The equations of magnetohydrodynamics (MHD) are written for
non-uniform viscosity and resistivity – the latter in the cases of Ohmic and
anisotropic resistivity. In the case of Ohmic (anisotropic) diffusivity, there is
(are) one (two) transverse components of the velocity and magnetic field
perturbation(s), leading to a second-order (fourth-order) dissipative Alfvén-
wave equation. In the more general case of dissipative Alfvén waves with
isotropic viscosity and anisotropic resistivity, the fourth-order wave equation
may be replaced by two decoupled second-order equations for right- and left-polarized waves, whose dispersion relations show that the first resistive
diffusivity causes dissipation like the viscosity, whereas the second resistive
diffusivity causes a change in propagation speed. The second resistive diffusivity
invalidates the equipartition of kinetic and magnetic energy, modifies the
energy flux through the propagation speed, and also changes the ratio of
viscous to resistive dissipation. If the directions of propagation and polarization
are equal (i.e. for right-polarized upward-propagating or left-polarized
downward-propagating waves), the magnetic energy increases relative to the
kinetic energy, the resistive dissipation increases relative to the viscous
dissipation, and the total energy density and flux increase relative to the case
of isotropic resistivity; the reverse is the case for opposite directions of
propagation, i.e. upward-propagating left-polarized waves and downward-propagating right-polarized waves, which can lead to the existence of a critical
layer. The role of the viscosity and first and second resistive diffusiveness on the
dissipation of Alfvén waves is discussed with reference to the solar atmosphere.
On global solutions to a nonlinear Alfvén wave equation
