## Abstract

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.

Original language | English |
---|---|

Article number | R1 |

Number of pages | 10 |

Journal | Journal of Fluid Mechanics |

Volume | 879 |

Early online date | 23 Sept 2019 |

DOIs | |

Publication status | Published - 25 Nov 2019 |

## Keywords

- Bifurcation
- Nonlinear Instability
- Stratified Flows

## ASJC Scopus subject areas

- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering