Markovian inhomogeneous closures for Rossby waves and turbulence over topography

Manifestly Markovian closures for the interaction of two-dimensional inhomogeneous turbulent flows with Rossby waves and topography are formulated and compared with large ensembles of direct numerical simulations (DNS) on a generalized $\unicode[STIX]{x1D6FD}$-plane. Three versions of the Markovian inhomogeneous closure (MIC) are established from the quasi-diagonal direct interaction approximation (QDIA) theory by modifying the response function to a Markovian form and employing respectively the current-time (quasi-stationary) fluctuation dissipation theorem (FDT), the prior-time (non-stationary) FDT and the correlation FDT. Markov equations for the triad relaxation functions are derived that carry similar information to the time-history integrals of the non-Markovian QDIA closure but become relatively more efficient for long integrations. Far from equilibrium processes are studied, where the impact of a westerly mean flow on a conical mountain generates large-amplitude Rossby waves in a turbulent environment, over a period of 10 days. Excellent agreement between the evolved mean streamfunction and mean and transient kinetic energy spectra are found for the three versions of the MIC and two variants of the non-Markovian QDIA compared with an ensemble of 1800 DNS. In all cases mean Rossby wavetrain pattern correlations between the closures and the DNS ensemble are greater than 0.9998.

Theory of particle diffusion in electrostatic turbulent plasma using extended direct-interaction approximation

The theory of particle diffusion in an electrostatic turbulent plasma is formulated by applying the direct-interaction approximation (DIA) to subensemble-averaged functions instead of conventional ensemble-averaged ones. This theory approximately incorporates the Lagrangian description into the DIA through decorrelation trajectories. The running diffusion coefficient is shown to be calculated by solving a nonlinear ordinary differential equation together with an equation for decorrelation trajectories and by averaging initial conditions at the starting point of trajectories.

Dynamical Subgrid-Scale Parameterizations from Direct Numerical Simulations

Dynamical subgrid-scale parameterizations of stochastic backscatter, eddy drain viscosity, and net eddy viscosity have been formulated and calculated for two-dimensional turbulent flows on the sphere based on the statistics of direct numerical simulations (DNSs) with the barotropic vorticity equation. A relatively simple methodology based on a stochastic model representation of the subgrid-scale eddies, but which takes into account the memory effects of turbulent eddies, has been employed. The parameterizations have a cusp behavior at the cutoff wavenumber of the retained scales and have closely similar forms to those based on eddy damped quasi-normal Markovian (EDQNM) and direct interaction approximation (DIA) closure models. Large-eddy simulations (LESs) incorporating DNS-based subgrid-scale parameterizations are found to have kinetic energy spectra that compare closely with the results of higher-resolution DNS at the scales of LES for both isotropic turbulence and Rossby wave turbulence. The methodology presented is general and should be equally applicable to parameterizations of baroclinic processes and convective processes. Applications of the parameterizations to climate models and prediction models are discussed.

KOLMOGOROV CONSTANT IN LARGE SPACE DIMENSIONS

A large d (space dimension) expansion together with the ∊-expansion is implemented to calculate the Kolmogorov constant from the energy equation of Kraichnan's direct-interaction approximation using the Heisenberg's eddy-viscosity approximation and Kraichnan's distant-interaction algorithm. The Kolmogorov constant C is found to be C = C0 d1/3 in the leading order of a 1/d expansion. This is consistent with Fournier, Frisch, and Rose. The constant C0 evaluated in the above scheme, is found to be C0 = (16/27)1/3.