BGK Modes Research Articles

Theory for two-dimensional electron and ion Bernstein–Greene–Kruskal modes in a magnetized plasma

AbstractA theory for two-dimensional electron and ion Bernstein–Greene– Kruskal (BGK) modes in a magnetized space plasma is presented. The BGK modes are constructed using the energy and the canonical angular momentum of the particles, which are conserved in a cylindrically symmetric potential. The typical length scale of the BGK modes is of the same order or larger than the thermal gyroradius of the particles. The results are relevant for understanding the properties of observed localized structures in the Earth's magnetosphere and auroral zone, as well as in laboratory magnetoplasmas.

Quantum Corrections to Classical BGK Modes in Phase Space

In this paper we study the quantum corrections to the Bernstein‐Greene‐Kruskal equilibria of plasma physics by using perturbation methods near the classical solution. We obtain the solution for the Wigner function and for the self‐consistent potential to first order in the smallness parameter, which is the square of Planck's constant (in dimensionless form), and investigate the structure of the quantum phase space.

A water bag theory of autoresonant Bernstein-Greene-Kruskal modes

The adiabatic water bag theory describing formation and passage through phase-space of driven, continuously phase-locked (autoresonant) coherent structures in plasmas [L. Friedland et al., Phys. Rev. Lett. 96, 225001 (2006)] and of the associated Bernstein-Greene-Kruskal (BGK) modes is developed. The phase-locking is achieved by using a chirped frequency ponderomotive drive, passing through kinetic Čerenkov-type resonances. The theory uses the adiabatic invariants (conserved actions of limiting trajectories) in the problem and, for a flat-top initial distribution of the electrons, reduces the calculation of the self-field of the driven BGK mode to solution of a few algebraic equations. The adiabatic multiwater bag extension of the theory for applications to autoresonant BGK structures with more general initial distributions is suggested. The results of the theories are in very good agreement with numerical simulations.

GRAND CANONICAL MIXED-STATE WIGNER FUNCTION IN A MAGNETIC FIELD: de HAAS–van ALPHEN OSCILLATIONS

Kohn proved in 1961 that interactions between electrons did not change the de Haas–van Alphen (dHvA) oscillation frequency for single electrons in the nondegenerate ground-state [Phys. Rev.123(4), 1242 (1961)]. It was proved recently that the pure-state Wigner function for an electron in a magnetic field carries this quantum and physical oscillation, and a quantum dielectric function, so the conductance can be calculated from the Wigner function [Int. J. Mod. Phys. B17(25), 4555 (2003)], [Int. j. Mod. Phys. B17(26), 4683 (2003)]. We present the first complete proof that at a finite temperature, the mixed-state Wigner function also shows dHvA oscillations with the same frequency. The Wigner function is a fundamental quantity, the fact that it carries observable physical information shows a great potential in the design of new quantum materials at the nanoscale. The definition of the mixed-state Wigner function involves a grand canonical partition function (GCPF). Although dHvA is a well-known phenomenon, we present the first complete proof of it happening in degenerate mixed-states, based on a GCPF, which requires reconciliation between the dHvA experimental condition of a fixed number of particles and the GCPF's sum over number of particles. The GCPF is applied to one of the two spin species, while both the spin and spin-magnetic moment interaction are considered. We show that the contour integration in ω(ε) leads to a non-oscillatory term that is much larger than an oscillatory term, in the dHvA experimental conditions of high fields and low temperatures. This dominance of the non-oscillatory term explains the constancy of the chemical potential, allowing it to reduce to the Fermi energy in the limit of zero temperature. The obtained mixed-state Wigner function shows a fundamental period of oscillation with respect to B-1 that reduces to the Onsager's period for dHvA oscillations. This indicates that in mixed-states, dHvA oscillations depend on electrons of one spin species, this means the population of electrons of each spin species oscillates with the magnetic field. The temperature dependence in the Wigner function will allow a combination of phase-space and thermodynamics information for mesoscopic structures, and the study of phase-space density holes such as BGK modes in the quantum domain.

