Solutions Of Vlasov Research Articles

The scaling of relativistic double-layer widths: Poisson–Vlasov solutions and particle-in-cell simulations

In this paper the study of relativistic plasma double layers is described through the solution of the one-dimensional, unmagnetized, steady-state Poisson–Vlasov equations and by means of one-dimensional, unmagnetized, particle-in-cell simulations. The thickness versus potential-drop scaling law is extended to relativistic potential drops and relativistic plasma temperatures. The transition in the scaling law for ‘‘strong’’ double layers suggested by analytical two-beam models by Carlqvist [Astrophys. Space Sci. 87, 21 (1982)] is confirmed, and causality problems of standard double-layer simulation techniques applied to relativistic plasma systems are discussed.

Application of the method of spectral deformation to the Vlasov-poisson system

The Vlasov equation is studied using the method of spectral deformation, a technique developed in the theory of Schrödinger operators. The method associates the linearized Vlasov operator L with a family of operators L (θ) which depend analytically on the complex parameter θ. The analysis of L (θ) leads to a novel solution to the linear initial value problem, a new eigenfunction expansion which can be applied to nonlinear problems, and a deeper understanding of the relation between the spectrum of L and Landau damping. For example, it follows from the theory of L (θ) that the embedded eigenvalues of L , which characteristically occur at the threshold of a linear instability, are generically simple. Extending the deformation construction to include the nonlinear terms leads to a family of evolution equations depending on the parameter θ. This family is shown to be time reversal invariant in a suitably generalized sense. The analyticity properties required of Vlasov solutions if they are to correspond to solutions of the new equations for complex θ are described. The eigenfunction expansion for L (θ) is applied to derive equations for the nonlinear evolution of electrostatic waves. This derivation and the resulting amplitude equations are unusual in that the standard assumptions of weak nonlinearity and separated time scales are not used. When these assumptions are made, the familiar form of the amplitude equations is recovered.

New classes of stationary solutions of vlasov's equation for an axially symmetrical beam of charged particles of constant density

The problem of determining the stationary selfconsistent field of a beam of charged particles described by the Vlasov equation in a longitudinal magnetic field is considered. It is assumed that the particles are uniformly distributed over the cross-section of the beam. For the selfconsistent particle density, a representation in the form of a certain integral with an arbitrary function is proposed, which generalizes previously known distributions. An integral equation is also obtained for determining the selfconsistent particle density. New classes of solutions of the Vlasov equation are obtained on the basis of this representation and an integral equation, special cases of which are certain previously well-known solutions.

Global strong solutions of Vlasov's equation---necessary and sufficient conditions for their existence

Global strong solutions of Vlasov's equation---necessary and sufficient conditions for their existence

Spatially homogeneous cosmological models with a collisionless gas

The authors pose problems of finding the simultaneous solutions of Vlasov's equation and Einstein's equations for all types of spatially homogeneous cosmological models and substantiate the method of counterflowing beams in the case of strong anistropy. The solutions obtained can be used in studying the question of the emergence of some anistropic solutions into Friedmann's regime. In this case, the anisotropy is small and the structure of the energy-momentum tensor is substantially simplified.

Some estimates of the solution of vlasov's equation

It follows from the estimates obtained that the solution of Vlasov's equation undergoes little variation when there is a small variation of the initial data and of the interaction potential.

Limits of applicability of the Bohm formula

A comparison is made between the densities of ion current on spherical and cylindrical probes calculated by Bohm's approximate formula and on the basis of a rigorous numerical solution of Vlasov's system of equations.

Exact time-dependent solutions of the Vlasov–Poisson equations

The Vlasov–Poisson equations are relevant to collisionless plasmas and to stellar dynamics. They can be solved in one spatial dimension for an interesting class of cases by using a recent result about exact invariants of the motion of a particle in a one-dimensional potential. Lewis and Leach have given the necessary and sufficient conditions on the potential energy V(x,t) that an invariant which is quadratic in the momentum exist. For such a V(x,t), they exhibit the invariant explicitly. This result can be used to find the solutions of the one-dimensional Vlasov–Poisson equations for which the distribution functions are functions of quadratic functions of the momenta. A special case is the class of locally Maxwellian time-dependent solutions. The solutions for a single-species plasma, or a multispecies plasma where the charge to mass ratios are all equal, can be obtained by translating stationary solutions of the Vlasov–Poisson equations rigidly with an arbitrarily time-dependent displacement. If the charge to mass ratios of a multispecies plasma are unequal, then the solutions can be obtained by translating stationary solutions of modified Vlasov–Poisson equations with a displacement that depends quadratically on time. The modified Vlasov–Poisson equations include a species-dependent pseudogravity or ponderomotive force. This technique can be extended to obtain solutions of the Vlasov–Poisson or Vlasov–Maxwell equations in three dimensions.

A method for the numerical solution of Vlasov's equation

A method for the numerical solution of Vlasov's equation

Migma-type solutions of the Vlasov equation in cylindrical coordinates

A rigourous solution of the stationary 2-dimensional Vlasov equation is presented for the case of a parallel beam of deuterium ions injected into the central zone of a Migma model. It is assumed that the system is only partially neutralized so that ions and electrons move in self-consistent electric and magnetic fields. These fields are calculated together with the resulting angular distributions and ion densities. The Vlasov solution is used to determine the power produced in the Migma model. Collision broadening is simulated by introducing a distribution in space and momentum in the beam of incident particles.

