A Lorenz Model for an Anelastic Oberbeck-Boussinesq System

TL;DR

This paper introduces a Lorenz model for an anelastic Oberbeck-Boussinesq system that accounts for compressibility, demonstrating increased stability of the rest state through a simplified Lorenz approximation.

Contribution

It develops a Lorenz approximation for a complex anelastic Boussinesq system, providing insights into stability and convection onset with compressibility effects.

Findings

Increased stability of the rest state in the anelastic system

Critical Rayleigh number likely higher due to compressibility

Lorenz model captures essential dynamics of the system

Abstract

In an Oberbeck-Boussinesq model, rigorously derived, which includes compressibility, one could expect that the onset of convection for the B\'enard problem occurs at a higher critical Rayleigh number. Since of the difficulties related to the new partial differential equations, with non constant coefficients and a non divergence-free velocity field, we show the increased stability of the rest state by exploring the related Lorenz approximation system.