Kelvin-Helmholtz billows above Richardson number 1/4

We study the dynamical system of a two-dimensional, forced, stratified mixing layer at finite Reynolds number , and Prandtl number . We consider a hyperbolic tangent background velocity profile in the two cases of hyperbolic tangent and uniform background buoyancy stratifications, in a domain of fixed, finite width and height. The system is forced in such a way that these background profiles are a steady solution of the governing equations. As is well known, if the minimum gradient Richardson number of the flow, , is less than a certain critical value , the flow is linearly unstable to Kelvin-Helmholtz instability in both cases. Using Newton-Krylov iteration, we find steady, two-dimensional, finite-amplitude elliptical vortex structures - i.e. 'Kelvin-Helmholtz billows' - existing above . Bifurcation diagrams are produced using branch continuation, and we explore how these diagrams change with varying . In particular, when is sufficiently high we find that finite-amplitude Kelvin-Helmholtz billows exist when 1/4> for the background flow, which is linearly stable by the Miles-Howard theorem. For the uniform background stratification, we give a simple explanation of the dynamical system, showing the dynamics can be understood on a two-dimensional manifold embedded in state space, and demonstrate the cases in which the system is bistable. In the case of a hyperbolic tangent stratification, we also describe a new, slow-growing, linear instability of the background profiles at finite , which complicates the dynamics.