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Showing papers in "Physics of Fluids in 1963"


Journal ArticleDOI
TL;DR: In this paper, the stability of a plane current layer is analyzed in the hydromagnetic approximation, allowing for finite isotropic resistivity, and the effect of a small layer curvature is simulated by a gravitational field.
Abstract: The stability of a plane current layer is analyzed in the hydromagnetic approximation, allowing for finite isotropic resistivity. The effect of a small layer curvature is simulated by a gravitational field. In an incompressible fluid, there can be three basic types of ``resistive'' instability: a long‐wave ``tearing'' mode, corresponding to breakup of the layer along current‐flow lines; a short‐wave ``rippling'' mode, due to the flow of current across the resistivity gradients of the layer; and a low‐g gravitational interchange mode that grows in spite of finite magnetic shear. The time scale is set by the resistive diffusion time τR and the hydromagnetic transit time τH of the layer. For large S = τR/τH, the growth rate of the ``tearing'' and ``rippling'' modes is of order τR−3/5τH−2/5, and that of the gravitational mode is of order τR−1/3τH−2/3. As S → ∞, the gravitational effect dominates and may be used to stabilize the two nongravitational modes. If the zero‐order configuration is in equilibrium, the...

2,314 citations


Journal ArticleDOI
TL;DR: In this paper, the stability of a liquid layer flowing down an inclined plane is investigated, and a new perturbation method is used to furnish information regarding stability of surface waves for three cases: the case of small wavenumbers, of small Reynolds numbers, and of large wavenifications.
Abstract: The stability of a liquid layer flowing down an inclined plane is investigated. A new perturbation method is used to furnish information regarding stability of surface waves for three cases: the case of small wavenumbers, of small Reynolds numbers, and of large wavenumbers. The results for small wavenumbers agree with Benjamin's result obtained by the use of power series expansion, and the results for the two other cases are new. The results for large wavenumbers, zero surface tension, and vertical plate contradict the tentative assertion of Benjamin. The three cases are then re‐examined for shear‐wave stability, and the results compared with those for confined plane Poiseuille flow. The comparison serves to indicate the vestiges of shear waves in the free‐surface flow, and to give a sense of unity in the understanding of the stability of both flows. The case of large wavenumbers also serves as a new example of the dual role of viscosity in stability phenomena.The topological features of the ci curves for...

851 citations


Journal ArticleDOI
Harold Grad1
TL;DR: In this article, a generalization of the Hilbert and Enskog expansions is described in terms of extended sets of macroscopic state variables, each governed by partial differential equations similar to those found in fluid dynamics but sufficiently general to approximate an arbitrary distribution function.
Abstract: The precise mathematical relation that the Hilbert and Chapman‐Enskog expansions bear to the manifold of solutions of the Boltzmann equation is described. These expansions yield inherently imprecise descriptions of a gas in terms of macroscopic fluid variables instead of a molecular distribution function. It is shown that these expansions are asymptotic to a very special class of solutions of the Boltzmann equation for sufficiently small mean free path. Next, a generalization of the Hilbert and Chapman‐Enskog expansions is described in terms of extended sets of macroscopic state variables, each governed by partial differential equations similar to those found in fluid dynamics, but sufficiently general to approximate an arbitrary distribution function. The generalized expansions are shown to be asymptotic to quite arbitrary solutions of the Boltzmann equation. It is then shown that the ordinary Hilbert and Chapman‐Enskog expansions can also be made asymptotic to very general solutions of the Boltzmann equation by reinterpreting the variables that enter these expansions as certain well‐defined replacements for the actual fluid state of the gas. In this way the scope of the Euler, Navier‐Stokes, Burnett equations, etc., is greatly extended by interpreting them as governing the artificial variables. Not only are general solutions of the Boltzmann equation shown to be approximated by fluid dynamics (in the limit of small mean free path), but the rapid decay of an arbitrary initial distribution function to a special Hilbert distribution function is also governed by sets of partial differential equations similar to those found in fluid dynamics.

