The dispersion properties of drift waves in a low β weakly collisional plasma, in the presence of ion acoustic or Langmuir waves parallel to magnetic lines, are investigated. It is shown that the non-linear interaction with an ion acoustic wave mainly leads to a shift of the drift wave spectrum towards lower frequencies (and only under very particular conditions affects the stability of the drift wave). The interaction with a Langmuir wave, on the contrary, yields a contribution to the imaginary part of the frequency. The conditions under which this stabilizes unstable drift modes are derived and discussed.

Dispersion relations are calculated and solved for frequencies near the first two harmonics (Ω/Ω+ = 2, 3) of perpendicular magnetosonic waves in relativistic anisotropic plasmas. In this, wavelengths larger than the ion Larmor radius and large plasma frequency (Ω2p, ≫ Ω2+) are assumed. It is found that, even for Tψ ≫ T the plasma is stable. This result has application to the steady-state, spherically symmetrical, expanding, solar-wind plasma.

Butler & Gribben (1968) showed how to generalize non-linear, uniform, wave type solutions of the Vlasov–Poisson system of equations to allow for slow variations in the properties of the plasma. Here one equation of the derived set (that obtained by ensuring that the appropriate solutions of the Poisson equation are non-secular) is rearranged (i) so as to bring out the physical importance of those particles, trapped and untrapped, travelling near the wave speed, and (ii) so as to yield a form particularly suited to the case when the electric field is small.

The resonant interaction of three monochromatic waves propagating in a uniform magnetoplasma is considered in the electrostatic approximation. It is shown that, for arbitrary directions of propagation, the coupling coefficients are in general complex, and are symmetric with respect to interchanges of the wave vectors.

The transfer of mass and heat through a diatomic gas bounded by two plane parallel surfaces is investigated by a kinetic model which includes translational, rotational and vibrational degrees of freedom. A full range moment method is employed in the solution. Numerical solutions obtained for small temperature and pressure differences show the effect of inelastic collisions on the mass and heat fluxes. Approximate analytical expressions are derived for the mass and heat transfer rates, which are valid over the entire density range and yield the correct free-molecule and continuum limits. The results agree well with existing heat transfer and density measurements.

Several types of two-dimensional solutions for the equations of magnetohydrodynamics are described. For all these solutions the magnetic field contains at least one hyperbolic neutral point. Two new magnetostatic equilibria are introduced for the ideally conducting case. The magnetic field associated with one of these is used to construct an exact time-dependent solution of the MilD equations where the fluid is necessarily at rest. In the case where the field lines are hyperbolae, it is demonstrated that retention of the energy equation (ordinarily decoupled for incompressible flows) implies that the flow beginning at rest, remains at trest

In this paper we employ similarity considerations to investigate the flow field produced by an electric discharge (mathematically a current source) and a jet of momentum emerging from the same hole of a plane wall bounding a viscous, incompressible conducting fluid, which extends to infinity. We also use similarity solutions in considering the axisymmetric configuration of a problem that has recently received some attention in connexion with field line reconnexion in conducting fluids. We derive the similarity equations for an incompressible, viscous conducting fluid, and discuss the solutions for the case when the fluid is perfectly conducting. When the electrical conductivity of the fluid is finite, we could not construct solutions, satisfying the appropriate boundary conditions.

Hamel & Willis (1966) first showed that anisotropy in temperature could be predicted theoretically in the expansion of a spherical source of gas into a vacuum, thus explaining anisotropies observed to occur in rarefied gas jets. Chou & Talbot (1967) carried out numerical calculations based on similai ideas for a plasma consisting of electrons, ions and neutral particles. These showed that the electron temperature remains isotropic, and little anisotropy occurring among the ions within the (rather extensive) calculations. Here, in the first of two papers, an analytic treatment is applied to the expansion of a fully ionized plasma source, with the particles interacting according to the Coulomb law.

