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

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3. Results

We present results of a parameter study of kink waves and spectral lines formed in their presence. The influence of the following quantities on the Stokes profiles and their parameters is investigated:

  1. The parameters determining the wave: These are the circular frequency [FORMULA] lying in the interval between [FORMULA] Hz and 0.150 Hz , and the velocity at the bottom of the computational domain: [FORMULA] km s [FORMULA] km s-1.
  2. The angle between the line-of-sight and the vertical: [FORMULA].
  3. The spectral line: Fe i [FORMULA], Fe i [FORMULA] and Fe i [FORMULA].

We first focus on the time dependence of the Stokes Profiles (Sect. 3.1) and then define and discuss the line profile parameters at 12 time steps (named phases 0 to 11 in the following) within a wave period (Sect. 3.2). Later, in Sect. 3.3, the time averaged profiles and parameters are discussed for all three spectral lines.

3.1. Time evolution of Stokes profiles

Fig. 4 provides an overview of the time evolution of Stokes V (left hand panels) and Stokes Q (right hand panels) of all three spectral lines over one wave period. The response to the distorting wave exhibited by each spectral line is different. For example, Fe i [FORMULA] shows little shift, but extreme variations in its [FORMULA] -component amplitudes and in particular in its blue-red asymmetry. In contrast, the amplitude and asymmetry of Fe i [FORMULA] varies only slightly. Instead, a considerable oscillatory line shift is present. Finally, Fe i [FORMULA] exhibits an intermediate behaviour. Note that the flat cores of Fe i [FORMULA] and Fe i [FORMULA] are not due to excessive Zeeman splitting, but are the result of a combination of saturation, finite FT width and horizontally homogeneous external and internal atmosphere. (See Solanki et al. 1996 for details.) Fig. 4 also shows that some features of Stokes V and Q profiles oscillate in phase, others in anti-phase. For example, their wavelength shifts and blue-red asymmetries evolve in phase, but their amplitudes in anti-phase, i.e. the V profile is strongest when the Q profile is weakest and vice versa. In the following we study the quantitative response of the above and other line parameters to kink waves.

[FIGURE] Fig. 4. Stackplots of the response of Stokes V (left-hand panels) and Stokes Q (right-hand panels) profiles of all three spectral lines to the same wave as in Fig. 1 are shown for 12 time steps covering one full wave period. All profiles are normalized to the continuum intensity [FORMULA]. The calculations correspond to a heliocentric angle [FORMULA]. The profiles are offset relative to each other for clarity. The time steps are marked at the right of each frame (time increases from 0 to 11)

3.2. Time evolution of line parameters

The evolution of a set of line parameters (defined below) with time over a full wave period is illustrated in Fig. 5a-j for the Fe i [FORMULA] line. We first discuss the results for this line. As an example we have chosen an upward propagating wave with [FORMULA] km s-1 and [FORMULA] Hz, i.e. the same wave as in Figs. 1, 2 and 4. This frequency is sufficiently high that (according to Fig. 3) velocity and FT inclination are nearly in antiphase.

[FIGURE] Fig. 5. The evolution of the line profile parameters of Fe i 5250 Å  over a single, full wave period is shown. The wave has frequency [FORMULA] Hz, wavelength [FORMULA] km and [FORMULA] km s-1, i.e. it is identical to the wave underlying Figs. 1, 2 and 4. Results are plotted for [FORMULA] (solid), [FORMULA] (dotted) and [FORMULA] (dashed). a to j present the response of various Stokes line parameters (left-hand panels Stokes V and right-hand panels Stokes Q). a and b show wavelength shifts, c and d line widths, e and f [FORMULA] -component amplitudes, g and h blue-red relative amplitude asymmetry and i and j relative area asymmetry. The definitions of each of the plotted parameters is given in the text. Time steps are indicated on the horizontal axes. Step 12 represents the same phase as step 0

Definitions of the line profile parameters: The line shift [FORMULA], where [FORMULA] is the wavelength of the red, respectively blue [FORMULA] -component peak (determined by placing a quadratic function through the 3 points closest to the peak) is plotted vs. time step or phase in Fig. 5a and b. We found this parameter to be superior to [FORMULA], the zero-crossing wavelength of Stokes V, and [FORMULA], the wavelength of the [FORMULA] -component maximum of Stokes Q, as can easily be confirmed by considering the profiles of, e.g., Fe i [FORMULA] in Fig. 4.

