Variational Closures for Two-Dimensional Turbulent Flows.

Unsteady two-dimensional flows satisfy two independent integral equations in the steady state: the balance equations for energy and enstrophy. In the present work we model these eddying flows with solutions that satisfy the same integral equations. In our closures we select one of the solutions of the integral equations by maximizing the eddy energy dissipation with respect to the time averaged flow. For that ad hoc principle, the total input must be specified to obtain an extremal solution with the same energetic properties as the simulated flow. In the case of barotropic flows, the eddy-dissipation terms in the energy balance equation can be expressed in terms of mean quantities by using the enstrophy equation and introducing a characteristic scale for the flow. A model solution can then be generated by extremizing the eddy-energy dissipation, for a given enery input. The solution is given by a second order linear differential equation with two parameters, that can be chosen to satisfy the constraints imposed on the flow (that is the given turbulence scale and energy input). We find good agreement between the extremal solution and the statistics of time -dependent calculations. We also find that in some flows, natural constraints, such as a lower bound on the dissipation of the time-averaged flow, can force the extremal solution close to the realized flow, without specifying the energy input. Extending the method to a barotropic ocean model in a square basin gives a turbulent, almost-inertial solution sustained by eddies. We extend the analysis to baroclinic flows, in a two-layer beta-plane channel forced by Newtonian cooling and bottom friction. After the introduction of an eddy scale to eliminate the eddy term in the energy equation with the enstrophy equation, we generate a solution by maximizing the eddy-energy dissipation, for specified energy input and dissipation on the mean surface zonal wind. The solution is given by two coupled differential equations governing the baroclinic and barotropic parts of the flow. The equations contain three parameters that can be determined to yield solutions that compare favorably with the statistics generated by a time-dependent model.