Quasi-neutral Limit Research Articles

Nonlinear Instability of Vlasov--Maxwell Systems in the Classical and Quasineutral Limits

We study the instability of solutions to the relativistic Vlasov--Maxwell systems in two limiting regimes: the classical limit when the speed of light tends to infinity and the quasineutral limit when the Debye length tends to zero. First, in the classical limit, $\varepsilon \to 0$, with $\varepsilon$ being the inverse of the speed of light, we construct a family of solutions that converge initially polynomially fast to a homogeneous solution $\mu$ of Vlasov--Poisson systems in arbitrarily high Sobolev norms, but become of order one away from $\mu$ in arbitrary negative Sobolev norms within time of order $|\log \varepsilon|$. Second, we deduce the invalidity of the quasineutral limit in $L^2$ in arbitrarily short time.

Quasineutral limit for Vlasov-Poisson with Penrose stable data

We study the quasineutral limit of a Vlasov-Poisson system that describes the dynamics of ions in a plasma. We handle data with Sobolev regularity under the sharp assumption that the profile of the initial data in the velocity variable satisfies a Penrose stability condition. As a by-product of our analysis, we obtain a well-posedness theory for the limit equation (which is a Vlasov equation with Dirac distribution as interaction kernel) for such data.

Convergence of the Two-Species Vlasov-Poisson System to the Pressureless Euler Equations

In this paper we study the quasineutral limit of the two-species Vlasov-Poisson system. For well-prepared initial data, we establish the convergence of the Vlasov-Poisson system to the pressureless Euler system as the Debye length tends to zero.

The combined non-relativistic and quasi-neutral limit of two-fluid Euler–Maxwell equations

We consider two-fluid Euler–Maxwell equations for magnetized plasmas composed of electrons and ions. By using the method of asymptotic expansions, we analyze the combined non-relativistic and quasi-neutral limit for periodic problems with well-prepared initial data. It is shown that the small parameter problems have a unique solution existing in a finite time interval where the corresponding limit problems (compressible Euler equations) have smooth solutions. The proof is based on energy estimates for symmetrizable hyperbolic equations and on the exploration of the coupling between the Euler equations and the Maxwell equations.

Cylindrical plasmas generated by an annular beam of ultraviolet light

We investigate a cylindrical plasma system with ionization, by an annular beam of ultraviolet light, taking place only in the cylinder's outer region. In the steady state, both the outer and inner regions contain a plasma, with that in the inner region being uniform and field-free. At the interface between the two regions, there is an infinitesimal jump in ion density, the magnitude approaching zero in the quasi-neutral (λD → 0) limit. The system offers the possibility of producing a uniform stationary plasma in the laboratory, hitherto obtained only with thermally produced alkali plasmas.

Quasi-neutral limit of the full Navier–Stokes–Fourier–Poisson system

The quasi-neutral limit of the full Navier–Stokes–Fourier–Poisson system in the torus Td (d≥1) is considered. We rigorously prove that as the scaled Debye length goes to zero, the global-in-time weak solutions of the full Navier–Stokes–Fourier–Poisson system converge to the strong solution of the incompressible Navier–Stokes equations as long as the latter exists. In particular, the effect of large temperature variations is taken into account.

Quasi-Neutral Limit, Dispersion, and Oscillations for Korteweg-Type Fluids

In the setting of general initial data and whole space we perform a rigorous analysis of the quasineutral limit for a hydrodynamical model of a viscous plasma with capillarity tensor represented by the Navier Stokes Korteweg Poisson system. We shall provide a detailed mathematical description of the convergence process by analyzing the dispersion of the fast oscillating acoustic waves. However the standard acoustic wave analysis is not sufficient to control the high frequency oscillations in the electric field but it is necessary to estimates the dispersive properties induced by the capillarity terms. Therefore by using these additional estimates we will be able to control, via compensated compactness, the quadratic nonlinearity of the stiff electric force field. In conclusion, opposite to the zero capillarity case \cite{DM12} where persistent space localized time high frequency oscillations need to be taken into account, we show that as $\lambda \to 0$, the density fluctuation $\rho^{\lambda}-1$ converges strongly to zero and the fluids behaves according to an incompressible dynamics.

