Turbulent thermal convection in a sphere

I. Cupall¹ - P. Hejda¹ - M. Reshetnyak^2,3

¹ Geophysical Institute, Acad. Sci, 141 31 Prague, Czech Republic
² Institute of the Physics of the Earth, Russian Acad.~Sci, 123995 Moscow, Russia
³ Research Computing Center of Moscow State University, 119899, Moscow, Russia

Abstract
Convection in a rotating spherical layer in the Boussinesq approximation is considered. A free rotating solid concentric inner sphere is included. The problem of developed turbulence is solved using the shell model approach. Two sets of equations are used. The Navier-Stokes equation and the thermal flux equation are solved on a large scale due to a coarse grid. The small-scale solution is described by the shell model, which generates another set of equations. This enables us to estimate the spectral energy flux on small scales. The turbulent coefficients, depending on the radial direction, are then calculated and used in the large-scale solution. The behavior of other characteristics (spectra, helicity) is also studied in time and space. The stabilized solution of the large-scale convection is obtained for the Rayleigh number \Ra = 10¹⁴ and the Ekman number \E = 10⁻⁶ based on the molecular values of viscosity and thermal diffusivity. The results correspond to the Reynolds number \Rn  ∼ 10⁹. Figs 5, Refs 22.