Asymptotic solutions for mean-field slab dynamos

Abstract

We discuss asymptotic solutions of the kinematic -dynamo in a thin disc (slab) surrounded by an electric insulator. Focusing upon the strong dynamo regime, in which the dynamo number satisfies , we resolve uncertainties in the earlier treatments and conclude that some of the simplifications that have been made in previous studies are questionable. Having abandoned these simplifications, we show, by comparing numerical solutions with complementary asymptotic results obtained for and , that the asymptotic solutions give a reasonably accurate description of the dynamo even far beyond their formal ranges of applicability. Indeed, our results suggest a simple analytical expression for the growth rate of the mean magnetic field that remains accurate across the wide range of values for that are typical of spiral galaxies and accretion discs. Finally, we analyse the role of various terms in the governing equations to clarify the fine details of the dynamo process. In particular, in the case of the radial magnetic field equation, we have shown that the term (where is the azimuthal magnetic field, is the mean-field dynamo coefficient and is measured across the slab), which is neglected in some of the earlier asymptotic studies, is essential for the dynamo as it drives a flux of magnetic energy away from the dynamo region towards the surface of the slab.