Small-scale turbulent dynamo for low-Prandtl number fluid: Comparison of the theory with results of numerical simulations

Context. During the past decades, significant progress has been made in numerical simulations of the turbulent dynamo and the theoretical understanding of turbulence. However, a quantitative comparison between simulations and the theory of the dynamo is still lacking. Aims. We investigate the generation of a magnetic field by the incompressible turbulent conductive fluid near the critical regime and compare the theoretical predictions of the Kazantsev model with results of recent direct numerical simulations. Methods. The Kazantsev equation was analyzed both analytically and numerically. Results. We studied the critical magnetic Reynolds number (Rmc) and the growth rate near the threshold in the limit of very high and for moderate Reynolds numbers. We argue that in the Kazantsev equation for magnetic field generation, the quasi-Lagrangian correlator of velocities should be used instead of the Eulerian, as is usually implied when theory and simulations are compared. The theoretical results obtained with this correlator agree well with the numerical results. We also propose that the decrease of Rmc can be explained as a function of the Reynolds number (Re) at intermediate to high Re. It is probably due to the Reynolds-dependent intermittency of the velocity structure function. We show that the scaling exponent of this function in the inertial range strongly affects the magnetic field generation, and it is known to be an increasing function of the Reynolds number. Conclusions. The quasi-Lagrangian correlator in the Kazantsev theory provides results that agree well with numerical simulations. An ideal way to compare them is to find the correlator that can be substituted in the Kazantsev equation and the generation properties in the same simulation. Universal parameters at least have to be used, regardless of the properties of the pumping scale. The Reynolds-dependent intermittency can explain the recently observed decrease in the critical magnetic Reynolds number at small Prandtl numbers.