Distribution of 2-D magnetic saddle points and morphology of flare kernels in solar active regions

We extrapolated the 3-D fields above the photosphere, taking the observed photospheric magnetic fields in the active regions NOAA 6659 and 7321 as the boundary conditions of a linear force-free held model, and detected the singular points of the 2-D fields in a plane at the chromospheric level. These singular points can be described with the Poincare index. Singular points with the index of +1 correspond to concentrations of magnetic flux, and those with the index of -1 to the saddle points in the plane. All of these singular points are connected by the lanes demarcating the 2-D magnetic cells in the plane. It has been confirmed that these saddle points are the intersections between separators and planes intersecting the 3-D fields. From comparisons between kernels of flares occurring in both regions and the saddle points, we found that there is a close morphological relationship between distributions of the saddle points and hare kernels. The main results are as follows: (a) The flare kernels tend to appear in areas with concentrating 2-D saddle points. (b) The morphology of the kernels is exactly confined by the lanes in the plane at chromospheric level. These facts seem favourable for the viewpoint that solar hares are closely related to magnetic separatrices and separators.