4 Results

4.1 Spatio-temporal behaviour

Sample images and timeslices.

Figure 5 illustrates the dynamical nature of the solar atmosphere sampled by the 1700Å passband in the form of a short sequence of small cutouts from the May 12 top subfield shown in Fig. 2. The cutout contains network at left and right, and internetwork in the center. The greyscale is logarithmic to show the grainy nature of both network and internetwork in these displays, which resemble Ca IIK line-center filtergrams. Each of the network grains results probably from multiple close-lying magnetic elements in the photosphere (we tried but did not succeed to obtain simultaneous G-band imaging with the Swedish Vacuum Solar Telescope to resolve the chromospheric network into photospheric magnetic element clusters). The internetwork area contains bright grains that appear only briefly, for example at x=-45, y=516 in the $\Delta t = 90$ panel. The network grains vary also with time, but are generally more stable.

The third column in Fig. 5 shows further enlargements of the cutouts in the first column with sign-reversed logarithmic greyscaling. The flip in contrast emphasises the spiderweb background pattern to which we return below.

Figure 6: Linear intensity variation along Y=512.5 arcsec in the first panel of Fig. 5. The same cut location is used for the X-t slices in Fig. 7. The short bars specify the small segments selected as network and internetwork by the mask in Fig. 4.

Figure 6 shows actual linear intensity along the cut defined by markers in the first panel of Fig. 5. The high peak at X=-65 and the peak at Y=-25 are network grains; the others are internetwork grains with the one at X=-33 a "persistent flasher'' (see below). The bright grains ride as enhancements on a general background, as shown already by Foing & Bonnet (1984). The severity with which the mask defines network and internetwork is illustrated by the shortness of the corresponding cut segments, with only the darkest background classified as internetwork. The flasher is located in the generally brighter intermediate zone.

Figure 7: Partial X-t slices from the May 12 sequence at Y=512.5 arcsec, illustrating temporal behaviour of network patches (bright stripes) and internetwork oscillations (grains on dark background). Axes: X in arcsec from disk center, elapsed time in minutes covering the full sequence duration. Panels: upper left 1700Å, upper right 1600Å, lower left 1550Å, lower right C IV construct. The snapshots in Fig. 5 sample the 7.5 min period between the white markers. Greyscale: logarithmic, clipped for C IV. Bright horizontal dashes result from cosmic ray hits.

The traditional way of displaying spatio-temporal variability is to construct space-time plots or "timeslices''. Figure 7 shows examples in the form of X-t graphs that cover the full May 12 sequence temporally but only the Y=512.5 cut marked in the first panel of Fig. 5 spatially. The sequence in Fig. 5 samples only the brief duration between the white markers in Fig. 7. The X-t slices in Fig. 7 resemble spectrometer data taken with the slit programmed to follow solar rotation (the format is directly comparable to many time evolution displays in the SUMER papers listed in the introduction). They typically show network as bright striping and internetwork oscillations (when present) as dappled grain sequences at lower brightness. Square-root or logarithmic greyscaling or subtraction of the mean intensity per pixel is usually required to enhance the visibility of the internetwork brightness modulation. Figure 7 illustrates this morphology with a bright network grain at left, some weaker network grains at right, and internetwork oscillations in the dark area around X=-50. The oscillatory internetwork patterning is roughly equal in the three TRACE passbands, but the network features become brighter and coarser at shorter wavelength. Additional emission contributed by the C IV lines shows up in the 1550Å panel and dominates in the C IV construct. The internetwork parts of the latter are very noisy.

A general comment with respect to Figs. 5 and 7 is that such cutouts and slices constitute a poor proxy for viewing the data cubes in fast on-screen computer display. We use an IDL "cube slicer'' developed by C.A. Balke to view concurrent (X,Y,t₀), (X,Y₀,t) and (X₀,Y,t) plane cuts through multiple data cubes simultaneously, with mouse control to select and move the X₀, Y₀ and t₀ cut location with co-moving cross-hairs in each, up to nine co-changing movie and timeslice panels in total. It enables inspection of such data sets using visual pattern recognition of variations, correlations, time-delay relationships, etc. Such slicing through TRACE data cubes makes one appreciate fully the richness and the variety of the solar scene - effectively sampling a manifold instead of the single cut plotted in a timeslice or observed by a spectrograph slit, and with displacements of one arcsec already showing significant pattern difference. The next-best option is to play TRACE sequences as movies, for example with SolarSoft Xstepper (URL http://www.lmsal.com/solarsoft/) or the ANA browser (URL http://ana.lmsal.com/). We urge the reader to do so in order to appreciate the dynamical nature of the quiet-Sun internetwork, and have made sample TRACE movies from this work available at URL http://www.astro.uu.nl/~rutten/trace1.

What do we see in our TRACE datacube visualizations? Just as on Ca II K_2V movies from ground-based telescopes, the network grains appear as islands of stability in an internetwork sea made up of fast-changing grainy emission features that are superposed on a spiderweb background mesh which is modulated with three-minute periodicity but evolves much slower. The internetwork background is best seen in negative, as in the third column of Fig. 5, because its mesh structure then stands out without visual contamination from the brighter network and internetwork grains. It then resembles reversed granulation with 2-5 arcsec cell size, but it cannot be just that since it shows large three-minute modulation. It was detected by Martic & Damé (1989) as a 8 Mm spatial power peak in a 4.5 min sequence of TRC 1560Å filtergrams that was Fourier-filtered passing three-minute modulations; the reverse display in Fig. 5 shows it directly without Fourier tuning. The pattern becomes fuzzier with height: the 1700Å sequences show it sharper than the 1550Å sequences.

The internetwork grains appear as localised enhancements of the background mesh pattern. They are not as sharp and bright as on K_2V filtergrams but show similar rapid morphology changes on a 2-4 min time basis. Sometimes, they reappear a few times at locations within soundspeed (7 km s^-1) travel distance. Fairly frequently, pairs of grains 5-10 arcsec apart brighten repeatedly in phase. Other grains appear only once. Many brightenings move with apparent supersonic speeds, up to 100 km s^-1, along the strands of the background mesh pattern. Such travel is more obvious for the darkest locations seen as bright in reversed-greyscale display, peaking when such traveling enhancements "collide'' at mesh strand junctions. Occasionally, the timeslices exhibit wavy patterns that indicate oscillatory coherence over spatial scales as long as 20 arcsec. The reversed-greyscale movies also display slower modulation in which the mesh junctions brighten (darken in reality) together in patches covering about 10-20 arcsec. Such large-area coherence was noted already by Cook et al. (1983).

Figure 8: Ca IIH segment of the disk-center spectrum atlas of Neckel (1999). The horizontal bars mark spectral windows used in Paper I for oscillation measurements shown below. The label C+S identifies the Fe I line used by CS1997 to derive subsurface piston driving in their numerical simulation of the Ca IIH line core behaviour in the same data.

Comparison with Ca II H & K.

It is of interest to compare the TRACE ultraviolet timeslices in Fig. 7 with Ca IIH & K results using similar display formats. Most Ca IIK filtergram movies are made with passbands between 0.5 and 3 Å, integrating over a fairly wide segment of the Ca IIK core and mixing K₃ line-center variations with the K_2V grain and inner-wing whisker phenomena (cf. Rutten & Uitenbroek 1991a). This implies both a wider response function including contributions from lower heights as well as phase averaging for upwards propagating waves. The difference in signature can be displayed from spectrogram sequences by integrating over appropriate spectral passbands. Such a comparison is shown between the Ca II H_2V feature and the passband of the Mount Wilson H & K photometer in Fig. 2 of Rutten (1994) and Fig. 1 of Rutten et al. (1999a), based on the Ca IIH spectrum sequence of Rutten et al. (1994) taken in 1991 at the NSO/Sacramento Peak Dunn Solar Telescope. It shows that limiting the bandpass to the K_2V feature itself produces sharper grains with higher contrast than from the integrated Ca IIH core, but also that the dynamical time-slice patterning is quite similar for the wider passband.

