Topological and statistical properties of nonlinear force-free fields

TL;DR

This paper investigates the topological and statistical properties of nonlinear force-free magnetic fields relevant to the solar corona, introducing new formulas and bounds to better understand magnetic braiding and energy estimates.

Contribution

It presents a semi-analytic solution for nonlinear force-free fields, new formulas for winding and crossing numbers, and analytical bounds for free energy and helicity based on topological measures.

Findings

New formula for winding number compared with crossing number

Calculated linkages as measures of magnetic braiding

Derived bounds for free energy and helicity in terms of linking number

Abstract

We use our semi-analytic solution of the nonlinear force-free field equation to construct three-dimensional magnetic fields that are applicable to the solar corona and study their statistical properties for estimating the degree of braiding exhibited by these fields. We present a new formula for calculating the winding number and compare it with the formula for the crossing number. The comparison is shown for a toy model of two helices and for realistic cases of nonlinear force-free fields; conceptually the formulae are nearly the same but the resulting distributions calculated for a given topology can be different. We also calculate linkages, which are useful topological quantities that are independent measures of the contribution of magnetic braiding to the total free energy and relative helicity of the field. Finally, we derive new analytical bounds for the free energy and relative helicity for the field configurations in terms of the linking number. These bounds will be of utility in estimating the braided energy available for nano-flares or for eruptions.