Eulerian Vlasov Code Research Articles

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.

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.

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.

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.

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.

Vlasov simulation of ion acceleration by an intense laser beam normally incident on a thin target

We use an Eulerian Vlasov code to study the radiation pressure acceleration of ions and the formation of plasma jets when a high-intensity circularly polarized laser beam is normally incident on a thin target. The code solves the one-dimensional (1D) relativistic Vlasov-Maxwell equations for both electrons and ions. We consider a deuterium plasma with a density n = 25ncr (ncr is the critical density, i.e. the cutoff density for the laser beam propagation). In the case we present, the target is sufficiently thin that a fraction of the incident laser beam is transmitted through it. Along with this, we observe first a leak or ejection of electrons in the forward direction from the back of the target, which is followed by the formation and ejection of a neutral plasma jet (a regime called “leaky light sail radiation pressure acceleration”). The Vlasov code allows to follow the evolution of the system, with an accurate representation of the phase-space structures of the distribution functions.

Ion acceleration and plasma jets driven by a high intensity laser beam normally incident on thin foils

AbstractWe study the problem of the radiation pressure acceleration of ions and the formation of plasma jets, driven by a high-intensity circularly polarized laser beam normally incident on thin plasma targets. We use an Eulerian Vlasov code to solve the one-dimensional relativistic Vlasov-Maxwell equations for both electrons and ions. We consider the case of a high density plasma with n/ncr = 100, where ncr is the critical density. Three cases are studied with different target thicknesses, to investigate the physical processes involved when decreasing the target thickness from several electron skin depths down to the order of one skin depth. The results show a more important acceleration of the ions when the thickness is decreased. Although we observe in all cases a neutral plasma jet ejected from the back of the target, the evolution of the system which leads to the formation of this neutral plasma jet is different in the three cases considered. In each case, this evolution will be studied in details. Also, a leak or ejection of electrons from the back of the target is observed in the thinnest case treated (thickness of the order of the skin depth), before the formation of the neutral plasma jet, a regime called leaky light sail radiation pressure acceleration. The absence of noise in the Eulerian Vlasov code allows an accurate representation of the phase-space structures of the distribution functions.

Ion acceleration and plasma jet formation in the interaction of an intense laser beam normally incident on an overdense plasma: a Vlasov code simulation

We use an Eulerian Vlasov code for the numerical solution of the one-dimensional relativistic Vlasov?Maxwell set of equations to study the acceleration of ions during the interaction of a high-intensity circularly polarized laser beam normally incident on an overdense plasma target. Both electrons and ions are treated with a kinetic equation. We consider the case when the laser free space wavelength ? 0 is greater than the scale length of the jump in the plasma density at the target plasma edge Ledge (? 0 Ledge). The laser beam intensity is a Gaussian-shaped pulse. We consider a density such that n/ncr=25 (ncr is the critical density). The ponderomotive pressure due to the incident high-intensity laser radiation pushes the electrons at the target plasma surface, producing a sharp density gradient, which gives rise to a charge separation. The resulting electric field accelerates the ions. Two cases are considered, namely the case of a relatively thin target (about five skin depth) and the case of a much larger target. The evolution of the charge separation, the associated electric field and the formation of a neutral plasma jet ejected toward the rear side of the target differ in the two cases considered. In the case of a thin target, we observe the ion population accelerated to higher velocities. The absence of noise in the Eulerian Vlasov code allows a detailed representation of the phase-space structures of the distribution functions associated with the two cases considered, which shows substantial differences to be discussed in this paper.

Numerical study of ion acceleration and plasma jet formation in the interaction of an intense laser beam normally incident on an overdense plasma

AbstractWe present a numerical study of the acceleration of ions in the interaction of a high intensity circularly polarized laser beam normally incident on an overdense plasma target, and the subsequent formation of neutral plasma ejected toward the rear side of the target. We compare the results obtained from two different numerical codes. We use an Eulerian Vlasov code for the numerical solution of the one-dimensional relativistic Vlasov-Maxwell set of equations, for both electrons and ions, and a particle-in-cell code applied to the same problem. We consider the case when the laser free space wavelength λ0 is greater than the scale length of the jump in the plasma density at the target plasma edge Ledge (λ0 ≫ Ledge), and the ratio of the plasma density to the critical density is such that n/ncr ≫ 1. The ponderomotive pressure due to the incident high-intensity laser radiation pushes the electrons at the target plasma surface, producing a sharp density gradient at the plasma surface, which gives rise to a charge separation. The resulting electric field accelerates the ions that reach a free streaming expansion phase, where they are neutralized by the electrons. A neutral plasma jet is thus ejected toward the rear side of the target. Two cases are studied: In the first case, the laser intensity rises to a maximum and then remains constant, and in the second case, the laser intensity is a Gaussian-shaped pulse. The results show substantial differences in the phase-space structure of the ions and the electrons between these two cases. There is good agreement between the quantitative macroscopic results obtained by the two codes, and good qualitative agreement between the results showing the kinetic details of the phase-space structures. The low noise level of the Eulerian Vlasov code allows a more detailed representation of the phase-space structures associated with this system, especially in the low density regions of the phase-space where ions are accelerated.

