Normal mode decomposition and dispersive and nonlinear mixing in stratified fluids

TL;DR

This paper analyzes internal wave propagation in stratified fluids, using modal decomposition to study dispersive and nonlinear mixing phenomena, with implications for oceanography and fluid dynamics.

Contribution

It introduces a modal decomposition approach for stratified Euler equations, revealing dispersive and nonlinear mixing effects in internal wave dynamics.

Findings

Linear approximation described by dispersive wave equations

Dispersive mixing occurs when Brunt-Väisälä frequency varies

Nonlinear mixing exhibits different behavior from dispersive mixing

Abstract

Motivated by the analysis of the propagation of internal waves in a stratified ocean, we consider in this article the incompressible Euler equations with variable density in a flat strip, and we study the evolution of perturbations of the hydrostatic equilibrium corresponding to a stable vertical strati-fication of the density. We show the local well-posedness of the equations in this configuration and provide a detailed study of their linear approximation. Performing a modal decomposition according to a Sturm-Liouville problem associated to the background stratification, we show that the linear approximation can be described by a series of dispersive perturbations of linear wave equations. When the so called Brunt-Vais{\"a}l{\"a} frequency is not constant, we show that these equations are coupled, hereby exhibiting a phenomenon of dispersive mixing. We then consider more specifically shallow water configurations (when the horizontal scale is much larger than the depth); under the Boussinesq approximation (i.e. neglecting the density variations in the momentum equation), we provide a well-posedness theorem for which we are able to control the existence time in terms of the relevant physical scales. We can then extend the modal decomposition to the nonlinear case and exhibit a non-linear mixing of different nature than the dispersive mixing mentioned above. Finally, we discuss some perspectives such as the sharp stratification limit that is expected to converge towards two-fluids systems.