Vlasov-Poisson Simulations Research Articles

Low Mach-number collisionless electrostatic shocks and associated ion acceleration

The existence and properties of low Mach-number () electrostatic collisionless shocks are investigated with a semi-analytical solution for the shock structure. We show that the properties of the shock obtained in the semi-analytical model can be well reproduced in fully kinetic Eulerian Vlasov–Poisson simulations, where the shock is generated by the decay of an initial density discontinuity. Using this semi-analytical model, we study the effect of the electron-to-ion temperature ratio and the presence of impurities on both the maximum shock potential and the Mach number. We find that even a small amount of impurities can influence the shock properties significantly, including the reflected light ion fraction, which can change several orders of magnitude. Electrostatic shocks in heavy ion plasmas reflect most of the hydrogen impurity ions.

Verification of 2D × 2D and Two-Species Vlasov-Poisson Solvers,

In [2], 1D×1D two-species Vlasov-Poisson simulations are performed by the semi-Lagrangian method. Thanks to a classical first order dispersion analysis, we are able to check the validity of their simulations; the extension to second order is performed and shown to be relevant for explaining further details. In order to validate multi-dimensional effects, we propose a 2D × 2D single species test problem that has true 2D effects coming from the sole second order dispersion analysis. Finally, we perform, in the same code, full 2D × 2D non linear two-species simulations with mass ratio 0:01, and consider the mixing of semi-Lagrangian and Particle-in-Cell methods.

Multidimensional Vlasov–Poisson Simulations with High-order Monotonicity- and Positivity-preserving Schemes

Abstract We develop new numerical schemes for Vlasov–Poisson equations with high-order accuracy. Our methods are based on a spatially monotonicity-preserving (MP) scheme and are modified suitably so that the positivity of the distribution function is also preserved. We adopt an efficient semi-Lagrangian time integration scheme that is more accurate and computationally less expensive than the three-stage TVD Runge–Kutta integration. We apply our spatially fifth- and seventh-order schemes to a suite of simulations of collisionless self-gravitating systems and electrostatic plasma simulations, including linear and nonlinear Landau damping in one dimension and Vlasov–Poisson simulations in a six-dimensional phase space. The high-order schemes achieve a significantly improved accuracy in comparison with the third-order positive-flux-conserved scheme adopted in our previous study. With the semi-Lagrangian time integration, the computational cost of our high-order schemes does not significantly increase, but remains roughly the same as that of the third-order scheme. Vlasov–Poisson simulations on mesh grids have been successfully performed on a massively parallel computer.

A high order semi-Lagrangian discontinuous Galerkin method for Vlasov–Poisson simulations without operator splitting

In this paper, we develop a high order semi-Lagrangian (SL) discontinuous Galerkin (DG) method for nonlinear Vlasov–Poisson (VP) simulations without operator splitting. In particular, we combine two recently developed novel techniques: one is the high order non-splitting SLDG transport method (Cai et al. (2017) [4]), and the other is the high order characteristics tracing technique proposed in Qiu and Russo (2017) [29]. The proposed method with up to third order accuracy in both space and time is locally mass conservative, free of splitting error, positivity-preserving, stable and robust for large time stepping size. The SLDG VP solver is applied to classic benchmark test problems such as Landau damping and two-stream instabilities for VP simulations. Efficiency and effectiveness of the proposed scheme is extensively tested. Tremendous CPU savings are shown by comparisons between the proposed SL DG scheme and the classical Runge–Kutta DG method.

Extreme driven ion acoustic waves

The excitation of large amplitude, strongly nonlinear ion acoustic waves from trivial equilibrium by a chirped frequency drive is discussed. Under certain conditions, after passage through the linear resonance in this system, the nonlinearity and the variation of parameters work in tandem to preserve the phase-locking with the driving wave via excursion of the excited ion acoustic wave in its parameter space, yielding controlled growth of the wave amplitude. We study these autoresonant waves via a fully nonlinear warm fluid model and predict the formation of sharply peaked (extreme) ion acoustic excitations with local ion density significantly exceeding the unperturbed plasma density. The driven wave amplitude is bound by the kinetic wave-breaking, as the local maximum fluid velocity of the wave approaches the phase velocity of the drive. The Vlasov-Poisson simulations are used to confirm the results of the fluid model, and Whitham's averaged variational principle is applied for analyzing the evolution of autoresonant ion acoustic waves.

