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

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3. Results

In this section we investigate the signature of torsional waves in polarized line profiles. We consider both time resolved and time averaged line profiles. To begin with (in Sect. 3.1) we discuss basic features of the line profiles generated in single, vertical planes cutting through the flux tube, such as the plane shown in Fig. 1. Because of the difference in behaviour we discuss Stokes V and Q (Sect. 3.2) separately from Stokes U (Sect. 3.3). The Stokes I profile is not discussed since the torsional wave mode only has a minute influence on it.

In Sects. 3.1 to 3.3 we consider the effect on the spectral line Fe i 5250 Å at the heliocentric angle [FORMULA] of a single type of wave having [FORMULA] Hz (which corresponds to a period of [FORMULA] Min and a wavelength of approximately 2000 km) and amplitude [FORMULA] km at [FORMULA]. Such a low frequency and long wavelength was chosen in order to ensure that the wave phase remains constant over the range of formation of Fe i 5250 Å. This spectral line has a Landé-factor [FORMULA] and was already employed in the study of kink waves in Paper I. Finally, the dependence of the signature of torsional waves on the characteristics of the wave ([FORMULA] and [FORMULA]), the location on the solar disc ([FORMULA]) and the chosen spectral line is discussed in Sect. 3.4. In that section we also consider the Fe i 5083 Å line, which is stronger than Fe i 5250 Å and which showed a larger influence of kink waves in Paper I.

3.1. Polarized line profiles

Fig. 4 shows a stack plot of Stokes V, Q and U generated in a flux tube supporting a torsional wave. The displayed profiles are formed in two planes lying at a distance of [FORMULA] km from the flux-tube axis (solid and dashed profiles in Fig. 4, respectively). From bottom to top the profiles correspond to 4 equally spaced phases or times covering a wave period T. We use the stellar convention in which phase runs from 0 to 1. Focus now on Stokes V generated in the plane at [FORMULA] (solid profiles in Fig. 4a). At phase 0.25 the profiles are seen to be blue-shifted and to have a larger blue than red lobe (leading to positive asymmetry, as defined in Appendix A). At phase 0.5 the profiles are more symmetric and almost unshifted. At phase 0.75 the profile has an asymmetry and shift opposite to phase 0.25 but with a larger total amplitude (see the end of Sect. 3.2.1). Finally, the situation at phase 1.0 is basically the same as that at 0.5 in the sense that both are near the unperturbed state. This description of the Stokes V evolution is also valid for Stokes Q (solid lines in Fig. 4b) with the exception that the Stokes Q amplitude is small when Stokes V is large and vice versa, i.e. Stokes Q is somewhat stronger at phase 0.25 than at phase 0.75. In summary, the change in asymmetry, line shift and broadening is in phase between Stokes V and Q whereas that of the total amplitude is in antiphase. (Stokes U is discussed later in Sect. 3.3). Note also that the line profiles exhibit an oscillatory behaviour with the same period as the wave.

[FIGURE] Fig. 4a-c. Stack-plot of Stokes V a , Stokes Q b and Stokes U c profiles of Fe i 5250.2 Å displayed at 4 phases spanning a wave period ([FORMULA] min[FORMULA] km and [FORMULA] km s-1). The thick solid lines represent the Stokes profiles formed along rays lying in the plane at [FORMULA] km whereas the thin dashed lines refer to [FORMULA] km. The numbers in brackets at the top of each frame are the maximum amplitudes reached by the signal in that frame. They corresponds to the amount by which the profiles at one phase are offset to the next in the figure. Phase 0.25 corresponds to the situations plotted in Figs. 2 and 3.

The time evolution of Stokes V and Q resembles the sequence generated by a kink wave, although the influence of the latter is larger (compare with Fig. 4 of Paper I). The similarity between the profiles generated by torsional and kink waves is not astonishing: along a single plane the line-of-sight components of the velocity and magnetic field perturbations due to the torsional wave are similar to the distortions produced by a kink wave. This can be seen approximately from Fig. 2. A kink wave (which shakes the flux tube in the y-direction) generates [FORMULA] and [FORMULA] which are constant in x and y. The corresponding distortions [FORMULA] and [FORMULA] due to a torsional wave are also constant along y, although not along x. The magnitude of both [FORMULA] and [FORMULA] is proportional to [FORMULA] and therefore depends strongly on the location of the plane. Consequently, the influence of torsional waves on Stokes V and Q increases with increasing [FORMULA]. This dependence is to be discussed in the next section.

