Small-Scale Kinematic Dynamo and Non-Dynamo in Inertial-Range Turbulence

TL;DR

This paper analyzes the conditions under which small-scale kinematic dynamo action occurs in turbulent plasma flows, revealing the importance of angular correlation of magnetic field lines and providing exact decay rates in non-dynamo regimes.

Contribution

It introduces a detailed analysis of the Lagrangian mechanism of magnetic dynamo in turbulent flows, especially at small magnetic Prandtl numbers, and extends the slow-mode expansion to non-Hermitian cases.

Findings

Dynamo effect requires sufficient angular correlation of line-vectors.

In the non-dynamo regime, the magnetic energy decay rate is precisely determined.

Exact results are obtained for the Kazantsev-Kraichnan model of kinematic dynamo.

Abstract

We investigate the Lagrangian mechanism of the kinematic ``fluctuation'' magnetic dynamo in turbulent plasma flow at small magnetic Prandtl numbers. The combined effect of turbulent advection and plasma resistivity is to carry infinitely many field lines to each space point, with the resultant magnetic field at that point given by the average over all the individual line vectors. As a consequence of the roughness of the advecting velocity, this remains true even in the limit of zero resistivity. We show that the presence of dynamo effect requires sufficient angular correlation of the passive line-vectors that arrive simultaneously at the SAME space point. We demonstrate this in detail for the Kazantsev-Kraichnan model of kinematic dynamo with a Gaussian advecting velocity that is spatially rough and white-noise in time. In the regime where dynamo action fails, we also obtain the precise rate of decay of the magnetic energy. These exact results for the model are obtained by a generalization of the ``slow-mode expansion'' of Bernard, Gawedzki and Kupiainen to non-Hermitian evolution. Much of our analysis applies also to magnetohydrodynamic turbulence.