Topological and statistical properties of nonlinear force-free fields

Abstract

We use our semi-analytic solution of the nonlinear force-free field equation to construct three-dimensional magnetic fields that areapplicable to the solar corona and study their statistical properties for estimating the degree of braiding exhibited by these fields. Wepresent a new formula for calculating the winding number and compare it with the formula for the crossing number. The comparisonis shown for a toy model of two helices and for realistic cases of nonlinear force-free fields; conceptually the formulae are nearly the samebut the resulting distributions calculated for a given topology can be different. We also calculate linkages, which are useful topologicalquantities 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 num-ber. These bounds will be of utility in estimating the braided energy available for nano-flares or for eruptions.Ó2017 COSPAR. Published by Elsevier Ltd. All rights reserved.