Definition of the Alfvén mode in inhomogeneous magnetic field

The article is methodological and defines the concept of the linear Alfvén mode. There are two definitions — electrodynamic and hydrodynamic. In the former, the Alfvén mode is considered a wave with a potential transverse electric field. In the latter, waves are more often identified with the Alfvén mode, plasma motion in which is purely vortex. While these definitions are equivalent for homogeneous plasma, they are incompatible if the field line curvature is taken into account: if the transverse electric field is purely potential, the plasma speed has not only a vortex component, but also a potential one, and vice versa. The electrodynamic and hydrodynamic definitions are equivalent only if the wave electric field completely lacks a component along the binormal to the external magnetic field. However, such waves do not exist in nature.