On the Application of the Boundary Element Method in Coronal Magnetic Field Reconstruction

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On the Application of the Boundary Element Method
in Coronal Magnetic Field Reconstruction
Yihua YAN National Astronomical
Observatories Chinese Academy of Sciences Beijing
100012, China
Work supported by CAS, NSFC MOST Grants.
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HSOS Huairou Solar Observing
Station NAOC National Astronomical
Observatories
HSOS
NAOC
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Tower of Solar Magnetic Field Telescope
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Solar Radio Telescopes at Huairou
1-2 GHz Radio spectrograph
2.84 GHz Radio flux telescope
2.6-3.8/5.2-7.6 GHz Radio spectrograph
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LHCP
RHCP
Solar Radio Bursts for 21 April 2002 Flare/CME
event (20 sec.) (Left Wavelet-processed spectra.
Right Observed)
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Calculated frequency drift for some bursts
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Outline

• Introduction
• Coronal Magnetic Field Modeling
• Boundary Integral Equation
• Numerical Implementation-BEM
• Applications
• Conclusions

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I. Introduction

• Solar flares CMEs etc. are believed due to
  re-organization of coronal magnetic field.
  Magnetic field plays a central role in the solar
  activities.
• At present reliable magnetic field measurements
  are still confined to a few lower levels, e.g.,
  at the photosphere and the chromosphere.
• IR technique may be applied to observe the
  coronal magnetic field (e.g., Lin et al 2000) but
  it is not well-established yet.
• Radio techniques may be applied to diagnose the
  coronal field (e.g. FASR) but assumptions on
  radiation mechanisms and propagations are needed.
  
• Coronal Observations in EUV/UV SXR ?s, etc.,
  provide information on coronal magnetic
  structures.

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Solar Atmosphere
Corona (in EUV) Chromosphere (in
H?) Chromosphere (in Ca II K) Photosphere (in
WL)
Swedish Solar Observatory on La Palma, Spain
(lower three), and with the TRACE (top).
( Credit TRACE web-site)
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Explosive TRACE Loops
Those loop or thread-like structures are believed
to resemble the coronal magnetic field.
However, what we observe are plasmas, not
magnetic field !
( Credit TRACE web-site)
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?Extrapolation from photospheric data upwards is
still the primary tool to reconstructing coronal
magnetic field.

• FF field can provide the required ?E (in excess
  of Epot) for energy release like flares, CMEs,
  etc. Linear models have undesirable properties.
• In the past 40 years, reconstructing the Non-P
  coronal field from boundary data assumes the
  magnetic field to be FF (but see recent attempts
  for non-FF field model, e.g., Gary 2001)
• Practical methods available at present for
  non-constant-? FF fields may be classified into 4
  types FEM (e.g.,Sakurai 1981, etc.), FDM (e.g.,
  Wu et al. 1990, Amari et al. 1999), MHD evolution
  (e.g., Mikic et al 1995, etc.), BEM (e.g., Yan
  Sakurai 2000)
• Here we present results mainly obtained by BEM of
  the non-constant-? FFF problem for flare/CME
  events.
• (cf. reviews by Sakurai 1989, Gary 1990,
  McClymont et al 1997 Wang 1999 etc.)

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II. Coronal Magnetic Field Modeling

• Three steps in dealing with any physical probelm
  by mathematical modelling(1) establish the
  mathematical model of the physical problem(2)
  solve the mathematical model(3) explain the
  physical meaning of the solution.

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Three issues in dealing with a mathematical
model, or the corresponding boundary value
problem (BVP)

• the existence of the solution
• the uniqueness of the solution
• the stability of the solution with respect to the
  boundary conditions.
• If we get positive answers to all these
  questions, the BVP is well-posed. otherwise, it
  is ill-posed.
• For NLFFF in open space above the Sun, it is,
  unfortunately, an ill-posed problem up to now.
• In the literature, however, a NLFFF model is
  often said well-posed, or not, only with respect
  to the third item, without knowing anything about
  the first two items.

