Fractal Structures of Solar Supergranular Cells
U.Paniveni, V.Krishan, Jagdev Singh and R.Srikanth†
Indian Institute of Astrophysics, Koramangala, Bangalore-560034, India
†Optics Group, Raman Research Institute, Bangalore-560080, India
Abstract. We employ fractal analysis to study the complexity of supergranulation structure using the Solar and Heliospheric
Observatory (SOHO) dopplergrams obtained during the solar minimum phase. Our data consists of 200 visually selected
supergranular cells, for which we find a broad, slightly asymmetric dispersion in the size distribution, with the most probable
size around 31:9 Mm. From the area-perimeter relation, we deduce a fractal dimension D of about 1:25. This is consistent
with that for isobars, and suggests a possible turbulent origin of supergranulation. By relating this to the variances of kinetic
energy, temperature and pressure, it is concluded that the supergranular network is close to being isobaric and that it has a
possible turbulent origin. Also we are exploring a possible dependence of fractal dimension of the supergranulation structure
on the solar maximum phase.
Keywords: turbulence- Sun:Supergranulation
Convection is the chief mode of transport of heat in the outer envelops of cool stars such as the Sun. The
convection zone which lies in the sub-photospheric layers of the Sun has a thickness of about 30% of the solar
radius. Here the opacity is so large that energy is carried by turbulent motions rather than by photon diffusion.
The convective motions on the Sun are characterized by two prominent scales: the granulation with a typical size
of 1000 km and the supergranulation with a typical size of 30000 km (Singh et. al 1994 ; Srikanth et. al 2000)
The supergranules are regions of horizontal outflows along the surface diverging from the cell centre and subsiding
flows at the cell borders. Such outflowing regions always show velocity of approach to the observer on the side
close to the centre of the disk and velocity of recession on the side towards the limb. Near the centre of the disk
where the horizontal outflows are transverse to the line-of-sight, there is less dopplershift and so the image is almost
uniformly grey. These high photospheric large convective eddies sweep up any shreds of photospheric magnetic fields
in their path from the declining active regions into the boundaries of the cell where they produce excess heating,
resulting in the chromospheric network. The approximate lifespan of a supergranular cell is 24 hours. Broadly speaking
supergranules are characterized by the three parameters namely length L, lifetime T and horizontal flow velocity vh.
The interrelationships amongst these parameters can shed light on the underlying convective processes.
A relationship between horizontal flow velocity and the size of a supergranular cell has been established as vh µ L1=3
by Krishan et al. (2002). The corresponding dependence of the lifetime T of the supergranular cell on its horizontal
flow velocity is found to be vh µ T0:5. Here T, also the eddy turn-over time is estimated from the relation T = L=vh
with L as the distance from the centre to the edge of the cell (Paniveni et. al 2004).
Fractal analysis is a valuable mathematical tool to quantify the complexity of geometric structures and thus gain
insight into the underlying dynamics. For example, statistical analyses like studies of the size distribution of active
regions or of the fractal dimension of solar surface magnetic fields in the photosphere are useful for comparing
observations and models. They can shed light on the turbulence of the magnetoconvective processes that generate
the magnetic structures (Stenflo, Holzreuter 2003a; Lawrence, Ruzmaikin, Cadavid 1993).
For our purpose, the fractal dimension D is characterized by the area-perimeter relation of the structures (Mandelbrot
1977). Self-similarity, or geometric scale-invariance, is expressed by a linear relationship between logP and logA (Eq.
1) over some range of scales.
Fractal analysis was first applied to a solar surface phenomena by Roudier and Muller (1987), who measured the
fractal dimension of granular perimeters. From Pic du Midi data, they find a fractal dimension D = 1:25 for granular
diameters of size d 100:37 and D 2 for larger granules.
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We analysed 33 hour data of full disc dopplergrams obtained during the solar minimum phase on 28th and 29th
June 1996 by the Michelson Doppler Interferometer (MDI) on board the Solar and Heliospheric observatory (SOHO)
(Scherrer et al. 1995). Also we have analysed 48 hour full disc GONG dopplergrams obtained on 4th and 5th August
2001 during the solar maximum phase.
Both the SOHO and the GONG full disc dopplergram data have been obtained with a resolution of 200 each, which
equals twice the granular scale. Further, the dopplergrams are time averaged over intervals of 10 min, which is
about twice the 5-minute period of oscillations. Thus the signal due to granular velocity is averaged out. Similarly
the contributions due to p-mode vibrations are reduced after the time averaging. Our analysis rests on the implicit
belief that time averaging removes noise significantly, as judged from visual inspection and also as seen in the typical
supergranular velocity profile for our data (cf.Fig. 1 of Paniveni et al.(2004)). After the averaging, the supergranular
network is brought out with a fair clarity. This procedure yielded usually six images per hour of the data. Corrections
due to solar rotation are applied to the dopplershifts. Two hundred well accentuated cells during the solar minimum
phase and 81 distinct cells during the solar maximum phase, lying between 15 and 60 angular distance from the disc
centre were selected.
Restricting to the above mentioned angular distance limits helps us discount weak supergranular flows as well as
foreshortening effects. The solar maximum data is analysed in a smaller range of cell area namely 200- 600 Mm2 for
the fractal dimension depends on the cell area and should be studied over short ranges of area especially in the solar
maximum phase (Meunier 2004).
The profile of a visually identified cell was scanned as follows: we chose a fiducial y-direction on the cell and
performed velocity profile scans along the x-direction for all the pixel positions on the y-axis. In each scan, the cell
extent is taken to be marked by two juxtaposed ‘crests’ (separated by a ‘trough’), expected in the dopplergrams. This
set of data points was used to determine the area and perimeter of a given cell, and of the spectrum for all selected
supergranules. The area-perimeter relation is used to evaluate the fractal dimension.
The main results pertaining to the maximum, minimum, mean, standard deviation and the skewness for cell area
A and cell perimeter P for both solar minimum and solar maximum data are summarized in Table 1 and Table 2
respectively. A large dispersion in the area and perimeter was obtained in both the phases of the solar cycle.
We analyzed planar shapes by analyzing the area-perimeter relation,
P µ AD=2 (1)
The log(A) vs log(P) relation is linear as shown in the lower frame of Figure (1). A correlation co-efficient of 0:92
indicates strong correlation. Fractal dimension D, calculated as (2/slope), is found to be D = 1:3450:082. If we
interchange the log(A) and log(P) axes (upper frame, Figure 1), fractal dimension D here is 2 slope and found to
be D = 1:1360:070. The small difference in D values thus obtained may be because error bars in P and A are not
symmetric. The average over the two methods is D = 1:240:076.
The log(A) vs log(P) relation is linear as shown in the lower frame of Figure (2) obtained for the solar maximum
data. A correlation coefficient of 0:94 indicates strong correlation. Fractal dimension D, calculated as (2/slope), is
found to be D = 1:5630:13. If we interchange the log(A) and log(P) axes (upper frame, Figure 2), fractal dimension.
We sincerely thank Dr. P. H. Scherrer and the SOHO consortium for providing us with the MDI/SOI data. The work
also utilises data obtained by the Global Oscillation Network Group (GONG) program. The particular data used were
from GONG+, for which we are very grateful to Dr.Thierry Corbard of the observatory of Cote0 de Azur.
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