The mean electromotive force generated by turbulence in the limit of perfect conductivity

Abstract

When homogeneous turbulence acts upon a non-uniform magnetic field B₀(x, t), a mean electromotive force [Escr ](x, t) is in general established owing to the correlation between the velocity field and the fluctuating magnetic field that is generated. The relation between [Escr ] and B₀ is linear, and it is generally believed that, when the scale of inhomogeneity of B₀ is sufficiently large, it can be expressed as a series involving successively higher spatial derivatives of B₀: \[ {\cal E}_i = \alpha_{il} B_{0l}+\beta_{ilm}\partial B_{0l}/\partial x_m+\ldots, \] where the tensors α_il, β_ilm,… are determined (in principle) by the statistical properties of the turbulence and the magnetic diffusivity Λ of the fluid. These tensors are of crucial importance in turbulent dynamo theory. In this paper the question of their asymptotic form in the limit Λ → 0 is considered. By putting Λ = 0 and assuming that the field is non-random at an initial instant t = 0, the developing form of the tensors α_l(t) and β_m(t) is determined. The expressions involve time integrals of Lagrangian correlation functions associated with the velocity field, which are comparable in structure with the eddy diffusion tensor in the analogous turbulent diffusion problem (Taylor 1921). Some doubts are expressed concerning the convergence of the time integrals as t → ∞, and it is concluded that a satisfactory treatment of the problem will require the inclusion of weak diffusion effects (as recognized by Parker 1955).