Second-order perturbation calculation shows that an explosive instability of three resonantly interacting coherent electrostatic waves can be limited, and converted into a multiple-periodic process by nonlinear terms of the same order as those that destabilize the waves in the first-order approximation. No higher- order nonlinearities are necessary. The method used is purely classical, and consists in transforming the Hamiltonian of the waves into angle-action variables, and canonical averaging of the Hamiltonian over the proper angles. The number of degrees of freedom is thus reduced to one, which permits one to analyse the wave interaction in the phase plane without using the usual equations for the complex wave amplitudes.

It is shown that an ion loss cone distribution function with m ≥ 1 becomes unstable against electrostatic waves with ω ≫ Ωp and k0 = 0 in the presence of a cold plasma population, in contrast with pure warm systems, which require m ≥ 3 for instability. This result is an extension to high frequencies, ω ≫ Ω of similar conclusions reached by Pearlstein et al. (1966) and Farr & Budwine (1968), for ω-values equal to the first few harmonics of the proton gyrofrequency.

The theory of ion heating in a Q-machine Barium plasma by the electrostatic ion cyclotron instability is considered in some detail. Linear and quasi-linear theories are considered. The effect on resistivity of the instability is also considered. The ion temperature was observed, optically, to increase to a value in excess of 25 000°K by a mechanism attributable to a collisionless process. Energy transfer from the drifting electron current to the ions is analyzed explicitly. Detailed comparisons with experiments are presented.

This paper uses the general solution of the linearized initial-value problem for an unbounded, exponentially-stratified, perfectly-conducting Couette flow in the presence of a uniform magnetic field to study the development of localized wave-type perturbations to the basic flow. The two-dimensional problem is shown to be stable for all hydrodynamic Richardson numbers JH, positive and negative, and wave packets in this flow are shown to approach, asymptotically, a level in the fluid (the ‘isolation level’) which is a smooth, continuous, function of JH that is well defined for JH < 0 as well as JH > 0. This system exhibits a rich complement of wave phenomena and a variety of mechanisms for the transport of mean flow kinetic and potential energy, via linear wave processes, between widely-separated regions of fluid; this in addition to the usual mechanisms for the absorption of the initial wave energy itself. The appropriate three-dimensional system is discussed, and the role of nonlinearities on the development of localized disturbances is considered.

We formulate, in the framework of MHD, a simple eigenvalue problem, capable of treating the stability of force-free magnetic fields curl B = αB in different geometries. We prove that a force-free field surrounded by a rigid wall is stable, if the eigenvalue α corresponds to the lowest value of compatible with the geometry considered. We extend this result to the case where α is a function of position; and we recliscuss it from the viewpoint of the first-order variation. We give various theorems and criteria for stability, for continuous as well as for admissible discontinuous or infinite displacements. A general upper limit for the growth rate is , where and and are respectively the maximum values of and υAthe Alfvén velocity inthe plasma volume.

We study five relativistic plane nonlinear waves: circularly polarized waves and electrostatic plasma oscillations propagating parallel to the magnetic field, relativistic Alfvén waves, linearly polarized transverse waves propagating in zero magnetic field, and finally the relativistic analogue of the extraordinary mode propagating at an arbitrary angle to the magnetic field. When the ions are driven relativistic, they behave like electrons, and the assumption of an ‘electron–positron’ plasma guides us to equations which have the form of a one dimensional potential well. Our solutions indicate that a large-amplitude super luminous wave determines the average plasma properties, and not vice versa. For example, linearly polarized waves impose a plasma number flux equal to the relativistic addition of Nc/β and NVE, where N is the density, c the speed of light, β (>1) the ratio of the phase speed to c and VE the E × B speed measured in the frame moving with speed c/β with respect to the frame in which the phase speed is measured. The implications for cosmic ray acceleration in pulsar magnetospheres are considered.

The nonlinear development of a plasma-mean system immersed in a static magnetic field B0 is investigated. The beam electrons are assumed to be streaming parallel to the direction of B0 and have a spread in the perpendicular component of the velocity. We restrict our consideration to the P instabilities which result in growing whistler waves aligned in a direction parallel to that of B0. Scaled equations describing the time evolution of these instabilities are derived under the assumptions that the beam density is small and that the resulting whistler wave is nearly sinusoidal. The perturbation of the perpendicular component of the electron velocity during the interaction can be excluded from the equation determining the wave dynamics, and then these scaled equations can be reduced to the simpler ones amenable to numerical solution which depends analytically on all the basic parameters of the problem. This solution shows that the amplitude of an unstable whistler wave first increases at the linear growth rate until the electrons are trapped and then, after overshooting, it approaches a steady state. The time evolution of the amplitude does not show an oscillation with a constant period; this results from the phase mixing enhanced by the spread in perpendicular velocities of the beam electrons.

