Direct Statistical Simulation and Generalized Quasilinear Approximation of Mean Flows and Convection in the Busse Annulus

Understanding the generation of mean flows from anisotropic turbulence is crucial to developing insight into a number of astrophysical and geophysical problems. The Busse Annulus is a common and simple 2D model for rotating convection. Direct numerical simulation has revealed that the system creates mean flows from the turbulence due to rotation-induced correlations. Here, we discuss the application of direct statistical simulation and the generalized quasilinear approximation to the convection in the Busse Annulus. Direct statistical simulation in the CE2 approximation uses the standard quasilinear approach to solve for the cumulants of the velocities and temperature perturbations, rather than the fields themselves. The generalized quasilinear approximation divides the flow into "low" and "high" wavenumbers, with the "low" modes taking the place of the mean flow. We will demonstrate that even a very small number of "low" modes significantly outperforms the standard quasilinear approximation in predicting mean flows in the Busse Annulus. Finally, we will compare generalized quasilinear results with CE2 direct statistical simulation and discuss the path to constructing a generalized CE2 method combining generalized quaslinear techniques with direct statistical simulation.