The most robust parameter describing the line broadening which we found is the difference between the centre-of-gravity wavelengths of blue and red [FORMULA] -components:

[EQUATION]

The function [FORMULA] stands for Stokes Q or V and [FORMULA] for the unsigned wavelength relative to line-centre. After isolating the effects of the wave by removing the width of the reference line according to [FORMULA] , the result is plotted in Fig. 5c and d. Since the magnetic field strength is not affected by the wave, we expect [FORMULA] to be mainly influenced by velocity gradients. Negative [FORMULA] (we take the square root of the positive number and change the sign), which is occasionally seen in our calculations (see Stokes Q for [FORMULA]), does not indicate that the line has been narrowed by the wave but rather signals a breakdown of the assumption of Gaussian profiles made above when removing the width of the reference profile. For example, the strength and width of the [FORMULA] -component, which is changed by the wave, has a significant effect on the [FORMULA] of the Q profile.

The unsigned [FORMULA] -component amplitudes are denoted by [FORMULA] and [FORMULA], (where b and r indicate the blue and red [FORMULA] -components, respectively). Fig. 5e and f show the variation of the total amplitude normalized to the reference amplitude [FORMULA]. The relative amplitude asymmetry [FORMULA], defined as

[EQUATION]

is plotted in Fig. 5g and h, while the time dependence of the relative area asymmetry,

[EQUATION]

is exhibited in Fig. 5i and j. Here, [FORMULA] and [FORMULA] are the unsigned areas of the blue and red [FORMULA] -components, respectively.

Discussion of the line parameters: With the exception of [FORMULA] all the line parameters plotted in Fig. 5 oscillate with the frequency of the wave. [FORMULA], [FORMULA] and [FORMULA] also react approximately linearly to the sinusoidal wave, whereas the [FORMULA] -amplitudes respond nonlinearly (the nonlinearity is more pronounced in Stokes V). In addition, most parameters possess maxima and minima around phases 1-2 and 7-8. Then the velocity around the estimated formation height of the [FORMULA] -peaks of Fe i [FORMULA] ([FORMULA] km above the quiet-sun [FORMULA] level, as estimated from contribution function calculations) is largest and directed towards, respectively away from the observer (Fig. 2). Note, that the magnetic field is inclined towards the observer (at time step 8) and away from the observer (at step 2). This is reflected in the amplitudes of Stokes Q and V. Whereas all other line parameters of Stokes V oscillate in phase with those of Q, the amplitude of V is in antiphase with the amplitude of Q.

We now discuss the temporal behaviour of the line parameters in greater detail, in particular their dependence on wave frequency, heliocentric angle and spectral line. The dependence on velocity [FORMULA] is discussed in Sect. 3.3.

Asymmetries and their production: In Fig. 5 the asymmetry parameters [FORMULA] and [FORMULA] of both Stokes Q and V reveal a nearly sinusoidal temporal behaviour. They all oscillate in phase and their oscillation amplitudes are roughly the same. In particular, the kink mode on its own does not enhance [FORMULA] relative to [FORMULA]. The asymmetries increase rapidly with increasing wave amplitude and even more rapidly with increasing [FORMULA] and can approach 100%. The formation of such large asymmetries needs to be discussed in greater detail. Asymmetric Stokes profiles are efficiently produced by co-spatial gradients of line-of-sight velocity and magnetic field vector (Illing et al., 1975; Makita, 1986; Grossmann-Doerth et al., 1988; Sánchez Almeida & Lites, 1992; Solanki, 1993). In the present geometry we must differentiate between the abrupt field strength and velocity gradients at the FT boundary (which is crossed at least twice by each ray for sufficiently large [FORMULA] values) and more gentle gradients in the field strength, inclination and velocity within the FT along each ray. For a kink wave the asymmetry produced at both the intersections of a slanted ray with the FT boundary has the same sign, as can easily be verified using the relation (e.g. Solanki, 1993)

