Relativistic Vlasov-Maxwell Research Articles

Higher-Order Accurate and Conservative Hybrid Numerical Scheme for Relativistic Time-Fractional Vlasov-Maxwell System

The historical analysis demonstrates that plasma scientists produced a variety of numerical methods for solving “kinetic” models, i.e., the Vlasov-Maxwell (VM) system. Still, on the other hand, a significant fact or drawback of most algorithms is that they do not preserve conservation philosophies. This is a crucial fact that cannot be disregarded since the Vlasov Maxwell system is associated with conservation rules and is capable of assessing after the accomplishment of certain helpful mathematical actions. To examine the fractional-order routine of charged particles, we constructed a fractional-order plasma model and proposed a higher-order conservative numerical approach based on operational matrices theory and Shifted Gegenbauer estimations. Numerical convergence is investigated to confirm its competence and compatibility. This concept may be used in problems involving variable order and multidimensionality, such as those involving Vlasov and Boltzmann systems.

An introduction to the relativistic kinetic theory on curved spacetimes

This article provides a self-contained pedagogical introduction to the relativistic kinetic theory of a dilute gas propagating on a curved spacetime manifold (M,g) of arbitrary dimension. Special emphasis is made on geometric aspects of the theory in order to achieve a formulation which is manifestly covariant on the relativistic phase space. Whereas most previous work has focussed on the tangent bundle formulation, here we work on the cotangent bundle associated with (M,g) which is more naturally adapted to the Hamiltonian framework of the theory. In the first part of this work we discuss the relevant geometric structures of the cotangent bundle T*M, starting with the natural symplectic form on T*M, the one-particle Hamiltonian and the Liouville vector field, defined as the corresponding Hamiltonian vector field. Next, we discuss the Sasaki metric on T*M and its most important properties, including the role it plays for the physical interpretation of the one-particle distribution function. In the second part of this work we describe the general relativistic theory of a collisionless gas, starting with the derivation of the collisionless Boltzmann equation for a neutral simple gas. Subsequently, the description is generalized to a charged gas consisting of several species of particles and the general relativistic Vlasov-Maxwell equations are derived for this system. The last part of this work is devoted to a transparent derivation of the collision term, leading to the general relativistic Boltzmann equation on (M,g). The meaning of global and local equilibrium and the stringent restrictions for the existence of the former on a curved spacetime are discussed. We close this article with an application of our formalism to the expansion of a homogeneous and isotropic universe filled with a collisional simple gas and its behavior in the early and late epochs. [abbreviated version]

A toy model for the relativistic Vlasov-Maxwell system

<p style='text-indent:20px;'>The global-in-time existence of classical solutions to the relativistic Vlasov-Maxwell (RVM) system in three space dimensions remains elusive after nearly four decades of mathematical research. In this note, a simplified "toy model" is presented and studied. This toy model retains one crucial aspect of the RVM system: the phase-space evolution of the distribution function is governed by a transport equation whose forcing term satisfies a wave equation with finite speed of propagation.</p>

On the 3D Relativistic Vlasov-Maxwell System with Large Maxwell Field

This paper is devoted to the study of relativistic Vlasov-Maxwell system in three space dimension. For a class of large initial data, we prove the global existence of classical solution with sharp decay estimate. The initial Maxwell field is allowed to be arbitrarily large and the initial density distribution is assumed to be small and decay with rate $(1+|x|+|v|)^{-9-}$. In particular, there is no restriction on the support of the initial data.

Magnetic confinement at a boundary approximates specular reflection

We conjecture that for a plasma in a spatial domain with a boundary, the specular reflection effect of the boundary can be approximated by a large magnetic confinement field in the near‐boundary region. In this paper, we verify this conjecture for the 1.5D relativistic Vlasov–Maxwell (RVM) system on a bounded domain with an external confining magnetic field.

Stimulated Brillouin scattering of backward stimulated Raman scattering

The rescattering of backward stimulated Raman scattering (BSRS) by stimulated Brillouin scattering (SBS) is found in the high electron density region by relativistic Vlasov-Maxwell simulation and particle-in-cell (PIC) simulation, where the BSRS is in the regime of absolute instability and dominates in all the scatterings. Both one dimension (1D) Vlasov simulation and two dimension (2D) PIC simulation have been given to verify that there exists SBS of BSRS in the regime of absolute instability for BSRS. The SBS of BSRS will be even stronger than forward stimulated Raman scattering (FSRS) and SBS in regime of absolute instability for BSRS. Thus, besides Langmuir decay instability and laser energy absorption, the SBS of BSRS is also an important saturation mechanism of BSRS in high electron density region.

