On the dissipation of Alfven waves in magnetic atmospheres and flux tubes

We consider the propagation of Alfven waves in a viscous and resistive atmospheric layer, in isothermal magnetohydrostatic equilibrium, with constant rate of ionization, under an horizontal magnetic field, either uniform or decaying with altitude. The wave equation relevant to these two cases is derived, and its solution in terms of hypergeometric functions is used to plot wave amplitude and phase versus altitude for several values of wave frequency, horizontal wavenumber and viscous and resistive damping. It is shown that: (i) the amplitudes initially grow due to reducing mass density and then decay due to dissipation, i.e. go through a maximum; (ii) the phases also go through a maximum, i.e. upward propagating waves are reflected downward; (iii) amplitude and phase changes are smaller in a flux tube than in an uniform magnetic field, because in the former case the magnetic field decays, as the flux tube fans-out with height