Thermally driven flows in spherical geometries

In this manuscript we describe an efficient numerical scheme for simu- lations of three-dimensional Navier-Stokes equations for incompressible viscous flows in spherical coordinates. The code is second order accurate in space and time and relies on a finite–difference discretization in space. The nonphysical singularities induced by the change of coordinates are addressed by exploiting a change of variables and special treatments of few discrete terms. Thanks to these precautions the time–step restrictions caused by the region around the po- lar axis are alleviated and the sphere center is source of limitations only in very unfavorable flow configurations. We test the code and compare results with literature, always obtaining an excellent agreement. The flexibility due to the structure of the code allows it to perform efficiently in several applications without requiring changes in the structure: the mesh can be stretched (in two of the three directions), complex boundary conditions can be implemented, and in addition to full spheres, also spherical shells and sectors can be easily simulated. Characterization of the behaviour of fluids between spherical shells is the focus of the second part of the manuscript. We firstly explored the low-Rayleigh number regime for non rotating Rayleigh-B ́enard convection. Various radial gravity profiles are analysed for both air and water. We observe how the effect of the different gravity can be reabsorbed by the introduction of an effective Rayleigh number, yielding a critical Rac ≈ 1750 for the onset of convection regardless of the specific gravity profile. The exploration of higher values of Ra shows that the system is subjected to hysteresis, i.e. the dynamic has a very strong dependence on initial conditions and flow parameters. We then explore the effect of an offset between the sphere center and the gravity center, which might be used to simulate the effect of a dishomogeneity in the Earth core. Even a small displacement between the two points gives rise to a distorted temperature profile, with a hot jet emerging from the inner sphere in the direction opposite to the shift. Nevertheless, while the local heat flux and temperature profile are greatly modified, the global heat flux seems to be mostly unaffected by these changes. Lastly, we analysed the diffusion–free scaling regime for slowly rotating Rayleigh- B ́enard convection between spherical shells. This regime is characterized by a bulk–dominated flow and its emergence, for the parameters used, is due to the peculiar properties of the spherical geometry.