Studies of the interaction of an intense laser beam normally incident on an overdense plasma

AbstractWe present two contrasting cases of the interaction of a high intensity laser beam with overdense plasma, namely the case of a circular polarization, and the case of a linear polarization of the laser beam. An Eulerian-Vlasov code is used for the numerical solution of the one-dimensional relativistic Vlasov-Maxwell set of equations, for both electrons and ions. The laser beam is incident normally on the plasma surface. We consider the case when the laser wave free space wavelength λ0 is greater than the scale length of the jump in the plasma density at the plasma edge Ledge (λ0 ≫ Ledge) and the ratio of the plasma density to the critical density is such that n/ncr ≫ 1. The incident high intensity laser radiation is pushing the electrons at the plasma surface through the ponderomotive pressure, producing a sharp density gradient at the plasma surface. There is a build-up of the electron density at this sharp edge that creates a space-charge, giving rise to a longitudinal electric field. The results obtained differ substantially in several aspects when circular or linear polarization for the incident laser wave is considered. In the case of a circular polarization, the radiation pressure is pushing the sharp edge in the forward direction, and the ions are accelerated and reach a free streaming expansion phase where they are neutralized by the electrons. For the case of a linear polarization, there is a standing structure with a sharp edge that forms at the wave front, and in this case, the electrons at the plasma edge oscillate nonlinearly in the field of the wave, which periodically goes to zero. This results in an important distorsion in the reflected electromagnetic wave that includes the generation of harmonics. We present two simulations to illustrate the differences between these two cases. The generation and propagation of collisionless shock waves in these systems are investigated. The results underline the importance of including the ion dynamics in the interaction of high intensity laser waves with overdense plasmas.

Oscillations of the collisionless sheath at grazing incidence of the magnetic field

An Eulerian Vlasov code is used to study the physics of a collisionless sheath with grazing angles of incidence of the magnetic field. The electrons are treated with a parallel-B kinetic equation. In the case where the ion gyroradius is large compared to the Debye length, the electrons, frozen by the magnetic field line, have to move rapidly along the magnetic field B in their attempt to follow the ions gyrating perpendicular to B. Below a critical angle of incidence of the magnetic field, the large gyroradius ions can be scraped off at the plasma-wall interface of the sheath. The electrons now determine the characteristic time for information propagation, with low frequency oscillations appearing in the system. In the case where the ion gyroradius is reduced to the same order as the Debye length, the results evolve closer to those of the classical sheath, and the amplitude of the low frequency oscillations is reduced.

Numerical simulation for ion acceleration in an intense laser wave incident on an overdense plasma

The acceleration of protons through the interaction of a circularly polarized ultrahigh intensity laser wave incident on an overdense plasma is studied using an Eulerian Vlasov code for the numerical solution of the one-dimensional (1D) relativistic Vlasov–Maxwell set of equations. The high power laser radiation is pushing the electrons through the ponderomotive pressure, producing a sharp edge at the plasma surface. There is a build-up of the electron density at this sharp edge which creates a charge separation, giving rise to a quasistatic longitudinal electric field which is responsible for the ion acceleration.

Oscillations of the collisionless sheath

An Eulerian-Vlasov code is used to study the properties of a collisionless sheath with grazing incidence of the magnetic field. In the case where the ion gyroradius is large compared to the Debye length, the electrons, frozen by the magnetic field lines, have to move rapidly along the magnetic field B in their attempt to follow the ions gyrating perpendicular to B. Below a critical angle, the electrons moving parallel to B can no longer follow the gyrating ions and low-frequency oscillations appear in the system.

AN EULERIAN VLASOV CODE FOR THE NUMERICAL SIMULATION OF WAKE-FIELD ACCELERATION

ABSTRACTWe use an Eulerian Vlasov code for the numerical solution of the fully relativistic one-dimensional (1D) Vlasov-Maxwell equations, to study the generation of plasma wake fields by intense laser pulses. The very low noiselevel of the Vlasov code allows to study accurately the physics of the particles acceleration by the wake-field, inthe very low density of the phase-space.