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Physical considerations for coronal fields

• Potential field has a minimum energy content. It
  cannot provide free energy for flares and CMEs,
  etc.
• FF field can provide the required excess ?E.
  Linear models, however, have the undesirable
  properties, e.g., infinite energy content.
• In the past 40 years, all methods reconstructing
  the Non-P coronal field from boundary data assume
  the magnetic field to be FF.
• With present computers instruments, it is now
  able to consider general non-constant-? FFF.
• There are some recent attempts for non-FF field
  modeling (e.g., Neukirch 1995, Gary 2001).
• We confine our discussions on practical methods.

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• Force-free field equations

By ? both sides of the FF equation, one has
It indicates that ? is constant along each field
line, though it is in general a function of
positions. It may not be arbitrarily specified in
space. Here it is assumed that the distribution
of ? consistent with such constraints is
known. To quote Prof. Eric Priest (1994)
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Motiviations Approach

• Non-constant-? force-free fields there are
  problems in the modeling (still mathematically
  ill-posed)
• We wish to provide a method that can take into
  account
• finite energy content in open space
• Solvable from data B directly without numerical
  difficulties
• gt Find an BIE representation of the NLFFF (Yan
  Sakurai 2000, Sol. Phys., v.195, p.89)

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Brief Introduction of Numerical Methods

• Finite Difference Method (FDM)
• Original differential equation Luf, is
  discretized directly by difference operator D to
  replace L Duifi
• Finite Element Method (FEM)
• Find a solution of the equivalent variational
  problem min ltLu,vgtltf,vgt
• In general Galerkin method is applied to
  approximate the solution
• Boundary Element Method (BEM)
• Find a solution of the equivalent boundary
  integral equation u??(F ?u/?n- ?F/?n u)d?,
  where F is fundamental solution

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III. Boundary Integral Equation
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B I E
where ?
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Evaluation of integrals
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• MHS Problem(Low,1992)
• ??B?B?F(B??,?) ? ??
• ? B0
• where ?gz is potential,F is a function of
  pressures in parallel and in perpendicular to B,
  ? is a function of position satisfying
• ? ? B0.
• When F is chosen as
• Ff(z) /g Bz, f(z)a exp(-kz)
• On has
• ??B?B f(z) ? Bz ?k
• ? B0

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The BIE Representation for NFF

• Introduce Ycos(?r)/4?r(Yan
  Sakurai,1997,2000),and apply Greens Theorem as
  before
• ?V (Y?2B-B?2 Y )dV
• ?S (Y ?B/?n-B?Y/?n)dS
• One has
• ciBi T (?) ?S (Y ?B/?n-B?Y/?n)dS
• ?V Y ?? f(z) ?Bz ?k dV
• Where
• T (?)?V Y ?2B- ?2B -?? ?B dV0

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IV. Numerical Implementation -BEM

• To solve Eq.(14) numerically, we first subdivide
  S into a number of elements and the approximation
  functions are assumed over each element.
• Both the boundary location, and B ?B/?n
  functions over each element arerepresented by
  the sums of products of nodal values and shape
  functions.

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• an element

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A detailed procedure of the BEM but for the
potential and linear problems can be found in a
tutorial paper in this edited book Yan, Y.,
Yu, Q., and Shi, H. (1993) in Advances in
Boundary Element Techniques, eds. J. H. Kane, G.
Maier, N. Tosaka and S. N. Atluri
(Springer-Verlag Berlin), pp.447 469.
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V. Applications

• Comparison with Low Lou (1990) solution for
  NLFFF (Yan Sakurai, SP, 2000Liu, Stanford
  web-site, 2002)
• NLFFF for AR7321 event in Oct 1992 (Yan
  Sakurai, SP, 1997 Wang et al. SP, 2000 Liu et
  al. SP, 1998)
• NLFFF for AR8100 event in Nov. 1997 (Yan et al.
  SP, 2001 Liu et al. SP, 2002)
• Global PF for AR8210 event in May 1998 (Wang et
  al. ApJ, 2002)
• Global NLFFF for CR1968 (Liu Zhao AGU spring
  meeting, 2002)
• NLFFF for AR9077 event in July 2000 (Yan et al.
  ApJ 2001 SP 2001)

