  The Chern-Simons magnetohydrodynamics (CSMHD) is introduced using a
Maxwell-Chern-Simons (MCS) Lagrangian including an axion-like field $\Theta$.
The MCS equation of motion derived from this Lagrangian consists of a modified
current, including a chiral magnetic (CM) and an anomalous Hall (AH) current,
in addition to the ordinary Ohm current of resistive magnetohydrodynamics
(MHD). The former consists of an axial chemical potential, which is given in
terms of the temporal comoving derivative of $\Theta$, and the latter arises
from the spatial gradient of $\Theta$. As it turns out, the existence of the
axial chemical potential is a nonequilibrium effect that plays no role in the
linear stability analysis, whereas the AH current arises as in the first-order
linear perturbation of the thermal equilibrium. We analyze the linear stability
and causality of the CSMHD in a resistive and chiral medium. We show that the
Alfven modes propagating sufficiently close to the direction of the magnetic
field are unstable but causal. They are also accompanied by a genuine nonhydro
mode. A stable mode in a particular direction can correspond to an unstable
mode propagating in the exact opposite direction. The AH instability is a
manifestation of a breakdown of the parity. A numerical analysis of the phase
velocity confirms these results.
