1. Fast Magnetic Reconnection B. Pang U. Pen E. Vishniac

2. Outline Ideal MHD Fast Magnetic reconnection: astrophysical settings, weak solutions? Petschek vs Sweet-Parker Numerical Experiments 2-D instability, 3-D stable?

3. Ideal MHD Conducting fluid: e.g. sun, earth core, interplanetary medium. Applies to most fluids in the universe. Analogous to Euler/Navier Stokes equations Derived from kinetic theory Perfect fluid with 8 variables: density, v, E, b Hyperbolic conservation law Some complications due to div B constraint Weak solutions with diverse shock structures Numerically tractable with shock capturing techniques

4. Resistive MHD Physics: ohmic resistive term allows field lines to slip, smoothes discontinuities on a resistive scale. Mathematically: diffusive term is parabolic, smoothes out weak solutions. Allows “Magnetic Reconnection”: the topological change of field lines. Analogous to viscosity in Navier-Stokes In practice, η always too small: need weak solutions?

5. Astrophysical reconnection Magnetic field topology change A wide range of settings has huge magnetic Reynolds numbers, but fast apparent reconnection ISM Solar flares Inter-planetary medium, magnetosphere Dynamo Requires at least 2-D to describe

6. Biskamp 1996 Stationary solution. What BC? Where?

7. Petschek Instability? Stationary solution determined by singular X point. Well posed boundary conditions? Numerical experiments in 2-D have shown Petschek solution unstable. Stability depends on the resistive limit taken at the X-point: unstable for Ohmic resistivity (leading order closure relation from BGK). Turns into slow Sweet-Parker solution

8. Sweet Parker Biskamp 1996

9. Problems S 0 is magnetic Reynolds number, often 10 8 or larger. Means reconnection just cannot happen on the observed timescales

10. Stalemate Numerical experiments show Petschek solution unstable: requires singular X point Sweet Parker almost inevitable from mass conservation and energetics viewpoint (?) BUT: fast reconnection is known to occur. Perhaps problem with: ideal MHD, boundary conditions, energetics?

11. Resolutions Two conceptual strategies: 1. non-ideal MHD effect (anomolous resistivity, etc), and 2. exploration of 3-D (Lazarian & Visniac 1999). The easily observable settings have long MFP, so ideal MHD might not apply. But plenty of settings (e.g. solar interior) are well into the ideal MHD limit. Our work looks at changing boundary conditions to a causal framework.

12. Conceptual Paradox Is it possible that infinitesimal fields (or points) can hold up flows? What is the cause and effect? The fluid pulling the field or the field pulling the fluid?

13. Global Structure Can dynamical processes drive the solution towards Petschek? Effects of 3-D?

14. 3-D Numerical Experiment

15. Numerical Laboratory TVD-MHD code (Pen, Arras & Wong 2003): solves ideal MHD equations using 2 nd order TVD, FCT (conserved div B). Sunnyvale cluster (1600 core CITA) Grids up to 800 3 Range of initial conditions, resolution, geometric ratios.

16. Initial Conditions

17. Pang et al, in prep

23. Discussion Fast reconnection ingredients: Periodic box with two interacting X points Dynamical moving X-points 3-D: allows loops to decay, X-points to bifurcate Constructive example for fast reconnection Robust to change in resolution, initial conditions, geometry Worth testing with other codes

24. Conclusions Constructive example of fast MHD reconnection. Global dynamics drives reconnection process. Puzzles: it is possible that 2-D unstable process is stable in 3-D? What determines the solution near X point? B.C. at X point, or far away? Weak solution of ideal MHD? Dependence on microscopic parameters?