COMPLEXITY IN SOLAR ACTIVE REGIONS

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COMPLEXITY IN SOLAR ACTIVE REGIONS

• Loukas Vlahos
• Department of Physics
• University of Thessaloniki, Greece

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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

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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

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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.

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THE CORONAL PART OF ACTIVE REGIONS RESPOND TO
THE EVOLUTION OF THEIR PHOTOSPHERIC BOUNDARY
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Active region formation
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Key observations to constrain the models

• Size distribution of active regions
• 1.9ltklt2.1 (see Howard 1996)

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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

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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)

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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.

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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

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Statistics of sub-photospheric evolution of
magnetic fields

• See Bogdan and Lerche (1985)
• There is considerable
• work published on the
• filamentary MHD
  

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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)

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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.

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A Cellular Automaton Model based on
percolation(models 2/3)

• See Wentzel and Seiden (1992), Seiden and Wenrzel
  (1996)

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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
  Dm(mobility) free motion on the grid.
• Diffusion due to submergence Dd (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

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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 (PxDmxDd
  (0-1)X(0-1)x(0-1)

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Results

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

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A basic portrait
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Size distribution

• k2.05

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Fractal correlation dimension

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

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Energy release

• Cancellation of flux due to collisions of
  opposite flux releases energy

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Peak flux frequency distribution

• a2.24

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Waiting Time Distribution
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Is the statistics of the size distribution
correlated to the energy release statistics?
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A movie on the active region evolution and
magnetic field cancellation
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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

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Magnetic field evolution in the corona(A 3-D MHD
simulation)

• Ake Nordlund and Klaus Galsgaard (1996)

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Similar results from the SOC theory

• Vlahos, Georgoulis, Isliker, Anastasiadis see
  also review by Charbonneau et al. (2001)

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Connection of CA to MHD

• Equations used

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A movie from the SOC and TRACE

• ..\..\..\movie_flare.mpg
• A TRACE movie

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Fractal properties of the unstable current regions

• McIntosh et al (2002) (DF?1.8-2.0)

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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

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Old paradigm

• Let us leave behind these nice historic cartoons
  and search for a new one to replace them

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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/phot
  ospheric evolution are imprinted on the evolving
  and changing current sheet in all levels of the
  corona

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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
  (1991-92)

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Levy flights in velocityan anomalous diffusion
in velocity space
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Combine magnetic turbulence and E-field

• Magnetic turbulence are trapping the particles
  for Energies EltEc
• Electric fields heat the particles up to Ec and
  freely accelerate them above Ec

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Velocity Distribution above cut off
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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)

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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