Mantle convection benchmarking in a 3D spherical shell: on the transitional behavior of polyhedral pattern formations

For 3-dimensional isoviscous thermal convection in a spherical shell, steady state patterns are predicted by analytical studies. Thus the tetrahedral and octahedral (or cubic) initial conditions are classically used to compare numerical codes and methods. However, polyhedral patterns with a higher order of symmetry and/or the stability of these convection patterns have not been studied. Here we performed a computational investigation on the stability of polyhedral steady states at various Rayleigh numbers (7,000 to 100,000). We compared two numerical codes: a finite element code CitcomS and a hybrid method that combines Chebyshev pseudo-spectral method and radial basis functions (RBF). The initial thermal perturbation is imposed by a combination of several spherical harmonics. A slight modification on the initial condition can destabilize the initial symmetry and can lead to a different final pattern. Three test cases are investigated. The first benchmark is based on the cubic initial condition (l=4). At a critical perturbation value, the symmetrical pattern is destabilized and leads to an axisymmetric convection pattern. The transition is well modeled by both methods. A second test case investigated the stability of the icosahedral initial condition (l=6). Although both methods converge first to the corresponding pattern, this structure is ultimately unstable and systematically degenerates to lower order of symmetries. The final convection pattern varies as a function of methods and Rayleigh number. Thus at Ra=10,000, CitcomS models converge to an axisymmetric pattern while RBF models converge to a tetrahedral pattern. The RBF method showed a longer stability period of the icosahedral structure than CitcomS. Lastly, we observed two new steady state patterns that have not been observed or predicted in the literature. We present the initial conditions as a combination of order 3 and 4 spherical harmonics leading to two different five cell patterns: a hexahedral and a pentahedral.