A new formulation of quasi-linear theory is presented, which allows for only resonant diffusion, caused by both growing and damped waves. Nonresonant terms do not appear in the diffusion equation, but contribute to wave momentum and energy, and ensure conservation of total momentum and energy.

The nonlinear behaviour of a realistic one-dimensional bounded plasma (specifically, the classical plasma slab–condenser system) is computed by an iterative perturbation method. The results indicate, somewhat unexpectedly, that the influence of the r.f. field on the static density profile and on the resonance spectrum is much smaller than would have been inferred from a previous analysis of an unbounded plasma. However, this approach is inherently limited by the fact that, even for not too high r.f. fields, the electron density perturbations can become of the same order of magnitude as the static density in the tenuous plasma sheath near the wall. The resonance curves obtained with this sophisticated model show quite remarkable agreement with existing experimental data.

The Herlofson paradox is exemplified by a capacitor containing cold, collisionless, inhomogeneous plasma as the dielectric: its response to a sinusoidal driving signal can exhibit continuous energy absorption, even though the system is lossless. The underlying mechanism has been explained generally by Barston in terms of the transient response of the system. In this paper, we confirm Barston 's conclusions by examining in detail several analytically tractable cases of delta-function and sinusoidal excitation, and consider the effects of collisions and non-zero electron temperature in determining the steady state fields and dissipation. Energy absorption without dissipation in plasmas is analogous to that occurring after application of a signal to a network of lossless resonant circuits. This analogy is pursued, and extended to cover Landau damping in a warm homogeneous plasma, in which the resonating elements are the electron streams making up the velocity distribution. Some of the practical consequences of resonant absorption are discussed, together with a number of paradoxical plasma phenomena which can also be elucidated by considering a superposition of normal modes rather than a single Fourier component.

The three important aspects of wave propagation in a stationary and a moving plasma (namely frequency, polarization and dispersion) are compared, taking the phase refractive index of the wave as the independent variable. Wave propagation and the motion of the plasma are taken to be along the magnetic field. The plasma is assumed to consist of one species only, and the effect of collisions is neglected. Wave propagation in a moving plasma has been shown to possess several important features, such as the absence of cyclotron resonance, reversal of the sense of polarization when the phase velocity becomes equal to the plasma velocity, and the existence of backward waves for very small and very large phase velocities.

The intensity of X-ray sources in a focused deuterium plasma produced by a coaxial accelerator has been analysed as a function of position, X-ray energy and time of emission. The X-ray source in the axial region can be resolved (by micro- densitometer readings on X-ray pinhole camera films) as a sequence of small sources (linear dimension ∼ 0.1–0.3 mm) of hard radiation ≳ 2 ke V inside a more diffused source (cylindrical region of 1–4mm diameter) of softer X-rays. In each discharge the point sources are distributed for the most part in the general axial region of the discharge and two or more sources with different radial positions can be frequently observed for one specific value of the axial co-ordinate. Images of localized X-ray sources are also observed in the off-axis halo region. Multiple repinching of the axial plasma column or emission from metal-vapour clouds (by anode bombardment) can be ruled out in this experiment (hollow central electrode, or anode, radius 3·4 cm). The source multiplicity can be related to a complex (filamentary) structure of the plasma.

From a basic discussion of the behaviour of VLF waves propagated almost perpendicularly to the earth 's magnetic field, the lower hybrid resonance (LHR) frequency is seen as a boundary between compressional Alfv én waves and electrostatic sound waves. The disparity of the wavelengths involved leads to a fundamental difference between the characteristics of the AC electric field far from and close to the source.

Some detailed observations of this field are presented, from an experiment that used both a mother-daughter technique for large distance measurements and a quadrupole probe technique for the local field, and that was performed in the equatorial ionosphere. The frequency dispersion of quasi-transverse (QT) waves received at a large distance and the amplitude-frequency response of the VLF quadrupole probe are compared with a theoretical analysis for the warm ionospheric plasma. From this analysis, the possibility is established of using the properties of fast and slow waves in active experiments, for measuring simultaneously the LHR, the electron density, a hybrid ion-electron thermal velocity and possibly the plasma drift velocity.

