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 $f(\ensuremath{\phi})$-coupling, an axion as massive ${\ensuremath{\phi}}^{2}$-coupling, helical electromagnetic field. $\ensuremath{\alpha}$-dynamo term appears naturally from coupling MHD formulation. In presence of coupling, however, Schr\"odinger hydrodynamic coherently oscillating not available conventional $\ensuremath{\phi}$ coupling; instead, ${\ensuremath{\phi}}^{2}$ allows successful preserving dark matter nature to nonlinear order. formulation, direct couplings between fields appear only nonideal MHD. study gravitational magnetic instabilities MHDs.