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Preprocessing of Vector Magnetograms for Extrapolation of Coronal Magnetic Fields

Preprocessing of Vector Magnetograms for Extrapolation of Coronal Magnetic Fields. General Discussion. Assumption: force-free:  with Force-free condition holds above 400km

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Preprocessing of Vector Magnetograms for Extrapolation of Coronal Magnetic Fields

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  1. Preprocessing of Vector Magnetograms for Extrapolation of Coronal Magnetic Fields

  2. General Discussion • Assumption: force-free:  with • Force-free condition holds above 400km • Measurements might be affected by non-magnetic effects • Extrapolation should take into account of plasma conditions

  3. Test case & Reality Low&Lou Bastille Event

  4. Preprocessing • Problems on real magnetograms • Force-free ? • Noise (lack of smootheness) • Hidden problems? • Philosophy: • Different mgms, close to the original one • Staying within given errorbars

  5. Preprocessing: Minimization • Force-freeness [Molodenskii (1969)] • Magnetic-field Stress Tensor T with ext. forces F • Force-free condition 

  6. Preprocessing: Minimization • Force-freeness • Transforming surface integral into an integral along contour L • Since the magnetic field weakens as the diameter of contour L increases, surface integral will approach to zero • Magnetic field decreases with height

  7. Preprocessing: Minimization • Torque-Freeness • Same procedure, but multiply vectorially with r • Then integrate over surface S, the z-component of the resulting equation • Seperating out the divergence part • Similar as before

  8. How to Preprocess: Minimization • Force-free • Torque-free

  9. Preprocessing: Minimization • Smootheness • Current Method: Laplacian 4th order • Pro: smoothes the Magnetogram, because of averaging the discontinuities • Contra: flattens the whole Magnetogram

  10. Numerical Expressions • L* : treshold to control smoothing • L4=0  no curvature

  11. Normalizations

  12. Minimizing functional

  13. Method of Minimization: Simulated Annealing • Usual Minimizing(Newton-Raphson, etc.) of a function leads to nearest local minimum, depends on starting point • Current Problem: • functional with several constraints • Several local minima • Simulated annealing finds also local minima, but leaves a chance to get out of it, in order to find better local minima

  14. How does Simulated Annealing Work? • Choose randomly new value of Bx,y,z • But within a small interval (~± 0.5%) • Calculate L, check if LNEW ≤LOLD • TRUE: Take Bnewx,y,z & forget Boldx,y,z, • FALSE: Take with probability ~ exp{-(LNEW - LOLD)} • Repeat • To scan the whole domain • Domain within given errorbars of the measurement (depends if transversal or normal component of Magnetogram)

  15. Results: preprocessing just force- & torque-freenessQuality of force- & torque-freeness

  16. Results: preprocessing just force- & torque-freenessMagnetogram

  17. Results: preprocessing both constraintsMagnetogram

  18. Results: preprocessing both constraints Quality of force-freeness

  19. Compared Results

  20. Conclusions • Extrapolation successful, but on real cases lack of force-freeness (σJ ~ 10-1) • Preprocessing: • successful to make magnetogram force-free  better agreement with initial assumption  slightly better results, but not enough • Difficulties to smooth without flattening  have to find a better smoothing term!

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