Asymptotical and mapping methods in study of ergodic divertor magnetic field in a toroidal system

Asymptotical and mapping methods to study the structure of magnetic field perturbations and magnetic field line dynamics in a tokamak ergodic divertor in toroidal geometry are developed. The investigation is applied to the Dynamic Ergodic Divertor under construction for the Torus Experiment for the Technology Oriented Research (TEXTOR-94) Tokamak at Jülich [Fusion Eng. Design 37, 337 (1997)]. An ideal coil configuration designed to create resonant magnetic perturbations at the plasma edge is considered. In cylindrical geometry, the analytical expressions for the vacuum magnetic field perturbations of such a coil system are derived, and its properties are studied. Corrections to the magnetic field due to the toroidicity are presented. The asymptotical analysis of transformation of magnetic perturbation into the Hamiltonian perturbation in toroidal geometry is carried out, and the asymptotic formulas for the spectrum of the Hamiltonian perturbations are found. A new method of integration of Hamiltonian equations is developed. It is based on a canonical transformation of variables that replaces the dynamics of a continuous Hamiltonian system by a symplectic mapping. The form of the mapping is established in the first order of perturbation theory. It is shown that the mapping well reproduces Poincaré sections of field lines, as well as their statistical properties in an ergodic zone obtained by the numerical integration of field line equations. The mapping is applied to study, in particular, the formation of a stochastic layer and the statistical properties of field lines at the plasma edge.