Phase Space Holes Research Articles

The magnetic structures of electron phase-space holes formed in the electron two-stream instability

It is well known that the parallel cuts of the parallel and perpendicular electric field in electron phase-space holes (electron holes) have bipolar and unipolar structures, respectively. Recently, electron holes in the Earth's plasma sheet have been observed by THEMIS satellites to have detectable fluctuating magnetic field with regular structures. Du et al. (2011) investigated the evolution of a one-dimensional (1D) electron hole with two-dimensional (2D) electromagnetic particle-in-cell (PIC) simulations in weakly magnetized plasma (Omega(epsilon) (x)) is along the x direction. Several 2D electron holes are formed during the nonlinear evolution, where the parallel cuts of E-x and E-y have bipolar and unipolar structures, respectively. Consistent with the results of Du et al. (2011), we found that the current along the z direction is generated by the electric field drift motion of the trapped electrons in the electron holes due to the existence of E-y, which produces the fluctuating magnetic field delta B-x and delta B-y in the electron holes. The parallel cuts of delta Bx and delta B-y in the electron holes have unipolar and bipolar structures, respectively.

A model of electromagnetic electron phase-space holes and its application

[1] Electron phase-space holes (EHs) are indicators of nonlinear activities in space plasmas. Most often they are observed as electrostatic signals, but recently Andersson et al. [2009] reported electromagnetic EHs observed by the THEMIS mission in the Earth's plasma sheet. As a follow-up to Andersson et al. [2009], this paper presents a model of electromagnetic EHs where the δE × B0 drift of electrons creates a net current. The model is examined with test-particle simulations and compared to the electromagnetic EHs reported by Andersson et al. [2009]. As an application of the model, we introduce a more accurate method than the simplified Lorentz transformation of Andersson et al. [2009] to derive EH velocity (vEH). The sizes and potentials of EHs are derived from vEH, so an accurate derivation of vEH is important in analyzing EHs. In general, our results are qualitatively consistent with those of Andersson et al. [2009] but generally with smaller velocities and sizes.

The evolution of the magnetic structures in electron phase-space holes: Two-dimensional particle-in-cell simulations

[1] Observations have shown that electron phase-space holes (electron holes) possess regular magnetic structures. In this paper, two-dimensional (2D) electromagnetic particle-in-cell (PIC) simulations are performed in the (x, y) plane to study magnetic structures associated with electron holes under different plasma conditions. In the simulations, the background magnetic field (B0 = B0x) is along the x direction. The combined actions between the transverse instability and stabilization by the background magnetic field lead to the generation of the electric field Ey. Then electrons suffer the electric field drift and produce the current in the z direction, which leads to the fluctuating magnetic field along the x and y directions. Meanwhile, the motion of the electron holes along the x direction and the existence of the electric field Ey generate the fluctuating magnetic field along the z direction. In very weakly magnetized plasma (Ωe ≪ ωpe, where Ωe and ωpe are the electron gyrofrequency and electron plasma frequency, respectively.), the transverse instability is very strong and the magnetic structures associated with electron holes disappear quickly. When Ωe is comparable to ωpe, the parallel cut of the fluctuating magnetic field δBx and δBz has unipolar structures in the electron holes, while the parallel cut of fluctuating magnetic field δBy has bipolar structures. In strongly magnetized plasma (Ωe > ωpe), electrostatic whistler waves with streaked structures of Ey are excited. The fluctuating magnetic field δBx and δBz also have streaked structures. The relevance between our simulation results and the magnetic structures associated with electron holes observed in the plasma sheet is also discussed.

Nonlinear dynamics of longitudinal structures in the electron cloud of a coaxial electron string ion source

The results of 2.5D particle-in-cell simulation of a coaxial electron trap with an internal anode are reported. It is found that, when the circulating current reaches the value of the ultimate vacuum current, first a virtual cathode arises in the trap and then the beam compresses (distributed virtual cathode). The transient preceding the compressed state exhibits complicated nonlinear dynamics, when compressed regions alternate with regions that are in a two-flow state (phase-space bubbles or phase-space holes). Physically, phase-space holes are similar to the well-known Bernstein-Greene-Kruskal plasma structures. Three types of phasespace holes with different dynamics (oscillating holes, flying holes, and chaotic holes) are revealed. Consideration of phase-space holes as quasi-particles makes it possible to find several channels of their interaction in pair collisions. The feasibility of the coaxial trap as a source of highly charged ions is analyzed. Although the compressed beam mode provides a larger amount of accumulated electrons compared with the conventional two-flow mode, the mean kinetic energy in the presence of a virtual cathode turns out to be much lower. A way of elevating the mean kinetic energy is suggested that consists in increasing the limit vacuum current in the axial configuration with an internal electrode.

