Dynamics Of Magnetic Field Lines Research Articles

General formulas for adiabatic invariants in nearly periodic Hamiltonian systems

While it is well known that every nearly periodic Hamiltonian system possesses an adiabatic invariant, extant methods for computing terms in the adiabatic invariant series are inefficient. The most popular method involves the heavy intermediate calculation of a non-unique near-identity coordinate transformation, even though the adiabatic invariant itself is a uniquely defined scalar. A less well-known method, developed by S. Omohundro, avoids calculating intermediate sequences of coordinate transformations but is also inefficient as it involves its own sequence of complex intermediate calculations. In order to improve the efficiency of future calculations of adiabatic invariants, we derive generally applicable, readily computable formulas for the first several terms in the adiabatic invariant series. To demonstrate the utility of these formulas, we apply them to charged-particle dynamics in a strong magnetic field and magnetic field-line dynamics when the field lines are nearly closed.

Dynamics of Field Line Mappings in Magnetic Flux Tubes

We study the topological constraints on the dynamics of magnetic field lines in flux tubes. Our approach is based on the application of the topological invariant: fixed point index. We consider periodic flux tubes and find various restrictions on the field lines that come from the sequence of fixed point indices of iterations. We also analyze the case of a tube with a cylindrical obstacle, deducing some special dynamical properties of the field line mapping from geometrical properties of the tube and the additional assumptions put on the field.

The essence of the Blandford–Znajek process

From a spacetime perspective, the dynamics of magnetic field lines of force-free electromagnetic fields can be rewritten into a quite similar form for the dynamics of strings, i.e., dynamics of "field sheets". Using this formalism, we explicitly show that the field sheets of stationary and axisymmetric force-free electromagnetic fields have identical intrinsic properties to the world sheets of rigidly rotating Nambu-Goto strings. Thus, we conclude that the Blandford-Znajek process is kinematically identical to an energy-extraction mechanism by the Nambu-Goto string with an effective magnetic tension.

Explosive Particle Dispersion in Plasma Turbulence.

Particle dynamics are investigated in plasma turbulence, using self-consistent kinetic simulations, in two dimensions. In the steady state, the trajectories of single protons and proton pairs are studied, at different values of plasma β (ratio between kinetic and magnetic pressure). For single-particle displacements, results are consistent with fluids and magnetic field line dynamics, where particles undergo normal diffusion for very long times, with higher β's being more diffusive. In an intermediate time range, with separations lying in the inertial range, particles experience an explosive dispersion in time, consistent with the Richardson prediction. These results, obtained for the first time with a self-consistent kinetic model, are relevant for astrophysical and laboratory plasmas, where turbulence is crucial for heating, mixing, and acceleration processes.

Notes on Magnetohydrodynamics of Magnetic Reconnection in Turbulent Media

Astrophysical fluids have very large Reynolds numbers and therefore turbulence is their natural state. Magnetic reconnection is an important process in many astrophysical plasmas, which allows restructuring of magnetic fields and conversion of stored magnetic energy into heat and kinetic energy. Turbulence is known to dramatically change different transport processes and therefore it is not unexpected that turbulence can alter the dynamics of magnetic field lines within the reconnection process. We shall review the interaction between turbulence and reconnection at different scales, showing how a state of turbulent reconnection is natural in astrophysical plasmas, with implications for a range of phenomena across astrophysics. We consider the process of magnetic reconnection that is fast in magnetohydrodynamic (MHD) limit and discuss how turbulence—both externally driven and generated in the reconnecting system—can make reconnection independent on the microphysical properties of plasmas. We will also show how relaxation theory can be used to calculate the energy dissipated in turbulent reconnecting fields. As well as heating the plasma, the energy dissipated by turbulent reconnection may cause acceleration of non-thermal particles, which is briefly discussed here.

Topology of magnetic field lines: Chaos and bifurcations emerging from two-action systems

Nonlinear dynamics of magnetic field lines generated by simple electric current elements are investigated. In general, the magnetic field lines show behavior similar to that of the Hamiltonian systems; in fact, they can be generally transformed into Hamiltonian systems with 1.5 degrees of freedom, obey the Kolmogorov-Arnold-Moser (KAM) theorem, and generate chaotic trajectories. In the case where unperturbed systems are described by two action (slow) and one angle (fast) variables, however, it is found that the periodic orbits of the unperturbed systems vanish for arbitrarily small symmetry-breaking perturbations (a breakdown of the KAM theorem) and drifting or periodic trajectories appear. The mechanism of this phenomenon is investigated analytically by weak nonlinear stability analysis. It is also shown numerically that scattering processes of the perturbed system exhibit typical features of chaotic dynamical systems.

On the accuracy of some mapping techniques used to study the magnetic field dynamics in tokamaks

The dynamics of magnetic field lines and of charged particles in toroidal chambers are commonly analysed by numerically solving the dynamical equations. They may also be analysed using deterministic reduced models, i.e. low-dimensional discrete time approximations (maps) of the Hamiltonian continuous time models. We report on the accuracy of the latter method by considering the mapping technique derived from the Hamilton–Jacobi equation. The optimum time stepping in some models for the study of the magnetic field in tokamaks is determined by using local criteria. Special attention is given to the analysis of stochasticity produced by time discretization.

