Constraints on dynamo action

Abstract

We discuss first geometrical constraints on dynamo action. Magnetic field decay is unavoidable under certain conditions of symmetry. 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 discuss criteria for the existence of and classification into two and three dimensional velocity results which are satisfied by a restricted set of geometries. Secondly, helicity conservation imposes constraints on dynamo action. The calculation of helicity is technically complicated because of open boundaries and the usual form for the MHD invariant is modified to take this into account. Here we present a simple application of the quasi linear model to the galactic dynamo.