We study several large (45-degree square) fields of supergranules for as long as they remain visible on the solar disk (about 6 days) to characterize the dynamics of the supergranular flow field and its interaction with surrounding photospheric magnetic field elements. These flow fields are determined by applying correlation tracking methods to time series of mesogranules seen in full-disk SOI-MDI velocity images. We have shown previously that mesogranules observed in this way are systematically advected by the larger scale supergranular flow field in which they are embedded. Applying correlation tracking methods to such time series yields the positions of the supergranular outflows quite well, even for locations close to disk center.

These long-duration datasets contain several instances where individual supergranules are recognizable for time scales as long as 50 hours, though most cells persist for about 25 hours that is often quoted as a supergranular lifetime. Many supergranule merging and splitting events are observed, as well as other evolving flow patterns such as lanes of converging and diverging fluid. By comparing the flow fields with the corresponding images of magnetic fields, we confirm the result that small-scale photospheric magnetic field elements are quickly advected to the intercellular lanes to form a network between the supergranular outflows. In addition, we characterize the influence of larger-scale regions of magnetic flux, such as active regions, on the flow fields. Furthermore, we have measured even larger-scale flows by following the motions of the supergranules, but these flow fields contain a high noise component and are somewhat difficult to interpret.

This research was supported by NASA through grants NAG 5-8133 and NAG 5-7996, and by NSF through grant ATM-9731676.

1. Motivation and Background

Measurements of the near-photospheric velocity field play a major role in efforts to understand how the turbulent motions within the convection zone transport energy and momentum throughout the interior. Such dynamic processes are responsible for sustaining global features, such as differential rotation and meridional circulation, as well as driving smaller-scale motions, including granulation, mesogranulation, and supergranulation.

This study examines the evolution of supergranular outflows as measured by applying correlation tracking methods to time series of mesogranulation observed by the MDI instrument on SOHO. Mesogranules visible in full-disk velocity images are used as effective tracers to deduce surface flows on supergranular scales and larger. We then use a pattern-recognition algorithm to identify the individual supergranules and trace their histories.

2. Data Processing

For this study, we use full-disk line-of-sight velocity images (2'' pixels) observed by MDI during the 1999 Dynamics Campaign at a cadence of one minute. Several 45-degree-square regions were tracked and remapped onto a lat-lon coordinate system for as long as they remained visible on the solar disk (about 6 days). After attenuating the velocity signals due to overall rotation and acoustic oscillations, each image now contains only the velocity signals of super- and mesogranulation. These steps in the data handling procedure are illustrated in Figure 1.

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Figure 1: The first three steps in the data processing. Figure 1(a) is a sample full-disk image of the sun observed by MDI. Figure 1(b) shows the 45-degree square region indicated in Figure 1(a) after remapping onto a lat-lon coordinate system and removing the overall rotation signal. After filtering the acoustic oscillations from this time series, we obtain the image shown in Figure 1(c). This image contains primarily the Doppler signals of supergranulation and mesogranulation.

After isolating the mesogranules present in these images, the local correlation tracking (LCT) technique is applied to yield the near-surface horizontal flow field. This scheme has been shown to measure supergranular outflows quite well, even for locations close to disk center (DeRosa, Duvall, & Toomre 2000), and especially after temporally averaging the flow fields to reduce the pixel-to-pixel noise generated by the LCT algorithm. To more easily identify the locations of individual supergranules, the horizontal divergence of the flow field is calculated, so that supergranules now appear as local maxima. Figure 2 shows a sample flow field with its associated divergence image. At this stage, we have time series of divergence images from which individual supergranules can be identified.

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Figure 2: The next three steps in the data processing. Figure 2(a) shows the image of Figure 1(c) after applying a high-pass Gaussian filter (FWHM = 10 Mm) to isolate the mesogranules. Figure 2(b) shows an excerpt of the flow map calculated by applying LCT to the time series of mesogranule images from which Figure 1(a) is taken. The flow maps have been 2-hr boxcar averaged to reduce the pixel-to-pixel noise generated by LCT. Figure 2(c) shows the horizontal divergence of this flow map.

3. Supergranule Evolution

To study the evolution of individual supergranules, we identify on each divergence image the pixels surrounding a local maximum as a "supergranular outflow core". These objects are contiguous groupings of pixels for which the local curvature in all directions is negative. The objects are also subject to size, strength, and lifetime cutoffs consistent with the effective spatial and temporal resolution of the time series. By identifying objects which fully or partially overlap on successive images, the evolution of individual supergranules can be studied. Figure 3 shows a flow map overlaid on an image where all objects recognized by the algorithm are indicated in blue.

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Figure 3: A flow map (red arrows) overlaid on supergranular outflow cores (in blue) as identified by the pattern recognition algorithm. Several prominent supergranules possessing well defined outflows are circled in black. The arrow scale is 400 m/s and the tick marks are spaced 20 Mm apart. This image is approximately 23 degrees square.

A typical 45-degree-square image of quiet sun contains 450 +/- 11 supergranular outflow cores identified by the pattern regonition algorithm, covering 20% of the available area. Assuming the supergranular pattern is space-filling, the average area per supergranule is 700 Mm² = (26.5 Mm)².

