Stochastic Transport of Magnetic Field Lines in the Symmetric Tokamap

Abstract

The topological structure and the statistical properties of stochastic magnetic fields are investigated on the basis of the so called tokamap. First, a monotonic safety factor (q-profile) is assumed. As it is demonstrated, the transition from the continuous model to the discrete mapping in its symmetric form is essential, not only for the symplectic structure, but also for the precise values characterizing the transition to chaos (e.g. the break-up of the KAM surfaces) in applications. Statistical properties of the symmetric tokamap, such as escape rates and anomalous diffusion properties, are being presented. By a systematic procedure the stable and unstable manifolds of the periodic hyperbolic fixed points and the resulting homoclinic tangles (stochastic layers) are determined. The latter are important for the magnetic field line transport. For a non-monotonic q-profile, the differences between the symmetric and non-symmetric revtokamap become also significant. The symmetric revtokamap represents an open nonlinear dynamical system which is characterized here with the relevant tools.

Escape patterns of chaotic magnetic field lines in a tokamak with reversed magnetic shear and an ergodic limiter

“…Plasmas 15, 092310 ͑2008͒ similar to those experimentally identified in tokamaks. 38,70 Figures 8͑a͒ and 8͑b͒ indicate a high concentration of regions with large connection lengths for the ͑4,1͒ perturbation mode. On the other hand, Figs.…”

“…Magnetic footprints have been obtained by using numerically 29,[38][39][40][41][42][43][44][45][46] and analytically [47][48][49][50] obtained maps of the perturbed magnetic fields. The observed fractal structure of magnetic footprints follows from the mathematical structure underlying the area-filling chaotic region of magnetic field lines-a chaotic manifold tangle, comprising the homoclinic and heteroclinic intersections of invariant manifolds of unstable periodic orbits embedded in the chaotic region.…”

“…The observed fractal structure of magnetic footprints follows from the mathematical structure underlying the area-filling chaotic region of magnetic field lines-a chaotic manifold tangle, comprising the homoclinic and heteroclinic intersections of invariant manifolds of unstable periodic orbits embedded in the chaotic region. 29,38,39 Another consequence of this manifold tangle is an involved structure of escape basins, which indicate the loci of points such that a field line passing through that point will escape through a given region at the tokamak wall. 51 In this paper we investigate the magnetic footprints and related escape basins for a tokamak with reversed magnetic shear, obtained through using a nonmonotonic plasma current profile.…”

Fractal structures in nonlinear plasma physics

“…An example of a fractal set one can mention is the escape basin, which is the set of initial conditions leading to chaotic orbits which hit the tokamak wall [20,31,35,36]. Besides the escape basin boundaries being fractal sets, they often have the so-called Wada property: every boundary point has an arbitrarily small neighbourhood containing points of all the basins [37][38][39][40].…”

Traces of stable and unstable manifolds in heat flux patterns

“…The main purpose of the present paper is to formulate, prove, and quantify selection rules for spatial heat flow patterns at the boundaries of stochastic plasmas. In order to analyze and classify the spatial structures of heat flux patterns, we propose the concept of magnetic footprints 5,6 together with an analysis of the stable and unstable manifolds of hyperbolic periodic points [7][8][9][10] of the last intact island chain. The latter is the last resonance in front of the wall at the transition from the ergodic zone to the laminar zone.…”