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Astron. Astrophys. 336, 743-752 (1998)

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4. Results and discussion

4.1. One-dimensional power, coherence, and phase spectra

The first approach to the chromospheric oscillations measured in the Na D2 line was made by plotting one-dimensional power spectra averaged over the whole field of view, but also separately for network and intra-network regions (see Fig. 1). However, as the same field of view was retained throughout the time series, this comparison may not yield a really representative result due to the relatively low statistical weight of the data. Yet, it is surprising that neither in the power spectrum derived from the Doppler shifts nor in another one derived from fluctuations of the line-centre intensity pronounced differences were found between the three curves.

[FIGURE] Fig. 1. One-dimensional mean power spectra - upper panel: derived from Doppler shifts, lower panel: derived from the fluctuation of the line-centre intensity (the dashed line refers to the intra-network, the dash-dotted line to network regions, the solid line represents the power spectrum for the whole field of view)

In the upper panel of Fig. 1, the power maximum lies around 3.3 mHz for the network, corresponding to the well-known 5-minute oscillations. It is remarkable, though, that for the intra-network, the maximum is just slightly shifted towards higher frequencies. The latter power curve does not rise for frequencies higher than 3.3 mHz to indicate maximum power for about 5 - 5.5 mHz (corresponding to periods between 180 and 200 s) which was reported by numerous observers from other chromospheric lines (Lites et al. 1993; Bocchialini et al. 1994; von Uexküll & Kneer 1995). An explanation for the comparatively weak power in the 5 - 5.5 mHz frequency range would be a lower line formation height of Na D2 than previously adopted. This hypothesis is supported by a similar finding of Kneer & von Uexküll (1993) for the Mg b2 line which is formed in the lower chromosphere (see e.g. Schmieder (1979), Staiger (1985) or Deubner et al. (1990)) and does not show maximum power from oscillations with periods of about 3 minutes, either. Besides, like for Na D2, the power curves did not indicate any marked differences between the network and the intra-network regions. Although as yet, the determination of the individual line formation heights is too complicated to yield definite results, there still seems to be an agreement on the relation of these heights: The Mg b2 line is formed lower in the solar chromosphere than Na D2, but the latter line formation height might not lie as high as about 1000 km above the level where [FORMULA] which was inferred e.g. from Gehren (1975) and from Caccin et al. (1980). From a number of further publications, one may conclude as well that Na D2 is formed somewhat lower in the chromosphere, but, of course, still higher than the Na D1 line. Fleck & Deubner (1989) also mentioned this relation and assumed a line formation height of about 800 km for [FORMULA] which Staiger (1987) previously stated to be 760 km. The results presented here give rise to some speculation about the exact formation height indeed. Although they are not suitable to settle this question, they may serve to narrow down the plausible height range. At least, they strongly suggest that its upper limit does not reach the highest estimates quoted above. Sect. 4.2 will provide some clues as to the equally important problem of the lower limit, from which not too precise a conclusion, yet a safe one, can be drawn: All results are consistent with the assumption that Na D2 is formed somewhere in the lower to middle chromosphere. As this piece of information is anticipated here, the question of the line formation height need not be resumed every time this topic is touched again in the discussion of further results.

The power spectra in the lower panel of Fig. 1 were obtained from the fluctuation of the line-centre intensity, but otherwise they correspond to those in the upper panel. As in the course of the Fast Fourier Transform (FFT) the average intensity or velocity, respectively, was subtracted from the data, all power spectra presented here drop to zero at 0 mHz, whereas a real common feature in both panels is the enhanced power in the frequency range from about 2 to 6 mHz for network as well as for intra-network areas. Yet the enhancement in the lower panel is small compared to the power maximum around 0.15 mHz, contrary to the situation shown in the upper diagram.

Even though the position of this low-frequency maximum in the power spectra in both panels is not related to any periodic process on the Sun but is given by the length of the underlying data set (a time series or "period" of nearly 2 hours is equivalent to a frequency of about 0.15 mHz), there is a physical explanation for the substantial increase of this maximum in the lower panel, which is illustrated by Figs. 2 and 3.

