Direct numerical simulations of incompressible channel flow have been performed that explore the effects of centrifugally stable differential rotation on thermal convection (with gravity and rotation axes aligned). In order to provide greater correspondence to the interior regions of astrophysical accretion disks, especially to the convective solar nebula, we consider a gravity that varies linearly with distance from midplane and Keplerian rotation. We are restricted, however, to unrealistically low Reynolds numbers. Our findings are: (1) Statistical thermal convective properties depend almost exclusively on Peclet number and epicyclic frequency, regardless of the anisotropy induced by the shear. (2) At low Reynolds numbers, Reynolds stresses show the remarkable behavior of changing sign with increasing rotation rate, going from positive to negative shear production rates. Higher-Reynolds-number simulations tend to retain positive shear production rates to more rapid rotation. (3) At very rapid rotation, independent of Reynolds number, the flow becomes quasi-two-dimensional by losing streamwise variation in one or more of its fluctuating variables (especially the vertical velocity). At this point the simulation results become unreliable. These results suggest that convection in accretion disks is characterized by very long azimuthal wavelengths, and that, in some circumstances, Reynolds stresses can feed turbulence kinetic energy to the mean flow in contradiction to the conventional eddy-viscosity ansatz.
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