A hybrid discrete exterior calculus and finite difference method for Boussinesq convection in spherical shells

TL;DR

This paper introduces a hybrid discrete exterior calculus and finite difference method for simulating three-dimensional Boussinesq convection in spherical shells, effectively addressing numerical challenges and demonstrating accurate, convergent results.

Contribution

The paper presents a novel hybrid DEC-FD numerical approach that preserves structure and avoids coordinate singularities in simulating spherical shell convection.

Findings

Accurate computation of critical Rayleigh numbers for various aspect ratios.

Successful simulation of complex convective patterns including giant spiral rolls.

Method achieves slightly better than second order convergence.

Abstract

We present a new hybrid discrete exterior calculus (DEC) and finite difference (FD) method to simulate fully three-dimensional Boussinesq convection in spherical shells subject to internal heating and basal heating, relevant in the planetary and stellar phenomenon. We employ DEC to compute the surface spherical flows, taking advantage of its unique features of structure preservation (e.g., conservation of secondary quantities like kinetic energy) and coordinate system independence, while we discretize the radial direction 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 the critical Rayleigh numbers for spherical shells characterized by aspect ratios ranging from 0.2 to 0.8, 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, and the quantification of Nusselt and Reynolds numbers for basally heated spherical shells. The method exhibits slightly better than second order error convergence with the mesh size.