Guiding center drifts in time-dependent meridional magnetic fields
Drift rate for trapped particles in meridional magnetic mirror field subjected to time dependent electric fields
Birmingham, T. J.↗
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Euler potentials and their application in the Hamiltonian formulation of guiding center motion
Scalar analysis and Euler potentials for Hamiltonian formulation of charged particle motion in magnetic field
Stern, D. P.↗
Guiding center drifts in time-dependent meridional magnetic fields.
Trapped particles drift velocity in time dependent meridional magnetic and perpendicular electric field
Birmingham, T. J.↗
Comments on 'Fluctuations in guiding center plasma in two dimensions'
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Vahala, G.↗
Theory of flux anisotropies in a guiding center plasma
The one particle distribution function f on the scale of the bounce motion of particles in a magnetic field B is considered. The Vlasov equation is expanded through O(epsilon) in the adiabatic parameter which is the ratio of particle gyroradius to scale length of the magnetic field. Because f is directly proportional to particle flux differential in kinetic energy and solid angle, f is in principle measurable in space experiments, and the analysis is tailored to be explicitly applicable to space problems. To O(1), f is gyrotropic; its first velocity moment is (if non-vanishing) parallel to B, and hence macroscopic parallel flow is included in this term. The O(epsilon) contribution is non-gyrotropic and macroscopic flow parallel to B plus additional parallel flow results from these terms. The degree of non-gyrotropy and the amount of cross-field macroscopic flow depend on the perpendicular component of the electric field, on curvature and shear in the magnetic field, and on the spatial gradient, pitch angle derivative, and speed derivative of the lowest order distribution function.
Birmingham, T. J.↗
Theory of flux anisotropies in a guiding center plasma
Assuming time stationarity of the one-particle distribution function f on the scale of the bounce motion of particles in a magnetic field, the paper expands the Vlasov equation through O(epsilon) in the adiabatic parameter epsilon, which is the ratio of particle gyroradius to scale length of the magnetic field. Since f is directly proportional to particle flux differential in kinetic energy and solid angle, f is in principle measurable in space experiments, and the present analysis is tailored to be explicitly applicable to space problems. It is shown that the usual expression for the electric field which produces plasma corotation in an axisymmetric system such as a dipole also holds for any nonaxisymmetric but rigidly rotating magnetic field pattern, provided the observed magnetic field is used in place of the dipole field. The analysis is applied to the electric field in a rigidly corotating magnetosphere.
Birmingham, T. J.↗
Kinetic theory of a two-dimensional magnetized plasma.
Several features of the equilibrium and nonequilibrium statistical mechanics of a two-dimensional plasma in a uniform dc magnetic field are investigated. The charges are assumed to interact only through electrostatic potentials. The problem is considered both with and without the guiding-center approximation. With the guiding-center approximation, an appropriate Liouville equation and BBGKY hierarchy predict no approach to thermal equilibrium for the spatially uniform case. For the spatially nonuniform situation, a guiding-center Vlasov equation is discussed and solved in special cases. For the nonequilibrium, nonguiding-center case, a Boltzmann equation, and a Fokker-Planck equation are derived in the appropriate limits. The latter is more tractable than the former, and can be shown to obey conservation laws and an H-theorem, but contains a divergent integral which must be cut off on physical grounds. Several unsolved problems are posed.