This paper considers electrostatic waves in a Vlasov plasma of unmagnetized ions and magnetized electrons undergoing an E x B drift. The linear dispersion relation is solved numerically for Te ΰTi. For a fixed ratio of drift velocity to electron thermal velocity, the growth rates of the E x B electron drift instability are smaller, and the waves are stabilized at much smaller values of k. B than in the Te ≫ Ti case.

A moment approach is used to find the thermal and collisional corrections to the ordinary wave in a plasma. Results are given that include like- and unlike- particle collisions.

The circumstances under which certain plasma dispersion equations may be used in electrostatic ray tracing problems connected with studies of ionospheric topside plasma resonances are investigated. Three plasma ‘models’ of increasing complexity are compared with each other. The models are based on the assumption of: (i) an adiabatic process with a scalar pressure, (ii) a fully adiabatic process (Buneman 1961), and (iii) an electrostatic kinetic description. It is found that near fN, and within 10° of the static magnetic field, all three models may be used with good accuracy, and collisionless damping is very small. Near fT, and for propagation nearly across the field, only the second model (ii) gives tolerably accurate results as judged by the kinetic equation. For frequencies within 10 kc s-1 of fT such as are encountered in ray tracing, collisionless damping may be neglected within a cone of ∽ 10° about the perpendicular to the field for fT < 2fH, and within ∽ 20° whenfT ≫ 2fH Model (iii) may be used to examine the dispersion equation near the electron gyroharmonics: this topic is discussed in Andrews & Fang (1971).

In periodic, inhomogeneous plasma, the dispertion relation w(k) can begeneralized to a relation w(k, K), where K is the wave-number of the periodic steady state. Two models are studied: the ‘water-bag’ model and the ‘two-stream’ model. The solution for these models is obtained easily by the introduction of a Lagrangian formulation. Results are presented. It is shown that, in the water-bag case, the appearance of narrow bands of allowed frequencies practically discretizes the lower part of the spectrum. In the two-stream case, while the long wavelengths have a smaller growth rate, an increased instability is found for k/K = ½

This paper introduces the effects of partial ionization and recombination into the steady supersonic spherical source expansion of a plasma. It presents an analytical solution for a problem previously solved only numerically, using moment equations derived directly from the Boltzmann equation under the assumption of an ellipsoidal distribution function. The recombination is assumed collisionalradiative, as considered by Bates, Kingston & MeWhirter (1962a, b) and an approximate rate constant is derived from their results. The plasma is taken to be optically thick. The results show that, in a highly ionized plasma, the species maintain equal isotropic temperatures for the main part of the flow, with the degree of ionization tending to a finite limit. Recombinations, however, remain sufficiently frequent to keep the temperature above that in an expansion with no recombination. In a weakly ionized plasma the temperature of the heavy particles is almost unaffected by recombinations and falls adiabatically, although the electron temperature remains high. The analytical results are compared with the numerical results of Chou & Talbot (1967).

The effects of thermal motions on the cyclotron radiation from test particles gyrating in a homogeneous magnetoplasma are studied. These effects take care of all singularities that exist in the theory of cyclotron radiation in cold magneto- plasma, e.g. the divergence in energy loss for small particle energies. Around the hybrid frequencies thermal corrections become of dominant importance. At these frequencies cold-plasma theory breaks down. Thermal effects arise in two ways: by modifying the wave modes known from cold plasma theory, and by the introduction of a new longitudinal wave mode, known as the Bernstein mode. All wave modes are damped (in stable plasmas), the damping increasing rapidly with the product (V/v|) (ω - s), where the wave frequency ω is measured in units of the gyro frequency; V is the thermal velocity; v| the parallel component of the particle velocity, and s is an integer.

The system studied is a plasma streaming parallel to a uniform magnetic field with a velocity which varies in a transverse direction. The flow is bounded at y =± a by perfectly conducting planes.

