Topology of Coronal Magnetic Fields: Extending the Magnetic Skeleton Using Null-like Points

Many phenomena in the Sun's atmosphere are magnetic in nature and study of the atmospheric magnetic field plays an important part in understanding these phenomena. Tools to study solar magnetic fields include magnetic topology and features such as magnetic null points, separatrix surfaces, and separators. The theory of these has most robustly been developed under magnetic charge topology, where the sources of the magnetic field are taken to be discrete, but observed magnetic fields are continuously distributed, and reconstructions and numerical simulations typically use continuously distributed magnetic boundary conditions. This article investigates the pitfalls in using continuous-source descriptions, particularly when null points on the z = 0 plane are obscured by the continuous flux distribution through, e.g., the overlap of non-point sources. The idea of null-like points on the boundary is introduced where the parallel requirement on the field B-parallel to = 0 is retained but the requirement on the perpendicular component is relaxed, i.e. B-perpendicular to not equal 0. These allow the definition of separatrix-like surfaces which are shown (through use of a squashing factor) to be a class of quasi-separatrix layer, and separator-like lines which retain the x-line structure of separators. Examples are given that demonstrate that the use of null-like points can reinstate topological features that are eliminated in the transition from discrete to continuous sources, and that their inclusion in more involved cases can enhance understanding of the magnetic structure and even change the resulting conclusions. While the examples in this article use the potential approximation, the definition of null-like points is more general and may be employed in other cases such as force-free field extrapolations and MHD simulations.