3 The analysis and the results

The phase difference maps have been obtained up to 8.33 mHz (see examples in Fig. 3). The $360^{\circ}$ range was divided into many intervals. The coverage of the phase values at any frequency was computed for an area out to 0.5 radii, and correspond to the probability of finding that phase in the part of the disk where the p-mode interference pattern dominates. In order to enhance the existence of different populations of phase values, an average of the "coverages'' was performed for the low frequency and five-minute bands respectively (Fig. 4). Two populations of phase values are significant on the disk: a positive (around $+140^{\circ}$) component and a negative (hereafter called $-144^{\circ }$ or background value), whose value we attribute to the solar background at the sodium D line formation layer (see also Fig. 2).

Figure 5: Top: the I-V phase differences (before crosstalk correction) corresponding to the high velocity power locations out to 0.5 R have been averaged at any frequency. The error bars are the sigma obtained from the spread over the half disk. Bottom: the difference between the values before and after crosstalk correction (Moretti et al., in preparation).

Figure 6: Left panels: the probability of finding a phase value (within $\pm 18^{\circ }$) computed at the locations where the velocity amplitude is lower (top) or greater (bottom) than a threshold (see text) is shown at three frequency samples. The x-axis represents the factor by which the threshold has been multiplied. Crosses, stars, diamonds and triangles represent $-144^{\circ }$, $0^{\circ }$, $+108^{\circ }$, $+126^{\circ }$ respectively. The velocity power acts as a selective filter at the five-minute band. Right panel: The frequency dependence of the phase probabilities at the high power locations (crosses, stars, diamonds and triangles for $-144^{\circ }$, $0^{\circ }$, $+108^{\circ }$, $+126^{\circ }$ respectively ( $\pm 18^{\circ }$), multiplying factor = 1.6).

In the sodium results the $-144^{\circ }$ value, clearly attributed to the background in the $\ell -\nu$ diagram at the low frequency band, is visible with the local analysis in the five-minute band too, but not in the $\ell -\nu$ diagram where the low temporal resolution does not permit us to distinguish the low power inter-ridges. On the contrary, in the local analysis, the low power sites are practically localised at any frequency and the $-144^{\circ }$ value can be seen.

In practice, what the spatial recognition performs in the local analysis, the high frequency resolution does in the $\ell -\nu$ diagram. The phase values for the modes have been selected as those corresponding to the locations where the velocity power is greater than a threshold. The threshold was chosen as the mean velocity amplitude computed from the power maps: it changes approximately from 1 m/s at low frequencies to 15 m/s in the five-minute band. The values for the phase at these locations have been successively averaged over the solar disk and their distribution is shown in Fig. 5.

The phase value corresponding to the high power locations, averaged between 3.2 and 5.2 mHz, is $+135^{\circ}\pm20^{\circ}$. The high power locations in the five-minute band are usually considered to be the places where the interference of the oscillations act constructively, and for this reason we also attribute these values to the "modes'' also in the local analysis.

While other phase values seem to be randomly distributed on the disk, at disk center in the five-minute band, the $-144^{\circ }$ value occurs where the velocity power is low and in the low frequency range. In order to make this behaviour visible, the probabilities as previously described have been computed for the locations at different velocity thresholds (see Fig. 6). It is clear that the positive values are the mode values and the power acts as a selective filter for the $-144^{\circ }$ value. The $-144^{\circ }$ is well visible at low frequencies, where no powerful phenomena dominate the disk. The frequency dependence of the phase difference is shown in Fig. 5 for the powerful locations: the five and three-minute bands show slightly different values. This difference, not visible in the $\ell -\nu$ diagram, can be attributed to a different dilution of the phenomena when they are spatially averaged. In fact, since the velocity amplitude in the three-minute band is small, a selection of the modes cannot be performed as clearly as in the five-minute band. As a result, the higher value of the p-modes phase is lowered (this effect could also justify the results in Khomenko 2001).

Even if the spatial resolution of the data does not permit us to distinguish unequivocally the magnetic network, a correlation with the magnetic field and its oscillations has been performed using a simultaneous data set of longitudinal magnetograms. The results do not show a one-to-one correspondence between the magnetic field and the $-144^{\circ }$ value, but this latter value is often shown in the magnetic locations, where the five-minute power is usually reduced (Fig. 7).

