Intermittency and anomalous scaling for magnetic fluctuations

The generation of magnetic fluctuations by turbulent flow of conducting fluid with a zero mean magnetic field for small magnetic Prandtl numbers 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 problem of anomalous scaling of magnetic fluctuations is discussed as well. The turbulent fluid velocity field is assumed to be homogeneous and isotropic with a power-law energy spectrum (proportional to $k^{-p})$ and with a very short scale-dependent correlation time. It is found that magnetic fluctuations can be generated when the exponent $p>\mathrm{}3/2.\mathrm{}$ In addition, the effect of compressibility (i.e., $\mathit{\nabla }\cdot \mathbf{u}\ne 0)$ of the low-Mach-number turbulent fluid flow $\mathbf{u}$ is studied. It is shown that the threshold for the generation of magnetic fluctuations by turbulent fluid flow with $\mathit{\nabla }\cdot \mathbf{u}\ne 0$ is higher than that for incompressible fluid. This implies that the compressibility impairs the generation of magnetic fluctuations.