Fast dynamo action-A critical problem for solar magnetism

Abstract

Dynamo action is the process by which kinetic energy is systematically converted to magnetic energy in electrically conducting fluids. This process may occur on a resistive time-scale (i.e. with a growth rate that tends to zero as the magnetic resistivity of the medium tends to zero) in which case the dynamo is described as “slow”; or it may occur on the convective time-scale, in which case the dynamo is described as “fast”. The conventional description of the solar dynamo (an αω dynamo in the terminology of mean-field electrodynamics) places it in the “fast” category, because the usual expressions for the regeneration parameter a and the turbulent resistivity β are both independent of molecular diffusivity η under the conditions obtained deep down in the solar convection zone. Current attempts to construct fast dynamos, and to understand their detailed structure, are therefore of critical importance in the context of solar magnetism.

It has been shown (Moffatt and Proctor, 1985) that in a fast dynamo, the length-scale of the magnetic field is necessarily nearly everywhere a factor of R ^−1/2_m smaller than the length-scale of the underlying convection process. In the case of the sun, with a magnetic Reynolds number R_m of order 10⁴, this implies that the scale of the magnetic field that is generated may be as small as 10km, a result that is not incompatible with observation.

The dynamo process being turbulent, it remains of the greatest importance to have a reliable theory for determination of the parameter α and β in the limit of large magnetic Reynolds number. A renormalisation group procedure involving repeated averaging over successively increasing length scales (as described by Moffatt, 1983) seems to offer the best hope. In this theory, molecular diffusivity is important in the early stages of averaging, but becomes less important as the process is repeated, and the ultimate values of a and β are quite independent of η.

These problems are discussed, and some of the outstanding difficulties in the understanding of fast dynamos are identified.