Fast Magnetic Reconnection

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

2. Outline • Ideal MHD • Fast Magnetic reconnection: astrophysical settings, weak solutions? • Petschekvs 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. Stationary solution. What BC? Where? Biskamp 1996

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 • S0 is magnetic Reynolds number, often 108 or larger. • Means reconnection just cannot happenon 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 2nd order TVD, FCT (conserved div B). • Sunnyvale cluster (1600 core CITA) • Grids up to 8003 • Range of initial conditions, resolution, geometric ratios.

16. Initial Conditions

17. Pang et al, in prep

22. 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

23. 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?