Calculation of stochastic broadening due to low mn magnetic perturbation in the simple map in action-angle coordinates

The simple map is the simplest map that has topology of divertor tokamaks [A. Punjabi, H. Ali, T. Evans, and A. Boozer, Phys. Let. A \textbf{364}, 140--145 (2007)]. Recently, the action-angle coordinates for simple map are analytically calculated, and simple map is constructed in action-angle coordinates [O. Kerwin, A. Punjabi, and H. Ali, Phys. Plasmas \textbf{15}, 072504 (2008)]. Action-angle coordinates for simple map cannot be inverted to real space coordinates (R,Z). Because there is logarithmic singularity on the ideal separatrix, trajectories cannot cross separatrix [\textit{op cit}]. Simple map in action-angle coordinates is applied to calculate stochastic broadening due to the low mn magnetic perturbation with mode numbers m=1, and n=$\pm $1. The width of stochastic layer near the X-point scales as 0.63 power of the amplitude $\delta $ of low mn perturbation, toroidal flux loss scales as 1.16 power of $\delta $, and poloidal flux loss scales as 1.26 power of $\delta $. Scaling of width deviates from Boozer-Rechester scaling by 26{\%} [A. Boozer, and A. Rechester, \textit{Phys. Fluids }\textbf{21}, 682 (1978)]. This work is supported by US Department of Energy grants DE-FG02-07ER54937, DE-FG02-01ER54624 and DE-FG02-04ER54793.