A co-axial plasma focus device is operated in deuterium gas at ambient pressures from 0·2 to 0·7 mm Hg and condenser bank voltages from 10 to 22 kV. Magnetic probing reveals an axisymmetric parabolic current sheath which propagates down the co-axial tube (steady propagation region) and collapses off the end of the co-axial gun forming a dense plasma focus. The state of the plasma just before the focusing action is very much dependent on the velocity in the steady propagation region. This velocity is experimentally studied and compared with a snow- plough theory. The collapsing phase of the current sheath is studied using high speed framing photography and current and voltage measurements. The results indicate a focus of length 1·5 cm and radius less than 1·7 mm.

The experimental modifications of the classical Landau mode (modification of the first pole, and occurrence of a new propagating mode below the plasma frequency) are due to a peculiar distribution function. This distribution function is approximated by a simple model (i.e. a Maxwellian function plus a water-bag function). A numerical calculation with this model gives new dispersion curves, which agree with experiment.

By linearizing about the upstream state, the characteristic length for the growth of magnetic field in a laminar wave front, propagating perpendicular to an external magnetic field, is found as a function of βi, the ratio of ion pressure to magnetic pressure. For values of βi, between 10–2 and 1 the thickness is found to scale as . This scaling is in good agreement with the observed scaling of turbulent shocks, suggesting that for such shocks the shock thickness is closely related to that of the corresponding laminar pulse.

The field of a pulsed beam (a ‘wave packet’), travelling through a medium with moderate absorption, is calculated by the saddle-point method. The packet velocity (i.e. the velocity of the spatial amplitude maximum) has the same direction as the velocity Re (∂ω/∂k), a generalization of the group velocity ∂ω/∂k in non-absorbing media. It differs from the absolute value of this velocity by a correction factor depending on the absorption, beam width and pulse duration. This factor is unity for vanishing absorption and infinite beam width. The velocity Im (∂ω/∂k) has no apparent physical meaning.

Measurements of sheath thickness in a low-pressure plasma are reported. The sheath is ion-rich and contains both ions and electrons; the sheath edge is taken as a boundary for ion acoustic waves. The thickness of the sheath is estimated from a frequency shift of the wave. Despite the actually continuous transition from plasma to sheath, a clear-cut sheath edge is observed. It should be emphasized that even a very small increase in thickness, such as 0·1mm, can be measured by this frequency-analysis method. The experimentally obtained sheath boundary is in agreement with the Tonks–Langmuir value as modified by Self.

The results of two experiments in a weakly ionized turbulent argon plasma are compared with a theory for the spacewisespectrum of the electron – ion density fluctuations of the plasma. It would appear that the theoretical predictions are in reasonable agreement with the results of these experiments. The theory, based on a quasi-static formulation of the Boltzmann equations, is a direct consequence of the Boltzman equations for the electrons, ions and neutrals in theweakly ionized plasma. A Heisenberg model is used for the turbulent diffusion coefficient of the plasma.

The equations of motion of an ideally conducting medium, in the magnetohydrodynamic approximation, are solved exactly under the hypotheses that (i) one component of the magnetic field is constant everywhere, (ii) the magnetic and hydrodynamic pressures are in equilibrium, and (iii) the solution is invariant under a continuous group of transformations which preserves the symmetry of the uniform magnetic field. It is seen that the invariants of the group of transformations form the basis for a parametric description of the full solution that describes the propagation of cylindrically symmetric magneto-sonic waves in the direction of the uniform field. A number of general features of the motion are deduced, and an integral expression is given for the amplitude of the component of velocity, perpendicular to the uniform magnetic field, which undergoes potential well oscillations.

Nonlinear stabilization of cold beam plasma instability are studied on the basis of Dupree's theory of strong plasma turbulence. Calculations are carried out to second order in the perturbed electric field, and it is found that, for a narrow spectrum satisfying δκ/κ≪1, stabilization occurs for the trapping frequency ωT ť ∣qeκΕk/me∣½ ť γL the linear growth rate. Nonlinear frequency correction, and also the changes in the kinetic and thermal energy densities of the beam, are calculated as the waves grow. Some points of difference between the resonant and non-resonant instabilities are discussed.