We add a similar timeslice comparison in Fig. 9 that is based on the Ca IIH spectrum sequence of Paper I taken with the Dunn telescope in 1984. These data have special interest for internetwork dynamics because they were numerically simulated in great detail by CS1997. Figure 8 identifies various spectral features discussed here and below. The first panel of Fig. 9 measures the peak brightness of the Ca II H_2V emission feature (following its wavelength shifts), the middle panel integrates brightness over a stable 0.16 Å band around its mean location (H_2V index), and the rightmost panel integrates brightness over a 0.9 Å wide band around mean line center. The noise decreases with bandpass width. The second panel gains grain contrast by sampling low intensity when the H₃ core shifts into the H_2V passband. The third panel shows the internetwork grains at lower contrast and appreciably fuzzier due to response smearing, but nevertheless, the dynamical morphology is similar in all three panels. Thus, wider-band Ca IIK movies are indeed suited to study the spatio-temporal patterns of the oscillations that cause K_2V and H_2V grains.

Figure 9: Ca IIH X-t slices from the data analysed also in Paper I. Left: Ca II H2V peak brightness. Middle: Ca II H2V index = brightness integrated over $3968.33 \pm 0.08$ Å. Right: Ca IIH index = brightness integrated over $3968.49 \pm 0.45$ Å. Bright column markers: solar locations simulated by CS1997, with the "slit positions'' specified by them added at the top. A Fourier decomposition of the middle slice is shown in Fig. 10.

CS1997 simulated the four solar locations that are marked in Fig. 9 and are labeled with the slit positions specified in their paper. Their simulations reproduced the spectral grain occurrence patterns at positions 110 and 132 remarkably well (their Plate 20). They computed too many and overly bright grains for positions 149 and 30 - which they attributed to a preceding computational excess of high-frequency (14 mHz) oscillations - but nevertheless, the overall agreement between the simulated and observed Ca IIH spectral behaviour is impressively good. This success has brought general acceptance of the notion that K_2V internetwork grains are primarily an acoustic weak-shock phenomenon as advocated by Rutten & Uitenbroek (1991a, 1991b).

The ultraviolet slices in Fig. 7 are very similar to the wider-band Ca IIH slice in the rightmost panel of Fig. 9. The TRACE images therefore appear to have adequate angular resolution to display grain patterns similarly to a 1Å Ca IIK filtergram sequence. The considerable spectral width of the TRACE passbands must cause considerable height-of-formation and propagative phase mixing, so that the 1 arcsec TRACE resolution is perhaps not too far from the intrinsic detail observable in these passbands. The direct comparison of a selected Ca IIK image with simultaneous TRACE images in Fig. 2 of Rutten et al. (1999a) demonstrates that TRACE is not resolving the grains as well as a 3 Å Ca IIK filter at excellent La Palma seeing. Note that the ultraviolet continua sampled by TRACE are affected by bound-free scattering which is similar to the bound-bound resonance scattering affecting Ca IIK (cf. Fig. 36 of Vernazza et al. 1981; Figs. 2-4 of Carlsson & Stein 1994; lecture notes by Rutten 2000) and produces similar smearing of the source function response to thermal fine structure along the line of sight.

Apart from its lower resolution the 1700Å image in Fig. 2 of Rutten et al. (1999a) is virtually identical to the Ca IIK image. The Ca II slices in Fig. 9 show several episodes of reduced seeing quality (greyish horizontal wash-outs) that do not occur in the TRACE slices. Thus, the TRACE ultraviolet passbands sample network and internetwork much as 1-3Å Ca IIK passbands do - but they do so seeing-free, without geometrical distortions or temporary degradations.

Internetwork background.

The grains represent the most striking feature in the internetwork parts of Ca IIK and ultraviolet filtergram sequences, but there is also the spiderweb emission pattern that makes up the greyish background in Figs. 5-7. It is observed most clearly in reversed-greyscale movies as the one at URL http://www.astro.uu.nl/~rutten/trace1, here emulated by the third column in Fig. 5, has 2-5 arcsec mesh size, is modulated with 2-4 min periodicity, and evolves at longer time scales. Enhancements (both brightenings and darkenings, the latter standing out more clearly in reversed-greyscale display) travel at large apparent speed along the strands (cf. Fig. 3 of Lites et al. 1999). Internetwork grains occur always as local, point-like enhancements of the mesh in its bright phase, as emphasised by Cram & Damé (1983) whose V/R timeslice shows the background as a "wavy curtain'' oscillatory pattern with considerable spatial extent (cf. Rutten & Uitenbroek 1991a).

Figure 10: Fourier-decomposed timeslices of the spectral data presented in Paper I. Upper panels: Ca II H2V decomposition. The H2V brightness slice at left is also shown in Fig. 9. The other panels display its decomposition into low-frequency modulation, five-minute modulation and three-minute modulation, obtained by temporal filtering of the sequence for each spatial position along the slit. The greyscale is set to maximum contrast for each panel independently. Lower panels: similar decomposition for the Dopplershift of Fe I 3966.82 Å, the blend from which CS1997 derived a subsurface piston to simulate the intensity in the upper panels at the locations marked along the top.

Figure 10 analyses the internetwork background and grain occurrence with another unpublished display from the 1984 Ca IIH data of Paper I. The upper part is a Fourier decomposition of the H_2V index slice in Fig. 9 into low-frequency, five-minute and three-minute oscillatory components, similar to the H_2V decompositions of 1991 data in Fig. 4 of Rutten (1994) and Fig. 6 of Rutten (1995). The split is obtained by temporal Fourier filtering per spatial x-position along the slit, so that adjacent columns are independently decomposed. As in any timeslice, horizontal structures imply instantaneous spatial coherence in the spatial direction while tilted structures suggest motion along the slit (real motion for a round feature, but for an elongated one moving at a small angle the apparent velocity of the slit-feature intersection exceeds the real motion). The lower part of Fig. 10 decomposes the simultaneously measured Dopplershifts of the photospheric Fe I line from which CS1997 obtained their subsurface piston excursions. The Fe I low-frequency panel (second in bottom row) presumably shows granular overshoot and/or internal gravity waves. The Fe I five-minute and three-minute panels show wavy-curtain patterns with considerable propagative spatial coherence and finer substructure at shorter periodicity. They extend across the network strip without obvious pattern change. The chromosphere (upper panels) shows qualitatively similar patterns, with appreciable correspondence with the underlying photospheric pattern at least for the five-minute components in internetwork areas. The H_2V low-frequency panel (second in upper row) is the only modulation panel in which the presence of network (strip at x=30-40 arcsec) is recognisable, having slower and larger-amplitude modulation than the internetwork areas on both sides.

The photospheric wavy-curtain patterns in the third and fourth lower panels represent interference between the ubiquitous acoustic waves of the global and local p-mode oscillations that dominate the dynamics of the middle photosphere where convective motions have died out. The similar patterns in the upper row indicate that the rapidly varying modulation of the internetwork background mesh in Ca IIK and ultraviolet filtergram movies is also dominated by acoustic wave interference. However, the background pattern in the low-frequency panels differs morphologically between the upper and lower rows, without obvious direct correspondence.

The internetwork grains in the first panel occur at spatio-temporal locations where the decomposed brightnesses to the right sum up to a bright total. For example, the bright grain at t=7 min in Carlsson-Stein column 110 marks coincidence of bright five-minute and bright three-minute phases as present in a broad-band impulse. A much weaker grain follows after five minutes because the three-minute modulation is then out of phase. A sequence of bright grains at three-minute periodicity, as the one at the top of Carlsson-Stein Col. 30, requires the combination of strong three-minute modulation, weak five-minute modulation, and a bright phase of the low-frequency modulation in the second panel. The component weighting follows the H_2V power spectrum which is similar to the H_2V index power spectra shown in Fig. 21 and is dominated by the three-minute modulation.

