Three-dimensional Phase Space Research Articles

Vlasov-Maxwell simulations of high-frequency longitudinal waves in a magnetized plasma.

The plasma response to the injection of a propagating purely electrostatic wave of finite amplitude is investigated by means of a kinetic code which solves the Vlasov equations for electrons and ions in the three-dimensional (one spatial and two in velocity, 1D2V) phase space, self-consistently coupled to the Maxwell equations. The plasma is uniformly magnetized, and the wave frequency close to the cold upper-hybrid resonance omega(0)=sqrt[omega(2)(pe)+omega(2)(ce)] is considered. Coherent structures are formed in the phase space that would be completely missed by a hydrodynamic analysis. In particular, in the early stage of the interaction, the initially unperturbed equilibrium electron distribution is strongly affected as a whole by the pump, taking a ringlike shape in the velocity plane transverse to the magnetic field. Then, a sort of instability occurs, leading to the broadening and flattening of the electron distribution.

A Cyclone Phase Space Derived from Thermal Wind and Thermal Asymmetry

Abstract An objectively defined three-dimensional cyclone phase space is proposed and explored. Cyclone phase is described using the parameters of storm-motion-relative thickness asymmetry (symmetric/nonfrontal versus asymmetric/frontal) and vertical derivative of horizontal height gradient (cold- versus warm-core structure via the thermal wind relationship). A cyclone's life cycle can be analyzed within this phase space, providing substantial insight into the cyclone structural evolution. An objective classification of cyclone phase is possible, unifying the basic structural description of tropical, extratropical, and hybrid cyclones into a continuum. Stereotypical symmetric warm-core (tropical cyclone) and asymmetric cold-core (extratropical cyclone) life cycles are illustrated using 1° Navy Operational Global Atmospheric Prediction System (NOGAPS) operational analyses and 2.5° NCEP–NCAR reanalyses. The transitions between cyclone phases are clearly illustrated within the phase space, including extratro...

Three-dimensional visualization of electron acceleration in a magnetized plasma

We examine wave-particle interactions in a magnetized plasma. We present snapshots of an animation of the three-dimensional electron phase space distribution produced by an electrostatic wave propagating across a magnetic field. The distribution function has been evolved by a particle in cell simulation. The electron phase space has been visualized by distributing the simulation electrons over an array representing phase space density and by volume rendering this array. The results are, due to the choice of initial plasma and wave parameters, of relevance for electron acceleration at astrophysical shocks.

Despiking Acoustic Doppler Velocimeter Data

A new method for detecting spikes in acoustic Doppler velocimeter data sequences is suggested. The method combines three concepts: (1) that differentiation enhances the high frequency portion of a signal, (2) that the expected maximum of a random series is given by the Universal threshold, and (3) that good data cluster in a dense cloud in phase space or Poincare maps. These concepts are used to construct an ellipsoid in three-dimensional phase space, then points lying outside the ellipsoid are designated as spikes. The new method is shown to have superior performance to various other methods and it has the added advantage that it requires no parameters. Several methods for replacing sequences of spurious data are presented. A polynomial fitted to good data on either side of the spike event, then interpolated across the event, is preferred by the authors.

Dynamics of the solar granulation

We investigate the attractor underlying the granular phenomenon by applying nonlinear methods to series of spectrograms from 1994 and 1999. In the three-dimensional phase space spanned by intensity, Doppler velocity, and turbulence (line broadening), the granulation attractor does not fill the entire phase space, as expected from the high Reynolds and Rayleigh numbers of the photospheric plasma, but rather shows a highly structured form. This could be due to the correlations between intensity, turbulence, and velocity, which represent also the Reynolds stress. To obtain insight into the dimensionality of the attractor, we use the time lag method, a nonlinear method that enables us to get information about the underlying attractor of a dynamical system (granulation) from the measurement of one physical quantity only. By applying this method to the observed Doppler velocities, we show that the granulation attractor can be described by three independent variables. The dimension of the granulation attractor seems to be independent of the appearance of big granules and shear flow. Furthermore, the power analysis of the Doppler velocity shows power down to the spatial resolution of the instrument (0.3 arcsec). In order to decide whether the power at the smallest scales is real or noise, we use again the time lag method in combination with either a high pass digital or wavelet filter, which filters out the large wave numbers. It appears that the power at the smallest scales represents a real signal.

