One-dimensional Vlasov Equation Research Articles

Study of inertial electrostatic confinement fusion using a finite-volume scheme for the one-dimensional Vlasov equation.

While the majority of fusion energy research is focused on magnetic confinement, there have been several alternative confinement methods aimed at the development of smaller and less expensive reactors. A number of these alternative reactors are based on a spherically convergent beam of recirculating ions and include designs such as inertial electrostatic confinement (IEC), multigrid IEC, and the periodically oscillating plasma sphere concept. Here, a fully time-dependent GPU-based Vlasov solver was developed in order to study these spherically convergent devices. This code solves the Vlasov equation for a spherically symmetric system using a finite-volume method with a modified flux to account for electrode transparency. The solver accounts for secondary electron emission, interactions between the charged particles, and collisional effects such as ionization and charge exchange. This code was used to investigate a system similar to the ion-injected device described by Hirsch (see [R. L. Hirsch, J. Appl. Phys. 38, 4522 (1967)10.1063/1.1709162]), who had reported a neutron production rate for deuterium-deuterium reactions in the range of 10^{6}to10^{7} neutrons per second, which was attributed to the formation of a virtual electrode structure near the center of the chamber. Attempts to reproduce this experiment [B. J. Egle, Ph.D. thesis, 2010] yielded similar fusion rates, though the majority of the reactions were found not to occur near the center of the chamber. The results of this Vlasov solver, considering only beam-beam and beam-background fusion reactions, show that beam-background reactions would be dominant in such an ion-injected device. This result is consistent with work by Baxter and Stuart, who proposed a simplified steady-state Boltzmann model. However, the result of both models are inconsistent with the experimental results, which indicate a higher neutron production rate, and an inverse pressure scaling trend. It is shown that the higher experimental rates may be explained by beam-target fusion between the ion beam and deuterium embedded on the inner surface of the cathode.

One-dimensional self-consistent kinetic models as Vlasov equation solutions

One-dimensional self-consistent kinetic models may describe some states of intense charged particle beams. In the collisionless approximation, which is appropriate for the short current pulse duration, the kinetic distribution function may be built as a function of the motion integrals. Two situations are considered corresponding to the wide charged particle flux with the sharp front and to the sheet continuous beam freely propagating in space. In both cases, one-dimensional Vlasov equation solutions are shown to exist, which are based on algebraic functions of the motion invariants.

The exact solution of one-dimensional nonrelativistic Vlasov equation: Antitropic electron beams and Landau damping

The exact stationary solution of one-dimensional non-relativistic Vlasov equation is obtained in the article. It is shown that in the energy exchange with the self-consistent longitudinal electric field, both wave trapped charged particles and the passing ones take part. It is proved that the trapped electron distribution is fundamentally different from distribution functions described by other authors, which used the Bernstein, Greene, and Kruskal method. So, the correct distribution function is characterized by its sudden change at the equality of wave and electrons' velocity but not on the edges of the potential well. This jump occurs for any arbitrary small value of wave potential. It was also found that the energy density of fast electrons trapped by the wave is less than the energy density of slow trapped electrons. This leads to the fact that the energy of the self-consistent electric field may both increase and decrease due to the nonlinear Landau damping. The conditions under which a similar effect can be observed are defined. Also for the first time, it is shown that the self-generated strong electric field always produces antitropic electron beams.

Hamiltonian reductions of the one-dimensional Vlasov equation using phase-space moments

We consider Hamiltonian closures of the Vlasov equation using the phase-space moments of the distribution function. We provide some conditions on the closures imposed by the Jacobi identity. We completely solve some families of examples. As a result, we show that imposing that the resulting reduced system preserves the Hamiltonian character of the parent model shapes its phase space by creating a set of Casimir invariants as a direct consequence of the Jacobi identity. We exhibit three main families of Hamiltonian models with two, three, and four degrees of freedom aiming at modeling the complexity of the bunch of particles in the Vlasov dynamics.

Monokinetic solutions to a singular Vlasov equation from a semiclassical perspective

Solutions to a singular one-dimensional Vlasov equation are obtained as the semiclassical limit of the Wigner transform associated to a logarithmic Schrodinger equation. Two frameworks are considered, regarding in particular the initial position density: Gaussian initial density, or smooth initial density away from vacuum. For Gaussian initial densities, the analysis also yields global solutions to the isothermal Euler system that do not enter the frame of regular solutions to hyperbolic systems by P. D. Lax.

On the neighborhood of an inhomogeneous stable stationary solution of the Vlasov equation—Case of an attractive cosine potential

We consider the one-dimensional Vlasov equation with an attractive cosine potential, and its non-homogeneous stable stationary states that are decreasing functions of the energy. We show that in the Sobolev space W1,p (p > 2) neighborhood of such a state, all stationary states that are decreasing functions of the energy are stable. This is in sharp contrast with the situation for homogeneous stationary states of a Vlasov equation, where a control over strictly more than one derivative is needed to ensure the absence of unstable stationary states in a neighborhood of a reference stationary state [Z. Lin and C. Zeng, Commun. Math. Phys. 306, 291-331 (2011)].

