On the Boussinesq equations with non-monotone temperature profiles

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 $T^{\prime }$ 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-Bénard 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.

arXiv comment30 pages; updated and added references

References in corpus (10)

• On Landau damping
• The Sobolev stability threshold for 2D shear flows near Couette
• Global well-posedness of the 2D Boussinesq equations with vertical dissipation
• Linear inviscid damping for monotone shear flows in a finite periodic channel, boundary effects, blow-up and critical Sobolev regularity
• Sharp decay estimates for an anisotropic linear semigroup and applications to the SQG and inviscid Boussinesq systems
• Convergence to Stratified Flow for an Inviscid 3D Boussinesq System
• Linear Inviscid Damping for Couette Flow in Stratified Fluid
• Stability of Couette flow for 2D Boussinesq system with vertical dissipation
• On the Smallness Condition in Linear Inviscid Damping: Monotonicity and Resonance Chains
• On Echo Chains in Landau damping: Self-similar Solutions and Gevrey 3 as a Linear Stability Threshold

Cited by in corpus (1)

• Stability threshold for 2D shear flows of the Boussinesq system near Couette