Multiscale Schemes for the BGK--Vlasov--Poisson System in the Quasi-Neutral and Fluid Limits. Stability Analysis and First Order Schemes

This paper deals with the development and the analysis of asymptotically stable and consistent schemes in the joint quasi-neutral and fluid limits for the collisional Vlasov--Poisson system. In these limits, the classical explicit schemes suffer from time step restrictions due to the small plasma period and Knudsen number. To solve this problem, we propose a new scheme stable for choices of time steps independent from the small scale dynamics and with comparable computational cost with respect to standard explicit schemes. In addition, this scheme reduces automatically to consistent discretizations of the underlying asymptotic systems. In this paper, we propose a first order in time scheme, and we perform a relative linear stability analysis to deal with such problems. The framework we propose will permit us to extend this approach to high order schemes in the near future. Finally, we show the capability of the method in dealing with small scales through numerical experiments.

Stability Issues in the Quasineutral Limit of the One-Dimensional Vlasov–Poisson Equation

This work is concerned with the quasineutral limit of the one-dimensional Vlasov-Poisson equation, for initial data close to stationary homogeneous profiles. Our objective is threefold: first, we provide a proof of the fact that the formal limit does not hold for homogeneous profiles that satisfy the Penrose instability criterion. Second, we prove on the other hand that the limit is true for homogeneous profiles that satisfy some monotonicity condition, together with a symmetry condition. We handle the case of well-prepared as well as ill- prepared data. Last, we study a stationary boundary-value problem for the formal limit, the so-called quasineutral Vlasov equation. We show the existence of numerous stationary states, with a lot of freedom in the construction (compared to that of BGK waves for Vlasov-Poisson): this illustrates the degeneracy of the limit equation.

From quantum Euler–Maxwell equations to incompressible Euler equations

The combined quasi-neutral and non-relativistic limit of compressible quantum Euler–Maxwell equations for plasmas is studied in this paper. For well-prepared initial data, it is shown that the smooth solution of compressible quantum Euler–Maxwell equations converges to the smooth solution of incompressible Euler equations by using the modulated energy method. Furthermore, the associated convergence rates are also obtained.

Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries II

In this paper, we study the quasineutral limit of the isothermal Euler-Poisson equation for ions, in a domain with boundary. This is a follow-up to our previous work \cite{GVHKR}, devoted to no-penetration as well as subsonic outflow boundary conditions. We focus here on the case of supersonic outflow velocities. The structure of the boundary layers and the stabilization mechanism are different.

Convergence of compressible Navier-Stokes-Maxwell equations to incompressible Navier-Stokes equations

The combined quasi-neutral and non-relativistic limit of compressible Navier-Stokes-Maxwell equations for plasmas is studied. For well-prepared initial data, it is shown that the smooth solution of compressible Navier-Stokes-Maxwell equations converges to the smooth solution of incompressible Navier-Stokes equations by introducing new modulated energy functional.

Convergence of the Euler–Maxwell two-fluid system to compressible Euler equations

The combined non-relativistic and quasi-neutral limit of two-fluid Euler–Maxwell equations for plasmas is rigorously justified in this paper. For well-prepared initial data, the convergence of the two-fluid Euler–Maxwell system to the compressible Euler equations is proved in the time interval where a smooth solution of the limit problem exists.

Study of a Finite Volume Scheme for the Drift-Diffusion System. Asymptotic Behavior in the Quasi-Neutral Limit

In this paper, we are interested in the numerical approximation of the classical time-dependent drift-diffusion system near quasi-neutrality. We consider a fully implicit in time and finite volume in space scheme, where the convection-diffusion fluxes are approximated by Scharfetter-Gummel fluxes. We establish that all the a priori estimates needed to prove the convergence of the scheme does not depend on the Debye length $\lambda$. This proves that the scheme is asymptotic preserving in the quasi-neutral limit $\lambda \to 0$.

Quasineutral limit of the pressureless Euler–Poisson equation

Abstract In this short paper, we consider the quasineutral limit for the pressureless Euler–Poisson system for ions. By applying the modulated energy method, it shows that the weak solutions for the Euler–Poisson system converge weakly to the strong solutions of the compressible Euler equation as the Debye length tends to zero.

The asymptotic behavior and the quasineutral limit for the bipolar Euler-Poisson system with boundary effects and a vacuum

In this paper, a one-dimensional bipolar Euler-Poisson system (a hydrodynamic model) from semiconductors or plasmas with boundary effects is considered. This system takes the form of Euler-Poisson with an electric field and frictional damping added to the momentum equations. The large-time behavior of uniformly bounded weak solutions to the initial-boundary value problem for the one-dimensional bipolar Euler-Poisson system is firstly presented. Next, two particle densities and the corresponding current momenta are verified to satisfy the porous medium equation and the classical Darcy’s law time asymptotically. Finally, as a by-product, the quasineutral limit of the weak solutions to the initial-boundary value problem is investigated in the sense that the bounded L∞ entropy solution to the one-dimensional bipolar Euler-Poisson system converges to that of the corresponding one-dimensional compressible Euler equations with damping exponentially fast as t → +∞. As far as we know, this is the first result about the asymptotic behavior and the quasineutral limit for the one-dimensional bipolar Euler-Poisson system with boundary effects and a vacuum.

