Phase Space Vortices Research Articles

Evolution of the electron phase space of the two-stream and bump-on-tail instabilities in collisional plasma

Particle collisions can have significant effects on plasma instabilities, especially in dense and/or low temperature plasmas. To understand the influence of collisional effects on the plasma waves, the Vlasov–Poisson system with Krook collisions is applied to study the long-term evolution of the two-stream (TS) and bump-on-tail (BOT) instabilities. The system is solved numerically with the fourth-order Runge–Kutta scheme and the Thomas algorithm. It is found that collisions can enhance the wave damping and mitigate the energy of the characteristic slow evolving nonlinear Landau damping oscillations associated with the wave-trapped electrons, especially if the collision rate ν is higher than 0.01ωp, where ωp is the plasma frequency of the background plasma. Collisions can also decrease the growth rate and saturation level of the TS and BOT unstable waves and tend to shrink the phase space vortex and narrow the phase-mixed region of the trapped electrons. However, our simulations show that collisions cannot readily prevent the nonlinear Landau damping oscillations. In fact, only with ν>0.001ωp for the TS instability and ν>0.01ωp for the BOT instability, as well as evolution times greater than several hundred ωp−1, the vortex structure of the wave-trapped electrons can be undetectable. The corresponding growth rates also drop dramatically, and the maximum wave energy can be one or two orders lower than that of the collisionless limits.

Ultrafast electron holes in plasma phase space dynamics

Electron holes (EH) are localized modes in plasma kinetic theory which appear as vortices in phase space. Earlier research on EH is based on the Schamel distribution function (df). A novel df is proposed here, generalizing the original Schamel df in a recursive manner. Nonlinear solutions obtained by kinetic simulations are presented, with velocities twice the electron thermal speed. Using 1D-1V kinetic simulations, their propagation characteristics are traced and their stability is established by studying their long-time evolution and their behavior through mutual collisions.

Landau damping in one dimensional periodic inhomogeneous collisionless plasmas

Landau damping in a collisionless plasma is a well-known example of wave particle interaction. In the past, this phenomenon was addressed for homogeneous equilibria in the linear and non-linear limit of the perturbation amplitude. However, in reality, equilibria are almost always inhomogeneous or non-uniform in space. Considering a one dimensional, collisionless, unmagnetized, electrostatic plasma with stationary ions and kinetic electrons in a periodic inhomogeneous exact equilibrium of scale k0−1 as the starting point, the fate of a small amplitude (linear) perturbation of scale k−1 is investigated using a Vlasov–Poisson solver. Three different spatial regimes, namely, k0 > k, k0 ∼ k, and k0 < k, are addressed. In the k0 > k regime, long wavelength perturbation k is found to generate (k ± Nk0) modes (where N is an integer), which allows damping of long wavelength perturbation in an inhomogeneous plasma and formation of phase-space vortices at phase velocities vϕ = ω/(k ± Nk0). Perhaps for the first time, novel phenomena such as “inhomogeneity induced Landau damping arrest” and “inhomogeneity induced plasma echo” are observed in k0 ∼ k and k0 < k regimes, respectively. New scaling laws as a function of inhomogeneity amplitude are also reported.

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.

Excitation of electrostatic solitary waves during neutralization of ion beam pulse by plasma

The possibility of exciting electrostatic solitary waves (ESWs) during the neutralization of ion beam pulses by plasmas is investigated using a two-dimensional (2D) electrostatic particle-in-cell (PIC) code. The simulation results show that if an axial magnetic field is applied, an ESW can be excited when the ion beam enters the plasma. The ESW is caused by the phase space vortex motion of electrons in the potential well formed by the plasma and the ion beam. The ESW is very stable and has a very long lifetime when it propagates in the plasma and the ion beam. The ESW has a 2D space structure, with about 3 cm length and more than twice the width of the ion beam. The splitting of the ESW is observed when it encounters a potential drop. The excitation conditions of the ESW are also discussed.

MMS Direct Observations of Kinetic-scale Shock Self-reformation

Abstract Studies of shocks have long suggested that a shock can undergo cyclical self-reformation on a timescale of ion cyclotron period. This process has been proposed as a primary mechanism for energy dissipation and energetic particle acceleration at shocks. Unambiguous observational evidence, however, has remained elusive. Here, we report direct observations for the self-reformation process of a collisionless, high Mach number, quasi-perpendicular shock using Magnetospheric Multiscale (MMS) measurements. We find that reflected ions by the old shock ramp form a clear phase-space vortex, which gives rise to a new ramp. The new ramp observed by MMS2 has not yet developed to a mature stage during the self-reformation, and is not strong enough to reflect incident ions. Consequently, these ions are only slightly slowed down and show a flat velocity profile from the new ramp all the way to the old one. The present results provide direct evidence of shock self-reformation, and also shed light on energy dissipation and energetic particle acceleration at collisionless shocks throughout the universe.

