Numerical Investigation of the Planar Velocity Antidynamo Theorem in a Sphere

A.A. Bachtiar, D.J. Ivers, R.W. James

Abstract

The Earth's main magnetic field is generally believed to be due to a dynamo process in the Earth's fluid outer core. A variety of antidynamo theorems exist that define conditions under which a magnetic field cannot be indefinitely maintained by dynamo actions against ohmic decay. One such theorem, the Planar Velocity Antidynamo Theorem, precludes field maintenance when the flow is everywhere parallel to some plane, e.g. the equatorial plane. This paper shows that the proof of the Planar Velocity Theorem fails when the flow is confined to a finite volume, so that then the theorem reverts to a conjecture. The paper also formulates the toroidal-poloidal spectral form of the magnetic induction equation for planar flows, as a basis for a numerical investigation. We have thus numerically determined the magnetic fields induced by various planar flows in spheres. In all but one flow the induced magnetic field has been found to decay in time, supporting a planar velocity conjecture. However one model is suggestive that field growth may occur. These results highlight the need for a mathematical proof of the conjecture, or alternatively, the determination of a functioning planar velocity dynamo.

Keywords: geodynamo, geomagnetic dynamo, planar flows.

AMS Subject Classification: Primary 86A25; secondary 76W05, 85A30.