Coronal magnetic fields

T. Wiegelmann

• Coronal magnetic field models• Potential and linear force-free fields• Non-linear force-free fields (NLFF):• A blindfold test of different NLFF-codes• Applications: Full Sun, Coronal holes,

Active Regions, Coronal Tomography• Conclusions

Helmet streamer,finite beta plasma,full solar corona

Vectormagnetogram+ TomographicInversion of density

MHSEquilibrium

Active regions,low beta plasmain low corona

Vectormagnetogram(3 times more data,ambiguities, noise)

Non Linear Force-Free

Local in activeregions, low-betaplasma

LOS magnetogram+ observations ofplasma structures

Linear Force-Free Fields

(Global) currentfree regions,quiet sun

Line of sightmagnetogram

PotentialFields

ValidityObservationsneeded

MathematicsModel

Coronal magnetic field models

0)()( 0

=⋅∇∇+∇=××∇

BpBB ψρµ

0)(

=⋅∇⋅=×∇

BBrB α

0=⋅∇⋅=×∇

BBB α

00

=⋅∇=×∇

BB

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 nofree 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 ⋅= )(α

Non-linear force-free field modeling:model techniques, boundary conditions, hares, and hounds

Karel Schrijvera, Marc DeRosaa, Tom Metcalfa, Yang Liub, Jim McTiernanc, Zoran Mikicd, Stepháne Régniere, Mike

Wheatlandf, Thomas Wiegelmanng.

a) Lockheed Martin Advanced Technology Center

b) Stanford University

c) University of California, Berkeley

d) Science Applications Int’l Corporation

e) European Space Agency

f) University of Sydney

g) Max Planck Institute for Solar System Research

Rationale and experiment

• Understanding the conditions under which solar magnetic fields can destabilize to erupt in flares and coronal mass ejections requires a quantitative understanding of the coronal magnetic field and currents.

• Non-linear force-free (NLFF) fields are likely a useful model to use when extrapolating the solar surface field upward into the coronal volume.

• Here, we report on a comparison of five algorithms for the computation of NLFF field by testing their performance on known force-free field modes for which boundary conditions are specified either for the entire surface area or for an extended lower boundary only.

• We compare three classes of algorithms: I) Green’s function like, II) optimization, III) Grad-Rubin like.

• We specified the vector field on the six sides of a volume (test case I ) or on the lower boundary only including a wide skirt around the volume to be modeled (test case II ). These were provided to five modelers (Liu, Régnier, McTiernan, and Wiegelmann, Wheatland) who then performed a best-effort computation of a nonlinear force-free field solution to match the specified boundary condition(s).

Known non-linear force free fields as test input for numerical models

• Axisymmetric force-free fields are generated by placing a source of multipolar field and current somewhere in space.

• Bury this source somewhere below the z=0 plane, choose an angle of orientation of the axis of symmetry, and define the region of interest (volume to be modeled) as x,y ∈ (–1,1), z ∈ (0,2).

B = (Bx,By,Bz) specified only on lower bound. x,y ∈ (–3,3),

n=3m=1

L=0.3Φ=4π/5

Low & Lou (1990 ApJ 352 343) discuss a class of force-free fields described by ordinary differential equations:

Conditions for a general nonlinear force-free magnetic field:

Parameters for the blind algorithm test:

I: Green’s function like method(as developed by Yan & Sakurai, and as run by Liu)

• This is a boundary-integral equation representation for finite- energy force-free magnetic fields. By introducing a reference function, the force-free magnetic field at any point in the space can be expressed in term of magnetic field and its gradient over the boundary surface. This model can be solved numerically by the boundary element method.

• The boundary condition used here in the model is vector magneticfield on the inner boundary surface, and the outer boundary is open to the infinity. The field is assumed to vanish at infinity as O(r--2).

References:• Yan, Y. & Sakurai, T., 1997, Solar Physics, 174, 65.

