“…In the ideal, magneto-hydrodynamic (MHD) limit Alfvén waves propagate exactly along the background magnetic field lines, but more realistic descriptions of these waves include the effects of either electron inertia or pressure which leads to a nonzero parallel electric field of the wave and cross-field propagation. This description has been invoked to explain the spatial structure of the shear wave and electron acceleration in phenomena ranging from auroral arcs [1][2][3], ionospheric density cavities [4,5], coronal loops [6], the solar wind [7], to tokamaks [8] and other laboratory plasmas [9]. When expanding on the MHD limit, it is important to consider the ratio of the electron thermal speed (v T e is the electron temperature in eV, m e the electron mass, B 0 the background field strength, n i the ion density, and m i the ion mass.…”