THE ROLE OF INERTIA IN MODELS OF THE GEODYNAMO

Abstract

Numerical studies of the geodynamo have taken different views on the importance of inertial effects. Some have neglected inertia, while others have boosted its strength, in much the same way as they have had to take an artificially high viscous force because of numerical considerations. Yet others have taken an intermediate view. In terms of the standard non-dimensional numbers, the Ekman number E measures the strength of viscous effects, the magnetic Ekman number E η measures the strength of inertia, and the magnetic Prandtl number $P m =E /E η $. Virtually all studies have $P m ≥ 1 (E η ≤E )$, even though geophysical values give $P m << 1$. Those studies that have undertaken parameter surveys have found no dynamo action when $P m < P mc $, where P mc is an O (1) number that depends on E. We have therefore been motivated to undertake a systematic study of the effect of inertia. In order to work with a manageable problem, we have used a non-linear mean-field dynamo driven by an α-effect $[α=α0 cos θ sin π(r -r i )]$. In this, a finite-amplitude field drives a flow through the Lorentz force in the momentum equation, and this flow feeds back on the field-generation process in the magnetic induction equation, equilibrating the field. This equilibration process is a key aspect of the full hydrodynamic dynamo. What we are not modelling here is the effect on convective-driven processes of changes in inertia; our forcing α-effect is fixed considered the system in the absence of inertia. Here, we include the full inertial term. For an Ekman number of $E = 2.5 × 10-4 $, we have investigated dynamo solutions for the magnetic Ekman number in the range $E η = 5.0 × 10-5 $ to 9.0 × 10-2 (corresponding to reducing P m from 5 to ~0.003). In this range we find three distinct types of solution. At the higher values of P m we find solutions very similar to those found in the absence of inertia. The addition of inertia damps out the rapid time dependence found in its absence. The major effect we have found is that the addition of inertia (decreasing P m ) facilitates dynamo action; for a given level of forcing (i.e. fixed α0 ), increasing E η results in an increased amplitude of the magnetic (and kinetic) energy. There is no shut-off of dynamo action with decreasing P m as found in hydrodynamic models. This difference gives an insight into the various aspects of the dynamo process. By focusing on the field-generation process, using a fixed α that is independent of E η, we have shown that inertia modifies the flow driven by the Lorentz force in a manner that is beneficial to field generation. The contrast between the present mean-field model and the results of hydrodynamic models shows that the effect of inertia on the driving process (thermal convection) is detrimental to field generation, more than compensating for the beneficial effect identified here.