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Topics addressed include: laboratory double layers; ion-acoustic double layers; pumping potential wells; ion phase-space vortices; weak double layers; electric fields and double layers in plasmas; auroral double layers; double layer formation in a plasma; beamed emission from gamma-ray burst source; double layers and extragalactic jets; and electric potential between plasma sheet clouds.
The evolution processes of double layers were studied in a series of laboratory experiments. It was found that the existence of virtual cathode-type potential wells at the electron injection boundary was the dominant triggering mechanism. The rapid growth of the potential wells led to collisionless ion trapping and the establishment of the necessary trapped ion population. For double layers with small potential drops, collisionless ion trapping actually induced ion-ion streaming instabilities and the formation of ion phase-space vortices. In this regime, the system often exhibited relaxation-type oscillations which correspond to the disruption and the recovery of the double layers.
Transversely localized double layers evolve randomly in turbulent regions of strongly magnetized plasma carrying current along the magnetic field. Results from numerical simulations and spacecraft observations in the auroral plasma indicate that the parallel electric field in such regions is microscopically intermittent or stochastic. The implications of stochastic double layer fields on electron acceleration will be discussed in terms of a statistical process involving ensemble averages over test particle motion. A Fokker-Planck equation can be derived for the electron phase space density, which depends on the mean and rms amplitudes of the double layers, the mean double layer density, and the initial electron velocity distribution. It is shown that the resulting electron acceleration is very sensitive to the ratio of the initial electron energy to the rms double layer amplitude. When this ratio is large, the acceleration process differs little from that expected in a dc electric field. When it is small, stochastic heating competes with directed acceleration. Evidence for both cases can be found in the auroral ionosphere in association with so-called inverted-V precipitation and collimated edge precipitation.
Previous work on the evolution of weak double layers in a hydrogen plasma was extended to include H(+) and O(+) with relative drift. The relative drift between hydrogen and oxygen ions due to a quasi-static parallel electric field gives rise to a strong linear fluid instability which dominates the ion-acoustic mode at the bottom of the auroral acceleration region. This ion-ion instability can modify ion distributions at lower altitudes and the subsequent nonlinear evolution of weak double layers at higher altitudes in the ion-acoustic regime. Ion hole formation can occur for smaller relative electron-ion drifts than seen in previous simulations, due to the hydrogen-oxygen two-stream instability. This results in local modification of the ion distributions in phase space, and a partial filling of the valley between the hydrogen and oxygen peaks, which would be expected at higher altitudes on auroral field lines. The observed velocity diffusion does not necessarily preclude ion hole and double layer formation in hydrogen in the ion-acoustic regime. These simulation results are consistent with the experimentally measured persistence of separate hydrogen and oxygen peaks, and the observation of weak double layers above an altitude of 3000 km on auroral field lines.
The rotational dynamics of irregularly shaped satellites and the origin of Kirkwood Gaps are discussed. The chaotic tumbling of Hyperion and the anomalously low eccentricity of Deimos are examined. The Digital Orrery is used to explore the phase space of the ellipic restricted three body problem near the principal commensurabilities (2/1, 5/2, 3/1, and 3/2). The results for the 3/1 commensurability are in close agreement with those found earlier with the algebraic mapping method. Large chaotic zones are associated with the 3/1, 2/1 and 5/2 resonances, where there are gaps in the distribution of asteroids. The region near the 3/2 resonance, where the Hilda group of asteroids is located, is largely devoid of chaotic behavior. Thus, there is a qualitative agreement between the character of the motion and the distribution of asteroids.
A Hamiltonian formulation of general relativity based on certain spinorial variables is introduced. These variables simplify the constraints of general relativity considerably and enable one to imbed the constraint surface in the phase space of Einstein's theory into that of Yang-Mills theory. The imbedding suggests new ways of attacking a number of problems in both classical and quantum gravity. Some illustrative applications are discussed.
