Subgrid-scale parameterizations of eddy-topographic force, eddy viscosity, and stochastic backscatter for flow over topography

General expressions for the eddy-topographic force, eddy viscosity, and stochastic backscatter, as well as a residual Jacobian term, are derived for barotopic flow over mean (single realization) topography. These subgrid-scale parameterizations are established on the basis of a quasi-diagonal direct interaction closure model, incorporating equations for the mean vorticity, vorticity covariance, and response functions. In general, the subgrid-scale parameterizations have a time-history integral representation, which reflects memory effects associated with turbulent eddies. In the Markov limit, the truncated equations for the ensemble mean and fluctuating parts of the vorticity have the same form as the full resolution equations but with the original 'bare' viscosity and bare mean and fluctuating forcings renormalized by eddy drain viscosities, eddy-topographic force, and stochastic backscatter terms. The parameterizations are evaluated at canonical equilibrium states for comparison with G. Holloway's heuristic expression for the eddy-topographic force, involving a product of the total viscosity and a canonical equilibrium expression for a mean vorticity. His functional form is recovered but with his total viscosity replaced by an eddy drain viscosity. For dynamical consistency, Holloway's parameterization also needs to be supplemented with a stochastic backscatter parameterization, even at canonical equilibrium. Implications of the results for subgrid-scale parameterizations of turbulent eddies in ocean and atmospheric circulation models are discussed.