Nonlinear force-free Nonlinear force-free coronal magnetic field coronal magnetic field extrapolation scheme extrapolation scheme

for solar active regionsfor solar active regions

Han He, Huaning Wang, Yihua YanHan He, Huaning Wang, Yihua YanNational Astronomical Observatories, Chinese AcadeNational Astronomical Observatories, Chinese Acade

my of Sciences, Beijing, Chinamy of Sciences, Beijing, China

Hinode WorkshopHinode WorkshopBeijing, ChinaBeijing, China

Dec. 8-10, 2007Dec. 8-10, 2007

Solar eruption events are Solar eruption events are connected with the coronal connected with the coronal

magnetic structuresmagnetic structures

Direct observation Calculation based on a physical modelOR

Nonlinear force-free field Nonlinear force-free field (NLFFF) extrapolation for solar (NLFFF) extrapolation for solar

active regionsactive regions

vector magnetogram observed in the phot

osphere

magnetic field configuration in

the corona

Calculation based on

NLFFF model

?iB

Solar active region x

yz

Given Bx, By, Bz at the bottom boundary

BrB )(

0 B)(~ BJ

field line

B

JB //

0)( BB orForce-free:

Divergence-free:

The field can be considered force-free roughly 400km (0.55 arc second) above the photosphere(Metcalf et al., 1995, ApJ, 439, 474)

0 BAlpha is a constant along one field line

Nonlinear force-free field Nonlinear force-free field (NLFFF) model(NLFFF) model

Direct Boundary Integral Equation Direct Boundary Integral Equation (DBIE) method for NLFFF (DBIE) method for NLFFF extrapolationextrapolation((Yan, Y., Li, Z. 2006, ApJ, Yan, Y., Li, Z. 2006, ApJ, 638638, 1162, 1162))

x

y

z

0BB :

:0 B

BB

)( 2 rOB

dd 00 BBBz

Y

n

Yi

0d)( 22 BBB Y

)( ir

4

)cos(

4

)cos(),(Y

irr irr

Infinite plane surface boundary

),,( iii zyx

),,( iii zyx

),,( zyx

n

Boundary condition of DBIE Boundary condition of DBIE methodmethod

Infinite plane surface boundary

d0BBz

Yi

)( 2 rOB

vector magnetogram

B is assumed to be zero out of the vector magnetogram

Parameter Parameter λλ► same dimension as the force-free fsame dimension as the force-free f

actor actor αα►defined locally at a field pointdefined locally at a field point

► three components three components λλxx, , λλyy, , λλz z correspocorrespo

nding to Bnding to Bxx, B, Byy, B, Bzz

4

)cos(

4

)cos(),(Y

Optimal method to determine Optimal method to determine λλ and B locall and B locallyy((Yan, Y., Li, Z. 2006, ApJ, Yan, Y., Li, Z. 2006, ApJ, 638638, 1162, 1162))

d0BBn

Yi

0BB :

),,( zyxii BB

),,( find ***zyx

||||

||),,(

BJ

BJzyxif

),,(min),,( ***zyxizyxi ff

),,( iii zyx

Best convergence property at the field points near the bottom Best convergence property at the field points near the bottom boundaryboundary

•The bottom boundary for applying the DBIE is moved upwardly layer by layer to achieve the best convergence property

•Enlarge the area for integration at higher layers•Keep the original number of pixels at each layer to save computing time

Upward boundary integration Upward boundary integration schemescheme

0Layer 1Layer

2Layer 3Layer

4Layer

5Layer

region active

Output region of the code

0B

Global field configuration

Test Case ITest Case I► Low and Lou (1990) analytical fieldLow and Lou (1990) analytical field► 4/,3.0,1,1 lmn

xz

Case I 3D-ViewCase I 3D-View

Analytical solution

Extrapolated field

TestTest Case IICase II5/4,3.0,1,3 lmn

Global field configuration

►

x

z

Case II 3D-ViewCase II 3D-View

Analytical solution

Extrapolated field

The degree of agreement between the extrapolated field and the analytical solution

2/122

vec

i iii

iii bBbBC

The extrapolated fields deviate from the analytical fields gradually with the increase in height

)a(

)d()c(

)b(height vsfLheight vsfL

height vsdL height vsdL

I Case

solution analytical field edextrapolat

solution analytical field edextrapolat

I CaseI Case

I Case

Internal consistency of the extrapolated Field Internal consistency of the extrapolated Field (Case I)(Case I)

Vf VBV

L d)(1 22 BB

vd V

VL d

1 2B

Force-free and divergence-free constraints are satisfied in the extrapolated field

)a(

)d()c(

)b(height vsfLheight vsfL

height vsdL height vsdL

II Case

solution analytical field edextrapolat

solution analytical field edextrapolat

II CaseII Case

II Case

Internal consistency of the extrapolated Field Internal consistency of the extrapolated Field (Case II)(Case II)

Force-free and divergence-free constraints are satisfied in the extrapolated field

NLFFF extrapolation for solar active NLFFF extrapolation for solar active regionregion

NOAA 9077NOAA 9077

2000.07.14 04:14UT

Vector magnetogram of NOAA 9077 was observed at 04:14 UT on 14 July 2000 bySolar Magnetic Field Telescope (SMFT) at Huairou Solar Observing Station (HSOS)

Bx By Bz0

Bx By Bz1

The noises in the boundary data can be suppressed by DBIE through the integration over the whole boundary