The many sources effect on the genuine multihadron correlations

Here we report on a study aimed to explore the dependence of the genuine multiparticle correlations on the number of sources while the influence of other possible factors affecting the multihadron production is avoided. The analysis utilised the normalised cumulants, calculated in three-dimensional phase space, of the reaction e +e −→Z 0 → hadrons using a large Monte Carlo event sample. The multi-sources reactions were simulated by overlaying a few independent single e +e − annihilation events. It was found that as the number of sources S increases, the cumulants do not change significantly their structure, but those of an order q > 2 (i.e. more than 2 pions) decrease fast in their magnitude. This reduction can be understood in terms of combinatorial considerations of source mixing which dilutes the correlations by a factor of about 1/ S q−1 which can also serve as a method to estimated the number of sources. This expected suppression is well reproduced by recent cumulant measurements in hadron and nucleus induced reactions both in one (rapidity) and two (rapidity vs. azimuthal angle) dimensions. The diminishing genuine correlations effect should also appear in other dynamical correlations like the Bose-Einstein in e +e −→W +W − → hadrons and in nucleus-nucleus reactions.

The three-dimensional transfer function and phase space mappings

Abstract For paraxial wave-fields in two dimensions, it is known that the defocused OTF is given by a section through the ambiguity function. This is no longer true for high aperture fields. It is shown that there is a simple relationship, even for high aperture fields, between the two-dimensional (one transverse and one longitudinal) generalized OTF and the spectral correlation function. The connection with the Wigner distribution function is also discussed. The generalization to three-dimensional wavefields is also considered.

Proper-Time Relativistic Dynamics on Hyperboloid

The dynamics of a point-like relativistic particle with respect of the proper time is formulated on the hyperboloid p20−p2=M2c2. The Hamilton-Jacobi equations on the hyperboloid are derived for the particles with mass (M2=m2, m>0), for the particles with zero-mass (M=0, m>0), and for the neutrino. It is shown, in a certain factorization of the momentum, the model can be identified with Nambu's three-dimensional phase space formalism. A first quantized version of the model is formulated according to a canonical scheme of quantization (Schrodinger quantization scheme).

Analysis of flow instabilities at the natural circulation loop DANTON with regard to non-linear effects

The natural circulation loop DANTON at the Dresden University of Technology was designed to investigate the thermohydraulic properties of integrated reactor concepts with a natural circulation driven primary loop. It is not possible to reach a stable two-phase flow in the loop without passing flow instabilities. At an equilibrium of heating and cooling power the flow oscillations can exist at nearly constant frequencies and amplitudes. The oscillating mass flow signal had been investigated by various methods: (a) autocorrelation function, (b) Fast Fourier transformation, (c) estimation of a temporal Liapunov-exponent and (d) reconstruction of the system attractor in a three-dimensional phase space. The selected time-series express a non-linear behaviour, however, they are not chaotic. In comparison to the usual methods the applied analysing methods yield additional information about system frequencies, sensitiveness to disturbances and properties due to non-linear and chaotic behaviour in a natural circulation loop.

Phase Space Representation and Characteristics of El Niño–La Niña

The variability of El Nino-La Nina events was analyzed in a low-dimensional phase space, a concept derived from dynamic system theory. The space-time extended EOFs derived from the observed monthly mean SST field over tropical Pacific were used as the basis of the phase space that describes the time evolution of ENSO signals. It was shown that the essential features of the ENSO variability, such as the irregular oscillation, the phase locking to the annual cycle, and the interdecadal changes in its propagation and onset, can be effectively represented by a three-dimensional phase space. The typical El Nino-La Nina life cycle is four years with its mature phases in boreal winter. The intensity of the ENSO signal within one life cycle is closely linked to the frequency of its occurrence (onset). The interdecadal variability of the ENSO signals is characterized by both the intensity and the frequency of occurrence, displaying an irregularity with the gross feature comparable to the regime behavior and intermittency of some low-dimensional chaotic systems.

