On the possibility of magnetic fields and fluid flows parallel to the X-line in a re-connexion geometry

We consider the possibility of modifying the Sonnerup solution for incompressible fluid flow about an X-type re-connexion line, to include fields and flows parallel to the X line. We find that such fields and flows may change across the discontinuities of the Sonnerup solution. By considering the requirements imposed by a proper matching across the various regions of flow, and by the integral conservation properties of the diffusion region, we seek to find the restrictions that are imposed on this parallel field and flow, and on the arrangement of the discontinuity planes around the diffusion region. We find that four types of such arrangements are possible, each corresponding to a different set of restrictions on the parallel field and flow. In one case, where all the discontinuity planes intersect at a common line, the ‘ parallel’ parameters of the in-flow and out-flow regions may be arbitrarily and independently chosen. Of the remaining three cases, one contains solutions with uniform parallel fields ad flows, while the other two depend for their existence on large fluid flow or magnetic field shears across the two in-flow regions.