One-dimensional Vlasov Equation Research Articles

Existence and stability of strong potential double layers

An elementary integral equation technique is used to construct strong and weak stationary shock solutions from the one-dimensional Vlasov equation. It is shown that the plasma is Penrose stable in all points in space under certain conditions.

Lagrangian formulation of the one-dimensional Vlasov equation

A new formulation of the one-dimensional Vlasov equation is derived which is analogous to the Kalman-transformed cold-plasma equations. The equations are shown to yield nonsecular, nonlinear approximations to a source or boundary-value problem. It is suggested that the formulation may have other applications in nonlinear plasma theory.

A difference scheme for Vlasov's equation

A difference scheme is developed for the solution of the initial-value problem for the one-dimensional Vlasov equation for a collisionless electron plasma with a homogeneous ion background. Instead of the distribution function itself, its Fourier transform with respect to velocity is used. Case studies of nonlinear Landau damping, a stationary equilibrium, and a two-stream instability are presented as preliminary applications.

Breaking of Large Amplitude Plasma Oscillations

The existence and structure of large amplitude, stationary, longitudinal plasma oscillations are studied for the case of a simple waterbag distribution of electrons and an immovable background of ions. The analysis employs the one-dimensional Vlasov equation for a plasma of infinite spatial extent. An expression for the maximum amplitude of the oscillations is derived. This maximum amplitude decreases monotonically as the ratio of the electron thermal velocity to the wave phase velocity increases. The structure of the oscillations is expressed analytically in terms of hyperelliptic integrals.

Formation of Ion-Acoustic Collisionless Shocks

Formation of ion-acoustic collisionless shocks was studied by numerically solving the one-dimensional Vlasov equation. The shock propagates at about the local ion-acoustic speed, and faster than the ion-acoustic speed ahead; the oscillations behind the shock show ion-acoustic dispersion. A precursor appears with a speed of 1.6 times the shock speed. Different temperature and density ratios produce qualitatively different flows, and they are analyzed in detail.

Phase Space Hydrodynamics of Equivalent Nonlinear Systems: Experimental and Computational Observations

The one-dimensional Vlasov equation describes the behavior of an incompressible self-interacting classical fluid which moves in the (q, p) phase plane. This type of phase fluid occurs in many physical problems and its hydrodynamic properties can be examined from a general point of view. A characteristic feature with initially unstable spatially homogeneous configurations is the development of stable nonlinear phase structures. Such examples occur as the result of the gravitational Jeans instability, or the two-stream and negative-mass instabilities of charged-particle beams. These structures can be related to one another by extending a duality principle due to Dory. The stable cavities in phase space which have been observed in numerical calculations on the two-stream instability are compared with stable proton clusters which develop from the negative-mass instability in the mirror experiment DCX-1.

Inhomogeneous Plasma Oscillations in Streaming Plasmas

By means of a Hermite-Hermite expansion of the single-particle distribution function, the one-dimensional Vlasov equation is solved for counterstreaming plasmas for several inverse plasma frequencies; all of the nonlinear terms are retained. From an analysis of the electric fields generated by the plasma motion, the natural resonant frequencies are deduced for several initial conditions. The frequencies found in the analysis are not harmonically related, and the deduced dispersion relation is in agreement with previous theoretical predictions. Also, it is shown that a coherent electric field cannot be maintained over large distances.

Non-linear quasi-neutral electrostatic plasma waves and shock waves

For the stationary one-dimensional Vlasov equation quasi-neutrality is used instead of the Poisson equation. This allows one to obtain explicit solutions for the density as a function of the potential and to prove the positiveness of the trapped particle distribution function in a potential interval. This method has obvious application to oscillatory waves and shock waves.

Fourier-Hermite Solutions of the Vlasov Equations in the Linearized Limit

The properties of the Fourier-Hermite transformation of the one-dimensional Vlasov equation for the motion of electrons against a uniform positive neutralizing background are examined in the linearized limit. The general dispersion relation is presented for the transformed equations. Numerical and analytical results are given which show the recovery of Landau damping for Maxwell's distribution even though the corresponding eigenfrequencies are always real. An example of double streaming is treated to show the appearance of imaginary eigenfrequencies in the analysis. Introduction of a small number of collisions is shown to improve the long-term behavior. Nonlinear numerical results for the nearly cold plasma are shown to agree with the exact treatment of Kalman.

Kinetische Theorie der Plasmaschwingungen im Plasmakondensator

AbstractStationary excitation of plasma oscillations by an external alternating electric field in a collisionless onedimensional plasma layer (plasma condenser) is examined. For this the linearized onedimensional Vlasov equation is integrated along the one‐electron characteristics, assuming the electrons are reflected specularly at the boundaries. In the case of a homogeneous plasma the resulting Fredholm integral equation for the electric field is solved exactly. The results agree with Shure's, received on the basis of distributional analysis à la Van Kampen and Case. In view of Tonks‐Dattner resonances the resonance properties of the homogeneous plasma condenser are discussed. Contrary to the hydrodynamical theory here we receive only a finite number of resonances, increasing with the ratio r of layer diameter to Debye length. The ratio of series limit to main resonance position is nearly constant for r not too small. This agrees with Dattner's qualitative statements about Tonks‐Dattner resonances found experimentally. However, the kinetical effect of series limit and another one of a second serie of resonances at lower frequencies are masked by strong Landau damping in this region of dispersion. The method of characteristics used here has the advantage of easily being generalized for the case of an inhomogeneous plasma condenser. For this case an integral equation for the electric field is derived in full analogy to the procedure employed for the homogeneous plasma. Finally the kernel of this equation is specialized for a parabolic static potential.

Plasma Wave Propagation in Hot Inhomogeneous Media

The WKB solution for plasma wave propagation is obtained for the one-dimensional Vlasov equation. In principle, a solution can be obtained to all orders in the smallness parameter.