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Low Lou (1990) Model

• Liu (2002) has compared PF, LFF NLFF field
  models with the analytical solution, and he
  concluded that
• Magnetic connectivity in those models appears
  similar and reasonable.
• Obviously, non-linear force-free field has better
  match with the solution in shape.
• Yan Sakurai (2000) gave comparison of the NLFFF
  
• Recently, we evaluate ?-distributions against Low
  Lou (1990) solutions.

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Comparisons of PF, LFF NLFF Fields with Low
Lou (1990) model by Liu (2002).
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Distributions of Field Lines (Yan Sakurai 2000)
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NLFFF for AR7321 event in Oct 1992

• (Wang et al. SP, 2001)

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NLFFF for AR8100 event in Nov. 1997

• (Yan et al. SP, 2001, Liu et al. SP, 2002)

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Formation of Sigmoid-Structure
(Liu et al. 2002, SP)
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Results Twist of EFR loop - 1.44? lt 2.5 ?
(Hood 1991) gt reconnection may transfer
mutual helicity to self helicity
when magnetic helicity is conserved (Priest 1998)
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Global Potential Field Model
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Potential Field Structure for May-2-1998 Event
by the Proposed Method (Wang et al. ApJ 2002)
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• Bald-patch was
• found (Wang
• et al. ApJ 2002)

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Global NLFFF for CR1968

• Liu Zhao (AGU spring meeting Stanford
  Web-site, 2002)
• Evident difference is found. Those areas are
  where the observed vector field replaces the
  calculated potential field. The force-free field
  appears sheared in areas A', D', E' and F' (
  flares and CMEs associated?). The potential
  magnetic field shows additional connectivity in
  areas B' and C', while such connectivities do
  not show up in force-free field configuration.
• Checked with vector magnetic field in those
  areas, we can see the transverse component has
  orientation evidently departure from the
  potential transverse component and this
  orientation implies that those connectivities may
  not exist instead, there are probably magnetic
  interfaces between them.
• This suggests that the vector field is essential
  to calculate magnetic topological structure and
  identify magnetic separatrix that are
  specifically important to understand flares and
  CMEs as such locations provide suitable
  conditions for occurrence of fast reconnection.

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• PF
• NLFFF

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NLFFF for AR9077 event in July 2000

• (Yan et al. ApJ 2001 SP 2001)

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Results TRACE Arcades
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Rope dimensions
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Field Intensity along Rope
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Magnetic Rope and H? Filament
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Rope and 1600A UV Bright Lane
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0119 UT Bz
044253 H?
092810 1600A
102025 H ?
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TRACE movies
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Evolution of Magnetic Field
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Evolution of the Flux Rope
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Evolution of the Flux Rope
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Some Post-Flare Field Lines
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Evolution of TRACE/EUV Loops(Achwanden
Alexander, Sol. Phys. 2001)
(AschwandenAlexander, 2001)
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Evolution of EUV Loops
(Yan,Aschwanden,Wang,Deng, Sol. Phys., 2001)
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Evolution of EUV Loops
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Evolution of EUV Loops
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Evolution of EUV Loops
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Evolution of EUV Loops
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Evolution of EUV Loop Heights
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Final RemarksHow to Understand Equivalent
Transform?
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VI. Conclusions

• The Boundary Integral Equation (BIE)
  representation is prospective in coronal field
  modeling and applications
• Vector magnetograms are essential in coronal
  magnetic field reconstruction
• Stereo observations of vector magnetograms are
  essential in resolving the 180-degree uncertainty
  in transverse fields.
• More realistic models are needed.
• Drs. Y. Liu, Wang H.N., Wang T.J. are
  acknowledged for their computation materials used
  here.