Noble internal transport barriers and radial subdiffusion of toroidal magnetic lines

Abstract

Internal transport barriers (ITB's) observed in tokamaks are described by a purely magnetic approach. Magnetic line motion in toroidal geometry with broken magnetic surfaces is studied from a previously derived Hamiltonian map in situation of incomplete chaos. This appears to reproduce in a realistic way the main features of a tokamak, for a given safety factor profile and in terms of a single parameter L representing the amplitude of the magnetic perturbation. New results are given concerning the Shafranov shift as function of L. The phase space ($\psi ,\theta$) of the "tokamap" describes the poloidal section of the line trajectories, where $\psi$ is the toroidal flux labelling the surfaces. For small values of L, closed magnetic surfaces exist (KAM tori) and island chains begin to appear on rational surfaces for higher values of L, with chaotic zones around hyperbolic points, as expected. Island remnants persist in the chaotic domain for all relevant values of L at the main rational q-values.
Single trajectories of magnetic line motion indicate the persistence of a central protected plasma core, surrounded by a chaotic shell enclosed in a double-sided transport barrier: the latter is identified as being composed of two Cantori located on two successive “most-noble” numbers values of the perturbed safety factor, and forming an internal transport barrier (ITB). Magnetic lines which succeed to escape across this barrier begin to wander in a wide chaotic sea extending up to a very robust barrier (as long as $L\precsim 1)$ which is identified mathematically as a robust KAM surface at the plasma edge. In this case the motion is shown to be intermittent, with long stages of pseudo-trapping in the chaotic shell, or of sticking around island remnants, as expected for a continuous time random walk.
For values of $L\succeq 1$, above the escape threshold, most magnetic lines succeed to escape out of the external barrier which has become a permeable Cantorus. Statistical analysis of a large number of trajectories, representing the evolution of a bunch of magnetic lines, indicate that the flux variable $\psi$ asymptotically grows in a diffusive manner as $(L^{2}t)$ with a L2 scaling as expected, but that the average radial position $r_{m}(t)$ asymptotically grows as $(L^{2}t)^{1/4}$ while the mean square displacement around this average radius asymptotically grows in a subdiffusive manner as $(L^{2}t)^{1/2}$. This result shows the slower dispersion in the present incomplete chaotic regime, which is different from the usual quasilinear diffusion in completely chaotic situations. For physical times $t_{\varphi }$ of the order of the escape time defined by $x_{m}(t_{\varphi })\sim 1$, the motion appears to be superdiffusive, however, but less dangerous than the generally admitted quasi-linear diffusion. The orders of magnitude of the relevant times in Tore Supra are finally discussed