639 citations


Journal ArticleDOI
TL;DR: In this paper, the rate of molecular dissociation behind strong shock waves is calculated with the assumption that dissociation can occur preferentially from the higher vibrational levels using an anharmonic oscillator model.
Abstract: The rate of molecular dissociation behind strong shock waves is calculated with the assumption that dissociation can occur preferentially from the higher vibrational levels. An exponential probability of dissociation from the various vibrational levels is employed using an anharmonic oscillator model. Results for the dissociation of oxygen in an argon diluent are presented. Vibrational non‐equilibrium introduces a T−3 temperature dependence into the oxygen dissociation rate constant in the range 4000°–8000°K. A dissociation lag‐time of the order of the extrapolated vibrational relax ation time is predicted immediately behind the shock front. The computed results are shown to be in agreement with available experimental results.

479 citations



Journal ArticleDOI
TL;DR: In this article, a weakly ionized plasma in a uniform magnetic field is considered and it is shown that this steady state is unstable if the plasma density is nonuniform in the direction of the applied electric field and if the applied potential is large enough.
Abstract: A weakly ionized plasma in a uniform magnetic field is considered. Application of a potential across the magnetic field results in a steady current flow, owing to the finite conductivity. It is shown that this steady state is unstable if the plasma density is nonuniform in the direction of the applied electric field and if the applied potential is large enough. It is necessary that the sign of the product of the electric field and the density gradient be positive.

363 citations


Journal ArticleDOI
TL;DR: In this paper, the reduction of test time in low pressure shock tubes, due to a laminar wall boundary layer, has been analytically investigated, and it was found that β is considerably larger than the estimates made by Roshko and Hooker except for very strong shocks.
Abstract: The reduction of test time in low pressure shock tubes, due to a laminar wall boundary layer, has been analytically investigated. In previous studies by Roshko and Hooker the flow was considered in a contact surface fixed coordinate system. In the present study it was assumed that the shock moves with uniform velocity, and the flow was investigated in a shock fixed coordinate system. Unlike the previous studies, the variation of free stream conditions between the shock and contact surface was taken into account. It was found that β, a parameter defined by Roshko, is considerably larger than the estimates made by Roshko and Hooker except for very strong shocks. Since test time is proportional to β−2, previous estimates of test time are too large, particularly for weak shocks. The present estimates for β appear to agree with existing experimental data to within about 10 percent for shock Mach numbers greater than 5. In other respects, the basic theory is in general agreement with the previous results of Roshko.

300 citations


Journal ArticleDOI
TL;DR: In this article, a continuum theory for spherical electrostatic probes in a slightly ionized plasma is developed, where the density of the plasma is taken to be sufficiently high such that both ions and electrons suffer numerous collisions with the neutrals before being collected by an absorbing probe.
Abstract: A continuum theory for spherical electrostatic probes in a slightly ionized plasma is developed. The density of the plasma is taken to be sufficiently high such that both ions and electrons suffer numerous collisions with the neutrals before being collected by an absorbing probe. A general discussion of probes at an arbitrary potential is given. It is found that for very negative probe potentials the sheath thickness can be comparable to the probe radius. Two explicit forms of current‐voltage characteristics are given; one for very negative probes, the other for probes at nearly plasma potential. Both of these are based on the assumption that the probe radius is large compared with the Debye length. Numerical computation is also given for negative probes of a wider range of probe sizes.

278 citations


Journal ArticleDOI
TL;DR: In this paper, the authors considered the plasma sheath problem for low pressure discharge in plane geometry and provided numerical solutions for various values of the parameter α, which is of the order of the ratio of the Debye length to the discharge width for 10−3 ≤ α ≤ 10−1; for three assumptions regarding the ion generation rate, namely generation uniform, proportional to electron density, and proportional to the square of electron density.
Abstract: The plasma‐sheath problem for the low‐pressure discharge in plane geometry is treated exactly, that is, with no arbitrary division into plasma and sheath regions. Numerical solutions are presented for various values of the parameter α, which is of the order of the ratio of the Debye length to the discharge width for 10−3 ≤ α ≤ 10−1; and for three assumptions regarding the ion generation rate, namely generation uniform, proportional to electron density, and proportional to the square of electron density.For the higher values of α, corresponding to weak laboratory discharges, there is a smooth transition from a quasi‐neutral plasma region to a thick sheath. At the smaller values of α, the conventional model of a quasi‐neutral plasma region passing rather abruptly into a narrow sheath region is substantiated. In all cases, accurate values for the potential profile throughout the plasma and sheath regions are given and compared with the separate asymptotic plasma and sheath solutions for α = 0. The ion curren...

