Abstract
The mean-field model is one of the basic models of the dynamo theory, which describes the magnetic field generation in a turbulent astrophysical plasma. The first mean-field equations were obtained by Steenbeck, Krause and Rädler for two-scale turbulence under isotropy and uniformity assumptions. In this article we develop the path integral approach to obtain mean-field equations for a short-correlated random velocity field in anisotropic streams. By this model we analyse effects of anisotropy and show the relation between dynamo growth and anisotropic tensors of helicity/turbulent diffusivity. Considering particular examples and comparing results with isotropic cases we demonstrate several mean-field effects: super-exponential growth at initial times, complex dependence of harmonics growth on the helicity tensor structure, when generation is possible for near-zero component or near-zero helicity trace, increase of the averaged magnetic field inclined to the initial current density that leads to effective Lorentz back-reaction and violation of force-free conditions.
Highlights
The mean-field approach was one of the first methods for studying the dynamo
The method of path integrals is one of the powerful approaches, which allows us to obtain mean-field equations without additional assumptions about spatial separation of turbulence scales or about small Reynolds number, which is usually assumed in traditional Krause–Radler approaches
The method is based on the assumption about short-correlated velocity field, but this assumption looks reasonable for astrophysical objects with rotational intervals larger than typical memory time
Summary
The mean-field approach was one of the first methods for studying the dynamo. This approach assumes averaging of the magnetic induction equation over uniform and isotropic random velocity field. Untying the averaging process for the magnetic and velocity field, it defines the anisotropic mean-field dynamo by helicity and diffusivity tensors and, what is not less important, in the statistically homogeneous and isotropic situation it restores the classical Equation (1). Basing on the obtained model, we find its isotropic/anisotropic solutions in an uniform field and analyze the dynamo growth rate depending on helicity and diffusivity tensors We apply it to two more specific problems. We demonstrate that for a statistically anisotropic (or isotropic) flow and suitable initial localized conditions a superexponential growth is possible This result can be important for the galactic magnetic field evolution because the mean-field dynamo time scale in galaxies is smaller comparable with the age of galaxies, see, e.g., [18]
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