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Astron. Astrophys. 325, 1199-1212 (1997)

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4. Conclusions

In the present study we have calculated linear, propagating kink-mode waves in thin flux tubes (FTs) and investigated the influence of the waves on the Stokes profiles and line parameters of three photospheric spectral lines. Most line parameters are strongly affected by kink waves. The profiles are considerably broadened and often exhibit large asymmetries, oscillations in amplitude and distorted [FORMULA] -components.

We now summarize the most important features of the time-resolved line parameters, proceed with time-averaged profiles and conclude with a comparison to related research.

It is found that the line shift and the blue-red asymmetry of the [FORMULA] -component amplitudes and areas of temporally resolved profiles quite clearly follow the wave (Fig. 5). These parameters oscillate in phase and have about the same magnitude for both Stokes Q and V. The Stokes Q and V amplitudes, however, oscillate in anti-phase. The response of the amplitudes is non-linear. For increasing wave frequency we observe increasing time-independent offsets around which all line parameters oscillate with increasingly smaller oscillation amplitudes. All the oscillation amplitudes are enhanced with increasing [FORMULA], except that of the Stokes Q amplitude, which exhibits the opposite centre-to-limb behaviour. A standard measure of the inclination of a FT, the ratio of the [FORMULA] -components of Stokes Q to V, gives a particularly clear signature of the wave.

Kink waves also affect time-averaged line profiles (which also correspond to snapshots of many FTs oscillating at random phases). Surprisingly, an upward propagating wave produces a positive asymmetry in Stokes Q but a negative Stokes V asymmetry. The Q and V line shifts also possess opposite signs. This seemingly contradictory behaviour is not seen in temporally resolved parameters, for which the two polarizations always respond in phase. The opposite asymmetries and shifts of Stokes Q and V are due to the antiphase of the oscillations of the Stokes Q and V amplitudes. The time-averaged asymmetries in Stokes V are generally larger than in Q.

The response of most line parameters of time averaged profiles is insignificant for waves with frequencies very close to the cut-off frequency - the line width, however, reacts equally strongly to waves of all frequencies. The reason for this dependence on frequency is the dependence of the phase relation between wave velocity and FT inclination on wave frequency. Velocity and inclination are [FORMULA] out of phase for [FORMULA], while for wave frequencies near the cut-off frequency the phase shift approaches [FORMULA] and remains there for standing waves. We conclude that standing waves or propagating kink modes with frequencies very near the cut-off hardly affect spatially unresolved measurements of asymmetries.

Consider now how the kink-mode signature differs from that of the tube mode investigated by Solanki & Roberts (1992). Most obviously the influence of the transverse kink mode increases towards the limb, while that of the longitudinal tube mode decreases. Another interesting difference is caused by the fact that the tube mode is compressible, whereas the kink mode is not. Due to the temperature variability accompanying compressibility the strength of a temperature-sensitive spectral line fluctuates over a wave period, while it remains essentially unchanged for kink-mode waves (excluding second-order effects). This causes the tube waves to have a much larger effect on the Stokes I profiles of temperature sensitive lines than kink waves.

Both wave modes produce an asymmetry in V and Q profiles. The sign of the blue-red asymmetry changes over a wave period. A net area and amplitude asymmetry can result even after averaging over a full wave period for both wave modes. In the case of the tube wave this is due to the fact that the velocity and temperature (or pressure) oscillate in phase for a propagating wave, while in the case of the kink wave it is the phase relation between the FT inclination and the velocity which is important. Note that in contrast to the tube wave the relevant phase shift of propagating kink waves changes dramatically with frequency near the cutoff.

Although both wave modes produce net blue-red asymmetric profiles, there are considerable differences between their signatures.

Firstly, the center-to-limb variation is expected to be very different. Secondly, the kink wave is more effective in producing asymmetric Stokes V and probably Q profiles. Thirdly, the ratio of amplitude to area asymmetry is much larger for the tube wave. Finally, whereas the two waves produce temporally averaged V with the same asymmetries for waves propagating in the same direction, say upwards, the sign of the Stokes Q asymmetry (for inclined lines of sight) is expected to differ. For the kink wave Q has the opposite asymmetry to V, while for the longitudinal wave we expect - from our understanding of the mechanisms producing the asymmetry - the Q profile to be asymmetric in the same sense as V. Note, however, that the kink-mode waves are not the only way of producing opposite asymmetry in V and Q (cf. Martínez Pillet et al. 1996).

One shortcoming of the present investigation is that we do not consider the combined effect of the wave and the surrounding granulation, or of different wave modes simultaneously present in the FT. Near the solar limb the granulation surrounding the FTs produces a Stokes V asymmetry of the same sign as an upward propagating kink wave, but the latter is more effective, particularly when we consider snapshots. Hence both mechanisms produce negative Stokes V asymmetry near the limb and thus enhance each other. They also produce approximately the same amplitude as area asymmetry, whereas the observations show a significant negative area asymmetry but little amplitude asymmetry near the limb. Granulation and kink waves alone do not appear capable of removing this discrepancy.

How do the results of our simple model compare with the line profiles resulting from the sophisticated simulations of Steiner et al. (1994, 1995, 1996), whose simulations include the effects of non-stationary, supersonic convection, longitudinal shock waves and non-linear kink waves? Steiner et al. (1995) have also calculated Stokes profiles along lines of sight with [FORMULA]. The main effect of the kink wave that they find is that it periodically produces large Stokes Q and V profiles (namely at the phases of maximum inclination of the FT, in agreement with our results). In addition, very close to the limb we also find Stokes V profiles with both [FORMULA] -lobes having the same sign (cf. Steiner et al., 1995, Fig. 8, profile d). In our model such profiles are created when the FT is nearly perpendicular to the line-of-sight. Our calculations obviously miss, however, the rest of the dynamic phenomena distorting the line profiles in the Steiner et al. simulations.

The observations of Martínez Pillet et al. (1996) show the Q and V amplitude asymmetries to have the same sign for averages over many individual measured line profiles. This obviously cannot be accounted for by kink waves alone, since these produce opposite signs of [FORMULA]. A mixture of kink waves and granulation, cannot, however, be ruled out, particularly since the temporally averaged Q asymmetry produced by the kink wave is significantly smaller than the V asymmetry (e.g. Fig. 8). Comparison of the relevant model calculation, which will be the subject of a future paper, with observations may be able to set limits on the energy flux transported into the upper atmosphere by kink-mode waves.

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© European Southern Observatory (ESO) 1997

Online publication: April 28, 1998