Time-dependent kinematic dynamos with stationary flows

Abstract

Numerical solutions to the magnetic induction equation in a sphere have been obtained for a number of stationary velocity models. By searching for non-steady magnetic fields and in some circumstances showing that all magnetic field modes decay, the inability of several earlier researchers to find convergent steady solutions is explained. Results of previous authors are generally confirmed, but also extended to cover non-steady fields, different values of magnetic Reynolds number and other parameters, and higher truncation limits. Some non-decaying fields are found where only decaying or non-convergent results have previously been reported. Two flows $\epsilon s^0_2 + t^0_2$ and $\epsilon s^0_2 + t^0_1$, each consisting of two very simple axisymmetric rolls are seen to sustain growing fields provided that (i) the magnetic Reynolds number R and the poloidal to toroidal flow ratio $\epsilon$ are of appropriate magnitudes, and (ii) the meridional s$^0_2$ flow is directed inwards along the equatorial plane and out towards the poles. An even simpler axisymmetric single roll flow $\epsilon s^0_1 + t^0_1$ is also seen to support growing fields for appropriate $\epsilon$ and R. These simple flows dispel the somewhat prevalent belief that dynamo maintenance relies on the supporting flow being complex, and having length scale significantly less than that of the conducting fluid volume.