I. L. Caldas

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

The existence of a reversed magnetic shear in tokamaks improves the plasma confinement through the formation of internal transport barriers that reduce radial particle and heat transport. However, the transport poloidal profile is much influenced by the presence of chaotic magnetic field lines at the plasma edge caused by external perturbations. Contrary to many expectations, it has been observed that such a chaotic region does not uniformize heat and particle deposition on the inner tokamak wall. The deposition is characterized instead by structured patterns called magnetic footprints, here investigated for a nonmonotonic analytical plasma equilibrium perturbed by an ergodic limiter. The magnetic footprints appear due to the underlying mathematical skeleton of chaotic magnetic field lines determined by the manifold tangles. For the investigated edge safety factor ranges, these effects on the wall are associated with the field line stickiness and escape channels due to internal island chains near the flux surfaces. Comparisons between magnetic footprints and escape basins from different equilibrium and ergodic limiter characteristic parameters show that highly concentrated magnetic footprints can be avoided by properly choosing these parameters.

A symplectic mapping for the ergodic magnetic limiter and its dynamical analysis

Abstract A model for a new bidimensional sympletic mapping describing magnetic field line trajectories in a tokamak perturbed by ergodic magnetic limiter coils is presented. Numerical examples of these trajectories, computed for plasma described by large aspect-ratio equilibria, simulate the main characteristics of trajectories in the toroidal geometry. Also the importance of the symplecticity of the new mapping regarding certain features of non-linear dynamical analysis, for which a large number of iterations is necessary, is shown. Thus, some standard algorithms, such as the Lyapunov exponents and the rotational transforms, are applied with precision in order to characterize regular and chaotic regions in the parameter space, improving the study of bifurcations, routes to chaos, and diffusion in this system.

Fractal structures in nonlinear plasma physics

Fractal structures appear in many situations related to the dynamics of conservative as well as dissipative dynamical systems, being a manifestation of chaotic behaviour. In open area-preserving discrete dynamical systems we can find fractal structures in the form of fractal boundaries, associated to escape basins, and even possessing the more general property of Wada. Such systems appear in certain applications in plasma physics, like the magnetic field line behaviour in tokamaks with ergodic limiters. The main purpose of this paper is to show how such fractal structures have observable consequences in terms of the transport properties in the plasma edge of tokamaks, some of which have been experimentally verified. We emphasize the role of the fractal structures in the understanding of mesoscale phenomena in plasmas, such as electromagnetic turbulence.

Magnetic field line mappings for a tokamak with ergodic limiters

Abstract An ergodic magnetic limiter is a device whose main effect in a tokamak is to create a cold boundary layer of chaotic magnetic field lines. In order to study its effect we have used two approaches. The first is a description of magnetic island formation through an analytical method to describe its dimensions, results being in accordance with numerical Poincare maps for magnetic field lines. The second is a model which simulates the ergodic limiter action as a sequence of impulsive perturbations, enabling the derivation of analytical formulae for the Poincare maps.

Disruptive instabilities in the discharges of the TBR-1 small Tokamak

Minor and major disruptions as well as sawteeth oscillations (internal disruptions) were identified in the discharges of the small Tokamak TBR-1, and their main characteristics were investigated. The coupling of a growing m=2 resistive mode with an m=1 perturbation seems to be the basic process for the development of a major disruption, while the minor disruption could be associated with the growth of a stochastic region of the plasma between the q=2 and q=3 islands. Measured sawteeth periods were compared with those predicted by scaling laws and good agreement was found. The time necessary for the sawteeth crashes also agrees with the values expected from Kadomtsevs model. However, there are some sawteeth oscillations, corresponding to conditions of higher plasma Zeff, which showed longer crashes and could not be explained by this model.

Torsion-adding and asymptotic winding number for periodic window sequences

Abstract In parameter space of nonlinear dynamical systems, windows of periodic states are aligned following the routes of period-adding configuring periodic window sequences. In state space of driven nonlinear oscillators, we determine the torsion associated with the periodic states and identify regions of uniform torsion in the window sequences. Moreover, we find that the measured torsion differs by a constant between successive windows in periodic window sequences. Finally, combining the torsion-adding phenomenon, reported in this work, and the known period-adding rule, we deduce a general rule to obtain the asymptotic winding number in the accumulation limit of such periodic window sequences.

Effective transport barriers in nontwist systems

In fluids and plasmas with zonal flow reversed shear, a peculiar kind of transport barrier appears in the shearless region, one that is associated with a proper route of transition to chaos. These barriers have been identified in symplectic nontwist maps that model such zonal flows. We use the so-called standard nontwist map, a paradigmatic example of nontwist systems, to analyze the parameter dependence of the transport through a broken shearless barrier. On varying a proper control parameter, we identify the onset of structures with high stickiness that give rise to an effective barrier near the broken shearless curve. Moreover, we show how these stickiness structures, and the concomitant transport reduction in the shearless region, are determined by a homoclinic tangle of the remaining dominant twin island chains. We use the finite-time rotation number, a recently proposed diagnostic, to identify transport barriers that separate different regions of stickiness. The identified barriers are comparable to those obtained by using finite-time Lyapunov exponents.

Chaotic transport in reversed shear tokamaks

For tokamak models using simplified geometries and reversed shear plasma profiles, we have numerically investigated how the onset of Lagrangian chaos at the plasma edge may affect the plasma confinement in two distinct but closely related problems. Firstly, we have considered the motion of particles in drift waves in the presence of an equilibrium radial electric field with shear. We have shown that the radial particle transport caused by this motion is selective in phase space, being determined by the resonant drift waves and depending on the parameters of both the resonant waves and the electric field profile. Moreover, we have shown that an additional transport barrier may be created at the plasma edge by increasing the electric field. In the second place, we have studied escape patterns and magnetic footprints of chaotic magnetic field lines in the region near a tokamak wall, when there are resonant modes due to the action of an ergodic magnetic limiter. A non-monotonic safety factor profile has been used in the analysis of field line topology in a region of negative magnetic shear. We have observed that, if internal modes are perturbed, the distributions of field line connection lengths and magnetic footprints exhibit spatially localized escape channels. For typical physical parameters of a fusion plasma, the two Lagrangian chaotic processes considered in this work can be effective in usual conditions so as to influence plasma confinement. The reversed shear effects discussed in this work may also contribute to evaluate the transport barrier relevance in advanced confinement scenarios in future tokamak experiments.

On a cellular automaton with time delay for modelling cancer tumors

In this work we considered cellular automaton model with time delay. Time delay included in this model reflects the delay between the time in which the site is affected and the time in which its variable is updated. We analyzed the effect of the rules on the dynamics through the cluster counting. According to this cluster counting, the dynamics behavior is investigated. We verified periodic oscillations same as delay differential equation. We also studied the relation between the time delay in the cell cycle and the time to start the metastasis, using suitable numerical diagnostics.

Replicate periodic windows in the parameter space of driven oscillators

In the bi-dimensional parameter space of driven oscillators, shrimp-shaped periodic windows are immersed in chaotic regions. For two of these oscillators, namely, Duffing and Josephson junction, we show that a weak harmonic perturbation replicates these periodic windows giving rise to parameter regions correspondent to periodic orbits. The new windows are composed of parameters whose periodic orbits have the same periodicity and pattern of stable and unstable periodic orbits already existent for the unperturbed oscillator. Moreover, these unstable periodic orbits are embedded in chaotic attractors in phase space regions where the new stable orbits are identified. Thus, the observed periodic window replication is an effective oscillator control process, once chaotic orbits are replaced by regular ones.