The statistical dynamics of Geophysical flows

Although the topography has long been recognized as playing a major role in determining ocean atmosphere circulation, aspects of this interaction remain poorly understood. Recent parametrizations have been used to argue that for ocean circulations the important interaction is not the topographic gravity drag but the interaction between turbulent vorticies and the topography. Although there exist tractable closure theories for flow over topography these have been for ensembles of random topography with zero mean value. These models unfortunately say little about the effect of the mean topography of the ocean or atmosphere on the structures of the mean flows nor can they comment on the problem of parametrizing the effects of the interaction of subgrid-scale turbulent eddies with the mean topography.

In this context we have developed a non-Markovian closure model consisting of general expressions for the eddy-topographic force, eddy viscosity, and stochastic backscatter derived for barotropic flow over mean (single realization) topography, with and without non-Gaussian restarts. The closure is compared with ensemble averaged direct numerical simulations (DNS) for severely truncated two-dimensional Navier-Stokes flows. The model, established on the basis of a quasi-diagonal direct interaction (QDIA) closure incorporates equations for the mean vorticity, vorticity covariance and response functions, formulated for discrete spectra relevant to flows on the doubly periodic domain. This procedure allows the unambiguous comparison between the closure and the DNS as well as allowing for the incorporation of all interactions both local and non-local. A significant computational efficiency is gained via the periodic truncation of the potentially long time-history integrals where the closures are restarted using both two and three-point cumulants as new non-Gaussian initial conditions. The closures and DNS are compared in 80-day integrations employing typical meteorological time and space scales in inviscid, viscous decay and forced dissipative experiments. The closure equations demonstrate the required conservation laws in the inviscid system ie. kinetic energy and potential enstrophy, as well as satisfying the canonical equilibrium relation.