  The presence of asymmetry between fermions of opposite handedness in plasmas
of relativistic particles can lead to exponential growth of a helical magnetic
field via a small-scale chiral dynamo instability known as the chiral magnetic
effect. Here, we show, using dimensional arguments and numerical simulations,
that this process produces through the Lorentz force chiral magnetically driven
turbulence. A k^{-2} magnetic energy spectrum emerges via inverse transfer over
a certain range of wavenumbers k. The total chirality (magnetic helicity plus
normalized chiral chemical potential) is conserved in this system. Therefore,
as the helical magnetic field grows, most of the total chirality gets
transferred into magnetic helicity until the chiral magnetic effect terminates.
Quantitative results for height, slope, and extent of the spectrum are
obtained. Consequences of this effect for cosmic magnetic fields are discussed.
