A comparison of methods for finding magnetic nulls in simulations and in situ observations of space plasmas

Context. Magnetic nulls are ubiquitous in space plasmas, and are of interest as sites of localised energy dissipation or magnetic reconnection. As such, a number of methods have been proposed for detecting nulls in both simulation data and in situ spacecraft data from Earth’s magnetosphere. The same methods can be applied to detect stagnation points in flow fields. Aims. In this paper we describe a systematic comparison of different methods for finding magnetic nulls. The Poincaré index method, the first-order Taylor expansion (FOTE) method, and the trilinear method are considered. Methods. We define a magnetic field containing fourteen magnetic nulls whose positions and types are known to arbitrary precision. Furthermore, we applied the selected techniques in order to find and classify those nulls. Two situations are considered: one in which the magnetic field is discretised on a rectangular grid, and the second in which the magnetic field is discretised along synthetic “spacecraft trajectories” within the domain. Results. At present, FOTE and trilinear are the most reliable methods for finding nulls in the spacecraft data and in numerical simulations on Cartesian grids, respectively. The Poincaré index method is suitable for simulations on both tetrahedral and hexahedral meshes. Conclusions. The proposed magnetic field configuration can be used for grading and benchmarking the new and existing tools for finding magnetic nulls and flow stagnation points.

Poincaré index of plane polynomial fields of third and fourth degree

The conditions of isolation of a zero singular point of plane polynomial fields of third and fourth degree are considered in terms of the coefficients of the components of these fields. The isolation conditions depend on the greatest common divisor of the components of polynomial fields: in some cases only on its degree, and in some cases, additionally, on the presence of nonzero real zeros. The reasoning, which allows one to write out the isolation conditions, is based on the concept of the resultant and subresultants of components of plane polynomial fields. If the zero singular point is isolated, its index is calculated through the values of subresultants and coefficients of components.

An index for Brouwer homeomorphisms and homotopy Brouwer theory

We use the homotopy Brouwer theory of Handel to define a Poincaré index between pairs of orbits for an orientation-preserving fixed-point-free homeomorphism of the plane. Furthermore, we prove that this index is almost additive.