One other important difference between kink and torsional waves is that whereas kink waves cause the whole flux tube to oscillate in phase, torsional waves cause the left and right halves of the flux tube as seen from the observer (i.e. the parts [FORMULA] and [FORMULA] of the flux tube, see Fig. 1) to oscillate in antiphase (see Eq. 8). The result of this is seen in Fig. 4 by comparing the dashed profiles (corresponding to [FORMULA]) with the solid ones ([FORMULA]). The dashed Stokes V and Q profiles at phase 0.25 are nearly identical to their solid counterparts at phase 0.75 (see Eq. 9). The profiles differ slightly due to the magneto-optical effects (see Sect. 3.2.2).

3.2. Time evolution of Stokes V and Q

In this section we discuss the evolution of Stokes V and Q on the basis of selected line-profile parameters. The choice of the line-profile parameters is the same as in Paper I. The definitions of [FORMULA] (line shift), [FORMULA] (line broadening), [FORMULA] (sum of the [FORMULA]-component amplitudes) and [FORMULA] and [FORMULA] (relative amplitude and area asymmetry, respectively) are given in Appendix A.

3.2.1. Spatially resolved line profile parameters

Fig. 5 displays [FORMULA] and [FORMULA] of Stokes V and Q formed within a single plane versus the location of that plane, [FORMULA], at the phases 0.25 and 0.75. Both Stokes parameters show no asymmetry at the central plane [FORMULA]. But with increasing distance [FORMULA] from the central plane the asymmetry reaches nearly 100% and reflects strongly distorted profiles. It is the presence of cospatial gradients of the magnetic field and line-of-sight velocity at the flux-tube boundary which is responsible for the production of the Stokes asymmetry (e.g. Grossmann-Doerth et al. 1989, see Paper I). Gradients of these quantities along the rays inside the flux tube are far smaller (certainly for the chosen wave frequency and heliocentric angle). The large asymmetry generated in planes with high [FORMULA] is due primarily to the increasing line-of-sight velocity component with [FORMULA]. This quantity vanishes in the plane [FORMULA] where no asymmetry is generated.

[FIGURE] Fig. 5a-d. Amplitude [FORMULA] and amplitude asymmetry [FORMULA] of Stokes V a and c and Q b and d vs. [FORMULA]. The solid curve refers to phase [FORMULA] and the dotted to [FORMULA]. The dashed curve displays the fractional area coverage of the magnetized plasma and is the same in all 4 frames (see text for details). The bullets mark the locations [FORMULA] of the planes containing the lines-of-sight. The underlying wave is the same as in Fig. 4. Note the increase of [FORMULA] with increasing [FORMULA], coupled with a rapid decrease of the amplitude. Also note that in the presence of the wave the amplitudes [FORMULA] at [FORMULA] are not the same as for [FORMULA].

The opposite dependence on [FORMULA] is found for [FORMULA] which decreases with increasing [FORMULA]. This reflects the fact that the larger the [FORMULA] the smaller the area of intersection of the flux tube with the vertical plane containing the lines-of-sight. In order to estimate the fraction of magnetized plasma we determined the intersection area of the flux tube with the vertical plane within the height range of line formation (between [FORMULA] km and [FORMULA] km). The ratio of this area to the corresponding total area in the computational domain is plotted versus [FORMULA] in Fig. 5 (dashed line) and agrees well with the decrease of [FORMULA].

Two relations are important to note. Firstly, at a given phase the V (and also Q) amplitudes at [FORMULA] differ from those at [FORMULA]. This effect can already be seen in Fig. 4 by comparing the solid and dashed profiles, in particular at phases 0.25 and 0.75. Secondly, at a phase at which Stokes V is stronger for [FORMULA] than for [FORMULA], the opposite is the case for Stokes Q: it is weaker for [FORMULA] than for [FORMULA]. The above described behaviour is due to the fact that a positive [FORMULA] increases [FORMULA] and consequently Stokes Q whereas a negative [FORMULA] similarly enlarges the V amplitude. We shall return to this point when discussing spatially averaged profiles. Note that line shift, and to some extent also line width, exhibits a similar dependence on [FORMULA] as the asymmetry (not plotted).

