Mean-field theory of the solar dynamo

Abstract

The generation of the solar magnetic field is generally ascribed to dynamo processes in the convection zone. The dynamo effects, differential rotation (Ω-effect and helical turbulence (α-effect are explained, and the basic properties of the mean-field dynamo equations are discussed in view of the observed properties of the solar cycle. Problems of the classical picture of a dynamo in the convection zone (fibril state of magnetic flux, field strength, magnetic buoyancy, polarity rules, differential rotation and butterfly diagram) are addressed and some alternatives to overcome these problems are presented. A possibility to make up for the missing radial gradient of rotation in the convection zone is an α² Ω-dynamo with an anisotropic a-tensor. Dynamo solutions then might have the characteristics of the butterfly diagram. Another approach involves meridional circulation as the cause of the migration of a dynamo wave. Another suggestion is that the solar dynamo operates in the overshoot region at the base of the convection zone where strong fields, necessary to explain the polarity rules, can be stored and radial gradients in the angular velocity occur. As an alternative to the turbulent α-effect a dynamic α-effect based on magnetostrophic waves driven by a magnetic buoyancy instability of a magnetic flux layer is introduced. Model calculations which use the internal rotation of the Sun as deduced from helioseismology only show solar cycle behaviour if the turbulent diffusivity is reduced in the layer and the a-effect is concentrated near the equator. Another possibility is a combined model. The non-uniform rotation and most of the azimuthal magnetic flux are confined to a thin layer at the bottom of the convection zone where turbulent diffusion is greatly reduced, with the convective region above containing only weak fields for which the α-effect and turbulent diffusion operate in the conventional manner. The dynamo takes on the character of a surface wave at the interface between the two regions. Another possibility is a fibril field approach where non-axisymmetric flux tube instabilities lead to an α-effect which, together with differential rotation and reconnection of flux tubes, forms the basis of a flux tube dynamo. Furthermore the stochastic excitation of magnetic fields by a fluctuating a-effect is addressed. This contributes to irregularities in the solar cycle and leads to the excitation of a spectrum of dynamo modes which can be compared with decompositions of surface fields into spherical harmonics. An interesting consequence is the generation of a north-south asymmetry in the butterfly diagram.