Figure 11: Compressed timeslices. Upper panel: the 1700Å data of Fig. 7 compressed in the time direction, expanded in the spatial direction, and clipped in greyscale so that the network becomes heavily saturated. Lower panel: similar format for the Ca II H2V index data in the middle panel of Fig. 9, including Carlsson-Stein markers. The internetwork background mesh shows up as 2-5 arcsec wide columns on which the three-minute oscillation is superimposed.

The chromospheric background pattern, its three-minute modulation, and the internetwork grain superposition on this pattern become more evident in timeslices when these are severely compressed in the time direction and clipped as in Fig. 11. The internetwork background pattern shows up as dark and bright structures of 20-40 min duration with three-minute modulations that often seem spatially coherent over lengths of 5-10 arcsec, occasionally as long as 20 arcsec - much longer than the coherence extents evident in Figs. 7 and 9-10. This is because the compression reduces the tilts caused by the apparent motion of brightenings and darkenings along the strands of the background pattern, so that the eye interprets the total travel distance as a single coherent feature. A similar pattern seen in Dopplershifts of C II and O VI lines from SUMER was reported by Wikstøl et al. (2000) as a new phenomenon with unexpectedly large spatial coherence, but their timeslice is similarly compressed.

Persistent flashers.

Some internetwork grains seem to mark a location with longer spatial memory than 5-10 min by reappearing repeatedly, sometimes with extended intervals of absence. An example is seen near X=-33 in the upper half of Fig. 7. A bright grain appears in the 1700Å passband at t=62 (also in the $\Delta t = 391$ panel of Fig. 5), seems to reappear at t=66 and t=72, and then to become part of a bright streak towards the end of the sequence. In the 1550Å panel this location has enhanced brightness all the time (at least on our screen). What looks like an isolated 1700Å grain at first sight is actually part of a long-lived (though modulated) brightness feature.

A particular K_2V grain of this type has been described by Brandt et al. (1992, 1994), who found it by playing a video movie of Ca IIK images very fast. It was occasionally absent for some time; fast movie display made its longer persistence recognisable. It flashed at 3-5 min periodicity when seen and it behaved as an actual solar "cork'', the chromospheric emission patch following the surface flows derived from local correlation tracking of granulation features and migrating from cell center to network within a few hours.

Such intermittently present persistent features are not easily identified without movie display. Only when stationary in Y do they become discernible in an X-t slice by leaving an extended trail. We find that many isolated bright features which, when seen on a single frame might be taken to represent acoustic shock grains, are in fact such flashers, reappearing over a longer period and often with slightly higher background emission producing a thin trail between flashes in 1550Å timeslices if the feature doesn't move "off the slit''. A shift of the slice position over one pixel (0.5 arcsec) often shortens or lengthens the apparent trail considerably. The sample slices in Fig. 7 contain a number of such weak long-duration streaks at right. The weakest ones fall in the intermediate pixel category in our mask definition and are not easily identifiable as network in Fig. 5. Only the dark area with dappled grain appearance around X=-50 in Fig. 7 seems to represent streak-less internetwork.

This split confirms the present conclusion from a long debate on magnetic internetwork grain anchoring (cf. Lites et al. 1999; Worden et al. 1999) that the internetwork areas may be locally dominated by grain-producing shock acoustics but may also contain features with a longer location memory that presumably mark magnetic field entities of sufficient strength to maintain an identity. We designate the latter "persistent flashers'' following Brandt et al. (1992, 1994). The three features marked in Fig. 2 of Lites et al. (1999) represent other examples in the form of isolated chromospheric Ca IIK grains that are clearly connected to photospheric G-band bright points of the type commonly believed to represent strong-field fluxtubes.

The flashers seem to appear more frequently near regular network. It is tempting to wonder whether such loose features build up the network or originate from it.

Figure 12: Mask-averaged brightness modulation spectra from the May 12 sequences, split between the network (NW), intermediate (IM), and internetwork (IN) pixel categories. Each curve shows linear Fourier amplitude scaled by the internetwork value for that wavelength at f=3 mHz, against temporal frequency. The mean value and trend (zero and first frequencies) are removed. Error bars: standard error of the average. Upper panels: 1700 and 1600 Å. Lower panels: 1550 Å and the C IV construct.

The Brandt et al. flasher migrated from cell center towards network as one might expect for a split-apart ephemeral region marking field emergence. Our TRACE movies also show such cell-crossing flashers (plus some brightness features which move very fast along chromospheric mottles or higher-lying loops and stand out as such in the C IV-construct movie, see below), but most flashers appear fairly stationary near network and produce intermittent streaks lasting from half an hour to multiple hours with drifts below 10 arcsec. These may portray products of network shredding rather than network assembly.

By having a large intermediate pixel category we have eliminated most of the persistent non-network features from what our masks define as internetwork, a very stringent selection that should limit the internetwork Fourier results below to the darkest streakless regions in our data, such as the one around X=-50 in Fig. 7. Presumably, their brightness modulation is acoustically dominated.

   
4.2 Spatial modulation variations

Mask-averaged modulation.

Figure 13: Mask-averaged 1700 and 1550 Å fractional brightness modulation spectra, as in Fig. 12 but normalised by the temporal mean (zero frequency value, subtracted) per curve.

Figure 12 shows spatially averaged power spectra per mask category for the three ultraviolet wavelengths and the C IV construct. Fourier amplitudes ($\sqrt P$) are plotted on linear scales that are scaled per panel (wavelength) to the internetwork value at f=3 mHz. This represents amplitude scaling analogous to Eq. (3). The heights of the different-category curves in each panel are directly comparable. Between panels, they are quantitatively comparable if the evanescent five-minute waves have similar brightness power at different heights in internetwork areas.

The curves in Fig. 12 show standard modulation behaviour (but with unprecedented precision) of network and internetwork. The brightness modulation power is highest at the lowest frequencies and reaches secondary maxima in the acoustic oscillation band, which grow in relative importance and in high-frequency extent with sampling height. Note that the acoustic peaks are less pronounced in these brightness modulation spectra than they appear in Dopplershift samples from the same atmospheric domain, largely because Dopplershift modulation lacks prominent low-frequency modulation peaks. The acoustic peak is less clearly defined for the C IV construct.

Network power exceeds internetwork power up to f=5 mHz but drops faster at higher frequencies. Higher up in the atmosphere, this distinction grows into fully disparate network-internetwork modulation behaviour, as illustrated below in Fig. 21. Around f=6 mHz the intermediate-pixel category is no longer intermediate between network and internetwork but has the largest amplitude. We return below to these "power aureoles''.

Choice of normalisation.

Figure 13 duplicates the 1700 and 1550 Å amplitude spectra in the first column of Fig. 12, but after normalisation by the mean amplitude, showing fractional modulation analogous to Eq. (4). The amplitude scales again have unity at the 3 mHz internetwork value. In the lefthand panel the fractional modulation of network and internetwork appears roughly equal over f=0.5-3 mHz. In the righthand panel the network category is lower than internetwork at all frequencies and the intermediate pixel class shows a "power shadow'' for f>4 mHz.

Comparison with Fig. 12 shows that the choice of normalisation affects the network/internetwork modulation ratio. It is not obvious which choice is physically preferable. The internetwork 3-6 mHz peak consists of acoustic waves (a mixture of the "photospheric five-minute'' and "chromospheric three-minute'' oscillations, where none of these names should be taken literally as meaning exclusive). The internetwork low-frequency peak is attributed to internal gravity waves (cf. Sect. 4.4). For these wave phenomena absolute power or amplitude specifies the importance of the wave itself, whereas fractional modulation scales the amplitude to the total time-averaged amount of photon emission which may be dominated by other processes. Even if the wave field inside network elements is the same as outside (fractionally or absolutely), the observed modulation does not need to be the same with either normalisation since density differences also affect the sampling height.