Saturation of trapped particle instability induced by vortex-merging in electron plasma waves

Phase space vortex merging in electron plasma waves is researched by applying a periodically changing electric field in plasma space with Maxwellian velocity distribution. The pump electric field excites the electron plasma wave with large amplitude and electron phase space vortices are formed. The electron plasma wave can be transited to longer wave modes through vortex merging processes. Excepting the fundamental mode of electron plasma waves, two or more modes appear and grow exponentially before vortex merging. After merging, another mode replaces the initial pumping wave as the strongest wave mode. It is verified that the growth of this new strong mode is related to trapping particle instability. Vortex merging is one of the saturation mechanisms of trapped particle instability.

Driven electrostatic phase space vortices in a 1D weakly dissipative Vlasov–Poisson system

The effect of collisions on driven electrostatic phase space vortices is analyzed by means of Eulerian simulation for two different collision models. It was demonstrated recently [P. Trivedi and R. Ganesh, Phys. Plasmas 23, 062112 (2016)] that in the absence of collisions, at late times, steady state phase space vortices manifest to form a plateau in the resonant region of the particle velocity distribution function, due to trapping of particles supporting multiextrema giant phase space vortices (PSVs). In the presence of collisions, over long time, this multiextrema plateau are found to smooth out, since collisions drive the velocity distribution toward Maxwellian, irrespective of how weak the collisions are as long as they are non-zero. In these conditions, kinetic processes and collisionality are found to be in competition, and the evolution of the plasma is found, therefore, to be a result of nontrivial combination of these two effects. An attempt has been made by means of numerical simulations to study the effect of weak collisionality on the electrostatic driven phase space vortices with two types of collision operators: (1) Bhatnagar–Gross–Krook (Krook) collision operator, where the colliding particles can be treated as isolated pairs and, (2) Fokker–Planck (FP) type collision operator (Zakharov–Karpman) in one dimension, where many weak collisions lead to particle diffusion in velocity space. It is shown that depending on the collision model used, the nature of smoothing in velocity space of giant PSVs results in qualitatively very different phase space structures. However, irrespective of the collision model used, excess density fractions over 10% are retained.

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.

Response to “Comment on ‘Symmetry in electron and ion dispersion in 1D Vlasov-Poisson plasma’” [Phys. Plasmas 26, 064701 (2019)

Treating electron and ion kinetics on the same physics footing or in a symmetric framework, dispersion relations (ωr, k) for ion and electron modes in a 1D Vlasov-Poisson (VP) plasma were obtained in the limit of weak flattening of the electron and ion distributions. Using this information of (ωr, k), a nonlinear VP solver with a low amplitude, harmonic external forcing has been solved. A possibility of simultaneous excitation of all the electrostatic normal modes was demonstrated in Trivedi et al. [Phys. Plasmas 25, 112102 (2018)]. The main focus of the paper is on the excitation of normal modes by applying a small amplitude external, monochromatic, electric field drive. In order to drive as well as to identify Phase Space Vortices (i.e., Bernstein-Greene-Kruskal/Cnoidal modes, etc.) formed out of the driven nonlinear VP system, assuming local flattening of electron and ion distributions, the real part of the plasma dispersion function for arbitrary ratios of ion to electron masses and temperatures is solved, which is known to be accurate in the low amplitude, harmonic limit, as pointed out by Schamel [Phys. Plasmas 19, 020501 (2012)].

Coherent phase space structures in a 1D electrostatic plasma using particle-in-cell and Vlasov simulations: A comparative study

We present a quantitative comparative study of the formation of coherent phase space structures in one dimension using two widely followed approaches in kinetic simulations of collision-less plasmas, namely, Monte Carlo based Particle-In-Cell (PIC) simulations and phase space grid based Eulerian Vlasov-Poisson simulations. Using a newly developed PIC solver, we demonstrate that, while for linear regimes, there is a ready quantitative agreement between Vlasov-Poisson and PIC solvers, whereas for weakly nonlinear regimes and late time simulations, for comparable field resolutions, Vlasov-Poisson simulation results are found to be relatively noise-free as compared to PIC results with a large number of PIC particles. As an extreme case, we address using high resolution PIC simulations, the formation of giant phase space vortices obtained recently using the Vlasov-Poisson method [P. Trivedi and R. Ganesh, Phys. Plasmas 23, 062112 (2016)] for an infinitesimal amplitude external drive. For identical parameters and numerical resolution, we present a qualitative and quantitative comparison between PIC results of driven giant phase space structures and those of the Vlasov method, for a Maxwellian plasma.

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.

