Solutions Of Vlasov Research Articles

Valis: A direct Vlasov solver for modelling laser plasma interaction

An accurate treatment of the relativistic Vlasov-Maxwell system is of fundamental importance to a broad range of plasma physics topics of direct relevance to inertial confinement fusion and fast ignition. With recent advances in high performance computing, the direct solution of Vlasov's equation is becoming increasingly viable as an additional modelling tool, alongside established particle in cell methods. We introduce VALIS, an algorithm for the numerical solution of the Vlasov-Maxwell system in two spatial dimensions with two momentum dimensions, and demonstrate its application to the problem of modelling short-pulse laser absorption in short-scale length, over-dense targets with system parameters comparable to those achieved on the HELEN laser system at AWE.

Invariant solutions of integro-differential Vlasov–Maxwell equations in Lagrangian variables by Lie group analysis

The Lie point symmetries of the Vlasov–Maxwell system in Lagrangian variables are investigated by using a direct method for symmetry group analysis of integro-differential equations, with emphasis on solving nonlocal determining equations. All similarity reduction forms for the system are obtained by using different approaches and some analytical and numerical solutions are presented.

A magnetospheric generator driving ion and electron acceleration and electric currents in a discrete auroral arc observed by Cluster and DMSP

Simultaneous observations on April 28, 2001 by Cluster and DMSP‐F14 reveal a stable discrete auroral arc and fluxes of field‐aligned accelerated electrons and ions coincident with a magnetospheric plasma interface at an altitude of 4.5 RE in the dusk sector. We compare satellite data with a quasi‐stationary magnetosphere‐ionosphere coupling model based on a Vlasov solution for the magnetospheric generator. The model provides a self‐consistent magnetospheric electric potential matching the Cluster observations. The ionospheric potential is derived from the current continuity equation and gives a field‐aligned potential drop and a flux of precipitating energy in agreement with the DMSP data. Model results and data analysis suggest a quasi‐stationary field‐aligned acceleration of auroral electrons and ions with a magnetospheric generator. We associate the generator with the convergent perpendicular electric field at the interface of the plasma sheet boundary layer with the lobe or at the inner edge of the low latitude boundary layer.

Solving Vlasov–Maxwell equations in singular geometries

This paper is devoted to the solution of the time-dependent Vlasov–Maxwell equations in singular geometries, i.e. when the boundary includes reentrant corners or edges. Indeed, computing the electromagnetic fields in this case is a challenge per se, as these geometrical singularities generate very strong solutions in their neighborhood. Moreover, they have also an influence over the solution of the Vlasov equation, through the coupling. We propose here a method to solve this problem, illustrated by numerical examples.

A MATHEMATICAL THEORY OF ISOLATED SYSTEMS IN RELATIVISTIC PLASMA PHYSICS

The existence and the properties of isolated solutions to the relativistic Vlasov–Maxwell system with initial data on the backward hyperboloid [Formula: see text] are investigated. Isolated solutions of Vlasov–Maxwell can be defined by the condition that the particle density is compactly supported on the initial hyperboloid and by imposing the absence of incoming radiation on the electromagnetic field. Various consequences of the mass-energy conservation laws are derived by assuming the existence of smooth isolated solutions which match the inital data. In particular, it is shown that the mass-energy of isolated solutions on the backward hyperboloids and on the surfaces of constant proper time are preserved and equal, while the mass-energy on the forward hyperboloids is non-increasing and uniformly bounded by the mass-energy on the initial hyperboloid. Moreover the global existence and uniqueness of classical solutions in the future of the initial surface is established for the one-dimensional version of the system.

