The inherent instability of axisymmetric magnetostrophic dynamo models

Recent studies have demonstrated the possibility of constructing magnetostrophic dynamo models, which describe the slowly evolving background state of Earth's magnetic field when inertia and viscosity are negligible. Here we explore the properties of steady, stable magnetostrophic states as a leading order approximation to the slow dynamics within Earth's core. For the case of an axisymmetric magnetostrophic system driven by a prescribed α-effect, we confirmed the existence of four known steady states: ±Bd, ±Bq, where Bd is purely dipolar and Bq is purely quadrupolar. Importantly, here we show that in all but the most weakly driven cases, an initial magnetic field that is not purely dipolar or quadrapolar never converges to these states. Despite this instability, we also show that there are a plethora of instantaneous solutions that are quasi-steady, but nevertheless unstable. If the dynamics in Earth's core are reasonably modelled by a strongly driven α-effect, this work suggests that the background state can never be steady. We discuss the difficulties in comparing our magnetostrophic models with geomagnetic timeseries.