3.2.2. Spatially averaged line profile parameters

Small flux tubes are generally not resolved by current telescopes. The wave signature in spatially averaged profiles is therefore also of interest. Consequently, we determine the parameters (Appendix A) of the spatially averaged Stokes profiles and study their time evolution over a wave period. Spatially averaged profiles are formed by averaging together the profiles from all planes (each of which is located at a different [FORMULA]). In general we have employed 9 planes. Tests based on the use of more planes indicate that this number is adequate.

The time evolution of the line profile parameters as seen at three positions on the disc ([FORMULA] and [FORMULA], represented by solid, dotted and dashed profiles, respectively) is plotted in Fig. 6. It shows, among other things, that all parameters evolve basically with double the wave frequency in both Stokes V and Q. This behaviour differs from the spatially resolved case (Fig. 4) and must therefore be a consequence of the spatial averaging. The following two points are of importance when considering this averaging.

[FIGURE] Fig. 6a-j. Parameters of spatially averaged Stokes V and Q profiles vs. phase. The plotted line profile parameters, indicated above each frame, are defined in Appendix A. The underlying wave is the same as in Fig. 5, but is now "observed" at three different disc positions corresponding to [FORMULA] (solid), [FORMULA] (dotted) and [FORMULA] (dashed). Note that all parameters oscillate with double the wave frequency.

Firstly, as evident from Fig. 5 only planes with [FORMULA] km give a significant contribution to the spatially averaged profiles (for the particular model flux tube chosen). This is the reason why, e.g., [FORMULA] and [FORMULA] of the spatially averaged profiles are not as large as for the kink wave studied in Paper I.

Secondly, as is evident from Figs. 4 and 5, profiles from opposite halves of the flux tube ([FORMULA] and [FORMULA]) display opposite shifts and asymmetries at a given phase. When adding the profiles from the two halves together the shift and asymmetries are further reduced. They do not disappear due to the difference in V and Q amplitudes between the two halves (see Fig. 4 and the discussion at the end of Sect. 3.2.1). These differences in amplitude are largest at the phases 0.25 and 0.75. At those phases the line shift and asymmetry in each half also have the largest magnitude (due to the correlation between [FORMULA] and [FORMULA] inherent to Alfvén waves, see Eq. 6). Both these facts conspire to produce a peak at phases 0.25 and 0.75 in shifts and asymmetries of V and Q.

That these two peaks have the same sign (i.e. that both are generally maxima or minima) within a wave period reflects the azimuthal symmetry of the wave. Note that to first order [FORMULA] and [FORMULA] and consequently both amplitudes do not depend on the sign of [FORMULA]. After half a wave period the left and right halves of the flux tube are basically interchanged. For the magnetic field and velocity contributions of the torsional wave this fact has been shown with Eqs. 8 and 9 (where for instance [FORMULA] has been neglected). The radial x-component of the background field, [FORMULA], changes sign from one half of the flux tube to the other. This leads to a corresponding change of sign in [FORMULA] which to first order, however, does not affect Stokes V and Q. Then, after spatial averaging (and neglecting magneto-optical effects, see below) the phases t and [FORMULA] are identical so that the resulting shift and asymmetry have the same sign at the peak values. Note the different origin of the doubled frequency in the line broadening. It is only affected by velocity magnitude whereas the sign of the velocity plays no role.

The evolution of the parameters differs between Stokes V and Q. The parameters [FORMULA], [FORMULA] and [FORMULA] of Stokes V have the opposite sign to those of Q. Note first that at a given phase in one half of the flux tube the field is inclined towards the observer (i.e. small [FORMULA] and large Stokes V), while in the other half it is inclined away (i.e. larger [FORMULA] and large Stokes Q). Consequently, at a given phase the dominant V and Q signals emanate from opposite halves of the flux tube (see Fig. 4). For a torsional wave the distortion of the magnetic field is in antiphase with that of the velocity which gives rise to opposite shifts at each phase in the different sides of the flux tubes. Because the dominating profiles of V and Q stem from opposite halves the antiphase between the field and velocity distortion gives rise to the opposite sign of the resulting shift of the spatially averaged profiles.