[FIGURE] Fig. 2. Temporal development of the fluctuation of the line-centre intensity for three distinct small areas from the network

[FIGURE] Fig. 3. Temporal development of a small network region in a ([FORMULA] [FORMULA] [FORMULA]) subfield of the minimum images

Evidently, the low-frequency power is much higher for network than for intra-network regions. Fig. 2 gives three examples of the temporal behaviour of the line-centre intensity of small network areas where the line-centre brightness slowly decreases while the amplitude of the intensity fluctuation remains nearly the same over 2 hours. It was checked by the white-light images of the time series, too, that seeing effects could only have had a minor influence on these curves. Therefore, the interpretation suggests itself that here, the (perhaps only short-term) fading of network regions is monitored, as the line-centre intensity of the network approaches the value of the surrounding intra-network. On the other hand, there were also network regions with slowly increasing (while "rapidly" fluctuating) minimum intensity which point to a strengthening of network boundaries. Thus, both the enhancement and the fading of network regions obviously occur only locally, and both processes leave their traces in the corresponding power spectrum of Fig. 1 forming low-frequency maxima.

Fig. 3 shows the temporal development within a subfield of the minimum images which contains the "pure" network region, whose fluctuation of the line-centre intensity is shown in the upper panel of Fig. 2, and the surrounding intra-network within a field of view of [FORMULA] [FORMULA] [FORMULA]. Here, time proceeds row by row with a distance of 56 s between successive images. The network boundary is (mostly) seen here as an almost vertical bright streak. At the beginning of the observing run and also during the second half of the time series, the appearance of this part of the network was changing significantly. As already mentioned, seeing effects did not contribute much to its rather diffuse appearance towards the end of the observation. Hence, this sequence of minimum images can be trusted to reflect the real changes within the solar atmosphere.

Fig. 4 represents the one-dimensional V-I coherence and phase spectra of the Na D2 line, obtained from the whole field of view. In the frequency range between 0 and 0.8 mHz, the phase reaches about -160o which might be due to supergranular flows, as there is an anti-correlation between blueshift and brightness. The 70o plateau between 0.8 and 2 mHz seems to be a new discovery in chromospheric V-I phase differences. Nevertheless, an explanation for this plateau was already found some time ago. As the two-dimensional phase spectrum yields more detailed information and the plateau is also seen there, it will be treated in the next subsection. A discussion of the change of sign with decreasing frequency at about 2.1 mHz from negative to positive values which was also found in the phase spectra of Schmieder (1976), Lites & Chipman (1979), and Staiger et al. (1984) will follow in Sect. 4.2 as well.

[FIGURE] Fig. 4. One-dimensional V-I coherence and phase spectra derived from the Na D2 line over the whole field of view

In the evanescent range, between 3 and 5 mHz, where the coherence is as high as 0.8, the phase difference is -90o, as predicted by theory. In the acoustic domain, between 5 and 7.5 mHz, the coherence decreases from high values down to 0.15. Here, the phase difference is still -90o on the average and thus suggests standing waves without necessarily being an indication of them. A similar behaviour was also found in the V-I phase spectra of the Ca ii line at 8498 Å and Na D1 and has been interpreted as the signature of standing waves indeed (Fleck & Deubner 1989). However, it is important to note that these one-dimensional spectra just represent the case of the corresponding two-dimensional k-[FORMULA] diagrams integrated over the horizontal wavenumber k. The phase differences found here thus result from a mixture of waves with various wavenumbers. The two-dimensional phase spectrum treated in Sect. 4.2 therefore provides a better basis for further analysis.

Contrary to other observations, a phase jump of [FORMULA]180o at about 8 mHz is missing here. It was observed for the first time by Staiger et al. (1984) and later on, Fleck & Deubner (1989) also found one in the Na D1 line at [FORMULA], whereas at the centre of the solar disk ([FORMULA]), they observed a phase jump of only about 90o in Na D1 which they considered to be "incomplete". However, in Fig. 4, the phase values just slowly decrease with some scattering from 7.5 mHz on. There is no indication of any phase jump at all. Yet, this does not necessarily mean that for Na D2, a phase jump really does not occur, as the Nyquist frequency for the data set presented here is only 8.9 mHz, and aliasing might have affected this result. At the expense of "true" two-dimensional recording of their data set, Deubner et al. (1996) obtained a time series in Na D2 with a much higher Nyquist frequency of 20 mHz, using the so-called lambdameter method for their analysis. Their V-I phase spectra displayed again a phase jump of 180o near 8 mHz. This discontinuity is best seen in two V-I phase spectra from two positions a little higher up in the Na D2 profile than the very line core. The lambdameter method also allowed them to derive V-V spectra from only one Fraunhofer line, i.e. from different positions within the line profile corresponding to different atmospheric heights. Surprisingly, the V-V spectra of Na D2 revealed another 180o phase jump at about 7 mHz, belonging to adjacent positions considerably higher in the line profile or, correspondingly, much lower in the solar atmosphere than in the case of the V-I phase spectra. The phase jump in the V-V spectra was claimed to be the first one of this kind ever seen. Deubner et al. explained their combined findings by a three-component wave field meeting several specific requirements.

