On Alfvén waves in a flowing atmosphere

The present paper considers Alfvén waves in an inhomogeneous medium, under a nonuniform external magnetic field and in the presence of a nonuniform flow, to obtain wave equations for the velocity and magnetic field perturbations, in some particular cases. The case which is chosen for solution is that of propagation along a vertical uniform external magnetic field, in an atmosphere with a vertical mean flow velocity, satisfying mass conservation. The wave equation has a critical layer where the mean flow velocity equals the Alfvén speed, and in the case of an isothermal atmosphere the wave equation for the magnetic field perturbation can be reduced to a Bessel type with complex order and imaginary variable. Thus it is possible to obtain exact solutions for all frequencies and distances, discussing the cases in which there are propagating or evanescent waves far below or far above the critical layer, i.e., the latter acts as a reflecting layer. The solution in the vicinity of the critical layer allows the amplitude and phase of the wave field to be plotted as a function of distance from the critical layer (measured in scale heights) for different values of the dimensionless frequency and initial Alfvén number (ratio of flow velocity to Alfvén speed). One application is the solar atmosphere, for which the magnetohydrodynamic (MHD) equations are appropriate, and the wavelengths are large compared with the density scale height, thus excluding the ray approximation.