Vlasov Code Research Articles

Development of a Global Toroidal Gyrokinetic Vlasov Code with New Real Space Field Solver

Development of a Global Toroidal Gyrokinetic Vlasov Code with New Real Space Field Solver

Coupled Vlasov and two-fluid codes on GPUs

We present a way to combine Vlasov and two-fluid codes for the simulation of a collisionless plasma in large domains while keeping full information on the velocity distribution in localised areas of interest. This is made possible by solving the full Vlasov equation in one region while the remaining area is treated by a 5-moment two-fluid code. In such a treatment, the main challenge of coupling kinetic and fluid descriptions is the interchange of physically correct boundary conditions between the different plasma models. In contrast to other treatments, we do not rely on any specific form of the distribution function, e.g. a Maxwellian type. Instead, we combine an extrapolation of the distribution function and a correction of the moments based on the fluid data. Thus, throughout the simulation both codes provide the necessary boundary conditions for each other. A speed-up factor of around 10 is achieved by using GPUs for the computationally expensive solution of the Vlasov equation. Additional major savings are obtained due to the coupling where the amount of savings roughly corresponds to the fraction of the domain where the kinetic equations are solved. The coupled codes were then tested on the propagation of whistler waves and on the GEM reconnection challenge.

Electron emission in a source-collector sheath system: A kinetic study

The classical source-collector sheath system describes a plasma that forms between a Maxwellian source and an absorbing wall. The plasma is assumed to be collisionless and without ionization. Two distinct areas are being formed: the collector sheath, an ion-rich region in contact with the absorbing boundary, and the source sheath, which is an electron-rich area near the Maxwellian source. In this work, we study a modified version of the classical source-collector sheath system, where the wall is no longer absorbing but emits electrons. As a result, we have two different types of collector sheath, one where a potential well is formed and one without a potential well. We examine the effect of electron emission for a range of conditions for the plasma and the emitted electrons. In the first part of this work, we study the problem analytically, and in the second, using our kinetic Vlasov code, Yggdrasil. The simulation results are in very good agreement with the predictions of our theoretical model.

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.

Relativistic Vlasov code development for high energy density plasmas

A newly developed relativistic Vlasov code is introduced. The governing Vlasov-Maxwell equation system is solved numerically in one-dimensional space and three-dimensional momentum space. Spherical coordinate system is adopted to characterize the momentum variables for its potential advantage on reducing computational cost. The resulting Vlasov equation is split into two advection equations with respect to position and momentum, respectively. They are solved with a conservative finite volume scheme, together with techniques suppressing numerical oscillations at sharp interfaces. Relativistic longitudinal plasma oscillations are investigated for different plasma temperatures and wave numbers. Results from code simulations are in good agreement with the existing theories.

Comparing the line broadened quasilinear model to Vlasov code

The Line Broadened Quasilinear (LBQ) model is revisited to study its predicted saturation level as compared with predictions of a Vlasov solver BOT [Lilley et al., Phys. Rev. Lett. 102, 195003 (2009) and M. Lilley, BOT Manual. The parametric dependencies of the model are modified to achieve more accuracy compared to the results of the Vlasov solver both in regards to a mode amplitude's time evolution to a saturated state and its final steady state amplitude in the parameter space of the model's applicability. However, the regions of stability as predicted by LBQ model and BOT are found to significantly differ from each other. The solutions of the BOT simulations are found to have a larger region of instability than the LBQ simulations.

Extended fluid models: Pressure tensor effects and equilibria

We consider the use of “extended fluid models” as a viable alternative to computationally demanding kinetic simulations in order to manage the global large scale evolution of a collisionless plasma while accounting for the main effects that come into play when spatial micro-scales of the order of the ion inertial scale di and of the thermal ion Larmor radius ϱi are formed. We present an extended two-fluid model that retains finite Larmor radius (FLR) corrections to the ion pressure tensor while electron inertia terms and heat fluxes are neglected. Within this model we calculate analytic FLR plasma equilibria in the presence of a shear flow and elucidate the role of the magnetic field asymmetry. Using a Hybrid Vlasov code, we show that these analytic equilibria offer a significant improvement with respect to conventional magnetohydrodynamic shear-flow equilibria when initializing kinetic simulations.

Study on longitudinal dispersion relation in one-dimensional relativistic plasma: Linear theory and Vlasov simulation

The dispersion relation of one-dimensional longitudinal plasma waves in relativistic homogeneous plasmas is investigated with both linear theory and Vlasov simulation in this paper. From the Vlasov-Poisson equations, the linear dispersion relation is derived for the proper one-dimensional Jüttner distribution. Numerically obtained linear dispersion relation as well as an approximate formula for plasma wave frequency in the long wavelength limit is given. The dispersion of longitudinal wave is also simulated with a relativistic Vlasov code. The real and imaginary parts of dispersion relation are well studied by varying wave number and plasma temperature. Simulation results are in agreement with established linear theory.

