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Modeling Magnetic Reconnection in a Complex
Solar CoronaDana Longcope
Montana State University
&
Institute for Theoretical Physics
The Changing Magnetic Field
TRACE 171: 1,000,000 K
8/10/01 12:51 UT
8/11/01 17:39 UT
8/11/01 9:25 UT (movie)
THE CORONAPHOTOSPHERE
Is this Reconnection?
TRACE 171: 1,000,000 K
8/10/01 12:51 UT
8/11/01 17:39 UT
8/11/01 9:25 UT (movie)
THE CORONAPHOTOSPHERE
Outline
1. Developing a model magnetic field2. A simple example of 3d reconnection3. The general (complex) case ---
approached via variational calculus.4. A complex example
Stress applied at boundary
Concentrated at X-point to form current sheet
Reconnection releases energy
How it’s done in 2 dimensions
The Magnetic ModelPoles Converging: v = 218 m/secPotential field: - bipole - changing
1.6 MegaVolts (on separator)
Mx 101.1 19
Mx/sec 106.1 14Mx 106.0 19
Reconnection Energetics
Flux transferred intermittently:
Current builds between transfers
Minimum energy drops @ transfer:
LI /LE /2
21
Post-reconnection Flux Tube
Flux Accumulated over
Releases stored Energy
Into flux tube just inside bipole (under separator)
Mx 108.1 17
min. 20t
ergs 106.1 26 IE Projected to bipole location
Post-reconnection Flux Tube
Flux Accumulated over
Releases stored Energy
Into flux tube just inside bipole (under separator)
Mx 108.1 17
min. 20t
ergs 106.1 26 IE
Minimum Energy: Equilibrium
• Magnetic energy
• Variation:
• Fixed at photosphere:
Potential field
V
xdW 32||2
1A
),()0,,( yxfyxBz
0)( A
)()()( xAxAxA 0W
Minimization with constraints
• Ideal variations only
force-free field
• Constrain helicity
( w/ undet’d multiplier constant- fff
BA
0)( BB
V
xdH 3)()( xBxA
BB
Domain fluxes
• Domain Dij connects sources Pi & Nj
• Flux in source i:• Flux in Domain Dij
• Q: how are fluxes related:
A: through the graph’s incidence matrix
�
A
iji
ReconnectionRelated to circuit in the domain graph
Must apply 1 constraint to every circuit in graph
Separators: where domains meet
4 distinct flux domains
Separator at interface
Closed loop encloses all flux linking P2N1
Minimum W subj. to constraint
Constraint on P2N1 flux
Current-free within each domain
current sheet at separator