Abstract
We investigate the Lagrangian mechanism of the kinematic ‘fluctuation’ magnetic dynamo in a 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 the dynamo effect requires sufficient angular correlation of the passive line vectors that arrive simultaneously at the same space point. We illustrate this in detail for the Kazantsev–Kraichnan model of the 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, Gawȩdzki and Kupiainen to non-Hermitian evolution. Much of our analysis applies also to magnetohydrodynamic turbulence.
Highlights
Turbulent magnetic dynamo effect is of great importance in astrophysics and geophysics [1]
Kinematic dynamo effect is due to the stretching of magnetic field lines as they are passively advected by a chaotic velocity field
In order to understand the turbulent dynamo process a crucial fact is that magnetic field lines are not frozen into the plasma flow, even in the zero-resistance limit κ → 0
Summary
Turbulent magnetic dynamo effect is of great importance in astrophysics and geophysics [1]. Important ideas have been contributed recently by Celani et al [13] They pointed out that the existence of dynamo effect in the KK model for space dimension d = 3 should be closely related to the angular correlation properties of material line-vectors. We show here that the decay of the magnetic field in the rough regime of the KK model is determined in the limit κ → 0, P r < P rc by the “slow modes” of the linear evolution operator M∗2 for pairs of infinitesimal line-elements. Unlike the scalar case determining the decay law of the magnetic energy requires an additional step of matching these self-similar solutions to explicit resistive-range solutions We shall use these results to discuss the physical mechanism of kinematic dynamo, and, in particular, to relate our dynamo “order parameter” to the process of “induction” by a spatially uniform initial magnetic field. Considerable insight can be obtained into the inner workings of the small-scale dynamo by considering the situations where it fails
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