Axion electrodynamics and magnetohydrodynamics
  We formulate axion-electrodynamics and magnetohydrodynamics (MHD) in the
cosmological context assuming weak gravity. The two formulations are made for a
general scalar field with general $f(\phi)$-coupling, and an axion as a massive
scalar field with $\phi^2$-coupling, with the helical electromagnetic field.
The $\alpha$-dynamo term appears naturally from the helical-coupling in the MHD
formulation. In the presence of the electromagnetic coupling, however, the
Schr\"odinger and hydrodynamic formulations of the coherently oscillating axion
are not available for the conventional $\phi$ coupling; instead, $\phi^2$
coupling allows successful formulations preserving the dark matter nature of
the axion to nonlinear order.