Simulation study of Landau damping near the persisting to arrested transition

A one-dimensional electrostatic filtered Vlasov–Poisson simulation study is discussed. The transition from persisting to arrested Landau damping that is produced by increasing the strength of a sinusoidal perturbation on a background Vlasov–Poisson equilibrium is explored. Emphasis is placed on observed features of the electron phase-space distribution when the perturbation strength is near the transition value. A single ubiquitous waveform is found perturbing the space-averaged phase-space distribution at almost any time in all of the simulations; the sole exception is the saturation stage that can occur at the end of the arrested damping scenario. This waveform contains relatively strong, very narrow structures in velocity bracketing $\pm v_{\text{res}}$ – the velocities at which electrons must move to traverse the dominant field mode wavelength in one of its oscillation periods – and propagating with $\pm v_{\text{res}}$ respectively. Local streams of electrons are found in these structures crossing the resonant velocities from low speed to high speed during Landau damping and from high speed to low speed during Landau growth. At the arrest time, when the field strength is briefly constant, these streams vanish. It is conjectured that the expected transfer of energy between electrons and field during Landau growth or damping has been visualized for the first time. No evidence is found in the phase-space distribution to support recent well-established discoveries of a second-order phase transition in the electric field evolution. While trapping is known to play a role for larger perturbation strengths, it is shown that trapping plays no role at any time in any of the simulations near the transition perturbation strength.

Kinetic simulations of ladder climbing by electron plasma waves.

The energy of plasma waves can be moved up and down the spectrum using chirped modulations of plasma parameters, which can be driven by external fields. Depending on whether the wave spectrum is discrete (bounded plasma) or continuous (boundless plasma), this phenomenon is called ladder climbing (LC) or autoresonant acceleration of plasmons. It was first proposed by Barth etal. [Phys. Rev. Lett. 115, 075001 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.075001] based on a linear fluid model. In this paper, LC of electron plasma waves is investigated using fully nonlinear Vlasov-Poisson simulations of collisionless bounded plasma. It is shown that, in agreement with the basic theory, plasmons survive substantial transformations of the spectrum and are destroyed only when their wave numbers become large enough to trigger Landau damping. Since nonlinear effects decrease the damping rate, LC is even more efficient when practiced on structures like quasiperiodic Bernstein-Greene-Kruskal (BGK) waves rather than on Langmuir waves per se.

Langmuir wave filamentation in the kinetic regime. I. Filamentation instability of Bernstein-Greene-Kruskal modes in multidimensional Vlasov simulations

A nonlinear Langmuir wave in the kinetic regime kλD≳0.2 may have a filamentation instability, where k is the wavenumber and λD is the Debye length. The nonlinear stage of that instability develops into the filamentation of Langmuir waves which in turn leads to the saturation of the stimulated Raman scattering in laser-plasma interaction experiments. Here, we study the linear stage of the filamentation instability of the particular family (H. A. Rose and D. A. Russell, Phys. Plasmas 8, 4784 (2001)) of Bernstein-Greene-Kruskal (BGK) modes (I. B. Bernstein et al., Phys. Rev. 108, 546 (1957)) that is a bifurcation of the linear Langmuir wave. Performing direct 2 + 2D Vlasov–Poisson simulations of collisionless plasma, we find the growth rates of oblique modes of the electric field as a function of BGK's amplitude, wavenumber, and the angle of the oblique mode's wavevector relative to the BGK's wavevector. Simulation results are compared to theoretical predictions.

Effects of ion motion on linear Landau damping

The effects of ion motion on Landau damping has been studied by the use of one-dimensional Vlasov-Poisson simulation. It is shown that the ion motion may significantly change the development of the linear Landau damping. When the ion mass is multiple of proton mass, its motion will halt the linear Landau damping at some time due to the excitation of ion acoustic waves. The latter will dominate the system evolution at the later stage and hold a considerable fraction of the total energy in the system. With very small ion mass, such as in electron-positron plasma, the ion motion can suppress the linear Landau damping very quickly. When the initial field amplitude is relatively high such as with the density perturbation amplitude δn/n0 > 0.1, the effect of ion motion on Landau damping is found to be weak or even ignorable.

Collisional effects on the numerical recurrence in Vlasov-Poisson simulations

The initial state recurrence in numerical simulations of the Vlasov-Poisson system is a well-known phenomenon. Here, we study the effect on recurrence of artificial collisions modeled through the Lenard-Bernstein operator [A. Lenard and I. B. Bernstein, Phys. Rev. 112, 1456–1459 (1958)]. By decomposing the linear Vlasov-Poisson system in the Fourier-Hermite space, the recurrence problem is investigated in the linear regime of the damping of a Langmuir wave and of the onset of the bump-on-tail instability. The analysis is then confirmed and extended to the nonlinear regime through an Eulerian collisional Vlasov-Poisson code. It is found that, despite being routinely used, an artificial collisionality is not a viable way of preventing recurrence in numerical simulations without compromising the kinetic nature of the solution. Moreover, it is shown how numerical effects associated to the generation of fine velocity scales can modify the physical features of the system evolution even in nonlinear regime. This means that filamentation-like phenomena, usually associated with low amplitude fluctuations contexts, can play a role even in nonlinear regime.

