Mathematical models of plasma
Nonlinear Vlasov equation treatment by Fourier- Hermite expansion techniques for plasma problems
Feix, M.↗
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Long wavelength ion-acoustic waves in a magneto-plasma in a gravitational field
Ion acoustic wave propagation in collisionless gravity-supported plasma in static magnetic field calculated from linearized Vlasov equation
Liu, C. H.↗
Interaction of an obliquely incident p- polarized plane electromagnetic wave with a hot plasma half-space
Obliquely incident p-polarized plane electromagnetic wave interaction with hot plasma half space, using linearized relativistic Vlasov equation and Laplace transform technique
Bolduc, P. E.↗
Propagators in strong plasma turbulence.
Straightforward relationships between Weinstock's (1969) propagator, U sub A, the Vlasov propagator, U, and the ensemble average Vlasov propagator, U (in carets) are derived. It is shown that U and U (in carets) are related to the characteristic trajectories of the Vlasov equation, and that U (in carets) can be related to various statistical correlations of the turbulent fields.
Birmingham, T. J.↗
Stability of a steady, large amplitude whistler wave.
Study of the behavior of weak electrostatic waves in a collisionless magnetoplasma supporting a steady large amplitude whistler wave. All waves are assumed to propagate parallel to a uniform background magnetic field B sub zero. In the presence of the whistler wave fields each particle executes an oscillatory motion parallel to B sub zero, in addition to a translation along B sub zero and transverse motions. This oscillation causes the Landau resonance to be replaced by a series of new resonances between particles and the electrostatic modes. A distribution function for the perturbed plasma is constructed by solving the Vlasov equation, linearized in the electrostatic wave amplitudes. A dispersion relation is obtained and solved approximately for the growth/damping rate of the perturbations. Growing electrostatic modes are found to be approximately uncoupled. Trapped particles have a strong influence on the stability of the system.
Palmadesso, P. J.↗
Investigation of resonance integrals occurring in cosmic ray diffusion theory
The method for calculating the pitch angle diffusion coefficient is investigated for cosmic rays in a static random magnetic field, using the resonance integral method. The pitch angle diffusion coefficient may be derived from the Vlasov equation via ensemble averaging.
Jones, F. C.↗
Reformulation of quasi-linear theory.
Standard plasma quasi-linear theory is reformulated on the basis of a classical quantum derivation proceeding from the Vlasov equation and dealing only with frequency, wavenumber, and velocity. The wave amplitudes are assumed to be weakly time-dependent, and no distinction is made between growing and decaying waves. The proposed method leads to no negative diffusivity of 'fake' diffusion. By appropraite treatment of nonresonant interaction, expressions are obtained for wave energy and momentum.
Kaufman, A. N.↗
Inertial magnetic field reconnection and magnetospheric substorms.
We describe and calculate the growth rate of a magnetohydrodynamic neutral-sheet instability due to electron-inertia terms in the infinite-conductivity Ohm's law. The results are compared with an approximate Vlasov-equation calculation, and are shown to be particularly germane to the geomagnetic-tail instability.
Van Hoven, G.↗
Computational study of nonlinear plasma waves
A low-noise plasma simulation model is developed and applied to a series of linear and nonlinear problems associated with electrostatic wave propagation in a one-dimensional, collisionless, Maxwellian plasma, in the absence of magnetic field. It is demonstrated that use of the hybrid simulation model allows economical studies to be carried out in both the linear and nonlinear regimes with better quantitative results, for comparable computing time, than can be obtained by conventional particle simulation models, or direct solution of the Vlasov equation. The characteristics of the hybrid simulation model itself are first investigated, and it is shown to be capable of verifying the theoretical linear dispersion relation at wave energy levels as low as .000001 of the plasma thermal energy. Having established the validity of the hybrid simulation model, it is then used to study the nonlinear dynamics of monochromatic wave, sideband instability due to trapped particles, and satellite growth.
Matsuda, Y.↗
Computational study of nonlinear plasma waves. I - Simulation model and monochromatic wave propagation. II - Sideband instability and satellite growth
A hybrid plasma simulation model is described and applied to the study of electrostatic wave propagation in a one-dimensional Maxwellian plasma with periodic boundary conditions. The model employs a cloud-in-cell scheme which can drastically reduce the fluctuations in particle simulation models and greatly ease the computational difficulties of the Vlasov equation approach. A grid in velocity space is introduced and the particles are represented by points in the x-v phase space. The model is tested first in the absence of an applied signal and then in the presence of a small-amplitude perturbation. The method is also used to study propagation of an essentially monochromatic plane wave. Results on amplitude oscillation and nonlinear frequency shift are compared with available theories.
Matsuda, Y.↗
Nonlinear Alfven waves in high-speed solar wind streams
A nonlinear proton distribution function that is an exact stationary solution of the nonlinear Vlasov equation and Maxwell's equations and which supports a single nonlinear transverse Alfven (ion cyclotron) wave that is circularly polarized and nondispersive is proposed for most of the observations during high-speed solar wind streams. This nonlinear distribution removes the strong Alfven wave instability, inconsistent with the persistence of the observed proton distribution functions in high-speed streams, found by the linear stability analysis. Model temperature anisotropies and drift velocities of the two spatially inhomogeneous bi-Maxwellian components are consistent with typical proton velocity distributions measured in high-speed streams at 1 AU. Two derived relations for each of the wave number and the phase velocity of the wave are obeyed within experimental uncertainties by two typical proton measurements. Our model also predicts that the alpha particle bulk flow velocity exceeds the proton particle bulk flow velocity, as is observed.
