Ill-posedness of the hydrostatic Euler-Boussinesq equations and failure of hydrostatic limit

TL;DR

This paper demonstrates the ill-posedness of the hydrostatic Euler-Boussinesq equations, showing that the hydrostatic approximation can fail due to inherent instabilities and nonlinear breakdown, challenging its validity in modeling stratified fluids.

Contribution

It provides a rigorous analysis revealing the nonlinear ill-posedness and instability of the hydrostatic limit for stratified fluids, highlighting limitations of the hydrostatic approximation.

Findings

Identification of stratified steady states violating Miles-Howard criterion

Demonstration of growing modes in linearized hydrostatic and non-hydrostatic equations

Proof of generic nonlinear ill-posedness in Sobolev spaces

Abstract

We investigate the hydrostatic approximation for inviscid stratified fluids, described by the two-dimensional Euler-Boussinesq equations in a periodic channel. Through a perturbative analysis of the hydrostatic homogeneous setting, we exhibit a stratified steady state violating the Miles-Howard criterion and generating a growing mode, both for the linearized hydrostatic and non-hydrostatic equations. By leveraging long-wave nonlinear instability for the original Euler-Boussinesq system, we demonstrate the breakdown of the hydrostatic limit around such unstable profiles. Finally, we establish the generic nonlinear ill-posedness of the limiting hydrostatic system in Sobolev spaces.