Calculation of fractal dimension of magnetic footprint in double-null divertor tokamaks

The simplest symplectic map that represents the magnetic topology of double-null divertor tokamaks is the double-null map, given by the map equations: x$_{1}$=x$_{0}$-ky$_{0}$(1-$y_0^2 )$, y$_{1}$=y$_{0}$+kx$_{1}$. k is the map parameter. The map parameter k represents the generic topological effects of toroidal asymmetries. The O-point is at (0,0). The X-points are at (0,$\pm $1). We set k=0.51763, and N$_{p}$=12. N$_{p}$ is the number of iterations of map that are equivalent to a single toroidal circuit of the tokamak. The width of stochastic layer near the upper and the lower X-points is exactly the same and equals 1.69 mm. We start 100,000 filed lines in the stochastic layer near the X-points and advance them for at most 10,000 toroidal circuits. We use the continuous analog of the map to calculate the magnetic footprints in the double-null divertor tokamaks. We calculate the area of the footprints and their fractal dimension. The area is A=0.0024 m$^{2}$, and fractal dimension is d$_{frac}$=1.0266. This work is supported by US Department of Energy grants DE-FG02-07ER54937, DE-FG02-01ER54624 and DE-FG02-04ER54793.