Kelvin–Helmholtz billows above Richardson number

Abstract

We study the dynamical system of a forced stratified mixing layer at finite Reynolds number Re, and Prandtl number P r = 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, Ri m , is less than a certain critical value Ri c , 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 Ri c . 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 Ri m > 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, slowgrowing, linear instability of the background profiles at finite Re, which complicates the dynamics. † Email address for correspondence: jpp39@cam.ac.uk

The viscous Holmboe instability for smooth shear and density profiles

“…This leads to the possibility that HWI exists even when R is of order one, when Ri g > 1/4, at finite Reynolds number. Such an instability was demonstrated, for a single specific choice of parameters, by the authors previously in Parker, Caulfield & Kerswell (2019). This could have profound implications for our understanding of geophysical processes, since HWI is known to have very different mixing properties to KHI (Salehipour et al 2016).…”

“…We consider the nonlinear evolution, which allows us to see whether the viscous Holmboe instability develops the classic counter-propagating vortices of HWI. We use the same DNS code as Parker et al (2019) to solve the full Boussinesq equations, which is pseudospectral in the streamwise direction and utilises finite differences in the vertical. In the present case, the background flow is allowed to diffuse.…”

The effects of Prandtl number on the nonlinear dynamics of Kelvin–Helmholtz instability in two dimensions

“…The pitchfork bifurcation is clearly supercritical, in agreement with weakly nonlinear theory. Figure 2 also shows the bifurcation curve at described in Parker et al (2019). This is close to the degenerate case between super- and sub-criticality; it can just be made out that this case is very slightly subcritical.…”

“…Stress-free boundary conditions are imposed at . Both the solution of these equations and the finding and tracking of states and bifurcations largely uses the procedures presented in Parker et al (2019). The key difference is that the non-uniform vertical grid has been modified to give a broader region of high resolution in the centre of the domain, in that we now use grid points located at States are converged using Newton-generalised minimal residual (GMRES), then followed as parameters vary using pseudo-arclength continuation.…”

“…Although the effect of on the sub/supercriticality of the bifurcation is well documented, this gives only a weakly nonlinear understanding beyond classical linear stability analyses, and cannot predict the full nonlinear effects. It could be the case that full turbulence is possible through subcritical transition for flows with high minimum Richardson numbers, substantially above , where turbulence is usually assumed to be suppressed (Thorpe 2005), or it could be that non-trivial, nonlinear states do not exist in flows with significantly larger than , and that the behaviour is simple and transient, as was found for (Parker, Caulfield & Kerswell 2019). Below, we argue for the former scenario by presenting direct evidence that two-dimensional finite-amplitude billow-like states exist for – i.e.…”