The one-dimensional V-I coherence and phase spectra for the intra-network and for the network regions (see Fig. 5) look similar to those for the entire field of view. Only the scattering of phase values for the network boundaries (e.g. between 1.1 and 2.4 mHz) is somewhat more pronounced, thus, in some frequency intervals, the coherence is lower than for the intra-network. This enhanced scattering might simply be accounted for by the smaller number of pixels belonging to network regions in the field of view (20 % of all pixels considered, cf. Sect. 3.4).

[FIGURE] Fig. 5. One-dimensional V-I coherence and phase spectra for the intra-network (asterisks) and for the network (diamonds)

4.2. Two-dimensional diagnostic tools

The two-dimensional power spectrum in Fig. 6 was derived from the oscillation of the Doppler shifts of Na D2. The solid line indicates the Lamb mode ([FORMULA], where 7 km s-1 was adopted for [FORMULA], and k = (horizontal) wavenumber) and the dashed line marks the f-mode ([FORMULA] with g = 274 m s-2).

[FIGURE] Fig. 6. Two-dimensional power spectrum of Na D2 oscillations derived from Doppler shifts; the solid line represents the Lamb mode, the dashed curve indicates the f-mode

The p-modes can be distinguished separately from p0 (or f-mode), extending to a horizontal wavenumber of about 4 Mm-1, to p5. The p-modes reach to higher frequencies than the acoustic cut-off frequency ([FORMULA] 4.7 mHz) and thus show "pseudo-ridges" (see Kumar & Lu (1991)) which are also seen in other chromospheric k-[FORMULA] diagrams (e.g. Fernandes et al. (1992)).

A most remarkable result of Fig. 6 is the missing hint of a chromospheric eigenmode which in case of existence would have shown as a horizontal bright streak in the frequency range from about 5 to 5.5 mHz (corresponding to 3-min oscillations). Ulrich & Rhodes (1977) predicted a frequency of 5.6 mHz for such a mode from theoretical calculations. Steffens et al. (1995) report enhanced power between the ridges near the cut-off frequency at 5.6 mHz which might be an indication of a "chromospheric ridge".

Fig. 7 displays a two-dimensional power spectrum obtained from the fluctuation of line-centre intensities. Here, too, the ridges corresponding to p0 up to p5 appear, but with somewhat poorer contrast than in Fig. 6. Yet, the extension of the f-mode to a wavenumber of 4 Mm-1 and of the pseudo-modes to about 6 mHz in frequency is still discernible. Fig. 7 gives no hint of a chromospheric eigenmode either, but it shows a striking region of enhanced power in the low-frequency range below 1.5 mHz and limited in wavenumber to below about 2 Mm-1. This enhancement might be due to the strengthening or fading of the network, and, possibly, to gravity waves.

[FIGURE] Fig. 7. Two-dimensional power spectrum of the line-centre intensity fluctuation of the Na D2 line

