Magnetohydrodynamic equilibria in the vicinity of an X-type neutral line specified by footpoint shear

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Abstract
Results pertaining to two‐dimensional (∂/∂z=0) magnetohydrodynamic equilibria in the presence of an X‐type neutral line are presented. Naive analyses indicate that there may be tangential discontinuities in B, specifically discontinuities in Bz across the separatrix connected to the X line. However, such analyses indicate an infinite z component of footpoint displacement (or safety factor q in the toroidal case) at the separatrix. The solutions presented here allow the specification of footpoint displacement (or safety factor q) that is finite as the separatrix is approached. These solutions are scale‐invariant, or similarity, solutions. They are appropriate near the X line on length scales intermediate between the boundary layer width because of resistivity (or other nonideal effects) and the macroscopic length scale. Force balance across the separatrix implies identical radial dependence in all four quadrants and continuity of B2z across the separatrix. The latter shows that there are two classes of solutions: those with Bz continuous across the separatrix and those with ‖Bz‖ continuous but with a sign change in Bz. The former class has fractional power‐law singularities at the separatrix. The latter class has, in addition, a sheet current along the separatrix in the x‐y plane associated with the jump in Bz. Detailed properties of these solutions are explored. In particular, sheetlike one‐dimensional solutions are found to be limiting cases of the general solutions. Except for one special case, these sheet solutions cannot have finite footpoint displacement if they are force free, but can in the presence of pressure gradient. The general solutions can have arbitrary separatrix angle and ratio of flux between the quadrants. Discussion is presented regarding the matching of these solutions to the physical conditions on boundaries where the field lines leave the system, e.g., the photospheric surface in the solar coronal context.