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Statistical studies of the evolution of
magnetic fields in the sun
Loukas Vlahos
Department of Physics,
University of Thessaloniki, Greece
([email protected])
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Outline
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Introductory remarks
Key observations
Sub-photospheric evolution of magnetic fields
Formation and evolution of active regions
(photosphere)
Coronal evolution of magnetic fields
Summary
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Introduction
(a few well accepted facts)

Active regions are diagnostics of sub-photospheric
activity
 Active regions reflect (heating and flaring) the
dynamic interaction of magnetic fields with the
turbulent convection zone
 Flux tubes generated initially at the base of the
convection zone rise to the surface by buoyant
forces.
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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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A working hypotheses
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Key observations to constrain the
models
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Size distribution of active regions
N ( A) ~ A k
 1.9<k<2.1 (see Howard 1996)
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Active regions form fractal structures
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The geometrical characteristics of the active
regions can be represented with a single
characteristic correlation dimension
1.3  DF  1.7
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See Meunier 1999 and references sited in this
article
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Statistics of the explosive events
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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)
a
N (E) ~ E
1.5  a  2.5
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Question?
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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 Santorini” apply to
all solar phenomena.
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Sub-photospheric evolution
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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 sphere 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)
N 1 
N

[ru (r , , t ) N ]    [ (r , , t ) N ]   Coll
t r r
T
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There is considerable
 work published on the
 filamentary MHD
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Vortex attraction and
formation of active regions
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“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)
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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)
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See Wentzel and Seiden (1992), Seiden and Wenrzel
(1996)
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The basic rules for Model-4
(Vlahos, Frangos,Isliker,Georgoulis)
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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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Results
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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
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k=2.05
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Fractal correlation dimension
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See also Meunier 1999 for similar results using a
variant of Wentzel and Seiden model.
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Energy release
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Cancellation of flux due to collisions of
opposite flux releases energy E ~ B 2
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Peak flux frequency
distribution

a=2.24
N (E) ~ E
a
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Waiting Time Distribution
P(t ) ~ (t )
2.14
 D 
exp  

 Dmax 
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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 basic rules for Model-4
(Vlahos, Fragos,Isliker,Georgoulis)
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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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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
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Vlahos, Georgoulis, Isliker, Anastasiadis see also review
by Charbonneau et al. (2001)
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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
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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)
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Wave propagation reinforces the current
sheet and the absorption coefficient of the
waves is enhanced by several orders of
magnitude
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The new paradigm
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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
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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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Photos from Skylab/SMM/Yohkoh seem to agree so well with this
cartoon?
Pictures some times may lead you to the wrong conclusions so be careful
how far you push the connection of the visual impression with the energy
release when you form cartoons
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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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Summary
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The turbulent convection zone, through the magnetic fields
drives the entire solar atmosphere.
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The complexity of our system (convection
zone/photosphere/chromosphere/corona) is such that only
statistical analysis and statistical models can capture its
dynamic 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
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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…
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