On Alfven waves convected by a radial flow in a non-uniform magnetic field

In the present paper are considered Alfven waves propagating in a radial external magnetic field, which must decay like the inverse square of distance, to be divergence-free in the presence of an uniform radial mean flow, the mass density must also decay like the inverse square of distance, to conserve the mass flux. These conditions lead to a second-order convected Alfven wave equation, for the velocity perturbation. The wave equation has a regular singularity at the critical layer, where the mean flow and Alfven speeds are equal. The other two singularities are the centre, which is a regular singularity, and the point-at-infinity, which is an irregular singularity. By considering the leading terms of the wave field at the centre and at infinity, it is possible to reduce the differential equation to a Gaussian hypergeometric type, for which all three singularities are regular. Thus the wave field can be calculated exactly at all distances, for all values of dimensionless frequency and Alfven number, including analytic continuation across the critical layer; these solutions are illustrated, as plots of amplitude and phase of the wave field, versus radial distance, for various choices of boundary and or radiation conditions. It is shown that the critical layer reflects and absorbs waves, i.e. inside the critical layer waves propagate inward and outward, and outside the critical layer they propagate only outward; the wave spectrum does not vary much with distance up to the critical layer, where wave absorption is greater for higher frequencies. Outside the critical layer, the amplitude of the lower frequency waves decays faster, so that the spectrum evolves so as to re-emphasize the higher frequencies, The overall result is an Alfvenic wave spectrum which steepens towards higher frequencies, as observed in the solar wind. (C) 2002 Elsevier Science B.V All rights reserved.