The Numerical Solution of the Kinetic Sheath Using an Eulerian Vlasov Code

The solution of the collisionless kinetic sheath at a plasma–wall transition is studied numerically using an Eulerian Vlasov code, for the one-dimensional (1D) phase-space problem. Because of its very low noise level, the Eulerian Vlasov code allows an accurate study of the non-neutral collisionless kinetic sheath, especially in the low density regions of the phase-space. Some of the basic macroscopic quantities associated with the sheath equilibrium are calculated.

Vlasov simulations of trapping and inhomogeneity in Raman scattering

We study stimulated Raman scattering (SRS) in laser-fusion conditions with the Eulerian Vlasov code ELVIS. Back SRS from homogeneous plasmas occurs in sub-picosecond bursts and far exceeds linear theory. Forward SRS and re-scatter of back SRS are also observed. The plasma wave frequency downshifts from the linear dispersion curve, and the electron distribution shows flattening. This is consistent with trapping and reduces the Landau damping. There is some acoustic ($\omega\propto k$) activity and possibly electron acoustic scatter. Kinetic ions do not affect SRS for early times but suppress it later on. SRS from inhomogeneous plasmas exhibits a kinetic enhancement for long density scale lengths. More scattering results when the pump propagates to higher as opposed to lower density.

Anomalous resistivity of current-driven isothermal plasmas due to phase space structuring

The anomalous electric resistivity of collisionless plasmas is an important issue in the physics of hot plasmas, e.g., in the context of auroral particle acceleration and of reconnection in the solar corona. The linear stability theory of isothermal current driven space plasmas predicts an ion-acoustic instability if the relative drift velocity of the current carrying particles exceeds a certain threshold, which, generally, depends on the plasma parameters. The spectrum of waves, excited by a marginal instability, is very narrow. Hence, the wave power at saturation and the corresponding electric resistivity due to wave-particle interaction cannot be obtained by means of a quasilinear, weak turbulence approach and the nonlinear single mode theory provides too small saturation amplitudes. To solve the nonlinear problem a newly developed unsplit conservative Eulerian Vlasov code is applied to simulate a strongly magnetized current driven plasma, which can be considered in 1D1V (one spatial, one velocity space direction). Instead of periodic boundary conditions, usually used as they are simpler to treat, open boundaries are implemented which allow to maintain a constant current flow. Simulated is a typical almost isothermal (Te=2Ti) hot (κTi=1keV) space plasma for the real mass ratio mi∕me=1836. The initial spontaneous instability is followed by a three-stage nonlinear evolution: First electron trapping leads to the formation of electron phase space holes. Due to a steepening of the leading edges of the potential wells the electron phase space holes gradually become asymmetric, they grow in size and deepen. The phase space holes accelerate until they move much faster than the initial ion-acoustic waves. The interaction of the current carriers with the asymmetric potential wells and causes a nonvanishing net momentum transfer between the particles and the self-generated electric field. After a few ion plasma periods ion trapping starts until, finally, an electrostatic double layer arises. It is found that the nonlinear saturated state of the system is dominated by the particle interaction with coherent phase space structures. The corresponding anomalous resistivity is slightly modulated with an oscillation period τ≈ωpi−1). For a macroscopic description its major part can be parameterized by means of an effective collision rate νeff of the order of 10−2ωpe≈0.5ωpi, where ωpe is the electron and ωpi the ion plasma frequency.

Vlasov simulations of plasma-wall interactions in a weakly collisional plasma

An Eulerian Vlasov code is used to model plasma-wall interactions in a weakly collisional plasma. The different numerical methods used to solve the Vlasov and Poisson equations are described in detail. The code is used to simulate measurements of the ion distribution and ion temperature in a low-pressure argon plasma. In particular, it is shown that the presence of material walls can lead to significant errors in the measurements, if the effect of the sheaths is not properly taken into account.

Study of the generation of a charge separation and electric field at a plasma edge using Eulerian Vlasov codes in cylindrical geometry

A one-dimensional (1D) Eulerian Vlasov code in cylindrical geometry is used to study the problem of the formation of a charge separation at a plasma edge, together with the self-consistent electric field, in the presence of a density gradient. The code applies a method of fractional steps for the solution of the Vlasov equation. The electric field calculated along the gradient is compared with the macroscopic values calculated from the same kinetic code for the gradient of the ion pressure and the Lorentz force term. We find that these quantities balance the electric field fairly well along the gradient. The extension of this method to a two-dimensional (2D) problem will be presented, together with the corresponding results.