For the time series containing quiet sun, the size and lifetime histograms of the 8416 objects identified in the six-day dataset are plotted in Figure 4. First, the peak in average areas is approximately 700 Mm², which is consistent with the estimate made earlier. However, there exists no such peak in the lifetime distribution, with lifetime monotomically decreasing with lifetime. In addition, while there are several objects which exist for ~50 hr, these constitute very few of the total spectrum of objects identified. The grand majority of the objects have lifetimes of 10 hr or shorter with the median object having a lifetime of 5 hr. While these objects do not exist as long as the prototypical supergranular lifetime of ~20 hr, they still possess areas large enough to be considered supergranular in size, as shown in Figure 5.

Figure 5 plots the average object size against the lifetime for the 8416 objects analyzed in Figure 4. The main feature is that the longest-lived objects possess a narrower range of areas than shorter-lived objects. There are no objects which exist for 20 hr or longer and which have average areas outside of the range 500 - 1200 Mm² = (22.4 Mm)² - (34.6 Mm)². It is also noted that the largest objects do not live the longest, though the lifetimes of these objects may be underestimated due to their merging and splitting as explained below.

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Figure 4: Average size and lifetime distributions for the 8416 supergranules identified by the pattern recognition algorithm from the six-day quiet sun dataset. The areas plotted here are the areas of the entire supergranule and not just the outflow core as averaged over the lifetime of the object.

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Figure 5: Average area plotted against time for the 8416 objects identified in the quiet sun region.

We believe the significant evolution of the supergranulation pattern, combined with the limitations of the pattern recognition algorithm, may explain this apparent inconsistency between the size and lifetime distributions shown here:

• Large supergranules may possess several cleavage planes which segment the overall supergranule into smaller regions. While such cleavage planes may divide individual supergranules in future images, they may take several hours to become established. Before such lanes become fully established, the algorithm will interpret these subsections as smaller supergranules which may undergo several mergings with and splittings from the main outflow.
• In a realted issue, the merging and splitting rates of the outflows detected in this algorithm isapproximately 50%. Of the 8416 objects analyzed here, only 3926 (47% of 8416) did not undergo a single merging or splitting during its lifetime.
• The pattern recognition algorithm may characterize weak outflows as several objects which appear and disappear because their strengths oscillate around the threshold. These objects show up in the lifetime distribution as several short-lived objects rather than single longer-lived objects. A better algorithm able to recognize such objects may yield a more realistic lifetime distribution.

4. Comparison with Modeling Efforts

Images of radial (vertical) velocity from a model of anelastic convection in a thin, rotating, spherical shell are shown in Figure 6. While such models do not (yet) possess adequate spatial resolution to capture all scales of motion and dissipation that exist in the real sun, the cellular nature of the surface layers is highly suggestive of supergranulation. However, the convective cells shown in the model are approximately twice the spatial size of real supergranules, as indicated by the area spectrum also shown in the same figure. In addition, while the area spectrum in Figure 6 shows no decrease in cell number at small spatial sizes (as does the spectrum of actual supergranulation in Figure 4), there may be evidence that the spectrum is turning over and that the maximum area is around 1000 Mm².

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Figure 6: Images (left and center) of radial velocity from the thin shell models and their associated object area spectrum (right) as calculated using the same method described above that was applied to the MDI data. The center image is a 90-degree-square excerpt taken from the full-sphere image on the left. The image on the left is tilted 35 degrees toward the observer, with the white dashed line indicating the equator. Upflow velocities are positive, and downflows negative. The central image is a 90-degree square region extracted from the left panel centered in latitude at the equator. The color table of the center panel is the same used for the left.

5. Conclusions

We conclude the following:

• A pattern recognition algorithm has been developed to study the evolution of supergranular outflows measured from MDI full-disk velocity images. (Figures 1 - 3)
• The spectrum of supergrnular areas peaks at approximately 700 Mm², with broad wings having a FWHM of 500 Mm². The spatial extent of such objects then corresponds to 19 Mm - 31 Mm across. (Figure 4)
• Lifetime estimates are more difficult to measure due to the evolution of the supergranular pattern. The spectrum measured here decreases monotonically with increasing lifetime. (Figure 4)
• The algorithm used here to catalog the evolution may not treat continual mergings or splittings appropriately. In addition, the algotihm may also mistreat weak outflows which oscillate around the strength threshold. Consequently, the number of cells possessing short lifetimes may be overestimated and the number of longer-lives calls may be underestimated.
• Numerical simulations of convection compare favorably in appearance with the supergranular pattern observed on the sun. While the global models shown here do not possess the spatial resolution high enough to resolve size scale on par with actual supergranulation, the convection in the models does possess the same cellular structure - upflow centers surrounded by downflow lanes - as actual supergranulation. Even the size spectrum may be similar in nature. (Figure 6)

• DeRosa, M., Duvall Jr., T.L., & Toomre, J. 2000, Sol. Phys., 192, 349.