Kinematic dynamo action in a sphere. II. Symmetry selection

The magnetic fields of the planets are generated by dynamo action in their electrically conducting interiors. The Earth possesses an axial dipole magnetic field but other planets have other configurations: Uranus has an equatorial dipole for example. In a previous paper we explored a two-parameter class of flows, comprising convection rolls, differential rotation (D) and meridional circulation (M), for dynamo generation of steady fields with axial dipole symmetry by solving the kinematic dynamo equations. In this paper we explore generation of the remaining three allowed symmetries: axial quadrupole, equatorial dipole and equatorial quadrupole. The results have implications for the fully nonlinear dynamical dynamo because the flows qualitatively resemble those driven by thermal convection in a rotating sphere, and the symmetries define separable solutions of the nonlinear equations. Axial dipole solutions are generally preferred (they have lower critical magnetic Reynolds number) for D > 0, corresponding to westward surface drift. Axial quadrupoles are preferred for D < 0, and equatorial dipoles for convection with little D or M. No equatorial quadrupole solutions have been found. Symmetry selection can be understood if one assumes that the flow concentrates flux in certain places without reference to sign. Fields with dipole symmetry must change sign across the Equator; if flux is concentrated at the Equator, as tends to be the case for D < 0, they have a small length-scale and consequent high dissipation, making them harder to generate than axial quadrupoles. If flux is concentrated nearer the poles (D > 0), axial dipoles are preferred. The equatorial dipole must change sign between east and west hemispheres, and is not favoured by any elongation of the flux in longitude (caused by D) or polar concentrations (caused by M): they are preferred for small D and M. Polar and equatorial concentrations can be related to dynamo waves and the sign of Parker's dynamo number. For the three-dimensional flow considered here, the sign of the dynamo number is related to the sense of spiralling of the convection rolls, which must be the same as the surface drift