BGK Modes Research Articles

High-resolution particle-in-cell simulations of two-dimensional Bernstein–Greene–Kruskal modes

We present two-dimensional (2D) particle-in-cell (PIC) simulations of 2D Bernstein–Greene–Kruskal modes, which are exact nonlinear steady-state solutions of the Vlasov–Poisson equations, on a 2D plane perpendicular to a background magnetic field, with a cylindrically symmetric electric potential localized on the plane. PIC simulations are initialized using analytic electron distributions and electric potentials from the theory. We confirm the validity of such solutions using high-resolutions up to a 20482 grid. We show that the solutions are dynamically stable for a stronger background magnetic field, while keeping other parameters of the model fixed, but become unstable when the field strength is weaker than a certain value. When a mode becomes unstable, we observe that the instability begins with the excitation of azimuthal electrostatic waves that ends with a spiral pattern.

Bernstein–Greene–Kruskal Ion Modes in Dusty Space Plasmas Application in Saturn’s Magnetosphere

Frequent observations of ion beams moving out from Saturn’s plasma environment hints at the generation of ion Bernstein–Greene–Kruskal (BGK) modes. As the plasma environments of Saturn and its moon Enceladus are characterized by the ubiquitous presence of massive negatively charged dust particles, the existing BGK theory for electron-ion plasma models cannot address this scenario. This manuscript develops a theoretical model for studying ion BGK modes in dusty plasmas. The analysis reveals that the presence of dust in the plasma enhances the stability of BGK modes. As the dust density increases, the effect of other parameters on stability, such as the electron temperature, becomes negligible. The model is developed by assuming that electrons and ions follow a kappa distribution, featuring a long tail trend in the superthermal component, in agreement with observations. Different scenarios with either electrons or ions obeying a Maxwell or kappa distribution function have been considered. A thorough analysis of the trapped ion distribution function considering various combinations indicates that a plasma where electrons are in thermal equilibrium and ions follow kappa distribution is the least favorable system for the generation of BGK modes.

Nonlinear adiabatic electron plasma waves. II. Applications

In this article, we use the general theory derived in Paper I [M. Tacu and D. Bénisti, Phys. Plasmas 29, 052108 (2022)] in order to address several long-standing issues regarding nonlinear electron plasma waves (EPWs). First, we discuss the relevance and practical usefulness of stationary solutions to the Vlasov–Poisson system, the so-called Bernstein–Greene–Kruskal modes, to model slowly varying waves. Second, we derive an upper bound for the wave breaking limit of an EPW growing in an initially Maxwellian plasma. Moreover, we show a simple dependence of this limit as a function of kλD, with k being the wavenumber and λD the Debye length. Third, we explicitly derive the envelope equation ruling the evolution of a slowly growing plasma wave, up to an amplitude close to the wave breaking limit. Fourth, we estimate the growth of the transverse wavenumbers resulting from wavefront bowing by solving the nonlinear, nonstationary, ray tracing equations for the EPW, together with a simple model for stimulated Raman scattering.

Trapped particle instability in : I homogeneous Vlasov plasmas

Kinetic instability due to trapped particles in non-linear evolution of large amplitude electrostatic waves i.e BGK (Bernstein-Greene-Kruskal) mode is investigated in a one dimensional, collisionless, homogeneous Vlasov-Poisson plasma with immobile ions and kinetic electrons. Considering a cosinusoidal density perturbation with non-linear perturbation amplitude α = 0.05 as starting point, we have demonstrated that long time BGK solutions with scale (where corresponds to the longest length scale in the system) are not stable due to Landau growth in the secondary sideband modes. This growth in the sideband modes is primarily related to trapped particles in the potential well of the perturbation which leads to trapped particle instability when the amplitude of the sidebands become comparable to that of the primary nonlinear mode. In the past, to explain trapped particle instability, two physical mechanisms were held responsible, namely, wave particle interaction phenomenon at resonance location v ϕ = ω/k and mode coupling interaction phenomenon between primary and secondary sideband modes. We have demonstrated and emphasized both the phenomena in our study. In addition, effect of primary mode perturbation amplitude on trapped particle instability is investigated and growth rates corresponding to secondary sideband modes are reported alongwith a new scaling law for detrapping time as a function of perturbation amplitude. Also, scaling of growth rates corresponding to lower and upper sideband modes with respect to perturbation amplitude is obtained which demonstrates the existence of a transition point at α = 0.075 between small and large perturbation amplitudes and growth rate scales as γ ∼ α 2 for α < 0.075 whereas γ ∼ α 1/2 for α > 0.075.