Figure 14: Fourier modulation over the top subfield of the May 12 data. Axes: solar X and Y at mid-observation in arcsec. Greyscale: logarithm of the temporal Fourier amplitude ($\sqrt P$) of the brightness per pixel, clipped and rescaled per panel to improve display contrast. Columns from left to right: average intensity over the full 14:30-16:00 UT sequence duration (also with clipped logarithmic greyscale), Fourier amplitude in the 0.1-1.2 mHz frequency bin, in the 2.6-3.6 mHz bin (periodicity $P \approx 5$ min) and in the 6-8 mHz bin ( $P \approx 2.5$ min). Graph rows: brightness modulation amplitude along Y=500 for the 1700, 1600, and 1550Å passbands, in linear units normalised by the mean modulation over the subfield. Image rows: brightness modulation maps for the 1700, 1600, and 1550Å passbands and the C IV construct.

Spatially-resolved modulation.

Figure 14 resolves the results in Fig. 12 spatially by displaying the brightness modulation per pixel for selected temporal frequency bands, with brighter greyscale indicating larger Fourier amplitude. Cuts along Y=500 are shown in the top panels and display linear modulation amplitudes normalised by the mean of the whole subfield.

The first column represents the sequence-averaged intensities (zero-frequency modulation). The averaging increases the contrast between the fairly stable network and the rapidly varying internetwork. The logarithmic greyscale permits identification of the darkest internetwork areas; they correspond closely to the internetwork pixel category (dark grey) of the mask in Fig. 4. The network grains appear coarser at shorter wavelength in these averages; the C IV construct looses detail. The network contrast increases towards shorter wavelength (cuts on top).

The second and third columns show modulation maps for slow variations, respectively with periodicities P > 13 min and $P \approx 5$ min. The network appears bright in the greyscale maps, with comparable modulation amplitudes at the different wavelengths (tracings at the top). Both the network and the internetwork areas are finely grained on a 1-2 arcsec scale in these long-duration modulation maps. There is no obvious morphology difference between high and small amplitude regions as in the short-duration modulation amplitude maps of Hoekzema et al. (1998b), probably due to pattern migration during the much longer durations that are Fourier-mapped here.

The normalised modulation amplitudes decrease with frequency (tracings at the top). At all three ultraviolet wavelengths the internetwork modulation gains amplitude with respect to the network modulation and exceeds the latter for frequencies $f \approx 4$ mHz. The rightmost column illustrates this in the form of dark patches that roughly correspond to the bright network areas in the first column. The high-frequency panel for the C IV construct (bottom right) does not show internetwork contrast reversal, but its internetwork areas are unreliable since these do not contribute C IV signal. The C IV construct images contain many artifacts from cosmic ray hits that have been removed.

Magnetic canopies.

The C IV low-frequency amplitude map (second from left in the bottom row) differs strikingly from the other amplitude maps. It resembles a H$\alpha$ filtergram by showing the onset of fibrils extending away from the network. There is also a long curved arch in the upper-right corner. When viewing this C IV subfield as a movie, bright blobs are seen to be traveling along the arch at apparent speeds of about 100 km s^-1. The map also shows bright modulation patches that are much more pronounced than in the longer-wavelength panels, for example around (-40,530). These non-photospheric features indicate substantial contribution by the C IV lines that appears to be visualised easiest through their slow modulation. The fibril-like extensions are barely visible in the C IV $P \approx 5$ min map.

The C IV fibrils clearly depict magnetic field topology in which the field lines fan out from the network in the form of low-lying bundles. Magnetic canopies are a standard feature of magnetostatic fluxtube modelling (e.g., Spruit 1976; Solanki & Brigljevic 1992; Solanki & Steiner 1990) and were first reported by Giovanelli (1980) and Giovanelli & Jones (1982, 1983) who concluded that the fibrils observed in chromospheric lines such as H$\alpha$ map the actual canopies only partially, showing only those low-lying magnetic loops that have sufficient population density in the H I n=2 level. We take the bright areas in the low-frequency C IV panel of Fig. 14 therefore as an incomplete indicator of canopy presence. The darkest areas in this panel correspond closely to the internetwork part of the mask in Fig. 4. This implies that the intermediate pixel category that we derived from the mean 1600Å brightness behaviour does not only correspond to more frequent occurrence of persistent flashers as discussed above, but also corresponds to the presence of overlying canopy fields that stretch out from the network over the adjacent internetwork at relatively low height.

Network aureoles.

The amplitude spectra in Fig. 12 around f=6 mHz and the rightmost column in Fig. 14 indicates the presence of weak "halos'' or "aureoles'' of enhanced high-frequency power around the network patches. They are seen as extended, though very patchy, bright rings around the dark areas marking network power deficits. They roughly cover the intermediate pixel zones and make the corresponding IM curves in the 1600Å and 1550Å panels of Fig. 12 exceed the other curves around f=6 mHz.

Power aureoles at 3-min periodicity around plage were first reported from a chromospheric Ca IIK image sequence by Braun et al. (1992) and from photospheric Dopplershift modulation maps by Brown et al. (1992) as patchy areas that contain small-scale 3-min modulation enhancements and are associated, but not cospatial, with magnetic regions. They were also seen in Ca IIK image sequences by Toner & LaBonte (1991), in MDI Dopplergram sequences by Hindman & Brown (1998), and in local-helioseismic images constructed from MDI Dopplershift data by Braun & Lindsey (1999) who renamed them "acoustic glories'' and suggested that the aureoles result from subsurface oscillation deficits underneath active regions as a result of wave refraction or scattering. Thomas & Stanchfield (2000) studied them in both the photosphere using a Doppler-sensitive iron line and the chromosphere using Ca IIK, finding that the photospheric aureoles favour pixels of intermediate field strength but are not clearly related to the Ca IIK aureoles. Hindman & Brown (1998) did not find aureoles in photospheric continuum intensity fluctuations in MDI data and use that to advocate that the halo oscillations are incompressible, a suggestion extended by Thomas & Stanchfield (2000) to the height of formation of their Fe I line.

Figure 15: Frequency-resolved aureole map, showing temporal power (grey scale) for the May 12 1600Å data as a function of distance r to the nearest network border, averaged over all non-network pixels in the whole May 12 field excluding edge pixels and normalised to the mean network power (value at r=0, grey) per frequency bin.

Figure 16: Spatially-averaged modulation aureoles for different pixel categories. Each curve measures the mean temporal modulation power of the May 12 1600Å data as a function of distance r to the nearest network border, excluding edge pixels and normalised to the mean network power (the value at r=0) per frequency bin. The shaded contours specify 3$\sigma$ error estimates for each mean. Curves with dark contours: averaged over all non-network pixels in the whole May 12 field. Lower curves with grey contours: averaged only over all internetwork pixels. Upper curves with grey contours: averaged only over all intermediate-category pixels (light grey in Fig. 4). Panels from left to right: 0.1-1.2 mHz frequency bin, 2.6-3.6 mHz bin (periodicity $P \approx 5$ min) and 6-8 mHz bin ( $P \approx 2.5$ min). At right, all three curves peak near $r \approx 4$ arcsec.

The network patches in our quiet fields of observation represent much smaller field concentrations than the active regions in the aureole literature above and our TRACE data do not sample Dopplershift, which displays 3-min modulation much clearer than brightness at chromospheric heights (e.g., Cram 1978), but nevertheless the TRACE images bring additional information because they sample intrinsic brightness modulation in the upper photosphere that result from temperature and/or opacity variations without crosstalk from Dopplershifts.

Figures 15, 16 attempt to quantify the visual impression of aureoles in Fig. 14 by plotting 1600Å Fourier power as a function of the distance of a pixel to the nearest network area with spatial averaging over the whole May 12 data set. Figure 15 is a frequency-resolved greyscale plot of this mean dependence for all pixels outside network. The average presence of aureoles shows up as a power peak (bright) between r = 3-7 arcsec and f=5-7 mHz.

Figure 16 displays the same power-versus-distance measurements for selected frequencies with spatial division between internetwork pixels, intermediate-category pixels, and their sum ("non-network'') according to the mask definition in Fig. 4. The first panel illustrates that at low frequencies network power generally exceeds non-network power. The five-minute power distributions in the middle panel show a remarkable split between the intermediate and true internetwork pixels. Both categories display an initial dip followed by a rise of mean power with distance from the nearest network. The intermediate category does so at a much higher value.

