SYMMETRY PROPERTIES OF THE DYNAMO EQUATIONS FOR PALEOMAGNETISM AND GEOMAGNETISM

Recently both palaeomagnetists and geomagnetists have searched for symmetries in their data which would give some guide to the nature of the Earth's dynamo, most frequently quoting analyses of mean-field (alphaomega) kinematic dynamos. The separable solutions of the fully nonlinear, convective dynamo with spherically symmetric buoyancy forces and boundary conditions arise from the group of symmetry operations that leave a rotating sphere unchanged; they are more general than the rather specialised solutions usually quoted in the geomagnetic literature. The full set of symmetry operations is an Abelian Lie group but two simple, finite subgroups contain all the symmetries we have found in the recent literature. The smaller subgroup contains both reflection in the equatorial plane, which gives rise to the so-called 'dipole/quadrupole' separation, and inversion, or field reversal. The full group also includes rotations about the polar axis; these rotations would not normally be significant but current interest in core-mantle interactions, which can make the core longitude sensitive, demands that we include them. This larger subgroup includes, in addition to field reversal and equatorial reflection, rotation by an angle pi about the polar axis and reflection through the origin. We give the spherical harmonic expansions for each separable solution and indicate the type of data required to discriminate between different symmetries.