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Astron. Astrophys. 330, 1136-1144 (1998)

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4. Temporal behavior of the mesocells

The temporal properties of the mesocale flows were investigated using two different approaches:

1) temporal correlation coefficients applied to the [FORMULA] and [FORMULA] flow components.

2) identification and tracking of individual features in the divergence pattern.

4.1. Temporal correlation

Following the method used by Brandt et al. (1994), we characterize the mesoscale persistence by the temporal correlation coefficient, applied to the measured [FORMULA] and [FORMULA] flow components as obtained with the windows [FORMULA] 40 min. [FORMULA], [FORMULA] and [FORMULA] 60 min. [FORMULA], [FORMULA]. As in Darvann (1991) and Brandt et al. (1994), we used the exponential form C(t)= C0 exp(-t/T) to get an 1/e decay estimation. Figs. 3a and b show, for each temporal window, the correlation coefficient "r" measured from various combinations of average flow maps as a function of the time difference between the maps. For example, the point at time difference = 10 min. in the Fig. 3 represents "r" measured between average flow maps no. 1 and 2, 2 and 3, 3 and 4, etc, and this for all the different time lags; the three plots in Fig. 3 represent the extrema (minima and maxima) and the average of the correlation coefficient values derived by this method.

[FIGURE] Fig. 3a and b. Decay of the correlation coefficient as a function of time for [FORMULA] a and [FORMULA] b components computed from a LCT temporal window of 40 min. and spatial window od [FORMULA]. The correlation coefficient were computed from various combinations of divergence maps as a function of the time difference between the maps (cf Darvann 1991 and the text). The middle line represents the mean correlation coefficient and the others two lines (top and bottom) are the extrema of the correlation coefficient.

In this correlation coefficient estimation, we selected the 40 and 60 minute temporal windows. This choice is a good compromise between the reduction of the seeing and granulation noise and the need to preserve of the intrinsic evolution of the flow pattern (see Brandt et al, 1994).

Taking, like Brandt et al. (1994), the highest values to represent the correlations which are less affected by residual seeing, we find a characteristic lifetime for [FORMULA] between 86-185 min and for [FORMULA] between 111-144 min. which is from 3 times to 40% less than the values previously obtained by the author cited above. The high scattering of the correlation coefficient values, as noted in Darvann (1991) and Brandt et al. (1994), is the major uncertainty in the estimation of mesogranulation lifetime by this method.


Table 1. Characteristic lifetime T deduced from the [FORMULA] and [FORMULA] flows gives, for two temporal windows 40 min. and 60 min and spatial window [FORMULA].

However, the means for [FORMULA] and [FORMULA] are found around 40 min. and 60 min. respectively, which indicates that the correlation technique is sensitive to the temporal window used in the LCT. This method is a global statistical approach in which the correlation coefficient is directly computed from frame to frame so that ([FORMULA], [FORMULA]) are located at exactly the same (x,y) coordinates of the correlated sequence. Thus, it is natural that this method, computed over the entire field of view, should be affected by the temporal window. As will be shown below, directly tracking positive divergence by following the non negligible motions of the mesocell (in a small field) reduces influence of the temporal window (shape of the lifetime distribution) because of the cells' displacements. The [FORMULA] and [FORMULA] components which are indirectly correlated via the divergence values, are found at different locations (x0,y0),(x1,y1)....(xn,yn) during the mesocells' evolution, which tends to reduce the temporal window effects. The local proper evolution of the mesocells also reduces also these effects.

Thus, the correlation coefficient is representative of both the cells' lifetime and their proper motions.

4.2. Direct tracking

To study the temporal characteristics of the mesogranules, we computed the divergence from the horizontal flows. We define mesocells as the positive diverging flows. The temporal and spatial windows used in the present study help to prevent the generation of "artificial" divergence cells (with a scale of mesogranules), produced by strong gradients in the flows. The animated sequence of divergence maps when superposed on the granulation pattern, reveals granular explosion in the positive divergence corresponding to the existence of mesogranules. This increases our confidence in the use of the previous mesocell definition when studying their temporal properties. In order to monitor the persistence of the divergence cells as accurately as possible, a time step of 10 minutes between two consecutive divergence maps was selected. There is some flow pattern overlapping between successive divergence maps, but the experiment revealed that some features evolve independently of the temporal window sizes. The divergence maps of the flow field are made up of cells with a [FORMULA] - [FORMULA] diameter, whose divergences values are found to be around:


Table 2. Computed divergences using different averaging time and spatial windows.

The divergence values resulting from the 40 and 60 min temporal windows are commensurate with the previous results published by G. Simon et al. (1994). In the case of the 20 min temporal window, the divergence value appears sensitive to the granulation expansion. Darvann (1991) has demonstrated that granular evolution is the dominant noise source in the measurement of large scale flows. A rough estimation of the divergence for a symmetrical expansion [FORMULA] -. - [FORMULA] with a horizontal velocity of 1 km/s gives, in our case, a value of 6.2 10 -3 sec-1. Thus, the 20 min temporal window does not seem suitable for our purposes because of the influence of granule expansion. Darvann (1991) demonstrates that a considerable reduction of granular evolution noise may be achieved by averaging proper motion maps over an extended period. This explains the choice, in some previous papers, of studying mesocell properties with temporal windows greater than 40 min (see Brandt et al. 1994), as it makes it possible to sample as many granulation realizations as possible and then to reduce the granulation evolution noise.

