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

Abstract

Abstract. 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.

Error estimate of a first-order time discretization scheme for the geodynamo equations

“…We mention that recently, Chan et al [2] studied the mathematical theory of the spherical interface dynamos, which is a similar model as ours, using different method. In this paper we concentrate on the following where = B r o (0)\B r i (0) ⊂ R 3 , 0 < r i < r o < ∞ is the physical domain of interest, n denotes the unit outer normal vector to the boundary of .…”

An iterative scheme for a geodynamo system

“…They use the finite-element method to simulate 3D dynamos in spherical systems. More recently, Chan et al [5] have reported the mathematical theory and finite-element approximation of mean-field spherical dynamos. In this paper, for real s ≥ 0, · s and · s,∂ denote the norms of H s ( ) (or H s ( ) for scalar functions) and H s (∂ ) 3 (or H s (∂ ) for scalar functions), respectively.…”

An efficient numerical scheme for a 3D spherical dynamo equation

“…There are also a few studies with numerical analysis on some numerical methods for these models, e.g., [4], [17], [23], [5] and [18]. In [4], Chan, Zhang and Zou studied the mathematical theory and its numerical approximation based on a finite element method, while Mohammad M. Rahman and David R. Fearn [18] developed a spectral approximation of some nonlinear mean-field dynamo equations with different geometries and toroidal and poloidal decomposition.…”

“…Using a finite-element method to deal with the above issues may be complicated and costly. We consider in this paper the model used in [4] and propose an efficient numerical scheme based on a semi-implicit discretization in time and a spectral method in space based on the divergence-free spherical harmonic functions. we first discretize the model in time using a semi-implicit approach such that at each time step one only needs to solve a linear system with piecewise constant coefficients.…”