Linear theory of fast reconnection at an X-type neutral point

Abstract

A linear theory is developed for the dissipation of the free magnetic energy in small disturbances imposed on the potential field of an X-type neutral point. An eigenmode analysis, using cylindrical coordinates centered on the neutral point, extends the work of Craig & McClymont (1991) to include non-azimuthally symmetric perturbations on the flux function. We demonstrate that all physically significant disturbances, both reconnective and nonreconnective, decay resistively on a "fast" time scale ∼|ln η|, where η is the nondimensional resistivity. Thus resistive diffusion is remarkably effective as a dissipation mechanism for all classes of perturbation not only those modes which induce changes in the field topology. The fundamental reason for the fast dissipation is the focusing of wave energy onto the neutral point by the gradient in Alfvén speed. This occurs for all types of disturbances launched from the outer boundary, but reconnective modes propagate purely radially while nonreconnective modes correspond to waves with an azimuthal motion and a smaller wave speed in the radial direction. Dissipation of reconnective disturbances takes place in a diffusion region of diameter ≈η1/2 around the neutral point, on a time scale ∝ |ln η|2, but nonreconnective perturbations decay before penetrating to the neutral point, because of the increased path lengths due to their azimuthal motion. These modes release their energy in an annulus of radius ≈[(m3/ 4k)η]1/2, where m and k are azimuthal and radial wavenumbers: because of their longer transit time, nonreconnective waves decay on a |ln η|3 time scale. We conclude by discussing the significance of the linear theory within the general context of steady state and dynamic reconnection studies. It is pointed out that, dynamically, disturbances can be expected to focus explosively in the vicinity of the neutral point. This suggests the formation of a "flux pile-up" current layer in which the bulk of the magnetic energy is released as heat rather than kinetic energy of mass motion.