Speaker
Description
We present a geometric theory of magnetic reconnection in which the global topology of a vacuum magnetic field is determined by boundary conditions. This program is the three-dimensional (3D) generalization of Syrovatskii's classical 2D work. Syrovatskii proved in 2D---under the assumption of a current-free outer field---that current singularities form at separatrices as a consequence of the boundary-value structure of the magnetic potential. His argument is implicitly a statement about the 2D kernel of the outer domain: in 2D, the Riemann mapping theorem makes the kernel computable via conformal maps, and the matching condition across a current singularity emerges naturally from the conformal Jacobian. In 3D, there is no Riemann mapping theorem, and the correct 3D global generalization is the so-called Bergman kernel, whose local realization is the Hessian tensor of the magnetic potential of the vacuum field. We derive upper bounds on the reconnection rate, which is in good accord with simulations and observations.
The framework offers a new perspective on the classical Parker problem of current singularities in line-tied magnetic fields, and is directly confirmed by published numerical simulations in which the Lagrangian Jacobian singularity leads to caustic (or ridge) formation, as anticipated by Parker’s optical analogy.