Internal wave resonances in strain flows

A simple mechanism of parametric excitation of internal gravity waves in a uniformly stably stratified flow under the inviscid Boussinesq approximation is presented. It consists in an oscillating planar irrotational strain field with frequency ω disturbed by three-dimensional plane waves. When the amplitude of the strain is weak, the problem is reduced to a Mathieu equation and a condition for parametric resonance is easily deduced. For a large-amplitude strain field equations are solved numerically with Floquet theory. In both cases, it is shown that parametric instabilities are excited when stratification is large enough, that is when N > 1/2ω , where N is the Brunt–Väisälä frequency of the flow. On the other hand, when N ≤ 1/2ω , the flow is shown to be stable for any periodic background excitation thanks to a theorem by Joukowski. Therefore, stratification promotes instability. In the strongly stratified case N\,\,ω , resonant waves satisfy the Billant–Chomaz self-similarity law and the resulting instabilities develop inside correlated quasi-horizontal layers. After discussion of the viscous effects, the theory of the paper is applied to the stability of an elliptical vortex in a rotating stratified medium.