Numerical method for geodynamo simulations based on Fourier expansion in longitude and finite difference in meridional plane
In three-dimensional geodynamo simulations, the most widely used method is the spectral transform method in which variables are expanded by spherical harmonics. This method is suitable because it has good accuracy and enables appropriate treatment of boundary conditions of the magnetic field. However, it causes a heavy drag especially on the high-resolution calculation because of the large amount of computation required for the spectral transform in the latitudinal coordinate (the Legendre transform). Therefore, we introduce a method for high-resolution geodynamo simulations to decrease the computation time by avoiding the Legendre transform and performing only the Fourier transform in longitude. In this Fourier transform method, the resulting spectral equations are solved in a meridional plane using a finite difference technique. We verify the applicability of the Fourier transform method for geodynamo simulations by solving the dynamo benchmark problem [Christensen, U.R., Aubert, J., Cardin, P., Dormy, E., Gibbons, S., Glatzmaier, G.A., Grote, E., Honkura, Y., Jones, C., Kono, M., Matsushima, M., Sakuraba, A., Takahashi, F., Tilgner, A., Wicht, J., Zhang, K., 2001. A numerical dynamo benchmark. Phys. Earth Planet. Inter. 128, 25–34] and some cases of higher Rayleigh (up to 4 times critical) and lower Ekman (10−4 at minimum) numbers than the benchmark problem. The results confirm the validity of this method, but suggest that its superiority over spherical harmonic models is limited to such a very high-resolution calculation as is realizable in near future.