Multiple spatial scaling is incorporated in a modified form of the Bogoliubov plasma cluster expansion; then this proposed reformulation of the plasma weak- coupling approximation is used to derive, from the BBGKY Hierarchy, a decoupled set of equations for the one- and two-particle distribution functions in the limit as the plasma parameter goes to zero. Because the reformulated cluster expansion permits retention of essential two-particle collisional information in the limiting equations, while simultaneously retaining the well-established Debye-scale relative ordering of the correlation functions, decoupling of the Hierarchy is accomplished without introduction of the divergence problems encountered in the Bogoliubov theory, as is indicated by an exact solution of the limiting equations for the equilibrium case. To establish additional links with existing plasma equilibrium theories, the two-particle equilibrium correlation function is used to calculate the interaction energy and the equation of state. The limiting equation for the equilibrium three-particle correlation function is then developed, and a formal solution is obtained.

Theorems and criteria on stability of force-free fields, developed in part 1 (Krüger 1976) are illustrated for continuous, as well as discontinuous, spectra of α. Previous and new results are understood from a unified viewpoint.

The paper investigates the effect, on the Bernstein-mode dispersion relation, of a small collision frequency v, such that , for the high-frequency regime. It is shown that the ion-acoustic mode dominates the Bernstein modes in this regime for homogeneous plasmas; and it is suggested that this also holds for inhomogeneous plasmas.

Nonlinear theory of Rayleigh—Taylor instability in plasma supported by a vacuum magnetic field shows that the growth rate of the mode, unstable in the linear theory, increases if the wavelength of perturbation π lies between

and 2πcrit. This might have an important bearing on the proposed thermonuclear MHD power generation experiments.

The nonlinear Landau damping of a broad-band spectrum of electron plasma waves propagating in a plasma column is considered.

Finite geometrical effects inherent to the boundary value problem are taken into account. It is shown that nonlinear Landau damping of such a spectrum can be easily observed on a plasma column with a density of the order of 2 × 108 particles cm−3 and an electron temperature of about 3eV.

The current-driven Alfvén instability is discussed analytically. The instability is driven by the resonant Landau and transit-time terms for propagation not parallel to B0. The critical current for instability is slightly less than the corresponding one for the ‘k-like’instability; thus, the instability discussed in this paper may be the more important for the parameter range in which our approximations are valid. The effect of current flow on the magnetosonic mode is also discussed, but no instability is found.

By using the concept of combined field, we present a simple and symmetric Hamiltonian formulation for an infinitely conducting fluid, moving in an electromagnetic field. The definition of the combined field potentials makes it possible to replace the interaction between the electromagnetic field and charges by an interaction between two fields. Some applications are made, to prove the theory (e.g. the Hamiltonian of a point charge, Fourier representation).

A new method using crossed laser and atom beams within a plasma is capable of determining the electron density with a spatial resolution of 1 mm and a time resolution of 2–5 nsec. The measurement is done in the forward direction of the atom beam and is to a first approximation independent of electric and magnetic fields inside and outside the plasma region. A calculation is carried out for a neutral cesium beam probing a fully ionized hydrogen plasma. It is pointed out that a laser beam can also be combined with available ion beam diagnostics.

The formation of an electrostatic double sheath at a magnetic separatrix is described, with particular reference to the polytron. Results are discussed for a collisionless plasma, and for a simple MHD model where the sheath thickness is determined by collisional effects. Expressions are given for the sheath thickness in the polytron. Of more general interest is the result that the formation of a sheath makes the resistance of the plasma to current flow across the separatrix proportional to ωτ, not ω2τ2 as expected for a homogenous magnetic field.

In a layer of magnetic field aligned current, waves corresonding to the slow mode in the limit of no current are absorbed and/or reflected as soon as they enter the layer, while, under certain conditions, the waves corresponding to the fast mode do propagate through the layer.

Sturrock's theory of stochastic acceleration and Dupree's theory of strong plasma turbulence are here derived in a very simple way, showing that trapping cannot correctly be taken into account by this kind of theory. A comparison of Sturrock and Dupree's diffusion coefficients D shows that they differ significantly only in the domain where these theories fail, and where the highest values of D are reached. The need for a theory taking trapped orbits into account appears clearly. Such an attempt is the object of a sequel.

The purpose of this paper is to demonstrate the possibility of measuring the drift velocity of a thermal plasma with an H.F. quadrupole probe. The plasma is assumed to be isotropic and collisionless. A hydrodynamic model is used without restrictive assumptions on the magnitude of the drift velocity, which can be made smaller or larger than the thermal velocity. Some curves are presented for velocities parallel or anti-parallel to the emission-reception axis of the probe.

The recurrence of ion waves has been examined experimentally in the region where the nonlinearity is so weak that recurrence length is determined oniy by dispersion. We observe that the recurrence length is longer than the theoretical prediction derived from the Korteweg–de Vries equation of hot ions, and that the recurrence is imperfect. The result is qualitatively explained by wave–particle interaction.