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 disturbances 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.
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Craig et al. (1996) studied this question.