The 6-8 mHz panel at right shows similar but steeper initial increases for all three pixel categories, peaks at r=3 - 5 arcsec, and average decreases further out. The intermediate-category pixels provide the largest contribution to this pattern, also at large distance r. Thus, the high-frequency aureoles seem to favour the relatively large time-averaged brightness that make up the intermediate-category pixels - possibly marking the presence of persistent flashers, or the presence of overlying canopy fibrils as evidenced by the low-frequency C IV modulation map in Fig. 14.

Network shadows.

Recently, Judge et al. (2001) have detected "magnetic shadows'' in time-slice projections from SUMER measurements which they attribute to mode conversion in canopy fibrils (cf. McIntosh et al. 2001). In contrast to the power aureoles, these shadows are near-network areas that lack both in brightness and oscillatory power. Judge et al. (2001) plotted fractional modulation as defined by Eq. (4) which suggests that the shadows may correspond to the drop of the intermediate-category curve below the network one in the righthand panel of Fig. 13. Spatially resolved examples of such power normalisation are given in the top panels of Fig. 17, selecting the 5-min and 2.5-min frequency bands of the 1600Å subfield that are shown in Fig. 14 without normalisation. The low power (darkness) of the network areas illustrates the relative lowering of the network curves from Fig. 12 to Fig. 13. These dark patches extend slightly beyond the brightest cores of the network (first panel of Fig. 4) and so overflow into our intermediate pixel category (light grey in the fourth panel of Fig. 4). Blinking these normalised-amplitude maps against the mean brightness maps shows that the dark patches correspond closely to the network at a somewhat less restrictive brightness contour, i.e. that their darkness derives primarily through the normalisation from the averaged network brightness. The aureoles quantified in Fig. 16 extend further and remain discernable as patchy bright borders in the upper-right panel.

Figure 17: Network shadows in normalised amplitude maps. Upper row: Fourier modulation maps for the top subfield of the 1600Å May 12 data, similar to the third and fourth panels in the second row of Fig. 14 but with the greyscale representing linear Fourier amplitude per pixel normalised by the temporal mean per pixel. Lower row: ratio of 1550 and 1700Å normalised amplitude maps. Left column: 2.6-3.6 mHz bin (periodicity $P \approx 5$ min). Right column: 6-8 mHz bin ( $P \approx 2.5$ min).

Blinking the normalised-amplitude maps between wavelengths shows that the dark patches darken and grow with decreasing wavelength. This is illustrated with the second row of Fig. 17 which displays the 1550/1700Å normalised-map ratio. Bright noise results from division by small numbers; greyish areas mark comparable modulation amplitude at the two wavelengths. The dark areas in the 2.5-min panel result from contrast increase with formation height, with appreciable dark-patch widening. The same areas darken slightly in the 5-min ratio map at left.

In the interpretation of Judge et al. (2001) the dark-patch increase should map magnetic canopy expansion with height as increasing loss of wave amplitude due to mode conversion, but it actually follows the growth of the temporal-mean network patches in the first column of Fig. 14 through the pixel-by-pixel normalisation. The apparent growth of the latter patches with formation height is indeed likely to mark canopy spreading, but the shadow behaviour in Fig. 17 cannot be attributed to mode conversion unless the wave amplitudes indeed scale with the mean brightness. Again, it is not clear a priori whether this is the case. Magnetic mode conversion seems better established at larger atmospheric height than sampled by the TRACE ultraviolet passbands (McIntosh et al. 2001).

Figure 18: Temporal Fourier spectra of the May 12 brightness data for 1700 and 1600Å, spatially averaged over internetwork (upper panel, 7838 pixels) and network (lower panel, 37712 pixels). Abscissae: temporal frequency, corresponding periodicity specified along the top. Solid curves in the lower half of each panel: power spectra on a logarithmic scale (at right), with 1$\sigma$ estimates (dash-dotted and dashed curves, the lower ones drop below the panel due to the logarithmic scaling). Solid curve in the upper half of each panel: coherence C between 1700 and 1600Å, with 1$\sigma$ estimates (dotted; scale at right, the extra tick indicates the C=0.45 random noise value). Scatter diagram in the middle of each panel: phase differences $\Delta \phi (1700 - 1600)$, scale at left, positive values corresponding to 1700Å brightness leading 1600Å brightness. Each spatial sample per frequency was weighted by the corresponding cross-power amplitude $\sqrt {P_1P_2}$ and then binned into a greyscale plot normalised to the same maximum darkness per frequency bin (column). The solid curve is the spatial vector average defined by Eq. (7); the dotted 1$\sigma$ curves are determined from the weighted phase differences. Pure noise produces $\Delta \phi$ scatter over all angles and correspondingly jagged behaviour of the mean $\Delta \phi$ curve.

Figure 19: Temporal Fourier spectra of the May 12 brightness data for 1600Å and 1550Å, spatially averaged over internetwork (upper panel) and network (lower panel). Format as for Fig. 18.

   
4.3 One-dimensional modulation spectra

Figures 18-22 provide a variety of Fourier spectra in a compact format, showing power spectra, coherence spectra and phase-difference spectra for different pairs of diagnostics with spatial averaging split between internetwork and network through masks as in Fig. 4. We first discuss Figs. 18-19 which are derived from the TRACE May 12 image sequences. Figure 18 compares 1700Å brightness modulation to 1600Å brightness modulation; Fig. 19 does the same for the 1600Å and 1550Å pair. The latter pair samples somewhat greater heights but the large similarity of the two figures indicates that the oscillations persist without major change of character over the increase. This is not the case for the Ca IIH diagnostics shown in Figs. 20-22 from the data of Paper I simulated by CS1997, which again provide a useful comparison.

Power and coherence.

The mask-defined spatial means of the temporal power spectra for each of the two passbands are shown at the bottom of each panel. They are the same as in Fig. 12 and are added here for reference, but now plotted on logarithmic scales to permit appreciation of the mean value and the high-frequency noise level. The units are the arbitrary TRACE data units. The same power spectra are shown with zero-frequency normalisation in Fig. 26. The combined network plus internetwork 1700 Å power spectrum is shown on a linear power scale in Fig. 23.

The mean brightness (zero frequency) and lowest-frequency modulation have much larger power than the acoustic oscillation peaks. The spectra show a gradual decline to the noise level at high frequency, with noisier behaviour of the network curves in the lower panel due to the smaller number of pixels.

The coherence C between the members of each pair is plotted at the top of each panel, also with $1\sigma$ error estimates (scale at right). We remind the reader that random noise gives C=0.45 with our method of frequency averaging plus spatially independent sampling. The coherences are significant for f < 15 mHz in all panels of Figs. 18-19 and are very high in the 5-3 min band.

Phase differences.

The phase-difference spectra are the most informative part of Figs. 18, 19. The pixel-by-pixel $\Delta\phi(x,y,f)$ values are shown as binned greyscale images in the format of Lites & Chipman (1979), weighting each (X,Y,f) sample by the cross-power amplitude. The greyscale is scaled to the same maximum blackness for each frequency bin (column) in order to appreciate the spread also at low power. The spatial averages are obtained vectorially as defined by Eq. (7) in Sect. 2 and are plotted as solid curves through the scatter clouds, with $1\sigma$ curves determined from the weighted and binned distributions.

The spatial samples bunch up tightly in the acoustic oscillation band where the coherence is high, especially for the internetwork. The large amount of spatial sampling in our large-field image sequences yields very good statistics so that the average phase differences (solid curve) are very well determined in this band. At higher frequency the scatter increases. Note that pure random noise causes random scatter over all angles which does not average out to zero but produces equally erratic excursions of the average curves, with 1$\sigma$ spread over 0.68 of the figure height. Such behaviour is seen at high frequency for the network averages, which are based on a smaller spatial sample.

