Vlasov Code Research Articles

Simulation of ion temperature gradient driven modes with 6D kinetic Vlasov code

With the increase in computational capabilities over the last few years, it becomes possible to simulate more and more complex and accurate physical models. Gyrokinetic theory has been introduced in the 1960s and 1970s in the need of describing a plasma with more accurate models than fluid equations but eliminating the complexity of the fast gyration about the magnetic field lines. Although results from current gyrokinetic computer simulations are in fair agreement with experimental results in core physics, crucial assumptions made in the derivation make it unreliable in regimes of higher fluctuations and stronger gradient, such as the tokamak edge. With our novel optimized and scalable semi-Lagrangian solver, we are able to simulate ion temperature gradient modes with the 6D kinetic model including the turbulent saturation. After thoroughly testing our simulation code against analytical computations and gyrokinetic simulations (with the gyrokinetic code GYRO), it has been possible to show first plasma properties that go beyond standard gyrokinetic simulations. This includes the explicit description of the complete perpendicular energy fluxes and the excitation of high-frequency waves (around the Larmor frequency) in the nonlinear saturation phase.

Global gyro-kinetic study of magnetic shaping effects on linear trapped electron mode instability in normal/reversed magnetic shear plasmas

A linear δf version of the gyro-kinetic Vlasov code GKNET (Gyro-Kinetic Numerical Experiment of Tokamak) is extended to the global cylindrical coordinates (R, φ, Z) and includes the kinetic electron response, to study the shaping effect on linear drift-wave instabilities in Tokamak plasmas. Cross-verifications are performed between three GKNET versions that use different electron models, i.e. the adiabatic electron model, hybrid electron model (only trapped electrons are calculated kinetically) and full-kinetic electron model (all electrons are calculated kinetically). A destabilizing effect of non-adiabatic passing electrons is observed in the full-kinetic case for the ion temperature gradient (ITG) mode in the short wavelength region and the trapped electron mode (TEM). The ion-to-electron mass ratio and the electron–electron collisionality have weak impacts on the destabilization of ITG/TEM. Utilizing magnetohydrodynamic equilibria designed with normal and reversed magnetic shear (NS and RS), the characteristics of linear TEMs and the corresponding impact of plasma shaping are studied based on the GKNET code with full-kinetic electrons. Due to the change in temperature/density gradient and magnetic shear either measured locally in the low field side (LFS) or averaged over the flux surface, plasma shaping shows different impacts on linear TEMs in NS and RS configurations. For the elongation κ, the increase in κ always stabilizes linear TEMs due to the reduction in the effective profile gradient over the flux surface. For the triangularity δ, in the NS case, the change in δ shows weak dependence on TEM linear growth rates due to the balance between variations of profile gradients and magnetic shear in the LFS. In the RS case, local magnetic shear in the peak gradient region is nearly zero. Thus, the variation in local profile gradient plays a dominant role on the linear growth in a plasma shaping scan. Consequently, the negative triangularity (δ < 0) has a destabilizing effect on the linear TEMs in the RS configuration mainly due to the upshift in local profile gradient in the LFS.

Cold dark matter protohalo structure around collapse: Lagrangian cosmological perturbation theory versus Vlasov simulations

We explore the structure around the shell-crossing time of cold dark matter protohaloes seeded by two or three crossed sine waves of various relative initial amplitudes, by comparing Lagrangian perturbation theory (LPT) up to the tenth order with high-resolution cosmological simulations performed with the public Vlasov code ColDICE. Accurate analyses of the density, the velocity, and related quantities such as the vorticity are performed by exploiting the fact that ColDICE can follow the phase-space sheet locally at the quadratic level. To test LPT predictions beyond the shell-crossing, we employ a ballistic approximation, which assumes that the velocity field is frozen just after the shell-crossing. In the generic case, where the amplitudes of the sine waves are all different, high-order LPT predictions match the exact solution very well, even beyond collapse. As expected, convergence slows down when going from quasi-1D dynamics, where one wave dominates over the two others, to the axial-symmetric configuration, where all the amplitudes of the waves are equal. We also notice that LPT convergence is slower when considering velocity-related quantities. Additionally, the structure of the system at and beyond collapse given by LPT and the simulations agrees very well with singularity theory predictions, in particular with respect to the caustic and vorticity patterns that develop beyond collapse. Again, this does not apply to axial-symmetric configurations, which are still correct from the qualitative point of view, but rather when multiple foldings of the phase-space sheet produce very high density contrasts and hence a strong back-reaction of the gravitational force.

