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.

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

Coronal magnetic Fields: Origin of Space weather

Question:Origin of coronal

eruptions

Solar magneticfield measured

routinely only inphotosphere

Aim: Extrapolate measured photospheric magneticfield into the corona under model assumptions.

Lorentz force

pressure gradient

gravity

How to model the stationary Corona?

Force-freeFields

Low plasmaBeta in corona

Neglect plasma pressure+gravity

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

Easy to computeRequire only

LOS-Magnetograms

Here: global constant linear force-free parameter

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

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.

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

• 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

Grad-Rubin methodSakurai 1981, Amari et al. 1997,2006,

Wheatland 2004,06,07

MagnetofrictionalChodura & Schlueter 1981,Valori et al. 2005

OptimizationWheatland 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.

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

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.

No net force

No net torque

Photosphere

Smoothness

Preprocessed boundary data

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).

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)

Prepro-cessing

We test preprocessing with Aad’s model

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

CoronalMagnetic Field

Nonlinear Force-free code

Preprocessing tool

Vectormagnetogram

H-AlphaImage

ChromosphericMagnetic Field

Optional

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.

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.

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)

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).

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

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

Comparison of two Active Regions

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.

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?