2 The solar background as a signature of the source of the solar oscillations

2.1 The I-V phase difference

The negative phase plateau, first discovered by Deubner et al. (1990), was confirmed by MDI data (Straus et al. 1998). Successively, using GONG data (Oliviero et al. 1999), the negative phase regime has been shown to extend to intermediate and low-$\ell$ values and, at high frequencies, positive values greater than those shown at the p-modes frequencies, have been found. Many of the differences in the values obtained from different data sets (GONG, MDI, VAMOS) can be attributed to different formation heights of the solar lines used and to different $\ell$ and $\nu$ resolution in the I-V phase difference spectra. To date, the experimental results have the following traits:

Figure 1: The I-V phase difference from the analysis of the GONG data in the Ni 676.8 nm solar line: stars for 15 hour, points for 1 month time-series. The velocity power of the shorter time-series is plotted (solid line) to show the coincidence of the measures only in the high power peaks.

Figure 2: For $\ell =100$, the I-V phase difference (left) and coherence (right) for the VAMOS (crosses) and Kanzelh $\ddot{\rm o}$he (diamonds) data. Both data refer to the Na D lines and were obtained from the analysis of 256 min time-series obtained through a Magneto-Optical Filter.

1) the phase values are approximately independent of the degree $\ell$;
2) the phases on the p-mode ridges depend on the height in the solar atmosphere;
3) the phases in the solar background show a step-like behaviour with negative values below about 3.3 mHz and positive values above about 4 mHz.

The value of the phase strongly depends on the spherical degree and frequency resolutions. As an example, we show in Figs. 1 and 2 the frequency dependence of the I-V phase difference for the Ni 676.8 nm data (36 days from GONG) and for the Na D lines (256 min from VAMOS and Kanzelh $\ddot{\rm o}$he, Moretti et al. 1997; Severino et al. 2001; Cacciani et al. 1999).

The measured phase values are related to the ability to distinguish between different phenomena: the resolution in $\ell$ (leakage) or in $\nu$ produces a mixing of the phases weighted by the different powers (Figs. 1 and 2 of Oliviero et al. 1998).

2.2 The local analysis

The background has been usually studied via its behaviour in the $\ell -\nu$ diagram: that is, the contribution of the signal filtered by the spherical harmonics is displayed as a function of the spatial scale and of the frequency. In the case of the p-modes, this representation shows the global resonance of the acoustic waves. In this framework, if the source of the oscillations is local and its scale outside the observed $\ell$ range (Skartlien & Rast 2000), any characteristic of the source should be rather independent of the spatial scales when analysed in a $\ell -\nu$ diagnostic. This is in fact the case in recent results on the phase difference between the intensity and velocity signals previously described. Therefore, it seems reasonable to use another technique to filter the data in order to enhance the local characteristics of the source of the solar oscillations. When the velocity and intensity oscillations are treated locally, they may be different due to mixing with local phenomena: their characteristic spatial distributions on the disk are not washed out by a filter, as the spherical harmonics decomposition.

The local analysis uses a three-dimensional representation, as the distribution on the disk (in x, y) has to be visualised as a function of the frequency (Lites et al. 1993). In fact, the time-series of the images are pixel-by-pixel fast fourier transformed (FFT) and the power and phase difference maps are obtained. Long time-series will produce a spatial average because of solar rotation and the evolution of structures on the solar surface. The granules and supergranules have time-scales of 10 minutes and hours with scales of $\simeq$1 $^{\prime \prime }$ and $\simeq$50 $^{\prime \prime }$ respectively. The structures would lose their identity after a lifetime and a compromise has to be reached, so as not to remove anything but the rotation. As a consequence, the temporal resolution we get is limited by an observing run of a few hours. Nevertheless, it is useful to perform this analysis since it permits us to investigate the spatial distribution of the oscillatory power and its relation to local phenomena. Moreover, in order to evaluate the contributions of the different phenomena to the signal, it is useful to produce the $\ell -\nu$ diagram to make the distinction between the oscillations and the convection easier. With our resolution, the convection contribution is negligible in comparison to that of the p-modes in the five-minute frequency range.

Figure 3: Examples of the phase maps for, from top to bottom, $+108^{\circ }$, $0^{\circ }$ and $-144^{\circ }$, $\pm 18^{\circ }$ each. From left to right: at 1.3, 3.3, 5.2 mHz.

2.3 The data

The data consist of three sets of dopplergrams, magnetograms and intensity images obtained with a sodium Magneto-Optical Filter (MOF) at Kanzelh $\ddot{\rm o}$he (Cacciani et al. 1999). The images were acquired every minute and 256 min from each day were selected for the analysis. Some days were analysed and we show the results relative to 30 January 1998. The spatial resolution is 4.3 $^{\prime \prime }$/pix. The dopplergrams were calibrated as described in Moretti & the MOF Development Group 2000; all the images were registered but no correction for the rotation has been applied. The full-disk data were treated locally (pixel-by-pixel) and globally (using the spherical harmonics decomposition developed for the VAMOS project, Oliviero et al. 1998) to obtain the power distribution on the disk and the $\ell -\nu$ diagrams. The daily trend was removed in the pixel-by-pixel time-series using a polynomial fit, while a differential filter was used in the spherical harmonics decomposition. The final spectra were corrected for the filtering of the data. Since the dopplergrams were not simultaneous to the intensity images, the correction for the time delay in the phase maps has been introduced.

2.4 The I-V phase difference when obtained through a narrow passband filter

A further correction in the phase difference values has to be applied when a narrow passband filter (like a MOF) is used (Moretti et al., to be submitted). In these systems, a velocity shift of the line profile induces an intensity fluctuation as the main undesirable effect. The contamination is larger where the velocity power is stronger and depends on the sensitivity of the signal to the velocity. That is, the contamination mainly changes along the solar disk according to the relative velocity to the Earth. The crosstalk has been modelled for this system and the correction to the phases applied. We note that the error in the corrected phase values may be large but, in this context, does not affect the discussion based on the spatial properties of the I-V phase difference. In fact, the phases' changes are well below the phase difference between the different phenomena.

Figure 4: The probabilities of finding the I-V phase values have been averaged in the low frequency (between 0.65 and 1.89 mHz) and five-minute (between 2.60 and 4.49 mHz) bands. Two populations are visible in the local analysis as well as in the $\ell -\nu$ diagram (see also Fig. 2). We associate the population with the positive value to the p-modes, and the negative to the solar background.