Lagrangian approach to reconnection and topology change

We employ well-known concepts from statistical physics, quantum field theories, and general topology to study magnetic reconnection and topology change and their connection in incompressible flows in the context of an effective field theory without appealing to magnetic field lines. We consider the dynamical system corresponding to wave packets moving with Alfvén velocity x[over ̇](t):=V_{A}(x,t) whose trajectories x(t) define pathlines, which naturally provides a mathematical way to estimate the rate of magnetic topology change. A considerable simplification is attained, in fact, by directly employing well-known concepts from hydrodynamic turbulence without appealing to the complicated notion of magnetic field lines moving through plasma, which may prove even more useful in the relativistic regime. Continuity conditions for magnetic field allow rapid but continuous divergence of pathlines, shown to imply reconnection, but not discontinuous divergence, which would change topology. Thus, topology can change only due to time-reversal symmetry breaking, e.g., by dissipative effects. In laminar and even chaotic flows, the separation of pathlines at all times remains proportional to their initial separation, argued to correspond to slow reconnection, and topology changes by dissipation with a rate proportional to resistivity. In turbulence, pathlines diverge superlinearly with time independent of their initial separation, i.e., fast reconnection, and magnetic topology changes by turbulent dissipation with a rate independent of small-scale plasma effects. The crucial role of turbulence in enhancing topology change and reconnection rates originates from its ability to break time-reversal invariance and make the flow superchaotic. In fact, due to the loss of Lipschitz continuity of the magnetic field in turbulence, pathlines separate superlinearly even if their initial separation tends to vanish, unlike deterministic chaos. This superchaotic behavior is an example of spontaneous stochasticity in statistical physics, sometimes called the real butterfly effect in chaos theory to distinguish it from the butterfly effect, in which trajectories can diverge exponentially only if initial separation remains finite. If 3D reconnection is defined as magnetic topology change, it can be fast only in turbulence where both reconnection and topology change are driven by spontaneous stochasticity, independent of any plasma effects. Our results strongly support the Lazarian-Vishniac theory of turbulent reconnection.