Mean-field dynamos: the old concept and some recent developments

TL;DR

This paper reviews the fundamental concepts and recent advances in mean-field dynamo theory, emphasizing the non-local, non-instantaneous nature of the mean electromotive force and discussing progress in modeling and experimental aspects.

Contribution

It highlights the importance of the test-field method, discusses the memory effect, and reviews recent developments in understanding alpha quenching and the Yoshizawa effect.

Findings

The mean electromotive force is generally non-local and non-instantaneous.

The test-field method significantly advances the calculation of dynamo coefficients.

Progress has been made in understanding alpha quenching and the Yoshizawa effect.

Abstract

This article reproduces the Karl Schwarzschild lecture 2013. Some of the basic ideas of electrodynamics and magnetohydrodynamics of mean fields in turbulently moving conducting fluids are explained. It is stressed that the connection of the mean electromotive force with the mean magnetic field and its first spatial derivatives is in general neither local nor instantaneous and that quite a few claims concerning pretended failures of the mean-field concept result from ignoring this aspect. In addition to the mean-field dynamo mechanisms of $\alpha^2$ and $\alpha$ $\Omega$ type several others are considered. Much progress in mean-field electrodynamics and magnetohydrodynamics results from the test-field method for calculating the coefficients that determine the connection of the mean electromotive force with the mean magnetic field. As an important example the memory effect in homogeneous isotropic turbulence is explained. In magnetohydrodynamic turbulence there is the possibility of a mean electromotive force that is primarily independent of the mean magnetic field and labeled as Yoshizawa effect. Despite of many efforts there is so far no convincing comprehensive theory of $\alpha$ quenching, that is, the reduction of the $\alpha$ effect with growing mean magnetic field, and of the saturation of mean-field dynamos. Steps toward such a theory are explained. Finally, some remarks on laboratory experiments with dynamos are made.