Trapped particle instability in : II inhomogeneous Vlasov plasmas

In Part I of the companion paper [], we have extensively investigated the instability due to trapped particle by large amplitude electrostatic waves i.e BGK (Bernstein-Greene-Kruskal) mode in one dimensional, collisionless, homogeneous Vlasov-Poisson plasma. In continuation of the work, in this Part II, we present the trapped particle instability in the presence of finite non-uniform background inhomogeneity of scale with equilibrium inhomogeneity amplitude A. In the process, we demonstrate and emphasize importance of the mode coupling phenomenon alongwith the wave-particle interaction phenomenon in order to understand the physical mechanisms, which are responsible for the growth of upper and lower sideband modes in inhomogeneous plasmas. It is shown that the larger the amplitude of equilibrium inhomogeneity A, smaller is the perturbation amplitude α for which destabilization sets in. Also, in addition, effect of equilibrium inhomogeneity amplitude on trapped particle instability, asymmetry in growth rates of upper and lower secondary sideband modes, alongwith novel scaling of detrapping time with respect to equilibrium inhomogeneity amplitude are reported.

Effect of background fluctuation on velocity diagnostics by Mach probe

The effect of background fluctuation on velocity diagnostics is discussed and studied. The kinetic theory of Mach probe (MP) and the theory of BGK mode are combined to evaluate how the measurement of MP is affected by electrostatics fluctuation. It is found that the quantity of speed by the MP model is closer to the effective velocity in the picture of momentum flux rather than the real mean velocity, while, with high fluctuation, the fitting parameter of MP’s exponential formula should be corrected.

Kinetic flux ropes: Bernstein–Greene–Kruskal modes for the Vlasov–Poisson–Ampère system

Electrostatic structures have been observed in many regions of space plasmas, including the solar wind, the magnetosphere, and the auroral acceleration region. One possible theoretical description of some of these structures is the concept of Bernstein–Greene–Kruskal (BGK) modes, which are exact nonlinear steady-state solutions of the Vlasov–Poisson system of equations in collisionless kinetic theory. We generalize the exact solutions of two-dimensional BGK modes in a magnetized plasma with a finite magnetic field strength [Ng et al., Phys. Plasmas 13, 055903 (2006)] to cases with azimuthal magnetic fields so that these structures carry the electric current as well as the steady electric and magnetic fields. Such nonlinear solutions now exactly satisfy the Vlasov–Poisson–Ampère system of equations. Explicit examples with either positive or negative electric potential structure are provided.

Generation of low frequency plasma waves after the wave-breaking

Spatio-temporal evolution of a non-relativistic electrostatic waves in a cold plasma has been studied in the wave-breaking regime using a 1D particle-in-cell simulation. It is found that, the plasma gets heated after the wave-breaking, but a fraction of initial energy always remains with the remnant wave in the form of BGK mode in the warm plasma. An interesting finding of this work is that the frequency of the resultant BGK wave, is found be below electron-plasma frequency, which decreases further with increasing initial amplitude. Moreover, the acceleration mechanism after the wave-breaking is also found to be different from the previous work. In order to explain the results observed in the numerical experiments, a simplified theoretical model is constructed, which exhibits a good agreement with the simulation. These investigations may have relevance for wake-field acceleration experiments.

