H. Matsui
Dept. of Research for Computational Earth Science, Research Organization for Information Science & Technology, Tokyo, JAPAN
We have developed a simulation code for MHD dynamo in a rotating spherical shell using the GeoFEM platform, which serves simulation platforms using the finite-element method (FEM). An advantage of FEM is that FEM is suitable for parallel computation because FEM consists of local operation while a spectral harmonics expansion requires a significant number of global computation. However, one difficulty exists in the treatment of boundary condition for the magnetic field between a conductive fluid and the electrical insulator under the FEM platform. To solve the problem, we consider a finite element mesh from the center to $r_{m}=14.8L = 5.1R_{e}$, where $L$ and $R_{e}$ are width of the fluid shell and the Earth's radius. Furthermore, the vector potential of the magnetic field with the Coulomb gauge is treated to solve the magnetic field. Because the vector potential and its differential have a continuity at the boundaries, the induction equation for the conductive fluid and the Laplace equation for the electrically insulator can be solved simultaneously without any special treatment for the vector potential at the boundaries. Only the boundary conditions for the vector potential at $ r = r_{m} $ instead of the real boundary condition for the vector potential at the infinite radius. To solve the velocity, temperature, and the vector potential, the fractional step scheme is applied, and the Crank-Nicolson Scheme is used for the diffusion terms and the Adams-Bashforth Scheme is chosen to solve the other terms. A parallel Conjugate Gradient solver served by GeoFEM is used for solving the diffusion terms and the Poisson equations to satisfy the conservation laws of the velocity and the vector potential. We performed a MHD simulation in a rotating hemispherical shell using the present code, and results are compared with the simulation using the spherical harmonics expansion to verify the present simulation code. The velocity, temperature, and vector potential are assumed to be symmetric with respect to the equator. The non-slip boundary was set at the boundaries of the fluid core, and the temperature was set to be 1.0 at the inner boundary of the fluid shell, and set to be 0.0 at outer boundary of the fluid shell. The vector potential was set to be 0.0 at the outer boundary of the simulation box. The Prandtl number was set to be 1.0, the Taylor number to be $9.0 \times 10^{4}$, Rayleigh number to be $ 1.2 \times 10^{4} $, and the magnetic Prandtl number to be 10.0 in the present simulation. The simulation was performed to 2.5 times of the magnetic diffusion time. The simulations represented same characteristics of the magnetic field and convection; that is, magnetic energy in the fluid shell is generated to approximately 4 times of the kinetic energy in both cases, and intense $z$-component magnetic fields ware generated in anti-cyclones. The characteristics of the magnetic field and convection patterns are also seen in other studies of MHD dynamo simulations. Differences between the two simulations are also seen in detail. Intensity of the generated magnetic energy by GeoFEM was approximately 90\% of that by the spectral method, and small scale patterns of the magnetic field were observed in the spectral method. We consider that these differences were caused by differences of the errors from low spacial resolution in both cases. Although some differences are seen in the results of the present simulation and those by the spectral harmonics expansion, our simulation code represents basic processes of the geodynamo processes. As a future study, we will perform the MHD simulation with rapid rotation and intense buoyancy with simple boundary conditions on the Earth Stimulator, which contains 640 nodes of the SMP-type 8 vector processors.