With the generalized group velocity vg = Re (∂ω/∂k) for media with moderate absorption, derived in a previous paper, the Hamilton equations for group propagation in non-absorbing media are generalized to include moderate absorption. They contain directional derivatives not only in the direction of v but also in the direction of Im (∂ω/∂k). Two iterative computational methods are presented for ray tracing with the generalized Hamilton equations.

The boundary-value problem for diffusion-controlled breakdown of gases in microwave cavities is converted to a variational principle for the square of the reciprocal of the ‘effective’ diffusion length. For a rectangular cavity excited in the TEαγ0 mode, the resultant variational principle was minimized by an iterativevariational procedure. This was done using the high-frequency ionization coefficient of Herlin & Brown. However, any other functional dependence on the r.m.s. value of the electric field could have been used for this purpose. The resultant hierarchy of approximations to the variational principle obtained by this iterative-variational procedure for a cavity of square transverse crosssection is applied to the TE110 and TE220 modes of excitation. Adequate convergence is obtained with the first six approximations. Resultant plots of the ratio of the ‘effective’ to the characteristic diffusion length as a function of the Herlin & Brown parameter β, for various ratios of longitudinal to effective transverse cavity dimensions, imply that diffusion loss is greater for the TE220 mode of excitation. Also, from this theoretical investigation it is felt that the first ten approximations to the variational principle should provide adequate convergence for most situations encountered in practice with regard to β, as well as cavity dimensions.

We study the resonant interaction of particles with a monochromatic wave in an inhomogeneous plasma. The effects of the inhomogeneity are represented by a variable phase velocity of the wave. This introduces, in the wave frame, an inertial force due to the non-Galilean transformation of co-ordinates. Asymptotic expressions are obtained for the spatial damping coefficient of the wave in two physically different cases.

(i) When the inertial force is greater than the force due to the trapping (strong inhomogeneity case) instead of the null asymptotic behaviour that one obtains in the homogeneous case, the spatial nonlinear damping coefficient behaves asymptotically like the linear Landau damping coefficient.

(ii) When the inertial force is lower than the trapping force (weakinhomogeneity case), the spatial nonlinear damping coefficient for short distances is proportional to the ratio of the inertial force to the trapping force. For long distances, this coefficient is proportional to the square of the preceding ratio multiplied by the number of trapping lengths. In both cases, the proportionality coefficient is the linear spatial Landau damping rate.

The large-amplitude plasma wave with a trapped-particle distribution monotonically increasing in energy is found to be unstable if the trapped-particle plasma frequency ωt = (4πnte2/m)½ is greater than the bounce frequency ωb = (eΕ0κ0/m)½.

The kinetic theory of a two-dimensional one-species plasma in a uniform d.c. magnetic field is investigated in the small plasma parameter limit. The plasma consists of charged rods interacting through the logarithmic Coulomb potential. Vahala & Montgomery earlier derived a Fokker –;Planck equation for this system, but it contained a divergent integral, which had to be cut-off on physical grounds. This cut-off is compared to the standard cut-off introduced in the two-dimensional unmagnetized Fokker –;Planck equation. In the small plasma parameter limit, it is shown (under the assumption that for large integer n, γn/γn+1 = O(np), with p < 2, where γn = ωn −nΩ. with ωn the nth. Bernstein mode and Q the electron gyro frequency) that the Balescu-Lenard collision term is zero in the long time average limit if one considers only two-body interactions. The energy transfer from a test particle to an equilibrium plasma is discussed and also shown to be zero in the long time average limit. This supports the unexpected result of zero Balescu-Lenard collision term.

We present a theory of the nonlinear damping of a plasma mode based on a solution of the Vlasov–Poisson system. The formulation is exempt from the objectionable separation of the particles into ‘resonant’ and ‘nonresonant’, and is valid for ion as well as for electron modes. The effect of the damping of the mode on particle motion is taken in account. In particular, we evaluate numerically the damping of an ion mode for a temperature ratio Te/Ti = 16. We also obtain a number of small new shifts in the damping of a plasma mode in general, including contributions from the second derivative of the stationary distribution function.