# Kelvin-Helmholtz billows above Richardson number $1/4$

## TL;DR

This study investigates the existence of finite amplitude Kelvin-Helmholtz billows above the classical Richardson number threshold, revealing complex bifurcation structures and new instabilities in stratified shear flows at finite Reynolds numbers.

## Contribution

It demonstrates that Kelvin-Helmholtz billows can exist at Richardson numbers greater than 1/4, challenging classical linear stability results, and explores their bifurcation behavior at finite Reynolds numbers.

## Key findings

- Kelvin-Helmholtz billows exist above Ri=1/4 at finite Re.
- Bifurcation diagrams reveal complex flow structures and bistability.
- A new slow-growing linear instability is identified in hyperbolic tangent stratification.

## Abstract

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, $Ri_m$, 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, slow-growing, linear instability of the background profiles at finite $Re$, which complicates the dynamics.