Coronal magnetic fieldsThomas Wiegelmann,

MPI for Solar-System Research, (Former: MPI für Aeronomie)

Katlenburg-Lindau

• Why are coronal magnetic fields important?

• Models for magnetic field reconstruction.

• Potential magnetic fields

• Linear force-free fields

• Non-linear force-free fields

• Magnetic fields and coronal tomography

• Conclusions

Why are coronal magnetic fields important?

• Magnetic fields couples the solar interior, photosphere and atmosphere.

• Magnetic field dominates in the solar corona. (Magnetic pressure >> Plasma pressure).

• Knowledge of the coronal B-Field is essentialto understand dynamic phenomena likecoronal mass ejections and flares.

How to obtain coronal B-Fields

• Direct measurements are extremely difficult.

• Measure B on the photosphere (line of sight Bor vector B) and extrapolate it into the corona.

• We need assumptions regarding coronal currents:- No currents Potential fields- Linear Force Free FieldsBJ - Non Linear Force Free FieldsBrJ )(

Model Mathematics Observationsneeded

Validity

Potential

Fields

Line of sightmagnetogram

(Global) currentfree regions,quiet sun

Linear Force-

Free Fields

LOS magnetogram

+ observations ofplasma structures

Local in activeregions, low-betaplasma

Non Linear Force-Free

Vectormagnetogram

(3 times more data,ambiguities, noise)

Active regions,low beta plasmain low corona

MHS

Equilibrium

Vectormagnetogram

+ Tomographic Inversion of density

Helmet streamer,finite beta plasma,full solar corona

Coronal magnetic field models

0

)()( 0

B

pBB

0

)(

B

BrB

0

B

BB 0

0

B

B

Global Potential Field reconstruction

Linear Force-Free Fields

Linear Force-Free Fields

Linear Force-Free Fields

Linear Force-Free Fields

EIT-image and projected magnetic field lines.(α · L=2)

3D magnetic field lines withKitt Peak magnetogram

Non-linear force-free fields

• Why do we need non-linear force-free fields?- In general alpha changes in space.- Potential and linear force-free fields have no free energy to be released during an eruption.

• The computation is much more difficult:- Mathematical difficulties due to non-linearity.- Vector magnetograms have ambiguities.- Transversal B-field is very noisy.- Limited field of view for current instruments. (Soon: Full disc vector magnetograph SOLIS.)

BrJ )(

Non-linear force-free fields

Potential field and non-linear force free reconstruction of a model active region regarding the same line of sight photospheric magnetic field. Our optimization code reconstructs the original analytic solution within the discretisation error.

Magnetic fields and coronal tomography

• Coronal tomography uses line of sight integrals of the coronal density from different viewpoints.

• Aim: 3D reconstruction of coronal density structure.• Density and magnetic field have to be reconstructed

selfconsistently (MHS-equations + observational data).

2,

,,

322

320

|)(|1

||

|)()(|),(

NIIxdBB

xdmNNTkBBNBL

Modelip

ip

obsip

V

V

B

0

)()( 0

B

pBB

Magnetic fields and coronal tomographya) Use only line of

sight density integrals.

b) Use only magneticfield data.

c) Use both line of sightdensity integrals andmagnetic field as regularization operator.

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)

• Nonlinear force free fields are necessary todescribe active regions exactly. More challenging both observational and mathematical.

• A consistent 3D model of the solar corona requirestomographic inversion and magnetic reconstructionin one model.