Abstract

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({\ell}){\prop}{\ell}$^{ϑ}$, where {\ell} and v({\ell}) 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 ({\thetav}=1/3), we find that the critical magnetic Reynolds number is Rm$_{crit}$$^{K}${\ap}110 and for Burgers turbulence ({\thetav}=1/2) Rm$_{crit}$$^{B}${\ap}2700. Furthermore, we derive that the growth rate of the small-scale magnetic field for a general type of turbulence is {$\Gamma$}{\prop}Re$^{(1-ϑ)/(1+ϑ)}$ in the limit of infinite magnetic Prandtl number. For decreasing magnetic Prandtl number (down to Pm{\gsim}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.