Dynamo Theory and the Solar Cycle

Abstract

In this paper solutions of the mean field induction equation in a spherical geometry are discussed. In particular, the 22-year solar magnetic cycle is considered to be governed by an axisymmetric, periodic solution which is antisymmetric with respect to the equatorial plane. This solution essentially describes flux tubes travelling as waves from mid-latitudes towards the equator. In a layer of infinite extent the period of such dynamo waves solely depends on the strength of the two induction effects, differential rotation and α-effect (cyclonic turbulence). In a spherical shell, however, mean flux must be destroyed by turbulent diffusion, so the latter process might actually control the time scale of the solar cycle.A special discussion is devoted to the question of whether the angular velocityincreaseswith increasing depth, as the dynamo waves seem to require, or whether itdecreases, as many theoretical models concerned with the Sun's differential rotation predict. Finally, theories for the sector structure of the large scale photospheric field are reviewed. These describe magnetic sectors as a consequence of the sectoral pattern in the underlying large scale convection, as non-axisymmetric solutions of the mean field induction equation, or as hydromagnetic waves, modified by rotational effects.