Yan, Y. & Sakurai, T., 2000, Solar Physics, 195, 89.

II: Optimization Method(as developed by Wheatland et al. (2000, ApJ 540, 1150), and run byWiegelmann and McTiernan)

The force-free equationsare defined as:

We define the functional L as a quadratic formof the force-free equations

L ideally converges to 0 if the magnetic field satisfies the force-free equations.We minimize L with respect to an iteration parameter t:

The surface integral vanishes, if themagnetic field is fixed on the boundariesof the computational box and L decreases monotonically for:

Method:1.) Compute potential field in 3D box.2.) Replace bottom boundary with

meassured vector magnetogram.3.) Iterate B until L reaches minimum.

III: Grad-Rubin-like method(following Grad & Rubin, and run by Régnier and Wheatland)

Wheatland (2004), and Régnier et al. (2002, 2004) use an iterative scheme to model the nonlinear force-free coronal fields based on Grad & Rubin (1958).

Wheatland models the current distribution in terms of field lines originating from a chosen polarity and then calculated the field at every grid points due to the currents using an exact integral solution to Ampere’s law. The bottom boundary conditions are the vertical magnetic field on the surface and the α distribution in the strong current-carrying region in one polarity.

Régnier models the nonlinear force-free field by injecting the electric current density inside the magnetic configuration. This method was developed by Amariet al. (1997, 1999). The bottom boundary conditions are the vertical magnetic field on the surface and the distribution of α in one chosen polarity. On the other side of the volume, we impose that the normal component of the field vanishes. Amari, T., Aly, J. J., Luciani, J. F., Boulmezaoud, T. Z., Mikic, Z. 1997, Solar Phys., 174, 129

Amari, T., Boulmezaoud, T. Z., Mikic, Z. 1999, A&A, 350, 1051

Grad, H., Rubin, H. 1958, in Proc. 2nd Intern. Conf. on Peaceful Uses of Atomic Energy, vol. 31, U.N., Geneva, p.190

Regnier, S., Amari, T., Kersale, E. 2002, A&A, 392, 1119

Regnier, S., Amari, T. 2004, A&A, 425, 345

Wheatland, M. S. 2004, Solar Phys.

Low & Lou Test Case 1( n=1, m=1, L=0.3, Φ=π/4 )

Low & Lou Test Case 2( n=3, m=1, L=0.3, Φ=4π/5 )

VIEW 1

Quantitative comparison metrics for vector fields v and w

Mean Vector Error:

NormalizedVector Error:

Cauchy-Schwarz:

Vector Correlation:

Quantitative Comparison Metrics

VC CS NVE MVE ME/1e5 RMEPotential 0.85 0.82 0.55 0.65 2.74 1.00Constant alpha 0.88 0.90 0.50 0.58 2.76 1.01Low & Lou 1.00 1.00 0.00 0.00 4.05 1.48Wiegelmann 1.00 1.00 0.02 0.02 4.07 1.49McTiernan 1.00 0.99 0.08 0.13 3.92 1.43Wheatland 0.98 0.83 0.36 0.58 3.18 1.16Regnier 0.93 0.49 0.59 0.91 3.22 1.18Liu 0.88 0.26 1.01 3.52 5.50 2.01

VC CS NVE MVE ME/1.e6 RMEPotential 0.92 0.35 0.54 1.60 0.58 1.00Constant alpha 0.93 0.08 1.80 38.0 1.49 2.60Low & Lou 1.00 1.00 0.00 0.00 0.64 1.12Wiegelmann 1.00 0.57 0.14 1.25 0.66 1.14McTiernan 1.00 0.52 0.16 1.40 0.65 1.14Wheatland 0.99 0.58 0.31 0.87 0.57 0.99Regnier 0.94 0.28 0.51 2.70 0.59 1.02Liu 0.97 0.41 1.02 15.0 1.07 1.87

Test Case 1

Test Case 2

Computing time

6+xGreen540 min (1 Step)

