Directed chaotic transport in the tokamap with mixed phase space

On the basis of the tokamap, characteristic features of magnetic field lines and zeroth-order guiding-center particle motion in the whole body of a magnetically confined plasma, e.g., a tokamak plasma, are investigated. It is shown that the tokamap exhibits a poloidal transport that can be considered as a Hamiltonian ratchet. In a situation with partially chaotic magnetic field lines the locking of the averaged poloidal velocity occurs to a value that does not depend precisely on the initial conditions. The so-called sum rule predicts the mean velocity in agreement with the observed magnitude. Possible consequences for the onset of poloidal plasma rotation in ergodized plasmas are elucidated.

Figure 1

(Color online) Geometry and coordinates relevant for the tokamap. From left to right: cylindrical coordinates $R$, $\Phi$, and $Z$ are used for the torus of a tokamak. Within a poloidal cross section the intrinsic coordinates $\rho$ and $\vartheta$ are thenew plane polar coordinates. The mapping is within a certain poloidal cross section at constant azimuthal angle $\Phi \sim \phi$ with $\psi \sim {\rho }^2$.

Figure 2

(Color online) (a) Phase space of the tokamap for $L=4.5/2\pi$. (b) Distribution of time-averaged velocities for the chains of islands 3, 5, and 7.

Figure 3

(Color online) Distribution of time-averaged velocities in the chaotic sea at (a) $t\sim {10}^3$, (b) $t\sim {10}^4$, (c) $t\sim {10}^5$ and (d) $t\sim {10}^6$. As $t\to \infty$ it evolves to a narrow peak around the asymptotic velocity, $\overline{v_{\theta }}\approx 0.50\pm 0.02$. We have $L=4.5/2\pi$.

Figure 4

(a) Phase space of the backward tokamap for $L=4.5/2\pi$. (b) Distribution of time-averaged velocities at time $t\sim 4\times {10}^6$. As $t\to \infty$ it evolves to a narrow peak around the asymptotic velocity, $\overline{v_{\theta }}\approx -0.474\pm 0.003$.

Figure 5

(a) Phase space of the bounded tokamap for $L=8/2\pi$. (b) Distribution of time-averaged velocities at time $t\sim 4\times {10}^6$ for the bounded tokamap.