Solar magnetic fields: force-free fields

1. Solar magnetic fields: force-free fields Anna (Ania) Malanushenko

2. Force-Free Fields • Know B in photosphere • Want to know B in the corona

3. Force-free field: definition Recall from the prev. lecture: neglect in the corona Equilibrium in the corona: Recall: Force-free field:

4. Force-free field: overview • – potential field • – “linear” or “const-” field • – non-linear field, most general case

5. Potential field – much like electrostatics! Need to solve Laplace’s equation: A boundary value problem:

6. Potential field Boundary value problem: Ways to solve(some): • – Point charge method (yes!)(in half space) – Green’s function method (in half space) • – Fourier transform method(in a box) • – Jacobi relaxation (in arbitrarily shaped volume)

7. Potential field: an application , potential field source surface model

8. Potential field: an application , potential field source surface model

9. Potential field: an application , potential field source surface model

10. Force-free field: overview • – potential field • – “linear” or “const-” field • – non-linear field, most general case • works generally well – especially on large scales • misses much in interesting regions

11. Constant- field Need to solvea system of three equations. Boundary value problem: Example: thin twisted cylinder Ways to solve(some): • – Point charge method(in half space) • – Green’s function method (in half space) • – Fourier transform method(in a box)

12. Constant- field • Meaning of   <0   >0 0

13. Constant- field Fourier method (Nakagawa & Raadu, 1972) Magnetic energy: • Coefficients from the lower boundary:

14. …at the times of old

15. Linear Force-Free Fields • Limited to Cartesian geometry (a.f.a.i.k.) • Field either limited to a box, or has infinite energy in half space • =const does not always work well • Easy to solve for • Existing & unique solution Fig. 1 from Burnette et. al., 2004

16. Non-Linear Force-Free Fields • For const, B=B is non-linear. • Side remarks: • B=0, i.e., =const along every field line. • Can obtain  from the photosphere: Fig. 1 from Burnette et. al., 2004

17. Non-Linear Force-Free Fields Problems solving B=B: • Data… azimutal ambiguity

18. Non-Linear Force-Free Fields Problems solving B=B: • Data… azimutal ambiguity • Noise in & B=0 • True in the corona, but not the photosphere B(z=0) (z=0)

19. Plasma:  Solar wind: low density, but field strength is even lower Corona: low density, magnetic field determines “what plasma does” Chromosphere: both plasma and field are equally important Photosphere & below: high density, plasma motions determine “what the field does” (Gary 2001)

20. Non-Linear Force-Free Fields Problems solving B=B: • Data… azimutal ambiguity • Noise in & B=0 • True in the corona, but not the photosphere • Non-linear system • The problem is ill-posed?.. • And nonetheless… B(z=0) (z=0)

21. Non-Linear Force Free Fields: Many Methods On The Market

22. NL-FFF: Magneto-Frictional (1/3) • We want B=B • Start with =0 & given B at the boundary • BB => JB there is Lorentz force, FL=JB • Move it! v(J B), B/t=(vB) • Hope it stops moving eventually… then FL=0

23. NL-FFF: Magneto-Frictional (1/3) Ballegooijen 2004 untwisted arcade w. cavity and non-uniformly twisted filament inside

24. NL-FFF: Magneto-Frictional (1/3) Ballegooijen 2004

25. NL-FFF: Optimization (2/3) • Want two things: B=B and B=0 • Minimize the integral: • Let B=B(t) => introduce “force” F(B): if B/t=kF, then dL/dt=0 when F=0. • B(n+1)=B(n)+F(n)t • Hope it converges…

26. NL-FFF: Optimization (2/3) Sun et al, 2012

27. NL-FFF: Grad-Rubin (3/3) • Pick a field B(0) (e.g., potential, =0) • Know “real”  from boundaries • Given B(n), find (n) from B(n) (n) =0 and B.C. on  • Find B(n+1) from B(n+1)=(n)B(n), keeping B(n+1)=0 • Repeat • Hope it converges…

28. NL-FFF: Do they work? Most recent: DeRosa et al, 2010 • AR 10953, 2007/04/30 22:20 UT • z=0: Bz(MDI), B(Hinode) • Coronal loops: SXR (XRT), EUV (STEREO)

29. NL-FFF: do they work? • There is a gridded 3D domain • There is B on the boundary • Find B in the volume, such that B=B and B=0 • Judge “how good” is B • Compare results for several methods

30. Results: pictures

31. Results: Numbers • E/Ep – Ep is the minimal energy for a FFF • <sin(J,B)>, current-weighted (that is, how well J=B works) • <|f(r)|>, |f(r)||B|/|B| (that is, how well B=0 works) • <> – between coronal loops (via STEREO) and field lines

32. Comparison XRT (core) STEREO (edges) B-J iteration Grad-Rubin Optimization Magneto-frictional

33. Results: favorite is Wh- yellow=good alignment, <5, red=bad alignment, >45

34. Results: “favorite” is Wh- • FL’s do not match  on the other end • “Censors” (=0) half of magnetogram! • Does not do better with EUV loops than the potential field

35. Non-linear force-free fields, morale: • None of the models did better than the potential field for EUV loops • All of them have issues with B and B-B • Fiddling with the data preprocessing did not help • Something completely new is needed

36. Basic Problems to Address • Data noise: • Azimutal ambiguity • Noise in  • Physics: • Photosphere is not force-free • Computational: • No data on the side & top boundaries Existing methods can’t deal with that. The “something new” has to address all these