1. Current sheets formation along magnetic separators in 3d Dana Longcope Montana State University Thanks: I Klapper, A.A. Van Ballegooijen NSF-ATM

2. The Coronal Field 1 MK plasma (TRACE 171A) Magnetic field @ photosphere (MDI) 8/10/01 12:51 8/11/01 9:25 Lower boundary: B z confined to source regions Corona: complex inter-connections between sources B z =0

3. The Coronal Field 8/10/01 12:518/11/01 17:39 8/11/01 9:25 1 MK plasma (TRACE 171A) Magnetic field @ photosphere (MDI) Evolution: lower boundary changes slowly (+30 hrs)

4. Outline I. Lowest energy magnetic field contains current sheets localized to separators (Flux-Constrained Equilibrium) II. Boundary motion drives field  singular equilibrium via repeated Alfven wave reflection

5. I. Equilibrium Force-Free Equilibrium: Minimizes Mag. Energy* Constraints: (minimize subject to…) None Ideal motion (line-tied to boundary)   potential general FFF *subject to BC: B z (x,y,0) = f(x,y)

6. A new type of constraint Boundary field B z (x,y,0) = f(x,y) : assume discrete sources B z =0 (Longcope 2001, Longcope & Klapper 2002)

7. A new type of constraint The domain graph Summarizes the Magnetic connectivity N5 N4 N6P3 P2 P1 Constrain coronal flux interconnecting sources

8. Structure of Constraint N5 N4 N6P3 P2 P1 Domain D 16 connects P1  N6 Flux in Domain D 16 : y 16 (want to specify this) Flux in source 6: F 6 (set by BC)

9. Structure of Constraint N5 N4 N6P3 P2 P1 Domain D 16 connects P1  N6 Flux in Domain D 16 : y 16 (want to specify this) Flux in source 6: F 6 (set by BC) Inter-related through incidence matrix of graph: i.e. - y 16 - y 26 - y 36 = F 6

10. A: N c = N d – N s + 1 Structure of Constraint N5 N4 N6P3 P2 P1 Q: How many domain fluxes y ab may be independantly specified? Number of domains Number of sources here N c = 7 – 6 + 1 = 2

11. Structure of Constraint N5 N4 N6P3 P2 P1 A: N c = N d – N s + 1 Specifying fluxes of Nc chords reduces graph to a tree y 14 =Y 2 y 34 =Y 1 Q: How many domain fluxes y ab may be independantly specified?

12. Structure of Constraint N5 N4 N6P3 P2 P1 A: N c = N d – N s + 1 Specifying fluxes of Nc chords reduces graph to a tree … all remaining fluxes follow from flux balance: y 36 =F 3 -Y 1 … etc. y 14 =Y 2 y 34 =Y 1 Q: How many domain fluxes y ab may be independantly specified?

13. How to apply constraints Topology of the potential field:* Extrapolate from bndry: Locate all magnetic null points B=0 Trace spine field lines to spine sources *same topology will apply to non-potential fields

14. The skeleton of the field Trace all fan field lines from null Form sectors Joined at separators A separator connects +  - null points

15. The skeleton of the field Trace all fan field lines from null Form sectors Joined at separators A separator connects +  - null points

16. The skeleton of the field Trace all fan field lines from null Form sectors Joined at separators A separator connects +  - null points

17. The skeleton of the field Trace all fan field lines from null Form sectors Joined at separators A separator connects +  - null points

18. Individual Domains Domain linking P a  N b must be bounded by sectors +’ve sectors: P a @ apex -’ve sectors: N b @ apex Sectors intersect @ closed separator circuit Circuit encircles domain

19. N5 N4 N6P3 P2 P1 y 34 =Y 1 Formulating the constraint Locate separator circuit Q i encircling domain D i : Constraint functional:

20. The Constrained Minimization Minimize: Lagrange multiplier Non-potential field: separator curve: Q i  annular ribbon x i ( x,h) Singular density d -function All con- straints

21. Vary Require stationarity: d C = 0 Get Euler-Lagrange equation The Variation Singular current density, confined to separator ribbon i

22. Flux Constrained Equilibria Potential field (w/o constraints): Y i = Y i (v) Non-potential field: Y i = Y i (v) + DY i i=1,…,N c Free Energy in flux-constrained field: General FFF: N4 N2 P3 P1 y 23 = Y 1

23. Flux Constrained Equilibria Min’m energy subject to N c constraints  N c fluxes are parameters: Y i Current-free within each domain Singular currents* on all separators N4 N2 P3 P1 Y 1 <Y 1 (v) * Always ideally stable!

