Guiding Center Motion Research Articles

A time‐dependent gyro‐kinetic model of thermal ion upflows in the high‐latitude F region

Ample evidence supports the significance of the high‐latitude ionospheric contribution to magnetospheric plasma. Assuming flux conservation along a flux tube, the upward field‐aligned ion flows observed in the magnetosphere require high‐latitude ionospheric field‐aligned ion upflows of the order of 108 to 109 cm−2 s−1. Since radar and satellite observations of high‐latitude F region flows at times exceed this flux requirement by an order of magnitude, the thermal ionospheric upflows are not simply the ionospheric response to a magnetospheric flux requirement. Several ionospheric ion upflow mechanisms have been proposed, but simulations based on fluid theory do not reproduce all the observed features of ionospheric ion upflows. Certain asymmetries in the statistical morphology of high‐latitude F region ion upflows suggest that the ion upflows may be generated by ion‐neutral frictional heating. We developed a single‐component (O+), time‐dependent gyro‐kinetic model of the high‐latitude F region response to frictional heating in which the neutral exobase is a discontinuous boundary between fully collisional and collisionless plasmas. The concept of a discontinuous neutral exobase and the assumption of a constant and uniform polarization electric field reduce the ion guiding center motion in the frame of a convecting flux tube to simple one‐dimensional ballistic trajectories. We thus are able to analytically calculate a time and height‐dependent ion velocity distribution function, from which we can compute the ion density, parallel velocity, parallel and perpendicular temperature, and parallel flux. Using our model, we simulated the response of a convecting flux tube between 500 km and 2500 km to various frictional heating inputs; the results were both qualitatively and quantitatively different from fluid model results, which may indicate an inadequacy of the fluid theory approach. The gyro‐kinetic frictional heating model responses to the various simulations were qualitatively similar: (1) initial perturbations of all the modeled parameters propagated rapidly up the flux tube, (2) transient values of the ion parallel velocity, temperature, and flux exceeded 3 km s−1, 2 × 104 K, and 109 cm−2 s−1. respectively, (3) a second transient regime developed wherein the parallel temperature drops to very low values (a few hundred Kelvins), and (4) well after heating ceased, large parallel temperatures and large downward parallel velocities and fluxes developed as the flux tube slowly returned to diffusive equilibrium. The ion velocity distributions during the simulation are often non‐Maxwellian and are sometimes composed of two distinct ion populations.

Kinetic and thermodynamic properties of a convecting plasma in a two‐dimensional dipole field

Charged particle guiding center motion is considered in the magnetic field of a two‐dimensional (“line”) dipole on which is superimposed a small, static, perpendicular electric field. The parallel equation of motion is that of a simple harmonic oscillator for cos θ, the cosine of magnetic colatitude θ. Equations for the perpendicular electric and magnetic drifts are derived as well as their bounce‐averaged forms. The latter are solved to yield a bounce‐averaged guiding center trajectory, which is the same as that obtained from conservation of magnetic moment µ, longitudinal invariant J, and total (kinetic plus electrostatic) energy K. The algebraic simplicity of the trajectory equations is also manifest in the forms of the invariants. An interesting result is that guiding centers drift in such a way that they preserve the values of their equatorial pitch angles and (equivalently) mirror latitudes. The most general Maxwellian form of the equilibrium one‐particle distribution function f is constructed from the invariants, and spatially varying density and pressure moments, parallel and perpendicular to the magnetic field, are identified. Much of the paper deals with the more restricted problem in which f is specified as a bi‐Maxwellian over a straight line of finite length in the equatorial plane of the dipole and perpendicular to field lines. This might be thought of as specifying a cross‐tail ion injection source; our formalism then describes the subsequent spatial development. The distribution away from the source is a scaled bi‐Maxwellian but one that is cut off at large and small kinetic energies, which depend on position. Density and pressure components are reduced from the values they would have if the total content of individual flux tubes convected intact. The equatorial and meridional variations of density and pressure components are examined and compared systematically for the isotropic and highly anisotropic situations. There appears to be little qualitative difference due to anisotropy. An anisotropy measure is defined, and its spatial variation determined as a signature of possible MHD instability. Extreme values are found, larger than at the source, but the plasma beta in such regions is probably so low as to render the effect inconsequential energetically. Finally, the possible consequence of “nonadiabatic” pressure profiles on electrostatic interchanges is considered, and a boundary delineating stabilizing and destabilizing regions determined.

