1. Extrapolation of magnetic fields
Thomas Wiegelmann
Why study coronal magnetic fields?
How to obtain the coronal magnetic field vector?
Linear and non-linear models.
Computational implementation and tests.
Recent problems and possible solutions.
Evolution of coronal fields and flare prediction.
Outlook: Coronal plasma and dynamics.

2. It is important to investigate the coronal magnetic field
Coronal mass ejections and flares
are assumed to occur due to
instabilities in the coronal magnetic field configuration.

3. Origin of coronal eruptions
Coronal magnetic Fields: Origin of Space weather
Question:
Origin of coronal eruptions

4. Solar magnetic
field measured
routinely only in
photosphere
Aim: Extrapolate measured photospheric magnetic
field into the corona under model assumptions.

5. How to model the stationary Corona?
Lorentz force
pressure gradient
gravity
Neglect plasma
pressure+gravity
Low plasma
Beta in corona
Force-free
Fields

6. Force-Free Fields Further simplifications
Equivalent
Relation between currents and magnetic field.
Force-free functions is constant along field lines, but varies between field lines.
=> nonlinear force-free fields
Further simplifications
Potential Fields (no currents)
Linear force-free fields (currents globally proportional to B-field)

7. Easy to compute Require only
Here: global constant
linear force-free parameter
Easy to compute Require only
LOS-Magnetograms

8. Potential Field Model
EUV-emission
Simple potential field models provide already a
reasonable estimate regarding the global magnetic
field structure. Mainly closed loops in active
regions and open field lines in coronal holes.

9. Active Regions
We use a linear force-free model with MDI-data and have the freedom
to choose an appropriate value for the force-free parameter α.
EIT-image and projections of magnetic
field lines for a potential field (α=0) .
(bad agreement)
Linear force-free field with α=+0.01 [Mm-1]
(bad agreement)

10. Linear force-free field with α=-0.01 [Mm-1]
(good agreement)
3D-magnetic field lines,
linear force-free α=-0.01 [Mm-1]

11. NonLinear Force-Free Fields
Equivalent
Compute initial a potential field (Requires only Bn on bottom boundary)
Iterate for NLFFF-field, Boundary conditions: - Bn and Jn for positive or negative polarity on boundary (Grad-Rubin) - Magnetic field vector Bx By Bz on boundary (Magnetofrictional, Optimization)

12. Grad-Rubin method Sakurai 1981, Amari et al
Grad-Rubin method Sakurai 1981, Amari et al. 1997,2006, Wheatland 2004,06,07

13. Magnetofrictional Optimization Chodura & Schlueter 1981,
Valori et al. 2005
Optimization
Wheatland et al. 2000,
Wiegelmann 2004,2007

14. Test: Model Active Region
(van Ballegooijen et al. 2007, Aad’s model)
Model contains the (not force-free) photospheric magnetic field vector and an almost force-free chromosphere and corona.

15. Optimization
Grad-Rubin
MHD-relaxation
Comparison paper, Metcalf et al., Sol. Phys
-Good agreement for extrapolations from chromosphere.
-Poor results for using photospheric data directly.
-Improvement with preprocessed photospheric data.

16. Force-Free
B-Field
Measurements,
non-force-free

17. Consistency criteria for vectormagnetograms (Aly 1989)
If these relations are NOT fulfilled on the boundary, then the photospheric data are inconsistent with the force-free assumption.
NO Force-Free-Field.

18. Preprocessed boundary data
No net force
No net torque
Photosphere
Smoothness

19. Chromospheric H-alpha preprocessing
H-alpha fibrils outline magnetic field lines.
With image-recognition techniques we get tangent to the chromospheric magnetic field vector (Hx, Hy).
Idea: include a term in the preprocessing to minimize angle of preprocessed magnetic field (Bx,By) with (Hx,Hy).

