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Astron. Astrophys. 357, 1093-1104 (2000)

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6. Phase difference and coherence spectra

In order to look for propagation characteristics of waves at different heights in the atmosphere, we computed phase difference ([FORMULA]) and phase coherence (C) spectra for many signature pairs (Straus 1995). We exclude in this analysis the NaD2 signature because, if the measured intensity oscillations are due essentially to velocity oscillations, we cannot establish the direction of the motion and hence assign the correct value to the phase difference.

Following Edmonds & Webb (1972), we adopted a smoothing width of about 1.5 mHz in the Fourier domain for the computation of phase and coherence spectra. Frequencies smaller than 0.7 mHz hence do not convey any significant information. Taking into account our smoothing width, a coherence smaller than [FORMULA]0.5 implies that phase differences at those frequencies are completely unreliable.

NSO and MDI data were analyzed independently, using their own frequency resolution and coverage. Finally, the study was performed separately for internetwork and network areas, to distinguish between features with different magnetic characteristics. We describe the characteristics of the spectra from lower photospheric layers through higher chromospheric ones.

In Fig. 5 left and central column, we show intensity (I-I) phase difference and coherence spectra for the pairs H[FORMULA] + 1.5 Å/H[FORMULA] - 1.5 Å, and H[FORMULA] + 1.5 Å / white light, originating at photospheric levels. We use here the white light signal (and not the NiI pseudo-continuum) in order to keep the maximum possible frequency resolution. Phase difference and coherence spectra between magnetic flux density and velocity (B-V) for the NBPs are shown in Fig. 5 right column.

[FIGURE] Fig. 5. Top and mid panels show the phase difference I - I, as a function of frequency, of photospheric signatures averaged for network and internetwork regions. Vertical bars, representing 1[FORMULA] error associated with the mean phase value, are clipped if higher than the y-axis limits. The bottom panels show the phase coherence spectra (computed with a smoothing width of 1.5 mHz, see text) averaged for network (solid line) and internetwork (dashed line) regions

To search for the relationship between the oscillations present at chromospheric and photospheric levels we computed the I-I phase difference and coherence spectra separately for the pairs H[FORMULA] center / H[FORMULA] -1.5 Å and H[FORMULA] center / H[FORMULA] [FORMULA] Å, shown in Fig. 6. At each considered atmospheric level, a general characteristic is that the coherence for NBPs is smaller than for internetwork, hence the phase values for NBPs are more uncertain.

[FIGURE] Fig. 6. Top and mid panels show the phase difference I - I, as a function of frequency, for the pairs H[FORMULA] center / H[FORMULA] -1.5 Å and H[FORMULA] center / H[FORMULA] [FORMULA] Å averaged for network and internetwork regions. Vertical bars, representing 1[FORMULA] error associated with the mean phase value, are clipped when higher than the y-axis limits. The bottom panels show phase coherence spectra (computed with a smoothing width of 1.5 mHz, see text) averaged for network (solid line) and internetwork (dashed line) regions

We examine different signature pairs separately in the two frequeny intervals 1.5-2.5 mHz and 3-4 mHz, disregarding the features with negligible power, and we summarize in Table 4 the phase and coherence values for each pair.


[TABLE]

Table 4. Phase difference [FORMULA] in degrees and coherence C for the considered pairs


6.1. Low frequency (1.5-2.5 mHz)

For both internetwork and network points the I-I phase difference between the H[FORMULA] red and blue wings is 0o. The two signals should be in phase if the observed oscillations are due only to temperature and in antiphase if due to velocity. We can then state that the observed oscillations at low frequencies are essentially due to temperature oscillations.

It follows from the previous considerations that the analysis of the phase difference spectra between the H[FORMULA] red wing and white light is essentially a study of the correlation between temperature oscillations at slightly different levels ([FORMULA] km in the quiet average photosphere). For internetwork areas the extremely high coherence makes the [FORMULA] value (5-10o) highly significant (see Table 4) and strongly suggests that the observed power is due to oscillations. A positive value of [FORMULA] between two layers with decreasing heights, indicates the presence of waves directed radially inward. For the internetwork areas, free from magnetic fields, at the spatial frequency of about 3Mm-1 used in our analysis and within the considered frequency range, these waves might be interpreted as gravity waves (Straus 1995). This same interpretation has been adopted for internetwork by Rutten (2000), who reported a similar value of [FORMULA] between the intensity of two continuum levels observed by TRACE. Downward directed gravity waves had been proposed earlier by Staiger et al. (1984) to explain similar phase differences for velocity signatures at photospheric levels.

At chromospheric levels, the most significant peak in the power spectrum of the H[FORMULA] center intensity appears at 2.2 mHz for NBPs and is not related to the oscillations present at photospheric levels (C [FORMULA] between H[FORMULA] center /H[FORMULA] [FORMULA] Å, see Table 4). This means that at chromospheric levels the NBPs experience a new regime of oscillations that seem to be independent from what happens in photosphere.

6.2. p-mode frequency (3.0-4.0 mHz)

In the p-mode range of frequencies, for internetwork there is a 10o phase lag between the two H[FORMULA] wings (see Table 4) that can be due to different coupling of velocity and intensity fluctuations in the two wings. This effect has been extensively studied for photospheric lines by Cavallini et al. (1987) and Alamanni et al. (1990).

As described in Sect. 5.1, the presence of a peak at these frequencies spectra could betray the presence of MHD waves in the network structures. In the limit of ideal MHD, a definite phase relation between velocity and magnetic field variations is expected as signature of Alfvèn waves ([FORMULA]) or magnetoacoustic waves ([FORMULA], with v leading B. See Ulrich 1996). However, we find a very low coherence ([FORMULA] 0.4) at all frequencies in the NBPs, i.e. the magnetic fluctuations are not related to the velocity ones, at least with the present sensitivity and resolution.

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© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000
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