  We study the dynamics of a plasma of charged relativistic fermions at very
high temperature $T\gg m$, where $m$ is the fermion mass, coupled to the
electromagnetic field. In particular, we derive a magneto-hydrodynamical
description of the evolution of such a plasma. We show that, as compared to
conventional MHD for a plasma of non-relativistic particles, the hydrodynamical
description of the relativistic plasma involves new degrees of freedom
described by a pseudo-scalar field originating in a local asymmetry in the
densities of left-handed and right-handed fermions. This field can be
interpreted as an effective axion field. Taking into account the chiral anomaly
we present dynamical equations for the evolution of this field, as well as of
other fields appearing in the MHD description of the plasma. Due to its
non-linear coupling to helical magnetic fields, the axion field significantly
affects the dynamics of a magnetized plasma and can give rise to a novel type
of inverse cascade.