[EQUATION]

When viewed near the limb (large [FORMULA]), therefore, gradients due to the kink waves at the flux tube boundaries are more efficient in producing [FORMULA] in the present case than internal gradients or external granular flows (compare with Bünte et al., 1993). This explains the extremely large [FORMULA] values produced by the wave (Fig. 5). The gradients at the FT boundaries also produce an amplitude asymmetry [FORMULA], which is generally of similar magnitude and sign as [FORMULA] (e.g. Solanki, 1989).

Let us now discuss the influence of internal gradients on the production of asymmetries. Internal gradients are expected to have a stronger influence at higher wave frequencies, because then, due to the shorter wavelength, the internal gradients are larger. Since the largest internal gradients of the velocity show a phase lag relative to maximum velocity we can test for the influence of internal gradients by checking whether the extrema in [FORMULA] and/or [FORMULA] occur at phases other than 2 and 8, at which the gradients across the FT boundary are largest. Fig. 5 shows that for small [FORMULA] such a shift is present. It increases with wave frequency (not plotted) and is more pronounced in the amplitude asymmetry. Thus the influence of internal gradients relative to the jumps at the boundaries is largest at small [FORMULA], possibly due to the comparatively long path of a given ray within the FT. (But note that both the velocity gradients and and the asymmetries get smaller for increasingly small [FORMULA]).

The influence of the wave frequency: There are two effects caused by the wave frequency. Firstly, the frequency determines the phase lag between velocity v and magnetic field inclination [FORMULA] (see Fig. 3). This phase lag controls the phase of maximum [FORMULA] with respect to the phase of extreme [FORMULA], [FORMULA] and [FORMULA]. For high frequencies, i.e [FORMULA], the maximum of [FORMULA] occurs at roughly the same time as the extrema of [FORMULA], [FORMULA] and [FORMULA] (cf. Fig. 5), whereas for frequencies close to the cut-off (not plotted) there is a significant temporal shift between the extrema of [FORMULA] and those of [FORMULA], [FORMULA] and [FORMULA]. At the lowest frequency we have considered, [FORMULA] Hz, we estimate this phase shift to be roughly a quarter of the wave period.

Secondly, as the wave frequency increases (and the wavelength accordingly decreases), the amplitudes of the oscillations exhibited by all line parameters decrease (for fixed wave velocity [FORMULA]). This is caused by the increasing ratio of the width of the line contribution (or response) function to the wavelength. In addition, with increasing wave frequency some parameters oscillate around an increasingly large, time independent offset. This is most obvious for the line shifts and the asymmetries.

Centre-to-limb variation: Whereas [FORMULA] (line shift) shows little dependence on [FORMULA] the other line parameters are significantly affected by it. For all parameters except the normalized amplitudes (Fig. 5e and f) Stokes Q and V show the same dependence on [FORMULA]. The [FORMULA] oscillation amplitude of Q, however, is largest at small [FORMULA], while that of V is largest at large [FORMULA]. We sketch out an explanation of this behaviour for an optically thin line (the results for an optically thick line are not expected to be too different). It is well known that Stokes V [FORMULA] and Stokes Q [FORMULA], where [FORMULA] is the angle between the line-of-sight and the magnetic vector. Now, [FORMULA] oscillates as the FT sways back and forth over a wave period. The relative derivatives of Stokes Q and V according to [FORMULA] are a measure of the change produced in the relative amplitudes of these profiles by changing [FORMULA]: [FORMULA] and [FORMULA]. Hence we expect [FORMULA] for Stokes V and [FORMULA] for Q. This difference in sensitivity to changes in [FORMULA] between Stokes Q and V amplitudes agrees well with Fig. 5e and f.

