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Astron. Astrophys. 329, 725-734 (1998)

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3. Results

3.1. Photospheric brightness

Fig. 1 displays time-delay correspondence between granular features. The various curves in the upper panel quantify the temporal alignment of granules with granules, lanes with lanes, and granules with lanes. The lower panel does the same for bright granules and dark lanes. The time resolution is 20 s. The sampling varies between 227 C values at [FORMULA] and 48 C values at [FORMULA]  min, so that the rms variations increase towards the right.


[FIGURE] Fig. 1. Persistence of granules and intergranular lanes, measured as spatial correspondence C between various G brightness classes as function of elapsed time [FORMULA]. The value [FORMULA] implies absence of spatial alignment, [FORMULA] preferential co-location, [FORMULA] spatial avoidance. The upper panel is for granules and lanes, the lower panel for the extreme subclasses made up by bright granules and dark lanes (see text). Thick solid curves: spatial correspondence between pixels belonging to granules and pixels that belong to granules [FORMULA] later (upper panel), and similarly for bright granules (lower panel). Thin solid curves: similarly between lanes and subsequent lanes (upper panel) and between dark lanes and subsequent dark lanes (lower panel). Thick dashed curves: similarly between lanes and granules (upper panel) or dark lanes and bright granules (lower panel), with the lanes observed before the granules. Thin dashed curves: similarly but with the granules observed before the lanes. One-sigma error estimates are added to each curve (dotted or dashed)

The curves show high correspondence (avoidance for cross-alignment) up to [FORMULA]  min, making that value a granular lifetime estimate. All curves reach [FORMULA] at [FORMULA]  min with the exception of the dark-lane curve (thin solid) in the lower panel. This value would represent the maximum lifetime of granular features if the curves would remain flat for longer [FORMULA]. They don't; instead, they display significant subsequent modulation. The solid and dashed curves in the upper panel show reverse behavior because the filling factors of granules and lanes are about equal (47% and 53% respectively).

The modulation indicates repetitive recurrence over extended duration. We have checked it by performing tests on another granulation sequence which covers a substantially larger field of view and which will be analysed in a forthcoming paper. It produces similar modulation, but only for small subfields; the modulation amplitude diminishes when the correspondence factor is determined over larger areas. We therefore suspect that the modulation results from the mesoscale flows present in the photosphere (e.g., November et al. 1981; November & Simon 1988; Simon et al. 1988; Title et al. 1989; Brandt et al. 1991; Muller et al. 1992; November 1994; Brandt et al. 1994; Wang et al. 1995). The flow cells measure about 4-8 arcsec and the flow speeds are of the order of 0.5 km s-1. Since our internetwork area samples only a few mesogranular cells (Fig. 1 of Paper I), pattern migration effects are not averaged out in our data. The modulation may therefore represent simple shifts of the granular brightness pattern over granules and lanes.

The larger-field test also reproduces the slow decay for dark lanes in the bottom panel of Fig. 1, indicating that these extreme features tend to persist longer than other granular structures and also tend to maintain their location better (cf. Roudier et al. 1997).

3.2. Chromospheric brightness

Fig. 2 is similar to Fig. 1 but concerns the chromospheric K brightness extremes. The bright K pixels show a time-lapse co-alignment probability that remains significantly above a random draw (upper curve in the upper panel). For example, the value [FORMULA] for the bright K pixels at [FORMULA]  min corresponds to 13.5% of the pixels being bright at both times, whereas only 9% would be bright in a random selection. This preferential co-alignment is larger for the bright K pixels than for the dark K pixels, especially at first. In addition, the curve shows short-period repetition. A Fourier decomposition (not shown) has a strong peak centered at [FORMULA]  mHz (3-4 min periodicity). It undoubtedly corresponds to the similar repetition rate seen in K [FORMULA] grain "trains", for example the one present at [FORMULA]  min in the righthand panel of Fig. 2 of Lites et al. 1993. The steep initial dip displays the characteristic grain lifetime of about a minute; the slow decay over the first 15 minutes shows that grains have slowly decreasing reappearance probability.


[FIGURE] Fig. 2. Ca II K brightness persistence, measured as spatial correspondence C between two K brightness classes as function of elapsed time [FORMULA]. The upper panel shows the persistence of bright K pixels (solid curve) and of dark K pixels (dashed curve). The lower panel measures the time-delay correspondence between bright K pixels and dark K pixels, with the solid curve for sampling bright after dark and the dashed curve for sampling dark after bright

The lower panel of Fig. 2 tests whether bright K and dark K pixels are sequentially co-located. Values [FORMULA] in this cross-correspondence plot would imply that bright and dark K pixels prefer to occupy the same locations in succession. Such co-location is not significantly present.

