Hamiltonian description for magnetic field lines: a tutorial

Under certain circumstances, the equations for the magnetic field lines can be recast in a canonical form, after defining a suitable field line Hamiltonian. This analogy is extremely useful for dealing with a variety of problems involving magnetically confined plasmas, like in tokamaks and other toroidal devices, where there is usually one symmetric coordinate which plays the role of time in the canonical equations. In this tutorial paper we review the basics of the Hamiltonian description for magnetic field lines, emphasizing the role of a variational principle and gauge invariance. We present representative applications of the formalism, using cylindrical and magnetic flux coordinates in tokamak plasmas

Crises in a dissipative Bouncing ball model

The dynamics of a bouncing ball model under the influence of dissipation is investigated by using a two dimensional nonlinear mapping. When high dissipation is considered, the dynamics evolves to different attractors. The evolution of the basins of the attracting fixed points is characterized, as we vary the control parameters. Crises between the attractors and their boundaries are observed. We found that the multiple attractors are intertwined, and when the boundary crisis between their stable and unstable manifolds occur, it creates a successive mechanism of destruction for all attractors originated by the sinks. Also, an impact physical crises is setup, and it may be useful as a mechanism to reduce the number of attractors in the system

Measure, dimension, and complexity of escape in open Hamiltonian systems

In this work, we introduce the escape measure, a finite-time version of the natural measure, to investigate the transient dynamics of escape orbits in open Hamiltonian systems. In order to numerically calculate the escape measure, we cover a region of interest of the phase space with a grid and we compute the visitation frequency of a given orbit on each box of the grid before the orbit escapes. Since open systems are not topologically transitive, we also define the mean escape measure, an average of the escape measure on an ensemble of initial conditions. We apply these concepts to study two physical systems: the single-null divertor tokamak, described by a two-dimensional map; and the Earth-Moon system, as modeled by the planar circular restricted three-body problem. First, by calculating the mean escape measure profile, we visually illustrate the paths taken by the escape orbits within the system. We observe that the choice of the ensemble of initial conditions may lead to distinct dynamical scenarios in both systems. Particularly, different orbits may experience different stickiness effects. After that, we analyze the mean escape measure distribution and we find that these vary greatly between the cases, highlighting the differences between our systems as well. Lastly, we define two parameters: the escape correlation dimension, that is independent of the grid resolution, and the escape complexity coefficient, which takes into account additional dynamical aspects, such as the orbit's escape time. We show that both of these parameters can quantify and distinguish between the diverse transient scenarios that arise.Comment: 12 pages, 12 figures, 2 table

Explaining a changeover from normal to super diffusion in time-dependent billiards

The changeover from normal to super diffusion in time dependent billiards is explained analytically. The unlimited energy growth for an ensemble of bouncing particles in time dependent billiards is obtained by means of a two dimensional mapping of the first and second moments of the velocity distribution function. We prove that for low initial velocities the mean velocity of the ensemble grows with exponent ~1/2 of the number of collisions with the border, therefore exhibiting normal diffusion. Eventually, this regime changes to a faster growth characterized by an exponent ~1 corresponding to super diffusion. For larger initial velocities, the temporary symmetry in the diffusion of velocities explains an initial plateau of the average velocity

Separation of particles leading to decay and unlimited growth of energy in a driven stadium-like billiard

A competition between decay and growth of energy in a time-dependent stadium billiard is discussed giving emphasis in the decay of energy mechanism. A critical resonance velocity is identified for causing of separation between ensembles of high and low energy and a statistical investigation is made using ensembles of initial conditions both above and below the resonance velocity. For high initial velocity, Fermi acceleration is inherent in the system. However for low initial velocity, the resonance allies with stickiness hold the particles in a regular or quasi-regular regime near the fixed points, preventing them from exhibiting Fermi acceleration. Also, a transport analysis along the velocity axis is discussed to quantify the competition of growth and decay of energy and making use distributions of histograms of frequency, and we set that the causes of the decay of energy are due to the capture of the orbits by the resonant fixed points

A nontwist field line mapping in a tokamak with ergodic magnetic limiter

For tokamaks with uniform magnetic shear, Martin and Taylor have proposed a symplectic map has been used to describe the magnetic field lines at the plasma edge perturbed by an ergodic magnetic limiter. We propose an analytical magnetic field line map, based on the Martin-Taylor map, for a tokamak with arbitrary safety factor profile. With the inclusion of a non-monotonic profile, we obtain a nontwist map which presents the characteristic properties of degenerate systems, as the twin islands scenario, the shearless curve and separatrix reconnection. We estimate the width of the islands and describe their changes of shape for large values of the limiter current. From our numerical simulations about the shearless curve, we show that its position and aspect depend on the control parameters

Sub-diffusive behavior in the Standard Map

In this work, we investigate the presence of sub-diffusive behavior in the Chirikov-Taylor Standard Map. We show that the stickiness phenomena, present in the mixed phase space of the map setup, can be characterized as a Continuous Time Random Walk model and connected to the theoretical background for anomalous diffusion. Additionally, we choose a variant of the Ulam method to numerically approximate the Perron-Frobenius operator for the map, allowing us to calculate the anomalous diffusion exponent via an eigenvalue problem, compared to the solution of the Fractional Diffusion Equation. The results here corroborate other findings in the literature of anomalous transport in Hamiltonian maps and can be suitable to describe transport properties of other dynamical systems.Comment: Submitted to EPJ-S