Impact of energetic particle orbits on long range frequency chirping of BGK modes

Long range frequency chirping of Bernstein–Greene–Kruskal modes, whose existence is determined by the fast particles, is investigated in cases where these particles do not move freely and their motion is bounded to restricted orbits. A nonuniform equilibrium magnetic field is included into the bump-on-tail instability problem of a plasma wave. The parallel field gradients account for the existence of different orbit topologies of energetic particles. With respect to fast particles dynamics, the extended model captures the range of particles motion (trapped/passing) with energy and thus represents a more realistic 1D picture of the long range sweeping events observed for weakly damped modes, e.g. global Alfven eigenmodes, in tokamaks. The Poisson equation is solved numerically along with bounce averaging the Vlasov equation in the adiabatic regime. We demonstrate that the shape and the saturation amplitude of the nonlinear mode structure depends not only on the amount of deviation from the initial eigenfrequency but also on the initial energy of the resonant electrons in the equilibrium magnetic field. Similarly, the results reveal that the resonant electrons following different equilibrium orbits in the nonuniform field lead to different rates of frequency evolution. As compared to the previous model (Breizman 2010 Nucl. Fusion 50 084014), it is shown that the frequency sweeps with lower rates. The additional physics included in the model enables a more complete 1D description of the range of phenomena observed in experiments.

Simulations of kinetic electrostatic electron nonlinear (KEEN) waves with variable velocity resolution grids and high-order time-splitting

KEEN waves are non-stationary, nonlinear, self-organized asymptotic states in Vlasov plasmas. They lie outside the precepts of linear theory or perturbative analysis, unlike electron plasma waves or ion acoustic waves. Steady state, nonlinear constructs such as BGK modes also do not apply. The range in velocity that is strongly perturbed by KEEN waves depends on the amplitude and duration of the ponderomotive force generated by two crossing laser beams, for instance, used to drive them. Smaller amplitude drives manage to devolve into multiple highly-localized vorticlets, after the drive is turned off, and may eventually succeed to coalesce into KEEN waves. Fragmentation once the drive stops, and potential eventual remerger, is a hallmark of the weakly driven cases. A fully formed (more strongly driven) KEEN wave has one dominant vortical core. But it also involves fine scale complex dynamics due to shedding and merging of smaller vortical structures with the main one. Shedding and merging of vorticlets are involved in either case, but at different rates and with different relative importance. The narrow velocity range in which one must maintain sufficient resolution in the weakly driven cases, challenges fixed velocity grid numerical schemes. What is needed is the capability of resolving locally in velocity while maintaining a coarse grid outside the highly perturbed region of phase space. We here report on a new Semi-Lagrangian Vlasov-Poisson solver based on conservative non-uniform cubic splines in velocity that tackles this problem head on. An additional feature of our approach is the use of a new high-order time-splitting scheme which allows much longer simulations per computational effort. This is needed for low amplitude runs. There, global coherent structures take a long time to set up, such as KEEN waves, if they do so at all. The new code’s performance is compared to uniform grid simulations and the advantages are quantified. The birth pains associated with weakly driven KEEN waves are captured in these simulations. Canonical KEEN waves with ample drive are also treated using these advanced techniques. They will allow the efficient simulation of KEEN waves in multiple dimensions, which will be tackled next, as well as generalizations to Vlasov-Maxwell codes. These are essential for pursuing the impact of KEEN waves in high energy density plasmas and in inertial confinement fusion applications. More generally, one needs a fully-adaptive grid-in-phase-space method which could handle all small vorticlet dynamics whether pealing off or remerging. Such fully adaptive grids would have to be computed sparsely in order to be viable. This two-velocity grid method is a concrete and fruitful step in that direction.

