Volume 3, Issue 1, January 1960
Index of content:

Test Particles in a Completely Ionized Plasma
View Description Hide DescriptionStarting from the Liouville equation, a chain of equations is obtained by integrating out the coordinates of all but one, two, etc., particles. One ``test'' particle is singled out initially. All other ``field'' particles are assumed to be initially in thermal equilibrium. In the absence of external fields, the chain of equations is solved by expanding in terms of the parameter g = 1/nL _{ D } ^{3}. For the time evolution of the distribution function of the test particle, an equation is obtained whose asymptotic form is of the usual Fokker‐Planck type. It is characterized by a frictional‐drag force that decelerates the particle, and a fluctuation tensor that produces acceleration and diffusion in velocity space. The expressions for these quantities contain contributions from Coulomb collisions and the emission and absorption of plasma waves. By consideration of a Maxwell distribution of test particles, the total plasma‐wave emission is determined. It is related to Landau's damping by Kirchoff's law. When there is a constant external magnetic field, the problem is characterized by the parameter g, and also the parameter λ = ω_{ c }/ω_{ p }. The calculation is made by expanding in terms of g, but all orders of λ are retained. To the lowest order in g, the frictional drag and fluctuation tensor are slowly varying functions of λ.
When λ ≪ 1, the modification of the collisional‐drag force due to the magnetic field, is negligible. There is a significant change in the properties of plasma waves of wavelength greater than the Larmor radius which modifies the force due to plasma‐wave emission. When λ ≫ 1, the force due to plasma‐wave emission disappears. The collisional force is altered to the extent that the maximum impact parameter is sometimes the Larmor radius instead of the Debye length, or something in between. In the case of a slow ion moving perpendicular to the field, the collisional force is of a qualitatively different form. In addition to the drag force antiparallel to the velocity of the particle, there is a collisional force antiparallel to the Lorentz force. The force arises because the particle and its shield cloud are spiralling about field lines. The force on the particle is equal and opposite to the centripetal force acting on the ``shield cloud.'' It is much smaller than the Lorentz force.

Experimental Study of Alfvén‐Wave Properties
View Description Hide DescriptionAlfvén hydromagnetic waves are propagated through a cylindrical plasma. The wave velocity, attenuation, impedance, and energy transfer are studied. The theoretical equations predict correctly the functional dependence of the velocity and attenuation, and from these quantities accurate measurements of plasma density and temperature can be obtained. A qualitative agreement between theory and experiment is obtained for the hydromagnetic coaxial waveguide impedance, and the energy transferred from an oscillating circuit to the hydromagnetic wave is measured to be 43 ± 10%.

Absorption of Plasma Waves
View Description Hide DescriptionThe propagation of waves through a plasma, wherein the density and/or magnetic field strength are slowly varying functions of position is discussed. If the local propagation constant, k_{x} , is a slowly varying function of x, the adiabatic approximation will be valid. However, k_{x} ^{2} may pass through zero as a function of x. Using the WKB linear turning point connection formulas, examination shows that an incoming plasma wave is totally reflected in the region where k_{x} ^{2} ≈ 0. A similar analysis for the case where k_{x} ^{2} is a singular function of x shows that absorption of an incoming wave occurs in the vicinity of the singularity. Such singular behavior in k_{x} ^{2} can occur for propagation along the magnetic field when the wave frequency is equal to the local ion or electron cyclotron frequency. For propagation transverse to the magnetic field, an apparent singularity occurs at a frequency somewhat below the ion cyclotron frequency, and at the two hybrid frequencies of Auer, Hurwitz, and Miller. A detailed examination, including higher order effects in electron mass ÷ ion mass, finite electron and ion temperatures, and ion‐ion and ion‐electron collisions shows that the absorption will take place at the apparent singularity only if the physical damping processes are strong enough to swamp the reactive effects of the higher order corrections. Otherwise the higher order reactive effects introduce a new propagation mode into the dispersion equation with a root which, in the vicinity of the apparent singularity, is conjugate to the root of the original mode. Partial or total reflection now occurs at the apparent singularity instead of absorption. It is, however, conjectured that some of the original mode energy may be reflected into the new mode. As the new mode recedes from the region of the apparent singularity, its wavelength can become comparable to the particle Larmor radius. Energy in this mode may then be absorbed by phase‐mixing processes which are of high order in the quantity (Larmor radius ÷ wavelength). Wave reflection from the apparent singularities will then heat ions in the case of the transverse ion cyclotron mode, and electrons in the case of the upper hybrid frequency.

