Non‐WKB Alfvén waves in the solar wind

Abstract

We treat, both analytically and numerically, small-amplitude, undamped, toroidal Alfv6n 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 Alfv6n 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 I 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 Alfv6n 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. INTRODUCTIONAlfv6n waves have long been known to be a conspicuous feature of the solar wind plasma [Belcher et al., 1969; Belcher and Davis, 1971]. They dominate the microscale structure of the solar wind at least 50% of the time; magnetoacoustic wave modes, if they occur, contribute much less power to the plasma fluctuations than does the Alfv6n mode [Belcher and Davis, 1971). The Alfv6n waves can contribute to the dynamics of the solar wind. They exert a pressure on the plasma and are capable, in principle, of driving the solar wind [Alazraki and Couturier, 1971; Belcher, 1971; Belcher and Olbert, 1975; Dewar, 1970; Hollweg, 1973a, b, 1...

Reflection of Alfvén waves in the corona and solar wind: An impulse function approach

“…[39] Following Heinemann and Olbert [1980], we have used the f and g variables, rather than the more common Elsässer variables. These variables follow sunward and antisunward characteristics in the local plasma frame.…”

“…[5] In this paper we will study non-WKB effects using the simple configuration discussed by Heinemann and Olbert [1980] (hereafter referred to as HO). They considered a nonrotating Sun with an axisymmetric background magnetic field B 0 and background flow velocity V 0 confined to meridional planes; V 0 and B 0 are locally aligned.…”

Wave‐driven Turbulent Coronal Heating in Open Field Line Regions: Nonlinear Phenomenological Model

“…For the coronal problem the background proÐles of velocity, magnetic Ðeld, and density determine the Alfven coefficients of the linear terms, and they can be complicated functions of the slow coordinate s. The study of linear waves in nonhomogeneous media has been done Alfven extensively in the past (Hollweg 1981(Hollweg , 1984(Hollweg , 1996Heinemann & Olbert 1980 ;An et al 1989An et al , 1990Moore et al 1991 ;Velli 1993) using di †erent one-dimensional (usually radial) proÐles. In some cases, the linear equations can be reduced to the Klein-Gordon form (Musielak, Fontenia, & Moore 1992), and so dispersive waves solutions are obtained, with evanescent and resonant type of phenomena, according to the relation between the frequency of a wave forcing imposed and the reÑection coefficient of the medium.…”

Estimating random transverse velocities in the fast solar wind from EISCAT Interplanetary Scintillation measurements

“…The Klinglesmith (1997) conclusion that the wave's amplitude agrees with WKB has low statistical significance, since the data show large scatter: Eq. (1) depends on two main parameters, R 0 and δV 0 , and on many other assumptions (Heinemann and Olbert, 1980;Klinglesmith, 1997) and so the errors in these parameters may be large.…”