Abstract
The growth and saturation of magnetic field in conductingturbulent media with large magnetic Prandtl numbers areinvestigated. This regime is very common in low-density hotastrophysical plasmas. During the early (kinematic) stage, weakmagnetic fluctuations grow exponentially and concentrate at theresistive scale, which lies far below the hydrodynamic viscousscale. The evolution becomes nonlinear when the magnetic energyis comparable to the kinetic energy of the viscous-scale eddies. A physical picture of the ensuing nonlinear evolution of the MHD dynamo is proposed. Phenomenological considerations are supplemented with a simple Fokker-Planck model of thenonlinear evolution of the magnetic-energy spectrum. It isfound that, while the shift of the bulk of the magnetic energy from the subviscous scales to the velocity scales may bepossible, it occurs very slowly - at the resistive, ratherthan dynamical, timescale (for galaxies, this means thatthe generation of large-scale magnetic fields cannot be explainedby this mechanism). The role of Alfvénic motions and theimplications for the fully developed isotropic MHD turbulenceare discussed.
Highlights
It has long been appreciated [1] that a generic three-dimensional turbulent flow of a conducting fluid is likely to be a dynamo, i.e. it amplifies an initial weak seed magnetic field provided the magnetic Reynolds number is above a certain instability threshold
If the magnetic Prandtl number (P r = ν/η, the ratio of the viscosity and the magnetic diffusivity of the fluid) is large and the hydrodynamic turbulence has Kolmogorov form, the scale of the magnetic field can decrease by a factor of P r1/2 below the viscous scale of the fluid
The Spitzer [42] value of the magnetic diffusivity is η ∼ 107 cm2 s−1, kν ∼ 10−16 cm−1 [8], so tη ∼ 1017 years, which exceeds the age of the Universe by seven orders of magnitude! The conclusion is that this mechanism cannot be invoked as a workable feature of the galactic dynamo—at least not if the dissipation of the magnetic field is controlled by the Spitzer magnetic diffusivity
Summary
It has long been appreciated [1] that a generic three-dimensional turbulent flow of a conducting fluid is likely to be a dynamo, i.e. it amplifies an initial weak seed magnetic field provided the magnetic Reynolds number is above a certain instability threshold (usually between 10 and 100). The growth of the magnetic energy established for the kinematic regime inevitably leads to the field becoming strong enough to resist further amplification and to modify the properties of the ambient turbulence by exerting a force (Lorentz tension) on the fluid This marks the transition from the kinematic (linear) stage of the dynamo to the nonlinear regime. The main issue is what happens to the coherence scale of the field during the nonlinear stage and, whether a mechanism could be envisioned that would effectively transfer the magnetic energy from small (subviscous) to velocity scales: perhaps as large as the outer scale of the turbulence and above. In this work, we study the possibility of the nonhelical, nonlinear, large-P r fluctuation dynamo saturating with magnetic energy concentrated at the outer scale of the turbulence.
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