Spherical Interface Dynamos: Mathematical Theory, Finite Element Approximation, and Application

Stellar magnetic activities such as the 11-year sunspot cycle are the manifestation of magnetohydrodynamic dynamo processes taking place in the deep interiors of stars. This paper is concerned with the mathematical theory and finite element approximation of mean-field spherical dynamos and their astrophysical application. We first investigate the existence, uniqueness, and stability of the dynamo system governed by a set of nonlinear PDEs with discontinuous physical coefficients in spherical geometry, and characterize the system by a saddle-point type variational form. Then we propose a fully discrete finite element approximation to the dynamo system and study its convergence and stability. For the astrophysical application, we perform some fully threedimensional numerical simulations of a solar interface dynamo using the proposed algorithm, which successfully generates the equatorially propagating dynamo wave with a period of about 11 years similar to that of the Sun.