Triadic resonance of internal wave modes with background shear

Abstract

In this work we study resonant triad interactions among discrete internal wave modes in a finite-depth, two dimensional uniformly stratified shear flow. The primary wave-field is considered to be a linear superposition of various internal wave modes. The weakly-nonlinear solution of the primary wave-field consists of a superharmonic (2ω) part and a mean-flow part (ω=0).  For a given modal interaction, we study the location in the frequency (ω) -Richardson number (Ri) parameter space where the amplitude of the superharmonic part attains a maximum i.e, where two primary internal wave modes of modenumbers 'm' and 'n' resonantly excite a secondary wave mode of modenumber 'q'. Using asymptotic theory we show that, unlike the case of no-shear, the presence of weak-shear, doesn't require the vertical wavenumber condition to be satisfied for resonance. This entails an activation of several new resonances in the presence of arbitrarily weak shear, where only the frequency and the horizontal wavenumber conditions are satisfied. This also leads to the possibility of self-interaction and resonances close to ω = 0. A similar asymptotic theory can be extended to other inhomogeneities (eg: non-uniform stratification) as well. For an exponential background shear flow, we track the location of these resonances in the (ω, Ri) parameter space and present their behaviour.