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 is expected to depend crucially on the
nature of turbulence. In this work, we present a model for the small-scale
dynamo that takes into account the slope of the turbulent velocity spectrum
v(l) ~ l^theta, 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 (theta=1/3), we find that the critical magnetic Reynolds number is
approximately 110 and for Burgers turbulence (theta=1/2) approximately 2700.
Furthermore, we derive that the growth rate of the small-scale magnetic field
for a general type of turbulence is Gamma ~ Re^((1-theta)/(1+theta)) in the
limit of infinite magnetic Prandtl numbers. For decreasing magnetic Prandtl
number (down to Pm approximately larger than 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.
