Magnetic fluctuations for small magnetic Prandtl numbers

Numerous studies of magnetic fluctuations with a zero mean-field for small magnetic Prandtl number (Pr_m 1) show that magnetic fluctuations cannot be generated by turbulent fluid flow with the Kolmogorov energy spectrum. In addition, the generation of magnetic fluctuations with a zero mean-field for Pr_m 1 were not observed in numerical simulations. However, in astrophysical plasmas the magnetic Prandtl numbers are small and magnetic fluctuations are observed. Thus a mechanism of generation of magnetic fluctuations for Pr_m 1 still remains poorly understood. On the other hand, in astrophysical applications (e.g., solar and stellar convection zones, galaxies, accretion disks) the turbulent velocity field cannot be considered as a divergence-free. The generation of magnetic fluctuations by turbulent flow of conducting fluid with a zero mean magnetic field for Pr_m 1 is studied by means of linear and nonlinear analysis. The turbulent fluid velocity field is assumed to be homogeneous and isotropic with a power law energy spectrum (∝ k^-p) and with a very short scale-dependent correlation time. It is found that magnetic fluctuations can be generated when the exponent p > 3/4. It is shown also that the growth rates of the higher moments of the magnetic field are larger than those of the lower moments, i.e., the spatial distribution of the magnetic field is intermittent. In addition, the effect of compressibility (i.e., u ≠ 0) of the low-Mach-number turbulent fluid flow u is studied. It is demonstrated that the threshold for the generation of magnetic fluctuations by turbulent fluid flow with u ≠ 0 is higher than that for incompressible fluid. This implies that the compressibility impairs the generation of magnetic fluctuations. Nonlinear effects result in saturation of growth of the magnetic fluctuations. Asymptotic properties of the steady state solution for the second moment of the magnetic field in the case of the Hall nonlinearity for the low-Mach-number compressible flow are studied.