FT inclination: The Stokes V and Q amplitudes are temporally in antiphase, as is clearly visible in Fig. 5e and f. This antiphase is expected due to their respective [FORMULA] and [FORMULA] dependence. In other words Q is largest near the phase at which the FT is inclined away from the observer, while V is largest when it is inclined towards the observer. The ratio of the amplitude of Stokes Q to that of Stokes V, [FORMULA], is plotted in Fig. 6. Note that the plotted curves have been divided by, i.e. normalized to, the Stokes Q to V ratio of the reference profile at the respective [FORMULA]. The ratio follows the wave very clearly and obviously reflects the periodic swaying motion of the FT. The 3 lines exhibit a similar behaviour. Note, however, the (small) phase difference in inclination between the infrared and visible lines arising from the higher formation of the latter. At large [FORMULA] this ratio exhibits extremely large fluctuations over a wave period. This has to do with the fact that the FT can become nearly perpendicular to the line-of-sight around phases 0-2, so that Stokes V (i.e. the denominator) becomes nearly zero at these phases.

[FIGURE] Fig. 6. The ratio of Stokes Q to V amplitude [FORMULA] is plotted vs. time step for the lines Fe i 5083 Å (Fig. 6a), Fe i 5250 Å (Fig. 6b) and Fe i 15648 Å (Fig. 6c) for [FORMULA] (solid), [FORMULA] (dotted) and [FORMULA] (dashed). The wave parameters are [FORMULA] and [FORMULA] Hz, i.e the same as in Figs. 1, 2, 4 and 5. The amplitude ratio has been divided by the corresponding ratio of the reference profiles at the same [FORMULA]

Dependence on spectral line: There are considerable differences between the line profile parameters of the 3 spectral lines. For example, the line shift and line width of Fe i [FORMULA] oscillates with a significantly smaller amplitude than the corresponding quantities of Fe i [FORMULA]. The [FORMULA] line itself shows smaller amplitudes of the [FORMULA] and [FORMULA] oscillations than the wave at this line's expected height of formation. This is in agreement with the expectation that the stronger line, Fe i [FORMULA], is formed over a larger range of height. For example, once the width of the contribution or velocity response function becomes of the order of the wavelength (the formation height range of [FORMULA] -200 km has to be compared with [FORMULA] km for [FORMULA] Hz) the line shift increasingly fails to reflect the true amplitude of the wave. For the two visible lines, the amplitude (but not necessarily the time evolution) of the wave is more accurately reflected in [FORMULA] than in [FORMULA], in particular for larger wave frequency.

The amplitudes of the asymmetry oscillations are greater than in Fe i [FORMULA], due to the larger saturation in Fe i [FORMULA] (Grossmann-Doerth et al., 1989; Solanki, 1989). In addition, Fe i [FORMULA] reacts more sensitively to changes in wave frequency.

The response of Fe i [FORMULA] differs substantially from that of the other two lines. The amplitude of the line shift corresponds closely to the wave amplitude around the level of line formation. The asymmetry produced by the wave is, in contrast, exceedingly small. This behaviour can be understood in terms of the relative weakness of the line and its large Zeeman splitting (Grossmann-Doerth et al., 1989).