The internetwork phase-difference behaviour in Figs. 18, 19 follows a well-defined pattern. It was observed earlier using optical lines (e.g., Fig. 7 of Lites & Chipman 1979; Fig. 7 of Lites et al. 1982; Fig. 7 of Paper I) and far-infrared continua (Kopp et al. 1992), but it is recovered here with much higher precision. At the lowest frequencies the internetwork shows a small but significant negative phase difference, meaning that the deeper-formed (longer wavelength) intensity lags the higher-formed intensity. This is attributed to internal gravity waves below. The 5-min oscillation (f = 3.3 mHz) has slightly positive I-I phase difference due to radiative damping (comparable V-V spectra as in the lower-left panel of Fig. 20 have zero phase difference as expected for evanescent waves). The phase difference then shows a gradual increase out to $f \approx 7$ mHz, implying upward acoustic wave propagation with the higher-formed 1600Å brightness lagging 1700Å brightness. The gradual increase follows the tilted line through $\Delta \phi = 0$ which would hold if the same height difference $\Delta h$ is sampled at each frequency and if all waves propagate up at the same speed $c_{\rm s}$ so that $\Delta \phi = (\Delta h/c_{\rm s}) f$ (cf. Lites & Chipman 1979). In this interpretation the increase has $\Delta h \approx 100$ km as characteristic value until f=7 mHz where a slow decrease to $\Delta \phi = 0$ sets in. We return to this pattern after a comparison with Ca IIH formation.

Comparison with Ca II H.

It is again useful to compare our TRACE results with the Ca IIH spectrogram sequence of Paper I. That analysis targeted network oscillations; Figs. 20-22 present unpublished but informative internetwork measurements from these data in a comprehehensive Fourier format similar to Figs. 18, 19. Figure 20 displays wing modulation sufficiently far from H-line center to portray acoustic oscillations in the middle photosphere where they behave linearly in well-known fashion. Figures 21, 22 use the H_2V brightness index which portrays nonlinear chromospheric kinetics in complex fashion. These figures bracket our TRACE data in formation height.

The upper panels of Fig. 20 compare I-I internetwork Fourier modulation of the brightness in three blend-free spectral windows in the blue wing of Ca IIH indicated by bars in Fig. 8. The oscillatory power peak becomes more pronounced at smaller separation from line center, i.e. higher formation. The I-I phase-difference patterns are very similar to the phase-difference patterns in Figs. 18, 19.

Figure 20: Photospheric comparison Fourier spectra from the Ca IIH data presented in Paper I. The format follows Lites et al. (1994) and is similar to that of Figs. 18, 19, but the power spectra (curves in lower half) are on linear scales normalised to their peaks with the mean removed, the coherence (curve in upper half) was evaluated with 9-point frequency averaging so that C=0.33 for random noise (dash at right), the vector averages of the phase differences are shown as solid dots, and the 1$\sigma$ estimates around the averages are plotted as bars. The lowest Dopplershift frequencies were removed in spectrograph drift correction (V spectra in lower panels). The various diagnostics are indicated in Fig. 8, are formed well away from the H-line center, and are measured from internetwork only (IN). First panel: I-I spectra for Ca IIH wing brightness at $\Delta \lambda = -2.17$ Å and -1.36 Å from H center. Second panel: the same for wing brightness at -1.36 Å and -0.75 Å. Third panel: V-V spectra for Fe I 3966.82Å and Fe I 3966.07Å Dopplershifts. Fourth panel: V-I spectra for Fe I 3966.82Å Dopplershift and wing brightness at -1.36 Å.

The lower panels of Fig. 20 add Dopplershift analysis for the weakest Fe I blend in the Ca IIH spectra, which is the line at $\lambda = 3966.82$Å used by CS1997 to define their simulation piston. It samples height $h \approx 250$ km above the $\tau_{5000}=1$ surface. The lefthand panel shows a V-V comparison with the strongest Fe I blend in the Ca IIH spectra (cf. Fig. 8) which was estimated to represent $h \approx 390$ km in Paper I from the phase difference at f = 8 mHz in this panel. The two power spectra have the same shape. Both lack the prominent low-frequency peak present in the brightness power spectra in the upper panels, but it should be noted that substantial spectrograph wavelength drift was removed from these Dopplershift measurements through nonlinear fit subtraction, a procedure which suppresses the small low-frequency peaks that do show up in the very similar (but more stable) data of Lites et al. (1994). The corresponding overshoot motions were therefore also not included in the simulation piston of CS1997, as shown by the piston power spectrum in Fig. 1 of Theurer et al. (1997a).

The V-V phase-difference spectrum shows evanescent behaviour for the five-minute waves and again displays upward propagation at higher frequencies up to $f \approx 10$ mHz (blueshift is positive in this paper). The V-V five-minute band shows very high coherence and a smaller spread than in the I-I upper panels but the spread becomes slightly larger for the propagating regime because Dopplershift measurement is noisier.

The lower-right panel of Fig. 20 shows a V-I comparison between Fe I 3966.82 Dopplershift and the Ca IIH wing intensity at $\Delta \lambda = -1.36$Å. Such V-I comparisons usually assume that brightness represents temperature, Dopplershift measures velocity, and that the two sample the same atmospheric height for a given spectral line. These assumptions are often erroneous and can be very misleading (cf. Skartlien et al. 1994; Stein & Carlsson 1997); are they valid here? The Ca IIH wing is not too much affected by coherent resonance scattering this far from line center (e.g., Uitenbroek 1989) and should obey LTE reasonably well, and the Fe I blends are also quite likely to have LTE source functions even when their opacity is depleted by radiative overionization (Athay & Lites 1972, cf. Rutten 1988). Figure 8 shows that the Fe I 3966.82 line core and the -1.36Å wing window have the same spatially-averaged emergent intensity, which with LTE and the Eddington-Barbier approximation implies formation at the same average temperature and so at the same height. Indeed, their acoustic power peaks coincide and their phase differences are roughly constant with frequency. Thus, the classical assumptions do hold in this case. At the coherence peak at f=5 mHz the V-I lag of about $-120^\circ$ fits the compilations from many lines in Figs. 1 and 3 of Deubner (1990) at the $h \approx 250$ km formation height and fits the classical interpretation of near-adiabaticity with some thermal relaxation (Schmieder 1976, 1977, 1978; Lites & Chipman 1979; Kopp et al. 1992).

Thus, all is well in Fig. 20. The TRACE Fourier spectra in Fig. 18 show comfortably similar I-I behaviour with much better measurement statistics. Although the ultraviolet continua are not formed in LTE they apparently sample the same oscillations with similar height discrimination.

Figure 21: Chromospheric comparison spectra from the Ca IIH data discussed in Paper I in the format of Fig. 20. Left: internetwork. Right: network. Upper panels: I-I spectra for wing brightness at $\Delta \lambda = -0.75$Å and H2V index brightness. Lower panels: V-V spectra for Fe I 3966.07Å Dopplershift and Ca II H3 line-center Dopplershift.

However, TRACE images and timeslices as in Figs. 5-7 display internetwork grains much like Ca IIK filtergrams, so that comparison should also be made with Ca IIH formation closer to line center. This is done in Figs. 21, 22. The upper panels of Fig. 21 show internetwork and network I-I comparisons using the H_2V brightness index of which the timeslice is shown in Fig. 9 and decomposed in Fig. 10. Its internetwork power spectrum is dominated by the f=5 mHz peak and its phase differences with the $\Delta \lambda = -0.75$Å brightness rise steeply with frequency, reaching $180^\circ$ at f=10 mHz. Spectrally, this upward propagation of the brightness modulation in the Ca II H & K wings was diagnosed from inward wing whisker contraction already by Beckers & Artzner (1974). It causes the fuzziness in the time direction of the H_2V grains in the wide-band timeslice at right in Fig. 9. The steep $\Delta \phi(I{-}I)$ gradient across the three-minute band also contributes apparent dissimilarity between the two three-minute slices of Fig. 10 because it washes out timeslice structure with sampling height even at high coherence. The larger similarity of the internetwork parts of the two five-minute slices in Fig. 10 corresponds to constancy across the five-minute band of the total $\Delta \phi$ sum of the two righthand panels of Fig. 20 and the upper-left panel of Fig. 21.