Particle trapping and ponderomotive processes during breaking of ion acoustic waves in plasmas

Recent fluid simulations show that the ponderomotive potentials and ponderomotive frequencies of electrons and ions can be used as proxies to identify steepening and breaking of the ion acoustic solitary waves (IASWs) in plasmas. However, the behavior of these proxies may deviate in the presence of kinetic effects such as particle trapping. We performed one-dimensional particle-in-cell (PIC) simulations to examine the effects of kinetic processes on the behavior of these proxies at the breaking of IASWs in plasmas. The electron and ion equilibrium densities were superimposed by a long-wavelength Gaussian type perturbation, which initially evolves into two IASWs observed as two phase space vortices due to the trapping of electrons in the ion acoustic (IA) potentials. These IASW structures grow due to the steepening of their trailing edges, and later they break into a chain of IA phase space vortices. Each of these vortices is associated with a bipolar electric field resulting in a positive potential structure. We examined the amplitude, width, and phase velocity of the IASWs at their breaking process to clarify their link with the trapping velocity. In addition, we estimated electron and ion ponderomotive potentials and frequencies from the PIC simulations to verify their applicability in identifying wave breaking limit under the kinetic regime. The present study shows that the behavior of the ponderomotive potential during the IA wave breaking process is similar to the one, which is proposed through fluid simulations. We find that IA wave breaking occurs when the maximum trapping velocity of the electron (Vtrap + Vs) exceeds its thermal velocity. The present simulation study shows that both maximum electron trapping velocity and ponderomotive potential can be used to identify the IA wave breaking processes in plasmas.

Evolution of electron phase space holes in inhomogeneous plasmas

Electron phase space holes or vortices (EHs) are electrostatic solitary waves with a bipolar parallel (magnetic field-aligned) electric field. They are formed in a nonlinear stage of electron streaming type instabilities and exist due to electrons trapped within the EH electrostatic potential. The background plasma density gradients, characteristic for both space and laboratory plasmas, can affect the evolution of EHs. In this paper, we use a one-dimensional electrostatic Vlasov–Ampère code (ions are immobile) with periodic boundary conditions to study the evolution of a single EH in inhomogeneous plasmas. We find that the EH propagating along a positive (negative) plasma density gradient is accelerated (decelerated) and narrowed (widened). EH propagating along a positive density gradient results in the acceleration of a relatively small population of trapped electrons to suprathermal energies. Interestingly, a decelerating EH is reflected at the point with the plasma density value dependent only on EH parameters, but independent of the average density gradient in the system. We show that the density gradients result in the development of a unipolar parallel electric field in a vicinity of the EH. A theoretical estimate of the corresponding potential drop along the EH is derived. The results are discussed in the light of EH observations in space plasma.

Driven phase space vortices in plasmas with nonextensive velocity distribution

The evolution of chirp-driven electrostatic waves in unmagnetized plasmas is numerically investigated by using a one-dimensional (1D) Vlasov-poisson solver with periodic boundary conditions. The initial velocity distribution of the 1D plasma is assumed to be governed by nonextensive q distribution [C. Tsallis, J. Stat. Phys. 52, 479 (1988)]. For an infinitesimal amplitude of an external drive, we investigate the effects of chirp driven dynamics that leads to the formation of giant phase space vortices (PSV) for both Maxwellian (q = 1) and non-Maxwellian (q≠1) plasmas. For non-Maxwellian plasmas, the formation of giant PSV with multiple extrema and phase velocities is shown to be dependent on the strength of “q”. Novel features such as “shark”-like and transient “honeycomb”-like structures in phase space are discussed. Wherever relevant, we compare our results with previous work.

Simulation study of the formation of a non-relativistic pair shock

We examine with a particle-in-cell (PIC) simulation the collision of two equally dense clouds of cold pair plasma. The clouds interpenetrate until instabilities set in, which heat up the plasma and trigger the formation of a pair of shocks. The fastest-growing waves at the collision speed $c/5$, where $c$ is the speed of light in vacuum, and low temperature are the electrostatic two-stream mode and the quasi-electrostatic oblique mode. Both waves grow and saturate via the formation of phase space vortices. The strong electric fields of these nonlinear plasma structures provide an efficient means of heating up and compressing the inflowing upstream leptons. The interaction of the hot leptons, which leak back into the upstream region, with the inflowing cool upstream leptons continuously drives electrostatic waves that mediate the shock. These waves heat up the inflowing upstream leptons primarily along the shock normal, which results in an anisotropic velocity distribution in the post-shock region. This distribution gives rise to the Weibel instability. Our simulation shows that even if the shock is mediated by quasi-electrostatic waves, strong magnetowaves will still develop in its downstream region.

