An optimal Galerkin scheme to solve the kinematic dynamo eigenvalue problem in a full sphere

Abstract

The kinematic dynamo approximation describes the generation of magnetic field in a prescribed flow of electrically-conducting liquid. One of its main uses is as a proof-of-concept tool to test hypotheses about self-exciting dynamo action. Indeed, it provided the very first quantitative evidence for the possibility of the geodynamo. Despite its utility, due to the requirement of resolving fine structures, historically, numerical work has proven difficult and reported solutions were often plagued by poor convergence. In this paper, we demonstrate the numerical superiority of a Galerkin scheme in solving the kinematic dynamo eigenvalue problem in a full sphere. After adopting a poloidal–toroidal decomposition and expanding in spherical harmonics, we express the radial dependence in terms of a basis of exponentially convergent orthogonal polynomials. Each basis function is constructed from a terse sum of one-sided Jacobi polynomials that not only satisfies the boundary conditions of matching to an electrically insulating exterior, but is everywhere infinitely differentiable, including at the origin. This Galerkin method exhibits more rapid convergence, for a given problem size, than any other scheme hitherto reported, as demonstrated by a benchmark of the magnetic diffusion problem and by comparison to numerous kinematic dynamos from the literature. In the axisymmetric flows we consider in this paper, at a magnetic Reynolds number of O(100), a convergence of 9 significant figures in the most unstable eigenvalue requires only 40 radial basis functions; alternatively, 4 significant figures requires 20 radial functions. The terse radial discretization becomes particularly advantageous when considering flows whose associated numerical solution requires a large number of coupled spherical harmonics. We exploit this new method to confirm the tentatively proposed positive growth rate of the planar flow of Bachtiar et al. [4], thereby verifying a counter-example to the Zel’dovich anti-dynamo theorem in a spherical geometry.