On an intrinsic approach of the guiding-center anholonomy and gyro-gauge-arbitrariness

In guiding center theory, the standard gyro-angle coordinate is associated with gyro-gauge dependence, the global existence problem for unit vectors perpendicular to the magnetic field, and the notion of anholonomy, which is the failure of the gyro-angle to return to its original value after being transported around a loop in configuration space. We analyse these three intriguing topics through the lens of a recently proposed, global, gauge-independent gyro-angle. This coordinate is constrained, and therefore necessitates the use of a covariant derivative. It also highlights the intrinsic meaning and physical content of gyro-gauge freedom and anholonomy. There are, in fact, many possible covariant derivatives compatible with the intrinsic gyro-angle, and each possibility corresponds to a different notion of gyro-angle transport. This observation sheds new light on Littlejohn's notion of gyro-angle transport and suggests a new derivation of the recently-discovered global existence condition for unit vectors perpendicular to the magnetic field. We also discuss the relationship between Cartesian position-momentum coordinates and the intrinsic gyro-angle

Hamiltonian description of a self-consistent interaction between charged particles and electromagnetic waves

The Hamiltonian description of the self-consistent interaction between an electromagnetic plane-wave and a co-propagating beam of charged particles is considered. We show how the motion can be reduced to a one-dimensional Hamiltonian model (in a canonical setting) from the Vlasov-Maxwell Poisson brackets. The reduction to this paradigmatic Hamiltonian model is performed using a Lie algebraic formalism which allows us to remain Hamiltonian at each step of the derivation

A Hamiltonian system for interacting Benjamin-Feir resonances

In this paper, we present a model describing the time evolution of two dimensional surface waves in gravity and infinite depth. The model of six interacting modes derives from the normal form of the system describing the dynamics of surface waves and is governed by a Hamiltonian system of equations of cubic order in the amplitudes of the waves. We derive a Hamiltonian system with two degrees of freedom from this Hamiltonian using conserved quantities. The interactions are those of two coupled Benjamin-Feir resonances. The temporal evolution of the amplitude of the different modes is described according to the parameters of the system. In particular, we study the energy exchange produced by the modulations of the amplitudes of the modes. The evolution of the modes reveals a chaotic dynamics

Creation of a Transport Barrier for the E x B drift in magnetized plasmas

International audienceWe modelize the chaotic dynamics of charged test-particles in a turbulent electric field, across the confining magnetic field in controlled thermonuclear fusion devices by a 1.5 degrees of freedom Hamiltonian dynamical system. The external electric field E is given by a some potential V and the magnetic field B is considered uniform. We prove that, by introducing a small additive control term to the external electric field, it is possible to create a transport barrier. The robustness of this control method is also numerically investigated

Enhancement of particle trapping in the wave-particle interaction

The saturated dynamics of a Single-Pass Free Electron Laser is considered within a simplified mean-field approach. A method is proposed to increase the size of the macro-particle, which is responsible for the oscillations of the intensity of the wave. This approach is based on the reconstruction of invariant tori of the dynamics of test particles. To this aim a dedicated control term is derived, the latter acting as a small apt perturbation of the system dynamics. Implications of these findings are discussed in relation to the optimization of the laser source

Gyro-gauge-independent formulation of the guiding-center reduction to arbitrary order in the Larmor radius

The guiding-center reduction is studied using gyro-gauge-independent coordinates. The Lagrangian 1-form of charged particle dynamics is Lie transformed without introducing a gyro-gauge, but using directly the unit vector of the component of the velocity perpendicular to the magnetic field as the coordinate corresponding to Larmor gyration. The reduction is shown to provide a maximal reduction for the Lagrangian and to work to all orders in the Larmor radius, following exactly the same procedure as when working with the standard gauge-dependent coordinate. The gauge-dependence is removed from the coordinate system by using a constrained variable for the gyro-angle. The closed 1-form d teta is replaced by a more general non-closed 1-form, which is equal to d theta in the gauge-dependent case. The gauge vector is replaced by a more general connection in the definition of the gradient, which behaves as a covariant derivative, in perfect agreement with the circle-bundle picture. This explains some results of previous works, whose gauge-independent expressions did not correspond to a gauge fixing but indeed correspond to a connection fixing. In addition, some general results are obtained for the guiding-center reduction. The expansion is polynomial in the cotangent of the pitch-angle as an effect of the structure of the Lagrangian, preserved by Lie derivatives. The induction for the reduction is shown to rely on the inversion of a matrix which is the same for all orders higher than three. It is inverted and explicit induction relations are obtained to go to arbitrary order in the perturbation expansion. The Hamiltonian and symplectic representations of the guiding-center reduction are recovered, but conditions for the symplectic representation at each order are emphasized

Efficient control of accelerator maps

Recently, the Hamiltonian Control Theory was used in [Boreux et al.] to increase the dynamic aperture of a ring particle accelerator having a localized thin sextupole magnet. In this letter, these results are extended by proving that a simplified version of the obtained general control term leads to significant improvements of the dynamic aperture of the uncontrolled model. In addition, the dynamics of flat beams based on the same accelerator model can be significantly improved by a reduced controlled term applied in only 1 degree of freedom

Control of stochasticity in magnetic field lines

We present a method of control which is able to create barriers to magnetic field line diffusion by a small modification of the magnetic perturbation. This method of control is based on a localized control of chaos in Hamiltonian systems. The aim is to modify the perturbation locally by a small control term which creates invariant tori acting as barriers to diffusion for Hamiltonian systems with two degrees of freedom. The location of the invariant torus is enforced in the vicinity of the chosen target. Given the importance of confinement in magnetic fusion devices, the method is applied to two examples with a loss of magnetic confinement. In the case of locked tearing modes, an invariant torus can be restored that aims at showing the current quench and therefore the generation of runaway electrons. In the second case, the method is applied to the control of stochastic boundaries allowing one to define a transport barrier within the stochastic boundary and therefore to monitor the volume of closed field lines

Reduction of the chaotic transport of impurities in turbulent magnetized plasmas

The chaotic transport of charged particles in a turbulent electrostatic potential sets the conditions of a severe limitation to the plasma confinement in devices such as tokamaks. In this chapter, we consider the motion of impurities driven by the ExB velocity where a strong magnetic field B (which allows for the guiding center approximation) is uniform and constant, and a turbulent electric field is obtained from models or from numerical fluid codes. Hamiltonian dynamics rule the transport properties of these impurities. Therefore a technique to reduce chaotic diffusion in Hamiltonian systems is able to address the issue of reducing the radial transport of impurities under some approximations. The general idea is to build barriers in phase space by a small and apt modification of the Hamiltonian. We show numerically that such perturbations are able to drastically reduce the diffusion of test-particles, and we discuss the robustness of such additional modifications