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::

Results for single granulation imagesResults for single granulation images

(a) (b)

(c) (d)

a b c d

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).