Guiding Center Equations Research Articles

Kinetic Guiding-Center Equations for the Theory of Drift Instabilities and Anomalous Transport

Littlejohn's guiding-center mechanics, with the polarization drift included, is rederived from a Lagrangian. The respective collisionless kinetic guiding-center theory is then established. While single-guiding-center energy is exactly conserved here in time-independent fields, the self-consistent kinetic guiding-center theory presented here conserves the total energy of the guiding-center plasma and its self-consistent electromagnetic fields only to leading order in ϵ ≡ rg/L. The present formalism seems to be relevant to the theory of drift waves and anomalous transport.

Guiding-center equations and characteristics for particles in a small-amplitude electromagnetic wave

Modified guiding-center equations are derived for particles experiencing a small-amplitude electromagnetic wave of arbitrary polarization. The resulting differential equations contain the first-order wave amplitude explicitly and are relativistically correct. A simple characteristic equation is obtained for the particle trajectory in energy and magnetic-moment phase space for cases where the Doppler frequency shift for drifts perpendicular to the magnetic field is negligible. Two examples are given to illustrate the behavior of particle trajectories in momentum space.

The adiabatic motion of charged dust grains in rotating magnetospheres

Dust grains in the ring systems and rapidly rotating magnetospheres of the outer planets such as Jupiter and Saturn may be sufficiently charged that the magnetic and electric forces on them are comparable with the gravitational force. The adiabatic theory of charged particle motion has previously been applied to electrons and atomic size particles. But it is also applicable to these charged dust grains in the micrometer and smaller size range. We derive here the guiding center equation of motion, drift velocity, and parallel equation of motion for these grains in a rotating magnetosphere. The effects of periodic grain charge‐discharge have not been treated previously and have been included in this analysis. Grain charge is affected by the surrounding plasma properties and by the grain plasma velocity (among other factors), both of which may vary over the gyrocircle. The resulting charge‐discharge process at the gyrofrequency destroys the invariance of the magnetic moment and causes a grain to move radially. The magnetic moment may increase or decrease, depending on the gyrophase of the charge variation. If it decreases, the motion is always toward synchronous radius for an equatorial grain. But the orbit becomes circular before the grain reaches synchronous radius, a conclusion that follows from an exact constant of the motion. This circularization can be viewed as a consequence of the gradual reduction in the magnetic moment. This circularization also suggests that dust grains leaving Io could not reach the region of the Jovian ring, but several effects could change that conclusion. Excellent qualitative and quantitative agreement is obtained between adiabatic theory and detailed numerical orbit integrations.

Finite-wavelength oscillations of a diffuse-profile, Vlasov-fluid plasma column

An equation of motion is derived which describes finite-wavelength (i.e., axial wavelength ∼ plasma radius) oscillations of a Vlasov-ion, fluid-electron plasma in a diffuse, near-ϑ pinch geometry. The equations include simultaneous geometric and ion kinetic effects. For a screw-pinch geometry, and with isothermal electrons, these equations are shown to be equivalent to the guiding-center equations. Boundary conditions and a closed-form solution for the sharp-profile case are discussed.

Extensions of guiding center motion to higher order

In a static magnetic field, some well-known guiding center equations maintain their form when extended to next order in gyroradius. In these cases, it is only necessary to include the next order term in the magnetic moment series. The differential equation for guiding center motion which describes both the parallel and perpendicular velocities correctly through first order in gyroradius is given. The question of how to define the guiding center position through second order arises and is discussed, and second order drifts are derived for one usual definition. The toroidal canonical angular momentum, Pφ, of the guiding center in an axisymmetric field is shown to be conserved using the guiding center velocity correct through first order. When second-order motion is included, Pφ is no longer a constant. The above extensions of guiding center theory help to resolve the different tokamak orbits obtained either by using the guiding center equations of motion or by using conservation of Pφ.

