Development of a Poisson solver in a global field aligned mesh in a tokamak edge geometry

A Poisson solver is developed to study macroscopic and microscopic instabilities in a diverted tokamak geometry. A global field aligned mesh [B.D.Scott, Phys. Plasmas 5, 2334 (1998).] provides us with computational efficiency to describe structures of toroidal eigenmodes, which follow the helically twisted magnetic field lines. The field aligned mesh can be applied to the open-field-line regions as well. However, on the separatrix, the magnetic field line is purely toroidal at the X-point (an infinite number of toroidal transits is required for the magnetic field to reach there). Inevitably, in the very vicinity of the separatrix, one needs to revert back to a mesh in a cylindrical coordinate. We employ a finite element elliptic solver (with the field aligned mesh) and analyze realistically how close we can approach to the separatrix.