Sensitivity of Horizontal Convection to Buoyancy and Wind-Stress Forcing

Horizontal Convection is a flow driven by differential buoyancy forcing across a horizontal surface. It has been considered as a simple model to study the influence of heating and cooling at the ocean surface on the Meridional Overturning Circulation. Here we consider the frequently-studied problem of horizontal convection driven by a step change in buoyancy, i.e. a transition from heating to cooling, along the surface.

We aim to compute the optimal buoyancy and wind stress perturbations which either maximize or minimize the overall circulation. We use the concept of local available potential energy as a measure of the circulation. The associated optimization problem is solved using the augmented Lagrangian approach. For instance this method allows for computing the sensitivity of the flow to the available potential energy flux inside the entire domain. Here the flow geometry consists of a rectangular box with an aspect ratio of 4 and the initial buoyancy distribution across the surface is given by a step function while enforcing zero wind stresses. Steady state solutions of the Navier-Stokes equations using the Boussinesq approximation at moderate Rayleigh numbers and Prandtl number close to unity have been considered.

The sensitivity analysis provides profiles for the optimal wind and buoyancy perturbations for a given Rayleigh number. We find that the response of the circulation to a small modification of the initial step profile is linear. However, further increasing the amplitude of the surface perturbation produces a nonlinear response and a dramatically increasing circulation. The impact of wind stresses and buoyancy are compared both concurrently and in isolation.