On the critical layer of Alfvn waves in the solar wind

Abstract

Alfvén waves are considered in a radial flow and external magnetic field, which relates to some features of the solar wind near the critical point. The Alfvén wave equation for the velocity perturbation is derived, showing that it has in general two singularities (besides the origin and infinity), namely a critical layer (at real distance), where the Alfvén speed equals the mean flow velocity, and a transition level (at imaginary distance), where the spatial derivative of the flow velocity equals the wave frequency. It is shown that in the case of mean flow velocity varying as a power of radial distance the wave field is specified at all distances by a combination of solutions of the Alfvén wave equation around three singularities: a regular singularity at the center, so that ascending power-series solutions exist, some with logarithmic terms; an irregular singularity at infinity, leading to the non-existence of any solution as an ascending Frobenius–Fuchs series, and the existence of two solutions as ascending–descending Laurent series; the region of validity of the preceding solutions is limited by a regular singularity at a finite, non-zero radial distance, which is the critical layer, where the flow velocity and Alfvén speed are equal. The wave field is singular at the critical layer, and has an amplitude jump, which is illustrated by plotting the wave field in the neighborhood of the critical layer, for several values of dimensionless frequency and Alfvén number, combined into a single parameter. When considering Alfvén waves in the solar wind, at least three kinds of boundary conditions could be applied: (i) an initial condition specifying the wave field at the surface of the Sun; (ii) an asymptotic condition excluding wave sources at infinity, by specifying an outward-propagating wave (radiation condition); (iii) a finiteness condition that the wave field be finite at the critical layer. Since the Alfvén wave equation is of second order, only two conditions can in general be applied. It is shown, for example, that (ii) and (iii) are generally incompatible. If the conditions (i) and (iii) are chosen, i.e. an initial wave field is given and the radiation condition of outward propagation at infinity is met, then (ii) will not in general be met; thus the wave field would be singular at the critical layer, in the absence of dissipation, corresponding to the resonance of a linear undamped system. It is shown that in the presence of dissipation, either by fluid viscosity or Ohmic resistivity, the wave field would be finite at the critical layer, corresponding to the resonance of a linear damped system.