Non‐WKB Alfvén waves in the solar wind

Abstract

We treat, both analytically and numerically, small‐amplitude, undamped, toroidal Alfvén waves in a model of axisymmetric solar wind flow in which solar rotation is neglected. There is no restriction to WKB waves; the waves may have any frequency. By transforming in simple ways the equations governing the waves we are able to obtain exact formal solutions to the general time‐dependent problem as well as to the Fourier‐analyzed problem. We discuss the equations and their solutions in terms of coupled inward and outward propagating waves. One integral of the equations for the Fourier amplitudes is obtained; it relates the amplitudes of the ingoing and outgoing waves. The integral is a special case of a general law of conservation of wave action, which we show to hold for finite wavelengths. The statement of the conservation of wave action is shown to be analogous to the conservation of particle‐antiparticle pairs in relativistic quantum theory. We obtain the condition required for WKB waves and show that it depends on the coupling of waves in a flowing medium. The solar wind problem is discussed in terms of the Fourier amplitudes. It is shown that there is a singularity in the equations, at the Alfvén point, which determines physically acceptable solar wind solutions. A qualitative account of the amplitudes far from the sun is given based on an exact solution for a model with constant solar wind flow speed. A conservation equation for the wave energy is obtained, and the relations among the wave energy density, energy flux density, force, and acceleration are stated. Numerical solutions, based on realistic solar wind profiles, are given. We show that non‐WKB waves with wave periods of about a day or two have somewhat greater wave energy densities, up to a factor of 2 or so, in the corona than do WKB waves with the same amplitude at 1 A.U. On the other hand, non‐WKB waves of any wave period are no more effective in accelerating the plasma than are WKB waves; they are much less effective for wave periods of a day or more. We conclude that, for conditions actually existing in the corona, WKB estimates quite accurately account throughout the corona for the wave energy density, energy flux density, and wave acceleration of the plasma for Alfvén waves with periods less than about 0.05, 1, and 0.01 day, respectively; the corresponding periods in the solar wind are about 1, 1, and 0.5 day.