276 citations


Journal ArticleDOI
TL;DR: In this article, the lowering of the ionization energy is derived by a statistical method, which is physically conceivable, and the results cover a wide density range below and above the so-called critical density.
Abstract: In a plasma the ionization energy is decreased due to the presence of the microfield. In the past several attempts have been made to calculate this effect. These calculations, which use statistical and thermodynamical procedures, give different results. They produce either a ``polarization term'' or a ``lattice term'' or both of them. Moreover there are quantitative discrepancies. The lowering of the ionization energy is here derived by a statistical method, which is physically conceivable. The results cover a wide density range below and above the so‐called critical density. The new results are compared with the results of all earlier calculations and reveal the cause of their discrepancies.

244 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that a weakly ionized plasma column in an axial magnetic field and an inward-directed radial electric field is unstable against a macroscopic m = 1 perturbation.
Abstract: It is shown that a weakly ionized plasma column in an axial magnetic field and an inward‐directed radial electric field—a Penning discharge at higher pressures—is unstable against a macroscopic m = 1 perturbation. The instability is analyzed by dimensional analysis. The applicability of this method is discussed. A steady‐state model of some experimental Penning discharges is developed. On the basis of this model, the critical magnetic field for the onset of instability as predicted by the theory agrees well with the values observed. The instability discussed here is caused by the presence of a third fluid, the neutrals, in an electron‐ion plasma. The same mechanism is also applicable to the initial phase of some rotating plasmas of the homopolar type.

Journal ArticleDOI
TL;DR: In this paper, an acoustical drop holder has been developed in which radiation pressure is used to support the drops at rest in the shock tube, and photographs showing new details in the breakup process are presented.
Abstract: The breakup of drops exposed to blasts of air is studied in a shock tube. Results are obtained for water, methyl alcohol, and three viscous oils. An acoustical drop holder has been developed in which radiation pressure is used to support the drops at rest in the shock tube. Photographs showing new details in the breakup process are presented.

Journal ArticleDOI
TL;DR: In this article, it has been observed experimentally that the application of a radio-frequency voltage (10 kc/sec−50 Mc/sec) to any one of several electrode configurations around the outside of a plasma discharge tube results in a constriction of the luminous portion of the plasma away from the inner walls of the glass tube.
Abstract: It has been observed experimentally that the application of a radio‐frequency voltage (10 kc/sec–50 Mc/sec) to any one of several electrode configurations around the outside of a plasma discharge tube results in a constriction of the luminous portion of the plasma away from the inner walls of the glass tube. This investigation has established that the phenomenon is basically a radio‐frequency rectification effect, leading to the formation of thick ion sheath. The interaction is described mathematically in terms of a differential equation which has an approximate solution that fits qualitatively all the observed characteristics of the phenomenon. The differential equation, in its most general form, has also been solved numerically and the solution is shown to quantitatively fit our experimental observations for both radio‐frequency sine and square wave signals. An application of this phenomenon as a possible external diagnostic probe technique is proposed.

Journal ArticleDOI
TL;DR: In this article, a method is described for the solution of time-dependent problems concerning the flow of viscous incompressible fluids in several space dimensions, using a high-speed computer for a finite-difference approximation to the partial differential equations of motion.
Abstract: A method is described for the solution of time‐dependent problems concerning the flow of viscous incompressible fluids in several space dimensions. The method is numerical, using a high‐speed computer for the solution of a finite‐difference approximation to the partial differential equations of motion. The application described here is to a study of the development of a vortex street behind a plate which has impulsively accelerated to constant speed in a channel of finite width; the Reynolds‐number range investigated was 15 ≤ R ≤ 6000. Particular attention was given to those features for which comparison could be made with experiments, namely, critical Reynolds number for vortex shedding, drag coefficient, Strouhal number, vortex configuration, and channel‐wall effects. The nature of the early stages of flow‐pattern development was also investigated.