The area asymmetry [FORMULA] is sensitive to the gradients along the line of sight of the magnetic field and velocity. The sign of the asymmetry is given by (Solanki & Pahlke 1988)

[EQUATION]

Large gradients occur at locations where the line-of-sight enters or leaves the magnetized plasma. Using Fig. 2 it is seen that the gradients at both piercing points along a line-of-sight induce the same sign of [FORMULA] but opposite signs in opposite halves of the flux tube, in accordance with Fig. 6.

Eye catching is the difference between the magnitudes at the two extremes of the Stokes Q parameters at phase 0.25 and 0.75, which is particularly pronounced in [FORMULA] and [FORMULA] (Figs. 6h and j). Stokes V parameters, in contrast, exhibit two almost equally strong peaks. As mentioned above, in the absence of magnetooptical effects the extrema at phases 0.25 and 0.75 are expected to be identical. This difference between the phases indeed vanishes if the radiative transfer is carried out without magneto-optical effects, as test calculations confirm. However, the largest Q profiles at phases 0.25 and 0.75 are generated in opposite halves of the flux tubes, i.e. at locations with opposite [FORMULA] (see Fig. 4). Although the absorption coefficient of Stokes Q is not affected by this, the magneto-optical effects give a term that is sensitive to the sign of [FORMULA] ([FORMULA]), so that the two phases of the wave affect Stokes Q differently. For Stokes V, however, both phases remain identical (except for possible small effects that may appear due to the coupling between the various Stokes parameters in a realistic numerical solution, such as ours, of the Unno-Rachkovsky equations). For a more detailed discussion in the Milne-Eddington approximation see Appendix B.

According to Figs. 6e and f the normalized [FORMULA] is below unity on the average, indicating that the profile amplitudes are decreased by the wave. Different processes play a role in determining [FORMULA] and [FORMULA]. The change of the inclination of the magnetic field vector due to the wave is one of them. However, a large part of the decrease in V and Q amplitudes is simply a compensation for the increased line width (Figs. 6c and d). The [FORMULA]-component area [FORMULA] (not shown) also oscillates, but with a considerably smaller relative amplitude, in support of this interpretation.

The line shift and the asymmetries of Stokes V show the opposite dependence on [FORMULA] than the corresponding parameters of Stokes Q. Note that without net fluctuations in [FORMULA] there would be no net fluctuations in the line shift and asymmetry (after averaging over the left and right halves of the flux tube) because all phases contribute equally to the spatial average. The larger the net fluctuations in [FORMULA] the larger the difference of the contribution of various phases. The dependence on [FORMULA] of the fluctuations of [FORMULA] is related to the relative sensitivity of Stokes Q and V: [FORMULA] and [FORMULA] (cf. Paper I). Consequently, changes in [FORMULA] within a wave period are large for V near the solar limb but near disc centre for Stokes Q.

3.3. Time evolution of Stokes U

The evolution of Stokes U profiles formed along rays lying in a fixed plane (Fig. 4c) does not differ substantially from that of of Stokes Q. In particular, the U-profile evolves in phase with Q. In contrast to Stokes Q, however, the sign of U corresponds to the sign of [FORMULA], i.e the U profiles coming from the right and left halves of the flux tube have opposite sign (see Appendix B for an explanation). The change of sign causes significant differences between the two Stokes parameters. Whereas spatially averaged Q profiles have a similar form to the spatially resolved profiles, this is not the case for Stokes U. Spatially averaged U profiles can be far more complex than their spatially resolved constituents.

In order to help understand the spatial average we display in Fig. 7a Stokes U profiles at phase 0.25 originating at [FORMULA]. The solid line in Fig. 7a denotes the signal at [FORMULA]. It is symmetric due to the vanishing line-of-sight velocity and positive since it is produced purely by magneto-optical effects. The profiles at [FORMULA] 50 km (dashed line) and 100 km (dash-dotted line) have larger amplitude than at [FORMULA] whereas profiles at [FORMULA] km (dash-triple-dotted line) decrease in magnitude. The increase in [FORMULA] from [FORMULA] to 100 km is due to [FORMULA], i.e. due to the increasing [FORMULA] with [FORMULA] (cf. Sect. 2.3, note that [FORMULA] never exceeds 45o). The decrease at larger [FORMULA] reflects, as for Q and V, the decreasing intersection area of the flux tube with the plane containing the lines-of-sight (cf. Sect. 3.2.2).

