The relaxation of a two-dimensional "X-type" neutral point magnetic field disturbed from equilibrium is considered. Perturbations are shown to possess well-defined azimuthal modes which allow an exact determi nation of the magnetic annihilation rate. Free magnetic energy is dissipated by oscillatory reconnection which couples resistive diffusion at the neutral point to global advection of the outer field. The decay of azimuthally symmetric (m = 0) modes-the only modes associated with topological reconnection-is limited by the dissi pation time scale of the "fundamental" (n = 0) mode with no radial nodes. This mode decays over typically 1 00 Alfven times. An analytic treatment shows that the oscillation and decay time scales couple according to ' osc � 2 lnS and ' decay � '� sc /(2n 2), where S is the Lundquist number (4n/c 2)v A R/r, and the times are in units of R/v A , with R the distance from the neutral point to the boundary, v A the Alfven speed at the boundary, and r, the resistivity.

The formation of current singularities at line-tied two-and three-dimensional ͑2D and 3D, respectively͒ magnetic null points in a nonresistive magnetohydrodynamic environment is explored. It is shown that, despite the different separatrix structures of 2D and 3D null points, current singularities may be initiated in a formally equivalent manner. This is true no matter whether the collapse is triggered by flux imbalance within closed, line-tied null points or driven by externally imposed velocity fields in open, incompressible geometries. A Lagrangian numerical code is used to investigate the finite amplitude perturbations that lead to singular current sheets in collapsing 2D and 3D null points. The form of the singular current distribution is analyzed as a function of the spatial anisotropy of the null point, and the effects of finite gas pressure are quantified. It is pointed out that the pressure force, while never stopping the formation of the singularity, significantly alters the morphology of the current distribution as well as dramatically weakening its strength. The impact of these findings on 2D and 3D magnetic reconnection models is discussed.

We consider the linear and nonlinear evolution of disturbed magnetic X-type neutral points. The problem is formulated within a unified analytic and computational framework which highlights the essence of the mag netic annihilation process, namely, the coupling of a global convection region to a localized diffusion region surrounding the neutral point. An analytic treatment is given for the case of small disturbances of the equilibrium field in the absence of gas pressure. This problem admits well-defined azimuthal modes which allow a formally exact determination of the magnetic annihilation rate. It is shown that reconnection can only occur in the case of purely radial (m = 0) disturbances: the reconnection process is oscillatory and "fast, " depending only logarithmically on the plasma resistivity (,,). We show that the linear theory supports the notion of an initial implosive stage which rapidly releases the bulk of the energy associated with reconnective field disturbances. This phase is initiated by the advective focusing of the perturbation energy into the neutral point and culminates in the formation of a cylindrical diffusion region of area A-17 and current density J-17i. This scaling provides a signature for fast linear reconnection. Next we consider the breakdown of the linear theory. Although fast reconnection is maintained for low amplitude disturbances in noncylindrical geometries, it is shown that finite gas pressure can stall the reconnec tion if sufficiently large. This effect, however, may not be critical in more complex X-point geometries. More seriously, for finite-amplitude displacements the cylindrical current structure close to the neutral point is dis torted into a quasi-one-dimensional current sheet whose thickness is limited by resistive diffusion. In this case fast reconnection is consistent with a flux pileup solution in which the bulk of the energy is released as heat rather than as the kinetic energy of mass motion.

The problem of steady state, incompressible magnetic reconnection in three dimensions is addressed. It is shown that exact reconnection solutions can be constructed by superposing nonlinear disturbances onto three-dimensional magnetic X-points. There are two distinct families of reconnection solutions. These can be understood in terms of the eigenstructure of the null, that is, in terms of the " spine" curves and "fan" surfaces that define the separatrices of the field. One family of solutions is driven by dis turbances in the fan and involves quasi-cylindrical current structures aligned to the axis of the spine; the other is associated with advection across the spine and a global current sheet aligned to the fan. Although both spine and fan solutions reduce to the two-dimensional analytic, shear-flow solutions of Craig & Henton, the three-dimensional spine current formulation allows far richer reconnective current structures.

The problem of steady state magnetic reconnection in incompressible, resistive plasmas is addressed. It is shown that families of exact analytic solutions can be derived by exploiting the formal symmetry between the magnetic field lines and the streamlines of the flow. These solutions, while incorporating previous models for antiparallel magnetic merging, assume no inherent topological symmetry for the reconnective geometry. It is demonstrated that fast reconnection occurs within a narrow X-type configuration defined by the intersection of two separatrix lines. One separatrix line is undistinguished physically, but the other is contiguous with a global current sheet across which there is no plasma flow. The implication is that fast reconnection is more likely to be associated with strong shearing motions close to the neutral point rather than with the conventional stagnation point flow topology.