On the Boussinesq equations with non-monotone temperature profiles

TL;DR

This paper investigates the stability of the 2D Boussinesq equations with partial dissipation and non-monotone temperature profiles, demonstrating conditions under which the system remains stable and how mixing can suppress instabilities.

Contribution

It establishes linear and nonlinear stability results for the Boussinesq equations with non-monotone temperature profiles and partial dissipation, extending understanding of flow stability.

Findings

Linear stability if temperature gradient is small enough

Enhanced dissipation can suppress Rayleigh-Bénard instability

Results extend to nonlinear forced equations with vertical dissipation

Abstract

In this article we consider the asymptotic stability of the two-dimensional Boussinesq equations with partial dissipation near a combination of Couette flow and temperature profiles $T(y)$. As a first main result we show that if $T'$ is of size at most $\nu^{1/3}$ in a suitable norm, then the linearized Boussinesq equations with only vertical dissipation of the velocity but not of the temperature are stable. Thus, mixing enhanced dissipation can suppress Rayleigh-B\'enard instability in this linearized case. We further show that these results extend to the (forced) nonlinear equations with vertical dissipation in both temperature and velocity.