Linear instabilities in the quasistatic guiding-center plasmas

A numerical scheme is developed to study the stability of the linearized three-dimensional guiding-center equations of a plasma of cold ions and hot electrons. The linear growth rates of the Kelvin-Helmholtz instabilities resulting from a velocity shear introduced by a non-uniform E*B drift are calculated numerically. In particular, the theory of neutrally stable waves in two dimensions is verified for a velocity profile of the form of a sine function, and the instability growth rates of the oscillations in the neighborhood of the neutrally stable waves are calculated numerically and are shown to be in very good agreement with theory. The effect of large wavelength oscillations along the magnetic field on these instabilities is studied. The numerical scheme is also used to study the linear instabilities of the density drift waves in an inhomogeneous plasma.

Kinetic theory of tearing instability

The guiding-center kinetic equation with Fokker-Planck collision term is used to study, in cylindrical geometry, a class of dissipative instabilities of which the classical tearing mode is an archetype. Variational solution of the kinetic equation obviates the use of an approximate Ohm’s law or adiabatic assumption, as used in previous studies, and it provides a dispersive relation which is uniformly valid for any ratio of wave frequency to collision frequency. One result of using the rigorous collision operator is the prediction of a new instability. This instability, driven by the electron temperature gradient, is predicted to occur under the long mean-free path conditions of present tokamak experiments, and has significant features in common with the kink-like oscillations observed in such experiments.

Mirror height lowering of trapped electrons by an inductive electric field

The effect of a ring current inductive electric field on the motion of electrons trapped in the magnetosphere is investigated. The lowering of mirror height of the electrons is of primary concern. The guiding center equations are used to describe the particle motion in a dipole field and an inductive electric field that is produced by a time-dependent symmetric ring current. The equations are solved numerically by means of a fourth-order Runge-Kutta technique. Results of the calculation show that a westward inductive electric field of about 0.6 mV/m can lower the mirror heights of electrons to the auroral height (∼100 km) from 200 km in about 290 s. The increase in the collision frequency accompanying such a lowering is 3 orders of magnitude or greater, so that the loss of electrons to the atmosphere should be greatly enhanced.

Effect of finite Larmor radius on the interchange instability in a cylindrical plasma

The eigenvalue equation for the interchange instability is derived from guiding-centre equations including finite-Larmor-radius corrections. Analytic solutions are found which are valid near the k⃗⋅B⃗ = 0 surface and away from this surface. Matching these two solutions over their common range of validity determines the eigenvalue and hence the mode frequency. The growth rate and radial profile of the mode may readily be evaluated explicitly. The analysis is applicable to the high-β pinch, since cylindrical effects dominate over toroidal. For typical parameters, FLR stabilization is found to increase the marginally stable pressure gradient by a factor two over that given by the Suydam criterion.

Guiding center motions of magnetosheath protons over the dayside polar cap in a model open magnetosphere

The guiding center equations are solved numerically to follow 200 ev protons with pitch angles near the loss cone from the dayside magnetosheath on the noon meridian to their mirror height in a model open magnetosphere. We examine the effect of magnetic field line curvature and electric field strength upon proton mirror heights. We find that when the magnetic field has dipole‐like curvature along a dayside proton trajectory an increase in the strength of the dawn‐dusk electric field will result in an increase of the proton mirror height. When there exists significant nondipole‐like magnetic field curvature along a dayside proton trajectory an increase in the dawn‐dusk electric field strength can result in lower proton mirror heights. In our model, when the interplanetary magnetic field is directed northward or southward the latitude dependence of changes in proton mirror height resulting from changed dawn‐dusk electric field strength is the same along the noon‐midnight meridian over the northern and southern polar caps, but when the interplanetary magnetic field vector lies in the noon‐midnight meridian plane at an angle to the earth dipole axis, proton mirror height behavior in response to electric field strength changes is different over the northern and southern polar caps. Proton drift perpendicular to the magnetic field but having a component parallel to the electric field can result in kinetic energy changes of up to 25% as a proton moves from the magnetosheath to its mirror height.

