On enhanced dissipation for the Boussinesq equations

In this article we consider the stability and damping problem for the 2D Boussinesq equations with partial dissipation near a two parameter family of stationary solutions which includes Couette flow and hydrostatic balance. In the first part we show that for the linearized problem in an infinite periodic channel the evolution is asymptotically stable if any diffusion coefficient is non-zero. In particular, this imposes weaker conditions than for example vertical diffusion. Furthermore, we study the interaction of shear flow, hydrostatic balance and partial dissipation. In a second part we establish stability and enhanced dissipation results for the nonlinear Boussinesq problem around flows combining Couette flow and hydrostatic balance for the setting of full dissipation. Here we adapt the methods used by Bedrossian, Vicol and Wang [4] in the Navier-Stokes problem and combine them with cancellation properties of the Boussinesq equations. (C) 2021 Elsevier Inc. All rights reserved.