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.

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.

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

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

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

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)

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

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.

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)

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

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)

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

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

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.

Optimization Grad-Rubin MHD-relaxation Comparison paper, Metcalf et al., Sol. Phys. 2008. -Good agreement for extrapolations from chromosphere. -Poor results for using photospheric data directly. -Improvement with preprocessed photospheric data.

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

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.

Preprocessed boundary data No net force No net torque Photosphere Smoothness

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).

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)

We test preprocessing with Aad’s model

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%

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

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

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.

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)

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. 2006 (Spy 235, 161), 2008 (ApJ 675, 1637), Metcalf et al. 2008 (SPh 247, 269), DeRosa et al. (2009, ApJ 696, 1780).

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

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

Comparison of two Active Regions

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.

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