I. Cupal $^{a}$, P. Hejda $^{a}$ and M. Reshetnyak $^{b,c}
$^{a}$ Geophysical Institute, Acad. Sci. Czech Rep., Prague, Czech Republic $^{b}$ Institute of Physics of the Earth, Russian Acad. Sci., Moscow, Russia $^{c}$ Research Computing Center of Moscow State University, Moscow, Russia
A model of thermally driven geodynamo in a rotating conductive spherical outer core in the Boussinesq approximation is considered. The free rotating solid and conductive concentric inner core is included. The resulting solution of the model has the typical geophysical features. The model provides a self-consistent magnetic field generation with the magnetic energy that is more than one order of magnitude larger than the kinetic energy. The inner core rotates eastward and the magnetic field is characterized by the leading dipole, which reverses several times during the calculated time. In this sense the presented model is comparable with the other geodynamos solved in last years. However, what is novel in this model is the numerical technique applied. The last decade revealed a quite dramatic progress in convection driven geodynamos. Many new models of self-consistent dynamo for compressible and incompressible thermal convection were developed. Models based on compositional convection appeared and attempts to include equations of thermodynamical state were also registered. Nevertheless, the numerical approach was mostly based on the decomposition of the quantities into poloidal and toroidal potentials and their expansion into spherical functions (spectral method). The full grid method was used only in 2D models. Though the pure spectral method is more accurate for the more or less smooth 3D dynamo solution than a comparable grid method, this advantage can disappear when the resolution of small irregular structures is required. Neither 2D grid combined with Fourier expansion in azimuthal direction brings essential progress in the solution of 3D models. The main problem in solving MHD equations on grids is how to cope with the singularities at the axis of rotation and in the polar region. Whereas various filtering techniques can result in unacceptable simplifications the control volume method offers adequate tools. The basic strategy of this numerical scheme (otherwise known as the finite volume method) is to write the differential equations in the conservative form and to integrate them over small spherical regions called control volumes. The volume integrals are then converted into the sum of integrals over the surfaces of the volumes using the Gauss' theorem. The area of surfaces is indirectly proportional to the singular coefficients near the axis of rotation and thus the resulting discretized equations are not singular. This approach demonstrates very stable behavior of the quantities and no problems appear near the axis of rotation.