7 - Mean-Field Electrodynamics

Summary

A two-scale approach is introduced, in which l0 is the dominant ‘energy-containing’ scale either of turbulence or of a random wave field, and L >> l0 is the scale of quantities averaged over the scale l0 . ‘ Mean-field electrodynamics’ obtains an equation for the mean magnetic field B0, by averaging the induction equation and by determining the ‘mean electromotive force’ < u x b > in terms of B0, so that the equation for B0 may be integrated. Here, u and b are the fluctuating parts of the velocity and magnetic fields, which are `cross-correlated’ only if the turbulence lacks reflection symmetry. The existence of the dominant ‘α-effect’ responsible for dynamo action is established for both isotropic and non-isotropic turbulence. At second order in l0/L, a generalised `eddy diffusivity’ is also established. Statistical properties of turbulence are defined, with emphasis on the energy and helicity spectrum functions. The α-effect is determined for a helical wave field and for turbulence under the `first-order smoothing approximation’ and is explicitly related to the helicity spectrum function. A Lagrangian approach to the weak-diffusion limit and a renormalisation approach based on successive averaging are also developed.