Dissipation of Alfv

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Dissipation of Alfvén Waves in Coronal Structures

• Coronal Heating Problem

Tcorona106 K
Tphotosphere6x103K
M.F. De Franceschis, F. Malara, P. Veltri
Dipartimento di Fisica
Università della Calabria
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In the Solar Corona Sgt109
very low dissipation coefficients
How to efficiently are waves dissipated before
they leave the corona?
Energy Dissipation Rate
l characteristic velocity and magnetic field
variation scale
An efficient dissipation is possible if small
scales are created
In a 3D-structured magnetic field small scales
can be efficiently creted by phase-mixing
mechanism Similon Sudan,1986
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The model
?Alfvénic perturbations propagating in a 3D
magnetic field equilibrium structure
?In the Corona
Cold Plasma
B must be a force-free field
?We assumed
(linear force-free field)
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xybase of the Corona zvertical
direction Lperiodicity lenght
?Planar geometry in which the curvature is
neglected
?Statistical homogeneity in horizontal
directions We assumed periodicity along x
and y directions
?
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Equilibrium Magnetic Field
is a superposition of several Fourier components
The choice of these parameters determines a
particular solution of the problem
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?
determines both the current density
and the maximum lenght
In order to respect the statistical homogeneity
so we used
Pommois et al.,1998
?
randomly chosen in the range 0,2p
? The magnetic field is generated by a turbulent
process. Assuming a spectral energy density
We get
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Wave evolution equations in a inhomogeneous plasma

• Alfvénic perturbations propagate in the above
  magnetic equilibrium.
• HYPOTESIS
• Cold plasma
• Small wavelenght with respect to the typical
  lenght scale
• WKB approximation
• Alfvénic perturbations are decomposed as a
  superposition of localized
• wave packets

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Magnetic field at the coronal base

• Red tones indicate the field lines flowing out
  the
• coronal base, while blue tones the flowing in
• Statistic homogeneity respected

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Magnetic field structure

• This figure is obtained by
• planning 70 packet trajectories
• Each line connects a positive
• polarity zone with a negative one
• Some lines follow a brief journey,
• other ones follow longer and more
• complicated trajectories

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• Magnetic Field Topology
• Flux tubes obtained by calculating the magnetic
  lines starting from a small circle at the coronal
  base

• broken flux tube The magnetic
  surface separates into various sheets At break
  points stretching of Alfvénic packets

• compact flux tube The initial
  circle is mapped in a closed curve onto the
  coronal base

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Packet Time Evolution

• The wavevector k
• as a function of time t,
• for a given packet
• Almost exponential growth

• The energy e
• as a function of time t,
• for a given packet, at S105
• Dissipation within few Alfvén times

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Dissipation Time Scaling Law

• The dissipation time td
• as a function of the Reynolds
• number S, for a given packet
• The scaling law
• is asymptotically verified for
• large S

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Conclusions

• Coronal heating due to Alfvén waves dissipation
• Linear force-free magnetic field in equilibrium
  configuration
• (statistic homogeneity hypotesis)
• Evolution equations for an Alfvén waves packet in
  a inhomogeneous
• cold plasma small scale generation
• Magnetic field topology sites of magnetic lines
  exponential separation
• Wave vector increase and energy decrease
• Scaling law of dissipation time
  recovered