Short-Wave Instability in a Three-Layer Stratified Shear Flow

Abstract

In inviscid fluid flows instability arises generically due to a resonance between two wave modes. Here, it is shown that the structure of the weakly nonlinear regime depends crucially on whether the modal structures coincide, or remain distinct, at the resonance point, where the wave phase speeds coincide. For short waves, the generic model is correspondingly either a nonlinear Klein–Gordon equation for the wave envelope, or a pair of coupled first-order envelope equations. A specific case, namely a three-layered stratified shear flow is examined to illustrate this, with a typical derivation.