Hexagonal generalisation of Van Siclen’s information entropy - Application to solar granulation Stefano Russo Università di Tor Vergata – Dipartimento di Fisica

Granulation 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 Technical data: wave lenght 468 ± 5 nm; exposure time 0.014s. The time series covers 35 min. the field of view is 10  10 Mm 2.

 thermal expansion coefficient d 3 convective cell volume cinematic dissipation coeff. k thermal diffusivity coeff. Convection   Lab experiments showed a new convective regime at high Rayleigh numbers (R>10 7 ). Parameters to describe the convective regime: 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): fully developed turbulence with a hierarchy of “eddies” fully developed turbulence with a hierarchy of “eddies” quasi-local, diffusion-like transport quasi-local, diffusion-like transport flows driven by local entropy gradient flows driven by local entropy gradient New paradigm (lab & numerical experiments): turbulent downdrafts, laminar isentropic upflows turbulent downdrafts, laminar isentropic upflows flows driven by surface entropy sink (radiative cooling) flows driven by surface entropy sink (radiative cooling) larger scales (meso/super granulation) driven by compressing and merging larger scales (meso/super granulation) driven by compressing and merging Spruit, H.C., 1997, MemSAIt, 68, 397 A new paradigm

Convection 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 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 spectrum The most known method to characterise regularities in a system is the power spectrum: This method is not usable in the granulation case: Å. Nordlund et al.: 1997, A&A 328, 229.

Geometrical properties of an hexagonal and square lattice Adjacency Adjacency Orientation Orientation Self-similarity Self-similarity

Hexagonal generalisation In order to utilise the isotropy properties of the hexagonal lattice, we have to: q represent the images with hexagonal pixels; q 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 L h horizontal dimension of the rosette.

512 images  t = 9.4 s exposure time: 8 ms 200 x 200 pixels Pixel scale: 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 Telescope The DST1996 series:

Results for single granulation images (a) (b) (c)(d) a b c d

Higher scales of clustering The average of the H’(r) shows a small bump near 7.5 Mm.

Granulation Entropy The Sun’s surface is like a newspaper page!!!

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).