Quasi-Separatrix layers in solar flares. I. Method.

Magnetic reconnection is usually thought to be linked to the presence of magnetic null points and to be accompanied by the transport of magnetic field lines across separatrices, the set of field lines where the mapping of field lines is discontinuous. In view of the variety of observed flaring configurations, we show that this view is too restrictive. Instead, Priest and Demoulin (1995) have explored a way of generalising the concept of separatrices to magnetic configurations without field-line linkage discontinuities. They propose that magnetic reconnection may also occur in 3D in the absence of null points at "quasi-separatrix layers" (QSLs), which are regions where there is drastic change in field-line linkage. In previous studies we have shown that solar flare kernels are linked to the topology of the active-region magnetic field. The observed photospheric field was extrapolated to the corona using subphotospheric magnetic sources and the topology was defined by the magnetic linkage between these sources, the method being called SM (for Source Method). In this paper we define a new method, called QSLM (for Quasi-Separatrix Layers Method), which finds the location of QSLs above the photosphere. It is designed to be applied to any kind of magnetic field representation, while, in the present paper, we apply it only to simple theoretical magnetic configurations in order to compare it with the SM. It generalises the concept of separatrices to magnetic configurations without field-line linkage discontinuities. The QSLM determines elongated regions that are in general located along small portions of the separatrices defined by the SM, and in the limit of very concentrated photospheric fields both methods give the same result. In bipolar magnetic configurations two QSLs are found at both sides of the inversion line, while in quadrupolar configurations four appear. We find that there is a wide range for the thickness of the QSLs, which is determined by the character (bipolar or quadrupolar) of the magnetic region and by the sizes of the photospheric field concentrations. We then show that smooth photospheric motions induce concentrated currents at the locations defined by the QSLM. We prove this only for initially potential configurations but, due to the form of the equations, we conjecture that it is also valid for any kind of initial magnetic equilibrium. We conclude that, even in bipolar configurations, there are localized places where current build-up can be induced by photospheric motions, leading to ideal MHD breakdown with strong flows and magnetic energy release.