Two-dimensional Vlasov solution for a collisionless plasma jet across transverse magnetic field lines with a sheared bulk velocity

We consider a two-dimensional (2D) stationary stream of a collisionless plasma injected across an external stationary magnetic field and a background stagnant plasma. The solution is found by solving the Vlasov equation for each species (electrons and protons), the Maxwell-Ampere equation for the magnetic vector potential, and the equation of plasma quasineutrality for the electrostatic potential. The solution of the stationary Vlasov equation is given in terms of two constants of motion and one adiabatic invariant. The partial charge and current densities are given by analytical expressions of the moments of the velocity distribution functions for each particle species. The 2D distribution of the plasma bulk velocity, Vx (y,z), is roughly uniform inside the jet. There is no plasma bulk flow in the direction of the magnetic field. Inside the boundary layer interfacing the jet and the stagnant plasma, the bulk velocity has gradients (i.e., shears) in the direction parallel as well perpendicular to the magnetic field. The parallel component of this gradient, inverted Delta parallel) inverted Delta perpendicular, produces a nonzero electric field component parallel to the magnetic field lines, E.B not equal to 0. The parallel electric field within the transition layer is a basic element allowing plasma elements to be transported across magnetic field lines in astrophysical systems as well as in laboratory experiments where plasmoids are injected across magnetic fields.

A cloudy Vlasov solution

We propose to integrate the Vlasov-Poisson equations giving the evolution of a dynamical system in phase-space using a continuous set of local basis functions. In practice, the method decomposes the density in phase-space into small smooth units having compact support. We call these small units ``clouds\'\' and choose them to be Gaussians of elliptical support. Fortunately, the evolution of these clouds in the local potential has an analytical solution, that can be used to evolve the whole system during a significant fraction of dynamical time. In the process, the clouds, initially round, change shape and get elongated. At some point, the system needs to be remapped on round clouds once again. This remapping can be performed optimally using a small number of Lucy iterations. The remapped solution can be evolved again with the cloud method, and the process can be iterated a large number of times without showing significant diffusion. Our numerical experiments show that it is possible to follow the 2 dimensional phase space distribution during a large number of dynamical times with excellent accuracy. The main limitation to this accuracy is the finite size of the clouds, which results in coarse graining the structures smaller than the clouds and induces small aliasing effects at these scales. However, it is shown in this paper that this method is consistent with an adaptive refinement algorithm which allows one to track the evolution of the finer structure in phase space. It is also shown that the generalization of the cloud method to the 4 dimensional and the 6 dimensional phase space is quite natural.

Kinetic susceptibility and transport theory of collisional plasmas.

A system of nonlocal electron transport equations for electrostatic perturbations in (omega,k) space in a high-Z plasma is derived from the Fokker-Planck equation for arbitrary relations between the time, space, and collisionality scales. The closed scheme for obtaining the longitudinal plasma susceptibility epsilon(omega,k) in the entire (omega,k) plane is proposed. Regions in the (omega,k) plane have been mapped for problems such as the relaxation of the local temperature enhancement with a time-dependent heat conductivity. The electron dielectric permittivity has been calculated over the entire range of parameters, including the transition region between Vlasov and Fokker-Planck equation solutions.

Io‐related Jovian auroral arcs: Modeling parallel electric fields

Recent observations of auroral arcs on Jupiter suggest that electrons are being accelerated downstream from Io's magnetic footprint, creating detectable emissions. The downstream electron acceleration is investigated using one‐dimensional spatial, two‐dimensional velocity static Vlasov solutions under the constraint of quasi‐neutrality and an applied potential drop. The code determines self‐consistent charged particle distributions and potential structure along a magnetic field flux tube in the upward (with respect to Jupiter) current region of Io's wake. The boundaries of the flux tube are the Io torus on one end and Jupiter's ionosphere on the other. The results indicate that localized electric potential drops tend to form at 1.5–2.5 RJ Jovicentric distance. A sufficiently high secondary electron density causes an auroral cavity to be produced similar to that on Earth. Interestingly, the model results suggest that the proton and the hot electron population in the Io torus control the electron current densities between the Io torus and Jupiter and thus may control the energy flux and the brightness of the aurora downstream from Io's magnetic footprint. The parallel electric fields also are expected to create an unstable horseshoe electron distribution inside the auroral cavity, which may lead to the shell electron cyclotron maser instability. Results from our model suggest that in spite of the differing boundary conditions and the large centrifugal potentials at Jupiter, the auroral cavity formation may be similar to that of the Earth and that parallel electric fields may be the source mechanism of Io‐controlled decametric radio emissions.

