Geodynamo and mantle convection simulations on the Earth Simulator using the Yin-Yang grid

We have developed finite difference codes based on the Yin-Yang grid for the geodynamo simulation and the mantle convection simulation. The Yin-Yang grid is a kind of spherical overset grid that is composed of two identical component grids. The intrinsic simplicity of the mesh configuration of the Yin-Yang grid enables us to develop highly optimized simulation codes on massively parallel supercomputers. The Yin-Yang geodynamo code has achieved 15.2 Tflops with 4096 processors on the Earth Simulator. This represents 46% of the theoretical peak performance. The Yin-Yang mantle code has enabled us to carry out mantle convection simulations in realistic regimes with a Rayleigh number of $10^7$ including strongly temperature-dependent viscosity with spatial contrast up to $10^6$.Comment: Plenary talk at SciDAC 200

Depletion potentials in highly size-asymmetric binary hard-sphere mixtures: Comparison of accurate simulation results with theory

We report a detailed study, using state-of-the-art simulation and theoretical methods, of the depletion potential between a pair of big hard spheres immersed in a reservoir of much smaller hard spheres, the size disparity being measured by the ratio of diameters q=\sigma_s/\sigma_b. Small particles are treated grand canonically, their influence being parameterized in terms of their packing fraction in the reservoir, \eta_s^r. Two specialized Monte Carlo simulation schemes --the geometrical cluster algorithm, and staged particle insertion-- are deployed to obtain accurate depletion potentials for a number of combinations of q\leq 0.1 and \eta_s^r. After applying corrections for simulation finite-size effects, the depletion potentials are compared with the prediction of new density functional theory (DFT) calculations based on the insertion trick using the Rosenfeld functional and several subsequent modifications. While agreement between the DFT and simulation is generally good, significant discrepancies are evident at the largest reservoir packing fraction accessible to our simulation methods, namely \eta_s^r=0.35. These discrepancies are, however, small compared to those between simulation and the much poorer predictions of the Derjaguin approximation at this \eta_s^r. The recently proposed morphometric approximation performs better than Derjaguin but is somewhat poorer than DFT for the size ratios and small sphere packing fractions that we consider. The effective potentials from simulation, DFT and the morphometric approximation were used to compute the second virial coefficient B_2 as a function of \eta_s^r. Comparison of the results enables an assessment of the extent to which DFT can be expected to correctly predict the propensity towards fluid fluid phase separation in additive binary hard sphere mixtures with q\leq 0.1.Comment: 16 pages, 9 figures, revised treatment of morphometric approximation and reordered some materia