Transverse Compression Waves in a Stabilized Discharge
View Description Hide DescriptionAn electric discharge which is compressed by its own magnetic field, and ``stabilized'' by means of an axial magnetic field, can have transverse wave motions which cause its periodic compression and expansion. This kind of motion can cause the heating of the ions in the discharge. The simplest of these wave modes are described and an estimate is given of the power available to the waves as a result of the interaction of the electrons in the discharge with an axial electric field. This interaction can cause the attenuation or spontaneous growth of the waves, depending upon the circumstances. It is likely that in high current gas discharge experiments there are examples of growing and decaying waves of this type.

Instability of Contra‐Streaming Plasmas
View Description Hide DescriptionThe problem of instabilities in colliding ionized hydrogen beams, which has been treated by Kahn and Parker in the special case of zero temperature, is solved for the nonzero temperature case by taking Maxwell distributions for the equilibrium density functions. At sufficiently high temperature it is found that the random thermal motion will prevent growing oscillations. The boundary between the stable and unstable regions is plotted as a function of energy and density parameters. Certain phenomena associated with solar particle streams are discussed in terms of these results.

Cyclotron Radiation from a Hot Plasma
View Description Hide DescriptionIn their Geneva paper, Trubnikov and Kudryavtsev calculated the cyclotron radiation from a hot plasma. In doing this, the assumption was made that the individual particles radiated as though they were in a vacuum. We have investigated this approximation by calculating the absorption length directly from the Boltzmann equation, and we find that indeed this assumption is correct whenever (ω_{ p }/ω_{ e })^{2} ≪ m ^{2}, where m is the harmonic number of the radiation in question, ω_{ p } is the plasma frequency, and ω_{ e } is the cyclotron frequency. For a contained plasma, the left hand side of this inequality is of the order of magnitude of one, and thus the inequality is well satisfied for the dominant radiation from a plasma at high temperature.
The angular independence of the absorption coefficient has been calculated, and this together with a more careful examination of the mechanism of thermonuclear energy transfer to the electrons, leads to a modification of the results presented by Trubnikov and Kudryavtsev at Geneva. In addition, it is shown that by the use of reflectors the critical size can be reduced by two orders of magnitude.

Irreversible Processes in Ionized Gases
View Description Hide DescriptionThe general theory of irreversible processes, developed by Prigogine and Balescu, is applied to the case of long range interactions in ionized gases. A similar diagram technique permits the systematic selection of all the contributions to the evolution of the distribution function, to an order of approximation equivalent to Debye's equilibrium theory. The infinite series which appear in this way can be summed exactly. The resulting evolution equations have a clear physical significance: they describe interactions of ``quasi particles,'' which are electrons or ions ``dressed'' by their polarization clouds. These clouds are not a permanent feature, as in equilibrium theory, but have a nonequilibrium, changing shape, distorted by the motions of the particles. From the mathematical point of view, these equations exhibit a new type of nonlinearity, which is very directly related to the collective nature of the interactions.

Asymptotic Value of the Pair Distribution Near a Wall
View Description Hide DescriptionWe calculate the asymptotic value of the pair probability density ρ_{2}(r _{2}, r _{1}) for finding a fluid particle at a point r _{2} far in the interior of a fluid, when it is known that there is a particle at r _{1} in contact with the walls (rigid) of the container. This value is different from the well‐known expression for the asymptotic value of ρ_{2}(r _{2}, r _{1}) when both r _{2} and r _{1} are in the interior of the fluid. Our derivation is based on the virial theorem for total momentum fluctuations in an equilibrium system and makes use of the assumption that there are no long range correlations in a fluid. Application is made of our result to re‐derive simply the expression for the second virial coefficient and the exact equation of state of a hard‐sphere gas in one dimension. Quantum systems are also treated.