Vlasov model using kinetic phase point trajectories for the study of BGK modes

Vlasov model using kinetic phase point trajectories for the study of BGK modes

Stabilisation of BGK modes by relativistic effects

We investigate the acceleration of electrons via their interaction with electrostatic waves, driven by the relativistic Buneman instability, in a system dominated by counter-propagating proton beams. We observe the growth of these waves and their subsequent saturation via electron trapping for a range of proton beam velocities, from 0.15c to 0.9c. We can report a reduced stability of the electrostatic wave (ESW) with increasing non-relativistic beam velocities and an improved wave stability for increasing relativistic beam velocities, both in accordance with previous findings. At the highest beam speeds, we find the system to be stable again for a period of ~160 plasma periods. Furthermore we observe a, to our knowledge, previously unreported secondary electron acceleration mechanism at low beam speeds. We believe that it is the result of parametric couplings to produce high phase velocity ESW's which then trap electrons, accelerating them to higher energies. This allows electrons in our simulation study to achieve the injection energy required for Fermi acceleration, for beam speeds as low as 0.15c in unmagnetised plasma.

Weakly collisional Landau damping and three-dimensional Bernstein-Greene-Kruskal modes: New results on old problems

Landau damping and Bernstein-Greene-Kruskal (BGK) modes are among the most fundamental concepts in plasma physics. While the former describes the surprising damping of linear plasma waves in a collisionless plasma, the latter describes exact undamped nonlinear solutions of the Vlasov equation. There does exist a relationship between the two: Landau damping can be described as the phase mixing of undamped eigenmodes, the so-called Case–Van Kampen modes, which can be viewed as BGK modes in the linear limit. While these concepts have been around for a long time, unexpected new results are still being discovered. For Landau damping, we show that the textbook picture of phase mixing is altered profoundly in the presence of collision. In particular, the continuous spectrum of Case–Van Kampen modes is eliminated and replaced by a discrete spectrum, even in the limit of zero collision. Furthermore, we show that these discrete eigenmodes form a complete set of solutions. Landau-damped solutions are then recovered as true eigenmodes (which they are not in the collisionless theory). For BGK modes, our interest is motivated by recent discoveries of electrostatic solitary waves in magnetospheric plasmas. While one-dimensional BGK theory is quite mature, there appear to be no exact three-dimensional solutions in the literature (except for the limiting case when the magnetic field is sufficiently strong so that one can apply the guiding-center approximation). We show, in fact, that two- and three-dimensional solutions that depend only on energy do not exist. However, if solutions depend on both energy and angular momentum, we can construct exact three-dimensional solutions for the unmagnetized case, and two-dimensional solutions for the case with a finite magnetic field. The latter are shown to be exact, fully electromagnetic solutions of the steady-state Vlasov-Poisson-Ampère system.

Particle-in-cell method for parallel dynamics in magnetized electron plasmas: Study of high-amplitude BGK modes

The present paper describes the numerical technique that has been developed, in the framework of the particle-in-cell (PIC) method, to study the dynamics of a nonneutral plasma along the magnetic field lines. In particular, the technique has been employed to simulate the formation and long-term evolution of large-amplitude electrostatic waves experimentally observed in electron plasmas confined in a Penning trap [W. Bertsche, J. Fajans, L. Friedland, Phys. Rev. Lett. 91 (2003) 265003]. Due to the peculiar features of the physical system, namely the existence of different time scales and the presence of a perturbative oscillating potential, ad hoc numerical techniques have been developed. In particular, with a suitable radial decomposition all important two-dimensional phenomena are fully taken into account while keeping the computational effort to that of a standard one-dimensional PIC codes. Moreover, a novel particle loading technique (ergodic loading) has been developed, which ensures a significant reduction of numerical noise. The results obtained with the present technique are in excellent agreement with the experiments [F. Peinetti, W. Bertsche, J. Fajans, J. Wurtele, L. Friedland, Phys. Plasmas 12 (2005) 062112]. Moreover, results presented here furnish clear evidences of the close relationship between the observed nonlinear structures and the Bernstein–Greene–Kruskal modes.

