Convection Driven Dynamos in Rotating Spheres

Abstract

Of the objects in the solar system the Earth, Mercury, Jupiter, Saturn, Uranus, Neptune, Ganymede, and the Sun exhibit a magnetic field. These magnetic fields are believed to be generated by the magnetohydrodynamic dynamo process, in which current, generated as electrically conducting fluid crosses magnetic field lines, regenerates the magnetic field. Although most of the bodies listed above are believed to consist of a fluid outer core with a solid inner core, i.e. a spherical shell geometry, the full sphere dynamo problem is of physical interest as the dynamo of the early Earth, the ancient dynamo of Mars, and possibly Venus, the Moon and (currently) Mercury, are believed to have had no solid inner core. In this thesis we consider numerically the problem of magnetic field generation in a full sphere of rotating uniformly conducting fluid driven by a volumetric heat source. In order to numerically integrate the governing system of equations we combine the poloidal-toroidal field representation of Elsasser (1946) and Bullard&Gellman (1954) with an implicit/explicit multi-step Gear timestepping method and finite differences in radius. For the implicit radial differencing we develop a generalised compact finite-difference method which results in high order/low bandwidth timestepping systems, and we demonstrate that this method is competitive with other finite-difference methods: standard finite differences, Padé finite-differences, and the combined compact finitedifference schemes of Chu&Fan (1998). The numerical integrator is applied to three physical problems of interest. The first is kinematic dynamo action in a sphere. We investigate the possibility of dynamo action for flows with a missing component in spherical polar coordinates and find the growth rates are highly sensitive to changes in the truncation level. Nevertheless, we do find a working kinematic dynamo with axisymmetric velocity with no azimuthal component which demonstrates convincing convergence. The second problem we consider is that of thermal convection in the absence of a magnetic field in a rotating sphere. We fix the Ekman and Prandtl number (E; Pr) = (5 10¿4; 0:7) and obtain an estimate of the critical Rayleigh number Rac for the onset of convection, and describe the main characteristic of the flow for the convection solutions for Ra 1:4 Rac and Ra 5 Rac. These solutions are primarily for comparison for solutions computed in the third problem: dynamical dynamo action in a rotating sphere. The primary aim is to survey dynamo solutions for the fixed Ekman and Prandtl numbers (E; Pr) = (5 10¿4; 0:7), for magnetic Prandtl number varied from 1 to 40 and the modified Rayleigh number varied up to a few times the critical value for the onset of convection. We consider the solutions through the lens of dynamo scaling laws, but find no universally satisfactory theoretical or numerical scaling law. We also consider a weak/strong field classification of the solutions, finding highly localised force balances. We finish by considering three solutions in detail which represent three distinct classes of dynamo solution: an oscillating dipolar solution, an oscillating quadrupolar solution and a chaotic solution which oscillates between two different hemispherical states. Finally, we develop a first approach to the problem of dynamo action in a fluid sphere as it cools (with no internal heat source), and we present some first convective solutions which function exactly as we expect: the convection dieing down as the fluid cools.