Semi-Lagrangian Method to Study Nonlinear Electrostatic Waves in Quantum Plasma

In this work, we used the semi-Lagrangian method and the Vlasov code to study the instabilities in a quantum plasma with 1-D temperature anisotropy. Taking the Fermi-Dirac type of distribution function (DF) and using the semi-Lagrangian Vlasov code, we studied the gradual evolution of density and kinetic energy in a degenerate plasma with quantum effects. In this collisionless plasma, adiabatic compression will be along the direction of wave propagation and will lead to temperature anisotropy of the electron distribution that varies along the direction of wave propagation. The findings will find application for dense plasma at finite temperature as in laser-produced plasmas, fusion plasma, as well as solar plasmas.

Simulation of ion temperature gradient mode in Chinese First Quasi-axisymmetric Stellarator

The Chinese First Quasi-axisymmetric Stellarator (CFQS) is now the only quasi-axisymmetric stellarator under construction in the world. In this work, ion temperature gradient (ITG) mode in CFQS is studied by using gyrokinetic Vlasov code GKV. The basic characteristics of the eletrtostatic ITG are separately given under the adiabatic condition and the non-adiabatic condition. There is a critical temperature gradient for ITG. The growth rate of ITG is proportional to the temperature gradient. Furthermore, the growth rate depends on not only the absolute value of density gradient, but also the plus or minus sign of the density gradient. The negative density gradient can strongly suppress the ITG. The kinetic electron can destabilize the ITG and the electron temperature gradient can also destabilize the ITG. For electromagnetic condition, the ITG modes can be suppressed by the finite plasma beta, and then a transition from ITG to Alfvenic ion temperature gradient mode/kinetic ballooning mode (AITG/KBM) comes into being. The maximum growth rate of KBM is linearly proportional to density gradient and temperature gradient when both gradients are large.

The dynamics of electron holes in current sheets

We present 1.5D Vlasov code simulations of the dynamics of electron holes in non-uniform magnetic and electric fields typical of current sheets and, particularly, of the Earth's magnetotail current sheet. The simulations show that spatial width and amplitude of electron holes do not substantially vary in the course of propagation, but there arises a double layer localized around the electron hole and manifested as a drop of the electrostatic potential along the electron hole. We demonstrate that electron holes produced around the neutral plane of a current sheet slow down in the course of propagation toward the current sheet boundaries. The leading contribution to electron hole braking is provided by the non-uniform magnetic field although electrostatic fields typical of the current sheets do provide a noticeable contribution. The simulations also show that electron holes with larger amplitudes are slowed faster. The simulation results suggest that some of the slow electron holes recently reported in the Earth's plasma sheet boundary layer may appear due to braking of initially fast electron holes in the course of propagation in the current sheet.

Suppression of auto-resonant stimulated Brillouin scattering in supersonic flowing plasmas by different forms of incident lasers**Project supported by the National Natural Science Foundation of China (Grant Nos.11875091 and 11975059) and the Science Challenge Project, China (Grant No. TZ2016005).

In supersonic flowing plasmas, the auto-resonant behavior of ion acoustic waves driven by stimulated Brillouin backscattering is self-consistently investigated. A nature of absolute instability appears in the evolution of the stimulated Brillouin backscattering. By adopting certain form of incident lights combined by two perpendicular linear polarization lasers or polarization rotation lasers, the absolute instability is suppressed significantly. The suppression of auto-resonant stimulated Brillouin scattering is verified with the fully kinetic Vlasov code.

Semi-Lagrangian Vlasov simulation of the interaction between femtosecond chirped and double laser pulses with a thin plasma slab

The wakefield generation and electrons acceleration in the interaction of femtosecond chirped and double laser pulses with an underdense thin plasma slab is examined by numerical solution of the relativistic Vlasov-Maxwell system of equations using the backward semi-Lagrangian method (BSL). Unlike particle-in-cell methods, the BSL method is free of numerical noise and provides an accurate representation of the distribution function, even in very low density regions of phase space where acceleration and trapping occur. The effect of density fluctuations in low density regions of phase space are important, since they affect the acceleration processes over the following interactions. Therefore, a detailed structure of the distribution function is needed to investigate this process in these regions, which only can be obtained by Vlasov codes. Excitation of wakefields by chirped, chirp free, and double pulse lasers in an underdense plasma slab are simulated using the BSL code and the results are compared. It is shown that by making use of chirped laser pulses, the mean kinetic energy of plasma electrons can be enhanced up to 40% compared with the chirp free case. The investigation of wakefield generation by double laser pulses with a time repetition equal to one plasma period, exhibits amplification of wakefield amplitude after the entrance of the second laser pulse into plasma slab.