Consideration is given to regimes of low-frequency variability in large-scale atmospheric dynamics. The model utilized is the fully-nonlinear, equivalent-baratropic vorticity equation on the sphere, with simplified forcing, dissipation and topography. It is found that certain limited regions in the system's phase space are visited repeatedly and for extended periods by model solutions. Flow patterns coupled with these regions correspond to synoptically-defined zonal and blocked Northern Hemisphere midlatitude flows. It is shown that the system's macrodynamics can be depicted by two or more planetary flow regimes, the expected residence time in each regime, and the transition properties from one regime to another. These model-derived ideas are also applied to a time series of atmospheric data from the Southern Hemisphere.
The origin of orbital resonances is shown in the demonstration of the evolution of a pair of planetary satellites through a commensurability of the mean motions by a sequence of diagrams of constant energy curves in a two-dimensional phase space; the closed curve corresponding to the motion in each successive diagram is identified by its adiabatically conserved area. It is found that two-body resonances serve as a basis in the solution of the problem of the origin and evolution of the three-body Laplace resonance among the Galilean satellites of Jupiter. The unusual rotation state of Saturn's satellite Hyperion which is expected to tumble chaotically for an indefinite amount of time is discussed.
An experiment is conducted on the transition from quasi-periodic to weakly turbulent flow of a fluid contained between concentric cylinders with the inner cylinder rotating and the outer cylinder at rest. Power spectra, phase-space portraits, and circle maps obtained from velocity time-series data indicate that the nonperiodic behavior observed is deterministic, that is, it is described by strange attractors. Various problems that arise in computing the dimension of strange attractors constructed from experimental data are discussed and it is shown that these problems impose severe requirements on the quantity and accuracy of data necessary for determining dimensions greater than about 5. In the present experiment the attractor dimension increases from 2 at the onset of turbulence to about 4 at a Reynolds number 50-percent above the onset of turbulence.
ISEE-1 observations of magnetospheric H(+) and O(+) ions at 0.01-17.4 keV/e during the recovery phase of the magnetic storm of December 11, 1977 are reported. The data are presented graphically and analyzed in detail. It is found that the phase-space density of H(+) in the magnetotail remained about the same during the recovery, while that of O(+) decreased steadily, and the loss of low-energy O(+) from the outer ring current is attributed to inward convection of the decreasing density in the tail.
Complete sets of orthonormal basis functions constructed according to a generalization of the quantum-mechanical WKB approximation can be used to generate a nearly-diagonal matrix representation of the radial-transport operator for ring-current ions in the presence of radial diffusion and charge exchange. The resulting eigenfunctions (constructed by weighting the basis functions in proportion to the respective components of the eigenvectors of the matrix representation) and eigenvalues provide a spatial and temporal description of the evolving phase-space density during and following a magnetospheric disturbance (e.g., a magnetic storm). A linear superposition of the basis functions can also be used to eliminate any discrepancy between the steady-state solution of the transport equation and the appropriate WKB approximation of this steady-state solution.
Persistent anomalies with recurrent spatial patterns play an important role in the atmosphere's low-frequency variability. A connection between statistical and dynamical methods of description and prediction of persistent anomalies is established by computing and analyzing the empirical orthogonal functions (EOFs) in a simple deterministic model, on the one hand, and in Southern Hemisphere geopotential heights, on the other. The dynamical model is governed by the fully nonlinear, equivalent-barotropic vorticity equation on the sphere, with simplified forcing, dissipation and topography. The Southern Hemisphere data consist in gridded daily maps of 500 mb heights from June 1972 to July 1983. Two types of persistent anomalies appear in this time series, both having a strong wavenumber-three component; they differ by the value of the constant phase of this wave and by the strength of the wavenumber-one component. The first two EOFs bear a striking resemblance to these two patterns. It is concluded that the dynamical interpretation of EOFs is their pointing from the time mean to the most populated regions of the system's phase space. Pursuing this interpretation, a Markov-chain formulation of transitions from one persistent anomaly regime to another is introduced, and the implications for long-range forecasting are discussed.