Algebraic damping in the one-dimensional Vlasov equation

We investigate the asymptotic behaviour of a perturbation around a spatially non-homogeneous stable stationary state of a one-dimensional Vlasov equation. Under general hypotheses, after transient exponential Landau damping, a perturbation evolving according to the linearized Vlasov equation decays algebraically with the exponent −2 and a well-defined frequency. The theoretical results are successfully tested against numerical N-body simulations, corresponding to the full Vlasov dynamics in the large N limit, in the case of the Hamiltonian mean-field model. For this purpose, we use a weighted particles code, which allows us to reduce finite size fluctuations and to observe the asymptotic decay in the N-body simulations.

DEPENDENCE OF THE ANOMALOUS RESISTIVITY ON THE INDUCED ELECTRIC FIELD IN SOLAR FLARES

Anomalous resistivity is a critical parameter for triggering the fast magnetic reconnection and interpreting the eruption of solar flares in the nearly collisionless coronal plasma. However, the mechanism for the production of anomalous resistivity and its evolution are weakly understood. In this paper, the one-dimensional Vlasov equation was numerically solved with the typical solar coronal parameters and realistic mass ratio in the presence of strong inductive electric field, and the relationship between the anomalous resistivity and the reconnecting electric field was inferred for the area near the center of reconnecting current sheets. Our principal findings are summarized as follows. (1) The relationship between the anomalous resistivity and the reconnecting electric field E 0 may be represented by ηeff = [10.82-10.99 exp (–0.36 E 0)]Ω m. (2) If E 0 is small enough, it may be described by ηeff = [4.02 E 0 – 0.18]Ω m, which is basically consistent with the early experimental results on the plasma response to the applied electric field. (3) In comparison with theoretical formulas for the current-driven ion-acoustic and Buneman anomalous resistivities, if E 0 is small, the anomalous resistivity may be due to the ion-acoustic instability; if E 0 is large, the anomalous resistivity may be due to the Buneman instability. These results are also basically consistent with early experiments.

Kinetic effects on the parametric decays of Alfvén waves in relativistic pair plasmas

Abstract Parametric decays of a circularly polarized wave propagating along a constant magnetic field in an electronpositron plasma are studied. Fully relativistic effects on the particle velocity in the wave field are considered, as well as kinetic effects in the parallel direction, by means of a one-dimensional relativistic Vlasov equation. In this approximation, a dispersion relation is found for the parametric decays which describes the coupling between normal modes of the system, namely electromagnetic sideband modes and Langmuir waves.

Phase space holes in non-periodic plasmas

We study the dynamics of phase space holes in a non-collisional plasma by numerical integration of the one-dimensional Vlasov equation. The plasma is bounded and connected to vacuum by two interfaces, which gives rise to non- periodic boundary conditions .B y choosing different initial conditions, we consider four different situations: development of a sinusoidal wave, two-stream instability, development of an ab initio Bernstein-Greene-Kruskal state and the expansion of a plasma into vacuum. For the latter situation ions move with a realistic mass of 1837 electron masses. The most prominent results of our investigation are: (a) a clockwise revolution of holes in phase space; (b) the appearance of stable oscillations of the total electron momentum, which, despite their striking linearity, cannot be related to electrostatic plasma oscillations. To the best of our knowledge none of these phenomena have been reported in kinetic simulations before: indeed we prove that the non-periodicity of the boundary conditions is an essential requirement for the sustainment of both hole revolution and momentum oscillations.

LONG TIME BEHAVIOR OF THE TWO-DIMENSIONAL VLASOV EQUATION WITH A STRONG EXTERNAL MAGNETIC FIELD

When charged particles are submitted to a large external magnetic field, their movement in first approximation occurs along the magnetic field lines and obeys a one-dimensional Vlasov equation along these field lines. However, when observing the particles on a sufficiently long time scale, a drift phenomenon perpendicular to the magnetic field lines superposes to this first movement. In this paper, we present a rigorous asymptotic analysis of the two-dimensional Vlasov equation when the magnetic field tends to infinity, the observation time scale increases accordingly. Techniques based on the two-scale convergence and the introduction of a second problem enable us to find an equation verified by the weak limit of the distribution function.

Vlasov equation with non-homogeneous boundary conditions

We study the one-dimensional Vlasov equation in a rod with non-homogeneous dissipative and conservative boundary conditions. The study is carried out by means of the theory of semigroups and affine operators. Existence, uniqueness and positivity of the solution are proved. The explicit form of the solution is derived and an approximating sequence is given.

