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Astron. Astrophys. 345, 986-998 (1999)

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4. Summary and conclusions

4.1. Summary of the results

In this study we investigate in detail the influence of torsional Alfvén waves in solar magnetic flux tubes on Stokes profiles. We have used basically the same methods as for our earlier investigations of longitudinal (Solanki & Roberts 1992) and kink waves (Ploner & Solanki 1997 called Paper I), i.e. we simply overlaid linear torsional waves calculated for isothermal, thin flux tubes onto realistic model atmospheres of the flux tube and its surroundings. At each time step over a wave period we then calculated line profiles along sets of inclined rays passing through the flux tube. In contrast to the kink wave it is extremely important to also include rays that do not pass through the flux-tube axis when considering torsional waves.

The shift, width and asymmetry of the Stokes profiles fluctuate according to the line-of-sight velocity. Their amplitude changes following the direction of the magnetic field vector. For profiles formed along the rays lying in a single vertical plane offset by [FORMULA] from the flux-tube axis (see Fig. 1) the variations are similar to those produced by kink waves (Paper I). The magnitude of the profile variations, however, depends strongly on [FORMULA] since the line-of-sight velocity [FORMULA] is proportional to [FORMULA] in our model. The line shift, broadening and asymmetry parameters vanish for [FORMULA] (i.e in a plane passing through the flux-tube axis) and increase rapidly with increasing [FORMULA]. The magnitude [FORMULA] of the profiles has the opposite dependence on [FORMULA] since the flux tube fills increasingly smaller parts of the atmosphere there: the intersection of the flux tube with the plane containing the lines-of-sight decreases with [FORMULA].

The spatially averaged (but temporally resolved) profiles of V and Q follow the wave with double the wave frequency because, due to the azimuthal symmetry of the wave perturbations, the left and right halves of the flux tube (as seen from an inclined observer) are exactly half a period out of phase. Half a wave period later the wave perturbations in the two halves are interchanged and lead to the same average line parameters (except for perturbations caused by magnetooptical effects).

Although profiles generated in outer (i.e. large [FORMULA]) planes are heavily distorted the spatially averaged profiles show only a moderate influence of the wave since they obtain their major contribution near the central plane. The wavelength shift and asymmetry of the spatially averaged Stokes V and Q have opposite signs. This has the same cause as that underlying the opposite signs of the same line parameters of the temporally averaged V and Q profiles in the presence of a kink wave (Paper I).

The behaviour of Stokes U differs from the other Stokes profiles because in U the [FORMULA]-components can be positive or negative according to the sign of [FORMULA] (and are therefore small near [FORMULA]). The spatially averaged U-profiles are found to be weak in amplitude, rather complex in shape and asymmetric.

Unsurprisingly, spatially and temporally averaged profiles are even less affected by the wave (except for line broadening). We find that all effects of the wave seen in the line-profile parameters are enhanced by the wave's amplitude [FORMULA], whereas the wave frequency plays only a minor role. The perturbations in Stokes V and Q due to the wave have opposite centre-to-limb variations. The asymmetries and line shift are largest at the limb for Stokes V, but closer to disc centre for Stokes Q.

4.2. Comparison between kink and torsional wave

Let us first consider temporally resolved but spatially averaged Stokes V and Q profiles. One major difference between the two wave modes is that the oscillations in Stokes V and Q reflect the frequency of the kink wave but double the wave frequency of torsional waves. In addition, line shift and asymmetry parameters influenced by torsional waves have a unique sign at all phases (positive for Stokes Q and negative for V). In contrast, the parameters affected by kink waves oscillate around zero. Also, for similar wave velocities, torsional waves shift the line profiles by less than half as much as kink waves do. The oscillation amplitudes and absolute values of the asymmetries are also significantly reduced (by up to a factor of 6). The temporal average does not alter the above points significantly. The dependence of the V and Q parameters on the wave amplitude and frequency and on the position on solar disc is basically the same for both waves.

That torsional waves affect polarized line profiles less strongly than kink waves has the following three reasons, which all root in the different nature of the waves.

  • 1. The phase velocity of torsional waves is [FORMULA] and is larger than that of the kink wave [FORMULA] because the latter is influenced by the density [FORMULA] of the external atmosphere which, for typical flux-tube parameters, is significantly larger than the density inside the tube [FORMULA]. The wave-induced field inclination is consequently larger for kink than for torsional waves if an equal wave-velocity amplitude is assumed. (Compare with Eq. 6.)

  • 2. The velocity induced by a kink wave is oriented in a single direction and has constant magnitude within a flux-tube cross-section. In contrast, the velocity induced by a torsional wave is azimuthal and its amplitude is proportional to the distance to the centre (to first order). Assume that the velocity amplitude v of the kink wave agrees with the velocity amplitude of the torsional wave at the flux-tube boundary [FORMULA], [FORMULA], where [FORMULA] is the angular velocity. In that case the average velocity within a cross section of the flux tube is 2/3 of that of a kink wave. Note that the maximum apparent velocity (assumed for [FORMULA]) is [FORMULA] (Eq. 10) and that consequently only [FORMULA] of a constant and isotropic velocity v can actually be seen by the radiative transfer equation. For the velocity field assumed above the energy flux (the product of kinetic energy and phase velocity) is roughly the same for kink and torsional waves (this assumes that the external density is four times the inernal density or that the phase velocity of kink waves is half of the Alfvén speed).

  • 3. The degree of polarization strongly depends on the path the light takes through the flux tube. The amount of magnetic material along a ray is largest if it intersects the flux-tube axis, but rapidly decreases as the shortest distance between the ray and the axis increases. Consequently, most of the polarized light stems from close to the flux-tube axis. The torsional velocity, however, is small there, so that the kink wave only has a small influence on the Stokes parameters formed there.

The largest consequence of the difference for the observational detection of these waves is that for a given wave energy flux it is far easier to detect a kink wave than a torsional wave by its signature in the Stokes parameters. Hence, the constraints set by observations on the wave flux (which will be the subject of another paper) is expected to be less tight for torsional waves than for kink or longitudinal waves.

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© European Southern Observatory (ESO) 1999

Online publication: April 28, 1999