Ion Phase Space Vortices Research Articles

The Impact of Turbulence on the Ionosphere and Magnetosphere

An important property associated with turbulence in plasmas and fluids is anomalous transport. Plasma, being a good conductor, can in addition be affected by turbulence causing an anomalous resistivity that can significantly exceed its classical counterpart. While turbulent transport may be adequately described in configuration space, some aspects of the anomalous resistivity are best accounted for in phase space. Kinetic phenomena like electron and ion phase space vortices can thus act as obstacles for the free flow of slow charged particles. Plasma instabilities and large amplitude plasma waves are candidates for contributions to the anomalous resistivity by generating such structures. Langmuir waves can be relevant, but also others, such as upper- as well as lower-hybrid waves in magnetized plasmas. Often these anomalous resistivity effects can be small, but due to the large spatial and temporal scales involved in space plasmas, planetary ionosphere and magnetosphere in particular, even such moderate effects can be important. This mini-review is discussing elements of the description of plasma turbulence with particular attention to wave phenomena that contribute to anomalous resistivity and diffusion. Turbulence effects can have relevance for space weather phenomena as well, where ground based and airborne activities relying on for instance Global Positioning and Global Navigation Satellite Systems are influenced by plasma conditions in geospace.

Shocks and phase space vortices driven by a density jump between two clouds of electrons and protons

We study with 1D PIC simulations the expansion of a dense plasma into a dilute one for density ratios 2.5 ≤ α ≤ 20. Both are unmagnetized and consist of electrons and protons. Shocks form in all cases. We determine how α affects the speed of the shock, that of the trailing velocity plateau and the proton beam instabilities in its upstream region. The speed of the velocity plateau relative to the upstream plasma increases significantly with α. Faster shocks reflect more upstream protons and fewer protons make it downstream, which slows down the shock in the downstream frame. This slow-down reduces noticably the increase with α of the shock speed in the upstream frame. All simulations demonstrate that an ion acoustic instability develops between the shock-reflected proton beam and the ambient protons. We solve the linear dispersion relation for ion acoustic waves that have wave vectors which are parallel to the beam velocity vector. Upstream conditions, for which their growth rate is largest, lead to the most unstable upstream plasmas also in the simulation. Even though linear theory predicts the growth of sine waves, which reach a small amplitude in the simulations, solitary waves become the dominant ones upstream of the shock. They enforce the formation of new shocks and ion phase space vortices. We discuss the relevance of our findings to laser-plasma experiments.

Evolution of ion–ion acoustic instability in multi-ion plasma sheaths

The generation of ion-acoustic solitary waves is investigated in a nonuniform multicomponent collisional plasma sheath containing cold ions and Boltzmann electrons to probe the formation and physics of modulation of nonlinear ion-acoustic waves. The new model for the plasma is adapted to include the effects of ion production-loss and momentum loss terms due to ion-neutral collisions, and implementation related to space and laboratory plasma applications are discussed. The discrete modes, the instability conditions and the growth rate of the streaming instability with the effect of the present plasma parameters are calculated based on the approximate but yet precise complex dispersion relation. The marginal stability curve, characterized by mode bifurcation, cutoff, and complex fold point, indicates a growth rate of a few percents of effective plasma frequency. The damping of ion-acoustic is affected by heavy neutrals, and its maximum rate found near the ion-neutral collision frequency. The variable-coefficient Korteweg-de Vries equation is derived via reductive perturbation method to govern the dynamics of small- as well as large-amplitude solitons. It is found that the propagating nonlinear coherent structures through the created ion phase-space vortices lead to ion trapping and acceleration, and the modulation of ion-acoustic instabilities in the turbulent region. The effect of ion streaming motion on the driven solitons and modulation instability for the variable-coefficient Korteweg-de Vries equation is numerically investigated in detail. The theoretical results can be applied to the observation of electrostatic waves in space plasmas, in industrial pair-ion plasmas as well as in laboratory dusty plasmas.

