Statistical Dynamics of Mean Flows Interacting with Rossby Waves, Turbulence, and Topography

Abridged statistical dynamical closures, for the interaction of two-dimensional inhomogeneous turbulent flows with topography and Rossby waves on a beta-plane, are formulated from the Quasi-diagonal Direct Interaction Approximation (QDIA) theory, at various levels of simplification. An abridged QDIA is obtained by replacing the mean field trajectory, from initial-time to current-time, in the time history integrals of the non-Markovian closure by the current-time mean field. Three variants of Markovian Inhomogeneous Closures (MICs) are formulated from the abridged QDIA by using the current-time, prior-time, and correlation fluctuation dissipation theorems. The abridged MICs have auxiliary prognostic equations for relaxation functions that approximate the information in the time history integrals of the QDIA. The abridged MICs are more efficient than the QDIA for long integrations with just two relaxation functions required. The efficacy of the closures is studied in 10-day simulations with an easterly large-scale flow impinging on a conical mountain to generate rapidly growing Rossby waves in a turbulent environment. The abridged closures closely agree with the statistics of large ensembles of direct numerical simulations for the mean and transients. An Eddy Damped Markovian Inhomogeneous Closure (EDMIC), with analytical relaxation functions, which generalizes the Eddy Dampened Quasi Normal Markovian (EDQNM) to inhomogeneous flows, is formulated and shown to be realizable under the same circumstances as the homogeneous EDQNM.