Method for creating a three-dimensional magnetic null point topology with an accurate spine axis

Three-dimensional magnetic null points (3D nulls) are sites of dynamic activity in a wide range of naturally-occurring and laboratory plasma environments. The topology of a 3D null is defined by a two-dimensional fan plane of radial field lines and a one-dimensional, collimated spine axis. Here, we build on previous work that was able to form an extended 3D null topology using an assembly of circular conducting coils, with each coil carrying a constant current. While that magnetic field design decayed from the mathematically pure form away from the central null, this paper introduces an algorithm for modulating the current through each coil to form a more mathematically pure spine axis along the entirety of the coil assembly. By the method of solving an inverse problem, we demonstrate that unique currents exist for any arbitrary distribution of axially-aligned circular coils for creating an accurate spine axis in a 3D null topology. Tests of this algorithm are performed on spherical, cylindrical, and cone-shaped coil assemblies. Vector magnetic field mapping of these small-scale demonstrators verifies that an accurate spine axis is maintained along the entire central axis of the coil assemblies. The magnetic field accuracy is roughly maintained along the fan plane but decays strongly toward the outer extents of the coils. The inverse method presented here is not limited to 3D null topologies but can be adapted to match any theoretical form of the magnetic field along a single axis.