Vahala, G.↗
Nonguiding Center Motion and Substorm Effects in the Magnetotail
Thick and thin models of the middle magnetotail were developed using a consistent orbit tracing technique. It was found that currents carried near the equator by groups of ions with anisotropic distribution functions are not well approximated by the guiding center expressions. The guiding center equations fail primarily because the calculated pressure tensor is not magnetic field aligned. The pressure tensor becomes field aligned as one moves away from the equator, but here there is a small region in which the guiding center equations remain inadequate because the two perpendicular components of the pressure tensor are unequal. The significance of nonguiding center motion to substorm processes then was examined. One mechanism that may disrupt a thin cross-tail current sheet involves field changes that cause ions to begin following chaotic orbits. The lowest-altitude chaotic region, characterized by an adiabaticity parameter kappa approx. equal to 0.8, is especially important. The average cross-tail particle drift is slow, and we were unable to generate a thin current sheet using such ions. Therefore, any process that tends to create a thin current sheet in a region with kappa approaching 0.8 may cause the cross-tail current to get so low that it becomes insufficient to support the lobes. A different limit may be important in resonant orbit regions of a thin current sheet because particles reach a maximum cross-tail drift velocity. If the number of ions per unit length decreases as the tail is stretched, this part of the plasma sheet also may become unable to carry the cross-tail current needed to support the lobes. Thin sheets are needed for both resonant and chaotic orbit mechanisms because the distribution function must be highly structured. A description of current continuity is included to show how field aligned currents can evolve during the transition from a two-dimensional (2-D) to a 3-D configuration.
Kaufmann, Richard L.↗
Cross-tail current, field-aligned current, and B(y)
Orbits of individual charged particles were traced in a one-dimensional magnetic field model that included a uniform cross-tail component B(sub yo). The effects of B(sub yo) on the cross-tail current distribution j(sub y)(z), the average cross-tail drift velocity(nu(sub y)z), and the average pitch angle change(delta alpha) experienced during current sheet encounters were calculated. The addition of a B(sub yo) that exceeded several tenths of one nanotesla completely eliminated all resonance effects for odd-N orbits. An odd-N resonance involves ions that enter and exit the current sheet on the same side. Pitch angles of nearly all such ions changed substantially during a typical current sheet interaction, and there was no region of large cross-tail drift velocity in the presence of a modest B(sub yo). the addition of a very large B(sub yo) guide field in the direction that enhances the natural drift produces a large j(y) and small (Delta alpha) for ions with all energies. The addition of a modest B(sub yo) had less effect near even-N resonances. In this case, ions in a small energy range were found to undergo so little change in pitch angle that particles which originated in the ionosphere would pass through the current sheet and return to the conjugate ionosphere. Finally, the cross-tail drift of ions from regions dominated by stochastic orbits to regions dominated by either resonant or guiding center orbits was considered. The ion drift speed changed substantially during such transitions. The accompanying electrons obey the guiding center equations, so electron drift is more uniform. Any difference between gradients in the fluxes associated with electron and ion drifts requires the presence of a Birkeland current in order to maintain charge neutrality. This plasma sheet region therefore serves as a current generator. The analysis predicts that the resulting Birkeland current connects to the lowest altitude equatorial regions in which ions drift to or from a point at which stochastic orbits predominate. The proposed mechanism appears only in analyses that include non-guiding-center effects.
Kaufmann, Richard L.↗
A tale of two polarization paradoxes: The diamagnetic polarization paradox
An accurate calculation of the total polarization charge density in a plasma is essential for a self-consistent determination of the electric field. Yet, the “diamagnetic polarization paradox” refers to the fact that there is a paradoxical factor of 1/2 difference between the pressure-driven “diamagnetic polarization” density calculated using real space drift theory vs action-angle space guiding center and gyrokinetic theory that has not been explained before. Here, we show that both results can be made consistent with one another. Half of the diamagnetic polarization is due to the transformation from the guiding center density to the real space density. The other half is due to the fact that, within the drift kinetic ordering assumptions, the guiding center density should be expressed as the gyroaverage of the density in the limit of vanishing Larmor radius. A comprehensive review of polarization is presented to complete the derivation, and then we derive results that are required for the polarization calculations to agree. Expressions for the diamagnetic polarization density are given that are accurate to first order in amplitude and all orders in gyroradius within the gyrokinetic theory for a constant magnetic field. Applications to Maxwell–Boltzmann particle distribution functions (PDFs), including anisotropic temperature, are presented. Local invariants, like total energy and toroidal momentum, do not generate net polarization effects; the electric and thermodynamic polarizations must precisely cancel. In contrast, anisotropic dependence on the magnetic moment generates a net polarization proportional to the temperature anisotropy. Finally, when sources are present, the equilibrium PDF is approximately the ratio of two orbit averages.