Journal ArticleDOI
TL;DR: In this paper, small perturbations of plane Couette flow in stably and unstably stratified fluid are considered, and the preferred mode of motion is roll-type convection, with the dimension in the wind direction much larger than in the transversal direction.
Abstract: Small perturbations of plane Couette flow in stably and unstably stratified fluid are considered. It is found that the system is more unstable when it is bounded both above and below than when its depth is infinite, but a finite negative Richardson number J is required to maintain the perturbation for both cases. For the former case, this limiting Richardson number is −3k12/4(k12 + k22), while for the latter it is −2k12/(k12 + k12), where k1 and k2 are wavenumbers in the mean flow and the transversal direction. These results show that in an unstably stratified layer of Couette flow, the preferred mode of motion is roll‐type convection, with the dimension in the wind direction much larger than in the transversal direction. The amplification factor σ for the perturbations has been determined as a function of the modified Richardson number J = gSzUz−2(1 + k1−2k22), and the dimensionless wavenumber α = h(k12 + k12)1/2. Four different regimes have been found, each corresponding to a different type of perturbation. An application of the theory is made to the formation of longitudinal cloud rolls observed in the earth's atmosphere and in certain laboratory experiments.

Journal ArticleDOI
TL;DR: In this article, the problem of an electrostatic probe in a dense, slightly ionized gas is treated by techniques of asymptotic analysis, in particular, ρp = rp/λd → ∞ and e = T+/T− → 0 are treated in considerable detail.
Abstract: The problem of an electrostatic probe in a dense, slightly ionized gas is treated by techniques of asymptotic analysis. In particular, the asymptotic limits ρp = rp/λd → ∞ and e = T+/T− → 0 are treated in considerable detail. (rp, λd, T−, T+ are probe radius, Debye length, electron, and ion temperature, respectively.) Sample integral curves for electrostatic potential, ion and electron density are given. Probe characteristic curves for three values of e (finite and small) and ρp (large) are also given. It is noted that these characteristics do not saturate for large probe potential because the influence of the probe is felt to very great distances from the probe; the shielding due to the space‐charge sheath is incomplete.

Journal ArticleDOI
TL;DR: In this article, a numerical quadrature using the exact analytical formula, which is obtained from the Gross-Bhatnagar-Krook model, was used to evaluate the slip coefficient.
Abstract: The slip coefficient is evaluated by a numerical quadrature using the exact analytical formula, which is obtained from the Gross‐Bhatnagar‐Krook model. The obtained value z=1.1466(5)l, which is exact to the fourth decimal figure, is compared with the previous results.

Journal ArticleDOI
TL;DR: In this article, the first two moments of the collisionless Boltzmann equation assuming a scalar pressure are used in conjunction with a physically reasonable radial electron density profile to describe the oscillations of a hot nonuniform plasma cylinder.
Abstract: A quantitative theory of resonance oscillations, such as observed by Dattner and others, is given. The first two moments of the collisionless Boltzmann equation assuming a scalar pressure are used in conjunction with a physically reasonable radial electron density profile to describe the oscillations of a hot nonuniform plasma cylinder. These equations coupled with Maxwell's equations assuming a scalar potential are solved numerically to yield the frequency spectrum of the plasma wave resonances. It is found that the frequency spectrum depends on the parameter r/w2/ where rw is the radius of the plasma column and is the mean square Debye length of the electron plasma. New experimental observations of dipole and quadrupole spectra for two plasma columns of differing radii are reported and the results of these observations are in good agreement with the theory. The physical mechanism of the resonances is described.

Journal ArticleDOI
TL;DR: In this article, the stability of a free laminar layer between parallel streams is examined and the neutral curve and the curves of constant amplification are obtained by a numerical method for Reynolds numbers ranging from 0 to 40 as well as for the inviscid case.
Abstract: The stability of a free laminar layer between parallel streams is examined. The neutral curve and the curves of constant amplification are obtained by a numerical method for Reynolds numbers ranging from 0 to 40 as well as for the inviscid case. No minimum Reynolds number is found. The eigenfunctions are discussed. The case of a layer of increasing thickness is considered, and it is concluded that turbulent transition will occur when the Reynolds number reaches 150.

Journal ArticleDOI
TL;DR: In this paper, the motion of an ideal collisionless plasma subject to periodic boundary conditions, with velocity distribution given initially, is studied, and the minimum value which the kinetic energy can subsequently attain is determined, on the assumption that the system is subject only to the constraint that phase space volume is conserved.
Abstract: The motion of an ideal collisionless plasma subject to periodic boundary conditions, with velocity distribution given initially, is studied. The minimum value which the kinetic energy can subsequently attain is determined, on the assumption that the system is subject only to the constraint that phase‐space volume is conserved.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the electron drift velocity in thermal cesium and potassium plasmas is about 10 times the ion thermal velocity, in agreement with the prediction of M. N. Rosenbluth.
Abstract: Oscillations near the ion cyclotron frequency have been excited in thermal cesium and potassium plasmas by drawing current in a filament along the axis of the plasma column. The oscillations appear to be electrostatic waves propagating radially from the filament. The waves are present if the electron drift velocity exceeds about 10 times the ion thermal velocity, in agreement with the prediction of M. N. Rosenbluth. The measured phase velocity is also in agreement with the phase velocity calculated from the fluid equations.

