extrapolation of magnetic fields thomas wiegelmann why study coronal magnetic fields? how to obtain...
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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.
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Coronal mass ejections and flares are assumed to occur due to instabilities in the coronal magnetic field configuration.
It is importantto investigatethe coronal
magnetic field
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Coronal magnetic Fields: Origin of Space weather
Question:Origin of coronal
eruptions
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Solar magneticfield measured
routinely only inphotosphere
Aim: Extrapolate measured photospheric magneticfield into the corona under model assumptions.
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Lorentz force
pressure gradient
gravity
How to model the stationary Corona?
Force-freeFields
Low plasmaBeta in corona
Neglect plasma pressure+gravity
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Force-Free Fields
Equivalent
• Potential Fields (no currents)• Linear force-free fields
(currents globally proportional to B-field)
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
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Easy to computeRequire only
LOS-Magnetograms
Here: global constant linear force-free parameter
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Simple potential field models provide already areasonable estimate regarding the global magneticfield structure. Mainly closed loops in activeregions and open field lines in coronal holes.
Potential Field Model EUV-emission
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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 RegionsWe use a linear force-free model with MDI-data and have the freedomto choose an appropriate value for the force-free parameter α.
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Linear force-free field with α=-0.01 [Mm-1](good agreement)
3D-magnetic field lines, linear force-free α=-0.01 [Mm-1]
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NonLinear Force-Free Fields
• Compute initial a potential field (Requires only Bn on bottom boundary)
• Iterate for NLFFF-field, Boundary conditions:- Bn and Jn for positive or negative polarityon boundary (Grad-Rubin)- Magnetic field vector Bx By Bz on boundary (Magnetofrictional, Optimization)
Equivalent
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Grad-Rubin methodSakurai 1981, Amari et al. 1997,2006,
Wheatland 2004,06,07
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MagnetofrictionalChodura & Schlueter 1981,Valori et al. 2005
OptimizationWheatland et al. 2000,Wiegelmann 2004,2007
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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.
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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.
Grad-Rubin
MHD-relaxation
Optimization
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Force-Free
B-Field Measurements,non-force-free
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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.
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No net force
No net torque
Photosphere
Smoothness
Preprocessed boundary data
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Chromospheric H-alpha preprocessing• H-alpha fibrils outline magnetic field lines.• With image-recognition techniques we get
tangent to the chromospheric magnetic fieldvector (Hx, Hy).
• Idea: include a term in the preprocessing tominimize angle of preprocessed magnetic field (Bx,By) with (Hx,Hy).
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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)
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Prepro-cessing
We test preprocessing with Aad’s model
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Angle <B,H> inChromosphere
Force-free coronalmagnetic Energy
No pre-processing 19o 65%
Classical pre-
processing 9o 97%
H-Alpha pre-processing 1o 100%
Results: Comparison with Aad‘s Model
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CoronalMagnetic Field
Nonlinear Force-free code
Preprocessing tool
Vectormagnetogram
H-AlphaImage
ChromosphericMagnetic Field
Optional
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Measured loops in a newly developed AR (Solanki, Lagg, Woch,Krupp, Collados, Nature 2003)
Potential field reconstruction
Linear force-free reconstruction Non-linear force-free reconstruction
Comparison of observed magnetic loops and extrapolationsfrom photospheric measurements
Nonlinear force-free Models are
superior.
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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-freeextrapolated field lines from Hinode/SOT with MDI-skirt.
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Stereoscopy vs. coronal field extrapolation
• Vector magnetogram data (here: Hinode/SOT) areessential for nonlinear force-free field modeling.
• Unfortunately Hinode-FOV covered only a smallfraction (about 10%) of area spanned by loopsreconstructed from STEREO-SECCHI images.
• Quantitative comparison was unsatisfactory,NLFFF-models not better as potential fields here.
• In other studies NLFFF-methods have shown to besuperior to potential and linear force-freeextrapolations. (Comparison with coronal images from one viewpoint, NLFFF-models from ground based data)
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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:1. large model volumes at high resolution that accommodate most of the
connectivity within a region and to its surroundings;2. accommodation of measurement uncertainties (in particular in the transverse
field component) in the lower boundary condition;3. '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).
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Temporal Evolution of Active RegionsUse time series of ground based vector magnetograms with sufficient large FOV (Solar Flare Telescope, SOLIS).
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Flaring AR-10540(Thalmann & Wiegelmann A&A 2008)
Active Region-10960
Solar X-ray flux. Vertical blue lines: vector magnetograms available
Magnetic field extrapolationsfrom Solar Flare telescope
Extrapolated from SOLISvector magnetograph
M6.1 Flare
Magneticenergy builds
up and isreleases during
flare
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Comparison of two Active Regions
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Conclusions• Potential and linear force-free fields are popular due to their
mathematic simplicity and because only LOS-magnetogramsare 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 vectoris measured, and the force-free corona.
• Coronal magnetic field models should be compared andvalidated by coronal observations.
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Time-dependentMHD-simulations
Self-consistentequilibrium
Artificialimages
LOS-integration
Where to go in corona modeling?
Force-free code
Vectormagnetogram
MHS code
3D Force-freemagnetic field
3D fieldlines
com
pare
Plasma along magnetic
loops
Scaling laws Tomograp
hy
Stereoscopy STEREOimages
3D EUVloops
consistent?
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