3.3. Chromospheric versus photospheric brightness

The amount of crosstalk between the photospheric and chromospheric brightness features is measured in Fig. 3. The curves in the upper panel describe the spatial correspondence between bright K pixels and the granular structure in the underlying photosphere. The curves in the lower panel do the same for the dark K pixels. The time delay [FORMULA] is plotted on a bilogarithmic scale in order to expand the center part which shows intricate structure. At [FORMULA] bright K pixels have nearly 30% excess probability to lie above a bright granule (thin solid curve in the upper panel), with corresponding avoidance of dark lanes (thin dashed curve) and comparable behavior above granules and lanes (at lower C because these have larger filling factor). The dark K pixels in the lower panel display opposite behavior.


[FIGURE] Fig. 3. Crosstalk between chromospheric internetwork intensity contrast and photospheric granulation, measured as spatial correspondence C between various G and K brightness classes as function of elapsed time [FORMULA]. The upper panel is for bright K pixels, the lower panel for dark K pixels. Thick solid curves: bright or dark K pixels measured [FORMULA] after granules. Thin solid: bright or dark K measured [FORMULA] after bright granules. Thick dashed: bright or dark K measured [FORMULA] after lanes. Thin dashed: bright or dark K measured [FORMULA] after dark lanes

The curves reverse sign rapidly for increasing [FORMULA] so that bright K pixels occur preferentially above dark lanes instead of bright granules already after [FORMULA]  min. On the other side (negative [FORMULA]) the decay is much more gradual. The dip at [FORMULA] corresponds to a similar co-location correlation in Fig. 15 of Rutten 1995 between Ca II H [FORMULA] grains, measured as a narrow-band phenomenon in high-resolution spectra, and intergranular lanes in the underlying photosphere a few minutes earlier. This correlation suggests a relationship between excess wave excitation in lanes and K [FORMULA] grain formation after the wave travel time to the overlying chromosphere. We return to this scenario in Sect.  4.

For large [FORMULA], both positive and negative, the curves in Fig. 3 show increasing undulations that resemble the modulation in Fig. 1 and may again be due to mesoscale pattern migration. Their amplitudes are indeed smaller for the dark K pixels (lower panel) which occur twice as frequent as bright K pixels. The dark K curves also show much smaller reversals at [FORMULA]  min. This difference with the curves in the upper panel illustrates that bright and dark K pixels do not occur in anti-symmetrical manner, as is also evident when comparing bright and dark K [FORMULA] behavior in spectral sequences (Cram & Dame 1983; Lites et al. 1993; Hofmann et al. 1996; Carlsson & Stein 1997).

3.4. Waves versus waves

We now turn to the Fourier amplitude maps constructed and displayed in Paper I. Such maps are available for the full set of frequencies defined by the 22 min segment duration and 20 s image sampling, so that the C measurements between Fourier amplitudes and other patterns may be displayed as two-dimensional charts plotting C as function of both Fourier frequency f and time delay [FORMULA]. This is the display format for the remaining figures, with the C values indicated by contours. The split of the data in ten partially independent segments again permits averaging per [FORMULA] combination and estimation of the corresponding rms variations. The latter are not shown in these charts but the contour intervals are chosen such that one interval corresponds roughly to one-sigma rms variation or less in the central regions of the charts where ten [FORMULA] pairs are averaged.

Fig. 4 compares the spatial distributions of the Fourier amplitudes at about 2-min, 3-min and 5-min periodicity in the photosphere (G) and chromosphere (K) with the spatial distributions of all such Fourier maps. The lefthand column charts time-delay Fourier crosstalk between different frequencies in the photosphere, the righthand column does the same for the chromosphere, and the middle column charts crosstalk between chromosphere and photosphere. The curves in Figs. 9 and 10 of Paper I represent horizontal cuts at [FORMULA] and vertical cuts at each of the three periodicities through these nine panels.


[FIGURE] Fig. 4. Time delay charts of the spatial correspondence C between Fourier amplitude maps for the photosphere (G) and the chromosphere (K) in different combinations. The contours specify the value of C between internetwork pixels with larger than average Fourier amplitude in both maps per [FORMULA] comparison pair, averaged over the available data segments (ten around [FORMULA]). Grey ([FORMULA]) implies absence of spatial correlation, bright ([FORMULA]) spatial co-alignment and dark ([FORMULA])) spatial avoidance. The values of C are specified along the contours. Lefthand column: (G,G) = three G amplitude distributions compared with all other G amplitude distributions. Middle column: same for (K,G) pairs. Righthand column: same for (K,K) pairs. Upper row: correspondence between the spatial distribution of Fourier amplitudes in the [FORMULA]  mHz (2-min periodicity) G or K maps with all other amplitude distributions for G or K. Middle row: same for [FORMULA]  mHz (3-min). Bottom row: same for [FORMULA]  mHz (5-min). Horizontal axes: Fourier frequency (bottom) and corresponding periodicity (top). Vertical: time delay [FORMULA] between 15-min data segments from which the Fourier maps are constructed, positive when the maps of the given periodicity cover a later 15-min segment than the maps indexed by their frequency along the horizontal axis

