Diffusion of weak magnetic fields by isotropic turbulence

Abstract

The diffusion of slowly varying, weak magnetic fields by a statistically isotropic and stationary velocity field in a perfectly conducting fluid is studied by Eulerian analysis. The characteristic wavenumber and variance of the velocity field are k₀ and 3v²₀, thus defining the eddy-circulation time τ₀ = 1/v₀k₀. The velocity field is assumed constant on intervals of duration 2τ₁ and statistically independent for distinct intervals. Thus the correlation time is τ₁. The α-effect dynamo mechanism in the quasi-linear approximation is corroborated. Both the quasilinear and the direct-interaction approximations give identical diffusion of magnetic and passive scalar fields in reflexionally invariant turbulence. This result is found to be exact for τ₁/τ₀ → 0 but is demonstrated to be incorrect in general for finite τ₁/τ₀ because of effects of helicity fluctuations. The nature of the failure of the direct-interaction approximation is exhibited by an exactly soluble model system. Analysis based on a double-averaging device shows that longrange, persistent helicity fluctuations in reflexionally invariant turbulence give an anomalous negative contribution to the magnetic diffusivity which depends on the helicity covariance function. We term this the α² effect. The magnitude of the effect depends sensitively on the turbulence statistics. If the characteristic scales of the helicity fluctuations are sufficiently larger than τ₀ and 1/k₀, the magnetic diffusivity is negative, implying unstable growth, while a passive scalar field diffuses normally. On the other hand, a crude estimate suggests that the α² effect is small in normally distributed turbulence.