Initial magnetic field amplification in the geodynamo model

In a hydromagnetic dynamo, the self-excitation mechanism generates the magnetic field by amplifying a weak initial field through specific configurations of conductive fluid flows. These flows sustain the field in a stationary or quasi-stationary state, preventing its attenuation. Demonstrating this amplification effect requires computational magnetohydrodynamics methods. In this paper, we analyze a geodynamo model with vacuum boundary conditions for the magnetic field at the outer boundary of a spherical layer. The dimensionless parameters that determine the problem are: $Pr$ is the Prandtl number, $Pm$ is the magnetic Prandtl number, $E$ is the Ekman number, and ${Ra}^*$ is the modified Rayleigh number. The initial level of the magnetic field depends on the parameter $\Lambda$, the Elsasser number. We conducted computational experiments using our MHD code, which we adapted for hybrid computing systems with graphics processors. Our code employs a finite-volume method to solve resistive magnetohydrodynamics problems for a viscous incompressible fluid. A weak ($\Lambda={10}^{-2}$) homogeneous magnetic field directed along the axis of rotation of the spherical layer was considered as the initial magnetic field. When conducting computational experiments, for the values of the parameters defining the problem $Pr=1$, $Pm=5$, $E=5 \cdot {10}^{-4}$, ${Ra}^*=200$, we obtained a quasi-stationary solution in which the initial level of the magnetic field energy increases by three orders of magnitude during the generation process. In the obtained solution, the temperature and velocity fields are symmetrical relative to the equatorial plane, and the dipole component predominates in the external magnetic field.