Generalised Sagdeev Potentials for Dusty Plasmas with Varying Grain Charges

Grain charges in a dusty plasma are determined by the random currents from the ambient plasma and vary with the local conditions. The charge on a slowly moving grain will be close to the locally determined equilibrium, given for zero net current to the grain. For a steady electrostatic structure (e.g., solitary wave, double layer) integrals of motion for grains with varying charge can then be found. (These integrals reduce to the total energy if the charge is constant, but in general the electrostatic term becomes an integral of the grain charge with respect to the potential.) Steady state solutions of Vlasov’s equation are piecewise given by arbitrary functions of these integrals of motion. A generalised Sagdeev (Classical) potential can be found, which is, to within an added constant, equal to minus the sum of the total particle pressures (including that of the grains). This extends the well known equivalence found for conventional plasmas and dusty plasmas with constant grain charges. The analysis of dust acoustic solitary waves is modified by additional terms proportional to potential derivatives of the charge. A grain size distribution may be incorporated. The second derivative of the Sagdeev potential (leading to the generalised Bohm condition) is then given in terms of the same effective distribution function as found for linear electrostatic modes. Comparisons are made with several analyses of nonlinear electrostatic structures including dynamical charging.

Possibility of a direct solution of Vlasov–Maxwell equations in tokamaks in the ion cyclotron range of frequencies

The possibility of a solution of the Vlasov–Maxwell equations as a time-dependent problem in a tokamak geometry at ω∼ωci is studied. The major approximations in the analysis are the following: small Larmor radius relative to the wavelength, particle orbits deviate little from a magnetic surface during the time of the dominant contribution to the orbit integral. This direct integration becomes possible due to a rapid growth of the computing power and the development of multiprocessing. A proper description of the plasma dielectric properties in this time-dependent approach is made. Due to a limited computing power and not completely optimized procedure applied in the analysis, the time evolution of Maxwell equations is limited to cases of relatively small perpendicular wave numbers. The results demonstrate that it is computationally feasible to perform a direct integration of the time-dependent Vlasov–Maxwell equations in a tokamak geometry on a substantially long time interval. The directions of the further optimization of this method are discussed and the suggestions for the further analysis are made.

Two‐fluid theory of the drift kink instability

A simple two‐fluid theory of the drift kink instability is developed. The validity of the theory is restricted to the regime where the ion gyroradius is small in comparison with the sheet thickness ρi ≪ L and wavelength ρiky ≪ 1. Analytic expressions are derived for the real frequency and growth rate. For the limited region of parameter space where two‐fluid theory is applicable, the analytic expressions are in excellent agreement with a full Vlasov solution [Daughton, 1999].

Nonlinear solutions of the Vlasov–Poisson equations

Based on the kinetic description, properties of collisionless unmagnetized plasma are studied. Time-dependent solutions of the nonlinear Vlasov–Poisson equations are found. Several dynamical structures of a Maxwellian type are presented.

Time Asymptotics for Solutions of Vlasov–Poisson Equation in a Circle

We prove that there exists a class of solutions of the nonlinear Vlasov–Poisson equation (VPE) on a circle that converges weakly, as t → ∞, to a stationary homogeneous solution of VPE. This behavior is called, in the linear case, Landau damping. The result is obtained by constructing a suitable scattering problem for the solutions of the Vlasov–Poisson problem. A consequence of this result is that a class of stationary solutions of the Vlasov–Poisson equation is unstable in a weak topology.

Nuclear dynamics below the Coulomb barrier

We discuss an approach towards the many-body description of sub-barrier fusion and spontaneous fission based on the semiclassical Vlasov equation and the Feynman path integral method. We define suitable collective variables from the Vlasov solution and use the imaginary time technique for the dynamics below the Coulomb barrier.

