Nonlinear Force-Free Field Extrapolations in Spherical Geometry

Magnetic field in the solar atmosphere plays a key role in various solar activities. However, it is still difficult to measure the coronal magnetic field. The force-free model is usually adopted to construct the 3D coronal magnetic structure. We test a nonlinear force-free field (NLFFF) optimization code in spherical geometry using an analytical solution from Low and Lou, which is well reconstructed when the boundary condition is properly managed. Analytical tests also show that NLFFF code in the spherical geometry performs better than that in the Cartesian one when the field of view of the bottom boundary is large. Additionally, we apply the NLFFF model to an active region observed by the Helioseismic and Magnetic Imager on board the Solar Dynamics Observatory (SDO) both before and after an M8.7 flare. By comparing the extrapolated magnetic field lines with the extreme ultraviolet (EUV) observations by the Atmospheric Imaging Assembly on board SDO, we find that the NLFFF performs better than the PFSS not only for the core field of the flare productive region, but also for large EUV loops higher than 50 Mm.