Sub-grid-scale parameterisations for large-scale eddy simulations

The accuracy of simulations of turbulent flows is strongly dependent on the appropriate treatment of subgrid-scale processes that cannot be explicitly resolved at the finite resolutions of the simulations. Subgrid-scale parameterizations of eddy viscosity, stochastic backscatter and net eddy viscosity are formulated for two-dimensional turbulence. based on eddy damped quasi-normal Markovian (EDQNM) and direct interaction approximation closures. The focus of the study is geophysical flows described by the barotropic vorticity equation but the results should also be relevant to the low frequency drift wave dynamics of magnetized plasmas described by the Hasegawa-Mima equation. The subgrid scale parameterizations are found to have a cusp behaviour at the cutoff wavenumber where they have their largest magnitudes. The conventional net eddy viscosity is shown to be the relatively small difference between the eddy drain viscosity and the backscatter viscosity. Large-eddy simulations (LES) with the barotropic vorticity equation have been performed incorporating these dynamic sub-grid scale 7 parameterizations and compared with higher-resolution direct numerical simulations (DNS), which are regarded as the benchmark or "truth" for comparisons. Good comparisons are found between kinetic energy spectra for the LES and DNS at the scales of the LES for both nonrotating two-dimensional turbulence and differentially rotating Rossby wave (drift wave) turbulence. This is contrasted with much poorer comparisons when using a number of ad hoc eddy viscosity parameterizations is LES. The applications of the results to more complex circulations models is discussed.