Solution of the drift-kinetic equation for global plasma modes and finite particle orbit widths

The response of a collisionless plasma to global electromagnetic perturbations of an axisymmetric toroidal equilibrium is derived. By adopting a variational formulation for guiding center motion, the perturbed distribution function is expressed in terms of the linearized guiding center Lagrangian. Finite orbit widths are retained. In particular, the high particle energy limit where mirror-trapped banana orbits are distorted into ‘‘potato-shaped’’ orbits is considered. In this limit, the time scales associated with the drift and bounce motions of a mirror-trapped orbit become comparable, yielding important consequences on plasma stability. Quadratic forms are constructed in the context of kinetic-magnetohydrodynamic (MHD) models of plasmas composed of a thermal component obeying fluid-like equations and a high-energy component described in terms of the collisionless drift-kinetic equation. Relevant applications include improved modeling of energetic ion effects on toroidicity-induced Alfvén gap modes and internal kinks.

Guiding-centre motion in the framework of Hamilton–Jacobi theory

Following renewed interest in the guiding-centre problem, we propose an alternative to previous approaches. It consists essentially in demonstrating the existence of a representative simple model for which the problem is rigorously solved in the Hamilton–Jacobi framework. It is shown that a perturbation method allows the extension of the model to more realistic cases. A further extension to cover both magnetic-mirror configurations and tokamaks can be achieved.

Geometric phase, rotational transforms, and adiabatic invariants in toroidal magnetic fields

The rotational transform associated with the magnetic surfaces of a toroidal magnetic field with a nonplanar axis is an example of the angle anholonomy which recently has been much discussed in quantum and classical dynamics (the Berry phase and Hannay angle). The same anholonomic angle appears in the phase of a charged particle spiraling around its guiding center in a strong magnetic field. This accounts for a contribution to the longitudinal invariant, associated with the guiding-center motion, which is different for guiding-center orbits that circulate in opposite directions and is absent for orbits that are reflected between mirrors.

Comparison of gyro-averaged and non-gyro-averaged non-linear analyses of cyclotron autoresonance masers

Abstract The non-linear evolution of the cyclotron autoresonance maser (CARM) is investigated numerically using two distinct three-dimensional formulations of the interaction. These formulations differ primarily in that one (CSPOT) relies upon a gyro-averaged orbit treatment, while the other (CHARM) imposes no gyro-average. Our purpose is to compare and contrast the results of these different formulations for CARMs with a range of physically interesting parameters. Both formulations employ a representation of the electromagnetic field based upon a superposition of the TE and TM modes of a loss-free cylindrical waveguide, and they are fully multi-mode. Waveguide losses are included by means of a constant loss rate. The distinction between these formulations is in the electron dynamics. The CSPOT formulation imposes a gyro-average on the electron orbit equations in which the guiding-centre motion is neglected. Thus, the electron motion is reduced to a set of three equations per particle. In contrast, the CH...

Unusual manifold structure and anomalous diffusion in a Hamiltonian model for chaotic guiding-center motion.

Unusual manifold structure and anomalous diffusion in a Hamiltonian model for chaotic guiding-center motion.

Numerical error in electron orbits with large ωceΔt

We have found that running electrostatic particle codes at relatively large ω ce Δt in some circumstances does not significantly affect the physical results. We first present results from a single particle mover finding the correct first order drifts for large ω ce Δt. We then characterize the numerical orbit of the Boris algorithm for rotation when ϵ ce Δt ⪢ 1. Next, an analysis of the guiding center motion is given showing why the first order drift is retained at large ω ce Δt. Lastly, we present a plasma simulation of a one dimensional cross field sheath, with large and small ω ce Δt, with very little difference in the results.

Guiding-center chaotic motion in three electrostatic waves.

