We study non-linear interaction between three monochromatic waves which propagate parallel to the direction of a magnetic field in a plasma. The approach to the problem is hydromagnetic, including temperature effects, and the method of solution is that of coupled mode theory. In particular drift motions of the particles along the magnetic field are considered taking into account relativistic effects. The interaction of two transverse waves and one longitudinal wave is treated as well as that of three longitudinal waves. Besides, the case of two longitudinal waves and one perpendicular wave has been studied in some detail assuming the latter to have a long wavelength.

The present paper represents a generalization of the problem of three-wave interaction in a plasma to situations more complex than have been treated before. The most essential limitation consists in the assumptions made for the directions of propagation of the waves.

The high frequency conductivity tensor of an isotropic plasma is derived taking into account particle correlations at the lowest consistent order in the parameter ωp/ω these correlations describe a weakly Langmuir turbulent plasma. Two special cases are investigated in which the two-particle correlation function is related to the turbulent electrostatic field spectrum. Particular distribution functions and spectra are considered and approximate dispersion relations are derived in both cases in ‘the cold plasma limit’. The importance of the corrective term is discussed in terms of three dimensionless parameters measuring the strength of the turbulence, the shape of the spectrum, and the frequency. The effect could be important for frequencies not too far from the plasma frequency.

Using a 337 μm gas maser (Gebbie, Stone & Findlay 1965; Gebbie, Stone, Slough, Chamberlain & Sheraton 1966) it is possible to obtain precise measurements of the submillimetre absorption and phase shift occurring in specimens of each of the phases of matter and results have already been reported for the soild, liquid and gaseous states (Chamberlain & Gebbie 1965; Gebbie, Stone, Findlay & Pyatt 1965; Chamberlain, Werner, Gebbie & Slough 1967;, Chamberlain, Findlay & Gebbie 1965).

The resonance oscifiations of low temperature, bounded, non-magnetized in-homogeneous plasmas are studied using different methods of solving the linearized Vlasov—Maxwell equations. A low-temperature expansion method of solving the orbit-integral solution of the Vlasov equation is compared to fluid descriptions of the plasma using the truncated moment equations. Numerical results for a one-dimensional plasma slab show qualitative agreement but some quantitative discrepancies.

We point out in this paper that when one increases the intensity of the second applied pulse or the time interval between the two pulses, the amplitude of the first echo saturates and multiple echoes of comparable amplitude appear. Moreover, we predict a damping of the echoes which is due to a phase mixing effect; that is collisionless.

Recently the question of existence and stability of laminar one-dimensional coffisionless shock waves has received some new attention (Montgomery & Joyce 1969). These electrostatic shock waves are usually thought of as being generated by ion acoustic waves (pulses) in a plasma with Te ≫ Ti Apart from the well- known class of shock solution with oscillatory structure (Moiseev & Sagdeev 1963) there exist, at least within the framework of Vlasov theory, monotonic transitions from one asymptotic state to another (Montgomery & Joyce 1969). For these shock-wave like solutions it is easy to see that the electrons cannot be in thermal equilibrium (have a Maxwellian distribution) in the downstream state, if they are so in the upstream state. Hence it appears to be natural to investigate first the stability of the homogeneous downstream state (which tells us if the transition is possible at all) by investigating the properties of the class of permitted distribution functions for the electrons, before attacking the much more involved problem of the possible instabilities arising from the inhomogeneitythe transition region of the shock.

The steady shock dicontinuity in the flow of a perfectly conducting gas with anarbitrarily oriented magnetic field is considered by taking as the parameters the Mach number, the Alfvén Mach number, the magentic field direction, and the shock angle. The shock solutions satisfying the conservation laws as well as the entrophy and evolutionary conditions are given by the roots of a simple cubic equation and the associated formulas. Several special cases are discussed analytically and numerically. The possible types of magnetohydrodynamic shock waves are shown on the speed-deflexion phase plane, and the possible shock configurations in the physical space are discussed on the basis of them.

