Starting from the Boltzmann equation, new boundary conditions are derived to be matched with the Navier—Stokes equations, that are supposed to hold in the main body of a gas. The idea upon which this method is based goes back to Maxwell and Langmuir. Since the distribution function is supposed to be completely determined by the Navier—Stokes equations, this new set of boundary conditions extends in some sense the validity of the macroscopic equations to the transition and free molecular régimes. In fact, it is shown that the free molecular and slip flow régimes are correctly described by this method; the latter is also supposed to give a reasonable approximation for the complete range of Knudsen numbers. The new procedure is applied to different problems such as plane Couette flow, plane and cylindrical Poiseuile flow, heat transfer between parallel plates and concentric cylinders. Results are obtained and compared with the exact numerical solutions for the above-mentioned problems.

The diochotron instability of a thin, tenuous, cylindrical layer of charged particles, whose gyro-radius is of the order of the mean radius of the cylindrical layer, is investigated using the Vlasov equation. Two distribution functions are considered which give practically the same density in the physical space, but are quite different in the velocity space: in the first the velocity spread is practically zero; in the second the particles oscillate around the mean radius of the cylindrical layer. It is shown that the first distribution exhibits the usual diochotron instability, while the second one does not. It is also shown that, however small the velocity spread, the thickness of the cylindrical layer cannot go to zero. Its minimum value is roughly determined by equating the plasma frequency to the cyclotron frequency.

The admittance of a plane grid capacitor immersed in an ion beam plasma has been measured, and the results are in general agreement with the kinetic theory of the admittance (Buckley 1968). With a steady bias potential applied to the grids, the results showed a deviation from the theoretical behaviour that is attributable to plasma depletion by the grids. The effect of varying the background pressure showed that electron-neutral collisions had a negligible effect on the measured admittance. A significant change in the admittance was found for a measuring voltage applied to the grids of the order of the electron temperature.

A pair of plane parallel grids is inserted in a hot plasma, and an oscillatory voltage is applied across them. The electric field excited in the plasma, and the complex admittance of the grid/plasma system, are computed for applied frequencies high enough to justify neglect of ion response. The grids are electrically, but not mechanically, coupled to the plasma, which is assumed to be in a spatially uniform collision free Maxwellian equilibrium state. The field is computed as a function of distance from the grid plates over a range of frequencies covering the plasma frequency, and the complex admittance of the system is computed as a function of frequency at various grid separation distances. The real component of the field and the capacitive component of the admittance are subject to three major effects: cold plasma dielectric behaviour, oscillatory Debye sheaths on the grids, and (above the plasma frequency), longitudinal plasma waves. The imaginary field component and the conductive admittance component are produced by spatial Landau damping. In an accompanying paper (Freeston 1968), the computed admittance is compared with laboratory measurements made in a situation approximating well to the idealized problem considered here.

The non-linear induced scattering by plasma particles of right-hand and left- hand polarized electromagnetic waves propagating along the magnetic field, is studied. The non-linear Landau damping for frequencies near the ion cyclotron frequency in a weak turbulent inhomogeneous plasma is determined.

The structure of modes for which k┴B0 and ω≪Ω_ (ion waves) has been studied qualitatively in the two limits κR+ ≪ 1 and κR+ ≫ 1, where R+ = (κT+/MΩ+)½ is the mean thermal Larmor radius, without the usual electrostatic approximation. Asymptotic forms of the dielectric tensor elements εij are developed in these two limits. The modes having appreciable k × E can be called ‘generalized’ Bernstein modes. The approximation which yields the familar electrostatic Bernstein modes is εxx = 0. This approximation is shown to be valid only for large κR+ and low β. However, for small and moderate values of κR+ the generalized Bernstein modes partake of a ‘mixed’ electromagnetic- electrostatic character. In particular, for κR+ ≲ 0(1) (but ω/κ < c) the electrostatic Bernstein modes are incorrect approximations. The warm plasma electromagnetic theory is discussed with reference to cold plasma theory for a low β plasma, and it is shown that: (1) the lower hybrid frequency is only an approximate resonance in the warm plasma; (2) electromagnetic cut-offs occur at all harmonics of the gyrofrequency as k → 0; (3) electrostatic resonances occur at all harmonics of the gyrofrequency as k→∞; (4) propagation can occur in warm plasmas at frequencies above the lower hybrid.

A uniform, unbounded, cold plasma stream, with relativistic velocity (0, U, 0), traverses a vacuum half-space y > 0 and a dielectric half-space y < 0. A plane electromagnetic wave in the plasma stream in y > 0 is incident obliquely on the face of the dielectric. The criterion for determining which characteristic waves are present in each half space is discussed, and it is shown that, for the polarization in which B is parallel to y = 0, there is one reflected wave and three transmitted waves. One of the latter can be an amplifying wave. A first approximation to the transmission and reflexion coefficients is found in the case when the density of the stream is low.

A numerical experiment is described in which a one-dimensional plasma model is used to simulate the resonance probe. The trajectories of a large number of plane electron sheets are followed in a self-consistent calculation which takes account of a spatially dependent but smeared-out ion background, and externally applied static and r.f. potentials. No differentiation between sheath and plasma regions is introduced, nor is there any linearization.

The typical resonant rectification curves, with resonance below ωp, are obtained. A comparison is made between the plasma behaviour at resonance and in frequency ranges above and below the resonance. For example, the phase of current collection w.r.t. probe potential is studied. Fourier analysis of potential and density variations is used to obtain further information on phase relationships. Orbit diagrams of electrons forming the current under resonant conditions are presented, and a model for the mechanism of resonant rectification in terms of electrons at the sheath edge put forward.

In this paper the electric fields of an infinitesimal dipole oscillating in a hot plasma are derived in the absence of an external magnetic field; explicit expressions are obtained by making an approximation on the background electron velocity distribution function. These results are applied to investigate the resonance observed at the plasma frequency, and to estimate the radiation resistance of the dipole.

The evolutionary condition for transverse and normal shock waves, and the fire- hose and mirror instability conditions for the associated flow, in a collisionless, anisotropic plasma having a strong magnetic field are determined using the theoretical representation of Chew, Goldberger & Low (1956) for such a medium. The results are expressed in terms of the Mach number, Alfvén Mach number, and the ratio of the temperatures parallel and perpendicular to the magnetic field in the flow approaching the shock wave, and applied to ascertain in what range of these parameters various types of instabilities may occur. The effect of the heat flux, which does not vanish generally in a collisionless plasma, on the shock stability is discussed.

In this work we evaluate the anomalous high-frequency conductivity arising from particle—wave interactions in a multi-species turbulent plasma with no external magnetic field. The calculation is made by using the full three-dimensional model of the quasi-linear theory of plasma turbulence for a wide range of frequencies embracing the electron plasma frequency. It is found that the conductivity tensor is anisotropic and slowly time dependent; this reflects the slow time dependence and the anisotropic character of the distribution functions of the velocities of the particles. It is also found that there is a very narrow range of frequencies (just above the electron plasma frequency) for which the components of the conductivity tensor are negative. The particular case of the two- stream instability is examined in detail. The anisotropic character of the components of the conductivity tensor and their dependence on the frequency of the external field is studied. Finally some possible means for an experimental check of our calculations are suggested.

J. Plasma Physics (1968) vol 2, part 2, pp. 105–118.

Page 115, legend to figure 4 should read as follows:

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(b) Plot of