## 4. Alfvén wave phase mixing and collisional dissipation in coronaConsider the evolution of an AW, excited at the magnetic field lines, where the length scale of the transverse inhomogeneity of the equilibrium is and the length scale of the field-aligned inhomogeneity of the equilibrium is . Both and can vary over a wide range in the solar corona, but usually . Take a wave frequency for which the parallel wavelength is much shorter than the length-scale of the field-aligned equilibrium inhomogeneity, . Then, as the wave propagates, the perpendicular length-scale of the wave decreases in time as ( is the wave period). Therefore, even if the initially excited AW has a smooth distribution in the direction of the plasma inhomogeneity, phase mixing creates short wave length-scales in a few wave periods. In open magnetic configurations, where waves propagate upward from the footpoints, this corresponds to heights of the order of a few field-aligned wavelengths. From now on we shall study the dynamics of Alfvén waves, in which short perpendicular wavelengths, , are either initially determined by a (unspecified) generator, or are created by phase-mixing. In this situation we consider the evolution of the -th harmonic of the wave field using a WKB-ansatz: The wave vector and the growth/damping rate have to be calculated from the local dispersion relation, and the integration is along the path of the wave propagation. The effects of plasma inhomogeneity for these waves come about through the spatial dependence of the plasma parameters along the path of the wave propagation. This path is determined by the ray equations for the wave vector and position : In accordance with the above, the wave frequency here is determined by the local dispersion relation found from the dynamic Eq. (31) for Alfvén waves. When we neglect the nonlinear terms in (30), we get the linear Alfvén wave frequency, , and the linear damping rate, , which accounts for the thermal and electron inertia effects, and for electron-ion collisions: where . Since we ignore the magnetic compressibility, . For a low- plasma, a good approximation to (35) and (36) in a wide range of parameters is When we neglect the electron inertia term, we find the AW dispersion to be close to the KAW dispersion, known from the kinetic theory (Hasegawa & Chen 1976), for all values of the dispersion parameter . In the limit they are identical. It is convenient to introduce the dispersion function : and to write the AW dispersion relation in the form The function is the factor by which the parallel phase velocity of the AWs with perpendicular wavelength exceeds . In the case of weak AW dispersion
(
),
, we recover by means of
Eq. (36) the asymptotic result of Heyvaerts & Priest (1983)
for the wave damping with height where is the characteristic height of the resistive (collisional) wave dissipation, However, in addition to collisional dissipation, the small-scale AWs undergo also nonlinear interaction, described by the right-hand side of (31). Therefore, the amplitude of the pump AW changes not only because of collisional dissipation, but also due to nonlinear interaction. Nonlinear AWs interaction and the question which process dominates for given plasma and wave parameters is investigated in the following sections. © European Southern Observatory (ESO) 2000 Online publication: June 5, 2000 |