Stability of a stratified boundary layer flow when shear and stratification are not aligned

The inviscid stability properties of a boundary layer flow with a tanh velocity profile in a stably stratified fluid with a constant Brunt-V\"ais\"al\"a frequency is examined when the direction of the shear is inclined with an angle $\alpha$ with respect to the vertical direction of stratification. We show that for all Froude numbers there exists a critical angle $\alpha_c$ above which the boundary layer becomes inviscidly unstable. The characteristics of the unstable modes are analysed. For small Froude numbers, unstable modes are shown to be 3D radiative modes with a internal wave structure that extends to infinity. For large Froude numbers, the modes are localized in the boundary layer and their frequency and growth rate are proportional to the Brunt-V\"ais\"al\"a frequency. Non-Boussinesq and compressibility effects on the stability properties are also considered. The results are discussed in the context of atmospheric applications.