Numerical study of the sharp stratification limit towards bilayer models

TL;DR

This paper compares continuous stratified and bilayer models of oceanic flows, demonstrating convergence of the former to the latter and analyzing the limitations of bilayer models due to Kelvin-Helmholtz instabilities through numerical dispersion relation computations.

Contribution

It provides a mathematical convergence analysis of stratified Euler equations to bilayer models and evaluates the stability limits of bilayer models under shear flow conditions.

Findings

Convergence of stratified Euler equations to bilayer models near piecewise constant profiles.

Kelvin-Helmholtz instabilities restrict the validity of bilayer models.

Numerical dispersion analysis highlights stability limits of bilayer models.

Abstract

In the study of oceanic flows at the geophysical scale, the phenomenon of density stratification plays a central role in the dynamics of the system. Two categories of mathematical models are commonly used to describe the role played by the density stratification: on the one hand, continuously stratified models - such as the stratified Euler equations in a strip, considered in the present article - offer an accurate description of vertical effects, but come with a high level of complexity, both at the theoretical and numerical levels. On the other hand, bilayer models approximate the stratification by a piecewise constant profile. In the latter case, the main point is to study the evolution of the free interface between both layers, which leads to a substantially simplified model. In the present article, we compare both approaches in the framework of the linearized stratified Euler equations around density profiles that are close to piecewise constant profiles, and prove the convergence towards the bilayer Euler equations. However, in the presence of a shear flow, bilayer models have a range of validity limited by the presence of Kelvin-Helmholtz instabilities. In this case, we use a suitable normal modes decomposition to compute numerically the dispersion relation of this linearized model, and provide numerical evidence that the Kelvin-Helmholtz instabilities limit the applicability of two widely used bilayer models, namely the bilayer Euler equations and the bilayer shallow-water equations.