The dynamics of guiding centers in a general configuration of three electrostatic plane waves is studied in the plane ({ital x},{ital y}) perpendicular to a strong magnetic field. The associated Hamiltonian is nonautonomous with 11/2 degrees of freedom, and the conjugated variables are the coordinates ({ital x},{ital y}). This is the simplest system that exhibits the onset of chaotic motion in electrostatic plasma turbulence. An explicit analytical solution is obtained for the unperturbed integrable system consisting of two low-amplitude waves, by solving a generalized Kepler equation. Chaotic diffusion of guiding centers becomes possible due to the existence of a third wave. As the amplitude of the perturbation increases, chaotic motion is first localized, then extended over a network spread over the whole phase space along separatrices, and finally densely covers the full space.

On the anomalous Doppler/inner Lindblad resonance

There are strong formal similarities between the condition for anomalous Doppler resonance in plasmas ( omega -k/sub ///v/sub ///=- Omega , where omega is the frequency of the perturbing wave field, k/sub ///v/sub /// measures the Doppler shift due to guiding-centre motion, and Omega is the cyclotron frequency) and the condition for inner Lindblad resonance in stellar discs ( Omega b- Omega 0=- kappa /m, where Omega b is the frequency of the perturbing wave field, whose azimuthal harmonic number is m, Omega 0 is the natural frequency of equilibrium guiding-centre circulation, and kappa is the epicyclic frequency). It is shown that this similarity reflects a close underlying correspondence between the two forms of resonance.

Chaotic diffusion across a magnetic island due to a single low-frequency electrostatic wave

It is shown that the guiding center motion around a single chain of magnetic islands in a sheared slab can become chaotic in the presence of a single prescribed low-frequency electrostatic wave. This process leads to radial diffusion across the islands without magnetic braiding. The chaotic diffusion appears to be selective in velocity space. Using realistic values of the physical parameters for a fusion plasma, it is concluded that this process can be effective in usual conditions: with the observed islands, and electronic field values corresponding to measured density fluctuations in the TFR Tokamak, this diffusion concerns ions with velocities higher than thermal, and almost all of the electron population. The consequences for radial diffusion are discussed.

The energy-momentum tensor for the linearized Maxwell–Vlasov and kinetic guiding center theories

A modified Hamilton–Jacobi formalism is introduced as a tool to obtain the energy-momentum and angular-momentum tensors for any kind of nonlinear or linearized Maxwell-collisionless kinetic theories. The emphasis is on linearized theories, for which these tensors are derived for the first time. The kinetic theories treated—which need not be the same for all particle species in a plasma—are the Vlasov and kinetic guiding center theories. The Hamiltonian for the guiding center motion is taken in the form resulting from Dirac’s constraint theory for nonstandard Lagrangian systems. As an example of the Maxwell-kinetic guiding center theory, the second-order energy for a perturbed homogeneous magnetized plasma is calculated with initially vanishing field perturbations. The expression obtained is compared with the corresponding one of Maxwell–Vlasov theory.

Guiding-center soliton.

We consider the behavior of solitons described by the nonlinear Schr\odinger equation with periodic perturbation (not necessarily small) whose period ${\mathit{Z}}_{\mathit{a}}$ is much shorter than the characteristic period of the solitons, ${\mathit{Z}}_{0}$. The soliton behaves like the guiding-center motion of a charged particle in a magnetic field, smooth and adiabatic. However, when ${\mathit{Z}}_{\mathit{a}}$ approaches ${\mathit{Z}}_{0}$, resonances appear between the periodic perturbations and the characteristic frequencies of the solitons which induce the generation of dispersive waves and/or splitting of solitons.

Canonical Hamiltonian guiding center variables

A simplification of the canonical Hamiltonian variables for the guiding center motion of a charged particle in a general toroidal field is obtained using the Lagrangian formalism.

Phase anholonomy in the classical adiabatic motion of charged particles.

The classical adiabatic motion of charged particles in strong magnetic fields, otherwise known as guiding center motion, exhibits an anholonomic phase similar to Berry's phase and the phase discovered in a general context by Hannay. Analysis of this effect reveals that there often is no best way to define the gyrophase when magnetic field lines are curved. Instead, a change in the definition of the phase is a kind of gauge transformation, the study of which leads to new insight into guiding center theory. The path-dependent phase that occurs in this problem is coupled with the metrical structure of physical space, giving rise to a transport process for vectors and frames which is similar to parallel transport in non-Euclidean geometry. Strong analogies with Fermi-Walker transport and Thomas precession in special relativity are pointed out.