Structure of current sheets in magnetic holes at 1 AU

Current density profiles in several types of interplanetary magnetic holes have been calculated using high‐resolution Imp 6 magnetic field data (12.5 vector measurements/s), assuming that the currents flow in planar sheets and that the magnetic field varies only in the direction normal to the sheet. The planarity was verified in four holes which were observed by two suitably spaced spacecraft. The structure of the current sheets ranges from very simple in some holes to very complex in others. Four types of simple magnetic holes are discussed, in which B varies nearly monotonically on each side of the hole. In two of the holes, B varies in intensity but not in direction as a result of currents normal to B. In the other two holes, B changes in both magnitude and direction as a result of currents both normal and parallel to B. The observed structures are found to be qualitatively consistent with the models of Burlaga and Lemaire, which are based on self‐consistent solutions of Vlasov's equation and Maxwell's equations. Examples of complex irregular magnetic holes are also presented, and they are shown to contain multiple current sheets in which currents flow parallel to one another at various angles with respect to B. There is no model of such magnetic holes at present.

Interplanetary magnetic holes: Theory

Magnetic holes in the interplanetary medium are explained as stationary nonpropagating equilibrium structures in which there are field‐aligned enhancements of the plasma density and/or temperature. Magnetic antiholes are considered to be associated with depressions in the plasma pressure. In this model the observed changes in the magnetic field intensity and direction are due to diamagnetic currents that are carried by ions which drift in a sheath as the result of gradients in the magnetic field and in the plasma pressure within the sheath. The thickness of the sheaths that we consider is approximately a few ion Larmor radii. An electric field is normal to the magnetic field in the sheath. Solutions of Vlasov's equation and Maxwell's equations are presented which account for several types of magnetic holes, including ‘null sheets,’ that have been observed.

Self-similar solutions for Vlasov and water-bag models

Using finite group theory, self-similar solutions for the one-dimensional Vlasov– Poisson system are considered, both for electron plasmas, with a fixed ion background, and for a single species beam. Difficulties arising from the Poisson equation are pointed out and handled by using an exponential group of transformations. Introducing, for the beam, a water-bag model, we find that the boundary condition problem, imposed by self-similarity, can be solved through the introduction of a ‘virtual particle’ concept. In order to test the analytical solution obtained, we perform a numerical simulation using a particle code. Complete agreement is found.

CONSTRUCTION OF A TURBULENT MEASURE FOR VLASOV'S SUSTEM OF EQUATIONS

A measure which is a generalization of the concept of a statistical solution and is useful for averaging functional nonlocal with respect to the time is constructed in the set of generalized solutions of Vlasov's system of equations.

Diamagnetic boundary layers: A kinetic theory

We present a kinetic theory for boundary layers associated with MHD tangential ‘discontinuities’ in a collisionless magnetized plasma such as those observed in the solar wind. The theory consists of finding self-consistent solutions of Vlasov's equation and Maxwell's equation for stationary, one-dimensional boundary layers separating two Maxwellian plasma states. Layers in which the current is carried by electrons are found to have a thickness of the order of a few electron gyroradii, but the drift speed of the current-carrying electrons is found to exceed the Alfven speed, and accordingly such layers are not stable. Several types of layers, in which the current is carried by protons are discussed; in particular, we considered cases in which the magnetic field intensity and/or direction changed across the layer. In every case, the thickness was of the order of a few proton gyroradii and the field changed smoothly, although the characteristics depended somewhat on the boundary conditions. The drift speed was always less than the Alfven speed, consistent with stability of such structures. Our results are consistent with the observations of boundary layers in the solar wind near 1 AU.

Kinetic theory of surface wave propagation along a hot plasma column

In this paper we propose an exact solution of Vlasov and Maxwell's equations for a bounded hot plasma in order to derive the dispersion relation of the axially-symmetric surface waves propagating along a plasma column. Assuming specular reflexion of plasma particles from the boundary, expressions for the components of the electric displacement vector are obtained on the basis of the Vlasov equation. Their substitution in Maxwell's equations, neglecting the spatial dispersion in the transverse plasma dielectric function, allows us to determine the plasma impedance. The equating of plasma and dielectric impedances gives the wave dispersion relation which, in different limiting cases, coincides with the well-known results.

Existence and uniqueness of the classical solution of vlasov's system of equations

Existence and uniqueness of the classical solution of vlasov's system of equations

Global existence of a weak solution of vlasov's system of equations

A PROOF is given of the theorem of the global existence of a weak solution of Vlasov's system of equations describing the evolution of the density distribution of a two-component plasma.

The recurrence of the initial state in the numerical solution of the Vlasov equation

The approximate recurrence of the initial state, observed recently in the numerical solution of Vlasov's equation by a finite-difference Eulerian model, is shown to be a property of three independent numerical methods. Some of the methods have exponentially growing modes (Dawson's beaming instabilities), and some others do not. The recurrence is in fact a manifestation of the finite velocity resolution of the numerical methods—a property which is independent of the approximation of a plasma by a finite number of electron beams. The recurrence is shown explicitly in the numerical simulation of Landau damping by three different methods: Fourier-Hermite, the recent variational method of Lewis, and the Eulerian finite-difference method.

On a family of exact nonlinear waves in plasmas

We describe a family of exact nonlinear solutions of Vlasov's equations; the linear counterpart being circularly waves. An exact solution for a sub-group of initial conditions is given. The plasmas treated are multi-species, infinite and homogeneous, embedded in a uniform magnetic field.