Magnetic field amplification by small-scale dynamo action: Dependence on turbulence models and Reynolds and Prandtl numbers

The small-scale dynamo is a process by which turbulent kinetic energy is converted into magnetic energy, and thus it is expected to depend crucially on the nature of the turbulence. In this paper, we present a model for the small-scale dynamo that takes into account the slope of the turbulent velocity spectrum v(l) proportional to l(v), where l and v(l) are the size of a turbulent fluctuation and the typical velocity on that scale. The time evolution of the fluctuation component of the magnetic field, i. e., the small-scale field, is described by the Kazantsev equation. We solve this linear differential equation for its eigenvalues with the quantum-mechanical WKB approximation. The validity of this method is estimated as a function of the magnetic Prandtl number Pm. We calculate the minimal magnetic Reynolds number for dynamo action, Rm(crit), using our model of the turbulent velocity correlation function. For Kolmogorov turbulence (v = 1/ 3), we find that the critical magnetic Reynolds number is Rm(crit)(K) approximate to 110 and for Burgers turbulence (v = 1/ 2) Rm(crit)(B) approximate to 2700. Furthermore, we derive that the growth rate of the small-scale magnetic field for a general type of turbulence is Gamma proportional to Re(1-v)/(1+v) in the limit of infinite magnetic Prandtl number. For decreasing magnetic Prandtl number (down to Pm greater than or similar to 10), the growth rate of the small-scale dynamo decreases. The details of this drop depend on the WKB approximation, which becomes invalid for a magnetic Prandtl number of about unity.