The Poisson-Vlasov equations are used to derive an integro-differential equation for ø the Fourier transform of the electrostatic potential. The kernel of this equation is expanded using a small Larmor radius expansion for ø and for the equilibrium distribution function f0.

The dispersion relation for electrostatic plasma waves propagating at frequencies near the electron cyclotron harmonics has been evaluated, and used to determine the refractive index curves for varying values of the angle θ between the k vector and the ambient magnetic field B0 in a warm magnetoplasma. It is shown that, under certain circumstances (which are defined), the (@, θ) curves have the characteristic ‘nose’ shapes that are necessary to provide a reflexion in the ambient plasma. Thus, the oblique echo reflexion mechanism (McAfee 1968, 1969a, b), which accounts for the appearance of the resonance signals or ‘spikes’ at the plasma and upper hybrid frequencies fN and fT on topside ionograms, may also be applied to the plasma waves propagating at frequencies near the electron gyro-harmonics. The relatively high damping that occurs for θ # 90° severely restricts the ranges of θ and wave-number k over which the reflexion mechanism applies. The restriction is such that the damping is severe for all values of θ when k is large, and for all values of θ outside a cone of ∽ 2° centred on θ = 90° when k is small.

A non-linear integro-differential equation is derived, governing the space time evolution of non-linear hydromagnetic waves in a finite β collisionless plasma. The theory is done for quasi-perpendicular propagation, for wave frequencies above ion cyclotron frequency, and wavelengths smaller than ion Larmor radius. In the low β limit, the derived equation reduces to a Korteweg de Vries equation.

A general description of cyclotron harmonic resonant pitch-angle scattering is presented. Quasi-linear diffusion coefficients are prescribed in terms of the wave normal distribution of plasma wave energy. Numerical computations are performed for the specific case of relativistic electrons interacting with a band of low frequency whistler-mode turbulence. A parametric treatment of the wave energy distribution permits normalized diffusion coefficients to be presented graphically solely as a function of the electron pitch-angle.

The diffusion coefficients generally decrease with increasing cyclotron harmonic number. Higher harmonic diffusion is insignificant at very small electron pitch-angles, but becomes increasingly important as the pitch-angle increases. One thus expected the rate of pitch-angle scattering to decrease with increasing electron energy, since the resonant value of the latter varies proportionately with harmonic number. This indicates that, in mirror-type magnet field geometrics, such as the Earth's radiation belts, the diffusion losses of high energy electrons are likely to be appreciably slower than those at low energy. Integration of the diffusion rates along a complete bounce orbit will be required to clarify this point, however, since the high-energy particles will be subject to more rapid first harmonic diffusion near their mirror points.

Small amplitude electrostatic oscillations in a cold plasma with continuously varying density have been investigated. The problem is the same as that treated by Barston (1964), and in part 1 of this paper, but, instead of starting from a differential equation, we base our treatment on an integro-differential equation for the transverse component of the electric field. This equation is very similar to the Vlasov equation, and we show this to be the cause of the striking similarities between the two theories pointed out in part 1. The normal-mode analysis utilizes the similarity of the integral equation for the eigenfunctions to the equation of anisotropic neutron transport. The continuum eigenfunctions are found in the form of a generalized van Kampen eigenmode, and their calculation is reduced to solving a non-singular integral equation. Equivalence of the eigenfimctions found in this way and of those found by solving the equivalent differential equation is proved, and the question of orthogonality and normalization is discussed.

A weak E layer in a non-uniform magnetic field will tend to precess as a rigid body in response to the radial focusing of external magnetic fields and fields due to wall currents. We study the interaction of this precessional mode with a background plasma, and we explicitly include dissipation mechanisms in the plasma, walls and external resistors. When the plasma background is treated in the MHD approximation, we find that the mode changes character from a precessional mode at low density to a compressional Alfvén wave at high density. For a very weak E layer, instability is found, even without dissipation, when a sufficiently high background plasma density is present. However, for moderate E-layer strengths, the modes are found to be stable, even with dissipation.