Vlasov code simulation of contact discontinuities

The stability of contact discontinuities formed by the relaxation of two Maxwellian plasmas with different number densities but the same plasma thermal pressure is studied by means of a one-dimensional electrostatic full Vlasov simulation. Our simulation runs with various combinations of ion-to-electron ratios of the high-density and low-density regions showed that transition layers of density and temperature without jump in the plasma thermal pressure are obtained when the electron temperatures in the high-density and low-density regions are almost equal to each other. However, the stable structure of the contact discontinuity with a sharp transition layer on the Debye scale is not maintained. It is suggested that non-Maxwellian velocity distributions are necessary for the stable structure of contact discontinuities.

Bernstein modes in a non-neutral plasma column

This paper presents theory and numerical calculations of electrostatic Bernstein modes in an inhomogeneous cylindrical plasma column. These modes rely on finite Larmor radius effects to propagate radially across the column until they are reflected when their frequency matches the upper hybrid frequency. This reflection sets up an internal normal mode on the column and also mode-couples to the electrostatic surface cyclotron wave (which allows the normal mode to be excited and observed using external electrodes). Numerical results predicting the mode spectra, using a novel linear Vlasov code on a cylindrical grid, are presented and compared to an analytical Wentzel Kramers Brillouin (WKB) theory. A previous version of the theory [D. H. E. Dubin, Phys. Plasmas 20(4), 042120 (2013)] expanded the plasma response in powers of 1/B, approximating the local upper hybrid frequency, and consequently, its frequency predictions are spuriously shifted with respect to the numerical results presented here. A new version of the WKB theory avoids this approximation using the exact cold fluid plasma response and does a better job of reproducing the numerical frequency spectrum. The effect of multiple ion species on the mode spectrum is also considered, to make contact with experiments that observe cyclotron modes in a multi-species pure ion plasma [M. Affolter et al., Phys. Plasmas 22(5), 055701 (2015)].

Formation of electron holes in the long-time evolution of the bump-on-tail instability

AbstractAn Eulerian Vlasov code is applied for the numerical solution of the one-dimensional Vlasov–Poisson system of equations for electrons, and with ions forming an immobile background. We study the non-linear evolution of the bump-on-tail instability in the case when the system length L is greater than the wavelength λ of the unstable mode, with a beam density of 10% of the total density, nb = 0.1. We follow the growth and the saturation of an initially unstable wave perturbation, and the formation of a traveling Bernstein–Greene–Kruskal (BGK) mode, which evolves out of the instability. This first stage is followed by sidebands growing from round-off errors which develop and disrupt the BGK equilibrium. In the excited spectrum, mode coupling is mediated by the oscillating resonant particles and results in the electric energy of the system flowing to the longest wavelengths (inverse cascade), and reaching in the asymptotic state a steady state with constant amplitude oscillation modulated by the persistent oscillation of the trapped particles. Coherent phase-space electron holes are formed, which are localized phase-space regions of reduced density on trapped electron orbits, where the electron density is lower than the surrounding plasma electron density. The distribution function evolves to a shape with stationary inflection points of zero slope, at the phase velocities of the excited waves. The longest wavelengths show oscillations at frequencies below the plasma frequency, with phase velocities higher than that of the injected beam, which can accelerate electrons to energies in excess of the initial beam energy. The present work makes a connection between the formation of electron holes, the existence of inflection points of zero slopes in the electron distribution function at the phase velocities of the dominant waves, and at frequencies below the plasma frequency. A fine resolution grid is used in the Eulerian Vlasov code in the phase space and time to allow an accurate calculation of the time history of the system and of the dynamic and oscillation of the trapped particles in the low-density regions of the phase space, and of those particles at the separatrix regions of the vortex structures which evolve periodically between trapping and untrapping states and which can only be accurately studied using a fine-resolution phase-space grid.

Evolution of electron phase space holes in inhomogeneous magnetic fields

AbstractElectron phase space holes (EHs) are electrostatic solitary waves that are widely observed in the space plasma often permeated by inhomogeneous magnetic fields. Understanding of the EH evolution in inhomogeneous magnetic fields is critical for accurate interpretations of spacecraft data. To study this evolution, we use 1.5‐D gyrokinetic electrostatic Vlasov code (magnetized electrons and immobile ions) with periodic boundary conditions. We find that EHs propagating into stronger (weaker) magnetic field are decelerated (accelerated) with deceleration (acceleration) rate dependent on the magnetic field gradient. Remarkably, decelerating EHs are reflected at the magnetic field dependent only on EH parameters (independent of the magnetic field gradient). A magnetic field inhomogeneity results in development of a net potential drop along EHs. Our simulations suggest that slow EHs recently observed in the plasma sheet boundary layer can appear due to braking of initially fast EHs by magnetic field gradients and that a large number of even fast EHs can contribute to macroscopic parallel potential drops.