24. II. Approach to Equilibrium Separator defined through footpoints  No locally distinguishing property* Singularity must build up through repeated reflection of information between footpoints * In contrast to 2 dimensions: B=0 @ X-point (Longcope & van Ballegooijen 2002)

25. Dynamics Illustrated Equilib. field maps sources to merging layer Long (almost straight) coronal field (RMHD) Maps between merging layers Sources on end planes

26. Dynamics Illustrated N3 N4 Sep’x from null on p-sphere S-S-

27. Dynamics Illustrated Sources move (rotation) N3 N4

28. Dynamics Illustrated Sources move (rotation) N3 N4

29. Dynamics Illustrated Sources move (rotation) vfvf BfBf Initiates Alfven pulse (uniform rotation) Dq N3 N4

30. Dynamics Illustrated Sources move (rotation) vfvf BfBf Initiates Alfven pulse (uniform rotation) Dq N3 N4

31. Dynamics Illustrated Sources move (rotation) vfvf BfBf Initiates Alfven pulse (uniform rotation) Dq N3 N4

32. Dynamics Illustrated Sources move (rotation) vfvf BfBf Initiates Alfven pulse (uniform rotation) Dq N3 N4

33. Dynamics Illustrated vfvf BfBf

34. Impact at Opposing End vfvf BfBf P2 P1 Motion at merging height mapped down to photosphere S+S+ c.c rotation

35. Impact at Opposing End P2 P1 S+S+ Photosphere: fixed source positions, moveable interiors Merging height: No motion across sep’x S + Free motion w/in source-regions

36. Impact at Opposing End P2 P1 Photosphere: fixed source positions, moveable interiors Vorticity sheet @ sep’x S+S+ Merging height: No motion across sep’x S + Free motion w/in source-regions c.clockwise motion in each region

37. Impact at Opposing End P2 P1 N3 N4 Image of opposing separator is distorted by boundary motion S+S+ S-S-

38. Impact at Opposing End P2 P1 N3 N4 Image of opposing separator is distorted by boundary motion S-S- S+S+

39. Impact at Opposing End P2 P1 N3 N4 Image of opposing separator is distorted by boundary motion S-S- S+S+

40. Impact at Opposing End P2 P1 N3 N4 Intersection of separatrices: The Separator Ribbon S-S- S+S+

41. The Reflected Wave Singular Alfven pulse: Voricity/Current sheet confined to S + P2 P1 S+S+

42. The Reflected Wave Separator ribbon left in wake of reflection S+S+ S-S-

43. Repeated Reflection S+S+ S-S- z=0z=L CS along S + reflects from z=0 CS along S - reflects from z=L Repeated reflection retains only current on  separator ribbon Wave w/ current on ribbon - perfectly reflected

44. The Final Current Sheet Interior CS (z=0) B^B^ Helical pitch, maps S +  S - z=0  z=L S+S+ S-S-

45. The Final Current Sheet Interior CS Helical pitch, maps S +  S - z=0  z=L S+S+ S-S- B^B^

46. The Final Current Sheet Interior CS Helical pitch, maps S +  S - z=0  z=L S+S+ S-S- B^B^

47. The Final Current Sheet Interior CS Helical pitch, maps S +  S - z=0  z=L S+S+ S-S- B^B^

48. The Final Current Sheet Interior CS Helical pitch, maps S +  S - z=0  z=L S+S+ S-S- Flux constrained equilib. I set by e.g y 23 B^B^

49. Conclusions New class of constraints: domain fluxes Flux constrained equilibria have CS on all separators Equilibrium is approached by repeated Alfven wave reflections from boundary

50. Implications Coronal field tends toward singular state Current sheets are ideally stable Magnetic reconnection can –Eliminate constraint –Decrease magnetic energy Free energy depends on flux in N C different fluxes within corona.

51. Individual Domains Field lines from one source fan outward… topological ball of field lines Opposing sources at surface of ball Sectors partition ball

52. Individual Domains Field lines from one source fan outward… topological ball of field lines Opposing sources at surface of ball Sectors partition ball

53. Individual Domains Field lines from one source fan outward… topological ball of field lines Opposing sources at surface of ball Sectors partition ball

54. Individual Domains Domain = Intersection of 2 balls… Intersection is a closed separator circuit Circuit girdles domain Negative sector in negative ball Positive sector in positive ball

55. A 3d example N s =6 sources N d =7 domains  N c =2 circuits 4 nulls (A 1 …B 2 ) 2 null-null lines C2C2 C1C1

56. A current sheet Isolating loop Q 1 links domain D 34 Current ribbon for    (v) -- vertical