Fourier space intermittency of the small-scale turbulent dynamo

The small-scale turbulent dynamo in the high Prandtl number regime is described in terms of the one-point Fourier space correlators. The second-order correlator of this kind is the energy spectrum and it has been previously studied in detail. We examine the higher order k-space correlators, which contain important information about the phases of the magnetic wave packets and about the dominant structures of the magnetic turbulence which cause intermittency. In particular, the fourth-order correlators contain information about the mean-square phase difference between any two components of the magnetic field in a plane transverse to the wave vector. This can be viewed as a measure of the magnetic field's polarization. Examining this quantity, the magnetic field is shown to become plane polarized in the Kazantsev-Kraichnan model at large times, corresponding to a strong deviation from Gaussianity. We derive a closed equation for the generating function of the Fourier correlators and find the large-time asymptotic solutions of these correlators at all orders. The time scaling of these solutions implies that the magnetic field has log-normal statistics, whereas the wave number scaling indicates that the field is dominated by intermittent fluctuations at high k.