The small-scale dynamo: Breaking universality at high Mach numbers
  (Abridged) The small-scale dynamo may play a substantial role in magnetizing
the Universe under a large range of conditions, including subsonic turbulence
at low Mach numbers, highly supersonic turbulence at high Mach numbers and a
large range of magnetic Prandtl numbers Pm, i.e. the ratio of kinetic viscosity
to magnetic resistivity. Low Mach numbers may in particular lead to the
well-known, incompressible Kolmogorov turbulence, while for high Mach numbers,
we are in the highly compressible regime, thus close to Burgers turbulence. In
this study, we explore whether in this large range of conditions, a universal
behavior can be expected. Our starting point are previous investigations in the
kinematic regime. Here, analytic studies based on the Kazantsev model have
shown that the behavior of the dynamo depends significantly on Pm and the type
of turbulence, and numerical simulations indicate a strong dependence of the
growth rate on the Mach number of the flow. Once the magnetic field saturates
on the current amplification scale, backreactions occur and the growth is
shifted to the next-larger scale. We employ a Fokker-Planck model to calculate
the magnetic field amplification during the non-linear regime, and find a
resulting power-law growth that depends on the type of turbulence invoked. For
Kolmogorov turbulence, we confirm previous results suggesting a linear growth
of magnetic energy. For more general turbulent spectra, where the turbulent
velocity v_t scales with the characteristic length scale as u_\ell\propto
\ell^{\vartheta}, we find that the magnetic energy grows as
(t/T_{ed})^{2\vartheta/(1-\vartheta)}, with t the time-coordinate and T_{ed}
the eddy-turnover time on the forcing scale of turbulence. For Burgers
turbulence, \vartheta=1/2, a quadratic rather than linear growth may thus be
expected, and a larger timescale until saturation is reached.
