Symplectic calculation of the outboard magnetic footprint from noise and error fields in the DIII-D

The backward symplectic DIII-D map and continuous symplectic analog of the map for magnetic field line trajectories in the DIII-D [10] (Luxon, J. L. and Davis, L. E. 1985 Fusion Technol. 8, 441) in natural canonical coordinates are used to calculate the magnetic footprint on the outboard collector plate of the DIII-D tokamak from the field errors and internal topological noise. The equilibrium generating function for the DIII-D used in the map very accurately represents the magnetic geometry of the DIII-D. The step-size of the map is kept considerably small so that the magnetic perturbation added from symplectic discretization of the Hamiltonian equations of the magnetic field line trajectories is very small. The natural canonical coordinates allow inverting to the real physical space. The combination of highly accurate equilibrium generating function, natural canonical coordinates, symplecticity, and small step-size then together gives a very accurate calculation of magnetic footprint. Radial variation of magnetic perturbation and the response of plasma to perturbation are not included. The footprint is in the form of toroidally winding helical strips. The area of footprint scales as 1st power of amplitude. The physical parameters as toroidal angle, length, and poloidal angle covered before striking, and the safety factor all have fractal structure. The average field diffusion near X-point for lines that strike and that do not strike differs by about four orders of magnitude. The flux loss decreases for high values of amplitude of perturbation.