When a correlation is performed between the maps of the $-144^{\circ }$ value averaged in the low frequency band and the magnetic power maps, the correspondence to the $H\alpha$ bright points is visible (Fig. 8, see also Moretti et al. 2000). The morphology is well recognisable, even if the structures are not co-spatial (due to the spread of the magnetic lines, due to the non-vertical propagation of the jets etc.). Nevertheless, this is not a proof of a physical relation between the phenomena at different layers (we note that the background is also localised where the magnetic power is larger, and that the magnetic signatures are typically related to the $H\alpha$ bright points).

Figure 7: From left to right: the $-144^{\circ }$ phase difference probabilities averaged between 1.6 and 2.6 mHz (white means high probability), the $-144^{\circ }$ phase difference probabilities averaged between 3.3 and 4.3 mHz, a longitudinal magnetogram, the magnetic power map at 3.3 mHz, the velocity power map at 3.3 mHz. The vertical size is approximately 0.75 solar radii.

Figure 8: A) A mask obtained as the product of the strongest magnetic power locations and the phase map at $-144^{\circ }$ at the low frequency band. B) The corresponding $H\alpha$ image (only the very bright points have been selected). C) The raw $H\alpha$ image. The marked area is shown in the next two panels. D) The zoom of the mask in A. E) The zoom of the mask of the $H\alpha$ bright points.

Figure 9: Let us assume these simplified spatial distributions for the coverages of the p-modes at the five-minute band (A), and for two possible background spatial distributions at low frequencies (B and C). A) The p-modes fill the area of a single cell of the constructive interference pattern (in black) whose radius is denoted by R. dR is the width of the circular corona representing the border lines separating different cells. From the coverage of the p-modes phase: 0.9 = 0.1 + $\frac {\pi R^2} {2 \pi R {\rm d}R + \pi R^2}$, we obtain $R/{\rm d}R=8$. Assuming ${\rm d}R = 1$ pix, $R \simeq 35$ $^{\prime \prime }$. B) The background is localised in the border line of the cell of the constructive interference pattern of the p-modes. C) The background uniformly covers the disk.

Is the velocity power really acting as a filter on the visibility of the background value whose distribution is indeed intrinsically uniform on the disk? If this is the case, we expect an increase in its coverage where the p-modes do not dominate, that is, in the low frequency band. In fact, the constructive interference patterns of the p-modes should not hide the background locations.

Since the $360^{\circ}$ phase range has been divided into 10 intervals each $36^{\circ}$ wide, a 0.1 coverage is expected for a uniform distribution. The increase of the background coverage from less than 0.1 in the five-minute band to 0.2 in the low frequency band can be interpreted in two ways, as described in Fig. 9 (see also Fig. 6). From the coverage of the p-mode phase in the five-minute band (0.9 = 0.1 + $\frac {\pi R^2} {2 \pi R {\rm d}R + \pi R^2}$), the ratio between a cell of the constructive interference pattern and the border lines can be deduced ( $R/{\rm d}R$ is equal to 8, that is an area of approximately 70 $\hbox{$^{\prime\prime}$ }$ of diameter if the border is chosen as one pixel wide). As a result, the 0.2 probability to find the $-144^{\circ }$ value in the low frequency band can be produced by a 0.5 non random coverage of the border lines or an uniform 0.2 coverage of the disk. The former case is suggested to be the case. In fact, the spatial distributions of the $-144^{\circ }$ maps at low frequencies seems not to be uniform but shows a clustering at scales of the order of 50 $^{\prime \prime }$ (see Fig. 10).

In addition, we computed the probability of finding a phase value (at the usual ten phase intervals $36^{\circ}$ wide each), in the same pixel, at different frequencies, in the low frequency (between 1.6 and 2.6 mHz) and five-minute (between 3.2 and 5.2 mHz) bands respectively. This corresponds exactly to the definition of the probability as the number of successes divided by the number of throws, where the throws are in our case the frequencies. The probability for a uniform distribution is 0.1, since ten intervals have been selected. We counted the number of pixels whose probability exceeds 0.333 (in the case of an uniform distribution, we expect a 8% of the points to show a probability larger than 0.333).

