  Magnetic fields in galaxies are believed to be the result of dynamo
amplification of initially weak seed fields, reaching equipartition strength
inside the interstellar medium. The small-scale dynamo appears to be a viable
mechanism to explain observations of strong magnetic fields in present-day and
high-redshift galaxies, considering the extreme weakness of seed fields
predicted by battery mechanisms or primordial fields. Performing
high-resolution adaptive mesh magneto-hydrodynamic simulations of a small mass,
isolated cooling halo with an initial magnetic seed field strength well below
equipartition, we follow the small-scale dynamo amplification from
supernova-induced turbulence up to saturation of the field. We find that
saturation occurs when the average magnetic pressure reaches only 3 % to 5 % of
the turbulent pressure. The magnetic energy growth transitions from exponential
to linear, and finally comes to halt. The saturation level increases slightly
with grid resolution. These results are in good agreement with theoretical
predictions for magnetic Prandtl numbers of order $\mathrm{Pr_M} \sim 1$ and
turbulent Mach numbers of order $\mathrm{M} \sim 10$. When we suppress
supernova feedback after our simulation has reached saturation, we find that
turbulence decays and that the gas falls back onto a thin disk with the
magnetic field in local equipartition.
  We propose a scenario in which galactic magnetic fields are amplified from
weak seed fields in the early stages of the Universe to sub-equipartition
fields, owing to the turbulent environment of feedback-dominated galaxies at
high redshift, and are evolved further in a later stage up to equipartition, as
galaxies transformed into more quiescent, large spiral disks.
