Magnetohydrodynamic flows sustaining stationary magnetic nulls

Abstract

Exact solutions of the resistive magnetohydrodynamic equations are derived which describe a stationary incompressible flow near a generic null point of a three-dimensional magnetic field. The properties of the solutions depend on the topological skeleton of the corresponding magnetic field. This skeleton is formed by one-dimensional and two-dimensional invariant manifolds (so-called spine line and fan plane) of the magnetic field. It is shown that configurations of generic null points may always be sustained by stationary field-aligned flows of the stagnation type, where the null points of the magnetic and velocity fields have the same location. However, if the absolute value |j∥| of the current density component parallel to the spine line exceeds a critical value jc, the solution is not unique—there is a second nontrivial solution describing spiral flows with the stagnation point at the magnetic null. The characteristic feature of these new flows is that they cross magnetic field lines but they do not cross the corresponding spine and fan of the magnetic null. Therefore these are nonideal but nonreconnecting flows. The critical value |j∥|=jc coincides exactly with a threshold separating the topological distinct improper radial and spiral nulls. It is shown that this is not an accidental coincidence: the spiral field-crossing flows of the considered type are possible only due to the topological equivalence of the field lines forming the fan plane of the spiral magnetic null. The explicit expression for the pressure distribution of the solution is given and its iso-surfaces are found to be always ellipsoidal for the field-aligned flows, while for the field-crossing flows there are also cases with a hyperboloidal structure.