Electron vortex critical points and the global magnetic reconnection rate

Magnetic reconnection is a universal and important plasma process that converts magnetic energy into particle energy in a way that redistributes particles' magnetic field line connections. It is still not fully understood what controls the reconnection rate and why it appears to converge, at least in the steady state, to a ubiquitous value regardless of external conditions. In this article, we derive a theoretical expression for a time-dependent reconnection rate in general magnetic reconnection. We show that, inside the diffusion region, the electron vorticity forms a critical point whose eigenvalue ratio controls the electron fluid's linkage to a magnetic field line. The principal of least action and the electron cross-helicity show that, at a threshold eigenvalue ratio, the electron parcel becomes unlinked from the magnetic field line. Simultaneously, the reconnection electric field satisfies the condition for global magnetic reconnection—i.e., where the topological changes are observable outside of the diffusion region. The eigenvalue ratio therefore serves as a 3D global, time-dependent reconnection rate derived analytically from physical principles and defined in terms of fundamental electron and ion time and length scales.