3.3. Temporally averaged profiles

Consider now Stokes Q and V profiles averaged over a full wave period. On the one hand this corresponds to time averaged measurements, on the other hand to a snapshot of many FTs caught at random phases, such as produced by observations with moderate spatial resolution. In Fig. 7 we plot Stokes V and Q profiles of Fe i [FORMULA] averaged over a wave period. In the upper panels (Fig. 7a and b) we illustrate the influence of wave frequency, in the lower panels (Figs. 7c and d) the influence of the wave amplitude. Figs. 7a and b show asymmetric profiles at high frequencies and a transition to more symmetric profiles at frequencies close to the cut-off. (We emphasize, however, that waves with frequencies near the cut-off do not satisfy the approximation of an isothermal atmosphere and, therefore, should be used with caution.) Note that in Figs. 7a and b the frequencies are not equidistant. The velocity obviously affects the amplitude, asymmetry, broadening and shift of the lines (Figs. 7c and d). Note in particular the opposite sense of the asymmetry of Stokes V (red wing stronger than blue wing) and Q (blue stronger than red). This contrasts strikingly with the time-resolved profiles plotted in Fig. 4. Those Stokes V and Q profiles show the same sense of asymmetry at practically every phase (cf. Fig. 5). The temporally averaged V profiles also appear to be shifted towards the red, whereas the Q profiles are blue shifted, again in contrast to the time resolved profiles. The cause of this difference between the time resolved and time averaged profiles is discussed later in this section.

[FIGURE] Fig. 7. Temporally averaged Stokes [FORMULA] (a) and [FORMULA] (b) profiles are plotted for waves with [FORMULA] km s-1 and different wave frequencies indicated in a in Hz. The radiative transfer was carried out for a heliocentric angle of [FORMULA]. The solid curves denote the reference profile, i.e. the Stokes profile calculated in the absence of a wave. c and d show the same for fixed [FORMULA] Hz and different values of [FORMULA] indicated in c in km s-1

The same profile parameters as in Fig. 5, but now of temporally averaged profiles of different lines, are plotted vs. [FORMULA] in Fig. 8a to j. This figure confirms that averaged over a wave period [FORMULA], [FORMULA] and [FORMULA] of Stokes V are opposite in sign to the respective Q parameters. Note, however, that in general the net wavelength shift due to the wave is relatively small. On the other hand, the net Stokes Q and V asymmetry produced by even a modest amplitude wave is extremely large, unless the wave frequency lies very close to the cut-off.

[FIGURE] Fig. 8. Parameters of temporally averaged Stokes V (left panels) and Q (right panels) profiles vs. [FORMULA] for Fe i 5083 Å (solid), Fe i 5250 Å  (dotted) and Fe i 15648 Å ( dashed). The parameters and their order in the figure are the same as in Fig. 5. The plotted line parameters are affected by a wave with frequency [FORMULA] Hz ([FORMULA] km) "observed" at [FORMULA] (thin lines) and [FORMULA] (thick lines). Due to the large width of the reference profiles of Fe i 15648 Å (caused by Zeeman splitting), changes in [FORMULA] due to the wave could not be well determined. The corresponding curves are not plotted in Fig. 8c and d

In order to understand the opposite senses of [FORMULA], [FORMULA] and [FORMULA] of Stokes V relative to Q we must bear in mind that these parameters change sign over a wave period, with [FORMULA] and [FORMULA] being positive when the lines are blue shifted and negative for red shifted lines (Fig. 5). If we simply averaged these line parameters over a wave period then their values would be exceedingly small. If the line profiles are averaged the sign and magnitude of the net asymmetry and shift of the temporally averaged profiles is then determined mainly by the absolute amplitude [FORMULA] of Stokes Q and V at different phases. The greater the amplitude of the Stokes profile at a certain phase the more it contributes to the average. The anti-phase of [FORMULA] of Stokes Q and V (cf. Fig. 5) is responsible for the opposite sense of the line shift and asymmetry of Stokes V relative to that of Stokes Q. The basic reason for this behaviour is the [FORMULA] phase difference between wave velocity and FT inclination (Fig. 3) for this wave. This means that maximum inclination away from the observer (maximum Q) is cotemporal with maximum velocity away from the observer (maximum positive asymmetry) while minimum inclination (maximum V) is cotemporal with maximum velocity towards the observer (maximum negative asymmetry).