The role of nonlinear Landau damping and the bounced motion of protons in the formation of dissipative structures in the solar wind plasma

Abstract. The present work examines the effects arising from the nonlinear Landau damping and the bounced motion of protons (trapped in the mirror geometry of the geomagnetic field) in the formation of nonlinear Alfvénic structures. These structures are observed at distances 1-5AU in the solar wind plasma (with ß ~ 1). The dynamics of formation of these structures can be understood using kinetic nonlinear Schrodinger (KNLS) model. The structures emerge due to balance of nonlinear steepening (of large amplitude Alfvén waves) by the linear Landau damping of ion-acoustic modes in a finite ß solar wind plasma. The ion-acoustic mode is driven nonlinearly by the large amplitude Alfvén waves. At the large amplitudes of Alfvén wave, the effects due to nonlinear Landau damping become important. These nonlinear effects are incorporated into the KNLS model by modifying the heat flux dissipation coefficient parallel to the ambient magnetic field. The effects arising from the bounced motion (of mirroring protons) are studied using a one-dimensional Vlasov equation. The bounced motion of the protons can lead to growth of the ion-acoustic mode, propagating in the mirror geometry of the geomagnetic field. The significance of these studies in the formation of dissipative quasistationary structures observed in solar wind plasma is discussed.

Numerical solution to the Vlasov equation: The 1D code

The code solves the one-dimensional Vlasov equation by an Eulerian Flux Balance algorithm (E. Fijalkow, A numerical solution to the Vlasov equation, Comput. Phys. Commun 116 (1999) 319–328). The self consistent electric field given by the solution of the Poisson equation is calculated by a method given by G Knorr et al. (G. Knorr, B. Joyce, A.J. Marcus, J. Comput. Phys. 38 (1980) 227–236). The output of the code is the distribution function in phase space and a listing of diagnostics as total mass conservation, total momentum and energy.

Longitudinal instability of space-charge dominated beams in transport channels with complex wall impedances

The longitudinal instability of space-charge dominated beams in transport channels with complex wall impedances is studied. The dispersion equation, derived from the one-dimensional Vlasov equation, characterizes the instability with all the relevant parameters including the space charge, the beam energy spread, the real and imaginary parts of the complex wall impedances. For a beam without energy spread, the growth rates of the instability in the different parameter ranges are discussed. The application of the theory to a resistive-wall channel and to a channel with induction gaps is presented. For a beam with energy spread, Lorentzian and Gaussian distributions are treated and the results presented illustrate the dependence of the stable regions on the beam perveance, the energy spread and the wall properties.

Lie transformation group solutions of the nonlinear one-dimensional Vlasov equation

The solution of the exact, nonlinear one-dimensional Vlasov equation with a space- and time-dependent electric field is reduced to the solution of a nonlinear, first-order ordinary differential equation with two subsidiary equations. The reduction holds for any electric field nonlinear in the spatial coordinate or for a subclass of electric fields linear in the spatial coordinate and is equivalent to the solution of a generalized Bernstein–Greene–Kruskal (BGK) Vlasov equation with a velocity-dependent Lorentz force. The Lie method for the solution of differential equations invariant under a transformation group has been used to calculate the group generator, the canonical variables, and the generalized BGK Vlasov equation. Analytical forms for the functional dependence of the Vlasov equation one-particle distribution function are given.

Small-signal gain of Free Electron Laser in a resonator Gaussian mode

We present an analytical expression for the small-signal gain of a Free Electron Laser (FEL) in the presence of a Gaussian mode. To describe the electron beam evolution we use the one-dimensional (1-d) Vlasov equation. Our perturbation result, valid for small values of the parameterq (length of the undulatorL divided by the Rayleigh rangeZR), can be extrapolated to values ofq≈4÷5.

Exact damped sinusoidal electric field of nonlinear one-dimensional Vlasov-Maxwell equations

An exact solution for a temporally damped sinusoidal electric field which obeys the nonlinear, one-dimensional Vlasov-Maxwell equations is given. The electric field is a generalization of the O'Neil model electric field for Landau damping of plasma oscillations. The electric field is a special case of the form found from the invariance of the one-dimensional Vlasov equation under infinitesimal Lie group transformations. The time dependences of the damping decrement, of the wave-number and of the angular frequency are derived. Use of a time-dependent BGK one-particle distribution function is justified for weak damping where, in general, it is necessary to carry out a numerical calculation of the invariant of which the distribution function is a functional.

A splitting scheme for the numerical solution of a one-dimensional Vlasov equation

A splitting scheme for the numerical solution of a one-dimensional Vlasov equation

The integration of the vlasov equation in configuration space

A convenient, fast, and accurate method of solving the one-dimensional Vlasov equation numerically in configuration space is described. It treats the convective terms in the x and v directions separately and produces a scheme of second order in Δt. The resulting free-streaming and accelerating equations are computed with Fourier interpolation and spline interpolation methods respectively. The numerical method is tested witth linear and nonlinear problems. The method is very accurate and efficient. A new method of smoothing the distribution function is given. It reduces the computational effort by artificially increasing the entropy of the system. As a result, the distribution function is smooth enough to be well represented on a given mesh. The methods can be generalized in a straightforward way to deal with more complicated cases such as problems with nonperiodic spatial boundary conditions, two- and three-dimensional problems with and without external magnetic and/or electric fields.