Application of the separatrix map to study perturbed magnetic field lines near the separatrix

In this paper a generalized separatrix map to study perturbed magnetic field lines in a confined magnetic system with the separatrix is discussed. The separatrix map is an iteration transform for the magnetic flux and a toroidal angle at a certain cross section after one period of the unperturbed trajectory. The map can be derived from magnetic field line equations and can be shifted along the toroidal angle, depending on the point of interest. Different asymptotic forms of the shifted separatrix map, with respect to the distance between the cross section of the toroidal angle and the X point, are considered, and their invariant properties are discussed in regard to the renormalization transformation of the perturbation amplitude. A method to find the map’s parameters for any specific magnetic system is proposed. It is shown that the shifted separatrix map can be directly applied to obtain magnetic footprints at the divertor plates, and to study chaotic scattering of field lines.