Vertically forced stably stratified cavity flow: instabilities of the basic state

Abstract

The linear stability of a stably stratified fluid-filled cavity subject to vertical oscillations is determined via Floquet analysis. Retaining the viscous and diffusion terms in the Navier–Stokes–Boussinesq equations, with no-slip velocity boundary conditions, no-flux temperature conditions on the sidewalls and constant temperatures on the top and bottom walls, we find that the instabilities are primarily subharmonic (as is typical in many parametrically forced systems), except for in a few low-forcing-frequency ranges where the instabilities are synchronous. When the viscosity is small, the Floquet modes resemble the inviscid eigenmodes of the unforced problem, except in boundary layers. We establish scaling laws quantifying how viscosity regularizes the degeneracy associated with the inviscid idealization, and how it scales the thickness and intensity of the boundary layers. The product of boundary layer thickness and intensity remains constant with decreasing viscosity, leading to a delta distribution of vorticity on the walls in the limit of zero viscosity. This is in contrast to the zero wall vorticity in the inviscid case.