Peter E. Williams1, W. Dean Pesnell2, John G. Beck3, and Shannon Lee4
1. Northern Virginia Community College, Annandale, VA, USA
2. NASA Goddard Space Flight Center, Greenbelt, MD, USA
3. UCLA Dept. of Astronomy & Physics, Los Angeles, CA, USA
4. San Francisco State University, San Francisco, CA, USA
Supergranules manifest themselves on the solar photosphere as cellular Doppler features that exhibit strong horizontal divergent flows (~300 m s-1) and weak radial upflows (~30 m s-1). Unlike their smaller counterparts, the granules, they tend not to be responsible for transporting thermal energy from within the convection zone to the surface where it is radiated to space. Supergranules exist for around 24-48 hours during which time they grow to around 35 Mm across before breaking up due to the emergence of newer cells. To study how the average supergranule size varies over time, 30 days of 15-minute cadence Doppler images from the Michelson Doppler Interferometer (MDI) were analyzed, which showed that the mean supergranulation diameter varies over time with a fluctuation period of 3-5 days, around 3 times the lifetime of a typical supergranule.
To determine whether the origin of these fluctuations is solar-based or an instrumental artifact, we also analyzed data from the Helioseismic Magnetic Imager (HMI). We selected full-disk Dopplergrams from both HMI and MDI covering June 1–10, 2010, providing nearly 1000 images each.
Figure 1. An example of a power spectrum representing the size distribution of supergranule cells subsequent to Doppler image analysis. This spectrum represents the average of all spectra derived from HMI data. The black dots, representing the power at a given wavenumber, trace out a feature between 100 and 200 wavenumbers relating to the distribution of supergranule sizes. By fitting this feature with a Lorentzian function, represented by the red line, the wavenumber at which it peaks can be deduced.
The analysis began by mapping the images to a heliographic (latitude-longitude) coordinate system, followed by the removal of known large-scale velocity fields, such as differential rotation and spacecraft motion, so that only cellular surface flow signals remained (see  for details). Each image was then analyzed with spherical harmonics to produce a cell size power spectrum, see Figure 1, which exhibited a distribution of power over a range of supergranule cell sizes. This distribution was fitted with a modified Lorentzian function. The wavenumber value, which is inversely proportional to cell size, at which this function peaks, was recorded. By analyzing all images in this way, we produced a time series of these peak wavenumber values, see Figure 2. While our previous work with a longer MDI data set showed that the average supergranule size tends to fluctuate with a period of 3-5 days, our process of comparing similar data using two shorter datasets from HMI & MDI displayed a strong correlation, suggesting that the observations are indeed solar in origin.
Figure 2. Peak supergranule wavenumbers calculated from each size-distribution spectrum, using the fitting method illustrated in Figure 1. This plot shows that while the wavenumber values of HMI (blue) and MDI (red) are offset due to the difference in resolution between the two instruments, the variation in time of both time series exhibits a strong correlation. This confirms that the observed 3-5 day fluctuations are indeed solar in origin.
An additional observation is that the derived mean diameter of supergranules was measured to be smaller by HMI than by MDI (33 Mm versus 38 Mm, respectively) due to the improved resolution of the former. Furthermore, we found the mean horizontal flow velocity within supergranulation tends to fluctuate at a similar rate as their size. This observation was also coherent across both instruments, although the mean velocity measured by HMI was stronger than that measured by MDI (154 m s-1 versus 141 m s-1, respectively).
We tested the influence of stochastic processes by performing similar analyses on simulated Doppler data and also by producing time series models of the growth and decay of supergranules. While both models exhibited size fluctuations, they did not tend to have a regular 3-5 day oscillation component. So while stochastic mechanisms do influence the variation of supergranule characteristics, our results suggest that an underlying harmonic element is also evident.
The mechanisms responsible for the size and kinematic scales of supergranulation are currently unknown. That these characteristics also fluctuate over time adds one more mysterious element to these phenomena.
 Williams, P.E., Pesnell, W.D., 2014, Solar Phys., 289, 1101.
 Williams, P.E., Pesnell, W.D., Beck, J.G., Lee, S., 2014, Solar Phys., 289, 11.
 Hathaway, D.H., 1992, Solar Phys., 137, 15.