  We study the dynamics of magnetic fields in chiral magnetohydrodynamics,
which takes into account the effects of an additional electric current related
to the chiral magnetic effect in high energy plasmas. We perform direct
numerical simulations, considering weak seed magnetic fields and
inhomogeneities of the chiral chemical potential mu_5 with a zero mean. We
demonstrate that a small-scale chiral dynamo can occur in such plasmas if
fluctuations of mu_5 are correlated on length scales that are much larger than
the scale on which the dynamo growth rate reaches its maximum. Magnetic
fluctuations grow by many orders of magnitude due to the small-scale chiral
dynamo instability. Once the nonlinear backreaction of the generated magnetic
field on fluctuations of mu_5 sets in, the ratio of these scales decreases and
the dynamo saturates. When magnetic fluctuations grow sufficiently to drive
turbulence via the Lorentz force before reaching maximum field strength, an
additional mean-field dynamo phase is identified. The mean magnetic field grows
on a scale that is larger than the integral scale of turbulence after the
amplification of the fluctuating component saturates. The growth rate of the
mean magnetic field is caused by a magnetic alpha effect that is proportional
to the current helicity. With the onset of turbulence, the power spectrum of
mu_5 develops a universal k^(-1) scaling independently of its initial shape,
while the magnetic energy spectrum approaches a k^(-3) scaling.
