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

Abstract

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 perturbation 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 subharmonic 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 = 𝑂(1) Floquet modes feature broadband spectrum. This broadening phenomenon is a manifestation of the advection of small-scale perturbations by the basic-wave velocity field. By working with a set of ‘streamline coordinates’ in the frame of the basic wave, this advection can be ‘factored out’. Importantly, when 𝛼 = 𝑂(1) PSI is replaced by a novel, multi-mode resonance mechanism which has a stabilising 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.