Direct tracking of the cells during their evolution seems more suitable for our study. In order to track the cells during their life as accurately as possible, we focused our analysis on the three pairs of temporal and spatial windows given above. The divergence cell lifetime histograms are shown in Fig. 4 for two time sequences, 6 h 40 min and 3 h 00 min (Muller et al. observation 1992), resulting from flow computations with [FORMULA] 20 min [FORMULA] and [FORMULA] (FWHM) windows. The two graphs are quite similar in shape peaking around 30 min with a distribution reaching up to 2 h 20 min. This reveals the existence of a predominent component, probably due to the exploding granules (68% of cells with lifetime [FORMULA] 50 min.), and of a flat component due to the mesoscale in the long lifetimes. Fig. 5 displays the cell lifetime histograms, for the [FORMULA] 40 min [FORMULA], [FORMULA], [FORMULA] 60 min [FORMULA], [FORMULA] and [FORMULA] 40 min [FORMULA], [FORMULA] windows. These distributions peak at around 30 to 40 min, with a respective maximum lifetime of 2 h20 min and 2 h 40 min. These graphs (and the associated integrated histograms), reveal that 70% of the cells have a lifetime [FORMULA]. The highest local maximun of the histogram ([FORMULA] 60 min [FORMULA], [FORMULA] pair) is probably a result of the convolution of the larger temporal window. In Fig. 5, the comparison between the cell lifetime histograms for the window pairs [FORMULA] 40 min [FORMULA], [FORMULA] and [FORMULA] 40 min [FORMULA], [FORMULA] shows that, contrary to our expectation, an increase of the spatial window from [FORMULA] to [FORMULA] shifts the lifetime histograms to the smaller values on the abcissa. It seems that the larger size of the spatial window tends to smooth the effect of a greater number of granules with their proper motions. Thus, the combination of the granule evolutions and motions with a large window seems to reduce the measured cell lifetime.

[FIGURE] Fig. 4. Histograms of the Divergence cell lifetime measured from two different temporal series: the 6 h 40 min solar granulation sequence(observation described in this paper) and the 3 h 00 min solar granulation sequence previously described in (Muller et al. 1992). The flows derived from the two sequences have been determined by using L. November's LCT with a temporal window of [FORMULA] 20 min [FORMULA] and a spatial window of [FORMULA] (FWHM). The solid line represents the distribution of the divergence lifetime obtained from the 6 h 40 min sequence. The dashed line represents the distribution of the divergence lifetime obtained from the 3 h 00 min sequence.

[FIGURE] Fig. 5. Example illustrating the mesocell lifetime histograms obtained using different spatial and temporal windows. The abscissa is the time in minutes. The dotted line is the mesocell lifetime histogram for a temporal window of [FORMULA] 40 min [FORMULA] and a spatial window of [FORMULA]. The dashed line is the mesocell lifetime histogram for a temporal window of [FORMULA] 60 min [FORMULA] and a spatial window of [FORMULA]. The solid line is the mesocell lifetime histogram for a temporal window of [FORMULA] 40 min [FORMULA] and a spatial window of [FORMULA].

We observe that, regardless of the spatial and temporal window sizes the positive divergences represent half of the field with a high stability (49% ([FORMULA] = 1.5)%) during the entire time sequence (6 h 40 min). A surprising result concerns the change in the number of new positive divergence cells over time during our sequence. Indeed, this number varies with a period of 90 to 100 minutes, regardless of the spatial and temporal window sizes (Fig. 6). This variation is less marked for the [FORMULA] 40 min. [FORMULA], [FORMULA] pair. A verification of our data reduction process (alignment, filtering, LCT) revealed that it does not seem to produce a variation on such a time scale. It is thus like that this variation has probably a solar origin, but it has to be confirmed by another long time sequence.

[FIGURE] Fig. 6. Change in the number of new positive divergence cells in the field of view over time during the 6 h 40 min sequence. The solid line represents the number of new diverging cells obtained using the horizontal flows computed with the following LCT parameters: temporal window of [FORMULA] 40 min [FORMULA] and spatial window of [FORMULA] The dashed line represents the number of new diverging cells obtained from the horizontal flows computed with the following LCT parameters: temporal window of [FORMULA] 60 min [FORMULA] and spatial window of [FORMULA]

The direct tracking method allows us to follow divergence cell motions with quite a high accuracy. Fig. 7 shows the measured displacements of the barycentre of cells with a lifetime [FORMULA], over the solar surface. We observe more random mesocell motions in the supergranule than were observed by Muller et al. 1992. These motions do not seem to tend in any particular direction with respect to supergranule boundaries, which have been delineated by NPBs present in our observation and by the flow field averaged over 6 h 40 min (Fig. 8). The horizontal velocities deduced from these proper motions are represented in the histogram (Fig. 9). The values of horizontal mesocell velocity lie between 0.1 to 0.9 km/sec, with a peaked distribution at 0.5 km/sec, which is consistent with previous determinations (Muller et al. 1992, Brandt et al. 1988, Simon et al. 1994). The distance covered by cells during their lifetime is between 1 to [FORMULA] (mean = [FORMULA] [FORMULA]) which corresponds to 25 to 50% of the cells' size.

[FIGURE] Fig. 7. Displacements, over the solar surface, of the barycenter for mesocells with a lifetime [FORMULA]. The tick marks are [FORMULA] apart. Supergranule boundaries have been delineated by the NPBs present in our observation during the sequence. Field of view: [FORMULA].

[FIGURE] Fig. 8. Flow-field calculated by averaging the displacement vector over the entire 6 h 40 min sequence. Note the location of the NPBs present in our observation during the sequence (delineated by the solid line) most of which correspond to the converging flow of the supergranule. The tick marks are [FORMULA] apart. The total field of view is [FORMULA].

[FIGURE] Fig. 9. Horizontal velocity histogram for mesocells with a lifetime [FORMULA]. These velocities are derived from the proper motions of these mesocells as shown in Fig. 7.

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© European Southern Observatory (ESO) 1998

Online publication: January 27, 1998