Three-dimensional simulations of turbulent compressible convection

Three-dimensional simulations of turbulent and fully compressible thermal convection in deep atmospheres are presented and analyzed in terms of velocity power spectra, mixing-length theory, and production of vorticity. Density contrasts across these convective layers are typically around 11. The fluid model is that of an ideal gas with a constant thermal conductivity. The piecewise parabolic method (PPM), with thermal conductivity added in, is used to solve the fluid equations of motion. No explicit viscosity is included, and the low numerical viscosity of PPM leads to a very low effective Prandtl number and very high effective Rayleigh number. Mesh resolutions range as high as 512 × 512 × 256, and the corresponding effective large-scale Rayleigh numbers range as high as 3.3 × 10¹². Compressional effects lead to intensely turbulent downflow lanes and relatively laminar updrafts, especially near the top boundary. The enstrophy contrast between downflows and upflows increases with mesh resolution (and hence with decreasing viscosity) and ranges as high as a factor of 30 in our highest resolution model. Vorticity is everywhere preferentially aligned with the principal direction of strain associated with the large-scale circulation. Near the top boundary, the strain field associated with the largest scale of convection dominates, which leads to a two-dimensional horizontal network of vortex tubes. For the same reason, both the upper portions of the downflow lanes and the lower portions of the updrafts contain many strong vertical vortex tubes with helicities of random sign. The horizontal vortex tubes near the very top of the downflow lanes tend to come in counterrotating pairs, with one on each side of the downflow lane. Therefore, as with observations of the Sun, there are upflows along each side of the prominent downflows in our simulations. Mach numbers in these convective layers are largest in the upper, more diffuse region. There they range as high as 0.8, which significantly modifies the pressure and gravitational force balance from that which would apply under static conditions. This effect is incorporated into our mixing-length analysis of the simulation data.