Finite amplitude Kelvin-Helmholtz billows at high Richardson number

We study the dynamical system of a stratified mixing layer at finite Reynolds number and unity Prandtl number with hyperbolic tangent profiles in streamwise background velocity and density, forced in such a way that these background profiles are a steady solution of the governing equations.

As is well-known, if the minimum Richardson number $Ri_m$ is less than a certain critical value $Ri_c$, the flow is linearly unstable to Kelvin-Helmholtz instability. We show that unstable, steady, two-dimensional, finite amplitude elliptical vortex structures , i.e. `Kelvin-Helmholtz billows', exist above $Ri_c$. Bifurcation diagrams are produced using branch continuation, and we explore how these diagrams change with varying Reynolds number. In particular, we examine whether such finite amplitude Kelvin-Helmholtz billows can exist at $Ri_m>1/4$, where the flow is linearly stable by the Miles-Howard theorem.