Abstract

The main magnetic fields of the larger planets, the Earth, possibly Mercury and the Sun are generated by the motions in electrically-conducting cores or shells. Motions which are sufficiently vigorous and asymmetric can act as self-exciting dynamos. Attempts are also underway by several research groups to develop laboratory rotating fluid dynamos. A pseudo-spectral dynamo code, which has been developed as a computational laboratory, is described herein. The prototype model underlying the code incorporates the dynamics of an electrically-conducting rotating liquid spherical shell surrounded by a stationary electrically-insulating mantle and enclosing a solid inner core. The Boussinesq approximation is made, in which density variations are retained only in the buoyancy force. The convection is thermally driven by prescribed temperatures at the inner and outer core boundaries. The gravitational field may be asymmetric. The magnetic field, the velocity, the pressure and the temperature in the shell are governed by the magnetic induction equation, the heat equation and the Navier-Stokes momentum equation in a uniformly rapidly rotating reference frame with inertia, including the non-linear advective term, Coriolis, buoyancy, viscous and magnetic Lorentz forces. The magnetic field and the velocity are solenoidal. The magnetic, viscous and thermal diffusivities are uniform.

The equations are discretised using solenoidal toroidal-poloidal representations of the magnetic field and the velocity, and scalar field expansions of Chebychev polynomials in radius and spherical harmonics in co-latitude and east-longitude. There are five scalar fields: the temperature, and the toroidal and poloidal potentials of the magnetic field and the velocity. The radial discretisation uses Chebychev collocation and the equations are time-stepped using finite differences. Differentiations are performed spectrally. All products are evaluated in physical space for efficiency. Fast Fourier and Gauss-Legendre methods are used to transform fields between physical and spectral spaces. The time-stepping method uses the implicit two-step Adams-Moulton method for the linear diffusion terms and the Coriolis force, and an Adams three-step predictor/two-step corrector method for the product and non-linear terms. Results are presented for three benchmark models: non-magnetic thermal convection with a non-rotating inner-core; a convective dynamo with a non-rotating electrically-insulating inner-core; and a convective dynamo with a rotating electrically-conducting inner core, which can rotate freely about rotation axis of the mantle under the control of the axial viscous and magnetic torques at the inner-core boundary.

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