Journal ArticleDOI
TL;DR: In this paper, a comparison of specific ionization rates was made with a simple extrapolation of the low-temperature rate constants according to the crossing point model of Bates and Massey for atom-atom ionizing collisions.
Abstract: The problem of spontaneous ionization (i.e., no externally applied electromagnetic fields, nor hard radiation) in the reaction zone behind strong normal shock waves in air has been treated concurrently with the problem of dissociation and vibrational relaxation. Through a comparison of specific ionization rates, one may conclude that up to a shock velocity of 9 km/sec (about 27 times the speed of sound at room temperature), the predominant electron production process would be atom—atom ionizing collisions. This would be followed in an approximately decreasing order of importance by photoionization, electron impact, atom—molecule collisions, and molecule—molecule collisions. The charge exchange reactions, while not contributing directly to the electron production process, were found to have a small but noticeable indirect effect on the resultant electron density distribution at some distance behind the shock due to their continuous shifting of the relative population between atomic and molecular ions (which recombine with the electrons at different rates). The specific rate constants for the atom—atom processes required to interpret all existing experimental results appear to be consistent with a simple extrapolation of the low‐temperature rate constants according to the crossing‐point model of Bates and Massey for atom—atom ionizing collisions.

Journal ArticleDOI
TL;DR: In this article, the equilibrium chemical composition and the thermodynamic properties of argon plasmas have been calculated for five pressures (0.1, 0.5, 1.0, 2.0 atm) at 100 K deg increments for the temperature range 5000° to 35 000°K.
Abstract: The equilibrium chemical composition and the thermodynamic properties of argon plasmas have been calculated for five pressures (0.1, 0.5, 1.0, 2.0, and 5.0 atm) at 100 K deg increments for the temperature range 5000° to 35 000°K. The argon plasma is assumed to be a perfect gas complex consisting of six components, namely electrons, argon atoms, and the first four argon ions. The partition functions for these have been calculated using tabulated data for observed atomic energy levels and estimated energies for energy levels which are predicted although not observed. The partition functions were terminated by application of the Debye cutoff criterion and a corresponding lowering of the ionization potential was included. The calculated data are presented in graphical form and the method followed in calculating the partition functions is evaluated.

Journal ArticleDOI
TL;DR: In this article, the angular distributions of ion and electron velocities must agree at each point in a collisionless plasma in a narrow axially symmetric magnetic mirror field, where the electric field vanishes identically along the magnetic field.
Abstract: Stationary states of a collisionless plasma in a narrow axially symmetric magnetic mirror field are considered. It is shown that if the electric field vanishes identically along the magnetic field, then the angular distributions of ion and electron velocities must agree at each point. In a general case, when this condition on the distributions is not fulfilled, there is an electric field parallel with the magnetic field.

Journal ArticleDOI
TL;DR: In this paper, the problem of equilibrium and stability of plasma confined in certain magnetic fields of combined mirror-cusp form is discussed, and conditions for plasma equilibrium are derived and interpreted; then by exploiting the existence of closed magnetic isobars certain low-β confined equilibria are constructed.
Abstract: The problem of equilibrium and stability of plasma confined in certain magnetic fields of combined mirror‐cusp form is discussed. These fields have the properties that they are nowhere zero and everywhere increase toward the periphery. Attention is drawn to the importance of the existence of closed surfaces of constant |B|—the magnetic isobars. The conditions for plasma equilibrium are derived and interpreted; then by exploiting the existence of closed magnetic isobars certain low‐β confined equilibria are constructed. These equilibria are shown to be stable according to the fluid (double adiabatic) energy principle and according to the small Larmor radius limit theory. A direct proof of stability against motions which preserve the magnetic moment is given. These equilibria have the property that there is no current along lines of force so that they are also immune to several drift instabilities.

