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A defining characteristic of the recent geomagnetic field is its dominant axial dipole which provides its navigational utility and dictates the shape of the magnetosphere. Going back through time, much less is known about the degree of axial dipole dominance. Here we use a substantial and diverse set of 3D numerical dynamo simulations and recent observation-based field models to derive a power law relationship between the angular dispersion of virtual geomagnetic poles at the equator and the median axial dipole dominance measured at Earth's surface. Applying this relation to published estimates of equatorial angular dispersion implies that geomagnetic axial dipole dominance averaged over 10 7 -10 9 years has remained moderately high and stable through large parts of geological time. This provides an observational constraint to future studies of the geodynamo and palaeomagnetosphere. It also provides some reassurance as to the reliability of palaeogeographical reconstructions provided by palaeomagnetism.

We investigate numerically the flow of an electrically conducting fluid in a rapidly rotating spherical shell where the inner boundary spins slightly faster than the outer one. The magnetic field evolves self-consistently from an initial dipolar configuration of weak amplitude, and a toroidal field is produced by winding this poloidal field through the internal differential rotation. First, we characterize the axisymmetric field solutions obtained at long times when the Lorentz force is negligible and the flow follows the steady, purely hydrodynamical solution. We then examine the stability of these solutions, focusing on the regime of large magnetic Reynolds numbers where the field is dominantly toroidal. When the ratio of the azimuthal Alfvén frequency to the rotation frequency exceeds a certain value, a nonaxisymmetric instability develops. We show that the instability properties are compatible with those expected for the magnetorotational instability. Finally, we compare the instability properties with predictions obtained from a local linear stability analysis. The linear analysis agrees well with the numerical simulation results, except in a number of cases where the discrepancies are attributed to shearing effects on the unstable modes.