adiabatic theory↗
Comparison of diffusion and particle drift descriptions of radial transport in the earth's inner magnetosphere
The paper compares two approaches for describing the radial motion of charged particles in the equatorial plane of the earth's magnetosphere: the radial diffusion and the guiding-center drift. For this purpose, a time-dependent observation-based model electrostatic field is computed for the August 1990 storm event, and this field is used to (1) calculate the guiding-center drift of a ring of monoenergetic test particles and (2) compute the specific radial diffusion coefficient and solve the relevant diffusion equation. Generally, only fair agreement was found between the guiding-center and the diffusion calculation approaches. The best agreement could be obtained for the case of a series of small storms rather than for one substantial storm.
Riley, Pete↗
Statistical mechanics of 'negative temperature' states
Consideration of the dynamics of a two-dimensional guiding center plasma, recently shown by Taylor and McNamara (1971) to be identical to the dynamics of the discrete vortex model of Onsager (1949). A semirigorous application of the methods of equilibrium statistical mechanics to the guiding center plasma (or equivalently, the line vortex system) is presented. An adaptation of the apparatus of the theory of probability is attempted, in the form given by Khinchin (1949) to obtain ensemble-average predictions for the states of the guiding center plasma. Interest focuses primarily on the regime in which the interaction energy is high enough to be above the Onsager 'negative temperature' threshold.
Montgomery, D.↗
The gravito-electrodynamics of charged dust in planetary magnetospheres
The orbital dynamics of small electrically charged dust grains within the corotating regions of planetary magnetospheres is considered. Equations are derived for the elliptical epicyclic motion of positively and negatively charged particles about the guiding center in an equilibrium circular orbit under the influence of small perturbations. Those orbits that are stable to the perturbations are found to have a ratio of the semiaxes of the epicycle between 1/2 and 1, depending on the specific charge, and a gyration frequency about the guiding center between the Kepler frequency and the grain gyrofrequency in a nonrotating frame. In the magnetospheres of Jupiter and Saturn, where the grains are expected to be negatively charged and move in a prograde sense, the guiding centers are predicted to have speeds intermediate to the Kepler speed and the corotation speed and thus may undergo a 1:1 magneto-gravitational resonance with a neighboring satellite. Results may be used in the interpretation of the waves in the F ring of Saturn in terms of the dust size distribution.
Mendis, D. A.↗
Scattering theory in noncanonical phase space: A Drift-Kinetic collision operator for weakly collisional plasmas
After developing a scattering theory for grazing collisions in general noncanonical phase spaces, we introduce a guiding center collision operator in five-dimensional phase space designed for plasma regimes characterized by long wavelengths (relative to the Larmor radius), low frequencies (relative to the cyclotron frequency), and weak collisionality (where repeated Coulomb collisions induce cumulatively small changes in particle magnetic moment). The collision operator is fully determined by the noncanonical Hamiltonian structure of guiding center dynamics and exhibits a metriplectic structure, ensuring the conservation of particle number, momentum, energy, and interior Casimir invariants. It also satisfies an H-theorem, allowing for deviations from an equilibrium Maxwellian distribution due to the nontrivial kernel of the noncanonical guiding center Poisson tensor, spanned by the magnetic moment. We propose that this collision operator and its underlying mathematical structure may offer valuable insight into the study of turbulence, transport, and self-organizing phenomena in both laboratory and astrophysical plasmas.
Hamiltonian mechanics↗
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 theta, the cosine of magnetic colatitude theta. 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 conversation of magnetic moment mu, 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 'nonadia- batic' pressure profiles on electrostatic interchanges is considered, and a boundary delineating stabilizing and destabilizing regions determined.