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.

Global Vlasov simulation on magnetospheres of astronomical objects

Space plasma is a collisionless, multi-scale, and highly nonlinear medium. There are various types of self-consistent computer simulations that treat space plasma according to various approximations. We develop numerical schemes for solving the Vlasov (collisionless Boltzmann) equation, which is the first-principle kinetic equation for collisionless plasma. The weak-scaling benchmark test shows that our parallel Vlasov code achieves a high performance and a high scalability. Currently, we use more than 1000 cores for parallel computations and apply the present parallel Vlasov code to various cross-scale processes in space plasma, such as a global simulation on the interaction between solar/stellar wind and magnetospheres of astronomical objects.

Nonlinear Brillouin amplification of finite-duration seeds in the strong coupling regime

Parametric plasma processes received renewed interest in the context of generating ultra-intense and ultra-short laser pulses up to the exawatt-zetawatt regime. Both Raman as well as Brillouin amplifications of seed pulses were proposed. Here, we investigate Brillouin processes in the one-dimensional (1D) backscattering geometry with the help of numerical simulations. For optimal seed amplification, Brillouin scattering is considered in the so called strong coupling (sc) regime. Special emphasis lies on the dependence of the amplification process on the finite duration of the initial seed pulses. First, the standard plane-wave instability predictions are generalized to pulse models, and the changes of initial seed pulse forms due to parametric instabilities are investigated. Three-wave-interaction results are compared to predictions by a new (kinetic) Vlasov code. The calculations are then extended to the nonlinear region with pump depletion. Generation of different seed layers is interpreted by self-similar solutions of the three-wave interaction model. Similar to Raman amplification, shadowing of the rear layers by the leading layers of the seed occurs. The shadowing is more pronounced for initially broad seed pulses. The effect is quantified for Brillouin amplification. Kinetic Vlasov simulations agree with the three-wave interaction predictions and thereby affirm the universal validity of self-similar layer formation during Brillouin seed amplification in the strong coupling regime.

Pulse shaping during Raman-seed amplification for short laser pulses

Raman-seed pulse amplification in a one-dimensional backscattering geometry is investigated with the help of numerical simulations and analytical estimates. The significant dependence of the initial amplification on the pulse form is revisited on the basis of a three-wave interaction as well as a kinetic Vlasov model. It is shown how the short duration of the input seed pulse influences its subsequent behavior, depending on plasma density and pump strength. The evolution during a "start-up period," which has been observed earlier, can be explained analytically. In the nonlinear (pump depletion) regime, the pulse generated in the start-up period will be further amplified and may evolve into a self-similar π-pulse solution. The Vlasov code predicts algebraic growth in time of the seed amplitude, similar to the findings based on self-similar solutions of the three-wave-interaction model. An initially very narrow pulse is shown to grow more slowly than an initially broad one.

A kinetic study of the source–collector sheath system in a drifting plasma

The source–collector sheath system describes a plasma that forms between a Maxwellian source and an absorbing wall. The plasma is assumed to be collisionless and without ionization. Two distinct areas are being formed: the collector sheath, an ion-rich region in contact with the absorbing boundary, and the source sheath, which is an electron-rich area near the Maxwellian source. Our work examines the effect that shifted Maxwellian distributions at the plasma source have on the characteristics of such a system. This is studied for a range of drift velocities and ratios of ion and electron temperatures. We study the problem both analytically and using our kinetic Vlasov code, Yggdrasil. The simulation results are in very good agreement with the predictions of our theoretical model.

Electron and ion kinetic effects on non-linearly driven electron plasma and ion acoustic waves

Fully non-linear kinetic simulations of electron plasma and ion acoustic waves (IAWs) have been carried out with a new multi-species, parallelized Vlasov code. The numerical implementation of the Vlasov model and the methods used to compute the wave frequency are described in detail. For the first time, the nonlinear frequency of IAWs, combining the contributions from electron and ion kinetic effects and from harmonic generation, has been calculated and compared to Vlasov results. Excellent agreement of theory with simulation results is shown at all amplitudes, harmonic generation being an essential component at large amplitudes. For IAWs, the positive frequency shift from trapped electrons is confirmed and is dominant for the effective electron-to-ion temperature ratio, Z Te/Ti ≳ 10 with Z as the charge state. Furthermore, numerical results demonstrate unambiguously the dependence [R. L. Dewar, Phys. Fluids 15, 712 (1972)] of the kinetic shifts on details of the distribution of the trapped particles, which depends in turn on the conditions under which the waves were generated. The trapped particle fractions and energy distributions are derived and, upon inclusion of harmonic effects, shown to agree with the simulation results, completing a consistent picture. Fluid models of the wave evolution are considered but prove unable to capture essential details of the kinetic simulations. Detrapping by collisions and sideloss is also discussed.

