Evolution of localized magnetic field perturbations and the nature of turbulent dynamo

Abstract

Kinematic dynamo in incompressible isotropic turbulent flows with high magnetic Prandtl number is considered. The approach interpreting an arbitrary magnetic field distribution as a superposition of localized perturbations (blobs) is developed. We derive a general relation between stochastic properties of an isolated blob and a stochastically homogenous distribution of magnetic field advected by the same stochastic flow. This relation allows us to investigate the evolution of a localized blob at a late stage when its size exceeds the viscous scale. It is shown that in three-dimensional flows, the average magnetic field of the blob increases exponentially in the inertial range of turbulence, as opposed to the late-batchelor stage when it decreases. Our approach reveals the mechanism of dynamo generation in the inertial range both for blobs and homogenous contributions. It explains the absence of dynamo in the two-dimensional case and its efficiency in three dimensions. We propose a way to observe the mechanism in numerical simulations.

Long-term properties of finite-correlation time isotropic stochastic systems

“…Comparing this with (10), (13) we see that the function w(η) defined in this subsection coincides with the GLE found in (11) and (19). From (11) and ( 25) it also follows that the function J(ξ) defined in (22) coincides with the rate function.…”

“…w(η) is the GLE of the process ξ(t). We note that, according to (10), the function w(η)T at large T coincides with the cumulant generating function of the integral…”

No feedback is possible in a small-scale turbulent magnetic field

“…In the exceptional case λ 2 = 0 the decrease is not exponential (G(ζ * 2 ) = 0) but a power law; one can check that for a Gaussian probability distribution of ζ 2 , P G ∝ √ te −ζ 2 2 t/2D , the statistical moments of f are proportional to 1/ √ t for any n. The values of S (−λ 2 ), S(−λ 2 ), as well as the whole shape of S, are determined by the statistics of velocity gradients. One can show (by means of the techniques developed in [17,20,21]) that for isotropic A ij (t), the possible value of S (−λ 2 ) is restricted by the boundaries |S (−λ 2 )| < 3/2. Thus, for these processes saturation of f n happens already at n < 3/2, so for all integer n ≥ 2 the n-order moments decrease with the same exponent.…”