Turbulent magnetic field amplification from the smallest to the largest
  magnetic Prandtl numbers
  The small-scale dynamo provides a highly efficient mechanism for the
conversion of turbulent into magnetic energy. In astrophysical environments,
such turbulence often occurs at high Mach numbers, implying steep slopes in the
turbulent spectra. It is thus a central question whether the small-scale dynamo
can amplify magnetic fields in the interstellar or intergalactic media, where
such Mach numbers occur. To address this long-standing issue, we employ the
Kazantsev model for turbulent magnetic field amplification, systematically
exploring the effect of different turbulent slopes, as expected for Kolmogorov,
Burgers, the Larson laws and results derived from numerical simulations. With
the framework employed here, we give the first solution encompassing the
complete range of magnetic Prandtl numbers, including Pm << 1, Pm ~ 1 and Pm >>
1. We derive scaling laws of the growth rate as a function of hydrodynamic and
magnetic Reynolds number for Pm << 1 and Pm >> 1 for all types of turbulence. A
central result concerns the regime of Pm ~ 1, where the magnetic field
amplification rate increases rapidly as a function of Pm. This phenomenon
occurs for all types of turbulence we explored. We further find that the dynamo
growth rate can be decreased by a few orders of magnitude for turbulence
spectra steeper than Kolmogorov. We calculate the critical magnetic Reynolds
number Rm_c for magnetic field amplification, which is highest for the Burgers
case. As expected, our calculation shows a linear behaviour of the
amplification rate close to the threshold proportional to Rm-Rm_c. Based on the
Kazantsev model, we therefore expect the existence of the small-scale dynamo
for any given value of Pm, as long as the magnetic Reynolds number is above the
critical threshold.
