The validity of the boundary The validity of the boundary integral equation for integral equation for

magnetic field extrapolation magnetic field extrapolation in open space above in open space above

spherical surfacespherical surface

He Han, Wang HuaningHe Han, Wang HuaningNAOC, BeijingNAOC, Beijing

2005-07-112005-07-11

magnetic field extrapolationmagnetic field extrapolation

• Potential field modelPotential field model

• Force-free field Force-free field modelmodel

BrB )(

0 B

)(~ BJ

field line

B

Extrapolation scheme:Extrapolation scheme:

• reliability and accuracyreliability and accuracy

in open space above spherical surfacein open space above spherical surface

Tool:Tool:• the axisymmetric nonlinear force-free magnthe axisymmetric nonlinear force-free magn

etic fields solutionsetic fields solutions ((Low, B.C. and Lou, Y.Q., 1990, Low, B.C. and Lou, Y.Q., 1990, Astrophys. J.Astrophys. J. 352352, 343 , 343 ))

Aim:Aim:

• the boundary integral equation representationthe boundary integral equation representation ((Yan Yihua, Sakurai, T. 2000, Yan Yihua, Sakurai, T. 2000, Solar Phys.Solar Phys., , 195195, 89, 89))

Boundary integral equation Boundary integral equation ((Yan Yihua, Sakurai, T. 2000, Yan Yihua, Sakurai, T. 2000, Solar Phys.Solar Phys., , 195195, 89, 89))

d)( 00

n

Y

nYi B

BB

0d)( 22 BBB Y

irr

4

)cos(),( Y )( ir

0R0BB :

:0 B

r

)( 2 rOB

BB

Variable pointVariable point

Fixed point i

),,( iii zyx),,( iiir

),,( zyx),,( r

rir

n)( ir

axisymmetric nonlinear force-free magnetic fieldsaxisymmetric nonlinear force-free magnetic fields solutionssolutions ((Low, B.C. and Lou, Y.Q., 1990, Low, B.C. and Lou, Y.Q., 1990, Astrophys. J.Astrophys. J. 352352, 343., 343.))

φθrB ˆ)1(

ˆ)1(

ˆd

d12/12

2

2/123 aPPP

r

01

)1()1(2

12

2

22

nP

n

naPnn

d

Pd

0n)(, mnPP ,3 ,2 ,1 ,0m

Field lines of the solutionsField lines of the solutions

0<n<1, m=1 force-free fieldN=1, m=0 potential field

Some consideration about field Some consideration about field lines selectionlines selection

• distance away from the center of the distance away from the center of the spherical surface: 1.1Rspherical surface: 1.1R00, 1.6R, 1.6R00, 2.0R, 2.0R00, , 2.5R2.5R00

• Observed average variation of Observed average variation of

6×106×10-11-11 +/- 3×10 +/- 3×10-9-9 m m-1-1

if we define the solar radius as the length if we define the solar radius as the length unit, the range of absolute value of isunit, the range of absolute value of is

0 ~ 2.10 ~ 2.1

0

(A.A. Pevtsov, R.C. Canfield, T.R. Metcalf, 1995, (A.A. Pevtsov, R.C. Canfield, T.R. Metcalf, 1995, Astrophys. J.Astrophys. J. 440440, L109), L109)

Field lines selectedField lines selected

n=1, m=0 potential fieldalpha= 0alpha_0= 0

n=0.999, m=1 force-free fieldalpha = 0.044alpha_0= 0.035

Field lines selectedField lines selected

n=0.9, m=1 force-free fieldalpha= 0.45alpha_0= 0.36

n=0.5, m=1 force-free fieldalpha = 1.26alpha_0= 1.01

results – surface integralresults – surface integral

• grid number 500(theta)×100(phi)grid number 500(theta)×100(phi)• appropriate lambda values can be appropriate lambda values can be

found for every field points examinedfound for every field points examined• lambda values corresponding to 3 lambda values corresponding to 3

components of B are not uniform components of B are not uniform except for potential field situationexcept for potential field situation

• appropriate lambda values are appropriate lambda values are generally not uniquegenerally not unique

results – surface integralresults – surface integral

potential field1.1R ii n

Y

nYY BB

B

d)()( 00

results – surface integralresults – surface integral

n=0.999, m=11.1R ii n

Y

nYY BB

B

d)()( 00

results – surface integralresults – surface integral

n=0.5, m=11.1R ii n

Y

nYY BB

B

d)()( 00

Results – volume integralResults – volume integral• Grid number: 500(theta)×100(phi)Grid number: 500(theta)×100(phi)

• Oscillation around zeroOscillation around zero

• Tendency to convergenceTendency to convergence

• Larger value of alpha means more Larger value of alpha means more computing timecomputing time

Results – volume integralResults – volume integralN=0.999, alpha_0= 0.035

Results of volume integral

X Axis: grid number (r)

Relative error (percent) with respect to B

Results – volume integralResults – volume integralN=0.5, alpha_0= 1.01

Results of volume integral

X Axis: grid number (r)

Relative error (percent) with respect to B

conclusionconclusion• The fields of Low’s solutions we used can be The fields of Low’s solutions we used can be

represented by the boundary integral represented by the boundary integral equation. equation.

This result is helpful to increase the reliability This result is helpful to increase the reliability of the method for force-free field of the method for force-free field extrapolation.extrapolation.

• this technique is valid at the large distance this technique is valid at the large distance from the spherical surface, field point that from the spherical surface, field point that locates at 2.5 radius has been checkedlocates at 2.5 radius has been checked

• For complicated force-free field with large For complicated force-free field with large range of alpha values, much more computing range of alpha values, much more computing time is needed to give meaning result.time is needed to give meaning result.

Thanks !Thanks !