[FIGURE] Fig. 7. a  Stokes U profiles arising from planes located at [FORMULA]0, 50, 100 and 150 km (solid, dashed, dash-dotted and dash-triple dotted lines, respectively). The plotted profiles corresponds to phase 0.25 of the same wave as underlies Fig. 4. b  The Stokes U profile spatially averaged over the whole flux tube. Solid curve: including magneto-optical effects, dashed curve: with magneto-optical effects switched off.

The average of the U profiles formed over the whole flux tube is displayed in Fig. 7b (solid curve). For comparison the spatially averaged profile calculated without magneto-optical effects is also plotted (dashed curve). Noteworthy are the small amplitudes of these profiles (compared to the amplitudes of some of the profiles in Fig. 7a), as well as their complex and asymmetric shapes. In particular, the profile calculated without magneto-optical effects is almost antisymmetric and appears more like a combination of two shifted Stokes V profiles. Both the reduced amplitude and complex shape are due to the addition of Stokes U profiles having opposite sign originating from the two halves of the flux tube. Their cancellation leads to the small amplitude. Also, profiles resulting from planes with opposite [FORMULA] are wavelength shifted in opposite directions and have different amplitudes. Therefore, they do not cancel each other exactly but build up complex profile shapes. When magneto-optical effects are neglected, Stokes U is proportional to [FORMULA] (Eq. B9, see Appendix B) and the spatially averaged U profile is nearly antisymmetric according to wavelength. Note that this signal is completely caused by the wave, since in the absence of a wave the U profile of a vertical flux tube is entirely generated by magneto-optical effects. If these were switched off U would disappear in an untwisted, static flux tube. The inclusion of the magneto-optical effects introduces terms proportional to [FORMULA], which produce a U signal having the same sign in both halves of the flux tube. These terms are responsible for the predominantly positive U profile in Fig. 7b (solid curve).

Due to the complex shape (which makes it difficult to define profile parameters that may be directly compared with those of Stokes V and Q) and the small amplitude of the spatially averaged U profiles we do not discuss them further, although Stokes U reveals the clearest signal of the torsional waves of all spatially averaged Stokes parameters. Hence we encourage low-noise observations of Stokes U near the limb. Note that in Fig. 7 we have concentrated on the phase 0.25 which, along with phase 0.75, produces the most asymmetric U profiles. Note also that spatially averaged Stokes U profiles fluctuate at the wave frequency in the sense that at all phases different profiles are generated. However, the U amplitude and width oscillate at twice the wave frequency, like the corresponding parameters of Stokes Q and V.

3.4. Temporally averaged parameters

In this section we discuss the signature of torsional waves of temporally (and spatially) averaged Stokes V and Q (Sect. 3.4.1) and U (Sect. 3.4.2) profiles. Note that averaging over time over a single flux tube corresponds approximately to spatially averaging over many flux tubes caught at random phases of the wave.

3.4.1. Stokes V and Q

Fig. 8 displays the dependence on [FORMULA] of the same line profile parameters as plotted in Fig. 6. The parameters are also shown for two heliocentric angles [FORMULA] and two spectral lines. In the following each of these dependences is briefly discussed. Note that all in all the influence of the torsional wave agrees qualitatively with the findings for the kink wave (Fig. 8 of Paper I). The dependence of the signature on the wave frequency and amplitude, the heliocentric angle and spectral line does not differ qualitatively from that of a kink wave.

[FIGURE] Fig. 8a-j. Temporally and spatially averaged line-profile parameters of Stokes V and Q vs. wave amplitude [FORMULA]. The parameters are displayed for two heliocentric angles ([FORMULA] solid, and [FORMULA] dashed) and two spectral lines (Fe i 5083 Å, thin lines and Fe i 5250.2 Å, thick lines).

