Simple Map in Action-Angle Coordinates.

The simple map is the simplest map that has the topology of a divertor tokamak. The simple map has three canonical representations: (i) the natural coordinates - toroidal magnetic flux and poloidal angle (\textit{$\psi $,$\theta $}), (ii) the physical coordinates - the physical variables ($R,Z)$ or ($X,Y)$, and (iii) the action-angle coordinates - (\textit{J,$\Theta $}) or magnetic coordinates (\textit{$\Psi $, $\Theta $}). All three are canonical coordinates for field lines. The simple map in the ($X,Y)$ representation has been studied extensively $^{1, 2}$. Here we analytically calculate the action-angle coordinates and safety factor $q$ for the simple map. We construct the equilibrium generating function for the simple map in action-angle coordinates. We derive the simple map in action-angle representation, and calculate the stochastic broadening of the ideal separatrix due to topological noise in action-angle representation. We also show how the geometric effects such as elongation, the height, and width of the ideal separatrix surface can be investigated using a slight modification of the simple map in action-angle representation. This work is supported by the following grants US Department of Energy - OFES DE-FG02-01ER54624 and DE-FG02-04ER54793 and National Science Foundation - HRD-0630372 and 0411394. [1] A. Punjabi, H. Ali, T. Evans, and A. Boozer, \textit{Phys Lett A,} \textbf{364} 140-145 (2007). [2] A. Punjabi, A. Verma, and A. Boozer, Phys.Rev. Lett. \textbf{69}, 3322 (1992).