Kinetic Alfvén waves and electron physics. II. Oblique slow shocks

One-dimensional (1D) particle-in-cell (PIC; kinetic ions and electrons) and hybrid (kinetic ions; adiabatic and massless fluid electrons) simulations of highly oblique slow shocks (θBn=84° and β=0.1) [Yin et al., J. Geophys. Res., 110, A09217 (2005)] have shown that the dissipation from the ions is too weak to form a shock and that kinetic electron physics is required. The PIC simulations also showed that the downstream electron temperature becomes anisotropic (Te‖>Te⊥), as observed in slow shocks in space. The electron anisotropy results, in part, from the electron acceleration/heating by parallel electric fields of obliquely propagating kinetic Alfvén waves (KAWs) excited by ion-ion streaming, which cannot be modeled accurately in hybrid simulations. In the shock ramp, spiky structures occur in density and electron parallel temperature, where the ion parallel temperature decreases due to the reduction of the ion backstreaming speed. In this paper, KAW and electron physics in oblique slow shocks are further examined under lower electron beta conditions. It is found that as the electron beta is reduced, the resonant interaction between electrons and the wave parallel electric fields shifts to the tail of the electron velocity distribution, providing more efficient parallel heating. As a consequence, for βe=0.02, the electron physics is shown to influence the formation of a θBn=75° shock. Electron effects are further enhanced at a more oblique shock angle (θBn=84°) when both the growth rate and the range of unstable modes on the KAW branch increase. Small-scale electron and ion phase-space vortices in the shock ramp formed by electron-KAW interactions and the reduction of the ion backstreaming speed, respectively, are observed in the simulations and confirmed in homogeneous geometries in one and two spatial dimensions in the accompanying paper [Yin et al., Phys. Plasmas 14, 062104 (2007)]. Results from this study conclude that the inability of the hybrid method to model the Landau resonance of KAW with electrons and the resulting time-evolving electron parallel heating lead to differences between the PIC and hybrid simulations.

Streaming instabilities driven by mildly relativistic proton beams in plasmas

The interaction between relativistic ion beams, electrostatic waves and particles in plasmas is studied numerically. This beam-particle interaction is important for the acceleration of electrons above the threshold, where they can perform Fermi acceleration across shocks perpendicular to the magnetic field direction. The electrons form Bernstein–Greene–Kruskal (BGK) modes, which are unstable due to a sideband instability. This process heats the electrons to relativistic temperatures and the electrons finally attain a flat-top momentum distribution with exponentially decreasing tails of the momentum distribution function. A scaling law, depending on the ion beam speed, is found for the electron distribution function. Furthermore, in the fully nonlinear phase, ion phase space vortices, or proton BGK modes are formed in the beam, which continue to interact with the electrons after the electron BGK modes have collapsed.

Phase space vortices in collisionless plasmas

Abstract. Results on the formation and propagation of electron phase space vortices from laboratory experiments are summarized. The electron phase space vortices were excited in a strongly magnetized Q-machine plasma by applying a pulse to a segment of a waveguide surrounding the plasma. Depending on the temporal variation of the applied pulse, one or more phase space vortices can be excited, and their interaction can be followed in space and time. We were able to demonstrate, for instance, an irreversible coalescence of two such vortices. These results are extended by numerical simulations, showing how electron phase space vortices can also be formed by beam instabilities. Furthermore, a study of ion phase space vortices is performed by numerical simulations. Both codes allow for an externally applied magnetic field in three spatial dimensions. Ion phase space vortices are formed by the nonlinear saturation of the ion-ion two-stream instability, excited by injecting an ion beam at the plasma boundary. By following the evolution of the ion distribution of the velocity perpendicular to the direction of propagation of the injected ion beam, we find a significant ion heating in the direction perpendicular to the magnetic field associated with the ion phase space vortices being formed. The results are relevant, for instance, for the interpretation of observations by instrumented spacecraft in the Earth's ionosphere and magnetosphere.

Guiding center theory for ion holes in magnetized plasmas.

A drift-kinetic theory for ion phase-space vortices in magnetized plasmas is developed, taking into account the effects of the ion polarization and anisotropic heating by ion beams. It provides a theoretical explanation for the bipolar electrostatic structures in the auroral zone of the Earth's magnetosphere and their spatial and temporal scales, as observed by S3-3, Viking, FREJA, Polar, and FAST spacecrafts. Several types of quasi-three-dimensional ion holes are obtained analytically, in the form of either cylinders or ellipsoids. Although topologically different, they produce similar signals on the spacecraft and cannot be distinguished on the basis of the existing satellite data.

Dynamics of periodic ion holes in a forced beam–plasma experiment

Experimental observation of the propagation of periodic ion phase space vortices is reported. Density fluctuation measurements in a double plasma device show an apparently spontaneous acceleration of these periodic structures from ion thermal to ion acoustic velocity. A nonlinear kinetic description explains this as a transition from a new type of nonlinear electrostatic mode (periodic ion holes) to an ion acoustic mode which is caused by trapped particle scattering.