Static Isolated Horizons: SU(2) Invariant Phase Space, Quantization, and Black Hole Entropy

We study the classical field theoretical formulation of static generic isolated horizons in a manifestly SU(2) invariant formulation. We show that the usual classical description requires revision in the non-static case due to the breaking of diffeomorphism invariance at the horizon leading to the non-conservation of the usual pre-symplectic structure. We argue how this difficulty could be avoided by a simple enlargement of the field content at the horizon that restores diffeomorphism invariance. Restricting our attention to static isolated horizons we study the effective theories describing the boundary degrees of freedom. A quantization of the horizon degrees of freedom is proposed. By defining a statistical mechanical ensemble where only the area aH of the horizon is fixed macroscopically—states with fluctuations away from spherical symmetry are allowed—we show that it is possible to obtain agreement with the Hawkings area law (S = aH /(4l 2p)) without fixing the Immirzi parameter to any particular value: consistency with the area law only imposes a relationship between the Immirzi parameter and the level of the Chern-Simons theory involved in the effective description of the horizon degrees of freedom.

Transverse instability and magnetic structures associated with electron phase space holes

Electron phase space holes (electron holes) are found to be unstable to the transverse instability. Two-dimensional (2D) electromagnetic particle-in-cell simulations are performed to investigate the structures of the fluctuating magnetic field associated with electron holes. The combined actions between the transverse instability and the stabilization by the background magnetic field (B0=B0e⃗x) lead a one-dimensional electron hole into several 2D electron holes which are isolated in both the x and y directions. The electrons trapped in these 2D electron holes suffer the electric field drift vE=E×B0/B02 due to the existence of the perpendicular electric field Ey, which generates the current along the z direction. Then, the unipolar and bipolar structures are formed for the parallel cut of the fluctuating magnetic field along the x and y directions, respectively. At the same time, these 2D electron holes move along the x direction, and the unipolar structures are formed for the parallel cut of the fluctuating magnetic field along the z direction.

The inversion layer of electric fields and electron phase-space-hole structure during two-dimensional collisionless magnetic reconnection

Based on two-dimensional fully kinetic simulations that resolve the electron diffusion layer in undriven collisionless magnetic reconnection with zero guide field, this paper reports the existence and evolution of an inversion layer of bipolar electric fields, its corresponding phase-space structure (an electron-hole layer), and the implication to collisionless dissipation. The inversion electric field layer is embedded in the layer of bipolar Hall electric field and extends throughout the entire length of the electron diffusion layer. The electron phase-space hole structure spontaneously arises during the explosive growth phase when there exist significant inflows into the reconnection layer, and electrons perform meandering orbits across the layer while being cyclotron-turned toward the outflow directions. The cyclotron turning of meandering electrons by the magnetic field normal to the reconnection layer is shown to be a primary factor limiting the current density in the region where the reconnection e...

Effect of dynamical friction on nonlinear energetic particle modes

A fully nonlinear model is developed for the bump-on-tail instability including the effects of dynamical friction (drag) and velocity space diffusion on the energetic particles driving the wave. The results show that drag provides a destabilizing effect on the nonlinear evolution of waves. Specifically, in the early nonlinear phase of the instability, the drag facilitates the explosive scenario of the wave evolution, leading to the creation of phase space holes and clumps that move away from the original eigenfrequency. Later in time, the electric field associated with a hole is found to be enhanced by the drag, whereas for a clump it is reduced. This leads to an asymmetry of the frequency evolution between holes and clumps. The combined effect of drag and diffusion produces a diverse range of nonlinear behaviors including hooked frequency chirping, undulating, and steady state regimes. An analytical model is presented, which explains the aforementioned diversity. A continuous production of hole-clump pairs in the absence of collisions is also observed.

Effects of Perpendicular Thermal Velocities on the Transverse Instability in Electron Phase Space Holes

A multi-dimensional electron phase-space hole (electron hole) is considered to be unstable to the transverse instability. We perform two-dimensional (2D) particle-in-cell (PIC) simulations to study the evolutions of electron holes in weakly magnetized plasma (Ωe < ωpe, where Ωe and ωpe are the electron gyrofrequency and plasma frequency, respectively), and the effects of perpendicular thermal velocities on the transverse instability are investigated. The transverse instability can cause decay of the electron holes. We find that with the increasing perpendicular thermal velocity tending to stabilize the transverse instability, the corresponding wave numbers decrease.

