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 (φ)-coupling, and an axion as a massive scalar field with φ 2 -coupling, with the helical electromagnetic field. The α-dynamo term appears naturally from the helical-coupling in the MHD formulation. In the presence of the electromagnetic coupling, however, the Schrödinger and hydrodynamic formulations of the coherently oscillating axion are not available for the conventional φ coupling; instead, φ 2 coupling allows successful formulations preserving the dark matter nature of the axion to nonlinear order.