Resistive instability in a uniformly rotating magnetoplasma

A closed set of guiding centre equations, derived for a rotating plasma in a static magnetic field, is applied to the problem of the stability of a plasma in a sheared field. The rotation is found to have a stabilizing effect in the absence of resistivity.A pair of coupled, linear differential equations is derived for the rotating plasma in a weakly sheared field. Dispersion relations are obtained by phase integral methods and, in the absence of finite Larmor radius effects and rotation, instability growth rates proportional to η½13 are found which become proportional to when either is included. The inclusion of both finite Larmor radius and rotation gives growing instabilities proportional to η which are stabilized by the rotation when the finite Larmor radius terms predominate.

Drift Instability in the Presence of Nonuniform Radial Electric Fields

An equation describing the drift instability in a collisionless low-β plasma, in the presence of a nonuniform radial electric field is derived from guiding center equations for the ions and the drift kinetic equation for the electrons. Numerical solutions to this equation indicate that the influence of a nonuniform radial electric field on the drift instability may be determined from the manner in which this field modifies the macroscopic electron rotation profile.

Finite Beta Equilibria in Axially Symmetric Toroidal Plasma Containment Devices

Numerical methods are applied to solve the guiding-center fluid equations, with isotropic and anisotropic pressure, in an axisymmetric toroidal configuration. The shape and structure of equilibrium magnetic flux surfaces with arbitrary plasma beta and arbitrary toroidal curvature are calculated. A comparison with the vacuum magnetic fields shows that for some pressure distributions the plasma markedly distorts the magnetic field. For other distributions the magnetic mirror instability limits the maximum allowable plasma beta. It is also found that with isotropic pressure the maximum allowable beta is much higher than with slightly anisotropic pressures.

Kinetic Theory of Plasma in a Magnetic Field

The Krylov-Bogoliubov transformation is used to simplify the guiding center equations for a charged particle in electric and magnetic fields. The resulting transformation of variables is then applied to the Vlasov distribution function, yielding a magnetic Vlasov equation which describes the low-frequency behavior of a system with no statistical effects. The equation is generalized to include effects of high- and low-frequency fluctuations by a procedure developed by Klimontovich and Dupree. A consistent treatment of the conservation laws and Maxwell's equations is given to complete the kinetic description.

Steady Flow in the Axially Symmetric Torus Using the Guiding-Center Equations

General steady flow in an axially symmetric torus is investigated using the guiding-center equations. Several constants of the macroscopic and microscopic motion are obtained, and a second-order quasilinear partial differential equation is derived for the self-consistent magnetic flux function. The condition specifying that this equation be hyperbolic or elliptic is given. For the former case, jump conditions are given when weak (discontinuous) solutions are admitted.

A Fundamental Solution for the Linearized Guiding-Center Equations

The linear one-dimensional propagation of an initial perturbation in an infinite, homogeneous, stable guiding-center plasma is studied. The dominant mode of decay after long periods of time is shown to be that of a reciprocal power of time.

Effect of Ion Collisions and Other Terms on the Resistive-g Instability in Plasmas

The treatment is based on guiding center equations, which include collisional diffusion across the magnetic field lines. The eigenvalues of the resulting differential equation are found using phase integral methods. The localized resistive-g mode becomes stable at sufficiently low collision frequencies, the dominant stabilizing mechanism depending on the shear. For strong magnetic shear [shear length < radius of curvature of field lines × ¼(M/m)¼] ion collisional effects are found to first produce stability, whereas for weak magnetic shear the dominant effect comes from the electron acoustic term in Ohm's law. Analytic stability criteria are derived which show satisfactory agreement with a recent numerical solution of the same problem, but underline the danger of extrapolation based on the numerical solution.

Flute instability in the presence of non-uniform electric fields

A general equation describing the flute instability in a low-β plasma, in the presence of a non-uniform electric field perpendicular to the magnetic field, is derived from guiding centre equations. The equations obtained by Rosenbluth and Simon and by Damm et al. correspond to two different limiting cases of this more general equation. The physical significance of various stabilizing terms is discussed. In the high density, zero gyro-radius limit, the shear flow resulting from the non-uniform electric field has a de-stabilizing effect. At low densities the imbalance between the equilibrium ion and electron densities has a significant stabilizing effect.