Stochastic subgrid parameterizations for atmospheric and oceanic flows

In this paper, we review recently developed closure-based and stochastic model approaches to subgrid scale modelling of eddy interactions motivated by advances in non-equilibrium statistical dynamical closure theory. We demonstrate how statistical dynamical closure models can be used to self-consistently calculate eddy damping and stochastic backscatter parameters, required in large eddy simulations (LESs), from higher-resolution simulations. A direct stochastic modelling scheme that is more generally applicable to complex models is then described and applied to LESs of quasigeostrophic turbulence of the atmosphere and oceans. We discuss the fundamental difference between atmospheric and oceanic LESs which is related to the difference in the deformation scales in the two classes of flows. We point out why the stochastic approach may be crucial when baroclinic instability is inadequately resolved. Finally, we discuss the application of inhomogeneous closure theory to the complex problem of flow over topography, and show that it can be used to understand the successes and limitations of currently used heuristic schemes and to provide a basis for further developments in the future.