Instabilities of rotating inclined buoyancy layers

The boundary layer near a cooled inclined plate, which is immersed in a stably stratified fluid rotating about an axis parallel to the direction of gravity, is a model for katabatic flows at high latitudes. In this paper the base flow of such an inclined buoyancy layer is solved analytically for arbitrary Prandtl numbers. By applying linear stability analyses, five unstable modes are identified for both the fixed temperature and the isoflux boundary conditions, i.e., the stationary longitudinal roll (LR) mode, the oblique roll with low streamwise wave-number (OR-1) and high streamwise wave-number (OR-2) modes, and the Tolmien-Schlichting (TS) wave with low streamwise wave-number (TS-1) and high streamwise wave-number (TS-2) modes. It is indicated that the Coriolis effect induced by the rotation leads the critical modes to be three dimensional, and a larger tilt angle of the plate and stronger Coriolis effect cause both TS wave modes to be more unstable for both thermal boundary conditions. When the Coriolis effect is considered, the OR-1 and OR-2 modes are the most unstable mode at low and high tilt angles, respectively, but the TS-1 wave mode may be the most unstable one when the plate is nearly vertical. In addition, the spanwise phase velocities of the TS wave modes change directions as the tilt angle passes some threshold values for both thermal boundary conditions except for the TS-1 wave mode with a fixed temperature boundary condition, which propagates in the same spanwise direction for all explored tilt angles.