The application of integration by the method of stationary phase to resonant oscillations excited by a small pulsed dipole is outlined. Both the growth and the decay of the oscillations near the plasma frequency are determined by this method at a fixed distance from the dipole, first in the absence and then in the presence of an external magnetic field. It is shown that Landau damping must be taken into account in the calculation of the growth but not of the decay. The oscillations are shown to spread out with a speed that is about half the mean thermal speed of electrons.

Only the decay, not the growth, of the oscifiations near harmonics of the cyclotron frequency can be calculated by the same method. It is shown, moreover, that the amplitude, calculated for an observation point that moves away with sateffite velocity in an ionospheric environment, is only valid for time delays longer than about a minute. Such a result is therefore of no practical interest because the resonances observed from sateffites only last a few milliseconds. The erroneous nature of using such a result and the need for a different approach, such as that used in earlier work by the first author, are thus demonstrated.

We discuss the propagation of wave packets of the form

in an infinite uniform plasma, where G(z, t) is a slowly varying function of space z and time t. One can very simply derive the equation of change of G(z, t) for the stable or unstable case. The terms in the equation are of physical interest and clearly define the limitations of linear theory. We then investigate the problem of whistler mode wave propagation in a collisionless Vlasov plasma in a given non-uniform magnetic field. We choose the electric field to be of a W.K.B. form and the particle distribution to be isotropic. We can express the perturbation in the particle distribution in terms of an integration along the zero-order par tide orbits (an integration overtime). These orbits can be found correct to a term linear in a smallness parameter ε (when ε equals zero we arrive back at a uniform magnetic field). The charge and current density due to the perturbation are related through Maxwell's equations to the electric and magnetic field of the wave in the usual self consistent Boltzmann—Vlasov description. We show that the contribution to the current arises from recent events in the history of a given particle because of the finite temperature of the plasma. This result leads to an expansion of slowly varying parameters which in turn gives rise to the equation governing the motion of the wave packet.

The kinetic equation for resonant three-wave coupling is solved exactly in the one-dimensional case assuming that only a single triplet of modes in interacting. No assumptions are made as to the froms of the interaction kernel or of the oscillation frequencies as functions of wave-number. Both positive- and negative- energy waves are allowed to be present simultaenously. In explosively unstable situations the time, t∞, for divergent behaviour to occur is calculated in terms of the interaction keral and intial conditions. In situations where the system is non-linearly stable, the density of quasi-particles is found to tend time-asymptotically to a steady state which depends in detail on the initial preparation of the system.

For the case of two or three dimensions, the asymptotic state is obtained when the system is stable, the k-space is finite, and only one triplet of modes interact. If these modes all have the same sign of energy, the Rayleigh—Jeans distribution is valid; but if one of the modes has the oppsoite sign of energy, an additional conservation law leads to a modification of the Rayleigh-Jeans distribution.

The Vlasov—Poisson equations are reformulated by applying an arbitrary transformation to the velocity variable in such a way that perturbation theory of the transformed equations does not exhibit the customary secularity in time (or space) in second or higher order. The lowest order approximation of the new formulation is discussed and compared with conventional results. The source of the non-uniformity appears to be the divergence of Particle trajectories as calculated by perturbation methods, from the exact ones after long times. The transformation which allows one to follow the particle trajectories is a transformation to a frame of reference moving with a plasma test particle in the self-consistent field.

The instabilities in a two stream plasma in which the density across the stream is variable are studied from the point of view of a scattering problem. Approximate dispersion relations are developed, one of which is identical with the infinite, homogeneous case. A second relation is obtained which depends upon the scale of density variation.

The approximations usually made to truncate the BBGKY hierarchy for a plasma are discussed; their failure at small inter-particle separations leads to divergence of the Balescu—Lenard collision integral. A number of authors have obtained convergent kinetic equations, often by rather complicated methods.

