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Astron. Astrophys. 360, 1157-1162 (2000)

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3. Results and discussion

3.1. Intensity and velocity maps

Fig. 2 gives in its upper part the speckle-reconstructed integrated white light (broad-band) image of the granulation in the quiet Sun. The intensity fluctuations have a contrast of [FORMULA]. This contrast value is low compared to those of reconstructed granular images described elsewhere (cf. references in Muller 1999) due to differences in the characteristics of the used spectral line.

[FIGURE] Fig. 2. Granular intensity and velocity at quiet Sun disc center. The velocity map shows the residuals after boxcar smoothing over [FORMULA]. Tickmarks are at 1" distance.

The lower part of Fig. 2 shows the velocity map. Here, to reduce the influence of the 5-min oscillations on the velocity we present the residual values after smoothing over [FORMULA]. The velocities range from -2.10 km s-1 (downflows) to +2.34 km s-1 (upflows) in the low photosphere. This is about a factor of 3 larger than those presented by Nesis et al. (1999) from excellent spectrograms taken with a slit spectrograph. Without smoothing, the velocity exhibits a rms value of [FORMULA] = 0.625 km s-1. The smoothing reduces this only a little to [FORMULA] = 0.575 km s-1, which is somewhat larger than the value obtained by Deubner (1988) from likewise excellent spectra from the VTT at Sacramento Peak Observatory. But it is substantially below the value of [FORMULA] = 1.5 km s-1 given by Espagnet et al. (1995) for the low photosphere from their MSDP data, which are uncorrected for seeing. Even for the middle and upper photosphere, the latter authors give larger rms values than we find here. Espagnet et al. (1995) demonstrate that most of the rms velocities are due to motions at scales [FORMULA]. However, all the way during data processing we took care to avoid leakage of the power as much as possible and do trust our reconstructed small band image at least down to [FORMULA]. Thus the reason for this discrepancy remains to be clarified.

High velocities occur only rarely and we do not see extreme downdrafts in intergranular lanes as expected for the low photosphere from numerical simulations (Nordlund et al. 1997). Instead, we see approximately the same number of fast upward flows as downward flows. Fast upward flows may occur in small structures. For instance, in the middle of the white box of Fig. 2, we see a small-scale bright point in the intensity image at the spatial resolution limit. The upward velocity coinciding with this structure amounts to about 2 km s-1. Close by, [FORMULA] towards the right, the downflow in the intergranular space is -2 km s-1. Further close inspection of Fig. 2 shows that many of the other fast upflow velocities coincide with the bright borders of granules (de Boer et al. 1992, Rast 1995 and Wilken et al. 1997). One example is clearly seen in the large granule just below the above mentioned bright point. This supports the identification of the cause of the bright granular rims as upcoming hot material at locations of reduced pressure, as suggested from numerical simulations by Steffen et al. (1994).

A word on the elimination of the signal of the 5-min oscillations is appropriate. Deubner (1988), Espagnet et al. (1995) and Hirzberger (1998) demonstrated how to properly disentangle oscillations in the 5-min period range and at shorter periods from granular flows, via filtering in the [FORMULA] plane. Yet here, we do not deal with a time sequence and thus only a filtering in space is possible. According to Espagnet et al. (1996) the 5-min oscillations occur at wavelengths [FORMULA] 4"-8" (see e.g. Fig. 8 in Espagnet et al. 1996). Thus, since we are interested in the very small scale dynamics, the influence of the 5-min oscillations after only spatial smoothing should be low, although shorter period waves at small scales may destroy the granular intensity - velocity correlations (see Espagnet et al. 1995 and the discussion below). In any case, time sequences are certainly needed in the future.

3.2. Statistical analysis

Fig. 3 gives histograms of the residuals of the [FORMULA] smoothed velocities (solid) and of the residuals of the likewise smoothed intensity fluctuations (dashed). The intensities are far from exhibiting a normal distribution, the granular intensity pattern is non-Gaussian. Although Fig. 2 shows also a clear distinction between isolated upflow regions and contiguous regions of downflows, the velocity histogram possesses only a small skewness. Apart from this it is well fitted by a Gaussian distribution [FORMULA] with V = 0.813 km s-1 = [FORMULA] and [FORMULA] = 0.575 km s-1 from above (dotted curve in Fig. 3). As already mentioned, we do not see, at heights of 50-200 km, fast downdrafts at intergranular regions expected from numerical simulations for the deep photosphere (Nordlund et al. 1997). The reason for this "non-detection" may be found from simulations of the granular convective dynamics by Solanki et al. (1996, Fig. 1 there) and Gadun et al. (1999, Fig. 2.). There it is seen that the difference between the mild, broad upflows and the strong, narrow downflows in deep layers is dissolved at larger heights where downward flows have become broader and cover a larger area.

[FIGURE] Fig. 3. Histograms of velocity (solid, lower abscissa) and intensity (dashed, upper abscissa). The dotted curve is a Gaussian with a "macroturbulent" velocity of V = 0.813 km s-1.

