Specifically, for different local Richardson numbers, we show the existence and stability of the resonant triad formed by three different internal waves having the wave vector and frequency pairs as ((k) over right arrow (m), omega), ((k) over right arrow (n), omega), and ((k) over right arrow (m) + (k) over right arrow (n), 2 omega). For each resonant triad, we solve the amplitude equations numerically as an initial value problem. The equilibrium solutions of the amplitude equations are obtained analytically. Furthermore, the linear stability analysis of the resonant triads, around the equilibrium solutions, is carried out for various interaction cases. The triads containing a wave with the lowest mode number are found to be linearly unstable. The exact solutions of the amplitude equations are presented under the pump-wave approximation, in which the amplitude of one of the waves in the triad is larger than the amplitudes of the other two waves initially. From the exact solutions, the amplitudes of the waves forming a resonant triad are found to be unbounded when the wave with the mode number 1 acts as the pump wave. The present study helps one to have a better understanding of the stability of resonant triads of internal waves formed in a stratified shear flow.

Automated summary

This study investigates the temporal evolution of internal waves forming a resonant triad in a two-dimensional inviscid stably stratified uniform shear flow. It derives amplitude equations, equilibrium solutions, and conducts linear stability analysis around equilibrium solutions for various interaction cases. Numerical solutions for different local Richardson numbers are also presented.