Liu

6Grad-RubinRegnier

6Grad-Rubin138 minWheatland

5Optimization888 minMcTiernan

5Optimization15 minWiegelmann

OrderMethodTime

Conclusions• All algorithms yield NLFF fields that agree well with the input field in the deep

interior of the volume, where the field and electrical currents are strongest.• The NLFF vector fields in the outer domains of the volume depends sensitively on the

details of the specified boundary conditions; best agreement is found if the field outside of the model volume is incorporated as part of the model boundary, either as potential field boundaries on the side and top surfaces, or as a potential field in a skirt around the main volume of interest. Requiring the field to close entirely within the volume, and ignoring weak currents in outlying field regions strongly affects the appearance of the field high over the core strong-field regions.

• For the best-fit model, the relative vector difference between input and model field has an average magnitude of 2%, and the average energy density in the field is approximated to within a few percent.

• The models converge towards the central, strong input field at speeds that differ by a factor of one million:

• The fastest-converging model is the Wheatland et al. (2000) optimization algorithm as implemented by Wiegelmann.

Workshop conclusions

Applications

• Global potential fields• Coronal holes• Active Regions• Coronal Tomography

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.

EIT 195 Fe XIIFormation temperature1.5 million K

EIT 304 He IIFormation temperature60,000-80,000 K

Coronal Holes (Wiegelmann & Solanki 2004)

Some statistic properties of coronal holes (CH)compared with the quiet Sun (QS)

CH QS Net flux 7.6 G +- 4.8 G 0.4 G +- 0.4 G Unbalanced flux 77% +- 14% 9% +- 9% Ave loop length 7.8+- 3.9 Mm 14.1+-4.1 Mm Ave loop height 0.9+- 0.4 Mm 3.0+-0.7 Mm

We investigated 12 CHs identified in He I (NSO/Kitt Peak coronal hole maps prepared byKaren Harvey and Frank Recely)

The emitting volume filled by gas at that temperaturecorresponds to the emitted radiation.In CH ~70% at low and ~10% at high temperaturescompared with the quiet Sun.

Use RTV scaling laws to approximate temperatures T ~ (pL)1/3

Closed (for Bz>30G) and open (for Bz>100G) magnetic field-lines in a coronal hole and the nearby quiet Sun. The red line correspondsto the FOV of SUMER (Ne VIII).

Coronal holes and SUMER (Wiegelmann, Xia & Marsch 2005)

Summary: coronal holes• Large net magnetic flux in coronal holes

(CH ~80 %, QS ~ 10%)• Long hot loops are almost absent in coronal holes• Number of small cool loops is only slightly reduced• Strongly reduced emission in CH for hot coronal lines

(~10% of the emission in QS)• Only slightly reduced emission in cooler transition

region radiation (~70% of QS)• Plasma almost at rest in closed loops• Significant outflow (up to 20 km/s) on open fieldlines

EIT-image and projections of magneticfield 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 2004)We 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 withα=-0.01 [Mm-1]

(better agreement)

3D-magnetic field lines, linear force-free α=-0.01 [Mm-1]

SUMER Dopplergram in NeVIII(λ 77 nm) and a 2-D-projection of some field lines.

(Marsch, Wiegelmann, Xia, A&A 2004)

Mass flux density inferred from Doppler-shift and intensity from SUMER observations

up

down

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 extrapolations(Wiegelmann, Lagg, Solanki, Inhester & Woch 2005)

We compared measurements of magnetic loops in a newly developed active region with extrapolations from the photosphere. We got the best agreement ofmeasured and extrapolated loops for a non-linear force-free magnetic field model.

Coronal tomography (Wiegelmann & Inhester 2003)

a) 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)

• Non-linear force-free fields are necessary todescribe active regions exactly. More challenging both observational (Vector magnetograms) and mathematical.

• Current vectormagnetograms have limited field ofview and only occasional available. Future:Solis and Solar B.