Convergence to Stratified Flow for an Inviscid 3D Boussinesq System

TL;DR

This paper investigates the stability of stratified solutions in a 3D inviscid Boussinesq system, revealing how dispersive effects and gravity influence the convergence to 2D stratified Euler equations.

Contribution

It demonstrates the convergence of the 3D Boussinesq system to a 2D stratified Euler system as dispersion strength increases, considering the effects of gravity and stratification profile.

Findings

Stability depends on anisotropic dispersive operator properties.

Dispersive decay is influenced by gravity and stratification profile.

System converges to 2D Euler equations with stratified density as dispersion increases.

Abstract

We study the stability of special, stratified solutions of a 3d Boussinesq system describing an incompressible, inviscid 3d fluid with variable density (or temperature, depending on the context) under the effect of a uni-directional gravitational force. The behavior is shown to depend on the properties of an anisotropic dispersive operator with weak decay in time. However, the dispersive decay also depends on the strength of the gravity in the system and on the profile of the stratified solution, whose stability we study. We show that as the strength of the dispersion in the system tends to infinity, the 3d system of equations tends to a stratified system of 2d Euler equations with stratified density.