Theory of relativistic phase-space holes in a hot-electron-positron-ion plasma

A theoretical and numerical study of phase-space holes in a relativistically hot-electron-positron-ion plasma is presented, and their potential and density profiles are calculated numerically for different sets of parameters. The phase-space holes are Bernstein–Greene–Kruskal modes in which particles are trapped in the self-consistent electrostatic potential. Relativistic effects increase the size of the phase-space hole and the amplitude of the associated electrostatic potential. In a pure electron-positron plasma, the phase-space holes must have a minimum speed close to the particle thermal speed. The presence of positively charged ions makes the holes smaller, and stabilizes the holes so that they can propagate with smaller speeds. A numerical Vlasov simulation demonstrates the stability of the holes and that they tend to interact and merge to form new holes.

Simulation study of surfing acceleration in magnetized space plasmas

We present a numerical study of the surfing mechanism in which electrons are trapped in Bernstein–Greene–Kruskal (BGK) modes, and are accelerated across the magnetic field direction by the Lorentz force in magnetized space plasmas. The BGK modes are the product of an ion-beam Buneman instability that excites large-amplitude electrostatic upper-hybrid waves in the plasma. Our study, which is performed with particle-in-cell (PIC) and Vlasov codes, reveals the stability of the BGK mode as a function of the magnetic field strength and the ion beam speed. It is found that the surfing acceleration is more effective for a weaker magnetic field owing to the longer lifetime of the BGK modes. The importance of our investigation to electron acceleration in astrophysical environments has been emphasized.

A Study of the Nonlinear Landau Damping in the Fourier Transformed Velocity Space

Abstract The collisionless plasma oscillations in the nonlinear regime (the nonlinear Landau damping) are investigated numerically in the Fourier transformed velocity space. A scattering process of a Maxwellian initial disturbance is again observed, as in the linear case. It is found that at the moment the initially excited Landau damped modes stop damping they are converted into Van Kampen‐Case eigenmodes. These modes create a quasi‐stationary wave pattern in the Fourier transformed phase plane, characteristic for a BGK mode. The scattered disturbance preserves its identity but exercises, on the background of the wave pattern, a complicated motion and undergoes a strong dispersion, up to complete disappearance.

Driven phase space holes and synchronized Bernstein, Greene, and Kruskal modes

The excitation of synchronized Bernstein, Greene, and Kruskal (BGK) modes in a pure electron plasma confined in Malmberg–Penning trap is studied. The modes are excited by controlling the frequency of an oscillating external potential. Initially, the drive resonates with, and phase-locks to, a group of axially bouncing electrons in the trap. These initially phase-locked electrons remain phase-locked (in “autoresonance”) during a subsequent downward chirp of the external potential’s oscillation frequency. Only a few new particles are added to the resonant group as the frequency, and, hence, the resonance, moves to lower velocities in phase space. Consequently, the downward chirp creates a charge density perturbation (a hole) in the electron phase space distribution. The hole oscillates in space, and its associated induced electric field constitutes a BGK mode synchronized with the drive. The size of the hole in phase space, and thus the amplitude of the mode, are largely controlled by only two external parameters: the driving frequency and amplitude. A simplified kinetic theory of this excitation process is developed. The dependence of the excited BGK mode amplitude on the driving frequency chirp rate and other plasma parameters is discussed and theoretical predictions are compared with recent experiments and computer simulations.

Instability of phase space structures

Adiabatic analysis of self-consistent dynamics of a phase space structure surrounded by passing particles under the assumption of a slowly changing amplitude and frequency has revealed that the system can reach a point where adiabatic theory breaks down. Linear perturbative analysis shows that an instability is triggered at precisely these points. Numerical runs were performed to test the adiabatic theory and the instability analysis of a Bernstein–Greene–Kruskal mode for the bump-on-tail problem. First a passing particle distribution function was used that has a constant slope with respect to the action variable. Then a flat passing particle distribution function, which has nearly the same instability criterion and growth rate as the first case was studied and it produces a precise comparison with the results from numerical simulations until instability sets in. Afterwards smaller phase space structures still persist and frequency sweeping continues at a lower rate.

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.

Some theoretical models for solitary structures of boundary layer waves

Abstract. Solitary electrostatic structures have been observed in and around auroral zone field lines at various altitudes ranging from close to the ionosphere, to the magnetopause low-latitude boundary layer (LLBL) and cusp on the dayside, and to the plasma sheet boundary layer on the night-side. In this review, various models based on solitons/double layers and BGK modes or phase space holes are discussed in order to explain the characteristics of these solitary structures.