Journal ArticleDOI
TL;DR: In this paper, the authors extended the elementary model to include the effects of ion correlations for arbitrary fixed ion distributions, and found that the ion shielding reduced the maximum effective impact parameter by a factor (1 + Z)−1.
Abstract: In an earlier work the ac conductivity of a plasma was investigated by means of an elementary model. The validity of this model has been borne out by a rigorous treatment of plasma at thermal equilibrium. The elementary model is now extended to include the effects of ion correlations for arbitrary fixed ion distributions. For thermal equilibrium correlations it is found that the ion shielding reduces the maximum effective impact parameter by the factor (1 + Z)½ (i.e., both ions and electrons contribute to the shielding) for frequencies low compared to the plasma frequency ωp. For frequencies high compared to ωp, the previous results obtain. The resistance due to the excitation of longitudinal waves at frequencies just in excess of ωp is reduced by the factor (1 + Z)−1. However, if large‐amplitude (nonthermal) ion fluctuations are present, the longitudinal wave contribution to the resistance may be greatly enhanced.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the magnitudes of the incident and reflected wavenumbers are generally unequal and the reflection process thus provides an interchange of energy among different scalar wenumbers k.
Abstract: If inertial waves in a uniformly rotating fluid reflect from a rigid plane surface, it is shown that the magnitudes of the incident and reflected wavenumbers are generally unequal. The reflection process thus provides an interchange of energy among different scalar wavenumbers k. In an inviscid fluid, the ratio α(k)/k is conserved on reflection, where α is the particle orbit speed associated with the wavenumber k. In a viscous fluid the reflection coefficient is generally 1 ‐ O(R1/2), where R is the wave Reynolds number 2Ω/νk2, though there is an exceptional case in which the incident energy flux is totally absorbed. When the fluid is contained in a large rotating box of general shape, the energy interchanges from repeated reflections can result in a statistical radiative equilibrium over the high wavenumbers. A dynamical equation is derived that specifies Eϑ(k), the energy distribution among wavenumbers inclined at an angle ϑ to the rotation vector. An approximate solution shows that Eϑ(k) is constant wh...

Journal ArticleDOI
TL;DR: In this paper, the energy and momentum relaxation of a nonequipartition gas mixture is considered, where each component of the mixture has a Maxwellian distribution at a temperature Ti, with the peculiar velocity of the Maxwellian measured relative to the mean velocity of ith species for the case in which a ''diffusion'' Mach number is not too large.
Abstract: The energy and momentum relaxation of a nonequipartition gas mixture is considered It is assumed that each component of the mixture has a Maxwellian distribution at a temperature Ti, with the peculiar velocity of the Maxwellian measured relative to the mean velocity of the ith species For the case in which a ``diffusion'' Mach number is not too large, the results have a particularly simple form The calculations were carried out for the hard sphere, Coulomb, and Maxwell force laws of interaction It is also noted how these results may be used to construct kinetic model equations for the case of hard sphere and Coulomb interaction, in a manner similar to that proposed by Sirovich

Journal ArticleDOI
TL;DR: Energy transfer from large to small eddies at three stations in turbulence behind a square mesh is determined by measuring the rates of change and viscous dissipation of the spectrum and the results are compared with a theoretical prediction.
Abstract: Energy transfer from large to small eddies at three stations in turbulence behind a square mesh is determined by measuring the rates of change and viscous dissipation of the spectrum and the results are compared with a theoretical prediction. Large eddies for which viscous dissipation is negligible satisfy a similarity relation which agrees with the fact that the total energy decays as some negative power of time. Small eddies which are in approximate statistical equilibrium satisfy local similarity according to Kolmogoroff. Various terms in the vorticity equation are also determined and the quantities representative of small scale motion are universal constants when expressed in terms of Kolmogoroff parameters.

Journal ArticleDOI
TL;DR: In this paper, an exact kinetic equation for plasma and the electromagnetic field is derived, which describes the fluctuations of the fields and particle distributions, and the solution is obtained by expanding in a parameter which characterizes the amplitude of these fluctuations.
Abstract: An exact kinetic equation for plasma and the electromagnetic field is derived. This equation describes the fluctuations of the fields and particle distributions. The solution is obtained by expanding in a parameter which characterizes the amplitude of these fluctuations. A systematic procedure is given for generating the solution to arbitrary order in the expansion. Some typical applications of the theory are presented. These include calculations of a collision integral, incoherent scattering, and bremsstrahlung emission and absorption.