The effect of many sources on the genuine multiparticle correlations

We report on a study aimed to explore the dependence of the genuine multiparticle correlations on the number of sources when the influence of other possible factors during multi-hadron production is avoided. The analysis utilised the normalised cumulants calculated in three-dimensional phase space of the reaction e +e −→Z 0→hadrons using a large Monte Carlo sample. The multi-sources reactions were simulated by overlaying a few independent single e +e − annihilation events. It was found that as the number of sources S increases, the cumulants do not change significantly their structure, but those of an order q>2 decrease fast in their magnitude. This reduction in the one-dimensional rapidity cumulants can be understood in terms of combinatorial considerations of source mixing which dilutes the correlations by a factor of about 1/ S q−1 . The diminishing of the genuine correlations is consistent with recent cumulant measurements in hadron and nucleus induced reactions and should also be relevant to other dynamical correlations like the Bose–Einstein one, in e +e −→W +W −→hadrons and in nucleus–nucleus reactions.

Generalized Newtonian Equations of Motion

We generalize Newtonian equations of motion to equations in three (and higher) dimensional phase space, which we denominate “Elliptic Equations of Motion.” The new system of equations contains the scale parameter of energy. We find that the total energy of the conservative system described by the elliptic equations is the product of the constants of motion related to the family of Hamilton–Nambu functions. For the case of a stationary potential we establish the equivalence between elliptic equations in n-dimensional phase space with the potential V(x) and Newtonian equations with the potential given by the polynomial of V(x). We show that the equations of motion for a relativistic particle in a stationary potential field can be reformulated via elliptic equations of motion in three-dimensional phase space.

BREAKDOWN OF LIBRATIONAL INVARIANT SURFACES

A numerical investigation of the stability of invariant librational tori is presented. The method has been developed for a model describing the spin-orbit coupling in Celestial Mechanics. Periodic orbits approaching the librational torus are computed by means of Newton's method. According to Greene's criterion, their stability is strictly related to the survival of invariant tori. We consider librational tori around the main spin-orbit resonances (1:1, 3:2). Their existence provides the stability of the resonances, due to the confinement properties in the three-dimensional phase space associated to our model. The results are consistent with the actual observations of the eccentricity and of the oblateness parameter. A different behavior of the Moon and Mercury around the main resonances is evidenced, providing interesting suggestions about the different probabilities of capture in a resonance.

Soliton switching in directional couplers

The mechanism of pulse switching is investigated analytically and numerically for a family of initial conditions with a solitonlike pulse in one channel and no signal on the other channel of the coupler. This investigation is performed directly in the coupled nonlinear Schroedinger equations that describe the directional coupler. This mechanism appears to be based on the simultaneous existence of the asymmetric and the symmetric soliton states: Switching then can take place for values of coupling constant and total energy for which the asymmetric soliton is stable and the symmetric soliton is unstable. The crucial features of the phase portrait in the infinite-dimensional space are described. In particular, near the switching threshold, a switching solution passes near the unstable symmetric soliton on its way to a new state with the energies in the channels reversed. On the basis of this mechanism a necessary condition for switching is found, which is confirmed numerically for some families of pulse shaped initial conditions. In the case of soliton switching with the energy as the switching parameter, this condition appears to give an accurate prediction of the actual switching threshold. The infinite-dimensional phase portrait has a simple structure, so that its features for a large part can be captured in a three-dimensional phase space.

北大西洋域における夏季の天候レジーム

Summertime weather regimes are examined over the North Atlantic sector based on a 44-year series of daily 700 hPa height anomalies for the period 1951-1994 in order to deepen our understandings of weather regime dynamics in the real atmosphere. Weather regimes are identified by examining the probability density function (PDF) in three-dimensional phase space, which is spanned by the three leading empirical orthogonal functions (EOFs) of the anomaly fields. Five weather regimes are obtained. Each weather regime corresponds to a distinct region with statistically significant large PDF values compared with multivariate Gaussianity. These include two extreme phases of the North Atlantic Oscillation (NAO), and flow patterns with intensified jet streams over the North Atlantic. The characteristics of weather regimes in summer are described in comparison with those in winter. The similarity in the circulation pattern of weather regimes in both seasons suggests the robustness of summertime weather regimes against the seasonal cycle. We discuss the persistence and transition of summertime weather regimes in connection with the bifurcation property of weather regimes elucidated in simple atmospheric models. The interannual variation of the occurrence frequency of weather regimes is also discussed.

