NLFFF Energy Measurement of AR8210

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NLFFF Energy Measurement of AR8210

• J.McTiernan
• SSL/UCB

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Some Jargon

• Force-free field JxB 0, current J is parallel
  to B
• Alpha constant of proportionality, J aB
• Linear Force-free field a is a spatial constant
• Non-linear Force-free field a is not constant
• Potential field a 0, J 0, also called
  current-free

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Optimization method Wheatland, Roumeliotis
Sturrock, Apj, 540, 1150
Minimize the Objective Function
With some math, we can write
If we vary B, such that dB/dt F in the volume ,
and dB/dt 0 on the boundary, then L will
decrease.
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Optimization method procedure

• Create a 3d B field in a volume. The bottom
  boundary is the magnetogram, the upper and side
  boundaries are the initial field typically a
  potential field or linear FFF, extrapolated from
  magnetogram.
• Calculate F, set new B B Fdt (typical dt
  1.0e-5). B is fixed on all boundaries. L is
  guaranteed to decrease -- the change in L
  decreases as iterations continue.
• Iterate until ?L approaches 0. The final
  extrapolation idepends on all boundary conditions
  and therefore on the initial conditions.
• Requires a vector magnetogram, with 180 degree
  ambiguity resolved.
• See Schrijver, et al. 2006, Solar Physics 235,
  161 for comparison of methods, and limitations of
  NLFFFs.

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Bx
By
Bz
IVM Vector Magnetograms For this work we used a
series of 21 IVM vector magnetograms, with 180
degree ambiguiuty resolved, supplied by K.D.
Leka. Times range from 1713 UT on 1-may-1998 to
2310 UT. Size is 210x166, with 1.1 arcsecond
pixels.
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Oops, I squared it off, Here is the full
magnetogram, the contours show Bz. Positive is
red and negative is blue. Small arrows show the
direction Bx, By.
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Here are some field lines for the initial
potential field model for the first magnetogram.
The potential field is found using a Fourier
method (Alissandrakis,1981, Gary, 1989),
Boundaries are periodic, but a guard ring of 0s
is put around the original magnetogram.
For the energy calculation, a volume of
210x166x166 was used. Grid spacing is constant,
1.1 arcsec or 800 km.
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Here are field lines, drawn from the same start
points, for the NLFFF extrapolation. They do not
look much different, except that the loops are
higher, and there are a couple of sheared lines,
at lower levels.
The start points are the left hand footpoint of
each field line.
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Given the field, the energy is easy to calculate.
The white line is the total energy in ergs for
the NLFFF extrapolation. The blue line is the
potential field energy. The red and green dashed
lines are start and end times for GOES
flares. The total energy increases by about 10
for this time period. The increase in energy is
the same as in Régnier and Canfield (2006)
although their total energy is 1e33 ergs.
Error bars are calculated using a Monte Carlo
method. The field in the initial magnetogram is
varied randomly using an error estimate obtained
from IVM software (9 G for Bz, 11 G for Bx, By),
and the calculation is repeated for a large
number of times. The error bars are the standard
deviation of the energy calculations. Uncertaintie
s are 1 to 2 in the NLFFF energy.
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This is a plot of the free energy. The free
energy is 22 to 24 of the potential energy.
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Effects of Preprocessing

• Weigelman (2006) algorithm changes the magnetic
  field to minimize JxB force in the magnetogram
  result is a more force-free magnetogram. (The
  photospheric magnetic field is not generally
  force-free.)
• Applied to 8210, the potential field energy is
  reduced, by approximately 11
• The NLFFF energy changes very little, by 1 to 2.
  Weird Coincidence?
• The free energy is larger, by 40 to 60, or 2.2
  to 2.4e33 ergs.

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Effects of Rebinning

• Often we rebin magnetograms, in order to speed up
  the calculation.
• Reducing the resolution of the magnetogram by a
  factor of X results in an increase in speed of
  X4. (This is very useful, for example, for the
  Monte Carlo error bars.)
• If you rebin the magnetogram by a factor of 4,
  from 210X166 to 52X41
• The estimated total energy increases by 24 to
  28, so that the total energy estimate ranges
  from 9.7e32 to 1.04e33 ergs. Also a similar
  increase in potential field energy.
• Similar results using AR9210, 10486.

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Spherical Coordinates

• The NLFFF code works in spherical coordinates
  too.
• For a test, take a PFSS model (Marc DeRosas SSW
  version), and insert the vector magnetogram.
• The vector magnetogram is rebinned, rotated,
  coaligned and poly_2dd to fit onto the PFSS grid
  (there are some as-yet-unquantified uncertainties
  in this process).
• The volume used is much larger, 37 X 37 X 1.5
  Rsun
• The NLFFF energy ranges from 9e32 ergs to 1.05e33
  ergs. Decreases with time
• The PFSS energy Is 7.25e32 ergs
• Free energy ranges from 2 to 3e32 ergs.
• Still working on uncertainties, binning issues

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PFSS on the left
NLFFF on the right The fieldlines
do look different. Particularly on the left
side of the AR.
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Conclusions

• NLFFF total energy ranges from 7.5 to 8.3e32
  ergs, increases overall during time range.
  Régnier and Canfield (2006) found 1e33 ergs, with
  similar 10 increase. Free energy is from 1.4 to
  1.6e32 ergs.
• The small decreases in total energy not well
  correlated with flares..
• Preprocessing decreases potential field energy,
  results in similar NLFFF energy, increased free
  energy.
• Rebinning magnetogram to lower resolution
  increases the energy estimate.
• Spherical coordinate calculation has larger
  energy, may be due to larger volume, rebinning?