This includes the fact that the sign of the wavelength shift and asymmetry of Stokes V profiles is opposite to that of Stokes Q profiles. This effect is already seen in the spatially averaged but temporally resolved profiles, as discussed in Sect. 3.2.2. Since these parameters maintain their sign over most of the wave period (Fig. 6) the temporally averaged profiles inherit this property. Compared to the peak values in Fig. 6 the influence of the same wave on the time averaged parameters is small, partly due to the averaging and partly because at phases where the shift, asymmetry and broadening are large (phase 0.25 and 0.75 in Fig. 6) the V and Q amplitude is reduced.

Dependence on wave amplitude and frequency: As expected, the wave amplitude, [FORMULA], plays a dominant role. The influence of the wave on all line parameters increases as [FORMULA] increases due to the increased velocity gradient. The role of the wave frequency [FORMULA] is less important (and therefore not displayed). The larger the frequency the larger the ratio between the height-range over which the line is formed to the wavelength of the wave. This increases the line-of-sight gradients somewhat, producing a slightly larger asymmetry, but decreases parameter fluctuations over the wave period.

Dependence on limb-distance: The heliocentric angle [FORMULA] determines firstly the line-of-sight velocity [FORMULA] (Sect. 2.3) and secondly the sensitivity of the Stokes profiles with respect to changes in magnetic inclination [FORMULA] (Sect. 3.2.2 and Paper I). Stokes V, whose parameters are displayed in the left panels of Fig. 8, shows the expected increase in shift, width and asymmetry from [FORMULA] to [FORMULA], because both the line-of-sight velocity and the sensitivity to [FORMULA]-changes increases towards the limb. The V amplitude also decreases more strongly at [FORMULA], partly as a compensation for the increased line width: The decrease in [FORMULA]-component area is much smaller. The behaviour of the Stokes Q line parameters reflects, on the one hand, the loss of sensitivity with respect to [FORMULA]-changes toward the limb, and on the other hand the increased [FORMULA]. Hence, the line width, which is mainly sensitive to [FORMULA], increases towards the limb (and the amplitude decreases). The line shift, however, decreases in magnitude towards the limb, while the asymmetry remains relatively unchanged.

Dependence on the spectral line. In Paper I we found that the line Fe i 5083 Å reacts more sensitively to the kink wave than Fe i 5250.2 Å. This is particularly true for [FORMULA] and [FORMULA]. We find that this is also the case for the torsional waves, as can be seen from Fig. 8. The main reason is again that Fe i 5083 Å is more saturated, which gives it a larger asymmetry (cf. Solanki 1989, Paper I).

3.4.2. Stokes U

Fig. 9 a shows the temporally and spatially averaged Fe i 5250 Å U profiles for [FORMULA] km s-1 (solid line), [FORMULA] km s-1 (dashed line) and [FORMULA] km s-1 (dash-dotted line). As expected from Sect. 3.3 the U amplitude is far smaller than that of Q or V, but nevertheless slightly larger than the U generated without the wave. Note the increasing asymmetry with increasing [FORMULA]. These temporally averaged U profiles are more symmetric than the profiles at phase 0.25 and 0.75 shown in Fig. 7. Two effects are responsible for this: 1) at most other phases the U profiles are more symmetric 2) in the course of a wave period the asymmetry of U changes sign, so that averaging over these profiles leads to far smaller net asymmetry. The opposite sign of the asymmetry to that of Stokes Q reflects the different dependence of these profiles on [FORMULA] in the presence of averaging.

[FIGURE] Fig. 9a and b. Temporally and spatially averaged line profiles of Stokes U for different [FORMULA] and [FORMULA]. The profiles displayed in frame a correspond to wave amplitudes [FORMULA] km s-1 (solid line), [FORMULA] km s-1 (dashed line) and [FORMULA] km s-1 (dash-dotted line), those in frame b to [FORMULA] (solid), [FORMULA] (dashed) and [FORMULA] (dash-dotted).

Fig. 9b shows line profile at [FORMULA] (solid line), [FORMULA] (dashed line) and [FORMULA] (dash-dotted line) for Fe i 5250.2 Å. At large [FORMULA] a residual effect of the wave is visible in the asymmetry of the profiles, whereas magneto-optical effects dominate the profiles at small [FORMULA].

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Online publication: April 28, 1999
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