A semi-implicit, energy- and charge-conserving particle-in-cell algorithm for the relativistic Vlasov-Maxwell equations

Conventional explicit electromagnetic particle-in-cell (PIC) algorithms do not conserve discrete energy exactly. Time-centered fully implicit PIC algorithms can conserve discrete energy exactly, but may introduce large dispersion errors in the light-wave modes. This can lead to intolerable simulation errors where accurate light propagation is needed (e.g. in laser-plasma interactions). In this study, we selectively combine the leap-frog and Crank-Nicolson methods to produce an exactly energy- and charge-conserving relativistic electromagnetic PIC algorithm. Specifically, we employ the leap-frog method for Maxwell's equations, and the Crank-Nicolson method for the particle equations. The semi-implicit formulation still features a timestep CFL, but facilitates exact global energy conservation, exact local charge conservation, and preserves the dispersion properties of the leap-frog method for the light wave. The algorithm employs a new particle pusher designed to maximize efficiency and minimize wall-clock-time impact vs. the explicit alternative. It has been implemented in a code named iVPIC, based on the Los Alamos National Laboratory VPIC code (https://github.com/losalamos/vpic). We present numerical results that demonstrate the properties of the scheme with sample test problems: relativistic two-stream instability, Weibel instability, and laser-plasma instabilities.

On propagation of higher space regularity for nonlinear Vlasov equations

This work is concerned with the broad question of propagation of regularity for smooth solutions to non-linear Vlasov equations. For a class of equations (that includes Vlasov-Poisson and relativistic Vlasov-Maxwell), we prove that higher regularity in space is propagated, locally in time, into higher regularity for the moments in velocity of the solution. This in turn can be translated into some anisotropic Sobolev higher regularity for the solution itself, which can be interpreted as a kind of weak propagation of space regularity. To this end, we adapt the methods introduced in the context of the quasineutral limit of the Vlasov-Poisson system in [D. Han-Kwan and F. Rousset, Ann. Sci. \'Ecole Norm. Sup., 2016].

Transition from convective to absolute Raman instability via the longitudinal relativistic effect by using Vlasov-Maxwell simulations

The longitudinal relativistic effect on stimulated Raman backscattering (SRBS) is investigated by using one-dimensional (1D) Vlasov-Maxwell simulations. Using a short backscattered light seed pulse with a very small amplitude, the linear gain spectra of SRBS in the strongly convective regime is presented by combining the relativistic and non-relativistic 1D Vlasov-Maxwell simulations, which is in agreement with the steady-state linear theory. More interestingly, by considering transition from convective to absolute instability due to electron trapping, we successfully predict the critical duration of the seed which can just trigger the kinetic inflation of the excited SRBS after the seed leaves the simulation box. The critical duration in the relativistic case is much shorter than that in the nonrelativistic case, which indicates that the kinetic inflation more easily occurs in the relativistic case than in the nonrelativistic case. In the weakly convective regime, the transition from convective to absolute instability for SRBS can directly occur in the linear regime due to the longitudinal relativistic modification. For the same pump, our simulations first demonstrate that the SRBS excited by a short and small seed pulse is a convective instability in the nonrelativistic case but becomes an absolute instability due to the decrease of the linear Landau damping from the longitudinal relativistic modification in the relativistic case. In more detail, the growth rate of the backscattered light is also in excellent agreement with theoretical prediction.

Relativistic Vlasov-Maxwell modelling using finite volumes and adaptive mesh refinement

The dynamics of collisionless plasmas can be modelled by the Vlasov-Maxwell system of equations. An Eulerian approach is needed to accurately describe processes that are governed by high energy tails in the distribution function, but is of limited efficiency for high dimensional problems. The use of an adaptive mesh can reduce the scaling of the computational cost with the dimension of the problem. Here, we present a relativistic Eulerian Vlasov-Maxwell solver with block-structured adaptive mesh refinement in one spatial and one momentum dimension. The discretization of the Vlasov equation is based on a high-order finite volume method. A flux corrected transport algorithm is applied to limit spurious oscillations and ensure the physical character of the distribution function. We demonstrate a speed-up by a factor of 7 × in a typical scenario involving laser pulse interaction with an underdense plasma due to the use of an adaptive mesh.Graphical abstract

Dense plasma heating by crossing relativistic electron beams.