Formation of double layers and evolution of the distribution functions during ion acceleration driven by a high-intensity short laser pulse normally incident on thin foils

AbstractWe use an Eulerian Vlasov code, which solves the one-dimensional relativistic Vlasov–Maxwell equations for both electrons and ions, to follow in details the evolution of the distribution functions and the mechanism of the formation and evolution of double layers during ion acceleration driven by a high-intensity circularly polarized short laser pulse (12 ω−1 where ω is the laser angular frequency) normally incident on a thin dense foil. We compare three cases with a high-density deuterium plasma target of total thickness 1.767 cω−1 and constant n/ncr = 100, where ncr is the critical density, and where the laser intensity is varied from a situation where the target is opaque to the laser pulse (normalized vector potential or quiver momentum a0 = 80), to a situation where, above a critical laser intensity, a very small fraction of the laser pulse is transmitted through the target (a0 = 90), and finally to a situation where a more important fraction is transmitted through the target (a0 = 100). The dynamics of ion and electron acceleration are quite different in the three cases, and are followed in detail by the Eulerian Vlasov code, which allows an accurate representation of the distribution function. In the intermediate case, the Vlasov code has revealed a remarkably well-developed spiral structure in the phase space of the electron distribution function, which is associated with large sawtooth modulations in the electron density profiles.

Cosmology in one dimension: Vlasov dynamics.

Numerical simulations of self-gravitating systems are generally based on N-body codes, which solve the equations of motion of a large number of interacting particles. This approach suffers from poor statistical sampling in regions of low density. In contrast, Vlasov codes, by meshing the entire phase space, can reach higher accuracy irrespective of the density. Here, we perform one-dimensional Vlasov simulations of a long-standing cosmological problem, namely, the fractal properties of an expanding Einstein-de Sitter universe in Newtonian gravity. The N-body results are confirmed for high-density regions and extended to regions of low matter density, where the N-body approach usually fails.

Numerical simulation of Raman and Brillouin laser-pulse amplification in a magnetized plasma

AbstractWe apply an Eulerian Vlasov code to study the amplification of an ultra-short seed pulse via stimulated Raman and Brillouin backscattering of energy from a long pump pulse, assumed at constant amplitude, in a plasma embedded in an external magnetic field. Detailed analysis of the spectra developed during the amplification process are presented, together with the evolution showing the pump depletion, accompanied by the counter-propagating seed-pulse amplification, compression and increased steepness of the waveform. In addition to the problem of the amplification of ultra-short seed pulses, there is an obvious academic interest in the study of problems of amplification of electromagnetic waves observed in many situations in laboratory plasmas and in the magnetosphere and other geophysical situations, such as in the environments of planets, where important variations in the presence and strength of magnetic fields are observed. The numerical code solves a one-dimensional relativistic Vlasov–Maxwell set of equations for a plasma in a magnetic field for both electrons and ions. We also apply the code to the problem of wakefield acceleration. The absence of noise in the Eulerian Vlasov code allows one to follow the evolution of the system with an accurate representation of the phase-space structures of the distribution functions.

A numerical study of ponderomotive ion acceleration in a dense plasma driven by a circularly polarized high-intensity laser beam normally incident on thin foils

AbstractWe use an Eulerian Vlasov code to study the efficient ion acceleration in dense targets by the ponderomotive force of a high-intensity circularly polarized laser beam, normally incident on a dense plasma. The code solves the one-dimensional relativistic Vlasov–Maxwell equations for both electrons and ions. We follow in details the mechanism of formation and evolution of a double-layer structure, where electrons are pushed steadily in the forward direction by the ponderomotive force of the laser beam, trapping an ion population, while an induced space charge electric field pulls ions behind them, forming a double-layer structure supported by the strong ponderomotive pressure of the intense laser beam. We consider the case of a high-density deuterium plasma with n/ncr = 100, where ncr is the critical density. Three cases are studied, by varying the width of the dense target and the intensity of the laser beam (with the normalized vector potential or quiver momentum a0 = 50 and a0 = 100), to follow the physical processes involved in the ion acceleration and the final formation of a neutral plasma jet ejected from the back of the target. We follow the transition from a situation where the laser pulse radiation pressure is acting on the double layer in the target, to a situation where below a given thickness a fraction of the laser energy is transmitted through the target. The absence of noise in the Eulerian Vlasov code allows us to follow accurately the evolution of the phase-space structures of the distribution functions.

