Hexagonal generalisation of Van Hexagonal generalisation of Van Siclen’s information entropySiclen’s information entropy
--Application to solar granulationApplication to solar granulation
Stefano RussoStefano RussoUniversità di Tor Vergata – Dipartimento di FisicaUniversità di Tor Vergata – Dipartimento di Fisica
GranulationGranulation
Evolution of an “exploding granule”. the dimension of each box are approximately of 5’’5’’, the whole sequence is of 13.5 min. Hirzberger et al. (1999)
Set of images obtained trough a fast frame selection system, at the SVST (La Palma) on the 5-6-1993. Technical data: wave lenght 468 ± 5 nm; exposure time 0.014s. The time series covers 35 min. the field of view is 10 10 Mm2.
thermal expansion coefficientd3 convective cell volume cinematic dissipation coeff.k thermal diffusivity coeff.
ConvectionConvection
Lab experiments showed a new convective regime at high Rayleigh numbers (R>107).
Parameters to describe the convective regime:
k
TdgR
3
1)0(
vF
FNu
diff
conv
F. Heslot et al.: 1987, Phys. Rev. A 36, 12.
Granule as classic convective cell
Convection guided by surface instability
Old paradigm (mixing-length model): Old paradigm (mixing-length model): fully developed turbulence with a fully developed turbulence with a
hierarchy of “eddies” hierarchy of “eddies” quasi-local, diffusion-like transport quasi-local, diffusion-like transport flows driven by local entropy flows driven by local entropy
gradientgradient
New paradigm (lab & numerical New paradigm (lab & numerical experiments): experiments):
turbulent downdrafts, laminar turbulent downdrafts, laminar isentropic upflows isentropic upflows
flows driven by surface entropy sink flows driven by surface entropy sink (radiative cooling) (radiative cooling)
larger scales (meso/super larger scales (meso/super granulation) driven by compressing granulation) driven by compressing and mergingand merging
Spruit, H.C., 1997, MemSAIt, 68, 397Spruit, H.C., 1997, MemSAIt, 68, 397
A new paradigmA new paradigm
Convection and orderingConvection and ordering
The resulting pattern after an average operation resembles that observed in Rayleigh-Bénard convection experiments.
Rast (2002) showed as, applying the same average operation on a random flux field, it is possible to derive the same geometrical shape.
It seems to be present a kind of self-organization in the photosphere. (Getling & Brandt, 2002)
Granular pattern averaged for 2 hours. The intensity rms contrast is of
2.9%
It is necessary to determine a objective criterion in order to
individuate a possible ordering of the granular structures
Segmentation and statistical methodsSegmentation and statistical methods
Structures individuation:
Da Prima lezione di Scienze cognitive – P. Legrenzi, 2002, Editori Laterza
It is necessary to individuate a statistical method to
correctly characterise the structures distribution
Segmentation based on the borders slope
Segmentation based on a dynamical threshold
Power spectrumPower spectrum
The most known method to characterise regularities in a system is the power spectrum:
2
)()(
dtetxS tj
This method is not usable in the granulation case:
Å. Nordlund et al.: 1997, A&A 328, 229.
Geometrical properties of an hexagonal Geometrical properties of an hexagonal and square latticeand square lattice
AdjacencyAdjacency OrientationOrientation Self-similaritySelf-similarity
Hexagonal generalisationHexagonal generalisation
In order to utilise the isotropy properties of the In order to utilise the isotropy properties of the hexagonal lattice, we have to:hexagonal lattice, we have to: represent the images with hexagonal pixels;represent the images with hexagonal pixels; modify the shape of the counting sliding boxes.modify the shape of the counting sliding boxes.
A more correct individuation of the lattice constant when the distribution of the structures follows a non-square disposition; higher intensity of the peaks for structures disposed randomly or on a hexagonal way.
Sliding box area: 3m(m-1)+1 with m equal to the side of the rosette.
Total area:
with Lh horizontal dimension of the rosette.
23( 1) 1
4 hL
13 3
3 ( 1) 1 ( 1) 1 (3 ( 1) 1) ( 1) 1( ) 4 4h h
i
m m L m m Lp m
iN i N
512 images
t = 9.4 sexposure time:
8 ms 200 x 200 pixels
Pixel scale:0.123 arcsec/pixel
Observation period:~80 min.
Field of view:18 Mm x 18 Mm
Wave lenght:550 nm FWHM10 nm
Observation: The R. B. Dunn Solar Observation: The R. B. Dunn Solar TelescopeTelescope
The DST1996 seriesThe DST1996 series::
Higher scales of clusteringHigher scales of clustering
The average of the H’(r) shows a small bump near 7.5 Mm.
Granulation EntropyGranulation Entropy
The Sun’s surface is like The Sun’s surface is like a newspaper page!!!a newspaper page!!!
Conclusions:Conclusions:
A more isotropic tool in image analysis has been developed. The peaks disposition of the H’(r) has shown a hierarchy of scales of clustering that we have interpreted as an ordering of the convective structures. A lattice constant has been measured (~1.5 Mm). Granulation images show a typical scale of clustering comparable to the mesogranular scale (~7.5 Mm).