Optimal perturbations and transition to turbulence at density fronts

Abrupt changes in fluid density, or fronts, are common features in a variety of industrial and environmental settings. Fronts with horizontal density gradients are associated with available potential energy which can be converted to kinetic energy and turbulent motions. At the same time, the release of potential energy and lowering of the centre of mass leads to a stable vertical density profile which suppresses turbulence. In rotating environments, the hydrostatic pressure gradient associated with horizontal density fronts can be balanced by the Coriolis acceleration. However, this equilibrium state is unstable to a variety of fluid dynamical instabilities. Previous work has almost exclusively focused on linearly unstable normal modes which is unable to make a direct connection between the unstable modes and turbulence. Here, we will take a different approach and look for nonlinear 'optimal perturbations' - those disturbances which grow fastest over a prescribed time interval. Because this approach permits nonlinear interacting modes, it can be used to study the transition from an unstable to a turbulent state.