Two-dimensional turbulence above topography: condensation transition and selection of minimum enstrophy solutions

We consider two-dimensional flows above topography, revisiting the selective decay (or minimum-enstrophy) hypothesis of Bretherton and Haidvogel. We derive a 'condensed branch' of solutions to the variational problem where a domain-scale condensate coexists with a flow at the (smaller) scale of the topography. The condensate arises through a supercritical bifurcation as the conserved energy of the initial condition exceeds a threshold value, a prediction that we quantitatively validate using Direct Numerical Simulations (DNS). We then consider the forced-dissipative case, showing how weak forcing and dissipation select a single dissipative state out of the continuum of solutions to the energy-conserving system predicted by selective decay. As the forcing strength increases, the condensate arises through a supercritical bifurcation for topographic-scale forcing and through a subcritical bifurcation for domain-scale forcing, both predictions being quantitatively validated by DNS. This method provides a way of determining the equilibrated state of forced-dissipative flows based on variational approaches to the associated energy-conserving system, such as the statistical mechanics of 2D flows or selective decay.