Scales of guide field reconnection at the hydrogen mass ratio

We analyze the signatures of component reconnection for a Harris current sheet with a guide field using the physical mass ratio of hydrogen. The study uses the fully kinetic particle in cell code IPIC3D to investigate the scaling with mass ratio of the following three main component reconnection features: electron density cavities along the separatrices, channels of fast electron flow within the cavities, and electron phase space holes due to the Buneman instability in the electron high speed channels. The width and strength of the electron holes and of the electron cavities are studied up the mass ratio proper of hydrogen, considering the effect of the simulation box size, and of the boundary conditions. The results compare favorably with the existing data from the Cluster and Themis missions and provide quantitative predictions for realistic conditions to be encountered by the planned magnetospheric multiscale mission.

Simulation of a collisionless planar electrostatic shock in a proton–electron plasma with a strong initial thermal pressure change

The localized deposition of the energy of a laser pulse, as it ablates a solid target, introduces high thermal pressure gradients in the plasma. The thermal expansion of this laser-heated plasma into the ambient medium (ionized residual gas) triggers the formation of non-linear structures in the collisionless plasma. Here an electron–proton plasma is modelled with a particle-in-cell simulation to reproduce aspects of this plasma expansion. A jump is introduced in the thermal pressure of the plasma, across which the otherwise spatially uniform temperature and density change by a factor of 100. The electrons from the hot plasma expand into the cold one and the charge imbalance drags a beam of cold electrons into the hot plasma. This double layer reduces the electron temperature gradient. The presence of the low-pressure plasma modifies the proton dynamics compared with the plasma expansion into a vacuum. The jump in the thermal pressure develops into a primary shock. The fast protons, which move from the hot into the cold plasma in the form of a beam, give rise to the formation of phase space holes in the electron and proton distributions. The proton phase space holes develop into a secondary shock that thermalizes the beam.

Electric field generation by the electron beam filamentation instability: filament size effects

The filamentation instability (FI) of counter-propagating beams of electrons is modelled with a particle-in-cell simulation in one spatial dimension and with a high statistical plasma representation. The simulation direction is orthogonal to the beam velocity vector. Both electron beams have initially equal densities, temperatures and moduli of their non-relativistic mean velocities. The FI is electromagnetic in this case. A previous study of a small filament demonstrated that the magnetic pressure gradient force (MPGF) results in a nonlinearly driven electrostatic field. The probably small contribution of the thermal pressure gradient to the force balance implied that the electrostatic field performed undamped oscillations around a background electric field. Here, we consider larger filaments, which reach a stronger electrostatic potential when they saturate. The electron heating is enhanced and electrostatic electron phase space holes form. The competition of several smaller filaments, which grow simultaneously with the large filament, also perturbs the balance between the electrostatic and magnetic fields. The oscillations are damped but the final electric field amplitude is still determined by the MPGF.

Self-consistent evolution of auroral downward-current region ion outflow and moving double layer

[1] Vlasov simulations have been performed to investigate the development and evolution of the parallel potential drop, often structured as a localized double layer (DL), electron phase-space holes, and ion perpendicular heating, resulting in ion outflow in the auroral downward-current region. We focus on the feedback between a moving DL and heated ions, suggesting a self-consistent model: the heated ions regulate the evolution of the DL which controls the level of ion heating and, consequently, the outflow rate. Strongly heated ions forming conics are found in agreement with in-situ observations. The intensification and widening of the DL develops with time, and the DL eventually weakens due to a reduction in the background population resulting from excessive ion heating.

Alternative dust acoustic solitary waves in a plasma consisting of superthermal electrons and nonthermal ions having kappa-vortex-like velocity distribution function

The existence of arbitrary amplitude dust acoustic solitary waves is investigated in an unmagnetized dusty plasma comprising a negatively charged dust fluid, superthermal electrons, and nonthermal ions having a κ-vortex-like velocity distribution function. It is shown that due to electron and ion superthermality, the present dusty plasma model may support subsonic as well as supersonic electrostatic solitary waves involving cusped potential humps. The interplay between different concepts of nonisothermality is highlighted. Our results should help to understand the role that phase-space holes may play in reconnection by scattering and energizing particles and to provide an explanation of the intense solitary structures observed by the FAST mission.

Publisher’s Note: New Features of Electron Phase Space Holes Observed by the THEMIS Mission [Phys. Rev. Lett.102, 225004 (2009)

Received 21 July 2009DOI:https://doi.org/10.1103/PhysRevLett.103.059901©2009 American Physical Society

New Features of Electron Phase Space Holes Observed by the THEMIS Mission

Observations of electron phase-space holes (EHs) in Earth's plasma sheet by the THEMIS satellites include the first detection of a magnetic perturbation (deltaB_{ parallel}) parallel to the ambient magnetic field (B0). EHs with a detectable deltaB_{ parallel} have several distinguishing features including large electric field amplitudes, a magnetic perturbation perpendicular to B0, high speeds ( approximately 0.3c) along B0, and sizes along B0 of tens of Debye lengths. These EHs have a significant center potential (Phi approximately k_{B}T_{e}/e), suggesting strongly nonlinear behavior nearby such as double layers or magnetic reconnection.