Multicomponent kinetic simulation of Bernstein–Greene–Kruskal modes associated with ion acoustic and dust-ion acoustic excitations in electron-ion and dusty plasmas

A series of numerical simulations based on a recurrence-free Vlasov kinetic algorithm presented earlier [Abbasi et al., Phys. Rev. E 84, 036702 (2011)] are reported. Electron-ion plasmas and three-component (electron-ion-dust) dusty, or complex, plasmas are considered, via independent simulations. Considering all plasma components modeled through a kinetic approach, the nonlinear behavior of ionic scale acoustic excitations is investigated. The focus is on Bernstein–Greene–Kruskal (BGK) modes generated during the simulations. In particular, we aim at investigating the parametric dependence of the characteristics of BGK structures, namely of their time periodicity (τtrap) and their amplitude, on the electron-to-ion temperature ratio and on the dust concentration. In electron-ion plasma, an exponential relation between τtrap and the amplitude of BGK modes and the electron-to-ion temperature ratio is observed. It is argued that both characteristics, namely, the periodicity τtrap and amplitude, are also related to the size of the phase-space vortex which is associated with BGK mode creation. In dusty plasmas, BGK modes characteristics appear to depend on the dust particle density linearly.

Numerical study of one-dimensional Vlasov–Poisson equations for infinite homogeneous stellar systems

We consider the gravitational Vlasov–Poisson (VP), or the so-called collisionless Boltzmann–Poisson equations for the self-gravitating collisionless stellar systems. We compute the solutions using a high-order discontinuous Galerkin method for the Vlasov equation, and the classical representation by Green’s function for the Poisson equation in the one-dimensional setting. We study both the case of damping and Jeans instability depending on the wavenumbers, which are taken to be greater than or less than the Jeans wavenumber, respectively. The method is shown to be stable, accurate and conservative. We report for the first time the BGK modes for the gravitational VP system, and we study the behavior of solutions associated with these various wavenumbers.

Nonlinear Mirror and Weibel modes: peculiarities of quasi-linear dynamics

Abstract. A theory for nonlinear evolution of the mirror modes near the instability threshold is developed. It is shown that during initial stage the major instability saturation is provided by the flattening of the velocity distribution function in the vicinity of small parallel ion velocities. The relaxation scenario in this case is accompanied by rapid attenuation of resonant particle interaction which is replaced by a weaker adiabatic interaction with mirror modes. The saturated plasma state can be considered as a magnetic counterpart to electrostatic BGK modes. After quasi-linear saturation a further nonlinear scenario is controlled by the mode coupling effects and nonlinear variation of the ion Larmor radius. Our analytical model is verified by relevant numerical simulations. Test particle and PIC simulations indeed show that it is a modification of distribution function at small parallel velocities that results in fading away of free energy driving the mirror mode. The similarity with resonant Weibel instability is discussed.

Averaged variational principle for autoresonant Bernstein–Greene–Kruskal modes

Whitham’s averaged variational principle is applied in studying dynamics of formation of autoresonant (continuously phase-locked) Bernstein–Greene–Kruskal (BGK) modes in a plasma driven by a chirped frequency ponderomotive wave. A flat-top electron velocity distribution is used as a model allowing a variational formulation within the water bag theory. The corresponding Lagrangian, averaged over the fast phase variable yields evolution equations for the slow field variables, allows uniform description of all stages of excitation of driven-chirped BGK modes, and predicts modulational stability of these nonlinear phase-space structures. Numerical solutions of the system of slow variational equations are in good agreement with Vlasov–Poisson simulations.

Nonlinear travelling waves in energetic particle phase space

An exact nonlinear solution is found for long-time behaviour of spontaneously formed phase space clumps/holes in dissipative plasmas with a population of energetic particles. This solution represents a Bernstein–Greene–Kruskal mode with slowly varying shape and velocity. It describes a continuous transformation of a plasma eigenmode excited just above the instability threshold into an energetic particle mode with a significantly different frequency. An electrostatic bump-on-tail instability is chosen to illustrate the analysis. However, generality of the resonant particle dynamics makes the described approach applicable to other resonance-dominated instabilities, including rapid frequency-sweeping events for Alfvénic modes in tokamaks.