AR 90772000.7.1404:14UT

Observed by SMFT at Huairou

FOV: 269x269 arcsec

64x64 grid4.2 arcsec/pixel~ 3000km/pixel

Red: field lines that leave Red: field lines that leave the modeling boxthe modeling box

Blue: closed field linesBlue: closed field lines

NS

(b)

(c) (d)

c

(a)

Internal consistency of the extrapolated field

height vsfL 9077NOAA height vsdL 9077NOAA

Vf VBV

L d)(1 22 BB

vd VV

L d1 2

B

Force-free constraint Divergence-free constraint

U-shaped field lines above the X-U-shaped field lines above the X-pointpoint

Side view

3D view

Compared with filament Compared with filament imagesimages

04:27UTTop view 04:14UT

(HSOS) filtergram H

04:12UT

TRACE 195A

Compared with TRACE imagesCompared with TRACE images

Before the flare During the flare (http://trace.lmsal.com)

maximum10:10UT

Onset of the flare

04:14UT

10:24UT

SOHO MDINOAA 10436 2003.08.22Vector magnetogram was observed at

2003.08.22 01:29:57UT bySolar Flare Telescope

NLFFF extrapolation for solar active NLFFF extrapolation for solar active regionregion

NOAA 10436NOAA 10436

AR 104362003.8.2201:29:57UT

Solar Flare Telescope

Lower layersLower layers

Red: field lines that leave the modeling boxRed: field lines that leave the modeling box Blue: closed field linesBlue: closed field lines

Compared with coronal imageCompared with coronal image NOAA 10436 NOAA 10436

GOES 12 SXI2003.08.2201:30:37UT

Solar Flare Telescope2003.08.22 01:29:57UT340 x 320 arcsec

64 x 64 grid5.3 arcsec/pixel

HHinode (Solar-B)inode (Solar-B)

SOT

XRT

EIS

Field of View

EIS (576”x512”)

XRT(2048”x2048”)

SOT:NFI/SP(328”x164”)SOT: BFI

(205”x102”)

N

E W

S

(From Hinode Team)

The instruments

46.8” X 162.3” (156 X 512 grids) 46.8” X 162.3” (156 X 512 grids)

Hinode Solar Optical Telescope – Spectro-polarimeterHinode Solar Optical Telescope – Spectro-polarimeter

Fast Map:Time per position: 3.2 sec

162.3” x 0.3”

Normal map

Fast map

Dynamics

Deep magnetogram

Sample dataSample data2007-6-4 08:49:04 - 08:58:54UT2007-6-4 08:49:04 - 08:58:54UT

Hinode Solar Optical Telescope (SOT)Hinode Solar Optical Telescope (SOT)Spectro-PolarimeterSpectro-Polarimeter (SP)(SP)-- -- level-0 datalevel-0 data

Level-0 dataLevel-0 data

I Q U V

Spectral coverage: 630.08nm - 630.32nm Fe I lines 630.15 and 630.25 nm

Hinode SOT SP level-1Hinode SOT SP level-1 data (by SSW code)data (by SSW code)

Level-1 dataLevel-1 dataI Q U V

Milne-Eddington inversion code (High Altitude ObservatorMilne-Eddington inversion code (High Altitude Observatory)y)

I Q

U V

NLFFF extrapolation for solar active NLFFF extrapolation for solar active regionregion

NOAA 10930NOAA 10930► Flare event: 2006.12.13 Flare event: 2006.12.13 0214 0240 0257UT X3.40214 0240 0257UT X3.4

SOHO MDINOAA 10930 2006.12.12

Vector magnetogram was observed at20:30UT on 12 Dec. 2006 by Hinode satellite

Images of Images of Hinode SOT Hinode SOT Spectro-Polarimeter Spectro-Polarimeter NOAA 10930 (630.15 nm)NOAA 10930 (630.15 nm)

I

Time of the observation ：2006.12.12 20:30-21:33UT

FOV ： 295.20 x 162.30 arcsecOriginal grid number ： 1000 x 512

Q

U V

Level-1 dataLevel-1 data

Remove 180 degree ambiguity of the Remove 180 degree ambiguity of the direction of the transverse field direction of the transverse field

componentcomponent

By a reference field with a force-free factor best fitting the observed fields Wang, Yan and Sakurai (2001)

Vector magnetogram of NOAA 1093Vector magnetogram of NOAA 10930 0 Hinode SOT SP data

Time ： 2006.12.12 20:30-21:33UTFOV ： 295.20 x 162.30 arcsec

Original grid number ： 1000 x 512Grid number for extrapolation ： 111 x 60 (2.7arcsec/pixel)

Milne-Eddington inversion code by High Altitude ObservatoryMilne-Eddington inversion code by High Altitude Observatory

Compared with coronal image observed by Compared with coronal image observed by XRTXRT

NOAA 10930NOAA 10930Hinode XRT image21:30:32UT2006.12.12

Extrapolated field lines based on vector magnetogram observed byHinode SOT SP

2006.12.12 20:30-21:33UTFOV:295.20 x 162.30 arcsec

Compared with coronal image observed by XRT NOAA 10930Compared with coronal image observed by XRT NOAA 10930

Central domain of the extrapolated Central domain of the extrapolated fieldfield

NOAA 10930NOAA 10930

During the flare

Before the flare

ThanksThanks