We derive the Alfvén-wave equation for an atmosphere in the presence of a non-uniform vertical magnetic field and the Hall effect, allowing for Alfvén speed and ion gyrofrequency that may vary with altitude; the pair of coupled second-order differential equations for the horizontal wave variables, namely magnetic field or velocity perturbations, is reduced to a single complex, second-order differential equation. The latter is applied to spinning Alfvén waves in a magnetic flux tube, in magnetohydrostatic equilibrium, in an isothermal atmosphere. The exact solution is found in terms of hypergeometric functions, from which it is shown that at ‘high altitude’the magnetic field perturbation tends to grow to a non-small fraction of the background magnetic field. By ‘high-altitude’ is meant far above the critical level, which acts as a reflecting layer for left-polarized waves incident from below, i.e. from the ‘low-altitude’ range. We also obtain the exact solution near the critical level, where the left-polarized wave has a logarithmic singularity, and the right-polarized wave is finite. The latter is plotted in this region of wave frequency comparable to ion gyrofrequency, and it is shown that the Hall effect can cause oscillations of wave amplitude and non-monotonic phases with slope of alternating sign. The latter corresponds to ‘tunnelling’, i.e. waves propagating in opposite directions or trapped in adjoining atmospheric layers; this could explain the appearance of inward- and outward-propagating waves, with almost random phases, in the solar wind beyond the earth, for which the Hall effect on Alfvén waves should be significant.
Numerical simulations of the lower solar atmosphere heating by two-fluid nonlinear Alfvén waves