COMPLEXITY IN SOLAR ACTIVE REGIONS Loukas Vlahos Department of Physics University of Thessaloniki, Greece

Active regions are open, non-linear dynamical systems Energy enters and escape from all boundaries but the most important boundary is the photosphere... The statistical properties of the formation and evolution of active regions at the photosphere are of importance for the flare energy release

SMALL SCALE VS LARGE SCALE ORGANIZATION AR are formed and developed gradually till they disappear Follow well defined statistical laws Size distribution of AR, fractal dimension have been studied AR made by N-mutually interacting loops, which are never stable and represent the eddy patterns of turbulence in the convection zone

Introduction (a few well accepted facts) The flux tubes during their buoyant rise to the surface are influenced by several physical effects e.g. Coriolis force, magnetic tension, drag and most importantly the convection motion.

THE CORONAL PART OF ACTIVE REGIONS RESPOND TO THE EVOLUTION OF THEIR PHOTOSPHERIC BOUNDARY

Active region formation

Key observations to constrain the models Size distribution of active regions 1.9<k<2.1 (see Howard 1996)

Active regions form fractal structures The geometrical characteristics of the active regions can be represented with a single characteristic correlation dimension See Meunier 1999 and references sited in this article

Statistics of the explosive events Peak intensity distribution of explosive events in the low chromosphere follow also a power law with index (see for example Ellerman bombs, Georgoulis et al. 2002)

Question? Are the sub-photospheric / photospheric / chromospheric/coronal characteristics of the magnetic field evolution independent? Basic working assumption: The Complexity of the magnetic field in active region suggest that all solar phenomena are interdependent and the well known say for the evolution of non-linear systems (attributed to Lorentz) “the sensitivity to the initial conditions in non-liner systems is such that the flopping of the winds of a butterfly in Brazil will influence the weather in New York” apply to all solar phenomena.

Sub-photospheric evolution Let us assume that the convection zone is penetrated with flux tubes (fibrils) with different size and magnetic strength all moving with different speeds towards the surface. Can we cut the 3-D box with a surface and consider that each magnetic tube is represented with a circle with diameter R. Almost 20 years ago Tom Bogdan in his Ph.D pose this question and try to develop the statistical evolution of the “dilute gas” consisted of 2-D fibrils

Statistics of sub-photospheric evolution of magnetic fields See Bogdan and Lerche (1985) There is considerable work published on the filamentary MHD

Vortex attraction and formation of active regions “The magnetic field emerging through the surface of the sun are individually encircled by one or more subsurface vortex rings, providing an important part of the observed clustering of magnetic fibrils..” Parker (1992)

A model based on transport on fractal support and percolation (Model-1) Carl Schrijver and collaborators (1992/1997) presented a model were magnetic field robes are filling a point in this lattice with probability p and then executing random walks on a structured lattice. The flux robe diffuse on a network already structured.

A Cellular Automaton Model based on percolation (models 2/3) See Wentzel and Seiden (1992), Seiden and Wenrzel (1996)

The basic rules for Model-4 (Vlahos, et al, ApJ Letters, 2002) We use a 200x1000 square grid with no magnetic flux (0) We star by filling 0.5 % (+1)positive magnetic flux a 0.5% (- 1) negative. Stimulation probability P: Any active point for one time step stimulate the emergence of new flux in the neighborhood. Newly emerged flux appear in dipoles. Diffusion due to unrestricted random walk D m :(mobility) free motion on the grid. Diffusion due to submergence D d : (submergence of flux) Fast disappearance if the neighboring points are non- active. Spontaneous generation of new flux E: (its value is not important) To keep the process going

The basic rules for Model-4 (Vlahos, et al., ApJ Letters, 2002) Comment: These models are based on two universal principals on the development of complex systems. (A) The continuous fight tendencies : Emergence vs diffusion and (B) Percolation The results are generic and independent on the exact values of the free parameters but the observations constrain their values to a subset of the available 3-D space (PxD m xD d ] [(0-1)X(0-1)x(0-1)]

Results The evolution of active points Are the values of P,D,E unique?

A basic portrait

Size distribution k=2.05

Fractal correlation dimension See also Meunier 1999 for similar results using a variant of Wentzel and Seiden model.

Energy release Cancellation of flux due to collisions of opposite flux releases energy

Peak flux frequency distribution a=2.24

Waiting Time Distribution

Is the statistics of the size distribution correlated to the energy release statistics?

A movie on the active region evolution and magnetic field cancellation

The standard SOC model for flares Loading phase-very important Rule-1: Instability threshold (criticality) Rule-2: Redistribution and energy release Reloading - Either continuous or after relaxation

Magnetic field evolution in the corona(A 3-D MHD simulation) Ake Nordlund and Klaus Galsgaard (1996)

Similar results from the SOC theory Vlahos, Georgoulis, Isliker, Anastasiadis see also review by Charbonneau et al. (2001)

Connection of CA to MHD Equations used

A movie from the SOC and TRACE..\..\..\movie_flare.mpg A TRACE movie

Fractal properties of the unstable current regions McIntosh et al (2002) (D F  )

Wave propagation in a structured active region ( filled with intermittent current sheets sitting on a fractal in 3-D space) Wave propagation reinforces the current sheet and the absorption coefficient of the waves is enhanced by several orders of magnitude

“Old” paradigm Let us leave behind these nice historic cartoons and search for a new one to replace them…

The new paradigm A new model for the energy release seems to be suggested This model has different characteristics from the “old” cartoons The current sheets are driven from the evolution of magnetic fields at the convection zone/photosphere level. Many characteristics of this sub- photospheric/photospheric evolution are imprinted on the evolving and changing current sheet in all levels of the corona

My favorite cartoon (it is time for change of paradigm) although here we must be careful on the same problems I have just mention. Vlahos(1992/1993), Vlahos and Anastasiadis ( )

Levy flights in velocity an anomalous diffusion in velocity space

Combine magnetic turbulence and E-field Magnetic turbulence are trapping the particles for Energies E<E c Electric fields heat the particles up to E c and freely accelerate them above E c

Velocity Distribution above cut off

Summary The turbulent convection zone, through the magnetic fields drives the entire solar atmosphere. The complexity of our system (convection zone/photosphere/chromosphere/corona) is such that only statistical analysis and statistical models can capture its dynamical evolution There is strong correlation between the evolution of photosphere patterns and chromospheric/coronal effects (this is indicated by my k-a dependence)

Summary We need a series of 3-D MHD studies to understand deeper the physical meaning of the free parameters of our CA models and restrict the rules further I believe that we need to start building global solar models using more techniques borrowed from complexity theory. We will make considerable progress only if we understand deeper the interconnection of the elements of our system, this new global understanding has to be reflected even on the drawing of new cartoons…