Comparison of inboard and outboard magnetic footprints from topological noise and field errors in the DIII-D

Abstract

Any canonical transformation of Hamiltonian equations is symplectic, and any area-preserving transformation in 2D is a symplectomorphism. Based on these, a discrete symplectic map and its continuous symplectic analog are derived for forward magnetic field line trajectories in natural canonical coordinates. The unperturbed axisymmetric Hamiltonian for magnetic field lines is constructed from the experimental data in the DIII-D. The equilibrium Hamiltonian is a highly accurate, analytic, and realistic representation of the magnetic geometry of the DIII-D. These symplectic mathematical maps are used to calculate the trajectories of magnetic field lines in the DIII-D. Internal statistical topological noise and field errors are irreducible and ubiquitous in magnetic confinement schemes for fusion. It is important to know the stochasticity and magnetic footprint from noise and error fields. The estimates of the spectrum and mode amplitudes of the spatial topological noise and magnetic errors in the DIII-D are used as magnetic perturbation. The discrete and continuous symplectic maps are used to calculate the magnetic footprint on the inboard and outboard collector plates of the DIII-D by inverting the natural coordinates to physical coordinates. Radial variation of magnetic perturbation and the response of plasma to perturbation are not included. The footprints are in the form of toroidally winding helical strips. The area of footprint scales linearly with amplitude. The outboard footprint has higher transverse extent, higher area, and higher fractal dimension. The field diffusion near the X-point for backward lines is higher than that for forward lines. Diffusion for lines that strike the plate is about three to four orders of magnitude higher than for lines that do not strike. Line loss and flux loss show a drop when the amplitude of perturbation is about 18–19 × 10−6, indicating the possible presence of a barrier. The physical parameters such as toroidal angle, length, and poloidal angle covered before striking and the safety factor all have a fractal structure.