A finite element method for nonlinear spherical dynamos

Many planetary and astrophysical bodies have intrinsic magnetic fields. To understand planetary magnetism, it is necessary to understand magnetohydrodynamic processes taking place in those magnetic bodies which share the same essential features: They are rotating rapidly and have a large core of electrically conducting liquid metal in the shape of a sphere or spherical shell. Because of spherical geometry and rotational effects, a numerical approach has to be employed to study the planetary dynamos.; Nearly all current spherical dynamo models employ spectral-type methods that use spherical harmonic expansions. However, the Legendre transform is computationally inefficient and severely limits the efficiency of the spectral methods and, furthermore, the global nature of the spectral methods causes difficulties in an efficient implementation on massively parallel computers. It is becoming increasingly clear that to reach a small Ekman number that is relevant to the Earth's dynamo, a dynamo model based on fundamental different numerical methods is necessary.; This thesis represents the first attempt to employ three-dimensional finite element techniques, which have inherited parallelism, for solving the problem of nonlinear spherical dynamos. We consider nonlinear spherical dynamo models involving the magnetic field and the fluid velocity, which are governed by the dynamo equation and the Navior-Stokes equation respectively.; The mathematical theories on the dynamo system in spherical dynamos are not available in the literature. This thesis will present the existence, uniqueness and stability of this system of nonlinear partial differential equations governing the mean-field dynamo problem. The finite element approximation to the model system and the convergence and stability of the fully discrete finite element methods are also developed successfully.; We have resolved the two fundamental problems in finite element spherical dynamo modeling: (1) the conflict between the local nature of the finite element method and the global boundary condition for the generated magnetic field; (2) the three-dimensional finite element discretization of the spherical system. The first difficulty is overcome by using an asymptotic boundary condition for the magnetic field. We demonstrate that our method is not only workable but also accurate and efficient. For the spherical domain decomposition, we apply a simple but efficient algorithm to divide the spherical domain into tetrahedral elements. The resulting high quality three-dimensional tetrahedralization of the spherical system produce a uniform mesh distribution on a spherical surface without the ‘pole problem’ and a small number of nodes in the neighborhood of the center without the origin ‘problem’.; We have investigated three fully three-dimensional, nonlinear, spherical dynamo problems using our finite element dynamo model: (a) a time-dependent kinematic α2 dynamo; (b) a time-dependent solar interface dynamo incorporating a time-dependent tachocline; and (c) a magnetohydrodynamic geodynamo.