The (G,G) charts in the lefthand column primarily display self-correspondence in the form of 100% cospatiality peaks around [FORMULA] at the three periodicities. They reach [FORMULA] because the higher-than-average amplitude filling factor is about 50%. The widths of the peaks is set by the frequency and time delay resolutions given by the 22-min (effectively 15-min) segment lengths. There is no outspoken preference for large amplitude to occur or not to occur at the same location at other times or other frequencies, except that there are weak ridges of slight excess co-location at high frequencies around [FORMULA], especially in the 2-min panel (top). We attribute these ridges to modulation by atmospheric seeing. It jitters small large-contrast features across their actual position and so causes broad-band brightness modulation of which the relative contribution is largest at high frequencies where the intrinsic solar power is smallest (Fig. 4 of Paper I).

The (K,K) panels in the righthand column show similar self-correspondence peaks with secondary high-frequency peaks at [FORMULA] which we again attribute to seeing. The 3-min peak has a tail towards the right. It is also present in the top panel of Fig. 9 in Paper I, where we speculated that shock steepening may produce nonlinear waveforms at this frequency. The whole (K,K) column, especially the 3-min panel, shows a surplus of [FORMULA] values. This indicates that high chromospheric amplitudes tend to be a broad-band phenomenon with relatively long persistence.

The (K,K) charts are very noisy outside the peaks, but a few other features seem significant. There is a patch of white correspondence in the 3-5 min band near the bottom of the 3-min (K,K) panel, suggesting that 3-min waves prefer locations where 4-min waves will occur 20-40 minutes later. The 2-min panel contains a similar patch. The time delay is long enough that this correspondence may again be set by mesoscale pattern migration; we return to this point in Sect.  4.

The 2-min (K,K) chart (top right panel) shows two dark bands along its lefthand side, symmetrically around [FORMULA], which are the only prominent zones of spatial avoidance in the (K,K) charts. These indicate that locations with above-average 2-min amplitude are preferentially dark well before and well after their 2-min excess. This may similarly point to migration in which internetwork areas that contribute to the self-correspondence peak move out of co-location over this time scale.

The (K,G) charts in the middle column show cross-correspondence peaks between 5-min and 3-min modulation in the chromosphere and the photosphere that are surprisingly low. One would rather expect that both the global p -modes and the local emission of sound waves at or below the surface would produce larger vertical correlation. The low height of these peaks was taken in Paper I (Fig. 9, measured by their [FORMULA] profile) as evidence of wave diffraction that is caused by subsurface convective inhomogeneities.

The remainder of the (K,G) panels seem without significant structure, except for the extended white blob of [FORMULA] values along the lower left side of the 3-min panel (middle). At [FORMULA] large photospheric amplitude implies the presence of granules, so that this blob suggests that chromospheric 3-min modulation possesses some preferential alignment with locations that subsequently contain granules in the underlying photosphere.

3.5. Photospheric brightness versus waves

The spatial correspondence tests so far have concerned more or less similar quantities, in the form of brightness distributions in Figs. 1-3 and wave amplitude distributions in Fig. 4. We now turn to dissimilar correspondence analysis by studying co-alignments between brightness patterning and wave amplitude patterning, again with two-dimensional charts in the format of Fig. 4. Fig. 5 shows such comparison between granular structuring and the wave amplitudes in the photosphere and chromosphere. This is again done for internetwork pixels only and concerns brightness extrema (bright granules and dark lanes only) and wave extrema (over twice and less than half the mean amplitude).


[FIGURE] Fig. 5. Time delay charts of the spatial correspondence C between K and G Fourier amplitude and granulation morphology. The format is the same as in Fig. 4, but the contours now specify the amount of alignment between bright granules or dark intergranular lanes in a 15-min averaged G image with the presence of high or low amplitude in K and G Fourier maps for the frequencies specified along the horizontal axis, where high means over twice the average value and low less than half. First column: amount of alignment between dark lanes and low Fourier amplitude. Second column: between dark lanes and high amplitude. Third column: between bright granules and low amplitude. Fourth column: between bright granules and high amplitude. Upper row: K amplitudes (chromosphere). Lower row: G amplitudes (photosphere). Horizontal axes: Fourier frequency (bottom) and periodicity (top). Vertical: time delay [FORMULA] between 15-min data segments, positive when the granulation morphology is sampled after the Fourier amplitude