Abraham-Shrauner, B.↗
Theory of flux anisotropies in a guiding center plasma
The one particle distribution function f on the scale of the bounce motion of particles in a magnetic field B is considered. The Vlasov equation is expanded through O(epsilon) in the adiabatic parameter which is the ratio of particle gyroradius to scale length of the magnetic field. Because f is directly proportional to particle flux differential in kinetic energy and solid angle, f is in principle measurable in space experiments, and the analysis is tailored to be explicitly applicable to space problems. To O(1), f is gyrotropic; its first velocity moment is (if non-vanishing) parallel to B, and hence macroscopic parallel flow is included in this term. The O(epsilon) contribution is non-gyrotropic and macroscopic flow parallel to B plus additional parallel flow results from these terms. The degree of non-gyrotropy and the amount of cross-field macroscopic flow depend on the perpendicular component of the electric field, on curvature and shear in the magnetic field, and on the spatial gradient, pitch angle derivative, and speed derivative of the lowest order distribution function.
Birmingham, T. J.↗
Impedance characteristics of coaxial and planar magnetoplasma capacitors
A theory has been developed for the impedance of a homogeneous magnetoplasma enclosed between two specular reflecting coaxial electrodes, with a static magnetic field parallel to the electrode axes. The parallel-plate magnetoplasma capacitor is treated as a sub-case. Starting with the Vlasov equation, an integral equation is derived for the electric field. Solving this equation, and integrating to obtain the voltage, gives the capacitor impedance. This includes a capacitive component, and a resistive component expressing the Landau damping associated with the open orbits of electrons reflected at the electrodes. A direct numerical solution of the field integral equation has been carried out for a range of values of magnetic field, plasma density, and signal frequency. The values of impedance so obtained are compared with the predictions of macroscopic theory, and of an approximate microscopic theory in which open orbits are ignored and solutions are obtained using finite Fourier transform methods. The mathematical relations between these theories are demonstrated.
Harker, K. J.↗
On the formation and evolution of clumps of galaxies in an expanding universe
Results are derived for the development of phase-space clumps of mass points in a background spectrum of gravitational-potential fluctuations. The Vlasov equation and the pair correlation equation (in the weak coupling limit) are solved exactly in an Einstein-de Sitter cosmology, and the plasma-clumping theory is used to identify terms that yield important collective effects. Various astrophysical implications are discussed, including the formation of large-scale inhomogeneity and the enhanced generation of correlations in the distribution of galaxies.
Norman, C. A.↗
Theory of flux anisotropies in a guiding center plasma
Assuming time stationarity of the one-particle distribution function f on the scale of the bounce motion of particles in a magnetic field, the paper expands the Vlasov equation through O(epsilon) in the adiabatic parameter epsilon, which is the ratio of particle gyroradius to scale length of the magnetic field. Since f is directly proportional to particle flux differential in kinetic energy and solid angle, f is in principle measurable in space experiments, and the present analysis is tailored to be explicitly applicable to space problems. It is shown that the usual expression for the electric field which produces plasma corotation in an axisymmetric system such as a dipole also holds for any nonaxisymmetric but rigidly rotating magnetic field pattern, provided the observed magnetic field is used in place of the dipole field. The analysis is applied to the electric field in a rigidly corotating magnetosphere.
Birmingham, T. J.↗
Two-dimensional nonstationary model of the propagation of an electron beam in a vacuum
A two dimensional nonstationary model of the propagation of a relativistic electron beam injected into a vacuum is considered. Collision effects are ignored and there are no external fields. Two types of the electron current propagation are shown from the computer simulation of the Maxwell-Vlasov equations.
Ginzburg, S. L.↗
Theory of scan plane flux anisotropies
When a spacecraft detector measures particle flux as a function of look direction in a plane (the scan plane), anisotropy is often seen. This anisotropy is caused by spatial gradients, by E x B particle drift, and by various spectral and geometric effects. This paper treats all of these effects systematically, starting from the nonrelativistic Vlasov equation. The general analysis is applied to a simple model of an anisotropic distribution to give a relation between the E x B drift, the gradient and the experimentally observed first, second, and third harmonics of the flux as a function of angle in the scan plane. Even with an assumed model, anisotropy observations in one plane alone do not suffice to determine the E x B drift velocity and the spatial gradient independently. If the E x B velocity is assumed (e.g., the corotational velocity in a rotating planetary magnetosphere), the spatial gradient may be deduced, and from it the time rate of change of flux in a nonrotating frame of reference.
Northrop, T. G.↗
Expansion of a multi-ion plasma into a vacuum
A numerical investigation of the expansion of a plasma with two ion species into a vacuum is presented. A set of Vlasov equations describe the ion behavior and the electrostatic potential is modelled by the Poisson equation. Electrons are assumed to follow Boltzmann's law. A plasma with H(+) and O(+) ions is considered, with the ions forming various combinations. Hydrodynamic calculations are performed for ions and electrons at equal temperatures, and for the presence of hot electrons. Self-similarity is shown to be valid where charge neutrality is dominant. An absence of significant quantities of ion-acoustic oscillations were observed.
Singh, N.↗