Ion phase-space vortices in 2.5-dimensional simulations

The formation and propagation of ion phase-space vortices are observed in a numerical particle-in-cell simulation in two spatial dimensions and with three velocity components. The code allows for an externally applied magnetic field. The electrons are assumed to be isothermally Boltzmann-distributed at all times, implying that Poisson's equation becomes nonlinear for the present problem. Ion phase-space vortices are formed by the nonlinear saturation of the ion-ion two-stream instability, excited by injecting an ion beam at the plasma boundary. We consider the effect of a finite beam diameter and a magnetic field, in particular. A vortex instability is observed, appearing as a transverse modulation, which slowly increases with time and ultimately breaks up the vortex. When many vortices are present at the same time, we find that it is their interaction that eventually leads to a gradual filling-up of the phase-space structures. The ion phase-space vortices have a finite lifetime, which is noticeably shorter than that found in one-dimensional simulations. An externally imposed magnetic field can increase this lifetime considerably. For high injected beam velocities in magnetized plasmas, we observe the excitation of electrostatic ion-cyclotron instabilities, but see no associated formation of ion phase-space vortices. The results are relevant, for instance, for the interpretation of observations by instrumented spacecraft in the Earth's ionosphere and magnetosphere.

Ion phase space vortices in ion beam-plasma systems and their relation with the ion acoustic instability: numerical and experimental results

Numerical simulations and experiments were performed in order to clarify the relation between ion phase space vortices, the ion acoustic instability and shock-like perturbations in ion beam-plasma systems. The simulations show that the formation of vortices in the ion phase space occurs only when the ion beam-plasma system is linearly unstable. The size and characteristic formation time of these vortices are in good agreement with predictions based on the wavelengths and growth rates of the linear one-dimensional ion acoustic instability; this confirms that vortices result from the saturation of the instability. Experiments on step-like perturbations in the ion beam-plasma system were performed in a DP device. They show the formation of extended 'hump and dip' regions behind the beam-induced shock front. Inside these regions localized occurrences of the ion acoustic instability have been seen. In contrast with the ion vortices, these structures do not result from the instability. These findings are in qualitative agreement with the results of numerical simulations designed after the experiment.

Coherent structures in numerically simulated plasma turbulence

Low level electrostatic ion acoustic turbulence generated by the ion-ion beam instability was investigated numerically. The fluctuations in potential were investigated by a conditional statistical analysis revealing propagating coherent structures having the form of negative potential wells which correspond to ion phase space vortices. The results demonstrate the importance of clump formation in ion phase space for the dynamics of the turbulence. The statistical analysis gives results in terms of averages over a conditionally selected subensemble. Because of the intermittent character of the turbulence it proved possible to devise a method, which permits recognition of essentially all coherent structures. With most of the structures recovered we are able to estimate their distributions of amplitude, width and velocity. A statistical evidence for interaction, i.e., binding, of phase space vortices was obtained.

A statistical analysis of numerically simulated plasma turbulence

Low-level electrostatic ion acoustic turbulence generated by the ion–ion beam instability was investigated numerically. It is demonstrated that a conditional statistical analysis can reveal the formation, propagation, and decay of negative potential wells which correspond to ion phase-space vortices. While the statistical analysis gives results in terms of averages over a conditionally selected subensemble, individual phase-space structures can be investigated by small clusters of test particles placed at selected positions in phase space. The predictions of the conditional statistical analysis can be tested and their accuracy estimated by this method. A discussion of various aspects of test particle dynamics in turbulent plasmas follows naturally, with particular emphasis on conditioned particle release and relative diffusion of particles. The statistical properties of the fluctuating electric fields sampled along Lagrangian particle orbits is analyzed.

Conditional eddies in plasma turbulence

Low-frequency electrostatic turbulence generated by the ion–ion beam instability was investigated experimentally in a double-plasma device. Real time signals were recorded and examined by a conditional statistical analysis. Conditionally averaged potential distributions reveal the formation and propagation of structures with a relatively long lifetime. Various methods for making a conditional analysis are discussed and compared. The results are discussed with reference to ion phase space vortices and clump formation in collisionless plasmas.

Ion phase-space vortices and their relation to small amplitude double-layers

The properties of ion phase-space vortices are reviewed with particular attention to their role in the formation of small amplitude double-layers in current-carrying plasmas. In a one-dimensional analysis, many such double-layers simply add up to produce a large voltage drop. A laboratory experiment is carried out in order to investigate the properties of ion phase-space vortices in three dimensions. Their lifetime is significantly reduced as compared with similar results from one-dimensional numerical simulations of the problem.

Formation mechanisms of laboratory double layers in triple plasma devices

The evolution processes of double-layers have been studied in a series of laboratory experiments using a triple plasma device. 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 well 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 corresponded to the disruption and the recovery of the double layers.

Experimental observations of ion phase-space vortices

Experimental observations of ion phase-space vortices are reported. The ion phase-space vortices form in the region of heated ions behind electrostatic ion acoustic shocks. The results are in qualitative agreement with numerical and analytic studies.