PARTICLE-IN-CELL SIMULATION OF A STRONG DOUBLE LAYER IN A NONRELATIVISTIC PLASMA FLOW: ELECTRON ACCELERATION TO ULTRARELATIVISTIC SPEEDS

Two charge- and current neutral plasma beams are modelled with a one-dimensional PIC simulation. The beams are uniform and unbounded. The relative speed between both beams is 0.4c. One beam is composed of electrons and protons and one out of protons and negatively charged oxygen (dust). All species have the temperature 9 keV. A Buneman instability develops between the electrons of the first beam and the protons of the second beam. The wave traps the electrons, which form plasmons. The plasmons couple energy into the ion acoustic waves, which trap the protons of the second beam. A proton phase space hole grows, which develops through its interaction with the oxygen and the heated electrons into a rarefaction pulse. This pulse drives a strong ion acoustic double layer, which accelerates a beam of electrons to about 50 MeV, which is comparable to the proton kinetic energy. The proton distribution eventually evolves into an electrostatic shock. Beams of charged particles moving at such speeds may occur in the foreshock of supernova remnant shocks. This double layer is thus potentially relevant for the electron acceleration (injection) into the diffusive shock acceleration by supernova remnants shocks.

Laboratory Observation of Electron Phase-Space Holes during Magnetic Reconnection

We report the observation of large-amplitude, nonlinear electrostatic structures, identified as electron phase-space holes, during magnetic reconnection experiments on the Versatile Toroidal Facility at MIT. The holes are positive electric potential spikes, observed on high-bandwidth ( approximately 2 GHz) Langmuir probes. Investigations with multiple probes establish that the holes travel at or above the electron thermal speed and have a three-dimensional, approximately spherical shape, with a scale size approximately 2 mm. This corresponds to a few electron gyroradii, or many tens of Debye lengths, which is large compared to holes considered in simulations and observed by satellites, whose length scale is typically only a few Debye lengths. Finally, a statistical study over many discharges confirms that the holes appear in conjunction with the large inductive electric fields and the creation of energetic electrons associated with the magnetic energy release.

The formation of relativistic plasma structures and their potential role in the generation of cosmic ray electrons

Abstract. Recent particle-in-cell (PIC) simulation studies have addressed particle acceleration and magnetic field generation in relativistic astrophysical flows by plasma phase space structures. We discuss the astrophysical environments such as the jets of compact objects, and we give an overview of the global PIC simulations of shocks. These reveal several types of phase space structures, which are relevant for the energy dissipation. These structures are typically coupled in shocks, but we choose to consider them here in an isolated form. Three structures are reviewed. (1) Simulations of interpenetrating or colliding plasma clouds can trigger filamentation instabilities, while simulations of thermally anisotropic plasmas observe the Weibel instability. Both transform a spatially uniform plasma into current filaments. These filament structures cause the growth of the magnetic fields. (2) The development of a modified two-stream instability is discussed. It saturates first by the formation of electron phase space holes. The relativistic electron clouds modulate the ion beam and a secondary, spatially localized electrostatic instability grows, which saturates by forming a relativistic ion phase space hole. It accelerates electrons to ultra-relativistic speeds. (3) A simulation is also revised, in which two clouds of an electron-ion plasma collide at the speed 0.9c. The inequal densities of both clouds and a magnetic field that is oblique to the collision velocity vector result in waves with a mixed electrostatic and electromagnetic polarity. The waves give rise to growing corkscrew distributions in the electrons and ions that establish an equipartition between the electron, the ion and the magnetic energy. The filament-, phase space hole- and corkscrew structures are discussed with respect to electron acceleration and magnetic field generation.

A water bag model of driven phase space holes in non-neutral plasmas

The formation and control of stable multiphase space hole structures and the associated Bernstein–Greene–Kruskal modes in trapped pure ion plasmas driven by an oscillating, chirped frequency perturbation are considered. The holes are formed by passing kinetic bounce resonances ωd=nπu∕L in the system, u and L are the longitudinal velocity of the plasma species and the length of the trap, and n is the multiplicity of the resonance (the number of the phase space holes). An adiabatic, quasi-one-dimensional water bag model of this excitation for an initially flat-top distribution of the ions in the trap is suggested, based on the isomorphism with a related problem in infinite quasineutral plasmas. A multiwater bag approach allows us to generalize the theory to other initial distributions. Numerical simulations yield a very good agreement with the theory until the coherent phase space structure is destroyed due to the resonance overlap when the decreasing driving frequency passes a critical value estimated within the water bag theory.