Diffusion and transport of a magnetic field in a turbulent medium with helicity

Abstract

An analysis is made of the relation between accurate formulas for the coefficients of turbulent diffusion DT and the alpha effect αT for a magnetic field in the Lagrange and Euler representations. It is shown that the quadratic term with respect to αT in the diffusion coefficient derived by Moffatt and Kraichnan is incorrect and should be dropped. First, a numerical solution of the nonlinear equation (DIA equation) for the Green function is presented, describing the transport of a magnetic field for the case of incompressible, uniform, isotropic, steady-state turbulence possessing helicity. These solutions are used to calculate the steady-state coefficients DT and αT for various values of the parameters ξ0=u0σ0/R0, a=H0/u02p0, σ0/σ1, and R0/R1, where u0, σ0, and R0 are the characteristic velocity, lifetime, and scale of the turbulent pulsations, and H0, σ1, and R1 are similar values describing the helicity of the medium h(1,2)=〈u(1)· (∇×u(2))〉, and the parameter α characterizes the degree of helicity. The DIA values of DT and αT and the self-consistent values of these quantities calculated using the Green tensor in the diffusion approximation are in qualitative agreement. It is shown that the coefficient of turbulent diffusion is always positive for all the types of turbulence studied. Nonsteady-state values of DT(t) and αT(t) calculated by a self-consistent method are given.