Critical density for Landau damping in a two-electron-component plasma

The asymptotic evolution of an initial perturbation in a collisionless two-electron-component plasma with different temperatures is studied numerically. The transition between linear and nonlinear damping regimes is determined by slowly varying the density of the secondary electron-component using high-resolution Vlasov-Poisson simulations. It is shown that, for fixed amplitude perturbations, this transition behaves as a critical phenomenon with time scales and field amplitudes exhibiting power-law dependencies on the threshold density, similar to the critical amplitude behavior in a single-component plasma.

A finite mass based method for Vlasov–Poisson simulations

A method for the numerical simulation of plasma dynamics using discrete particles is introduced. The shape function kinetics (SFK) method is based on decomposing the mass into discrete particles using shape functions of compact support. The particle positions and shape evolve in response to internal velocity spread and external forces. Remapping is necessary in order to maintain accuracy and two strategies for remapping the particles are discussed. Numerical simulations of standard test problems illustrate the advantages of the method which include very low noise compared to the standard particle-in-cell technique, inherent positivity, large dynamic range, and ease of implementation.

Dory–Guest–Harris instability as a benchmark for continuum kinetic Vlasov–Poisson simulations of magnetized plasmas

The Dory–Guest–Harris instability is demonstrated to be a well-suited benchmark for continuum kinetic Vlasov–Poisson algorithms. The instability is a special case of perpendicularly-propagating kinetic electrostatic waves in a warm uniformly magnetized plasma. A complete derivation of the closed-form linear theory dispersion relation for the instability is presented. The electric field growth rates and oscillation frequencies specified by the dispersion relation provide concrete measures against which simulation results can be quantitatively compared. A fourth-order continuum kinetic algorithm is benchmarked against the instability, and is demonstrated to have good convergence properties and close agreement with theoretical growth rate and oscillation frequency predictions. Second-order accurate simulations are also shown to be consistent with theoretical predictions, but require higher resolution for convergence. The Dory–Guest–Harris instability benchmark extends the scope of current standard test problems by providing a substantive means of validating continuum kinetic simulations of magnetized plasmas in higher-dimensional 3D (x,vx,vy) phase space. The linear theory analysis, initial conditions, algorithm description, and comparisons between theoretical predictions and simulation results are presented.

Noiseless Vlasov–Poisson simulations with linearly transformed particles

We introduce a deterministic discrete-particle simulation approach, the Linearly-Transformed Particle-In-Cell (LTPIC) method, that employs linear deformations of the particles to reduce the noise traditionally associated with particle schemes. Formally, transforming the particles is justified by local first order expansions of the characteristic flow in phase space. In practice the method amounts of using deformation matrices within the particle shape functions; these matrices are updated via local evaluations of the forward numerical flow. Because it is necessary to periodically remap the particles on a regular grid to avoid excessively deforming their shapes, the method can be seen as a development of Denavit's Forward Semi-Lagrangian (FSL) scheme (Denavit, 1972 [8]). However, it has recently been established (Campos Pinto, 2012 [20]) that the underlying Linearly-Transformed Particle scheme converges for abstract transport problems, with no need to remap the particles; deforming the particles can thus be seen as a way to significantly lower the remapping frequency needed in the FSL schemes, and hence the associated numerical diffusion. To couple the method with electrostatic field solvers, two specific charge deposition schemes are examined, and their performance compared with that of the standard deposition method. Finally, numerical 1d1v simulations involving benchmark test cases and halo formation in an initially mismatched thermal sheet beam demonstrate some advantages of our LTPIC scheme over the classical PIC and FSL methods. Benchmarked test cases also indicate that, for numerical choices involving similar computational effort, the LTPIC method is capable of accuracy comparable to or exceeding that of state-of-the-art, high-resolution Vlasov schemes.

Excitation and control of chirped nonlinear ion-acoustic waves.

Large-amplitude ion acoustic waves are excited and controlled by a chirped frequency driving perturbation. The process involves capturing into autoresonance (a continuous nonlinear synchronization) with the drive by passage through the linear resonance in the problem. The transition to autoresonance has a sharp threshold on the driving amplitude. The theory of this transition is developed beyond the Korteweg-de Vries limit by using the Whitham's averaged variational principle within the water bag model and compared with Vlasov-Poisson simulations.