Self-similar space-time evolution of an initial density discontinuity

The space-time evolution of an initial step-like plasma density variation is studied. We give particular attention to formulate the problem in a way that opens for the possibility of realizing the conditions experimentally. After a short transient time interval of the order of the electron plasma period, the solution is self-similar as illustrated by a video where the space-time evolution is reduced to be a function of the ratio x/t. Solutions of this form are usually found for problems without characteristic length and time scales, in our case the quasi-neutral limit. By introducing ion collisions with neutrals into the numerical analysis, we introduce a length scale, the collisional mean free path. We study the breakdown of the self-similarity of the solution as the mean free path is made shorter than the system length. Analytical results are presented for charge exchange collisions, demonstrating a short time collisionless evolution with an ensuing long time diffusive relaxation of the initial perturbation. For large times, we find a diffusion equation as the limiting analytical form for a charge-exchange collisional plasma, with a diffusion coefficient defined as the square of the ion sound speed divided by the (constant) ion collision frequency. The ion-neutral collision frequency acts as a parameter that allows a collisionless result to be obtained in one limit, while the solution of a diffusion equation is recovered in the opposite limit of large collision frequencies.

Quasineutral plasma expansion into infinite vacuum as a model for parallel ELM transport

An analytic solution for the expansion of a plasma into vacuum is assessed for its relevance to the parallel transport of edge localized mode (ELM) filaments along field lines. This solution solves the 1D1V Vlasov–Poisson equations for the adiabatic (instantaneous source), collisionless expansion of a Gaussian plasma bunch into an infinite space in the quasineutral limit. The quasineutral assumption is found to hold as long as λD0/σ0 ≲ 0.01 (where λD0 is the initial Debye length at peak density and σ0 is the parallel length of the Gaussian filament), a condition that is physically realistic. The inclusion of a boundary at x = L and consequent formation of a target sheath is found to have a negligible effect when L/σ0 ≳ 5, a condition that is physically plausible. Under the same condition, the target flux densities predicted by the analytic solution are well approximated by the ‘free-streaming’ equations used in previous experimental studies, strengthening the notion that these simple equations are physically reasonable. Importantly, the analytic solution predicts a zero heat flux density so that a fluid approach to the problem can be used equally well, at least when the source is instantaneous. It is found that, even for JET-like pedestal parameters, collisions can affect the expansion dynamics via electron temperature isotropization, although this is probably a secondary effect. Finally, the effect of a finite duration, τsrc, for the plasma source is investigated. As is found for an instantaneous source, when L/σ0 ≳ 5 the presence of a target sheath has a negligible effect, at least up to the explored range of τsrc = L/cs (where cs is the sound speed at the initial temperature).

Modulation instability of an intense laser beam in the hot magnetized electron-positron plasma in the quasi-neutral limit

The aim of the present study is to investigate the problem of modulation instability of an intense laser beam in the hot magnetized electron-positron plasma. Propagation of the intense circularly polarized laser beam along the external magnetic field is studied using a relativistic fluid model. A nonlinear equation describing the interaction of the laser pulse with the magnetized hot pair plasma is derived based on the quasi-neutral approximation, which is valid for the hot plasma. Also, the nonlinear dispersion equation for the hot plasma is obtained. The growth rate of the instability is calculated and its dependence on temperature and external magnetic field are considered.

Self-focusing of circularly polarized laser pulse in the hot magnetized plasma in the quasi-neutral limit

In this paper the problem of self-focusing of an intense laser beam in the hot magnetized plasma is studied. Using a relativistic two-fluid model, the propagation of intense circularly polarized laser beam along the external magnetic field is considered. The nonlinear equation describing the interaction of laser beam with magnetized hot plasma in the quasi-neutral approximation is derived. Expanding the nonlinear terms of current density and saving only the parabolic nonlinear terms, we investigated the self-focusing phenomenon for right- and left-hand circularly polarized laser beam. The evolution of laser beam spot size with Gaussian profile for both polarizations is considered and the effect of the temperature and external magnetic field on the self-focusing quality is studied. It is shown that for the right-hand polarization, the effect of temperature and external magnetic field on the self-focusing property is more considerable.