Symplectic approach to calculation of magnetic field line trajectories in physical space with realistic magnetic geometry in divertor tokamaks

Abstract

A new approach to integration of magnetic field lines in divertor tokamaks is proposed. In this approach, an analytic equilibrium generating function (EGF) is constructed in natural canonical coordinates (ψ,θ) from experimental data from a Grad–Shafranov equilibrium solver for a tokamak. ψ is the toroidal magnetic flux and θ is the poloidal angle. Natural canonical coordinates (ψ,θ,φ) can be transformed to physical position (R,Z,φ) using a canonical transformation. (R,Z,φ) are cylindrical coordinates. Another canonical transformation is used to construct a symplectic map for integration of magnetic field lines. Trajectories of field lines calculated from this symplectic map in natural canonical coordinates can be transformed to trajectories in real physical space. Unlike in magnetic coordinates [O. Kerwin, A. Punjabi, and H. Ali, Phys. Plasmas 15, 072504 (2008)], the symplectic map in natural canonical coordinates can integrate trajectories across the separatrix surface, and at the same time, give trajectories in physical space. Unlike symplectic maps in physical coordinates (x,y) or (R,Z), the continuous analog of a symplectic map in natural canonical coordinates does not distort trajectories in toroidal planes intervening the discrete map. This approach is applied to the DIII-D tokamak [J. L. Luxon and L. E. Davis, Fusion Technol. 8, 441 (1985)]. The EGF for the DIII-D gives quite an accurate representation of equilibrium magnetic surfaces close to the separatrix surface. This new approach is applied to demonstrate the sensitivity of stochastic broadening using a set of perturbations that generically approximate the size of the field errors and statistical topological noise expected in a poloidally diverted tokamak. Plans for future application of this approach are discussed.

Homoclinic tangle of the ideal separatrix in the DIII-D tokamak from (30, 10) + (40, 10) perturbation

“…5 We combine our method with our new recent symplectic approach. 26,27 At the heart of the new approach is a set of new canonical coordinates called the natural canonical coordinates (NCC). [26][27][28] Here, our motivation is to calculate the tangle of ideal separatrix.…”

“…[26][27][28] Magnetic coordinates are averages over surfaces and have a singularity on the separatrix. For this reason, we cannot use the magnetic coordinates to integrate across the separatrix.…”

An accurate symplectic calculation of the inboard magnetic footprint from statistical topological noise and field errors in the DIII-D

“…Failing this requirement will violate the symplectic invariance of the Hamiltonian mechanics. 7,8 There is no magnetic confinement device that is perfectly axisymmetric. Perfect axisymmetry is an idealization.…”

“…In our two previous papers, we have shown how we developed this approach, and how it can be applied to tokamaks in general. 7,8 In this paper, we apply this approach to the DIII-D tokamak, 9 and calculate the trajectories of field lines in the DIII-D from the magnetic perturbation of internal topological noise and magnetic field errors. 9 An important issue that we address is: Of the different possible canonical formulations of field line Hamiltonian, which is the best suited to help us calculate the threedimensional magnetic structure in real physical space of magnetic confinement schemes, that is both symplectic and accurate.…”

“…These natural coordinates are new and were developed. 7,8 The natural coordinates ͑ , ͒, generalized time , and the Hamiltonian ͑ , , ͒ are generally well-behaved near magnetic separatrix, such as the separatrix of a divertor tokamak. In an axisymmetric field, such as equilibrium field in divertor tokamaks, we can transform to action-angle coordinates.…”