E. Fijalkow

A Vlasov code for the numerical simulation of stimulated Raman scattering

Numerical simulations of the stimulated Raman scattering are presented using an Eulerian relativistic Vlasov code. Such a code allows a finer resolution in phase space than a particle code and provides a better understanding of the acceleration process for the particles at relativistically high energy. Forward Raman scattering as well as backward Raman scattering are considered to illustrate the possibilities of the Eulerian Vlasov code.

Stability of Bernstein–Greene–Kruskal plasma equilibria. Numerical experiments over a long time

Bernstein–Greene–Kruskal (BGK) equilibria for a Vlasov plasma consisting of a periodic structure exhibiting depressions or ‘‘holes’’ in phase space are under consideration. Marginal stability analysis indicates that such structures are unstable when the system contains at least two holes. An Eulerian numerical code is developed allowing noiseless information on the long time phase space behavior (about 103ω−1p) to be obtained. Starting with equilibria with up to six holes, it is shown that the final state is given by a structure with only one large hole, the initial instability inducing coalescences of the different holes. On the other hand, starting with a homogeneous two‐stream plasma it is shown that, in a first step, a BGK periodic structure appears with a number of holes proportional to the length of the system, followed, in a second step, by a coalescence of the holes to always end up with the above mentioned one large hole structure.

Solution of the multidimensional quantum harmonic oscillator with time-dependent frequencies through Fourier, Hermite and Wigner transforms

Abstract A group of transformations preserving the hamiltonian formalism is introduced to transform the problem to free particle motion. Moreover, introducing the Wigner distribution function allows a description through a particle picture reducing the problem to the solution of decoupled differential equations and avoiding all the difficulties connected with Fourier or Hermite transform calculations.

Self-similar and asymptotic solutions for a one-dimensional Vlasov beam

Rescaling transformations bringing friction terms in the new equation are used to obtain the asymptotic solution of a one-dimensional, one-species beam. It is shown that for all possible initial conditions this asymptotic solution coincides with the self-similar solution.

The numerical integration of the Vlasov equation possessing an invariant

A new method for the numerical integration of the Vlasov equation is presented, which can be applied whenever its characteristics possess an exact invariant. It consists in expressing the distribution function in terms of the invariant itself. The dimensionality of the phase space is thus reduced of one unity, since the invariant only appears as a label of the Vlasov equation and can be coarsely discretized. This technique is applied to the study of the Kelvin-Helmoltz instability, with a very limited number of invariants. Subsequently an example of ion-temperature-gradient instability is analyzed. Although a larger number of invariants are required to describe the temperature profile, qualitatively correct results can be obtained with fewer invariants. Test particles are used to illustrate stochastic diffusion in the phase space and to calculate the diffusion coefficients.

Numerical simulation of a negative ion plasma expansion into vacuum

The expansion into vacuum of a one-dimensional, collisionless, negative ion plasma is investigated in the framework of the Vlasov–Poisson model. The basic equations are written in a “new time space” by use of a rescaling transformation and, subsequently, solved numerically through a fully Eulerian code. As in the case of a two species plasma, the time-asymptotic regime is found to be self-similar with the temperature decreasing as t−2. The numerical results exhibit clearly the physically expected effects produced by the variation of parameters such as initial temperatures, mass ratios and charge of the negative ions.

BGK structures as quasi-particles

Abstract The Vlasov-Poisson system describing the stability of Bernstein-Greene-Kruskal equilibrium is solved by a direct integration in phase space. The nonlinear evolution of this system, in a one-dimensional and collisionless plasma, is strongly influenced by an initial breaking of the symmetry, which leads to an acceleration of the evolution. New physical results of strong nonlinear BGK instability are also reported, such the tendency of holes, in phase space, to behave as quasi-particles.

Group Transformation for Phase Space Fluids

We want to study the time behavior of systems where long distance forces are predominant. Such is the case of plasmas, accelerator beam (where we are dealing with Coulomb forces plus electromagnetic confining external fields) and self-gravitating gas (galaxy, cluster of stars, etc.) where the Newtonian attraction competes against the thermal (ballistic) expansion. In many cases we can disregard the small irregularities due to the grain structure of the matter (with grain as big as a star in a galaxy!) and describe the interaction through a continuous field obtained by the solution of the Poisson equation. This is the well-known Vlasov Poisson system where the global description is obtained by considering the distribution function \(f(\vec x,\vec v,t)\) in the six-dimensional phase space in contrast to a regular gas where we can usually deal with the first moments of f with respect to \(\vec x\) (particle density, momentum, energy density, etc.). Consequently, we call such systems phase space fluids. A discussion of the relative properties of these a priori very different fluids is given by Feix [1975]. From the model maker’s point of view adopted here they present great similarities.

GENERATION OF ELECTROMAGNETIC RADIATION IN A RELATIVISTIC BEAM-PLASMA INTERACTION

Abstract A 1D periodic, electromagnetic, relativistic Vlasov code, with a fluid approximation in the normal direction, is used to study the interaction of a relativistic beam with a plasma, in the absence of an external magnetic field. We study the coupling and growth of an electromagnetic radiation during and after the beam-plasma interaction.

Nonlinear evolution of the beam-plasma instabilities

Abstract We study numerically the nonlinear evolution of the beam-plasma instabilities using the one-dimensional Vlasov-Poisson system. When the ratio of the beam density to the background density is about 10%, a strong beam-plasma instability develops, causing rapid diffusion of the particles, which are accelerated considerably beyond the initial beam velocity, resulting in a two-temperature maxwellian distribution function with a high energy tail having a smooth negative slope. The relevance of these results to current drive experiments via injection of an electron beam is discussed.