3D magnetic field configuration of the 2006 December 13 flare extrapolated with the optimization method

The photospheric vector magnetic field of the active region NOAA 10930 was obtained with the Solar Optical Telescope (SOT) on board the Hinode satellite with a very high spatial resolution (about 0.3 ''). Observations of the two-ribbon flare on 2006 December 13 in this active region provide us a good sample to study the magnetic field configuration related to the occurrence of the flare. Using the optimization method for nonlinear force-free field (NLFFF) extrapolation proposed by Wheatland et al. and recently developed by Wiegelmann, we derive the three-dimensional (3D) vector magnetic field configuration associated with this flare. The general topology can be described as a highly sheared core field and a quasi-potential envelope arch field. The core field clearly shows some dips supposed to sustain a filament. Free energy release in the flare, calculated by subtracting the energy contained in the NLFFF and the corresponding potential field, is 2: 4; 1031 ergs, which is similar to 2% of the preflare potential field energy. We also calculate the shear angles, defined as the angles between the NLFFF and potential field, and find that they become larger at some particular sites in the lower atmosphere, while they become significantly smaller in most places, implying that the whole configuration gets closer to the potential field after the flare. The Ca II H line images obtained with the Broadband Filter Imager (BFI) of the SOT and the 1600 8 images with the Transition Region and Coronal Explorer (TRACE) show that the preflare heating occurs mainly in the core field. These results provide evidence in support of the tether-cutting model of solar flares.