Instabilities of internal gravity waves in the two-dimensional Boussinesq system

TL;DR

This paper rigorously demonstrates the modulational instability of internal gravity waves in a 2D inviscid Boussinesq system, providing the first proof of the Parametric Subharmonic Instability mechanism in this context.

Contribution

It offers a novel mathematical proof of the instability of internal gravity waves in an inviscid setting, using Floquet-Bloch analysis and Kato's transformations.

Findings

Proves eigenvalues with positive real part bifurcate from double eigenvalues.

First rigorous justification of the Parametric Subharmonic Instability in inviscid internal waves.

Uses Floquet-Bloch decomposition and Kato's transformations for eigenvalue analysis.

Abstract

We consider a two-dimensional, incompressible, inviscid fluid with variable density, subject to the action of gravity. Assuming a stable equilibrium density profile, we adopt the so-called Boussinesq approximation, which neglects density variations in all terms except those involving gravity. This model is widely used in the physical literature to describe internal gravity waves. In this work, we prove a modulational instability result for such a system: specifically, we show that the linearization around a small-amplitude travelling wave admits at least one eigenvalue with positive real part, bifurcating from double eigenvalues of the linear, unperturbed equations. This can be regarded as the first rigorous justification of the Parametric Subharmonic Instability (PSI) of inviscid internal waves, wherein energy is transferred from an initially excited primary wave to two secondary waves with different frequencies. Our approach uses Floquet-Bloch decomposition and Kato's similarity transformations to compute rigorously the perturbed eigenvalues without requiring boundedness of the perturbed operator - differing fundamentally from prior analyses involving viscosity. Notably, the inviscid setting is especially relevant in oceanographic applications, where viscous effects are often negligible.