Power, coherence, and phase spectra of the white light image are depicted in Fig. 4a and b. They are azimuthal averages in the (horizontal) [FORMULA] plane, from the non-smoothed data, giving high statistical significance. The straight lines in the log-log representations of the power spectra (Fig. 4a)) indicate the exponential decays for an inertial convective range (-5/3) and an inertial conductive range (-17/3) in isotropic turbulence (cf. Espagnet et al. 1993 and Muller 1999). The vertical positions of the straight lines were chosen freely to come close to the power spectra measured here. While the overall agreement to these theoretical slopes is not too bad, the smooth decrease of the power spectra in Fig. 4a does not necessarily suggest such kind of turbulence and looks different to the results of Muller (1999). Also, a distinct change of the slope near [FORMULA], as in Espagnet et al. (1993) does not appear in our high spatial resolution data. If at all, a change of slope may occur for the velocity power spectrum at structural wavelengths of about [FORMULA]. Nordlund et al. (1997), from considerations of fluid dynamics and from numerical simulations, have presented reasons that granular convection of the stratified solar gas should be essentially laminar and not compatible with isotropic turbulence. The sharp decline of the power below [FORMULA] is due to our optimum filter and does not reveal any physical relevance.

[FIGURE] Fig. 4. a  Power spectra of velocity (solid) and intensity (dashed) fluctuations on a log-log scale. For the straight lines (-5/3 and -5/17) see the text. b  Coherence (solid) and phase difference (dashed) between intensity and velocity.

In Fig. 4b the coherence between velocity and intensity fluctuations drops below 0.5 at horizontal wavenumbers [FORMULA] = 11 Mm-1 ([FORMULA]). Thus the coherence between the granular intensity pattern and the flow stays high to much smaller structures than observed hitherto, e.g. with the MSDP by Espagnet et al. (1995) or from excellent slit spectrograms by Deubner (1988). The coherence of motions with intensity at small scales is also much higher than in the data presented earlier by Wiehr & Kneer (1988) also from one-dimensional spectrograms. Likewise, the phase difference stays stable at 0o up to [FORMULA] 17.5 Mm-1. This corresponds to [FORMULA], which is approximately the estimated spatial resolution of the velocity map. We agree with Deubner's (1988) notion that the loss of coherence may be caused by a change of character of the flow due to the appearance of internal gravity waves. Yet, if so, our observations show that this occurs at substantially smaller scales than [FORMULA]-[FORMULA] proposed by Deubner.

Turbulence in a gravitationally stratified atmosphere is in itself an interesting phenomenon. Even if it is difficult to see directly turbulence on the Sun from velocity measurements, at least with the limitations posed by the spatial resolution of today's telescopes, we may still ask under which conditions and where turbulent flows occur.

Thus, finally, we search for large velocity differences at short spatial distances. The places where these occur are the most likely regions where turbulence and short period waves are generated (see e.g. Nesis et al. 1997, 1999, for discussions of turbulence and Musielak et al. 1994, for wave generation). From the reasoning by Nordlund et al. (1997) it is suggested that the regions of intergranular downflows are the best candidates for turbulence generation due to strong velocity gradients. Therefore, we start with the downflows and divide the field of view into squares of [FORMULA] and simply locate in each square the minimum velocity, omitting the according positions if the velocity at any pixel next to the square border is still smaller. We then ask for the number of occurrences of velocity differences, in steps of 0.5 km s-1, in circular areas of [FORMULA] and [FORMULA] radius about the minimum positions. Still larger scales are easy to implement but of little interest since we search for large velocity changes on short distances. Fig. 5 gives the corresponding histograms. It is seen that large differences up to 4-5 km s-1 do occur, but only rarely, only 10 times for the search within the [FORMULA] radius circles. They are a very intermittent phenomenon in the [FORMULA] field of view. Smaller differences of 2-3 km s-1 are seen more often, 930 times within the small [FORMULA] circle search, or about 1.25 times per 1"[FORMULA]1".

[FIGURE] Fig. 5. Occurrences of velocity differences between velocity minima in squares of [FORMULA] and ambient velocities in circles of [FORMULA] radius (solid-thick) and [FORMULA] radius (solid-thin) centered at the minimum position. The dashed-thick and dashed-thin distributions are the occurrences after reshuffling randomly the positions within the field of view of the velocity data for circles of 1:000 radius and 0:005 radius, respectively. (See the text).

We show also in Fig. 5 the occurrence of velocity differences after randomly reshuffling the positions of the velocities, for the same circles (0:005 and 1:000). This is a test on how much the occurrence is influenced by the data and by the method. The differences vs. the result with the original flow pattern are seen to be large. Randomly reshuffled positions of the velocities, but otherwise with the identical distribution as in Fig. 3 (solid curve) result in an order of magnitude larger numbers of occurrence in the middle range of velocity differences than the original data. The distribution of velocity differences is thus an intrinsic property of the granular flow.

The relevance of such velocity gradients must be further investigated in comparison with numerical simulations. And it will certainly be interesting and important to observe with still higher spatial resolution, that is with a telescope with larger aperture than that of the VTT.

If we search for velocity differences from the maximum velocities, the histograms look very similar, which is not obvious a priori . The reasons are very likely the same as given above for the nearly Gaussion velocity distribution: Numerical simulations show that at layers of 50-200 km both up- and downflows occur more equally distributed than at the bottom of the photosphere.

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© European Southern Observatory (ESO) 2000

Online publication: August 23, 2000
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