Calculating and Testing Nonlinear Force-Free Fields

Improvements to an existing method for calculating nonlinear force-free magnetic fields (Wheatland 2006, Solar Phys. 238, 29) are described. In particular a solution of the 3-D Poisson equation using 2-D Fourier transforms is presented. The improved nonlinear force-free method is demonstrated in application to linear force-free test cases with localized non-zero values of the normal component of the field in the boundary. These fields provide suitable test cases for nonlinear force-free calculations because the boundary conditions involve localized non-zero values of the normal components of the field and of the current density, and because (being linear force-free fields) they have more direct numerical solutions. Despite their simplicity, fields of this kind have not been recognised as test cases for nonlinear methods before. The examples illustrate the treatment of the boundary conditions on current in the nonlinear force-free method, and in particular the limitations imposed by field lines which connect outside of the boundary region.