It is shown here that, if the standard truncation procedure is modified in a way which makes it less obviously inconsistent for close approaches, the standard methods maybe closely followed in deriving a convergent collision integral which agrees to dominant order with the ‘cutoff’ Balescu—Lenard integral and with the other work on the problem. In fact, the kinetic equation obtained is identical with the Balescu—Lenard equation except that the Coulomb potential is replaced by another that is non-singular at the origin. A physical interpretation of this result is suggested.

The effect of Hall current and finite electrical resistivity has been studied on the Rayleigh—Taylor instability of superposed incompressible fluids in the presence of a uniform horizontal magnetic field. It is found that Hall current has a negligible stabilizing influence in the high resistivity limit whereas it has a destabilizing influence in the low resistivity limit.

We have studied the behaviour of a charged particle in an axially symmetric magnetic field having a neutral point, so as to find a possibility of confining a charged particle in a thermonuclear device. In order to study the motion we have reduced a three-dimensional motion to a two-dimensional one by introducing a fictitious potential. Following Schmidt we have classified the motion, as an ‘off-axis motion’ and ‘encircling motion’ depending on the behaviour of this potential. We see that the particle performs a hybrid type of motion in the negative z-axis, i.e. at some instant it is in ‘off-axis motion’ while at another instant it is in ‘encircling motion’. We have also solved the equation of motion numerically and the graphs of the particle trajectory verify our analysis. We find that in most of the cases the particle is contained. The magnetic moment is found to be moderately adiabatic.

The effect of weak collisions on ion-acoustic wave instabifities in a plasma with large ratios of electron to ion temperatures is investigated. The critical current for instability is shown to be signfficantly reduced by collisions. At the critical current the instability first occurs at non-zero values of kλe. As kλe increases the collisional effects become less and less significant.

In plasma, where collective motions are made possible by the long range of the Coulomb interaction, there is properly no general kinetic equation for the one-particle distribution function alone. In part 1, we gave the foundations of an enlarged kinetic theory of plasma where the time evolution of the electric field fluctuations Ik is considered simultaneously with the time evolution of the one-particle distribution F. Here we improve these equations in a systematic fashion to include all processes which are relevant in stable and weakly unstable plasma. More precisely, the equation for the waves now includes, besides Landau damping and Cerenkov emission, the effect of collisional damping and nonlinear Landau damping, as well as emission via particle-particle scattering (bremsstrahlung), wave-particle and wave-wave scattering. The particle kinetic equation is also improved accordingly so that we obtain a kinetic description which is uniformly valid for all plasma réegimes, if one excludes strong turbulence (for which there is at present no completely satisfactory statistical theory).

The problem is to determine the electrical and thermal conductivities of high pressure are plasmas from measurements of the current—voltage characteristics of the are and a single radial temperature profile. A new numerical method is described together with the corresponding computer program. The latter is applied to some recent measurements on wall-stabilized nitrogen ares, covering the temperature range 4500—11,000 °K, for which radiation can be neglected, and the results are compared with those of other workers.

Ions reflected from the sharp potential rise in a magneto-plasma, perpendicular shock wave result hi a broad, small-amplitude ‘foot’ in the magnetic field proffle. A theory is given for the structure of this foot, which shows its shape to be a rather flat parabolic curve commencing at about half a Larmor radius in front of the shock front proper. The theory also predicts the amplitude of the foot just in front of the sharp rise as a function of Alfvén—Mach number. Close agreement with experiment is found.

In this note the problem of stability of two hot collisionless streams of charged particles is considered. The masses, charges, densities, and temperatures are arbitrary and the distribution functions are modelled by one ‘resonance’ function for each stream. The problem of stability is resolved by Nyquist diagrams, and, for the case of equal plasma frequencies, also by solving the dispersion relation in ω. A comparison with two Maxwellians on the one hand, and a two step function model on the other, is given. Step functions appear to be too crude for this problem.