Figure 22: Chromospheric V-I spectra from the Ca IIH data in Paper I for internetwork. Left: Fe I 3966.07Å Dopplershift and H2V index. Right: H3 Dopplershift and H2V index. The rms $\Delta \phi$ bars wrap around from below $-180^\circ$ to above $+180^\circ$ and vice-versa.

The steep $\Delta \phi(I{-}I)$ increase is in conflict with the small amount of propagative difference in Fig. 19 if one assigns H_2V brightness formation to temperature behaviour at temperature-minimum formation height. This would be the case if one assumes that equal time-averaged emergent intensity implies the same formation height, as proven above for the Fe I 3966.82Å and $\Delta \lambda = -1.36$Å pair. One would then assign Fe I 3966.82Å and the H_2V index to the same layer on the basis of their equal mean intensity in Fig. 8, but the assumption is wrong as indicated by the Fe I 3966.82 - Ca II H₃ V-V comparison in the lower-left panel which shows a similar $\Delta \phi(f)$ gradient but spans a much larger height interval. The similarity illustrates that H_2V brightness is strongly influenced by the Dopplershift excursions of the H₃ core well above the layer where the H_2V photons escape, with H_2V enhancements corresponding to H₃ redshifts. This well-known relationship, initially proposed and modelled by Athay (1970), Cram (1972), and Liu & Skumanich (1974), is explained in detail in the informative H-line formation break-down diagrams in Figs. 4-7 of CS1997 and is here demonstrated observationally by the V-I diagrams in Fig. 22. In the righthand panel the coherence is higher, the two three-minute power peaks are virtually identical, and $\Delta \phi (V{-}I)$ is roughly constant with frequency. Thus, the steep $\Delta \phi$ increases in the I-I and the V-V internetwork panels of Fig. 21 are both set by H₃ Dopplershift modulation. The same increase was observed in Fig. 5 of Lites & Chipman (1979) and the grain-rich V-V diagram from 1991 data in Fig. 5 of Rutten (1994), and it is also present (with a sign-definition flip) in Fig. 4 of Skartlien et al. (1994) which evaluates V-V phase difference for the Ca II H₃ - 8542Å line pair from the CS1997 simulation, confirming that the increase is a characteristic of acoustic shock dynamics.

We conclude from this comparison that the internetwork Fourier behaviour diagnosed in Figs. 18-19 samples oscillations comparably to the inner wings of the Ca II H & K lines, but without the brightness enhancement that internetwork grains gain when observed at the K_2V and H_2V wavelengths from the large-amplitude shock modulation of the H₃ Dopplershift in higher layers.

High-frequency waves.

Phase-difference measurements with TRACE should permit recognition of oscillation signatures to higher frequencies than is possible in ground-based observations. The interest in diagnosing high-frequency oscillations is large, for example to settle the current debate on chromospheric heating by high-frequency acoustic waves and their presence or absence in the Carlsson & Stein piston (Musielak et al. 1994; Theurer et al. 1997a; Theurer et al. 1997b; Ulmschneider 1999; Kalkofen et al. 1999) and its stellar ramifications (Cuntz et al. 1999).

The phase-difference spectra in Figs. 18, 19 show gradual decrease above $f \approx 7$ mHz. In older oscillation studies such declines have generally been attributed to atmospheric seeing (e.g., Endler & Deubner 1983; Deubner et al. 1984; Deubner & Fleck 1990; see Rutten et al. 1994 for a discussion). In TRACE observations, however, there is no seeing apart from the small-scale guiding jitter that has mostly been removed by our alignment procedures. However, also from space acoustic oscillations are harder to measure at higher frequency because the wavelength shortens: at soundspeed propagation it drops to 350 km for f=20 mHz. An upwards propagating wave may fit from maximum compression to maximum rarefaction within the response function width, so that much less power is detected than actually present - but even in that case phase difference may yet be measurable.

In this context the phase-difference behaviour for f=10-20 mHz in the two lefthand panels of Fig. 21 should be noted. The noise is large (the 1$\sigma$ bars nearly reaching their maximum extent of 68% of the panel height) but the averages seem to display a pattern which we deem possibly significant because similar behaviour is seen in Fig. 5 of Lites & Chipman (1979) and Fig. 7 of Rutten (1994).

Figures 18, 19 show no high-frequency behaviour analogous to the internetwork behaviour to Fig. 21 at all, only the gradual $\Delta \phi$ decrease. The latter seems significant even though the power drops strongly towards f=10 mHz. We suggest that the phase difference decline results primarily from non-linear wave steepening. On their way to become weak acoustic shocks in the low chromosphere, the higher-frequency components travel faster and arrive earlier at the higher sampling location. In the TRACE brightness measurements the steepening is sampled far below the level that produces the patterns seen in Fig. 21, which must both be attributed to K₃ Dopplershift behaviour as discussed above. Thus, the $\Delta \phi$ decreases in Figs. 18, 19 sample nonlinear wave steepening at its onset, well below the level where shocks result and produce H_2V and K_2V grains as well as observable sawtooth behaviour. The question to what maximum frequency these $\Delta \phi$ decreases are significant is addressed in Sect. 5 where we conclude that the non-simultaneity of the TRACE imaging in the different passbands sets the $\Delta \phi$ reliability limit to f = 15 mHz at best.

Network-internetwork differences.

The network panels in Figs. 18, 19 differ only slightly from the internetwork panels. The differences in power spectra are exhibited more clearly in Figs. 12, 13 and correspond to the relative lack of network power in the 2.5-min maps in Fig. 14. The coherences drop a bit faster with frequency in the network and the phase-difference scatter is slightly larger even in the five-minute band. The large similarity with the internetwork behaviour shows that the acoustic oscillations pervade network at these heights without much modification, as they do in the photosphere (plots as in Fig. 20 for network, not shown, are identical apart from small-sample noise).

The picture is drastically different for the chromospheric diagnostics in Fig. 21 where power spectra and phase difference behaviour are very different between network and internetwork. The fourth panel illustrates the conclusion of Paper I that the long-period modulation peak that is particular to chromospheric network sampled by H₃ Dopplershift has no connection (low coherence) to the dynamics of the underlying photosphere. This disconnection is demonstrated here by the large disparity in internetwork-network differences between Fig. 19 (upper and lower panels nearly the same) and Fig. 21 (left and right panels very different).

Figure 23: Partial two-dimensional power spectra of the 1700 Å brightness in the complete May 12 sequence (full field and duration). Only the significant part is shown. Axes: horizontal wavenumber kh and temporal frequency f. The corresponding spatial wavelengths and temporal periodicities are specified along the top and at right. Upper panel: absolute brightness power on logarithmic greyscale. The solid curves show the temporally and spatially integrated power spectra on linear inverted scales with the mean and trend removed. Lower panel: the same after subtraction of the background variation to enhance the p-mode and pseudo-mode ridges, and with the greyscale inverted to display the contours better. The latter define the ridge mask. The vertical fringing is an artifact from TRACE data compression.

Figure 24: Two-dimensional $\Delta \phi (1700 - 1600)$ phase-difference spectrum from the May 12 data. Axes: horizontal wavenumber kh and temporal frequency f. The corresponding wavelengths and periodicities are specified along the top and at right. Greyscale: phase difference $\Delta \phi$ coded as indicated at the top.

Figure 25: Upper panel: absolute power in 1700Å brightness as function of horizontal wavenumber at different frequencies, from top to bottom 4.7, 5.6, 6.6, 7.5 mHz. Triangles: ridge locations as in Fig. 23. Dotted curves: background estimates underlying the ridges. Lower panel: fraction of the summed ridge power (above the dotted backgrounds in the upper plot) of the total power, as function of frequency. The dotted curve results from the longer-duration October 14 sequence.

   
4.4 Two-dimensional modulation spectra

Figures 23, 24 complete our suite of Fourier diagnostics for the May 12 data in the form of two-dimensional power and phase-difference spectra, resolving the oscillations in both temporal frequency f (vertical) and in spatial wavenumber k_h through annular averaging over the horizontal (k_x,k_y) plane. These spatially-averaged diagrams make no distinction between internetwork and network. Only the lower-left (k_h,f) quadrants with significant structure are shown.

