1. Can we use nonlinear and selfconsistent models for data analysis? We learned from Prof. Schindler about the importance of nonlinear and selfconsistent models. Karl Schindler also introduced me to observational data (Helmetstreamer and coronal mass ejections observed with SOHO/LASCO) and encouraged me to develop a corresponding 2D MHD model under his supervision. New aspects: - Feed models directly with measured data. - Computations in 3D. Thomas Wiegelmann

2. We cannot measure coronal magnetic fields directly. Extrapolate the coronal magnetic fields from photospheric (vector) magnetograms. Needed: Model assumptions regarding the coronal plasma. Use the reconstructed magnetic field model to support data analysis, e.g., - Coronal images (SOHO, Solar-B) - Doppler maps (SOHO/Sumer) - Stereoscopy (STEREO-mission) - Tomography (STEREO-mission) Example: Coronal magnetic fields

3. ModelMathematicsObservations needed Validity Potential Fields Line of sight magnetogram (Global) current free regions, quiet sun Linear Force- Free Fields LOS magnetogram + observations of plasma structures Local in active regions, low-beta plasma Non Linear Force-Free Vectormagnetogram (3 times more data, ambiguities, noise) Active regions, low beta plasma in low corona MHS Equilibrium Vectormagnetogram + Tomographic Inversion of density Helmet streamer, finite beta plasma, full solar corona Coronal magnetic field models

4. Active regions contain mainly closed magnetic loops. Coronal plasma is trapped in closed loops and causes bright emission. The large scale magnetic field structure in coronal holes is open. The coronal plasma escapes along open field lines (solar wind). and the emissivity in coronal lines is strongly reduced here.

5. EIT 195 Fe XII Formation temperature 1.5 million K EIT 304 He II Formation temperature 60,000-80,000 K Coronal Holes (Wiegelmann & Solanki, Sol. Phys. 2004)

7. The emitting volume filled by gas at that temperature corresponds to the emitted radiation. In CH ~70% at low and ~10% at high temperatures compared with the quiet Sun. Use RTV scaling laws to approximate temperatures T ~ (pL) 1/3

8. 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) Active Regions (Marsch, Wiegelmann & Xia, A&A 2004) 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.

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

10. SUMER Dopplergram in NeVIII ( 77 nm) and a 2-D-projection of some field lines. Mass flux density inferred from Doppler- shift and intensity from SUMER observations. up down

11. Nonlinear force-free fields Why do we need nonlinear force-free fields? - Non magnetic forces are small in the corona. - Potential and linear force-free fields are to simple to estimate free energy and magnetic topology. The computation is much more difficult: - Mathematical difficulties due to non-linearity. - Vector magnetograms are not force-free. - Transversal B-field is very noisy. - Limited field of view for current instruments (SFT/Tokyo, VTT/Tenerife, IVM/Hawaii)

12. Grad-Rubin Method: ( Grad & Rubin 1958; Sakurai 1981) - Compute potential field, and then specify currents along some of the fieldlines. - These currents generate additional fields, so recompute B using Ampère’s Law. - Iterate until the currents and fields reach a stable equilibrium. Magnetofrictional Method : ( Choudra & Schlüter 1981) - Solves a reduced MHD system of equations. -When combined with the Lorentz force term in the momentum equation, a velocity is prescribed which drives the system toward a force-free state. Optimization Method: (Wheatland et al. 2000) - Minimize a functional of quadratic forms. Greens function method: (Yan & Sakurai 2000) - Boundary-integral equation representation. Blind algorithm test (Schrijver et al.2006): Optimization approach most accurate and fast. Methods to extrapolate nonlinear force-free fields from boundary data.

14. Preprocessing of non-force-free and noisy magnetograms. (Wiegelmann, Inhester, Sakurai 2006, using force-free consistency criteria developed by Aly 1989)

16. Test the nonlinear force-free extrapolations with Low&Lou 1990 equilibrium. a: Reference, b: Potential field, c: Reconstruction from noisy data. d: Reconstruction from preprocessed noisy data

17. Preprocessing of vector magnetograms helps to diminish inconsistencies and noise. (Wiegelmann, Inhester, Sakurai, Sol. Phys. 2006)

18. Measured loops in a newly developed AR (Solanki, Lagg, Woch, Krupp, Collados, Nature 2003) Potential field reconstruction Linear force-free reconstructionNon-linear force-free reconstruction Comparison of observed magnetic loops and extrapolations from the photosphere with different models.

19. We compared measurements of magnetic loops in a newly developed active region with extrapolations from the photosphere. We got the best agreement of measured and extrapolated loops for a non-linear force-free magnetic field model. (Wiegelmann, Lagg, Solanki, Inhester, Woch, A&A 2005)

21. Magnetic fields and coronal tomography a)Use only line of sight density integrals. (Wiegelmann and Inhester, Sol. Phys., 2003) b)Use only magnetic field data. c)Use both line of sight density integrals and magnetic field as regularization operator.

22. Launched 25.October 2006. Near Future: STEREO-mission Two almost identical spacecrafts will observe the Sun and solar wind Spacecrafts have different orbits due to Swing-by at moon. The two STEREO spacecrafts separate about 44 o every year. More than 2GB data every day. PhD students at the MPS in Lindau: - Li Feng: Stereoscopy of Active Regions. - Peng Ruan: Global model of the corona. Markus Aschwanden: Since the STEREO mission is our first extensive multi-spacecraft 3D exploration of our heliosphere, its importance might be compared with the first determination of the true 3D geometry of our Earth globe, thaught by Thales of Milet and Pythagoras around 600 BC.

23. 2000-3-1 14:22 3-2 17:44 Li Feng: Stereoscopy of Active Regions TRACE EUV images Classical Stereoscopy Magnetic Stereoscopy Optimal linear force-free model (black) and coronal loops (red) from two viewpoints.

24. Peng Ruan: Global coronal modeling Temperature distribution in the solar corona cut through longitude Ø=40 o and 220 o Compute global magnetic field with different models: - Potential and LFF fields -MHS (Neukirch 95 model) Compute the coronal plasma with the help of Scaling Laws (Schrijver et al 2004) or selfconsistently from MHS model. Compare artificial coronal images with real images from 2 Stereo viewpoints. Aim: Find optimal model parameter set.

25. Conclusions Potential magnetic fields and linear force-free fields are popular due to their mathematic simplicity and available data. (e.g. from MDI on SOHO, Kitt Peak) Non-linear force-free fields are necessary to describe active regions exactly. More challenging both observational (vector magnetographs) and mathematical. Vector magnetograms with high spatial and temporal resolution become available soon (SOLIS, Solar-B, SDO, Solar Orbiter).

26. Thank you very much to Prof. Karl Schindler

27. Thank you very much to Prof. Karl Schindler