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A Constrained Tectonics Model for Coronal
Heating

• C. S. Ng A. Bhattacharjee,
• The Astrophysical Journal, March 2008

An introduction to this paper and summary of
results, by Alexander Russell
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Talk Outline

• Authors, Ng Bhattacharjee. lt2gt
• Coronal heating (general). lt5gt
• Coronal tectonics model. lt2gt
• Introducing the paper. lt1gt
• Constrained tectonics model (description). lt1gt
• Constant footpoint drive. lt3gt
• Random footpoint drive. lt2gt
• Energy dissipation rate and h. lt3gt
• Role of reconnection and secondary instabilities.
  lt2gt
• Can this heat the corona? lt1gt

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The Authors

• Chung-Sang Ng
• Research Assistant Professor at Uni. of New
  Hampshire
• Ph.D. Auburn Uni., Alabama (1994)
• B.S. Chinese Uni. Of Hong Kong (1986)
• 40 refereed papers, many with A. Bhattacharjee
• Amitava Bhattacharjee
• Holds 2 professorships at Uni. Of New
  Hampshire
• Ph.D. Princeton, New Jersey (1981)
• B.S. Indian Institute of Technology, Kharagpur
  (1975)
• Hundreds of refereed papers
• One time director of The Center for Magnetic
  Recon. Studies
• Fellow of American Association for the
  Advancement of Science

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Coronal Heating

• The question of what heats the solar corona
  remains one of the most important problems in
  astrophysics.
• -Klimchuk 2006
• Subject of SOHO 15 workshop (St Andrews 2006).
• ADS search for coronal heating returns 1909
  results.
• Problem is to identify understand the
  mechanisms responsible for heating the corona to
  multi-million degree temperatures particularly
  finding the dominant mechanism, in general and in
  specific situations.

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Coronal Heating is a complex problem with many
elements. We shall briefly look at energy source
and conversion mechanism.
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Energy Source for Coronal Heating
Main contender Photospheric motions
granular flows or acoustic waves
Movie Credit Luc Rouppe van der Voort and
Michiel van Noort Swedish Solar Telescope, Royal
Swedish Academy of Sciences
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AC and DC Models

• Can think of photospheric motions as causing two
  broad classes of effect AC or DC.
• AC caused by motions with periods of order less
  than loop transit times wave-like.
• DC caused by motions with longer periods build
  up stresses in magnetic field.
• Both will be present, but question as to which is
  more important for coronal heating still hotly
  disputed!

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Twisting and Braiding of Flux

• DC component of flows acts to twist and braid
  flux ropes.
• Proposed as method of coronal heating by Parker
• The footpoints of the field are continually
  manipulated by the subphotospheric convection in
  such a way that the lines of force are
  continually wrapped and rotated around one
  another.
• Magnetic neutral sheets form, and dynamical
  reconnection of the field takes place.
  - Parker (1983)
• the X-ray luminosity of the Sun is a
  consequence of a sea of small reconnection events
  nanoflares in the local surfaces of
  tangential discontinuity throughout the bipolar
  magnetic fields of active regions
• - Parker (1994)

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Coronal Tectonics

• Parkers idea refined by Priest, Heyvaerts
  Title (2002), to produce an analytical model.
• Recognised importance of magnetic carpet.
• Motion of photospheric flux elements leads to
  formation of current sheets at coronal separatrix
  surfaces named coronal tectonics after
  analogy with tectonic plate motions (and energy
  release - earthquakes) on Earth.

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Ng Bhattacharjee (2008)

• Looking at constrained model 2D.
• Interested in dissipation of energy due to the
  effects of resistivity and viscosity (no
  reconnection or secondary instabilities).
• Highlighted Results
• When coherence time of random footpoint motions
  much less than the resistive diffusion time
  (tcohltlttr), then heating due to ohmic and viscous
  dissipation becomes independent of h.
• Scaling relations suggest that if reconnection
  and/or secondary instabilities were to limit
  buildup of magnetic energy, overall heating rate
  will still be independent of h.

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The Constrained Model

• Rectangular coordinates.
• Straight loops, line-tied at z0,L start from
  B B ez.
• Low b.
• Strong assumption that dynamics depends only on
  x-coordinate shear flows.
• Assume system lies in a periodic box of width
  unity.

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Constant Footpoint Drive

• Paper first examines constant footpoint drive
  limited relevance to coronal heating, but
  analytically tractable and helpfully instructive.
• Analytic steady state
• Obtain potentials

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Constant Footpoint Drive
Paper first examines constant footpoint drive
limited relevance to coronal heating, but
analytically tractable and helpfully
instructive. Analytic steady state Ohmic
dissipation rate is
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Supporting Numerical Simulation
(Constant footpoint motion)
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Random Footpoint Drive

• The velocity field is given a random walk aspect
  through a time-dependent factor.

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Supporting Numerical Simulation
(Random footpoint motion)
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Energy Dissipation Rate and h
For large tcoh, Wd decreases as h
increases. Dependence on h is much weaker than
1/h (corresponds to tcoh infinite).
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Energy Dissipation Rate and h
For small tcoh, Wd is independent of h.
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Understanding Through Scaling Estimates

• Random walk behaviour at zL moves field lines.
• N steps in time t where Nt /tcoh.
• Footpoint moves rms distance lrms(N)1/2vLtcohvL(
  t tcoh)1/2.
• On timescale tr, this becomes lrmsvL(tr
  tcoh)1/2.
• Can use this to estimate average perpendicular
  magnetic field and ohmic dissipation rate

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Will the result change?

• In introducing the constrained model, commented
  that there is no reconnection and no secondary
  instabilities.
• These work to limit the buildup of magnetic
  energy (otherwise By huge for realistic h).
• Would including them reintroduce an h dependence?

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Scaling Estimate Assume buildup of By continues
until By f Bz, in a time t tE. Magnetic
energy built up during this process is then
quickly dissipated in a time much shorter than
tE. Using scaling estimates one obtains,
No dependence on either h or f !
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Finally Can This Heat the Corona?

• Average energy dissipation rate, just obtained,
  corresponds to a heating rate per unit area,
• Which, introducing, By,eff is same as effective
  Poynting flux,
• This rate can be sufficient to match observations
  if vLtcoh/L is in approximate range of 0.25-0.5.
  Equivalently,
• tan-1(By,eff /Bz) in the range around 20 degrees,
  consistent with observations (Klimchuk 2006).

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