Rotating Convection inf‐Planes: Mean Flow and Reynolds Stress

Abstract

We study turbulent compressible convection under the influence of rotation in an f-plane configuration using numerical simulation. Our focus is on the behaviors of the mean flows and the Reynolds stress. A parametric study is conducted, using 36 computed cases with different energy fluxes, rotation rates (Ω), and inclinations of the rotation vector. The flux varies over a factor of 8; the Coriolis number ranges from 0 to approximately 7; the inclination of the rotation vector covers the range from 0 (at the pole) to π/2 (at the equator). The coverage of this piece of parameter space is rather full and dense, so that we do not need to base our discussions on extrapolations of sparsely distributed cases. Special attention is paid to obtaining statistical convergence of the Reynolds stress, a very slow process that consumes much computer time. The numerical results show that: (1) Even though the properties of the convection zones are different (efficient versus inefficient convection), the behavior of our cases has considerable similarity to that of the turbulent cases explored by Brummell et al. (2) Between the two studies, the most significant difference in flow behavior occurs in the mean zonal velocity at low Rossby numbers. While Brummell et al. found a "constant-with-depth profile in the bulk of the layer" (and two "boundary layers"), we find that the profile develops a prominent retrograde dip at the top of the convection zone. (3) We offer an explanation for the dip based on the vertical distribution of the vertical-meridional component of the Reynolds stress. This may have relevance for understanding the radial drop of the solar angular velocity near the Sun's surface. (4) When the rotation vector is perpendicular to the vertical direction (at the equator), the behavior of the system undergoes a qualitative jump from those with other rotation vector inclinations. A shear with a strain rate of -2Ω develops in the mean zonal velocity, independent of the size of the energy flux. (5) We have tabulated the maxima and minima of the mean horizontal velocities, the rms velocities, and correlations between different velocity components for all the cases. Behavioral trends of these quantities can be readily identified. (6) The correlation of the meridional and zonal velocity fluctuations, important in both theory and observation, is found to change its sign from negative to positive when the Rossby number drops past 1. (7) After the meridional momentum equation (which ignores minor effects due to a shell geometry) is averaged over longitude, depth, and time, the resulting equation relates the vertically averaged Reynolds stress to the rotation rate of a whole conical sheet of fluid at a specific latitude. Using our f-plane results to estimate the amount of driving generated by the turbulent Reynolds stress in the solar convection zone, we find that its role is quite insignificant and inadequate for producing the observed differential rotation. We conclude that the latitudinal gas pressure gradient and the global circulation play more important roles.