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Astron. Astrophys. 318, 957-962 (1997)

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1. Introduction

Phase mixing has been proposed as a means of efficiently dissipating Alfvén waves in the solar corona by Heyvaerts & Priest 1983, who suggested that shear Alfvén waves with amplitudes perpendicular to the background magnetic field would decay away with height above the basal excitation region.

Much work has been done on coronal heating by resonant absorption (Kappraff & Tataronis 1977; Ionson 1978; Hollweg 1987; Poedts & Kerner 1992; Ruderman & Goossens 1993; Goossens 1994; Karpen et al. 1994; Wright & Rickard 1995) but less on the related problem of phase-mixing (Browning & Priest 1984; Sakurai & Granik 1984; Nocera et al. 1984; Cally 1991).

Fig. 1 shows the basic model we are considering. Coronal holes can be modelled by this geometry, at least to a first approximation. The background Alfvén speed [FORMULA] is a function of x only and has gradients in the [FORMULA] -direction. This inhomogeneity in the background field causes Alfvén waves on neighbouring field lines to have different wavelengths. Hence, as height increases, waves on neighbouring field lines move out of phase with each other, causing large gradients in the Alfvén wavefront, in the direction of the inhomogeneity. Dissipation then comes into play, allowing the energy in the wave to heat the plasma. Consider a wave equation for phase-mixing in a non-dissipative system, i.e.,

[EQUATION]

[FIGURE] Fig. 1. Heyvaerts & Priest 1983 model of phase mixing.

The solution is

[EQUATION]

for a wave of frequency [FORMULA], where [FORMULA]. Note that,

[EQUATION]

i.e., gradients in the [FORMULA] direction increase with height. The appearance of sharp gradients is also mediated by the gradient of [FORMULA]. Hence large gradients in the Alfvén wavefront will appear at lower heights when the plasma is more inhomogeneous. It is these large gradients that will be substantially affected by the presence of dissipation in the plasma (see Sect. 3.2 and Figs. 5, 4).

Using the ansätz,

[EQUATION]

and assumptions of weak damping,

[EQUATION]

and strong phase mixing,

[EQUATION]

Heyvaerts & Priest obtained the form

[EQUATION]

for the velocity, where

[EQUATION]

and [FORMULA] is the magnetic diffusivity and [FORMULA] is the kinematic viscosity.

The main feature to note is the [FORMULA] decay with height, which indicates a strong decay for a large enough height. However, this solution is only valid under the conditions described above. In particular, it is not valid at small z where it breaks condition 6. Nonetheless, the decay is fast enough to warrant further investigation to see under what conditions it can be reproduced.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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