Remark on Dilute Bose Systems
View Description Hide DescriptionSome results are given on the connection existing between the Lee‐Huang‐Yang theory for the interacting Bose systems, and the Bogoliubov theory.

One‐Dimensional Equilibrium Spectra in Isotropic Turbulence
View Description Hide DescriptionThe one‐dimensional equilibrium spectra in isotropic turbulence are given for the physical transfer theories of Heisenberg, Kovásznay, and Obukhoff. These results are then compared with the experimental measurements of the spectrum of ∂^{3} u _{1}/∂x _{1} ^{3} fluctuations. For two of the theories (Heisenberg's and Kovásznay's), reasonable agreement is obtained for k _{η} < 0.04, but for larger values of k _{η} there is considerable divergence between the theoretical and experimental results. The relationship between the equilibrium and similarity spectra are also discussed for these two theories.

Boundary Layer Over a Flat Plate in Presence of Shear Flow
View Description Hide DescriptionThe governing equations of an incompressible boundary layer over a flat plate in the presence of a shear flow with finite vorticity are derived. For large vorticity, a similarity solution is obtained. For moderate vorticity, one of the governing equations is replaced by an approximate one for which similarity solutions exist.

Heat Transport by Convection
View Description Hide DescriptionThe character of heat transport by cellular convection, which arises beyond the marginal state of stability in a layer of fluid bound between two constant temperature surfaces is examined. It is shown that a simple equation characterizes the heat transport in the neighborhood of the marginal state of stability when the convection is steady, and its cellular pattern of motion is represented by the solutions of the linear theory. The results of the study include all three cases of boundary conditions, namely, when the bounding surfaces of the layer are both free, are both rigid, or one is free and the other is rigid.

Heat Transport by Convection in Presence of an Impressed Magnetic Field
View Description Hide DescriptionThe effect of an impressed magnetic field on heat transport by convection, which arises from instability in a layer of an electrically conducting fluid, bounded between two constant temperature surfaces, is examined. It is shown that such a field reduces the amount of heat transported by convection, and that when the strength of the magnetic field is increased, such reduction becomes proportional to (π^{2} Q)^{−1}, where Q = σμ^{2} cos^{2}ϑH ^{2} d ^{2}/ρν, d is the depth of the layer, ρ the density, H the strength of magnetic field, ϑ the inclination of the direction of H to the vertical, and σ, μ and ν are the coefficients of electrical conductivity, magnetic permeability, and kinematic viscosity, respectively, for all types of the boundary conditions. It is also shown that in the neighborhood of the marginal state of stability, a simple formula characterizes the heat transport by convection.

Growth of Instabilities on Displacement Fronts in Porous Media
View Description Hide DescriptionThe problem investigated is that of the penetration of a fluid into a porous medium containing a more viscous liquid. In order to do this, the flow potentials for a displacement front which is just about to become unstable are calculated. For such a displacement front it is possible to linearize the differential equations, and to give a description in terms of Fourier analysis. The law of growth for each spectral component of the front is deduced, and it is shown how the time dependence of the whole front can be represented by a superposition of elemental solutions. Subsequently, the effect of the heterogeneities contained in the porous medium is accounted for by introducing a random velocity perturbation term into the differential equation for each spectral component. In this fashion one arrives at an equation describing the growth of each spectral component of the fingers with time. It is shown that, under given external conditions, fingering should be independent of the speed with which the displacement proceeds. This is, in fact, what has been observed experimentally.

Isotope Separation by a Hot‐Wire Thermal Diffusion Column
View Description Hide DescriptionThe performance of a hot‐wire thermal diffusion column is discussed. The role of spacers used for centering the hot wire has been investigated experimentally for a glass column of 9‐mm i.d. cold wall, and hot wall 20‐mil tungsten wire; the results suggest that the maximum separation is obtained when the spacers (20‐mil nickel wire) are installed every 70 cm along the hot wire. The separation falls when the spacers come either closer together or farther apart. An explanation of this optimum dependence of separation on spacer distance is advanced.

