Accessing the dipole-multipole transition in rapidly rotating spherical shell dynamos

Earth’s magnetic field has exhibited erratic polarity reversals over much of its history; however, the processes that cause polarity transitions are still poorly understood. Dipole reversals have been found in many numerical dynamo simulations and often occur close to the transition between dipole-dominated and multipolar dynamo regimes. However, the physical conditions used in reversing simulations are necessarily far from those in Earth’s liquid iron core because of the long runtimes needed to capture polarity transitions and because a systematic exploration of parameter space is needed to find the dipole-multipole transition. Here, we use the theory of distinguished limits in an attempt to simplify the search for the dipole-multipole transition at increasingly realistic physical conditions. We consider three limits that are all built from the requirements of a constant magnetic Reynolds number Rm; one limit further attempts to impose balance between Magnetic, Coriolis, and Archimedean forces (a QG-MAC balance) while the other two seek to constrain solutions to an inertia-MAC, or QG-IMAC, balance. The presence of inertia, although not geophysically realistic, allows us to build limits that more closely follow the conditions where simulated reversals have been found to date. Numerical simulations along paths in parameter space defined by these limits show some consistencies with the assumed dynamical balances within the accessible parameter space, but also important discrepancies from predicted behaviour for certain diagnostic quantities, particularly the magnetic field strength and the magnetic/kinetic energy ratio. Furthermore, the paths do not follow the dipole-multipole transition; starting from reversing conditions, simulations move into the dipolar non-reversing regime as they are advanced along the path. By increasing the Rayleigh number, a measure of the buoyancy driving convection, above the values predicted by the distinguished limit, we are able to bound the dipole-multipole transition down to an Ekman number E ~10⁻⁶, comparable to the most extreme conditions reported to date. Our results, therefore, demonstrate that using distinguished limits is an efficient method for seeking the dipole-multipole transition in rapidly rotating dynamos. However, the conditions under which we bound the dipole-multipole transition become increasingly hard to access numerically and also increasingly unrealistic because Rm rises beyond plausible bounds inferred from geophysical observations. Future work combining the theory of distinguished limits with variations in the core buoyancy distribution, as suggested by recent studies, appears a promising approach to accessing the dipole-multiple transition at extreme physical conditions.