Vlasov simulation of laser-driven shock acceleration and ion turbulence

We present a Vlasov, i.e. a kinetic Eulerian simulation study of nonlinear collisionless ion-acoustic shocks and solitons excited by an intense laser interacting with an overdense plasma. The use of the Vlasov code avoids problems with low particle statistics and allows a validation of particle-in-cell results. A simple, original correction to the splitting method for the numerical integration of the Vlasov equation has been implemented in order to ensure the charge conservation in the relativistic regime. We show that the ion distribution is affected by the development of a turbulence driven by the relativistic ‘fast’ electron bunches generated at the laser-plasma interaction surface. This leads to the onset of ion reflection at the shock front in an initially cold plasma where only soliton solutions without ion reflection are expected to propagate. We give a simple analytical model to describe the onset of the turbulence as a nonlinear coupling of the ion density with the fast electron currents, taking the pulsed nature of the relativistic electron bunches into account.

Stimulated Raman scattering in nonextensive statistics

Stimulated Raman scattering (SRS) in a collisionless plasma was analytically and numerically investigated in the context of nonextensive statistics proposed by Tsallis. The generalized wavenumbers and growth rates in both backward and forward scattering were derived as a function of different values of the nonextensive parameter q, which quantified the degree of nonextensivity of the system. It was shown that the increased numbers of superthermal particles and low velocity particles could affect the evolution of stimulated Raman scattering. A relativistic Eulerian Vlasov code for conditions similar to single-hot-spot experiments was used in order to compare the analytical results for the frequency and wavenumber with the quantities obtained from the numerical simulations. A nonextensive q-distribution by the restriction q<1 was used as the initial velocity distribution. The results of numerical simulation showed a strong reflection by SRS. According to the simulation, the peak of the main components in both wavenumber and frequency spectra were found at positions estimated from the phase-matching conditions and the analytical data.

Vlasov versus N-body: the Hénon sphere

We perform a detailed comparison of the phase-space density traced by the particle distribution in Gadget simulations to the result obtained with a spherical Vlasov solver using the splitting algorithm. The systems considered are apodized H\'enon spheres with two values of the virial ratio, R ~ 0.1 and 0.5. After checking that spherical symmetry is well preserved by the N-body simulations, visual and quantitative comparisons are performed. In particular we introduce new statistics, correlators and entropic estimators, based on the likelihood of whether N-body simulations actually trace randomly the Vlasov phase-space density. When taking into account the limits of both the N-body and the Vlasov codes, namely collective effects due to the particle shot noise in the first case and diffusion and possible nonlinear instabilities due to finite resolution of the phase-space grid in the second case, we find a spectacular agreement between both methods, even in regions of phase-space where nontrivial physical instabilities develop. However, in the colder case, R=0.1, it was not possible to prove actual numerical convergence of the N-body results after a number of dynamical times, even with N=10$^8$ particles.

Relativistic Eulerian Vlasov simulations of the amplification of seed pulses by Brillouin backscattering in plasmas

We apply an Eulerian Vlasov code to study the amplification by Brillouin scattering of a short seed laser pulse by a long pump laser pulse in an underdense plasma. The stimulated Brillouin backscattering interaction is the coupling of the pump and seed electromagnetic waves propagating in opposite directions, and the ion plasma wave. The code solves the one-dimensional relativistic Vlasov-Maxwell set of equations. Large amplitude ion waves are generated. In the simulations we present, the density plateau of the plasma is ne=0.3 nc (nc is the critical density), which excludes spurious stimulated Raman scattering amplification (which can occur only if ne&amp;lt;nc/4). We also varied the duration and/or amplitude of the short input seed pulse to study how these influence its subsequent behaviour. An initially broad pulse grows more rapidly than an initially narrow pulse. Furthermore, for an initially broader seed pulse, towards the end of the simulation, it is seen to become narrower and to gradually detach from the trailing signal. On the contrary, initially very narrow seed pulses are seen to broaden. The absence of noise in the Vlasov simulations allows to simulate long plasma amplifier lengths, and to follow the evolution of the system with a fully kinetic description and with an accurate representation of the phase-space structures of distribution function.