A symplectic map for trajectories of magnetic field lines in double-null divertor tokamaks

The coordinates of the area-preserving map equations for integration of magnetic field line trajectories in tokamaks can be any coordinates for which a transformation to ($\psi $,$\theta $,$\phi )$ coordinates exists [A. Punjabi, H. Ali, T. Evans, and A. Boozer, Phys. Lett. A \textbf{364}, 140 (2007)]. $\psi $ is toroidal magnetic flux, $\theta $ is poloidal angle, and $\phi $ is toroidal angle. This freedom is exploited to construct a map that represents the magnetic topology of double-null divertor tokamaks. For this purpose, the generating function of the simple map [A. Punjabi, A. Verma, and A. Boozer, Phys. Rev. Lett. \textbf{69}, 3322 (1992)] is slightly modified. The resulting map equations for the double-null divertor tokamaks are: x$_{1}$=x$_{0}$-ky$_{0}$(1-$y_0^2 )$, y$_{1}$=y$_{0}$+kx$_{1}$. k is the map parameter. It represents the generic topological effects of toroidal asymmetries. The O-point is at (0.0). The X-points are at (0,$\pm $1). The equilibrium magnetic surfaces are calculated. These surfaces are symmetric about the x- and y- axes. The widths of stochastic layer near the X-points in the principal plane, and the fractal dimensions of the magnetic footprints on the inboard and outboard side of upper and lower X-points are calculated from the map. This work is supported by US Department of Energy grants DE-FG02-07ER54937, DE-FG02-01ER54624 and DE-FG02-04ER54793.