1. NLFFF Energy Measurement of AR8210
J.McTiernan
SSL/UCB

2. Some Jargon:
Force-free field: JxB = 0, current J is parallel to B
Alpha: constant of proportionality, J = αB
Linear Force-free field: α is a spatial constant
Non-linear Force-free field: α is not constant
Potential field: α = 0, J = 0, also called “current-free”

3. 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.

4. 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 + F*dt (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 ΔL 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 NLFFF’s.

5. 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.

6. 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].

7. 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 0’s 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.

8. 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.

9. Given the field, the energy is easy to calculate
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.
Uncertainties are 1 to 2% in the NLFFF energy.

10. This is a plot of the free energy
This is a plot of the free energy. The free energy is 22 to 24% of the potential energy.

11. 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 NLFF 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.

12. 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,

13. Spherical Coordinates:
The NLFFF code works in spherical coordinates too.
For a test, take a PFSS model (Marc DRosa’s SSW version), and insert the vector magnetogram.
The vector magnetogram is rebinned, rotated, coaligned and poly_2d’d 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…

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

17. The small decreases in total energy not well correlated with flares..
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?