Magnetic-field-aligned electric fields associated with Debye-scale plasma structures

Quasi-static, magnetic-field-aligned (parallel) potentials have been considered the primary source of charged particle acceleration in the aurora where precipitating electrons create a visible display. This finding has been controversial since, at one time, it was widely believed that parallel potentials could not be supported by a collisionless plasma. We present observations from the fast auroral snapshot (FAST) satellite which strongly support this acceleration mechanism and, moreover, show evidence of a second plasma regime region which supports quasi-static parallel potentials. The uncovering of parallel potentials in two plasma regimes suggests that they may be fundamental in astrophysical plasmas, supplementing the classical mechanisms of Fermi and betatron acceleration. We summarize the observations that demonstrate this acceleration mechanism. We also summarize evidence of Debye-scale plasma structures which are associated with these parallel potentials. These small-scale structures appear to be three-dimensional electron phase space holes, a new type of plasma structure.

A zero-dimensional climate-vegetation model containing global carbon and hydrological cycle

A model of a hypothetical zero-dimensional planet containing a global carbon cycle, which describes the fundamental interaction between climate and biosphere, is used as a basis to formulate a new model incorporating a hydrological cycle. Using the conservation law for carbon and water in the system the number of differential equations is thereby reduced by two from five to three. The number and positions of the stable equilibria in the three-dimensional phase space is determined using numerical methods. This numbers is related to the value of the maximum productivity P max., the total amount of carbon A and water B in the system as bifurcation parameters of the system.

Bifurcating bright and dark solitary waves for the perturbed cubic-quintic nonlinear Schrödinger equation

The existence of bright and dark multi-bump solitary waves for Ginzburg–Landau type perturbations of the cubic-quintic Schrodinger equation is considered. The waves in question are not perturbations of known analytic solitary waves, but instead arise as a bifurcation from a heteroclinic cycle in a three-dimensional ODE phase space. Using geometric singular perturbation techniques, regions in parameter space for which 1-bump bright and dark solitary waves will bifurcate are identified. The existence of N-bump dark solitary waves (N ≧ 1) is shown via an application of the Exchange Lemma with Exponentially Small Error. N-bump bright solitary waves are shown to exist as a consequence of the work of Kapitula and Maier-Paape.

Quasi-two-dimensional waves in three-dimensional magnetic confinement systems

A reduced, scalar magnetohydrodynamic (MHD) description for linear perturbations of arbitrary aspect-ratio, fully three-dimensional plasma configurations is derived for perturbations with short scale-length perpendicular to the magnetic field. The application of the scalar wave equation to ballooning modes is presented. It is shown that coexistence of toroidally extended and toroidally localized unstable ideal hydromagnetic ballooning modes occurs in an interchange–unstable equilibrium case modelling the Large Helical Device (LHD) with a broad pressure profile. The toroidally extended modes can be understood on the basis of a ripple-averaged ballooning equation, whereas the toroidally localized branch corresponds to modes localized along a flux tube. A new three-dimensional ray phase space is introduced in which the periodicity properties are simplified.

New Results for an Asymmetric Rigid Body with Constant Body-Fixed Torques

The rotational motion of an asymmetric rigid body under the ine uence of constant body-e xed torque is investigated. A set of nondimensional equations of motion is introduced for the stability analysis of equilibrium points. In particular, three-dimensional phase space trajectories are developed for two different cases: 1 ) constant torque along the major or minor axis and 2 ) constant torque along the intermediate axis. For both cases, some new analytical as well as simulation results are presented in terms of equilibrium manifolds, separatrix surfaces, periodic or nonperiodic solutions, and extrema of the periodic solutions.