Transport Barriers and Diffusion Phenomena for the Magnetic Field Lines in Tokamak

The aim of this study is to localize and to describe the transport barriers and the diffusion phenomena that appear in tokamak in two Hamiltonian models for the study of magnetic field line dynamics: a twist map (tokamap) and a non-twist one (rev-tokamap). New results obtained for rev-tokamap will be compared to the already known ones on tokamap and with those concerning the non-twist standard model.The analysis will be done in terms of a stochasticity parameter describing the perturbation amplitude of the ideal magnetic configuration, and of the q-profiles.

Compressible dynamics of magnetic field lines for incompressible magnetohydrodynamic flows

It is demonstrated that the deformation of magnetic field lines in incompressible magnetohydrodynamic flows results from a compressible mapping associated with the transverse motion of fluid particles. Appearance of zeros for the Jacobian of this mapping corresponds to the breaking of magnetic field lines and the local blowup of the magnetic field intensity. The occurrence of such events is found to be unlikely in two dimensions but possible in three dimensions.

Reconnection in a global model of Poincaré map describing dynamics of magnetic field lines in a reversed shear tokamak

Magnetic field lines behaviour in a reversed shear tokamak can be described by a one and a half degree of freedom Hamiltonian system. In order to get insights into its dynamics we study numerically a global model for a Poincaré map associated to such a system. Mainly we investigate the scenario of reconnection of the invariant manifolds of two hyperbolic orbits of the same type n/ m and show that it is a generic one. When the two Poincaré–Birkhoff chains involved in this process are aligned in phase, i.e. they are in a nongeneric position, a sequence of two saddle–center bifurcations occur in one of the chains, interfering with the former elliptic orbit of that chain, such that at the reconnection threshold the two chains are in a generic position. Dynamics around the new created configuration at the reconnection appears to vary from a regular motion to a chaotic one. In connection with the study of this global bifurcation we give the first example of region of instability in the dynamics of a nontwist map.

Asymptotical and mapping methods in study of ergodic divertor magnetic field in a toroidal system

Asymptotical and mapping methods to study the structure of magnetic field perturbations and magnetic field line dynamics in a tokamak ergodic divertor in toroidal geometry are developed. The investigation is applied to the Dynamic Ergodic Divertor under construction for the Torus Experiment for the Technology Oriented Research (TEXTOR-94) Tokamak at Jülich [Fusion Eng. Design 37, 337 (1997)]. An ideal coil configuration designed to create resonant magnetic perturbations at the plasma edge is considered. In cylindrical geometry, the analytical expressions for the vacuum magnetic field perturbations of such a coil system are derived, and its properties are studied. Corrections to the magnetic field due to the toroidicity are presented. The asymptotical analysis of transformation of magnetic perturbation into the Hamiltonian perturbation in toroidal geometry is carried out, and the asymptotic formulas for the spectrum of the Hamiltonian perturbations are found. A new method of integration of Hamiltonian equations is developed. It is based on a canonical transformation of variables that replaces the dynamics of a continuous Hamiltonian system by a symplectic mapping. The form of the mapping is established in the first order of perturbation theory. It is shown that the mapping well reproduces Poincaré sections of field lines, as well as their statistical properties in an ergodic zone obtained by the numerical integration of field line equations. The mapping is applied to study, in particular, the formation of a stochastic layer and the statistical properties of field lines at the plasma edge.

Contribution to the study of the non–Gaussian dynamics of stochastic magnetic field lines in a toroidal geometry (tokamak)

In this work, we study the non–Gaussian dynamic of the magnetic field lines of a plasma confined in a toroidal configuration. We have used the Fourier paths technique in order to calculate the kurtosis of the magnetic field lines. We have also calculate numerically this quantity. The magnetic field lines are described by the standard mapping. A comparison between our analytical and numerical calculations has been done.

Twist mapping for the dynamics of magnetic field lines in a tokamak ergodic divertor

Symplectic twist mapping is proposed to model magnetic field line dynamics in the ergodic divertor at the tokamak plasma edge. The relationship between a perturbation function in the mapping and magnetic field perturbation in the tokamak is found. The mapping is specified for the Dynamic Ergodic Divertor being proposed for the Torus Experiment for Technology Oriented Research (TEXTOR-94) [Fusion Eng. Design, 37, 337 (1997)]. The spectrum of the poloidal harmonics of perturbation is assumed to be localized around the harmonics m=12. It creates the stochastic layer near the resonant magnetic surface q=3. The mapping is applied to the formation of the stochastic layer and field line diffusivity at the plasma edge. For the moderate magnetic field perturbations, the ergodic layer consists of a stochastic sea with regular Kolmogorov–Arnold–Mozer (KAM) -stability islands. The radial profiles of the Kolmogorov lengths and the field line diffusivity are studied for different perturbations. It is shown that the behavior of open field lines at the lower boundary of the stochastic layer is subdiffusive. For large perturbations a regular convective behavior of open field lines dominates over their diffusion at the large region of the ergodic layer.

Peripheral Stochasticity in Tokamaks.The Martin-Taylor Revisited

Abstract We analyse the effect of an Ergodic Magnetic Limiter on the magnetic field line dynamics in the edge of a large aspect-ratio Tokamak. We model the limiter action as an impulsive perturbation and use a peaked-current model for the Tokamak equilibrium field. The theoretical analysis is made through the use of invariant flux functions describing magnetic surfaces. Results are compared with a numerical mapping of the field lines