Current Sheet and Vortex Singularities: Drivers of Impulsive Reconnection

1. Current Sheet and Vortex Singularities: Drivers of Impulsive Reconnection A. Bhattacharjee, N. Bessho, K. Germaschewski, J. C. S. Ng, and P. Zhu Space Science Center Institute for the Study of Earth, Oceans, and Space University of New Hampshire Isaac Newton Institute, Cambridge University, August 9, 2004

2. High-Performance Computing Tools • Magnetic Reconnection Code (MRC), based on extended two-fluid (or Hall MHD) equations, in a parallel AMR framework (Flash, developed at the University of Chicago). • EPIC, a fully electromagnetic 3D Particle-In-Cell code, with explicit time-stepping. MRC is our principal workhorse and two-fluid equations capture important collisionless/kinetic effects. Why do we need a PIC code? • Generalized Ohm’s law • Thin current sheets, embedded in the reconnection layer, are subject to instabilities that require kinetic theory for a complete description.

3. Adaptive Mesh Refinement

4. Effectiveness of AMR High effective resolution Example: 2D MHD/Hall MHD Efficiency of AMR log Level # grids # grid points 0 1 70225 1 83 146080 2 103 268666 3 153 545316 4 197 1042132 5 404 1926465 6 600 1967234 Grid points in adaptive simulation: 6976118 Grid points in non-adaptive simulation: 268730449 Ratio 0.02

5. Impulsive Reconnection: The Trigger Problem Dynamics exhibits an impulsiveness, that is, a sudden change in the time-derivative of the reconnection rate. The magnetic configuration evolves slowly for a long period of time, only to undergo a sudden dynamical change over a much shorter period of time. Dynamics is characterized by the formation of near-singular current sheets in finite time. Examples Sawtooth oscillations in tokamaks and RFPs Magnetospheric substorms Impulsive solar and stellar flares

6. Sawtooth crash in tokamaks (Yamada et al., 1994)

7. Sawtooth events in MST (Almagri et al., 2003)

8. Magnetospheric Substorms

10. Current Disruption in the Near-Earth Magnetotail (Ohtani et al., 1992)

11. Impulsive solar/stellar flares

12. 2D Hall MHD: m=1 sawtooth instability Two-Field Reduced Model for large guide field and low plasma beta (Schep/Pegoraro/Kuvshinov 1994) (Grasso/Pegoraro/Porcelli/Califano 1999) Equilibrium:

13. Resolving the current sheet zoom zoom

14. current density Current sheet collapse, s = 0 1/current sheet width t

15. magnetic flux function Island width t

16. Island equation or If dx and dL attain constant values and are of order of de (or ), the island equation becomes (Ottaviani & Porcelli 1993 for rs = 0 ): with ,

17. Island equation c.f. simulation Solid: simulation,dashed: island equation, cJ = 0.025,

18. Scaling of the reconnection rate: Is it Universal? Consider scaling of the inflow velocity: It has been argued that f ~ 0.1 [Shay et al., 2004], in a universal asymptotic regime, independent of system size and dissipation mechanism. Using the island equation in the asymptotic regime: Note that dL is of the order of de (or ). f depends on parameters de, and k. It also depends weakly on time through l(t) ~ 1 in the nonlinear phase. Numerically f is seen to be of the order of 0.1 for certain popular cloices of simulation parameters, but this is not universal.

19. Magnetospheric Substorms

20. Observations (Ohtani et al. 1992) Hall MHD Simulation Cluster observations: Mouikis et al., 2004

21. J y y

23. Ux Uz Highly Compressible Ballooning Mode in Magnetotail (Voigt model) x = -1 to -16 RE z = -3 to 3 RE ky= 25*2 e = 126 growth rate:0.2 t=5.8 t=29

24. ky and Dependence

27. High-symmetry flow (Pelz 1997) t=.49 t=0 t=.33

28. Vorticity in the high-symmetry flow Vorticity 2D and 1D cuts

29. Growth of vorticity Distortion of vortices

30. pressure vorticity

33. Resistive Tearing Modes in 2D Geometry Equilibrium Magnetic field assumed to be either infinite or periodic along z x Neutral line at y=0 y Time Scales Lundquist Number Tearing modes (Furth, Killeen and Rosenbluth, 1963)

34. Collisionless Tearing Modes at the Magnetopause (Quest and Coroniti, 1981) • Electron inertia, rather than resistivity, provided the mechanism for breaking field lines, but the magnetic geometry is still 2D. • Growth rates calculated for “anti-parallel” ( ) and “component” ( ) tearing. It was shown that growth rates for “anti-parallel” merging were significantly higher. • Model provided theoretical support for global models in which global merging lines were envisioned to be the locus of points where the reconnecting magnetic fields were locally “anti-parallel” (e.g. Crooker, 1979)

35. Magnetic nulls in 3D play the role of X-points in 2D Spine Fan (Lau and Finn, 1990)

36. Towards a fully 3D model of reconnection • Greene (1988) and Lau and Finn (1990): in 3D, a topological configuration of great interest is one that has magnetic nulls with loops composed of field lines connecting the nulls. • The null-null lines are called separators, and the “spines” and “fans” associated with them are the global 3D separatrices where reconnection occurs. • For the magnetosphere, the geometrical content of these ideas were already represented in the vacuum superposition models of Dungey (1963), Cowley (1973), Stern (1973). However, the vacuum model carries no current, and hence has no spontaneous tearing instability. The real magnetopause carries current, and is amenable to the tearing instability.

37. Equilbrium B-field

38. Perturbed B-lines

39. Equilibrium Perturbed Field lines penetrating the spherical tearing surface

40. Equilbrium Current Density, J

42. Features of the spherical tearing mode • The mode growth rate is faster than classical 2D tearing modes, scales as S-1/4 (determined numerically from compressible resistive MHD equations). • Perturbed configuration has three classes of field lines: closed, external, and open (penetrates into the surface from the outside). • Tearing eigenfunction has global support along the separatrix surface, not necessarily localized at the nulls. • The separatrix is global, and connects the cusp regions. Reconnection along the separatrix is spatially inhomogeneous. Provides a new framework for analysis of satellite data at the dayside magnetopause. Possibilities for SSX, which has observed reconnection involving nulls.

43. Solar corona astron.berkeley.edu/~jrg/ ay202/img1731.gif www.geophys.washington.edu/ Space/gifs/yokohflscl.gif

44. Solar corona: heating problem photosphere corona Temperature Density Time scale Magnetic fields (~100G) --- role in heating? ~ ~  Alfvén wave  current sheets

45. Parker's Model (1972) Straighten a curved magnetic loop Photosphere

46. Reduced MHD equations low  limit of MHD

47. Magnetostatic equilibrium (current density fixed on a field line) c. f. 2D Euler equation A, J, zt Existence theorem: If is smooth initially, it is so for all Time. However, Parker problem is not an initial value problem, but a two-point boundary value problem.

48. Footpoint Mapping Identity mapping: or e. g. uniform field For a given smooth footpoint mapping, does more than one smooth equilibrium exist?

49. A theorem onParker's model For any given footpoint mapping connected with the identity mapping, there is at most one smooth equilibrium. A proof for RMHD, periodic boundary condition in x (Ng & Bhattacharjee,1998)

50. Implication An unstable but smooth equilibrium cannot relax to a second smooth equilibrium, hence must have current sheets.