From Order to Chaos - Chaos in Tokamaks Due to Tearing Modes

In this and the next paper, we show how tearing modes create chaos in the ohmically heated tokamaks with standard q-profile, and how we can build barriers inside the chaos in these tokamaks. We have constructed a new symplectic map to calculate trajectories of magnetic field lines in generic tokamaks. The map is given by \[ \psi _{n+1} =\psi _n -{k\partial \chi (\psi _{n+1} ,\theta _n )} \mathord{\left/ {\vphantom {{k\partial \chi (\psi _{n+1} ,\theta _n )} {\partial \theta _n }}} \right. \kern-\nulldelimiterspace} {\partial \theta _n },\theta _{n+1} =\theta _n +k{\partial \chi (\psi _{n+1} ,\theta _n )} \mathord{\left/ {\vphantom {{\partial \chi (\psi _{n+1} ,\theta _n )} {\partial \psi _{n+1} }}} \right. \kern-\nulldelimiterspace} {\partial \psi _{n+1} }. \] Poloidal flux, $\chi $, is the generating function for the map, the toroidal flux, $\psi $, is the action, and the poloidal angle, $\theta $, is the angle. We use the standard safety factor profile for the tokamaks. We apply the magnetic perturbations (m,n)={\{}(3,2),(2,1){\}}, each with the same amplitude $\delta $. When $\delta $=0, we see invariant tori. For $\delta $ from 1X10$^{-4}$ to 7.5X10$^{-4}$, tori are destroyed and islands are formed. For $\delta \quad >$ 7.5X10$^{-4}$, islands overlap, and finally create full-scale chaos. In the next paper, we show how we can erect a barrier inside this chaos to control transport. This we do by adding a term of order $\delta ^{2}$ to the generating function. This work is supported by the US DOE DE-FG02-02ER54673 and NASA SHARP PLUS.