The 2D Boussinesq equations with vertical dissipation and linear stability of shear flows

Abstract

This paper studies the linear stability of a steady-state solution with the velocity being a shear flow to the 2D Boussinesq equations with only vertical dissipation. The Boussinesq equations model many fluid phenomena when the Boussinesq approximation applies such as the Rayleigh-Benard convection, atmospheric fronts and oceanic circulation. The vertically dissipative 2D Boussinesq equations model geophysical fluids in certain physical regimes. Whether or not the vertical dissipation can damp perturbations near the equilibrium with the velocity being a shear and the temperature being zero is an important but difficult problem. Assuming the spatial domain is periodic in the horizontal direction and half-line in the vertical direction with no flux boundary condition, we show that any perturbation satisfying the linearized equation around this equilibrium is infinitely smooth in the x−variable and decays exponentially in time and in the horizontal Fourier mode, even though the linearized system involves only vertical dissipation.