Slow electrostatic fluctuations generated by beam-plasma interaction

Eulerian simulations of the Vlasov-Poisson equations have been employed to analyze the excitation of slow electrostatic fluctuations (with phase speed close to the electron thermal speed), due to a beam-plasma interaction, and their propagation in linear and nonlinear regimes. In 1968, O'Neil and Malmberg [Phys. Fluids 11, 1754 (1968)] dubbed these waves “beam modes.” In the present paper, previous analytical results on the beam modes in both linear and nonlinear regimes have been revisited numerically, pointing out that, when an electron beam is launched in a plasma of Maxwellian electrons and motionless protons and this initial equilibrium is perturbed by a monochromatic density disturbance, the electric field amplitude grows exponentially in time and then undergoes nonlinear saturation, associated with the kinetic effects of particle trapping and phase space vortex generation. Moreover, new numerical results give evidence that, when the initial density perturbation is setup in the form of a low amplitude random phase noise, the whole Fourier spectrum of wavenumbers is excited. As a result, the electric field profile appears as a train of isolated pulses, each of them being associated with a phase space vortex in the electron distribution function. At later times, these vortical structures tend to merge and, correspondingly, the electric pulses collapse, showing the tendency towards a time asymptotic configuration characterized by the appearance of electric soliton-like pulses. This dynamical evolution is driven by purely kinetic processes, possibly at work in many space and laboratory plasma environments.

Chirp-driven giant phase space vortices

In a collisionless, unbounded, one-dimensional plasma, modelled using periodic boundary conditions, formation of steady state phase space coherent structures or phase space vortices (PSV) is investigated. Using a high resolution one-dimensional Vlasov-Poisson solver based on piecewise-parabolic advection scheme, the formation of giant PSV is addressed numerically. For an infinitesimal external drive amplitude and wavenumber k, we demonstrate the existence of a window of chirped external drive frequency that leads to the formation of giant PSV. The linear, small amplitude, external drive, when chirped, is shown to couple effectively to the plasma and increase both streaming of “untrapped” and “trapped” particle fraction. The steady state attained after the external drive is turned off and is shown to lead to a giant PSV with multiple extrema and phase velocities, with excess density fraction, defined as the deviation from the Maxwellian background, Δn/n0≃20%−25%. It is shown that the process depends on the chirp time duration Δt. The excess density fraction Δn/n0, which contains both trapped and untrapped particle contribution, is also seen to scale with Δt, only inhibited by the gradient of the distribution in velocity space. Both single step drive and multistep chirp processes are shown to lead to steady state giant PSV, with multiple extrema due to embedded holes and clumps, long after the external drive is turned off.

Nonlinearly interacting trapped particle solitons in collisionless plasmas

The formulation of collective waves in collisionless plasmas is complicated by the kinetic effects produced by the resonant particles, capable of responding to the smallest of the amplitude disturbance. The dispersive plasma manifests this response by generating coherent nonlinear structures associated with phase-space vortices, or holes, at very small amplitudes. The nonlinear interaction between solitary electron phase-space holes is studied in the electron acoustic regime of a collisionless plasma using Vlasov simulations. Evolution of the analytic trapped particle solitary solutions is examined, observing them propagate stably, preserve their identity across strong mutual interactions in adiabatic processes, and display close correspondence with observable processes in nature.

Vlasov simulations of kinetic Alfvén waves at proton kinetic scales

Kinetic Alfvén waves represent an important subject in space plasma physics, since they are thought to play a crucial role in the development of the turbulent energy cascade in the solar wind plasma at short wavelengths (of the order of the proton gyro radius ρp and/or inertial length dp and beyond). A full understanding of the physical mechanisms which govern the kinetic plasma dynamics at these scales can provide important clues on the problem of the turbulent dissipation and heating in collisionless systems. In this paper, hybrid Vlasov-Maxwell simulations are employed to analyze in detail the features of the kinetic Alfvén waves at proton kinetic scales, in typical conditions of the solar wind environment (proton plasma beta βp = 1). In particular, linear and nonlinear regimes of propagation of these fluctuations have been investigated in a single-wave situation, focusing on the physical processes of collisionless Landau damping and wave-particle resonant interaction. Interestingly, since for wavelengths close to dp and βp ≃ 1 (for which ρp ≃ dp) the kinetic Alfvén waves have small phase speed compared to the proton thermal velocity, wave-particle interaction processes produce significant deformations in the core of the particle velocity distribution, appearing as phase space vortices and resulting in flat-top velocity profiles. Moreover, as the Eulerian hybrid Vlasov-Maxwell algorithm allows for a clean almost noise-free description of the velocity space, three-dimensional plots of the proton velocity distribution help to emphasize how the plasma departs from the Maxwellian configuration of thermodynamic equilibrium due to nonlinear kinetic effects.