II.2.d. PPM Simulations of Intensely Turbulent Compressible Convection

The PPM gas dynamics code was used to solve the Euler equations of inviscid fluid flow in a 3D rectangular region of a convectively unstable atmosphere. A deep atmosphere, with the horizontally averaged density increasing by about a factor of 14 from the top to the bottom, was confined between two impenetrable, horizontal, friction free walls. The top wall was maintained at a constant, relatively cool temperature, while a constant heat flux was maintained through the bottom wall. A rectangular region was studied which was twice as wide in each horizontal dimension as its height. Periodic boundary conditions were applied in the horizontal dimensions. This region was wide enough only for a single convection cell in the lower portion of the unstable layer, where the local pressure scale height was largest. Studies at lower grid resolution of much wider regions suggest that typical convection cells would prefer to be about 25\% wider than the ones in our fine grid simulations. However, the focus of this work was the detailed interaction of the convective motions with the turbulence which they generate. Therefore the region of the simulation was set to contain only a single convection cell so that all possible grid resolution permitted by the size and speed of the computing resources available could be concentrated on this single convection cell in order to give an accurate representation of these interactions.

The supercomputer simulations in this study were mainly carried out at PSC on the T3D. The voluminous data which was generated about these convection flows was analyzed in detail during the last year and is presented in a preprint by Porter and Woodward. The PPM code was used in order to keep viscous effects to a minimum. The viscosity introduced by this code has been studied in detail in previous work. It acts almost exclusively on length scales below 16 grid cells in wavelength, where it very effectively dissipates kinetic energy into heat, conserving the total energy exactly. Using the formal characterization of the PPM viscosity appropriate to the length scale of the convective layer depth, we estimate that our highest resolution simulation reached a Rayleigh number of 2.0e+15 and a Prandtl number of 7.5e-05. These dimensionless numbers characterizing the physics of the problem are still several orders of magnitude short of their values in the convection zone of the sun, but they are well into the relevant regime of very large Rayleigh and very small Prandtl numbers. This high resolution run on the Cray T3D at Pittsburgh used a grid of 512x512x256 cells. This simulation was begun in 1994 literally with the arrival of the 512 node T3D at PSC, and it continued, off and on as time on this machine became available, for over a year. This PPM convection code on the T3D began at a performance level of about 7.3 Gflop/s on 512 processors, and through heroic and bizarre modifications to the code which should have been made by the compiler, David Porter managed to increase this performance to 9.7 Gflop/s. Initial results from this simulation were shown on our PowerWall display at Supercomputing '94 and at Supercomputing '95. An image from this simulation also appeared on the cover of Science magazine in Sept 1995. Images from this simulation can also be found at http://www.lcse.umn.edu/nsf. Our 512x512x256-cell PPM simulation, together with the lower resolution runs which preceded it, has provided us with our best insights to date into the interactions of convection and turbulence in the regime of very low viscosity which, of course, is the regime of practical interest for solar convection. This series of simulations has also served to establish quantitatively the grid resolution requirements for accurate representation of these phenomena in the high Reynolds number regime. The comparison of the spectra of the horizontal velocities at the middle height in the convective layer for 4 simulations in this series spanning a factor of 8 in grid resolution is perhaps the most compelling evidence that the simulations have converged at the finest grid resolution to an accurate description of the larger fluid motions, with no need for additional mesh refinements. The horizontally averaged temperature gradient within the convective layer, a critical quantity for comparison with standard mixing length theories, also shows close agreement between the simulations in this series, with higher grid resolution giving accurate results increasingly close to the domain boundaries. In the highest resolution run, the flow is accurately represented, essentially free from boundary effects, over about 2 pressure scale heights, so that, finally, quantitative comparisons with simplified theoretical models can be made. Such comparisons are made in the preprint of Porter and Woodward and continue to be extended and refined. It should be noted that the highest grid resolution used allows the coefficient of thermal diffusivity to be reduced, without giving up simulation accuracy, to such an extent that 80\% of the heat flux in the layer is carried by convective motions. This is the important regime of dominance of convective heat transport which is most relevant for the sun and which allows the clearest tests of conventional mixing length theories.