  Using direct numerical simulations (DNS), we study laminar and turbulent
dynamos in chiral magnetohydrodynamics (MHD) with an extended set of equations
that accounts for an additional contribution to the electric current due to the
chiral magnetic effect (CME). This quantum phenomenon originates from an
asymmetry between left- and right-handed relativistic fermions in the presence
of a magnetic field and gives rise to a chiral dynamo. We show that the
magnetic field evolution proceeds in three stages: (1) a small-scale chiral
dynamo instability; (2) production of chiral magnetically driven turbulence and
excitation of a large-scale dynamo instability due to a new chiral effect
(alpha_mu effect); and (3) saturation of magnetic helicity and magnetic field
growth controlled by a conservation law for the total chirality. The
$\alpha_\mu$ effect becomes dominant at large fluid and magnetic Reynolds
numbers and is not related to kinetic helicity. The growth rate of the
large-scale magnetic field and its characteristic scale measured in the
numerical simulations agree well with theoretical predictions based on
mean-field theory. The previously discussed two-stage chiral magnetic scenario
did not include stage (2) during which the characteristic scale of magnetic
field variations can increase by many orders of magnitude. Based on the
findings from numerical simulations, the relevance of the CME and the chiral
effects revealed in the relativistic plasma of the early universe and of
proto-neutron stars are discussed.
