The Minimum Energy Fit

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The Minimum Energy Fit

• The Slowest Motion Required by Induction

Dana Longcope Montana State University
Work supported by DoD MURI grant
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The data
Time-resolved vector magnetograms
000316
000405
slices of MHD simulation (Tetsuya Magara)
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The Induction equation
Vertical induction Eq.
Known from data
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The Induction equation
Vertical induction Eq.
1 Equation 3 unknowns
To be found
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The Induction equation
Vertical induction Eq.
1 Equation 3 unknowns
Horizontal induction Eq.
5 more unkowns
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Solving Induction for v
Introduce unknown scalar potentials
Induction Eq. ? Possion Eq. for f
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Processing data
t2
t3/2

t1
Region w/ info for velocity
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Processing data
t2
t3/2
-
t1
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Solve Poisson Eq.

• f(x,y) inside
• Region
• f0 on bndry
• Inductn eq.
• is exactly
• satisfied

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Finding other components
free fields
Define function to optimize
Solution will have smallest v consistent w/ data
Bonus v .B0
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The Minimization
known
held fixed
dvz
known
dy
Elliptic operator
held fixed
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Magnetogram grid
Bz
Bh
Bz
Bh
Bz
Bh
Bz
Bh
Bz
Bh
Bz
Bh
Bz
Bh
Bz
Bh
Bz
Bh
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Solving for y
vy
y
y
y
y
vx
f
f
f
vz
vz
vz
vy
y
y
y
y
Finite Difference Elliptic operator
f
f
vz
vz
vx
f
vz
y
y
y
y
f
vz
f
f
Variables on staggered mesh
vz
vz
y
y
y
y
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Solving for y
Solve elliptic equation Within strong-field bounda
ry
Current implementation Relaxation method
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Minimizing the Energy
Alternate dy dvz minimization
Relaxation steps
Actual flow
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Comparison of Results
MHD
MEF
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Comparison of Results
MHD
MEF
Fup 3.3 X 1022 cm3/s
Fup 1.5 X 1022 cm3/s
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Real data AR8210

• IVM (U.Hawaii)
• 3 min cadence
• 1.1 resolution
• Ambigty res.
• Canfield et al. 1993
• avg. 5 grams
• Dt 30 min.

Boundary Bz60 G
(Courtesy KD Leka)
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Real data AR8210
dBz/dt (grey) f(x,y) (contours)
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Real data AR8210
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Doppler Flows

• Find uz(x,y) by other means
• (e.g. Doppler measurments)
• Incorporate using new functional

Mismatch w/ Doppler
i.e. find consistent flow with small horizontal
velocities which best matches observations
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Doppler Flows
Use vz(x,y) from solution as Doppler signal
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Data Considerations

• Cadence
• Large Dt ? problems (Dt 15-30 min)
• Co-alignment
• Seeing produces spurious v(x)
• Ambiguity resolution
• tn ? tn1 consistency - important
• Doppler measurements (easily used)
• Best v for magnetized plasma

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Implementing the Algorithm

• Define outer boundary (necessary?)
• (trickiest step at present)
• Solve elliptic equation(s) relaxation?
• (could be made much faster)
• Artifacts near PIL
• (smoothing, lotsa relaxin)
• Use v(x) from previous step(s) as additional
  constraint