### abstract

- Numerous studies of magnetic fluctuations with a zero mean-field for small magnetic Prandtl numbers (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 applications the magnetic Prandtl numbers are small and magnetic fluctuations are observed. Thus the mechanism of generation of magnetic fluctuations for Pr m ≪ 1 still remains poorly understood. The generation of magnetic fluctuations by turbulent flow of a conducting fluid with a zero mean magnetic field for Pr m ≪ 1 is studied. The equation for the high-order correlation functions of the magnetic field is derived. It is shown 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. 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 > 3/2. In addition, the effect of compressibility (i.e., ∇ · u ≠ 0) of the low-Mach-number turbulent fluid flow u is studied. It is shown that the threshold for the generation of magnetic fluctuations by turbulent fluid flow with V · u ≠ 0 is higher than that for incompressible fluid. This implies that the compressibility impairs the generation of magnetic fluctuations. Problems of anomalous scalings for vector (magnetic) field is discussed as well. The anomalous scaling for the second moment of the magnetic field passively advected by an turbulent fluid flow with very short scale-dependent correlation time is found for the case 3/2 < p < 3. This anomalous scaling has the complex exponent which depends on degree of compressibility of a low-Mach-number turbulent flow. When 1 < p < 3/2 the anomalous exponent is real. 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 turbulent flow are studied.