  At high energies, the dynamics of a plasma with charged fermions can be
described in terms of chiral magnetohydrodynamics. Using direct numerical
simulations, we demonstrate that chiral magnetic waves (CMWs) can produce a
chiral asymmetry $\mu_5 = \mu_\mathrm{L} - \mu_\mathrm{R}$ from a spatially
fluctuating (inhomogeneous) chemical potential $\mu = \mu_\mathrm{L} +
\mu_\mathrm{R}$, where $\mu_\mathrm{L}$ and $\mu_\mathrm{R}$ are the chemical
potentials of left- and right-handed electrically charged fermions,
respectively. If the frequency of the CMW is less than or comparable to the
characteristic growth rate of the chiral dynamo instability, the magnetic field
can be amplified on small spatial scales. The growth rate of this small-scale
chiral dynamo instability is determined by the spatial maximum value of $\mu_5$
fluctuations. Therefore, the magnetic field amplification occurs during periods
when $\mu_5$ reaches temporal maxima during the CMW. If the small-scale chiral
dynamo instability leads to a magnetic field strength that exceeds a critical
value, which depends on the resistivity and the initial value of $\mu$,
magnetically dominated turbulence is produced. Turbulence gives rise to a
large-scale dynamo instability, which we find to be caused by the magnetic
alpha effect. Our results have consequences for the dynamics of certain
high-energy plasmas, such as the early Universe.