Here we investigate, using relativistic fluid theory and Vlasov-Maxwell simulations, the local heating of a dense plasma by two crossing electron beams. Heating occurs as an instability of the electron beams drives Langmuir waves, which couple nonlinearly into damped ion-acoustic waves. Simulations show a factor 2.8 increase in electron kinetic energy with a coupling efficiency of 18%. Our results support applications to the production of warm dense matter and as a driver for inertial fusion plasmas.

Discontinuous Galerkin Methods for Relativistic Vlasov–Maxwell System

The relativistic Vlasov–Maxwell (RVM) system is a kinetic model that describes the dynamics of plasma when the charged particles move in the relativistic regime and their collisions are not important. In this paper, we formulate and investigate discontinuous Galerkin (DG) methods to solve the RVM system. When standard piecewise polynomial functions are used to define trial and test spaces, the methods conserve mass as expected. However the energy conservation does not hold due to the specific form of the total energy of the system. In order to obtain provable mass and energy conservation, we take advantage of the flexibility of DG discretizations and enrich the discrete spaces with some non-polynomial function. For the semi-discrete DG methods with standard and enriched spaces, stability and error estimates are established together with their properties in conservation. In actual implementation with the enriched space, special care is needed to reduce the loss of significance for better numerical stability. Numerical experiments, including streaming Weibel instability and wakefield acceleration, are presented to demonstrate the performance of the methods. Positivity-preserving limiter is also used in simulating wakefield acceleration to obtain physically more relevant solutions.

Dispersion relations in cold magnetized plasmas

Starting from kinetic models of cold magnetized collisionless plasmas, we provide a complete description of the characteristic variety sustaining electromagnetic wave propagation. As in [ 4 , 13 , 17 ], the analysis is based on some asymptotic calculus exploiting the presence at the level of dimensionless relativistic Vlasov-Maxwell equations of a large parameter: the electron gyrofrequency. The method is inspired from geometric optics [ 29 , 33 ]. It allows to unify all the preceding results [ 8 , 12 , 38 , 31 , 37 , 40 ], while incorporating new aspects. Specifically, the non trivial effects [ 5 , 9 , 10 , 24 ] of the spatial variations of the background density, temperature and magnetic field are exhibited. In this way, a comprehensive overview of the dispersion relations becomes available, with important possible applications in plasma physics [ 7 , 28 , 30 ].

Can One Hear Whistler Waves?

The aim of this article is to propose a mathematical framework giving access to a better understanding of whistler-mode chorus emissions in space plasmas. There is presently a general agreement that the emissions of whistler waves involve a mechanism of wave-particle interaction which can be described in the framework of the relativistic Vlasov-Maxwell equations. In dimensionless variables, these equationsinvolve a penalized skew-symmetric term where the inhomogeneity of the strong exterior magnetic field plays an essential part. The description of the related phenomena is achieved in two stages. The first is based on a new approach allowing to extend in longer times the classical insights on fast rotating fluids; it justifies the existence and the validity of long time gyro-kinetic equations; it furnishes criterions to impose on a magnetic field in order to obtain the long time dynamical confinement of plasmas. The second stage is based on a study of oscillatory integrals implying special phases; it deals with the problem of the creation of light inside plasmas.

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<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.

On a Canonical Form for Maxwell Equations and Convergence of Finite Element Schemes for a Vlasov-Maxwell System

This work is a swift introduction to the nature of governing laws involved in the Maxwell equations. We then approximate a “one and one-half” dimensional relativistic Vlasov-Maxwell (VM) system using streamline diffusion finite element method. In this geometry d’Alembert representation for the fields functions guarantees the existence of a unique solution of the Maxwell equations. The VM system is then approximated using the streamline diffusion finite element method. In this part we derive some stability inequalities and optimal a priori error estimates due to the maximal available regularity of the exact solution.

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.

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.

Linear stability and instability of relativistic Vlasov‐Maxwell systems

AbstractWe consider the linear stability problem for a symmetric equilibrium of the relativistic Vlasov‐Maxwell (RVM) system. For an equilibrium whose distribution function depends monotonically on the particle energy, we obtain a sharp linear stability criterion. The growing mode is proved to be purely growing, and we get a sharp estimate of the maximal growth rate. In this paper we specifically treat the periodic 1½D case and the 3D whole‐space case with cylindrical symmetry. We explicitly illustrate, using the linear stability criterion in the 1½D case, several stable and unstable examples. © 2006 Wiley Periodicals, Inc.