Turbulent Convection in Stellar Interiors. I. Hydrodynamic Simulation

(Abridged) We describe the results of three-dimensional (3D) numericalsimulations designed to study turbulent convection in the stellar interiors,and compare them to stellar mixing-length theory (MLT). Simulations in 2D aresignificantly different from 3D, both in terms of flow morphology and velocityamplitude. Convective mixing regions are better predicted using a [dynamicboundary condition] based on the bulk Richardson number than by purely local,static criteria like Schwarzschild or Ledoux. MLT gives a good description ofthe velocity scale and temperature gradient for a mixing length of $\sim 1.1H_p$ for shell convection, however there are other important effects that itdoes not capture near boundaries. Convective "overshooting" is best describedas an elastic response by the convective boundary, rather than ballisticpenetration of the stable layers by turbulent eddies. We find that the rate atwhich material entrainment proceeds at the boundaries is consistent withanalogous laboratory experiments as well as simulation and observation ofterrestrial atmospheric mixing. In particular, the normalized entrainment rateE=$u_E/\sigma_H$, is well described by a power law dependence on the bulkRichardson number $Ri_B = \Delta b L/\sigma_H^2$ for the conditions studied,$20\lesssim Ri_B \lesssim 420$. We find $E = A Ri_B^{-n}$, with best fitvalues, $\log A = 0.027 \pm 0.38$, and $n = 1.05 \pm 0.21$. We discuss theapplicability of these results to stellar evolution calculations.