Numerical Simulations of Compressible Convection

Abstract

We outline the results of a series of three-dimensional computations of thermally driven, compressible, and turbulent convection in deep atmospheres[1, 2, 3, 4, 5, 6, 7] performed with the PPM algorithm[8, 9, 10]. We use these numerical simulations to examine the nature of turbulence which is driven by convection in deep atmospheres. Boundary conditions are free slip impenetrable walls at top and bottom, and periodic in the two horizontal directions. The top boundary is kept at a constant temperature, and a uniform heat flux is imposed along the lower boundary. The flow follows a γ—law with an adiabatic index of γ = 5/3. A uniform gravitational acceleration is imposed in the vertical, which is normal to the impenetrable walls. The entire layer is convectively unstable and the resulting convection is efficient. Hence the layer is well mixed and nearly adiabatic. The total heat content of the system is chosen so that the resulting gravitationally stratified atmosphere has a density contrast of 11. Runs are performed on uniform grids ranging from 64 × 64 × 32 to 512 × 512 × 256 computational cells. All of the simulations have aspect ratios of 2 by 2 by 1, with the short dimension being in the vertical. Effective large scale Rayleigh and Prandtl numbers range from Ra = 4.354 × 1012 and Pr = 3.865 × 10−2 to Ra = 2.229 × 1015 and Pr = 7.549 × 10−5. Stellar convection, in stars like the Sun, is characterized by high Rayleigh number and low Prandtl number. All four of these simulations are run long enough to be in convective equilibrium with the imposed heat flux. The computation on the largest grid is run for 40 large eddy turnover times.