The Small-Scale Dynamo at Low Magnetic Prandtl Numbers
  The present-day Universe is highly magnetized, even though the first magnetic
seed fields were most probably extremely weak. To explain the growth of the
magnetic field strength over many orders of magnitude fast amplification
processes need to operate. The most efficient mechanism known today is the
small-scale dynamo, which converts turbulent kinetic energy into magnetic
energy leading to an exponential growth of the magnetic field. The efficiency
of the dynamo depends on the type of turbulence indicated by the slope of the
turbulence spectrum v(l) \propto l^{theta}, where v(l) is the eddy velocity at
a scale l. We explore turbulent spectra ranging from incompressible Kolmogorov
turbulence with theta = 1/3 to highly compressible Burgers turbulence with
theta = 1/2. In this work we analyze the properties of the small-scale dynamo
for low magnetic Prandtl numbers Pm, which denotes the ratio of the magnetic
Reynolds number, Rm, to the hydrodynamical one, Re. We solve the Kazantsev
equation, which describes the evolution of the small-scale magnetic field,
using the WKB approximation. In the limit of low magnetic Prandtl numbers the
growth rate is proportional to Rm^{(1-theta)/(1+theta)}. We furthermore discuss
the critical magnetic Reynolds number Rm_crit, which is required for
small-scale dynamo action. The value of Rm_crit is roughly 100 for Kolmogorov
turbulence and 2700 for Burgers. Furthermore, we discuss that Rm_crit provides
a stronger constraint in the limit of low Pm than it does for large Pm. We
conclude that the small-scale dynamo can operate in the regime of low magnetic
Prandtl numbers, if the magnetic Reynolds number is large enough. Thus, the
magnetic field amplification on small scales can take place in a broad range of
physical environments and amplify week magnetic seed fields on short
timescales.
