Convergence to stratified flow for an inviscid 3D Boussinesq system

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.

Keywords

inviscid 3D Boussinesq, singular limit, stratification

2010 Mathematics Subject Classification

35Q35, 76B15, 76B70

Full Text (PDF format)

Received 11 November 2017

Accepted 28 May 2018

Published 7 February 2019