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

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4. Discussion

We have here discovered a two-dimensional, self-similar solution for phase mixing in an inhomogeneous magnetic field. It depends on a parameter, [FORMULA], which can be written as

[EQUATION]

where the Lundquist number, [FORMULA], A is the size of the coronal hole region and a is the lengthscale of the plasma inhomogeneity. The parameter [FORMULA], therefore, combines the relative strengths of phase mixing (through the value of [FORMULA] and dissipation (through the Lundquist number S).

Using this solution, we have verified that an exp([FORMULA]) height-dependence does exist, as predicted by Heyvaerts & Priest 1983, over a height range given by

[EQUATION]

Fig. 8 shows that the height of maximum dissipation depends on the value of [FORMULA] and therefore on the values of a, [FORMULA] and [FORMULA], all of which are highly uncertain: the dissipation height decreases with both a and [FORMULA].

[FIGURE] Fig. 8. Height (z) of the maximum ohmic heating as a function of [FORMULA], expressed in units of H. This has been calculated using the full expression for [FORMULA], Eq. .

Consider the following example. For [FORMULA] and [FORMULA] (corresponding to a high speed solar wind blowing from a coronal hole) we require [FORMULA] to place the maximum ohmic heating in the range [FORMULA]. With these values, the wave dissipates within a few wavelengths H which agrees with Heyvaerts & Priest.

Limits on [FORMULA] can also be used to place limits on the Lundquist number S and inhomogeneity lengthscale a. For a coronal hole size [FORMULA] (Ofman & Davila 1995) we obtain

[EQUATION]

If we further assume that at worst [FORMULA] and at best [FORMULA] (Woo 1996) then [FORMULA]: this implies that we can dissipate Alfvén waves at reasonable heights in the corona over a wide range of possible Lundquist number, given a particular inhomogeneity scale. If either the background Alfvén speed [FORMULA] or the period of the phase-mixed Alfvén wave is reduced, H decreases, allowing more oscillations and hence more phase mixing to occur for a given height. For a fixed Lundquist number this implies that the wave will be dissipated at a lower height.

It should be noted that Alfvén waves with periods about [FORMULA] and wavelengths on the order of [FORMULA] should be observable by the CDS (Coronal Diagnostic Spectrometer) instrument on board the recently launched SoHO (Solar and Heliospheric Observatory) platform and that studies are in place to look for Alfvén waves in coronal holes (Harrison and Hassler 1995). Future theoretical work will concentrate on similar phase-mixing equations for different background Alfvén speed structures.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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