On the adjustment to the Bondi-Gold theorem in a spherical-shell fast dynamo

Abstract

We present a numerical solution of the magnetic induction equation in a spherical fluid shell, with an insulator inside and outside. Prescribing an axisymmetric, time-dependent, chaotic flow, we find that the magnetic field appears to grow on the fast advective, rather than on the slow diffusive time scale. We demonstrate how this may be reconciled with the theorem of Bondi and Gold (1950), that the potential field in these insulators inside and outside the shell cannot be amplified on the fast time scale, by having the field become increasingly contained within the shell with increasing magnetic Reynolds number. Thus, as the Bondi-Gold theorem becomes more and more applicable, there is indeed less and less external field being amplified. This is in precise agreement with the conjecture of Rädler (1982) that the resolution would be to have an “invisible dynamo,” one having no external field. Finally, we consider some of the implications of this adjustment for the different symmetries of the field (dipolar versus quadrupolar) and the flow (u versus—u).