Highly efficient numerical algorithm based on random trees for accelerating parallel Vlasov–Poisson simulations

An efficient numerical method based on a probabilistic representation for the Vlasov–Poisson system of equations in the Fourier space has been derived. This has been done theoretically for arbitrary dimensional problems, and particularized to unidimensional problems for numerical purposes. Such a representation has been validated theoretically in the linear regime comparing the solution obtained with the classical results of the linear Landau damping.The numerical strategy followed requires generating suitable random trees combined with a Padé approximant for approximating accurately a given divergent series. Such series are obtained by summing the partial contributions to the solution coming from trees with arbitrary number of branches. These contributions, coming in general from multi-dimensional definite integrals, are efficiently computed by a quasi-Monte Carlo method. It is shown how the accuracy of the method can be effectively increased by considering more terms of the series.The new representation was used successfully to develop a Probabilistic Domain Decomposition method suited for massively parallel computers, which improves the scalability found in classical methods. Finally, a few numerical examples based on classical phenomena such as the non-linear Landau damping, and the two streaming instability are given, illustrating the remarkable performance of the algorithm, when compared the results with those obtained using a classical method.

Kappa Distribution Function Effects on Landau Damping in Electrostatic Vlasov Simulation

The non-thermal high-energy electron effects on Langmuir wave-particle interaction are investigated using an initial value approach. A Vlasov-Poisson simulation is employed based on the splitting scheme by Cheng and Knorr (1976). The kappa distribution function is taken as an example of non-thermal electrons. The modification is manifested as an increase in the Landau damping rate and a decrease in the real frequency for a long wavelength limit. A part of the analyses using the modified plasma dispersion function (Summers and Thorne 1991) is reproduced for l = 2, 3 and 6. The dispersion relation from the initial value simulation and the plasma dispersion function compare favorably. (PACS numbers: 52.35.Fp, 52.35.Sb, 52.65.Ff).

Undamped electrostatic plasma waves

Electrostatic waves in a collision-free unmagnetized plasma of electrons with fixed ions are investigated for electron equilibrium velocity distribution functions that deviate slightly from Maxwellian. Of interest are undamped waves that are the small amplitude limit of nonlinear excitations, such as electron acoustic waves (EAWs). A deviation consisting of a small plateau, a region with zero velocity derivative over a width that is a very small fraction of the electron thermal speed, is shown to give rise to new undamped modes, which here are named corner modes. The presence of the plateau turns off Landau damping and allows oscillations with phase speeds within the plateau. These undamped waves are obtained in a wide region of the (k,ωR) plane (ωR being the real part of the wave frequency and k the wavenumber), away from the well-known “thumb curve” for Langmuir waves and EAWs based on the Maxwellian. Results of nonlinear Vlasov-Poisson simulations that corroborate the existence of these modes are described. It is also shown that deviations caused by fattening the tail of the distribution shift roots off of the thumb curve toward lower k-values and chopping the tail shifts them toward higher k-values. In addition, a rule of thumb is obtained for assessing how the existence of a plateau shifts roots off of the thumb curve. Suggestions are made for interpreting experimental observations of electrostatic waves, such as recent ones in nonneutral plasmas.

Critical Initial Amplitude of Langmuir Wave Damping

By one-dimensional Vlasov-Poisson simulation, the critical initial state marking the transition between the Landau scenario, in which the electric fields definitively damped to zero and the O'NEIL scenario, in which the Landau damping is stopped after a certain damping stage, is studied. It is found that the critical initial amplitude ∊* can only exist when the product of the wave number (km) and the electron thermal velocity (vth) is moderate, that is, 0.2 < kmvth < 0.7. Otherwise, no critical initial amplitude is found. The value ∊* increases with the increase in km for a fixed vth, and also increases with the increase in vth for a fixed km. When kmvth is fixed, the value ∊* also changes with the wave number and the electron thermal velocity, even though the damping rate and the oscillation frequency are the same in this case.

A non-oscillatory and conservative semi-Lagrangian scheme with fourth-degree polynomial interpolation for solving the Vlasov equation

A conservative semi-Lagrangian scheme for the numerical solution of the Vlasov equation is developed based on the fourth-degree polynomial interpolation. Then, a numerical filter is implemented that preserves positivity and non-oscillatory. The numerical results of both one-dimensional linear advection and two-dimensional Vlasov–Poisson simulations show that the numerical diffusion with the fourth-degree polynomial interpolation is suppressed more than with the cubic polynomial interpolation. It is also found that inherent conservation properties of the Vlasov equation can be improved by combining numerical fluxes of the upwind-biased and central fourth-degree polynomial interpolations.