Transport of ionospheric ions in the magnetosphere: Theory and observations

A Monte Carlo transport code is described that follows the distribution functions of upward flowing ions (UFI) from a source location in the ionosphere into the magnetosphere. Examples of the guiding-center motion and the full gyration motion of ions from the cusp ionosphere are shown to describe the characteristics of the ions expected in the outer magnetosphere, and these characteristics are compared with measurements of ion streams in the lobes and plasma sheet. This comparison indicates that the ionospheric ions observed in the outer magnetosphere originate from the cusp ionosphere. Moreover, a comparison of ISEE-1 measurements of the distribution functions of H + and O + in the ring current during a magnetic storm with the distribution functions of ions from the ionosphere, computed with an earlier version of the Monte Carlo transport code, indicates that the UFI from the cusp ionosphere may fully account for all the H + and O + in the near-earth plasma sheet and ring current during stormtime conditions. It is also shown that if O + ions from the cusp ionosphere cross the center plane of the magnetotail at distances greater than about 9 R E, their magnetic moments are not conserved.

Two- and three-dimensional magnetoinductive particle codes with guiding center electron motion

A magnetoinductive (Darwin) particle simulation model developed for examinig low frequency plasma behavior with large time steps is presented. Electron motion perpendicular to the magnetic field is treated as massless keeping only the guiding center motion. Electron motion parallel to the magnetic field retains full inertial effects as does the ion motion. This model has been implemented in two and three dimensions. Computational tests of the equilibrium properties of the code are compared with linear theory and the flunctuation dissipation theorem. This code has been applied to the problems of Alfvén wave resonance heating and twist-kink modes.

Hamiltonian theory of guiding centre motion revisited

The classical problem of the motion of a charged particle in a slowly varying electromagnetic field is reconsidered in the framework of ‘pseudo-canonical transformations’ in a Hamiltonian formalism. As compared with Littlejohn's important recent work, we develop a method which we believe to be more transparent. It consists, in essence, of exploiting directly the requirement that the Lie brackets of the guiding centre variables be independent of the (new) gyrophase. By using this method, we construct explicitly the complete set of functional relationships between the guiding centre variables and the particle variables, up to second order in the drift parameter є These expressions are valid for magnetic and electric fields of arbitrary geometry (within the drift approximation), which may be slowly varying in time.

Hamiltonian and non-Hamiltonian perturbation theory for nearly periodic motion

Kruskal’s asymptotic theory of nearly period motion [M. Kruskal, J. Math. Phys. 4, 806 (1962)] (with applications to nonlinear oscillators, guiding center motion, etc.) is generalized and modified. A new more natural recursive formula, with considerable advantages in applications, determining the averaging transformations and the drift equations is derived. Also almost quasiperiodic motion is considered. For a Hamiltonian system, a manifestly Hamiltonian extension of Kruskal’s theory is given by means of the phase-space Lagrangian formulation of Hamiltonian mechanics. By performing an averaging transformation on the phase-space Lagrangian for the system (L → L̄) and adding a total derivative dS/dτ, a nonoscillatory Lagrangian Λ=L̄+dS/dτ is obtained. The drift equations and the adiabatic invariant are now obtained from Λ. By truncating Λ to some finite order in the small parameter ε, manifestly Hamiltonian approximating systems are obtained. The utility of the method for treating the guiding-center motion is demonstrated in a separate paper.

Thermal equilibrium of a cryogenic magnetized pure electron plasma

The thermal equilibrium correlation properties of a magnetically confined pure electron plasma (McPEP) are related to those of a one-component plasma (OCP). The N-particle spatial distribution ρs and the Helmholtz free energy F are evaluated for the McPEP to O(λ2d/a2), where λd is the thermal de Broglie wavelength and a is an interparticle spacing. The electron gyromotion is allowed to be fully quantized while the guiding center motion is quasiclassical. The distribution ρs is shown to be identical to that of a classical OCP with a slightly modified potential. To O(λ2d/a2) this modification does not affect that part of F which caused by correlations, as long as certain requirements concerning the size of the plasma are met. This theory is motivated by a current series of experiments that involve the cooling of a magnetically confined pure electron plasma to the cryogenic temperature range.