Guiding Center Equations Of Motion Research Articles

Particle transport induced by electrostatic wave fluctuations

Particle transport driven by electrostatic waves at the plasma edge is numerically investigated, for large aspect ratio tokamaks, by considering a kinetic model derived from guiding-center equations of motion. Initially, the transport is estimated for trajectories obtained from differential equations for a wave spectrum generated by a dominant spatial mode and three time modes. Then, in case of infinite time modes, the differential equations of motion are used to introduce a symplectic map that allows to analyze the particle transport. The particle transport barriers are observed for spatial localized dominant perturbation and infinite modes. In presence of infinite spatial modes, periodic islands arise in between chaotic trajectories at the plasma edge.

Computer simulation of ion clouds in a Penning trap.

In a recent series of experiments, ' a collection of N ions (where N=100-1000) were stored in a Penning trap and cooled to very low temperature (T= 1-10 mK). One expects these ions to be strongly correlated since the coupling parameter I is larger than unity. The relatively small number of ions used in the experiments makes the system ideal for molecular-dynamics (MD) simulations with realistic boundary conditions. We have performed such simulations for N 100 and N 256 in the large-I regime and have seen behavior quite unlike that observed in simulations of an unbounded homogeneous system of ions. These latter simulations predict a liquid phase for I =2 and a transition to a body-centered cubic lattice for I =170. In contrast, we observe that at large I the system of ions arranges itself into concentric spheroidal shells. However, the ions wander randomly over the surface of the shells. The system might therefore be characterized as a crystal in the direction perpendicular to the shells and as a liquid on the shells; similar behavior is observed in smectic-liquid crystals. As I is further increased, diffusion decreases and a 2D hexagonal lattice forms on the outer shells. However, the lattice is imperfect and diffusion persists even for I =300-400. A 2D hexagonal lattice on cylindrical shells was observed previously by Rahman and Schiffer in a simulation designed to model a system of ions in a storage ring. These authors also considered spherically symmetric potentials as a test of the effect of boundary conditions on their model. While space does not allow for a detailed comparison, their results are consistent with those presented here. Our MD code is novel in that it is based on guidingcenter equations of motion. We will 6rst discuss the advantages and range of applicability of the code, and then we present the results of the simulations. In the experiments, a strong magnetic 6eld is applied to con6ne the ion cloud, and this 6eld makes a straightforward simulation difficult by introducing a small time scale and a small length scale: the cyclotron period and the cyclotron radius. To overcome this difficulty, we average out the cyclotron dynamics, replacing the exact equations of motion by guiding-center equations of motion. The idea here is that for a sufficiently strong magnetic field the cyclotron motion decouples from the motion associated with the collision dynamics. For I ) 1, this decoupling requires the cyclotron frequency to be large compared to the plasma frequency. This strong-magnetic-field limit is often achieved in the experiments, and in this case the guiding-center equations of motion provide a good approximation to the exact dynamics of the system. Furthermore, we will see that for N large, the spatial properties of the guiding-center system in equilibrium are the same as those of an equivalent system undergoing exact dynamics. In the guiding-center approximation the state of each ion is specified by its guiding-center position x and its velocity U parallel to the magnetic field B. We take B to be uniform and directed along the z axis of a cylindrical-coordinate system (p, p, z). The equations of motion are then

Local conservation laws for the Maxwell-Vlasov and collisionless kinetic guiding-center theories.

expressions for the and flux densities were determined. Here the full energy-momentum tensor and the angular momentum tensor are obtained via Noether's theorem. The appropriate symmetries of these tensors are shown and the local conservation laws are proven. These results are of importance for applications; e.g., knowledge of the total angular momentum combined with the can lead to an energy principle for linear stability analysis. The tensors are obtained in the usual way by first deter­ mining the general expression for an arbitrary variation of the total· Lagrangian density in normal position space x. We note, however, that there is a slight complication be­ cause the particle part of the Lagrangian is primarily defined on an extended space Y=(YY2) where y, is iden­ tical to x and Y2 is an additional coordinate that is needed in order to describe guiding centers. By means of transla­ tional invariance in x space and time, and rotational in­ variance in x space· the canonical tensors are obtained. These tensors are not gauge invariant, but each can be split into a divergence-free gauge-invariant part and a divergence-free non-gauge-invariant part. The gauge­ invariant energy-momentum tensor turns out to be sym­ metric in its spatial components. This is shown to follow from its relationship to the gauge-iI,lVariant part of the an­ gular momentum tensor. All of these expressions are also applicable to relativistic theories. Two applications are considered: first we treat the Maxwell-Vlasov equations and show that the gauge­ invariant parts of our tensors reduce to the usual well­ known expressions. Following this we treat the Maxwell­ kinetic guiding-center theory based on Littlejohn's guiding-center equations of motion.4