Stability of internal gravity wave modes: from triad resonance to broadband instability

A theoretical study is made of the stability of propagating internal gravity wave modes along a horizontal stratified fluid layer bounded by rigid walls. The analysis is based on the Floquet eigenvalue problem for infinitesimal perturbations to a wave mode of small amplitude. The appropriate instability mechanism hinges on how the perurbation spatial scale relative to the basic-state wavelength, controlled by a parameter μ, compares to the basic-state amplitude parameter, ε << 1. For μ = O(1), the onset of instability arises due to perturbations that form resonant triads with the underlying wave mode. For short-scale perturbations such that μ << 1 but α = μ/ε >> 1, this triad resonance instability reduces to the familiar parametric sbharmonic instability (PSI), where triads comprise fine-scale perturbations with half the basic-wave frequency. However, as μ is further decreased holding ε fixed, higher-frequency perturbations than these two subharmonics come into play, and when α = Ο(1) Floquet modes feature broadband spectrum. In this regime, PSI is replaced by a novel, multi-mode resonance mechanism which has a stabilizing effect that provides an inviscid short-scale cut-off to PSI. The theoretical predictions are supported by numerical results from solving the Floquet eigenvalue problem for a mode-1 basic state.