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    • Abstract: vector magnetogram. B is assumed to be zero out of the vector magnetogram. Properties of ... vector magnetogram. The height of the extrapolation can be specified in the code. Recent ...

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Nonlinear force-free field modeling of
force-
coronal magnetic field using the data
obtained by Hinode satellite
Han He, Huaning Wang, Yihua Yan
National Astronomical Observatories, Chinese Academy of Sciences
The Dynamic Solar Corona
CAS-
CAS-IAU Joint Solar Eclipse Meeting
July 24, 2009, Suzhou, China
Solar eruptive events are connected with the
coronal magnetic structures
AR 10930
Understand the 3-D structures Better prediction of the
of the coronal magnetic fields solar eruptive events
Direct observation OR Calculation based on a physical model using the
(difficult) photospheric magnetic field data as the bottom
boundary condition
Non-
Non-linear force-free field (NLFFF)
force-
extrapolation for solar active regions
magnetic field
configuration in the
corona
Bi = ?
Calculation
based on
NLFFF model
z y
vector magnetogram solar active region x
observed in the Given Bx, By, Bz
photosphere at the bottom boundary
Nonlinear force-free field (NLFFF) model
force-
Force-free:
B (∇ × B) × B = 0 or
∇ × B = α (r )B
J (~∇× B) Divergence-free:
B// J ∇⋅B = 0
field line ∇α ⋅ B = 0
Alpha is a constant along one field line
(force-free factor)
The field can be considered force-free roughly 400km (0.55 arc second)
above the photosphere
(Metcalf et al., 1995, ApJ, 439, 474)
Direct Boundary Integral Equation (DBIE)
representation for non-linear force-free field
non- force-
(Yan, Y., Li, Z. 2006, ApJ, 638, 1162)
ApJ, 638, 1162)
B = O ( r −2 )
∂Y ∂Y
Bi = −∫ B 0dΓ = ∫ B 0 dΓ
z
Γ
∂n Γ
∂z
Ω : ∇ × B = αB cos(λρ ) cos(λρ ′)
( xi , yi , zi ) Y (λ , ρ ) = −
∇⋅ B = 0 4πρ 4πρ ′
ρ
y ρ = r − ri ρ ′ = r − ri′ λ = λ (ri )
( x, y , z )
Γ : B = B0
ρ′ ∫

Y (λ2 B − α 2 B − ∇ α × B )d Ω = 0
x n ( xi , yi ,− zi )
Infinite plane surface boundary
Boundary condition of the DBIE-NLFFF
DBIE-
extrapolation method for solar active region
B = O(r −2 )
∂Y
Bi = ∫ B 0 dΓ
Γ
∂z
vector magnetogram
Infinite plane surface boundary
B is assumed to be zero out of the vector magnetogram
Properties of parameter λ
(Li, Z., Yan, Y., & Song, G. 2004, MNRAS, 347, 1255)
347, 1255)
(Yan, Y., & Li, Z. 2006, ApJ, 638, 1162)
ApJ, 638, 1162)
(He, H., & Wang, H.N. 2006, MNRAS, 369, 207)
369, 207)
► same dimension (1 / length) as the force-free factor α
force-
► same order of magnitude as the force-free factor α
force-
► The sign of λ (positive or negative) does not influence
the value of B (even function)
► defined locally at every field point (may be different
along a field line)
► three components λx, λy, λz corresponding to Bx, By, Bz
cos(λρ ) cos(λρ ′)
Y (λ , ρ ) = −
4πρ 4πρ ′
Optimal method to determine λ locally
(Yan, Y., Li, Z. 2006, ApJ, 638, 1162)
ApJ, 638, 1162)
∂Y find (λ* x , λ* y , λ* z )
Bi = −∫ B 0 dΓ
∂n
Γ B i = B i (λ x , λ y , λ z )
| J×B |
f i (λ x , λ y , λ z ) =
( xi , yi , zi ) | J || B |
f i (λ* x , λ* y , λ* z ) = min f i (λx , λ y , λ z )
Γ : B = B0
Iterative calculation for each field point (need much computing time)
Upward boundary integration scheme for nonlinear
force-
force-free field (NLFFF) modeling of the coronal
magnetic field (He, H., & Wang, H.N. 2008, JGR, 113, A05S90)
113, A05S90)
• The bottom boundary for applying the DBIE is moved upwardly layer by
layer to achieve the best convergence property and accuracy
• Enlarge the area for integration gradually layer by layer to fit the
expanding field
• Keep the original number of pixels at each layer to save computing time
Output region of the code
Layer 5
Layer 4
B=0 Layer 3
Layer 2
boundary Γn +1
Layer 1
Γ Layer 0
boundary Γn
active region
Calculation of J in the new scheme
J is calculated in a small square pyramid to fully utilize the boundary data
information and thus save the computing time
Test Case
► Low and Lou (1990 ApJ, 352, 343) analytical field
ApJ, 352,
► n = 1, m = 1, l = 0.3, Φ = π / 4
z x
Γ
Global field configuration Field lines in modeling box
Improvement of the new scheme compared with
the original scheme of Yan and Li [2006]
(tested by the analytical solution of Low and Lou [1990])
Original scheme New scheme
Comparing the new computational scheme with the
original scheme (Cvec curves for the test Case)
(C
1/ 2
 