Electrostatic solitary waves as collective charges in a magnetospheric plasma: Physical structure and properties of Bernstein–Greene–Kruskal (BGK) solitons

The electrostatic pulses recorded by the Geotail spacecraft and labeled electrostatic solitary waves (ESW) are considered within the framework of Bernstein–Greene–Kruskal (BGK) solitons. The main goal of this paper is to develop sufficiently general and simple theoretical models of the nonlinear wave perturbations under study. Such models are necessary for understanding the physical structure and properties of the observed waves as well as for the treatment of the extensive experimental material, which has been accumulated on the ESW waveforms and for meaningful data processing. Examples of localized BGK modes described previously in literature are difficult to compare with the experimental data due to their particular nature. In contrast to these particular models, the general approach developed in the article applies to arbitrary particle distributions of the background plasma, velocities of the BGK solitons and wide variety of the recorded ESW waveforms. Relying on the theory of BGK solitons, with an emphasis on their general properties, the work contains a consistent analytical description and physical interpretation of ESW as effective spatial charges shielded by a collisionless plasma. The general analysis of localized BGK waves leads to a number of new results, such as unified physical models of ESW based on the additivity of the distribution function, as well as energetic relations and universal restrictions on the parameters and waveforms of the solitons. The new models and physical interrelations reveal universal features of the BGK soliton structure and allow a direct juxtaposition with the observations. The established interconnections between the physical characteristics of the waves agree well with the Geotail data on ESW waveforms. We also discuss less studied nonstationary phenomena, among which are wave emission by the BGK solitons, mechanisms of electron phase density hole interaction and certain questions of stability. Some of the most typical examples of the various BGK solutions described in the article are useful for a correlation between the theory and measurements as well as for an identification of the ESW waveforms observed in the magnetosphere.

Nonlinear dust Bernstein–Greene–Kruskal modes in charge-varying dusty plasmas

Nonlinear dust Bernstein–Greene–Kruskal (BGK) modes are investigated in a charge-varying dusty plasma. It is found that highly localized structures, solely due to the dust charge variation, can exist. The dust BGK soliton suffers the well-known anomalous damping, the importance of which is roughly proportional to the dust grain velocity. This dissipation causes the soliton amplitude to decay algebraically and the conservation of “soliton mass” leads to the development of a noise tail.

Nonlinear oscillations in dusty plasmas with variable charges on dust particles

Recent analysis of the excitation of dust Bernstein–Greene–Kruskal modes [Tribeche et al., Phys. Plasmas 7, 4013 (2000)] is extended to include self-consistently the dust charge variation. The grain charge becomes a new self-consistent dynamical variable, leading to some new and interesting results such as threshold lowering, mode damping, and spatially localized nonlinear structures.

A self-consistent trapping model of driven electron plasma waves and limits on stimulated Raman scatter

A Vlasov equation based model is used to determine various regimes of electron plasma wave response to a source appropriate to stimulated scatter in a laser hot spot. It incorporates trapped particle effects such as the standard nonlinear frequency shift, extended beyond the weak regime, and a reduction of damping à la Zakharov and Karpman [V. E. Zakharov and V. I. Karpman, JETP 16, 351 (1963)]. The results are consistent with those of Holloway and Dorning [J. P. Holloway and J. J. Dorning, Phys. Rev. A 44, 3856 (1991)] for small amplitude Bernstein–Greene–Kruskal modes. This leads to the prediction that as long as kλD⩾0.53 for a background Maxwellian distribution function, e.g., a 5 keV plasma with ne/nc⩽0.075, anomalously large backward stimulated Raman scatter can be excluded. A similar analysis leads to density limits on stimulated Brillouin scatter.

A NUMERICAL STUDY OF LINEAR AND NONLINEAR SUPERPOSITIONS OF BGK MODES

In this paper we investigate numerically a class of solutions of the one-dimensional Vlasov-Poison system evolving from initial conditions that correspond to nonlinear superpositions of BGK modes, of the type introduced by Buchanan and Dorning (see M. Buchanan and J. J. Dorning, Phys. Rev. Letters 70(1993) 3732). The results show that these nonlinear superpositions give nonlinear steady-state oscillations that are usually observed only asymptotically in time. Small scale structures and chaotic layers are observed in phase space near the separatrices of the trapped and untrapped particle regions.