Acoustic oscillations.

Figure 23 shows the (k_h,f) power spectrum for the May 12 1700Å data. The solid curves are the spatially and temporally averaged power spectra, respectively. The one at right is the combination of the internetwork and network 1700Å power spectra in Fig. 18 but is here plotted on a linear scale with removal of the mean and trend to emphasise the low-frequency and acoustic modulation peaks.

The (k_h,f) power spectrum contains p-mode (f= 2-5 mHz) and pseudo-mode (above the cutoff frequency $f_{\rm AC}=5.3$ mHz) power ridges out to f = 8 mHz and to k_h = 3.2 arcsec^-1 or spatial wavelength $\Lambda_h = 2$ arcsec (scale on top; the latter limit may be set by the 1.5 arcsec boxcar image smoothing of the May 12 sequences). The pseudo-mode ridges do not originate from global mode selection through multi-pass interference, as is the case for the evanescent five-minute modes used in helioseismology, but come from waves that are excited near the surface in downward directions and are reflected only once at the inward temperature increase before propagating upwards out of the photosphere. They add power to the directly excited outgoing wave at (k_h,f) locations corresponding to the one-bounce travel length and duration from the source. Thus, they do not represent global wave interference but local power addition. They correspond to first-bounce power ridges in time-distance plots in which the appearance of only one ridge for three-minute waves established the up-and-out loss after the first reflection (Duvall et al. 1993), and they roughly obey the Duvall dispersion law for a fixed waveguide with n nodes along its length (Duvall 1982; cf. Eq. (2.14) of Deubner & Gough 1984; see also Kumar et al. 1994; Kumar 1994; Jefferies 1998; Vorontsov et al. 1998).

Figure 24 shows the (k_h,f) diagram for the May 12 1700-1600Å phase differences. The pseudo-mode ridges stand out by having larger $\Delta \phi (1700 - 1600)$ values than between the ridges, representing a ridge-interridge split of the $\Delta \phi$ averages in Fig. 18. Since the ridges correspond to single-bounce power enhancements in (k_h,f) selection, they are likely to correspond to locations where acoustic waves propagate vertically upwards. The on-ridge $\Delta \phi (1700 - 1600)$ values then correspond to vertical travel time between the two sampling heights for such waves. They do not stand out through interference between spatially different locations but because outward propagation is enhanced at the (k_h,f) ridge locations.

The interridge background has smaller phase differences, implying partial non-propagative signature (oblique propagation would increase the delay). We suspect that these smaller differences are standing-wave contributions from partial reflections at larger height, possibly at shocks (e.g., Deubner et al. 1996; Carlsson & Stein 1999), or from deep penetration of ballistic post-shock downfall. In the Carlsson-Stein simulation, shocks are sometimes stopped in their upward propagation by such downdrafts after prior shocks. Such a case is observationally present in the form of the triple-grain sequence at the top of C+S Col. 30 (x=63) in Figs. 9-11. When blinking the upper and lower rightmost panels of Fig. 10 against each other, the upward propagation of the three-minute waves is immediately evident as up-and-down jumping patterns - except for this particular grain train which appears standing still when blinked, so simultaneously present at the two sampling heights, with zero phase difference.

Not only does the interridge background have distinctly different phase behaviour at pseudo-mode frequencies but it also has appreciable power in the whole acoustic regime. This is quantified in Fig. 25. The upper panel shows power distributions along horizontal cuts through the (k_h,f) diagram in Fig. 23. The ridges defined by the ridge mask (lower panel of Fig. 23) are indicated by triangles. The dotted curves are background estimations. The lower panel shows the fraction of ridge power above the local background summed over all mask ridges along horizontal cuts. The ridge power is most important below the cutoff frequency but even there the background contribution is sizable. The latter rapidly gains importance with frequency above the cutoff frequency.

In the p-mode regime below the cutoff frequency the on-ridge power enhancements represent constructive interference of global oscillatons with long-duration mode persistence, whereas the p-mode background represents shorter-duration wavetrains from local excitations without global mode selection. Long-duration sampling as in helioseismology increases the ridge/interridge power contrast because shorter wavetrains then cancel through having random phases. In contrast, pseudo-mode ridges do not correspond to such long-duration persistence and may actually be dominated by local excitations, whereas the pseudo-mode background may have a larger interference contribution from wave reflections. This intrinsic difference is illustrated by the difference between the two curves in the lower panel of Fig. 25. The twice longer duration of the October 14 sequence (dotted curve) produces higher contrast for the p-mode ridges but lower contrast for the pseudo-mode ones.

Low-frequency oscillations.

At lower frequency, the dark wedge in Fig. 24 indicates downward propagation which corresponds to the initial $\Delta \phi <0$ dip in the upper panels of Figs. 18 and 20. The wedge shape, the location of the upper limit at the Lamb line, and the larger-than-granule wavelengths suggest strongly that this wedge is due to internal gravity waves (cf. Straus & Bonaccini 1997). It was previously observed in similar (k_h,f) spectra from ground-based filtergram sequences by Kneer & von Uexküll (1993) who tentatively made the same identification, as did Schmieder (1976), Cram (1978), Brown & Harrison (1980), Deubner & Fleck (1989), Bonet et al. (1991) and Komm et al. (1991) from various other diagnostics. The same tentative identification was made in Paper I for the low-frequency power peaks shown here in Fig. 20. These are much higher, relatively to the acoustic peak, than the low-frequency Doppler modulation by convective overshoot (which is artifically suppressed in the lower panels of Figs. 20, 21 as noted above but remains well below the acoustic peak height in other data). We now confirm the identification as gravity waves from the (k_h,f) phase-difference signature in Fig. 24.

It is no surprise to find gravity waves in the upper photosphere since they should be abundantly excited by convective overshoot. However, their detection is not easy because they combine long periods and short wavelengths with small Dopplershifts and because they travel preferentially in slanted directions, so that sampling widely different heights along radial lines of sight suffers loss of coherency (see the detailed descriptions and predictions in the fundamental papers of Mihalas & Toomre 1981, 1982).

Our TRACE $\Delta \phi (1700 - 1600)$ measurement represents a particularly sensitive gravity-wave diagnostic because (i) - it samples brightness rather than velocity, (ii) - does so sufficiently high (the intensity modulation grows considerably with height), but (iii) - yet below the wave breaking height above which the waves vanish, and (iv) - compares brightness phase at sufficiently close heights with sufficient spatial integration (1.5 arcsec boxcar smoothing) to detect slanted waves. The negative-phase signature is smaller for the higher-located sampling in Fig. 19; the corresponding (k_h,f) diagram (not shown) has similar ridge-interridge difference at smaller amplitude.

The gravity wave signature is similarly present in the Ca IIH wing brightness modulation spectra in the upper panels of Fig. 20, for the same reasons: gravity waves have much larger brightness modulation than Dopplershift modulation, the brightness modulation is sampled at heights where the amplitude grows large before reaching breaking height, and the Ca IIH wing windows sample a sufficiently small height difference to avoid spatial mismatch from oblique propagation. In addition, since the H wings are formed in LTE they may respond more faithfully to temperature modulation than the ultraviolet continua which suffer from scattering. The low-frequency power peak is still weakly present in the internetwork H_2V index spectrum in the upper-left panel of Fig. 21.

Note that both $\rm H - 0.75$ power and H_2V index power have a (relatively) more pronounced low-frequency peak in the network (upper-right panel of Fig. 21) and that this peak seems to survive as the only peak in internetwork H₃ Dopplershift power (lower-right panel). The tantalising suggestion is that the H₃ Dopplershift peak may result from fluxtube interaction at mid-photospheric heights with gravity waves impinging sideways from the surrounding internetwork that get enhanced in the fluxtube. An alternative interpretation is that the network H₃ Dopplershift peak represents mottle flow dynamics as mapped by the second bottom panel of Fig. 14.