A modified cluster analysis method was developed to identify spatial patterns of planetary flow regimes, and to study transitions between them. This method was applied first to a simple deterministic model and second to Northern Hemisphere (NH) 500 mb data. The dynamical model is governed by the fully-nonlinear, equivalent-barotropic vorticity equation on the sphere. Clusters of point in the model's phase space are associated with either a few persistent or with many transient events. Two stationary clusters have patterns similar to unstable stationary model solutions, zonal, or blocked. Transient clusters of wave trains serve as way stations between the stationary ones. For the NH data, cluster analysis was performed in the subspace of the first seven empirical orthogonal functions (EOFs). Stationary clusters are found in the low-frequency band of more than 10 days, and transient clusters in the bandpass frequency window between 2.5 and 6 days. In the low-frequency band three pairs of clusters determine, respectively, EOFs 1, 2, and 3. They exhibit well-known regional features, such as blocking, the Pacific/North American (PNA) pattern and wave trains. Both model and low-pass data show strong bimodality. Clusters in the bandpass window show wave-train patterns in the two jet exit regions. They are related, as in the model, to transitions between stationary clusters.
Attention is given to solar system cases in which chaotic solutions of Newton's equations are important, as in chaotic rotation and orbital evolution. Hyperion is noted to be tumbling chaotically; chaotic orbital evolution is suggested to be of fundamental importance to an accounting for the Kirkwood gaps in asteroid distribution and for the phase space boundary of the chaotic zone at the 3/1 mean-motion commensurability with Jupiter. In addition, chaotic trajectories in the 2/1 chaotic zone reach very high eccentricities by a route that carries them to high inclinations temporarily.
The applicability of eigenfunction methods to the theory of magnetospheric radial diffusion is investigated analytically, with a focus on the operator (Lambda) for radiation-belt and ring-current particles. It is shown that the eigenfunctions of Lambda can be expressed in terms of a linear combination of ordinary Bessel functions of the first and second kinds, and that the solution for the drift-averaged phase-space density can be expressed as the sum of a quasi-static solution and a time-dependent superposition of the eigenfunctions of Lambda. Results for illustrative and practical sample problems are presented in extensive tables and graphs and discussed in detail.
The new kinetic ion-cyclotronlike (IC-like) instability driven by a localized electric field perpendicular to the ambient magnetic field is investigated using particle simulation. The existence of the new mode is demonstrated and the mode is distinguished from the well-known Kelvin-Helmholtz branch and the kinetic modification of this branch. The issue of the initial loading of particles in phase space in the simulation is clarified. The simulation results show that the growing ion waves are associated with the small vortices in the linear state, which evolve into a nonlinear stage dominated by large vortices with lower frequencies.
The generalized first adiabatic invariant, an extension of the magnetic moment for regions of large field gradients, has been applied to the study of particles in the magnetotail current sheet. Boundaries in phase space are obtained which form a generalized loss cone and separate particles drifting into and out of the layer from particles trapped within the layer, and the boundaries can be employed in the moment integrals for densities and currents when the drifting particles differ in temperature. It is shown that the parallel pressure of the drifting particles must exceed the transverse pressure for self-consistent solutions to exist. Fully self-consistent solutions are obtained using bi-Maxwellian ion and Maxwellian electron distributions.
A quadratic nonlinear generalization of the linear Rotta model for the slow pressure-strain correlation of turbulence is developed. The model is shown to satisfy realizability and to give rise to no stable non-trivial equilibrium solutions for the anisotropy tensor in the case of vanishing mean velocity gradients. The absence of stable non-trivial equilibrium solutions is a necessary condition to ensure that the model predicts a return to isotropy for all relaxational turbulent flows. Both the phase space dynamics and the temporal behavior of the model are examined and compared against experimental data for the return to isotropy problem. It is demonstrated that the quadratic model successfully captures the experimental trends which clearly exhibit nonlinear behavior. Direct comparisons are also made with the predictions of the Rotta model and the Lumley model.