A non-linear, 3-D spherical α2 dynamo using a finite element method

Abstract

The problem of non-linear planetary dynamos with a prescribed two- or three-dimensional, time-dependent α is investigated using a finite element method. Magnetic fields are generated in a turbulent electrically conducting fluid spherical shell of constant electric conductivity surrounded by an electrically insulating uniform mantle or an electrically conducting heterogeneous mantle. The inner sphere is assumed to be a solid electric conductor that has the same conductivity as that of the fluid shell. Equilibration of the generated magnetic field is achieved by the non-linear process of α-quenching. The key numerical features of the finite element method for solving the dynamo equation are discussed in detail. The conflict between the local nature of the finite element method and the global boundary condition of the generated magnetic field is resolved by using approximate boundary conditions for the magnetic field. Tests and comparisons between analytical or semi-analytical solutions based on asymptotic boundary conditions and finite element solutions based on approximate boundary conditions have been conducted. A variety of non-linear solutions, including two-and three-dimensional stationary dynamos and two-and three-dimensional time-dependent dynamos, are obtained with the finite element method. Implications for planetary dynamos are discussed. © 2001 Elsevier Science B.V. All rights reserved.