The low-A columns in Fig. 5 differ strikingly from the high-A columns. The first are bland; the second display marked structure. The dark lanes and the bright granules themselves are evident as dark and bright blobs at [FORMULA] along the lefthand sides of the two high-A G panels because at low frequency the Fourier amplitude describes the average brightness over the 15-min segment duration. The low-A panels show weak reverse contrast along their lefthand sides. For both bright granules and dark lanes, seeing jitter of these small high-contrast structures produces excess power at the highest frequencies (white blobs for [FORMULA]  mHz at [FORMULA] in the high-A G panels).

The dark lanes show large spatial correspondence with high wave amplitudes over the 4-2 min regime (second G panel). This alignment shows fairly large persistence, appreciably longer than the [FORMULA] width of the self-correspondence peaks in Fig. 4. The rightmost G panel shows alignment between bright granules and enhanced amplitudes only at [FORMULA]  mHz, but avoidance for 3-6 min periodicities which implies that bright granules lack somewhat in high-A 5-min co-location.

The K panels in the upper row are also bland for low wave amplitude. The second panel shows a minor 4-min peak near [FORMULA]  min, adjacent to a zone of avoidance at 3-min periodicity. This steep transition corresponds to the steep drop of the dark-lane curve in the upper panel of Fig. 5 in Paper I, where it was taken to suggest diffraction of propagating waves ([FORMULA]  mHz) away from the lane location over their travel length from photosphere to chromosphere. The large disparity between the second K and G panels indeed indicates that the excess of waves above lanes in the photosphere (lower panel) does not survive as vertical co-location in the overlying chromosphere (upper panel).

The fourth K panel shows a similar avoidance of 5-min enhancement as the corresponding G panel. In addition, it has a significant 4-5 min peak around [FORMULA]  min, implying that bright granules favor locations where the overlying chromosphere showed excess 4-5 min wave amplitudes half an hour earlier. Since it is rather unlikely that chromospheric wave modes control subsurface convection, we again seek the explanation for this delayed correspondence in mesoscale pattern migration (Sect.  4).

3.6. Chromospheric brightness versus waves

Fig. 6 is similar to Fig. 5 in that brightness structuring is compared with wave amplitudes, but in this case it is the internetwork K brightness that is tested for spatial correspondence with excesses in the photospheric and chromospheric amplitude distributions. The definition of the bright and dark K pixels is modified from the one used in Figs. 2-3 in that only those pixels that are over 130%, respectively under 70%, of the average K internetwork brightness in at least 9 of the 18 filtergrams in six-minute intervals at the mid-point of the 22 min data sequences are retained. This procedure produces temporal compatibility with the effective 15-min duration of the Fourier maps and favors K brightness recurrency, i.e., K [FORMULA] grain "trains", over more temporally isolated events.


[FIGURE] Fig. 6. Time delay charts as in Fig. 5, but the contours now specify the spatial correspondence C between bright pixels or dark pixels in the internetwork parts of the K images and high or low K or G Fourier amplitudes. First column: amount of alignment between dark K pixels and low Fourier amplitude. Second column: dark K pixels and high amplitude. Third column: bright K pixels and low amplitude. Fourth column: bright K pixels and high amplitude. Upper row: K amplitudes (chromosphere). Lower row: G amplitudes (photosphere). Horizontal axes: Fourier frequency (bottom) and periodicity (top). Vertical: time delay [FORMULA], positive when the K pixel is measured as bright or dark after the Fourier amplitude sampling

The (bright K, high K amplitude) correspondence chart at the upper right displays much structure. The high ridge at [FORMULA] shows that K [FORMULA] grain formation is a broad-band phenomenon in wave terms. Its rise to the left corresponds to the tail of persistence over the first 15 min in Fig. 2. The long downward extent of the [FORMULA] area, most notably at 7 min periodicity, indicates that the longer-period waves that contribute to bright K [FORMULA] grain formation at [FORMULA] persist over multiple periods.

There are three blobs of high-frequency correspondence along the righthand side of the upper-right chart that we again attribute to seeing jitter. The one at [FORMULA] illustrates that K [FORMULA] grains are small. The other two lie at about twenty minutes before and after the K brightness sampling and similarly indicate enhanced probability for the presence of a small bright structure in the chromosphere at the same location. They may illustrate grain recurrency.

The other charts show much less structure. The most significant feature is the extended white ridge at the top of the (bright K, high G amplitude) panel at the lower right, to which we return below.

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© European Southern Observatory (ESO) 1998

Online publication: December 8, 1997
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