Kelvin-Helmholtz billows above Richardson number 1/4

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Parker, JP 
Caulfield, CP 
Kerswell, RR 

We study the dynamical system of a forced stratified mixing layer at finite Reynolds number Re, and Prandtl number Pr=1. We consider a hyperbolic tangent background velocity profile in the two cases of hyperbolic tangent and uniform background buoyancy stratifications. 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, Rim, is less than a certain critical value Ric, 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 Ric. Bifurcation diagrams are produced using branch continuation, and we explore how these diagrams change with varying Re. In particular, when Re is sufficiently high we find that finite amplitude Kelvin-Helmholtz billows exist at Rim>1/4, where the flow 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 Re, which complicates the dynamics.

bifurcation, stratified flows, nonlinear instability
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Journal of Fluid Mechanics
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Cambridge University Press (CUP)
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Engineering and Physical Sciences Research Council (EP/K034529/1)
EPSRC (1940773)