Cvec = ∑ Bi ⋅ bi  ∑ Bi ∑ 
2 2
bi 
i  i i
Recent progress of DBIE-NLFFF modeling code
The code can be directly applied to non-
non-
square (rectangle) magnetograms observed
by Hinode satellite
vector magnetogram
Infinite plane surface boundary
The height of the extrapolation can be specified in the code.
Recent progress of DBIE-NLFFF modeling code
Parallel Computing is realized through
Fortran 95 + OpenMP
Layer-1 Layer-2 … Layer-n
multi-threaded, shared memory parallelism
GNU Fortran compiler with OpenMP in Linux System
Recent progress of DBIE-NLFFF modeling code
Computing time of different codes
IDL Fortran 95 Fortran 95+OpenMP
8 threads
64x64x64 10.03 hours 4.25 hours 42 minutes
grid (2.36 times faster) (14.3 times faster)
Platform for test calculation:
CPU 1: Intel Xeon E5410 @ 2.33GHz (4-core)
CPU 2: Intel Xeon E5410 @ 2.33GHz (4-core)
Memory: 2G
Fedora 9 Linux System, GNU Fortran compiler (gfortran –O3)
Three vector magnetograms observed by
Hinode/SOT Spectro-Polarimeter within the interval
Spectro-
of 26 hours are selected for NLFFF modeling
SP field of view:
295.20 x 162.30 arcsec
(1) (2) (3)
(1) 2006-12-10T21:00:07.290 ~ 2006-12-10T22:03:41.199 LMSAL Hinode Data Center
(2) 2006-12-11T20:00:05.927 ~ 2006-12-11T21:03:17.090 http://sot.lmsal.com/sot-data
(3) 2006-12-11T23:10:05.750 ~ 2006-12-12T00:13:16.509 HINODE (SOLAR-B) Data Center
http://darts.isas.jaxa.jp/solar/hinode/
Remove 180 degree ambiguity of the direction of the
transverse field component
By a reference field with a force-free factor best fitting the observed fields
Wang, Yan and Sakurai (2001)
Vector magnetograms of NOAA 10930
(1) 2006-12-10 21:00:07
(2) 2006-12-11 20:00:05
(3) 2006-12-11 23:10:05
Field lines of the extrapolated field
(1) 2006-12-10 21:00:07
(2) 2006-12-11 20:00:05
(3) 2006-12-11 23:10:05
Modeling box: 160x88x16
white lines: closed in the modeling box
black lines: open in the modeling box
Closed field lines of the extrapolated field
(1) 2006-12-10 21:00:07
(2) 2006-12-11 20:00:05
(3) 2006-12-11 23:10:05
Modeling box: 160x88x16
Field lines overlying XRT coronal image
(1) 2006-12-10 21:00:07
(2) 2006-12-11 20:00:05
(3) 2006-12-11 23:10:05
Closed field lines compared with XRT coronal image
20061210 210026UT
(1) 2006-12-10 21:00:07
Closed field lines compared with XRT coronal image
20061211 210006UT
(2) 2006-12-11 20:00:05
Closed field lines compared with XRT coronal image
20061211 234555UT
(3) 2006-12-11 23:10:05
Field lines compared with EIS coronal image
20061211 201843UT 284A
(2) 2006-12-11 20:00:05
3D view of the field lines
(3) 2006-12-11 23:10:05
Thanks !


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