The V-I coherence and phase spectra of Na D2 are represented by Figs. 8 and 9. In the first one, a region of very high coherence extends from the domain of 5-min oscillations over the acoustic cut-off frequency to about 6.5 to 7 mHz. In the inter-ridges, the coherence drops a little, thus revealing the ridges from p0 to p2. Below the Lamb frequency in the domain of gravity waves, the coherence decreases from about 0.8 to even below 0.4. Only in a small area below 0.5 mHz and between 1.3 and 2.3 Mm-1 it takes high values again. In another area limited by the Lamb mode and by the f-mode where frequencies are lower than about 2 mHz, the coherence is also remarkably high. The phase values corresponding to this region are positive, as can be seen in Fig. 9. This outcome is baffling enough as an evanescent wave domain is expected to show a phase difference of -90o. The one-dimensional V-I spectrum also shows a "plateau" with phase values of about 70o (cf. Sect. 4.1) whose origin may now be understood as being due to the positive phase differences in the region between the Lamb and the fundamental mode as well. The transition from this plateau to the evanescent region causes the phase discontinuity seen at 2.1 mHz. Some similar results showing phase discontinuities in V-I spectra at about 2 mHz, though in photospheric lines, have been reported previously (Lites & Chipman 1979; Staiger et al. 1984; Deubner et al. 1990). As thus the same problem has been encountered with lines which are supposed to have formed in quite different layers of the solar atmosphere, it is once again tempting to explain this finding by a lower line formation height of Na D2 than was first assumed here. There are indeed some authors who consider Na D2 to possibly originate from the upper photosphere (e.g. Babij & Stodilka (1987), Espagnet et al. (1995) or Kariyappa (1996)), but these new results do not support such a low formation height. Several typical features found in the V-I phase spectra of lines formed in the upper photosphere like, for instance, the extension of the plateau showing positive phases into the inter-ridge area between the p-modes and others quoted by Deubner et al. (1996) cannot be seen in Fig. 9. Therefore, the lower chromosphere seems to be a good estimate of the lower limit of the line formation height of Na D2.

[FIGURE] Fig. 8. Two-dimensional V-I coherence spectrum (solid line: Lamb mode, dashed curve: f-mode); the colour bar to the right of the diagram indicates the coherence values

[FIGURE] Fig. 9. Two-dimensional V-I phase spectrum; for each colour, the corresponding phase (degree) is given by the colour bar

Hence, it was not expected beforehand that the above-mentioned phase discontinuities would also appear in the line under consideration. A solution to this problem proposed by Marmolino & Severino (1991) was actually based on an earlier work by Souffrin (1966) and a related contribution by Mein (1966). According to their investigation, to compensate for the loss of energy by radiative damping, running waves, which are capable of propagating energy, appear in the evanescent regime of the k-[FORMULA] diagram. This explanation may well apply here.

There is another small region deserving attention which was already mentioned above. Below 0.5 mHz and between 1.3 and 2.3 Mm-1 high coherence (0.8 to 0.9) is found. Phase differences in this area marking the domain of gravity waves are about -160o. At first sight, one might suspect convective supergranular flows to be responsible, but the typical size of a supergranule (about [FORMULA]) corresponds to a wavenumber of 0.216 Mm-1 and thus fails to explain the high coherence between 1.3 and 2.3 Mm-1. Mesogranules do fit in this wavenumber range, but they have not been observed in the minimum images (showing the line-centre intensities) and thus cannot account for the coherence, either. Therefore, it seems plausible that gravity waves left their mark in these diagrams. Kneer & von Uexküll (1993) found a region of high coherence in their I-I phase spectrum between Mg b2 and Ca K below 1.59 mHz and between 1 and 2 Mm-1 which they also attributed to gravity waves.

The high coherence values below about 0.5 Mm-1 showing in the low-frequency range of Fig. 8, though, may well be connected with convective flows.

The two-dimensional phase spectrum exhibits a remarkable behaviour in the decrease of phase differences from about [FORMULA] for the f-mode to about [FORMULA] for higher modes. This finding still lacks an explanation. It should be mentioned in this context that according to Kumar & Lu (1991) pseudo-modes (i.e. running acoustic waves) are supposed to show a V-I phase difference of 0o. Here, as a possible interpretation one may assume the signals for the fluctuation of the Doppler shifts (velocity) and for the intensity to originate from different atmospheric heights. For spectral lines like Na D2 and similar chromospheric lines which are not formed in the local thermodynamic equilibrium (LTE) the velocity signal represents higher layers than the intensity signal. As a consequence, acoustic waves running upwards show an increasing "distance" between the intensity and the velocity signal with increasing frequency. This explanation may well be valid for either one of the Na D lines which are radiation-dominated, but the situation is different for collision-dominated chromospheric lines like, for example, Ca ii K, 8542, and 8498. Mein & Mein (1980) have computed the formation heights of velocity and intensity for the latter lines and found the distance of the two heights decreasing in the order of decreasing line strength, i.e. as collisions become more frequent for Ca ii 8542 and 8498 than for Ca ii K (which means that the weaker Ca ii lines get closer to LTE conditions).

It appears to be useful to recall that these results are not applicable to the Na D2 line whose behaviour, however, might finally be explained by models of wave propagation which also consider radiative transport.

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© European Southern Observatory (ESO) 1998

Online publication: July 20, 1998