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He Han, Wang Huaning NAOC, Beijing 2005-07-11

The validity of the boundary integral equation for magnetic field extrapolation in open space above spherical surface. He Han, Wang Huaning NAOC, Beijing 2005-07-11. magnetic field extrapolation. Potential field model Force-free field model. field line. Extrapolation scheme:.

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He Han, Wang Huaning NAOC, Beijing 2005-07-11

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  1. The validity of the boundary integral equation for magnetic field extrapolation in open space above spherical surface He Han, Wang Huaning NAOC, Beijing 2005-07-11

  2. magnetic field extrapolation • Potential field model • Force-free field model field line

  3. Extrapolation scheme: • the boundary integral equation representation (Yan Yihua, Sakurai, T. 2000, Solar Phys., 195, 89) • reliability and accuracy in open space above spherical surface Aim: Tool: • the axisymmetric nonlinear force-free magnetic fields solutions (Low, B.C. and Lou, Y.Q., 1990, Astrophys. J.352, 343 )

  4. Boundary integral equation (Yan Yihua, Sakurai, T. 2000, Solar Phys., 195, 89)

  5. Variable point Fixed point i

  6. axisymmetric nonlinear force-free magnetic fieldssolutions (Low, B.C. and Lou, Y.Q., 1990, Astrophys. J.352, 343.)

  7. Field lines of the solutions N=1, m=0 potential field 0<n<1, m=1 force-free field

  8. Some consideration about field lines selection • distance away from the center of the spherical surface: 1.1R0, 1.6R0, 2.0R0, 2.5R0 • Observed average variation of 6×10-11 +/- 3×10-9 m-1 if we define the solar radius as the length unit, the range of absolute value of is 0 ~ 2.1 (A.A. Pevtsov, R.C. Canfield, T.R. Metcalf, 1995, Astrophys. J.440, L109)

  9. Field lines selected n=0.999, m=1 force-free field alpha = 0.044 alpha_0= 0.035 n=1, m=0 potential field alpha= 0 alpha_0= 0

  10. Field lines selected n=0.5, m=1 force-free field alpha = 1.26 alpha_0= 1.01 n=0.9, m=1 force-free field alpha= 0.45 alpha_0= 0.36

  11. results – surface integral • grid number 500(theta)×100(phi) • appropriate lambda values can be found for every field points examined • lambda values corresponding to 3 components of B are not uniform except for potential field situation • appropriate lambda values are generally not unique

  12. results – surface integral potential field 1.1R

  13. results – surface integral n=0.999, m=1 1.1R

  14. results – surface integral n=0.5, m=1 1.1R

  15. Results – volume integral • Grid number: 500(theta)×100(phi) • Oscillation around zero • Tendency to convergence • Larger value of alpha means more computing time

  16. Results – volume integral N=0.999, alpha_0= 0.035 Results of volume integral X Axis: grid number (r) Relative error (percent) with respect to B

  17. Results – volume integral N=0.5, alpha_0= 1.01 Results of volume integral X Axis: grid number (r) Relative error (percent) with respect to B

  18. conclusion • The fields of Low’s solutions we used can be represented by the boundary integral equation. This result is helpful to increase the reliability of the method for force-free field extrapolation. • this technique is valid at the large distance from the spherical surface, field point that locates at 2.5 radius has been checked • For complicated force-free field with large range of alpha values, much more computing time is needed to give meaning result.

  19. Thanks !

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