Nonlinear Force-Free Field
Extrapolation
Kanya Kusano
Earth Simulator Center/JAMSTEC
Satoshi Inoue
STE Lab., Nagoya University
Saku Tsuneta, and SOT/NAOJ Team1

OutlookIntroduction
Why extrapolation
 Why NLFF
extrapolation methods (review) examples
 problems and issues
2

Why extrapolation?
The 3D structure of magnetic field is
crucially important to understandthe stability of active region
 magnetic helicity & free magnetic energy magnetic topology, current sheet and quasiseparatrix layer direct comparison with theoretical modelsHowever, magnetogram can provide only
the 2D data mainly on photospheric surface.

2D phospheric data should be extrapolated to 3D space.
3

Nonlinear Force-Free
Plasma beta in the solar corona is usuallymuch less than unity.
 Force-free field could be a good
approximation for the magnetic field in low
corona. (∇ × B ) × B − ∇p = 0

∇ × B = α (r ) B∇ ⋅B = 0( B ⋅ ∇ )α = 0

boundary
conditionα (r ) ⊂ ∂Ω

±Bz ⊂ ∂Ωα is constant on each magnetic field line.

conjugate points4NLFF Solvers

Direct Integration

Grad-Rubin-type methods

(Wu 1990, Cuperman et al. 1991)(Sakurai 1981; Amari et al. 1997, 1999; Régnieretal. 2002; Wheatland 2004)

Relaxation-type methods

stress & relax method (Roumeliotis 1996;
McClymont, Jiao & Mikic; Valori et al. 2006)
 optimization method(e.g. Wheatland et al. 2000, Wiegelmann &Neukirch 2003, McTiernan, ...)


Boundary integral method(Yan & Sakurai 1997, 2000, Yan & Li 2006)
5Grad-Rubin method
α ( n ) |∂Ω = (∇ × B) z / Bz+B(n)⋅ ∇α∇×B( n +1)(n)J=0=α B(n)(n)in Ω∇ ⋅ B ( n +1) = 0Bz( n +1) |∂Ω = Bz

B(n)αBz |B(n+1)

boundary conditions

α |∂Ω+Bz |∂Ω

fit to the force-free equation.
6

Regnier et al. 2002

7

Relaxation MethoddV= (∇ × B) × B −νVρdt
∂B= ∇ × ( V × B − η∇ × B )∂tη = η (V )
η → 0 as V → 0V = ν (∇ × B) × B−1V

boundary conditions

Bx , B y , Bz |∂Ω8

Relaxation method (test by Kusano)exact solution

relaxationnumerical solution

9Optimization method
Wheatland-Sturrock-Roumeliotis 2000
L = ∫ [BVrr∂B= μF,(μ > 0)∂t interr∂B=0∂t bound−2r 2r r 2| j × B | + | ∇ ⋅ B | ]dVrrr
∂B
∂B r
dL
= −2 ∫⋅ FdV − 2 ∫⋅ GdS
VSdt∂t∂tdL= −2 ∫ μ F 2 dVVdt
monotonically decrease10Wheatland et al. 2000

Low and Lou’s solution

11Nonlinear force-free (AR8100)

PotentialLinear force-free<JxB>

Nonlinear force-freeNLFFLFFF

iteration loop12Problems
inconsistency with force-free condition
methodconsistency with F-F

dataGrad-Rubin○Bz & α (not Bxy)

Relaxation-type△

Bx By Bz

<JxB>

inconsistent data (Blow&Lou+BLFF)

compromise between dataand force-free condition

degraded
NLFF (Blow&Lou)
relaxation step13

Wiegelmann, Inhester & Sakurai 2006


Aly’s virial theorem (1989)
force-free condition

total force balancedata reliability∫ B B dxdy = ∫ B B dxdy = 0xzySzx

222()
B+Bdxdy=B∫ x y∫ z dxdySS

total torque balance
222()
x
B+Bdxdy=xB∫ x y∫ z dxdy
Spreprocessingof magnetogramx222()
yB+Bdxdy=yB∫ x y∫ z dxdy
Sx∫ yB B dxdy = ∫ xB B dxdyx

Szyz

NLFF solverx

14

Reconstruction of conjugate point+−∂Ω→∂Ω field tracing from→ conjugate point→ mapping α→ Jz=aBz−∂Ω→ Poisson eq. onBJ

∇ 2 Bx = −∂ x J z − ∂ yz Bz∇ B y = −∂ y J z + ∂ xz Bz2VRelaxation of B
15

Summary
Magnetic extrapolation is crucial to reveal the 3D
structure and the stability of active region
magnetic field.
 Several methods have been recently developed,
and they can provide the solution as long as thedata are consistent with the force-free equation.
 Issuescompromise between data and force-free condition
 non-force free effects (pressure, gravity, flow)→ full-MHD modeling condition on the other boundaries→ combination with global model convergence speed→ parallel computation
16

Earth Simulator

Solar-B

40 TFlops
10 TB(5120 cpu)

• datapreprocessing• NLFFextrapolation• multi-resolution
modeling

Numerical Solar Corona17