Modeling Magnetic
Reconnection in a Complex
Solar Corona
Dana Longcope
Montana State University
&
Institute for Theoretical Physics
The Changing Magnetic Field
PHOTOSPHERE
THE CORONA
TRACE 171: 1,000,000 K
8/10/01 12:51 UT
8/11/01 9:25 UT (movie)
8/11/01 17:39 UT
Is this Reconnection?
PHOTOSPHERE
THE CORONA
TRACE 171: 1,000,000 K
8/10/01 12:51 UT
8/11/01 9:25 UT (movie)
8/11/01 17:39 UT
Outline
1. Developing a model magnetic field
2. A simple example of 3d reconnection
3. The general (complex) case --approached via variational calculus.
4. A complex example
The Sun and its field
Focus on the
p-phere
And the corona
just above
Modeling the coronal field
Example: X-ray bright points
EIT 195A image of
“quiet” solar corona
(1,500,000 K)
Example: X-ray bright points
Small specks occur above
pair of magnetic poles
(Golub et al. 1977)
Example: X-ray bright points
When 2 Poles Collide
All field lines from
positive source P1
All field lines to
negative source N1
When 2 Poles Collide
Regions overlap when poles approach
How it’s done in 2 dimensions
Stress applied at
boundary
Concentrated at
X-point to form
current sheet
Reconnection
releases energy
A Case Study
TRACE & SOI/MDI observations 6/17/98
(Kankelborg & Longcope 1999)
The Magnetic Model
19


1
.
1

10
Mx
Poles

Converging: v = 218 m/sec
Potential field:
19
- bipole
  0.6 10 Mx
- changing 
  1.6 1014 Mx/sec
 1.6 MegaVolts
(on separator)
Reconnection Energetics
Flux transferred intermittently:

Current builds between transfers
I   / L
Minimum energy drops @ transfer:
E   / L
1
2
2
Post-reconnection Flux Tube
17


1
.
8

10
Mx
Flux
Accumulated over
t  20 min.
Releases stored Energy
E  I   1.6 10 ergs
26
Into flux tube just inside
bipole (under separator)
Projected
to bipole
location
Post-reconnection Flux Tube
17


1
.
8

10
Mx
Flux
Accumulated over
t  20 min.
Releases stored Energy
E  I   1.6 10 ergs
26
Into flux tube just inside
bipole (under separator)
A view of the model
More complexity
Defines
connectivity
Find coronal
coronal field
From p-spheric
field (obs).
Minimum Energy: Equilibrium
• Magnetic energy W  1  |   A |2 d 3 x
2V
• Variation:
A(x)  A(x)  A(x)
 W 0
• Fixed at photosphere: Bz ( x, y,0)  f ( x, y)
 Potential field
  (  A)  0
Minimization with constraints
• Ideal variations only
 force-free field
 A    B
(  B)  B  0
H   A ( x)  B ( x) d x
• Constrain helicity
V
( w/ undet’d multiplier a )
 constant-a fff
B a B
3
A new type of constraint…
…flux in each domain
Photospheric field: f(x,y) -- the sources
Domain fluxes
•
•
•
•
Domain Dij connects sources Pi & Nj
Flux in source i: 
i
Flux in Domain Dij  ij
Q: how are fluxes related:
A: through the graph’s incidence matrix
  
A   
The incidence matrix
•Ns Rows:
sources
•Nd Columns: domains
 Nc = Nd – Ns + 1 circuits
The incidence matrix
3   34  36
  
A   
Reconnection
possible allocation of flux…
Reconnection
… another possibility
Reconnection
Related to circuit in the domain graph
Must apply 1 constraint to
every circuit in graph
Separators: where domains meet
4 distinct flux domains
Separators: where domains meet
4 distinct flux domains
Separator at interface
Separators: where domains meet
4 distinct flux domains
Separator at interface
Closed loop encloses
all flux linking
P2N1
Minimum W subj. to constraint
Current-free within
each domain
Constraint on P2N1 flux
 current sheet at separator
Minimum W subj. to constraint
2d version: X-point
@ boundary of 4 domains
becomes current sheet
A complex example
Ns = 20
A complex example
Ns = 20  Nc = 33
The original case study
Approximate p-spheric field using
discrete sources
The domain of new flux
Emerging bipole
P01-N03
New flux connects
P01-N07
Summary
• 3d reconnection occurs at separators
• Currents accumulate at separators
 store magnetic energy
• Reconnection there releases energy
• Complex field has numerous separators