Diffusion Coefficients in Flames and Detonations With Constant Enthalpy
View Description Hide DescriptionIt is found that the entropy per gram of mixture remains constant in a flame or a one‐dimensional chemically reacting gaseous flow system if all the binary diffusion coefficients are equal to each other and , where C̄_{p} is the specific heat at constant pressure per gram of the mixture, m is the average molecular weight of the mixture, and n is the number of moles per gram. This value for each of the binary diffusion coefficients corresponds to setting each of the Lewis numbers equal to unity. For detonations, or systems having large kinetic energy, the enthalpy (per gram) including kinetic energy remains constant if, in addition to the diffusion coefficients having this special value, the Prandtl number is equal to ¾. It is clear that the assumption of constant enthalpy should not be applied to hydrogen‐bromine or hydrogen‐oxygen flames where some coefficients of diffusion are very large and others very small. The constant enthalpy assumption is applied to unimolecular decomposition flames supported by the reaction A → sB′. It is found that, to a rough approximation, the flame velocity varies as the 1/12th power of s.

Statistical Study of Accelerating Flames
View Description Hide DescriptionStudy of velocity fluctuations observed by means of ionization probes during the development of detonation reveals its significance as an indicator of physical characteristics of the flame. The scatter in time of arrival (the reciprocal of the velocity) was found normally distributed at an 85% probability level. The means and standard deviation were determined within 5% and 20%, respectively, at a confidence level of 90%. The intensity of scatter is interpreted consequently as indicative of the combustion front fluctuation that can be considered to delineate the ``effective flame thickness.'' It is found then that, as the flame accelerates, its effective thickness first increases, reaching a maximum in the vicinity of velocity overshoot, and then decreases, attaining finally a minimum, constant value when the steady detonation wave is established. An interesting bimodal distribution of scatter for the 2H_{2}–0_{2} mixture has been observed, indicating a possible existence of two alternative, independent modes for the development of the process.

Boltzmann Equation for an Electron Guide Field Accelerator. I. Quasi‐Stationary Solutions for an Electron Beam
View Description Hide DescriptionA simplified model for dePackh's version of an electron guide field accelerator is set up by substituting for the actual external quadrupole or solenoidal focusing magnetic field an azimuthally symmetric focusing field, and by replacing the actual toroidal geometry by a cylindrical geometry with periodic boundary conditions. The Boltzmann equation for an electron beam in this system is studied, and a set of solutions is obtained which contain just enough parameters to represent the quasi‐stationary behavior of the beam realistically. The values of these parameters are related to the initial conditions of the beam by the adiabatic invariance of linear charge density and of the radial and azimuthal action integrals in the absence of collisions and radiation. Thus, the quasi‐stationary development of the beam in time is determined without an explicit time dependence in the Boltzmann equation. While the electron energy is being increased by a betatron field, the beam passes from a condition in which its electrons are in almost neutral equilibrium with respect to displacement from the axis (low temperature or `` '' regime) to a condition in which each electron is hardly affected by the other electrons (high temperature or betatron or ``½'' regime), as predicted by dePackh. If the beam is initially isothermal, the temperature becomes a monotonic decreasing function of distance from the axis in the course of electron energy increase.

Steady State Solutions of the Radiofrequency Discharge with Flow
View Description Hide DescriptionThe electron density distribution and diffusion length have been investigated for a steady‐state, diffusion‐controlled radiofrequency discharge acting over a finite portion of an infinite cylinder in which there is a uniform axial gas flow. This model simulates to some extent the flow in a plasma wind tunnel. A qualitative relationship is obtained for the influence of active cylinder length and gas velocity parameter on the diffusion length. The effect of these parameters on the electric field necessary to sustain the discharge is also discussed. It is shown that the peak of the electron density distribution shifts downstream with increasing gas velocity, but never leaves the region of production. A numerical example is calculated for the case of helium, indicating that while there is moderate effect on breakdown parameters, the ambipolar case may be changed considerably by the presence of flow.
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RESEARCH NOTES

N‐Wave Propagating into a Stratified Atmosphere
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