Convected Scheme Simulations of the Electron Distribution Function in a Positive Column Plasma

A numerical procedure that provides an accurate and efficient solution of the Boltzmann equation in cylindrical geometry with coordinates (ρ, v) is discussed. The procedure, based primarily on the convected scheme, propagates the particle phase-space density on a computational mesh. A combination of the choice of independent variables, the design of the mesh structure and the use of a moving mesh can provide an exact solution of Vlasov's equation. A variant of the method is also presented which first iterates the scattering rate throughout phase space then finds the distribution function from the scattering rate. We describe the methods and apply them to a kinetic model of a dc positive column. Results are compared to Monte Carlo simulations.

Spontaneous Fission: A Kinetic Approach

We present an attempt towards the many-body description of spontaneous fission based on the semiclassical Vlasov equation and the Feynman path integral method. We define suitable collective variables from the Vlasov solution and use the imaginary time technique for the dynamics below the Coulomb barrier. We demonstrate (1) the stability of the Vlasov solution, (2) the ability of this model to calculate the basic features of the whole spontaneous fission process, (3) the sensitivity of the dynamics below the barrier to the mean field potential, and (4) the important role of collective excitations and of particle emission during the fission process.

The permittivity tensor and bounce resonance effects on wave dissipation in toroidal plasmas

The asymptotic solution of Vlasov's equation in the drift approximation for a toroidal plasma geometry with circular magnetic surfaces is presented. The method of solution introduces Jacobi functions as variables, making it possible to obtain results in a relatively simple manner. The analytic evaluation of the parallel components of the dielectric permittivity tensore3j is shown and the resulting strong influence of the bounce resonance of untrapped electrons on the wave dissipation is discussed. Additionally, a numerical calculation ofe33, together with a comparison between the toroidal and the cylindrical (Landau damping) results, are presented.

The solution of Vlasov's equation for complicated plasma geometry. I. Spherical type

The asymptotic solution of Vlasov's equation for complicated plasma geometry (spherical type) using the computer algebra system Maple is presented. The approximation of small ion and electron Larmor radii (drift approximation) for a dipole magnetic field configuration modeling the earth's plasmasphere was used. The method of solution introduces elliptic integrals as variables, which makes it possible to obtain results in a relatively simple manner. A bounce resonance factor (connected with trapped particles) different from the one appearing in the Landau damping case is found.

Vlasov solution describing nonadiabatic plasma heating by magnetic compression

An analytical Vlasov description of a fast magnetic compression is developed with use of the Lewis invariant for the time-dependent harmonic oscillator. Simple formulas for the nonadiabatic heating of an initial rigid rotor Maxwellian distribution are derived, providing a low β kinetic description of the oscillatory state after compression. The role of diamagnetic shielding and orbit stochasticity to relax the oscillations in the plasma after compression is considered. In addition to the adiabatic heating (which is equal to the field ratio B1/B0), a much stronger heating, proportional to (B1/B0)2, can be obtained in a nonturbulent fast compression. The ultimate need to use low initial β to achieve strong nonadiabatic heating and avoid turbulent development of instabilities follows from a formula describing the increase in β during the implosion. An evolution profile for B(t) with a constant nonadiabaticity parameter is considered for optimization of compression time (of order 5–10 μsec) and nonadiabatic heating. The radial extension of the compressed plasma is calculated. Space charge effects can arise from the differential radial motions of electrons and ions in a nonadiabatic compression, which can have important consequences. In straight devices, space charge can be removed by letting magnetic field lines connect to a conducting plate. Special attention is paid to the possibility of creating a ‘‘Migma pinch’’ with self-colliding orbits in the MeV range. The model indicates that this high energy can be reached in a compression with large field ratio, i.e., B1/B0∼103. The derived diamagnetic shielding limit, which corresponds to a compressed beta value below which shielding of the induction field can be ignored, gives a particle number per unit length of about 1017 m−1 for the Migma case. This number exceeds by several orders of magnitude the particle number achieved in the Migma IV experiment by a different beam trapping technique.