Discrete kinetic eigenmode spectra of electron plasma oscillations in weakly collisional plasma: A numerical study

It has been demonstrated that in the presence of weak collisions, described by the Lenard-Bernstein (LB) collision operator, the Landau-damped solutions become true eigenmodes of the system and constitute a complete set [C.-S. Ng et al., Phys. Rev. Lett. 83, 1974 (1999) and C. S. Ng et al., Phys. Rev. Lett. 96, 065002 (2004)]. We present numerical results from an Eulerian Vlasov code that incorporates the Lenard-Bernstein collision operator [A. Lenard and I. B. Bernstein, Phys. Rev. 112, 1456 (1958)]. The effect of collisions on the numerical recursion phenomenon seen in Vlasov codes is discussed. The code is benchmarked against exact linear eigenmode solutions in the presence of weak collisions, and a spectrum of Landau-damped solutions is determined within the limits of numerical resolution. Tests of the orthogonality and the completeness relation are presented.

A direct Vlasov code to study the non-stationary current collection by a cylindrical Langmuir probe

The time-dependent current collection by a cylindrical Langmuir probe, whose bias is suddenly changed from zero to a positive or negative finite value, is studied with a novel direct Vlasov code. The numerical algorithm is based on finite-difference formulas to approximate spatial and velocity derivatives and the time integration is carried out with an explicit Runge-Kutta method, or in the case of probe radius small compared with the Debye length, by using the unconditionally stable backward Euler scheme. Both electrons and ions are treated kinetically by the code, which implements initial and boundary conditions that are consistent with the presence of the probe. Within the considered parameter range, the plasma sheath around the probe exhibited an overshoot and it later recovered a steady state. Phase space diagrams of the particle trajectories revealed the presence of a trapped population of particles. The dependence of this population as a function of the probe radius is presented as well as a comparison with the stationary theory. The performance of the code and a comparison with previously used particle-in-cell algorithms are discussed.

Threshold for electron trapping nonlinearity in Langmuir waves

We assess when electron trapping nonlinearity is expected to be important in Langmuir waves. The basic criterion is that the inverse of the detrapping rate νd of electrons in the trapping region of velocity space must exceed the bounce period of deeply trapped electrons, τB≡(ne/δn)1/22π/ωpe. A unitless figure of merit, the “bounce number” NB≡1/νdτB, encapsulates this condition and defines a trapping threshold amplitude for which NB=1. The detrapping rate is found for convective loss (transverse and longitudinal) out of a spatially finite Langmuir wave. Simulations of driven waves with a finite transverse profile, using the 2D-2V Vlasov code LOKI, show trapping nonlinearity increases continuously with NB for transverse loss, and is significant for NB≈1. The detrapping rate due to Coulomb collisions (both electron-electron and electron-ion) is also found, with pitch-angle scattering and parallel drag and diffusion treated in a unified manner. A simple way to combine convective and collisional detrapping is given. Application to underdense plasma conditions in inertial confinement fusion targets is presented. The results show that convective transverse loss is usually the most potent detrapping process in a single f/8 laser speckle. For typical plasma and laser conditions on the inner laser cones of the National Ignition Facility, local reflectivities ∼3% are estimated to produce significant trapping effects.

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.

Hybrid simulation of stimulated Brillouin scattering in laser fusions

In laser plasmas interactions, the simulation of stimulated Brillouin scattering takes a long time. In order to decrease the simulation time, a hybrid Vlasov code is developed, in which the electrons are treated as Boltzmann fluid. The two-ion beam instability is simulated, and the simulation results are compared with the theoretical results. Stimulated Brillouin scattering is also simulated, the results show the code correctness.

On kinetic description of electromagnetic processes in a quantum plasma

A nonlinear kinetic equation for nonrelativistic quantum plasma with electromagnetic interaction of particles is obtained in the Hartree’s mean-field approximation. It is cast in a convenient form of Vlasov-Boltzmann-type equation with “quantum interference integral”, which allows for relatively straightforward modification of existing classical Vlasov codes to incorporate quantum effects (quantum statistics and quantum interference of overlapping particles wave functions), without changing the bulk of the codes. Such modification (upgrade) of existing Vlasov codes may provide a direct and effective path to numerical simulations of nonlinear electrostatic and electromagnetic phenomena in quantum plasmas, especially of processes where kinetic effects are important (e.g., modulational interactions and stimulated scattering phenomena involving plasma modes at short wavelengths or high-order kinetic modes, dynamical screening and interaction of charges in quantum plasma, etc.) Moreover, numerical approaches involving such modified Vlasov codes would provide a useful basis for theoretical analyses of quantum plasmas, as quantum and classical effects can be easily separated there.