20. Preprocessing of vector magnetograms (Wiegelmann, Inhester, Sakurai, Sol. Phys. 2006)
Use photospheric field vector as input.
Preprocessing removes non-magnetic forces from the boundary data.
Boundary is not in the photosphere (which is NOT force-free).
The preprocessed boundary data are chromospheric like.
Preprocessing can be improved by including chromospheric observations.
(Wiegelmann, Thalmann, Schrijver, DeRosa, Metcalf,
Sol. Phys. 2008)

21. We test preprocessing with Aad’s model

22. 19o 65% 9o 97% 1o 100% Results: Comparison with Aad‘s Model
Angle <B,H> in Chromosphere
Force-free coronal magnetic Energy
No pre- processing
19o
65%
Classical pre-
processing 9o
97%
H-Alpha pre-processing 1o
100%

23. Nonlinear Force-free code
Vector
magnetogram
H-Alpha Image
Optional
Preprocessing tool
Chromospheric
Magnetic Field
Nonlinear Force-free code
Coronal
Magnetic Field

24. Nonlinear force-free Models are superior.
Comparison of observed magnetic loops and extrapolations
from photospheric measurements
Potential field reconstruction
Nonlinear
force-free Models are
superior.
Measured loops in a newly developed AR (Solanki, Lagg, Woch,
Krupp, Collados, Nature 2003)
Linear force-free reconstruction
Non-linear force-free reconstruction

25. Stereoscopy vs. coronal field extrapolation
Hinode FOV
From DeRosa et al. 2009: Blue lines are stereoscopic reconstructed
loops (Aschwanden et al 2008), Red lines nonlinear force-free
extrapolated field lines from Hinode/SOT with MDI-skirt.

26. Stereoscopy vs. coronal field extrapolation
Vector magnetogram data (here: Hinode/SOT) are essential for nonlinear force-free field modeling.
Unfortunately Hinode-FOV covered only a small fraction (about 10%) of area spanned by loops reconstructed from STEREO-SECCHI images.
Quantitative comparison was unsatisfactory, NLFFF-models not better as potential fields here.
In other studies NLFFF-methods have shown to be superior to potential and linear force-free extrapolations. (Comparison with coronal images from one viewpoint, NLFFF-models from ground based data)

27. Results of NLFFF-workshop 2008
When presented with complete and consistent boundary conditions, NLFFF algorithms succeed in modeling test fields.
For a well-observed dataset (a Hinode/SOT-SP vector-magnetogram embedded in MDI data) the NLFFF algorithms did not yield consistent solutions. From this study we conclude that one should not rely on a model-field geometry or energy estimates unless they match coronal observations.
Successful application to real solar data likely requires at least:
large model volumes at high resolution that accommodate most of the connectivity within a region and to its surroundings;
accommodation of measurement uncertainties (in particular in the transverse field component) in the lower boundary condition;
'preprocessing’ of the lower-boundary vector field that approximates the physics of the photosphere-to-chromosphere interface as it transforms the observed, forced, photospheric field to a realistic approximation of the high-chromospheric, near-force free field.
See: Schrijver et al (Spy 235, 161), 2008 (ApJ 675, 1637), Metcalf et al (SPh 247, 269), DeRosa et al. (2009, ApJ 696, 1780).

28. Temporal Evolution of Active Regions
Use time series of ground based vector magnetograms with
sufficient large FOV (Solar Flare Telescope, SOLIS).

29. Magnetic energy builds up and is releases during flare
Flaring AR-10540
(Thalmann & Wiegelmann A&A 2008)
Active Region-10960
M6.1 Flare
Magnetic
energy builds up and is
releases during
flare
Solar X-ray flux. Vertical blue lines: vector magnetograms available
Magnetic field extrapolations
from Solar Flare telescope
Extrapolated from SOLIS
vector magnetograph

30. Comparison of two Active Regions

31. Conclusions
Potential and linear force-free fields are popular due to their mathematic simplicity and because only LOS-magnetograms are needed as input.
Non-linear force-free fields model coronal magnetic fields more accurately [energy, helicity, topology etc.].
Nonlinear models are mathematical very challenging and require high quality photospheric vector magnetograms as input.
We still need to understand the physics of the interface-region between high beta photosphere, where the magnetic field vector is measured, and the force-free corona.
Coronal magnetic field models should be compared and validated by coronal observations.

32. Where to go in corona modeling?
Vector
magnetogram
STEREO
images
3D field
lines
3D EUV
loops
Stereoscopy
Force-free
code
consistent?
Plasma
along magnetic
loops
Scaling
laws
Tomography
compare
3D Force-free
magnetic field
Artificial
images
LOS-integration
Self-consistent
equilibrium
MHS
code
Time-dependent
MHD-simulations