# Topological structures of velocity and electric field in the vicinity of a cusp-type magnetic null point

## TL;DR

This paper investigates the topological properties of magnetic and electric fields near a cusp-type magnetic null point, revealing the hyperbolic nature of stagnation points and the saddle point structure of resistivity and heating functions.

## Contribution

It provides a topological analysis of the magnetic and electric field structures around cusp null points, highlighting the non-monotonous relationship between current density and resistivity.

## Key findings

- Stagnation point must be hyperbolic.
- Resistivity and Ohmic heating functions have saddle point structures.
- No monotonic relation between current density and resistivity.

## Abstract

Topological characteristics reveal important physical properties of plasma structures and astrophysical processes. Physical parameters and constraints are linked with topological invariants, which are important for describing magnetic reconnection scenarios. We analyze stationary non-ideal Ohm's law concerning the Poincare classes of all involved physical fields in 2D by calculating the corresponding topological invariants of their Jacobian (here: particularly the eigenvalues) or Hessian matrices. The magnetic field is assumed to have a cusp structure, and the stagnation point of the plasma flow coincides with the cusp. We find that the stagnation point must be hyperbolic. Furthermore, the functions describing both the resistivity and the Ohmic heating have a saddle point structure, being displaced with respect to the cusp point. These results imply that there is no monotonous relation between current density and anomalous resistivity in the case of a 2D standard magnetic cusp.