 

 

Table 1: The coverage of points where a phase value ( $\pm 18^{\circ }$, first column) occurs at least once between 1.6 and 2.6 mHz (top) and 3.2 and 5.2 mHz (bottom) have been computed (second column). Those points whose probability in that frequency interval exceeds 0.333 have been selected and the probability of occurring is shown in the third column. For an uniform distribution, the probability of finding a point with a probability larger than 0.333 is 8%.

Averaged between 1.6 and 2.6 mHz
phase $\pm 18^{\circ }$	coverage (%)	high prob. points (%)
$-162^{\circ}$	67.4	0.7
$-126^{\circ }$	96.1	19.9
$-90^{\circ}$	87.9	4.4
$-54^{\circ}$	67.7	0.7
$-18^{\circ}$	53.5	0.2
$+18^{\circ}$	51.1	0.2
$+54^{\circ}$	61.0	0.4
$+90^{\circ }$	84.9	2.9
$+126^{\circ }$	97.2	27.4
$+162^{\circ}$	65.1	0.7
Averaged between 3.2 and 5.2 mHz
phase $\pm 18^{\circ }$	coverage (%)	high prob. points (%)
$-162^{\circ}$	50.2	0.0
$-126^{\circ }$	86.1	0.0
$-90^{\circ}$	54.6	0.0
$-54^{\circ}$	31.4	0.0
$-18^{\circ}$	28.6	0.0
$+18^{\circ}$	38.1	0.0
$+54^{\circ}$	70.1	0.0
$+90^{\circ }$	99.8	18.3
$+126^{\circ }$	100.0	98.3
$+162^{\circ}$	65.9	0.4

Figure 10: The probability of finding the $-144^{\circ }$ phase at the low frequency band, in the case of an uniform distribution, is 0.1 (since 10 intervals in phase have been chosen). The pixels where this probability is larger than 0.333 have been selected and shown (in black). The background locations seem to cluster around cells of about 50 $^{\prime \prime }$.

These numbers (shown in Table 1) clearly show that the $-144^{\circ }$ locations are not randomly distributed at any frequency but seem to be associated with a spatially located phenomenon. In fact, since the "throws" are the frequencies, this means that something has occurred in a place with a temporal behaviour whose transform is approximately flat or confined in a $36^{\circ}$ interval at low frequencies. The pixels where the probability of finding the $-144^{\circ }$ value is high over the whole low frequency band have been selected and their time-series compared to those where the p-modes value dominate. Some darkenings, lasting few minutes, at intervals of about 100 min, are visible. These results suggest the presence of localised events, whose detection in the time-series is difficult due to their short duration and to their possible small scales.

If the same results were interpreted as the filtering of p-modes on the background produced by the atmospheric cavity, as proposed by Deubner et al. (1996), why do the background locations correspond to the low velocity p-modes power locations in the low frequency range too? Why does the background maintain the characteristic of small signatures clustered in larger structures?

To confirm the hypothesis of the background generated by localised phenomena, the autocorrelation image of the coverage maps at disk center (out to 0.5 R) at any phase interval has been performed. The east-west and north-south standard deviations for a 2-D Gaussian fit have been computed. In Fig. 11, their difference is shown versus the frequency. The p-modes locations show the east-west elongation, that is, the associated structures (the constructive interference patterns of the five-minute oscillations, lasting more than the 4 h duration of the observing run) rotate, while the $-144^{\circ }$ locations, whose typical scale is one pixel, do not. This means that the background locations are related to rotating structures well below the resolution or to structures at the limit of the resolution but lasting a period whose trace during the rotation is confined in one pixel, that is they last less than 30 min.

In the case of the cavity hypothesis, if the interference patterns related to the cavity itself are supposed to last at least 4 h, the results limit their spatial dimension to subarcsec scales.

Figure 11: The east-west and north-south sigmas for a two-dimensional Gaussian fit were computed for the autocorrelation image of the coverage maps at disk center (out to 0.5 R) at any phase interval. The difference between the sigmas along the east-west and north-south directions is plotted. The rotation imprinting is visible in the mode phase at the five minute band. Symbols: squares for $+90^{\circ }$, diamonds for $54^{\circ }$, stars for $-126^{\circ }$, points for the other intervals, $+36^{\circ }$ wide each.