Huang, T. S.↗
Strongly magnetized classical plasma models
The class of plasma processes for which the so-called Vlasov approximation is inadequate is investigated. Results from the equilibrium statistical mechanics of two-dimensional plasmas are derived. These results are independent of the presence of an external dc magnetic field. The nonequilibrium statistical mechanics of the electrostatic guiding-center plasma, a two-dimensional plasma model, is discussed. This model is then generalized to three dimensions. The guiding-center model is relaxed to include finite Larmor radius effects for a two-dimensional plasma.
Montgomery, D. C.↗
Stormtime ring current and radiation belt ion transport: Simulations and interpretations
We use a dynamical guiding-center model to investigate the stormtime transport of ring current and radiation-belt ions. We trace the motion of representative ions' guiding centers in response to model substorm-associated impulses in the convection electric field for a range of ion energies. Our simple magnetospheric model allows us to compare our numerical results quantitatively with analytical descriptions of particle transport, (e.g., with the quasilinear theory of radial diffusion). We find that 10-145-keV ions gain access to L approximately 3, where they can form the stormtime ring current, mainly from outside the (trapping) region in which particles execute closed drift paths. Conversely, the transport of higher-energy ions (approximately greater than 145 keV at L approximately 3) turns out to resemble radial diffusion. The quasilinear diffusion coefficient calculated for our model storm does not vary smoothly with particle energy, since our impulses occur at specific (although randomly determined) times. Despite the spectral irregularity, quasilinear theory provides a surprisingly accurate description of the transport process for approximately greater than 145-keV ions, even for the case of an individual storm. For 4 different realizations of our model storm, the geometric mean discrepancies between diffusion coefficients D(sup sim, sub LL) obtained from the simulations and the quasilinear diffusion coefficient D(sup ql, sub LL) amount to factors of 2.3, 2.3, 1.5, and 3.0, respectively. We have found that these discrepancies between D(sup sim, sub LL) and D(sup ql, sub LL) can be reduced slightly by invoking drift-resonance broadening to smooth out the sharp minima and maxima in D(sup ql, sub LL). The mean of the remaining discrepancies between D(sup sim, sub LL) and D(sup ql, sub LL) for the 4 different storms then amount to factors of 1.9, 2.1, 1.5, and 2.7, respectively. We find even better agreement when we reduce the impulse amplitudes systematically in a given model storm (e.g., reduction of all the impulse amplitudes by half reduces the discrepancy factor by at least its square root) and also when we average our results over an ensemble of 20 model storms (agreement is within a factor of 1.2 without impulse-amplitude reduction). We use our simulation results also to map phase-space densities f in accordance with Liouville's theorem. We find that the stormtime transport of approximately greater than 145-keV ions produces little change in f-bar the drift-averaged phase-space density on any drift shell of interest. However, the stormtime transport produces a major enhancement from the pre-storm phase-space density at energies approximately 30-145 keV, which are representative of the stormtime ring current.
Lyons, Larry R.↗
Measurement of ion temperature and toroidal flow during magnetic reconnection with a large guide field
Here, ion temperature and toroidal flow along the guide field direction are measured using a new ion tomographic diagnostic on the Magnetic Reconnection eXperiment (MRX) during magnetic reconnection with a guide field strength of about 1.4 and 2.1 times the strength of the reconnecting component. Strong toroidal flows, beyond what has been measured in anti-parallel and lower guide field conditions on MRX, are observed. Sustained ion heating with no discernible structure within the measurement region is also observed. Probe measurements including Langmuir and Mach probe measurements are made to support the tomographic inversion of line-integrated measurements, as well as to provide local measurements of plasma parameters. Measurements of toroidal velocity and ion temperature are supported with time series data. Energy flow into and out of the X-line region is estimated using a guiding center framework and presented in the Appendix of this manuscript, suggesting an outsized role played by parallel electric field in energizing ions. The guiding center approximation is not well satisfied in the region of interest; however, the estimates provide a springboard for future, further experimentation.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