Next, let us compare the three spectral lines. As expected from the behaviour of the time resolved line parameters Fe i [FORMULA] reacts most strongly to the kink mode, with the exception of the net line shift, which is smallest for this line (it also showed the smallest time-resolved shift). Note, in particular, that the Stokes V asymmetry of Fe i [FORMULA] is more negative than that of Fe i [FORMULA] at every [FORMULA]. This is particularly interesting since the observations away from disk centre suggest that such is actually the case on the sun (Pantellini et al., 1988; Bünte et al., 1993), whereas the simulations of Bünte et al. (1993) show that using a purely granular model it is difficult to obtain the correct relative asymmetries.

Finally, note the almost linear dependence of the time-averaged line broadening on the wave velocity. [FORMULA] is expected to scale as [FORMULA]. The calculated [FORMULA] usually lies somewhat above this estimate, partly we expect due to the difference between the line formation height and the lower end of the calculation domain, to which [FORMULA] refers.

In Fig. 9 we plot the centre-to-limb variation of the line parameters of Fe i [FORMULA] for waves with a fixed velocity amplitude [FORMULA] but different frequencies. The centre-to-limb variation of Stokes V is relatively easy to predict: [FORMULA], [FORMULA], [FORMULA] and [FORMULA] all increase monotonically towards the limb, exactly as expected for a kink wave running on a vertical FT. The behaviour of Stokes Q is at first sight more enigmatic. Only [FORMULA] steadily increases towards the limb, whereas [FORMULA] decreases and [FORMULA] and [FORMULA] initially increase with increasing [FORMULA], before decreasing again.

[FIGURE] Fig. 9. The same parameters of temporally averaged Stokes V and Q profiles of Fe i 5250 Å as in Figs. 5 and 8, but now plotted vs. [FORMULA]. The parameters are calculated in the presence of waves with [FORMULA] km s-1 and four different frequencies, [FORMULA] Hz (solid curves), [FORMULA] Hz (dotted), [FORMULA] Hz (dashed) and [FORMULA] Hz (dot-dashed). Note that the Stokes V parameters for [FORMULA] are not reliable since so close to the limb the wave causes the longitudinal component of the field and thus Stokes V to change sign periodically. Consequently, they have not been plotted

In order to understand this behaviour recall that the time-averaged parameters plotted in Fig. 9 are large when (among other things) there is a large difference in the strength or [FORMULA] -amplitude of the Stokes V, respectively Q profiles at different phases in the wave (Fig. 5e and f). This difference in strength is produced by the oscillation of [FORMULA]. We must thus consider how Stokes Q and V react to changes in [FORMULA], which has been discussed in Sect. 3.2. The sensitivity of the Stokes V relative amplitude to [FORMULA] is largest near the limb, while the Stokes Q relative amplitude become increasingly independent of [FORMULA] near the limb. The decrease (at large [FORMULA]) in the Stokes Q asymmetry is consequently due to this insensitivity (Fig. 5). The initial increase of [FORMULA] and [FORMULA] with [FORMULA] reflects the increase of the line-of-sight velocity gradients.

The dependence on wave frequency produces no major surprises. As expected, the smallest frequency dependence is exhibited by [FORMULA] (which does not depend on the phase difference between velocity and inclination), whereas the asymmetries react most strongly to [FORMULA] (particularly Stokes Q asymmetry). Also as expected, changes in [FORMULA] values near [FORMULA] have the largest influence on the line parameters. In addition, these low [FORMULA] values produce the smallest line shift and asymmetries, due to the nearly [FORMULA] phase shift between velocity and FT inclination at these frequencies.

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© European Southern Observatory (ESO) 1997

Online publication: April 28, 1998

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