Comment on “Weakly collisional Landau damping and three-dimensional Bernstein–Greene–Kruskal modes: New results on old problems” [Phys. Plasmas 13, 055903 (2006)

In their paper on two-dimensional/three-dimensional 2D/3D Bernstein–Greene–Kruskal BGK modes, Ng et al. made a couple of very brief references to my 2005 paper on BGK modes in a magnetized plasma Ref. 2 , referring it with others as one with gyrokinetic solutions of 3D BGK for a “strong B” their characterization of the fact that the solution breaks down when B is sufficiently weak . They did not reference my 2002 work, which earlier analyzed BGK modes in a magnetized plasma with unexplicable typographical error subsequently noted , an analysis which was expanded in my 2005 paper. Furthermore, while Ng et al. did not specify the meaning of strong B, in both references to Ref. 2 they immediately lumped it in with their discussion of the published B→ analyses. In fact, the strong-B solutions presented in Ref. 2 are clearly more general than the B→ case and only reduce to that one-dimensional 1D case when the limit B→ is specifically taken in the general solutions derived. The claim is incorrect in the later discussion of their paper when, in referring to these ”strong-B” solutions, they claimed that the “physical reason such solutions exist is related to the 1D aspect of the theory.” Furthermore, for the 3D strong-B solutions I derived an angular criterion with respect to the magnetic field direction for the existence of magnetized BGK modes,

Response to “Comment on ‘Weakly collisional Landau damping and three-dimensional Bernstein–Greene–Kruskal modes: New results on old problems’ ” [Phys. Plasmas 17, 054701 (2010)

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Nonlinear stationary solutions of the Wigner and Wigner–Poisson equations

Exact nonlinear stationary solutions of the one-dimensional Wigner and Wigner–Poisson equations in the terms of the Wigner functions that depend not only on the energy but also on position are presented. In this way, the Bernstein–Greene–Kruskal modes of the classical plasma are adapted for the quantum formalism in the phase space. The solutions are constructed for the case of a quartic oscillator potential, as well as for the self-consistent Wigner–Poisson case. Conditions for well-behaved physically meaningful equilibrium Wigner functions are discussed.

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.

Damping of Bernstein–Greene–Kruskal modes in collisional plasmas

In this paper, the effect of Coulomb collisions on the stability of Bernstein–Greene–Kruskal (BGK) modes [I. B. Bernstein, J. M. Greene, and M. D. Krukal, Phys. Rev. 108, 546 (1957)] is analyzed by comparing the numerical results of collisional particle-in-cell (PIC) simulations with the theoretical predictions by Zakharov and Karpman [V. E. Zakharov and V. I. Karpman, Sov. Phys. JETP 16, 351 (1963)], for the collisional damping of nonlinear plasma waves. In the absence of collisions, BGK modes are undamped nonlinear electrostatic oscillations, solutions of the Vlasov–Poisson equations; in these structures nonlinearity manifests as the formation of a plateau in the resonant region of the particle distribution function, due to trapping of resonant particles, thus preventing linear Landau damping. When particle-particle Coulomb collisions are effective, this plateau is smoothed out since collisions drive the velocity distribution towards the Maxwellian shape, thus destroying the BGK structure. As shown by Zakharov and Karpman in 1963, under certain assumptions, an exponential time decay with constant damping rate is predicted for the electric field amplitude and a linear dependence of the damping rate on the collision frequency is found. In this paper, the theory by Zakharov and Karpman is revisited and the effects of collisions on the stability of BGK modes and on the long time evolution of nonlinear Landau damping are numerically investigated. The numerical results are obtained through a collisional PIC code that reproduces a physical phenomenology also observed in recent experiments with trapped pure electron plasmas.