The stability of a stratified shear layer

The stability of a stratified shear layer is investigated using an exponential density profile and a laminar shear flow with a continuous velocity distribution. It is shown that an exact stability boundary can be obtained for an inhomogeneous inviscid fluid under the action of gravity without the need to impose the Boussinesq approximation. The stability boundary is given by J=k̂2(1−β2/4−k̂2), where β is the ratio of the velocity and density gradient scale sizes, J is the Richardson number, and k̂ is the perpendicular wavenumber normalized to the velocity gradient scale size; this reduces to the stability boundary derived by Drazin [J. Fluid Mech. 4, 214 (1958)] in the limit β=0. The solution allows for c=β/2, where c is the normalized phase velocity.