Alfv�n waves in stellar winds

Abstract

A mathematical model for undamped, toroidal, small-amplitude Alfven waves in a spherically-symmetric or equatorial stellar wind is developed in this paper. The equations are reduced to a very simple form by using real Fourier amplitudes and the ratio of the inward and outward propagating wave amplitudes, which is interpreted as a measure of the relative influence of wave reflection in the flow, on the solution at a given point. Asymptotic solutions at large distances are found to depend only on one parameter, α = ω/ω P - the ratio of wave frequency and critical (or cutoff) frequency which is a flow characteristic; a = 1 divides solutions into two qualitatively different groups. When α≤ 1 the asymptotic (r-→∞) ratio of the inward and outward propagating wave amplitudes does not depend on wave frequency and is equal to unity, while the phase shift between them changes; in this case the wave pattern is a standing wave. If α > 1 the converse occurs with the ratio of the amplitudes decreasing rapidly as the frequency increases, and the phase shift equals to -1/2π, corresponding to a propagating wave pattern. The result is also expressed in terms of velocity and magnetic field perturbations. Existence of a finite incoming wave amplitude solution at the Alfven critical point indicates that this point is stable with respect to the perturbations which originate at the critical point and spend an infinite time in its vicinity. Special attention is paid to the applicability of the WKB approximation. It is argued that it can be used only in finite intervals which do not contain the Alfven critical point, with inward propagating waves taken into account through the boundary conditions. It is shown that despite the presence of reflection, the outward propagating wave amplitude can be described reasonably well by the WKB formula, perhaps with different constants in different regions. In this context α = 1 divides solutions which cannot be approximated by the WKB estimate at all at large distances (the first group), from those which can with any given accuracy. As an illustration of the analytical behaviour some numerical results are shown using a cool wind model. These are likely to express qualitatively the features of the Alfven waves in any stellar wind, since the only assumptions about the flow used in the analytical study of the wave equations were that: the flow has small velocity at the base of the corona; it then passes through the critical point, and reaches its finite non-zero limit at infinity.