A unified description of anti-dynamo conditions for incompressible flows

Abstract

A general type of mathematical argument is described, which applies to all the cases in which dynamo maintenance of a steady magnetic field by motion in a uniform density is known to be impossible. Previous work has demonstrated that magnetic field decay is unavoidable under conditions of axisymmetry and in spherical or planar incompressible flows. These known results are encompassed by a calculation for flows described in terms of a generalized poloidal-toroidal representation of the magnetic field with respect to an arbitrary two dimensional surface. We show that when the velocity field is two dimensional, the dynamo growth, if any, that results, is linear in one of the projections of the field while the other projections remain constant. We also obtain criteria for the existence of and classification into two and three dimensional velocity results which are satisfied by a restricted set of geometries. In addition, we discuss the forms of spatial variation of the density and the resistivity that are allowed so that field decay still occurs for this set of geometries.