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(), where and v() 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 (=1/3), we find that the critical magnetic Reynolds number is RmcritK110 and for Burgers turbulence (=1/2) RmcritB2700. Furthermore, we derive that the growth rate of the small-scale magnetic field for a general type of turbulence is ΓRe ⁽1 ^{-) /(}1 ⁺⁾ in the limit of infinite magnetic Prandtl number. For decreasing magnetic Prandtl number (down to Pm10), 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.