M. Reshetnyak $^{a,b}$, I. Cupal $^{c}$, and P. Hejda $^{c}$
$^{a}$ Institute of Physics of the Earth, Russian Acad.~Sci, Moscow, Russia $^{b}$ Research Computing Center, Moscow State University, Moscow, Russia $^{c}$ Geophysical Institute, Acad. Sci. Czech Rep., Prague, Czech Republic
The turbulent phenomena appearing on small scales complicate the computer simulation of the convection in planetary interiors. The control parameter for flows is the Reynolds number $Re = VL/\nu^M$, where $V$, $L$ are the characteristic velocity and length, respectively and $\nu^M$ is the molecular kinematic viscosity. Geodynamo models considering flows with the Reynolds number $Re = 10^9$ would required a grid with $\approx Re^{9/4} \approx 10^{20}$ nodes for small-scale turbulence to be resolved. When solving large-scale models due to a spectral method the artificial hyperdiffusivity must be introduced to avoid the influence of the small-scale turbulence causing instabilities due to higher harmonics. Hyperdiffusivity can be the solution of the problem for a spectral method. However, it is difficult to introduce something similar in the numerical process based on the space grid. In this paper we attempt to solve only the hydrodynamic problem, however, still bearing in mind a later application in magnetohydrodynamics. Convection in a rotating spherical layer in the Boussinesq approximation is considered and a free rotating solid concentric inner sphere is included. The problem of developed subgrid turbulence is solved using the shell model approach. In principle, the shell model is a finite-dimensional model of turbulent motions based on the Fourier expansion, which gives a correct description of the spectral properties of the turbulent motions. 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 the second set of equations. This enables us to estimate the spectral energy flux on small scales. The turbulent coefficients depending on radial direction are then calculated and they are 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 $R_a = 10^{14}$ and Ekman number $E = 10^{-6}$ based on the molecular values of viscosity and thermal diffusivity. The results correspond to Reynolds number $Re \sim 10^{9}$.