Boussinesq Convection in Spherical Shells using a Hybrid Discrete Exterior Calculus and Finite Difference Method

We present a new hybrid discrete exterior calculus (DEC) and finite difference (FD) method to simulate Boussinesq convection in spherical shells. We discretize the surface spherical operators using DEC, taking advantage of its unique features including coordinate system independence to preserve the spherical geometry, while we discretize the radial operators using FD method. The grid employed for this novel method is free of problems like the coordinate singularity, grid non-convergence near the poles, and the overlap regions. We have developed a parallel in-house code using the PETSc framework to verify the hybrid DEC-FD formulation and demonstrate convergence. We have performed a series of numerical tests which include quantification of Nusselt and Reynolds numbers for basally heated spherical shells, quantification of the critical Rayleigh numbers for spherical shells characterized by aspect ratios ranging from 0.2 to 0.8, and the simulation of robust convective patterns in addition to stationary giant spiral roll covering all the spherical surface in moderately thin shells near the weakly nonlinear regime.