The stability of an anisotropic plasma jet has been investigated using equations of Chew, Goldberger & Low (1965) in both plane and cylindrical geometries. The main conclusions are as follows:

(i) For waves of small wavelength, when λ (the ratio of plasma density to the density of surrounding non-conducting medium) is much greater than unity, the plasma jet is stable only if 1/6(1 + σ) <p∥/p⊥ < 1 + 2σ, where p∥ and p⊥ denote the components of pressure parallel and perpendicular to the direction of magnetic field in the steady state, and σ denotes the ratio of magnetic pressure to the perpendicular component of plasma pressure.

(ii) When λ ≫ 1, the plasma jet is unstable if the parallel component of pressure exceeds the sum of the perpendicular component of pressure and twice the magnetic pressure of the plasma.

(iii) For long waves, we find that there exists an infinite number of growing and decaying waves. Expressions for the growth rates have been obtained.

A model is proposed to explain the observed electron heating in perpendicular collisionless shock waves. This relies on trapping of the electrons by the turbulent electric field to modify their orbits in such a way that they gain energy from the steady potential jump across the shock. With simple assumptions about the nature of this trapping, reasonable agreement with the experimentally observed heating is obtained. In addition, the model provides qualitative agreement with a number of other experimental observations.

Transverse waves and instabilities propagating along the magnetic field in a homogeneous plasma are discussed analytically and numerically for frequencies of the order of the ion cyclotron frequency and below. The free energy driving the instabilities is assumed to be provided by thermal anisotropies, with the parallel temperature exceeding the perpendicular temperature, a situation appropriate to the solar wind near the earth and to the downstream conditions in collisionless shocks propagating approximately parallel to the magnetic field. It is shown that in the case where the ion β is of order one the long wavelength Firehose instability is not stabilized by finite Larmor radius effects, but that for smaller wavelengths it goes over smoothly into the resonant proton mode, discussed by Kennel & Scarf (1968).

The instability of right-handed, circularly polarized electromagnetic waves, propagating along an external magnetic field (whistler mode), is studied for electron plasmas with distribution functions peaked at some non-zero value of the transverse velocity. Based on the linearized Vlasov-Maxwell equations, the criteria for instability are given both for non-resonant instabilities arising from distribution functions with no thermal spread parallel to the magnetic field, and for resonant instabilities arising from distribution functions with Maxwellian dependence in the parallel velocities. It is found that, in general, the higher the average perpendicular energy, the more is the plasma susceptible to the whistler instability. These criteria are then applied to a sharply peaked ring distribution, and to loss-cone distributions of the Dory, Guest & Harris (1965) type.

This paper is an investigation of the effect of a magnetic field on transverse waves in a perfectly conducting gas which is rotating like a Rankine vortex about the axis of a cylinder. The magnetic field is assumed to be in the axial direction. There are three waves: two waves are rotating in the same direction as the gas, one faster and the other slower than the core speed of the gas, and one wave rotates in the opposite direction.

In many plasmas the ions are cold relative to the electrons but have a directed, subsonic velocity (the sonic velocity is (kTe/M)½). The behaviour of this flow around an obstacle is considered and solutions obtained for cylindrical geometry. The results are of direct interest in the interpretation of probe characteristics, especially when relative density measurements are involved, as the effects of the flow are typically of the same order of magnitude as the relative variations expected. Plasma flow past an obstacle may be of interest in other contexts (e.g. the case of dust particles in a gas discharge).

The statistical properties of non-linearly interacting waves are considered from a physically motivated viewpoint. A Fokker-Planck wave transport equation is derived. In certain circumstances an H theorem can be proved.

A wide-ranging investigation is rendered possible by a judicious combination of products of electric field components and electric polarization components for two distinct modes of propagation of electromagnetic waves in an anisotropic, ionized, stratified medium. The differential equations, governing oblique propagation in these two distinct modes in such a medium, are combined to yield various integral identities when integrated throughout the medium. These lead to a large number of relations between the reflexion and transmission coefficients (for incidence from below and from above) and the fields throughout the medium, each containing as a factor just one of the components of the external magnetic field pervading the medium.

The spectral distribution of radiation emitted by a charged particle moving nonrelativistically in a prescribed electromagnetic field, consisting of a strong uniform magnetic induction on which is superimposed a weak inhomogeneity, is determined. The calculation is confined to trajectories characterized by small Larmor radii, and to the azimuthal average of the full angular distribution of the radiation, the azimuthal angle being measured in a co-ordinate system for which the uniform component of the field defines the polar axis. If the particle gyrates for a time long compared to the gyro-period, the spectrum separates into several independent lines. Quantitative expressions for the distribution of energy within each line are presented, and then evaluated for selected inhomogeneities. Qualitatively, it is found that inhomogeneities (i) may distort the distribution of energy within the spectral line at the gyro-frequency by broadening, displacing, destroying the symmetry of, and/or splitting that line, and (ii) may also generate spectral lines at aipproximate harmonics of the gyro-frequency. These harmonics are second order in the inhomogeneity, are of successively higher order in the Larmor radius, involve successively higher-order spatial derivatives of the inhomogeneities, may occur at frequencies whose ratios are not precisely ratios of integers, and may be split into two or more closely spaced components.

Owing to the complexity of the Vlasov–Maxwell equations for inhomogeneous plasmas, it is of general interest to investigate simpler approximate models. We compare three specific models, proposed in the literature in connexion with the high frequency oscillations of such plasmas, by means of a singular perturbation expansion in powers of E = (λD/L)⅔, where λD and L are appropriate electron Debye and equilibrium scale lengths, respectively. Explicit formulas are obtained for the eigenfrequencies (to second order) and for the electric field eigenfunctions (to leading order), and the computation of these quantities for various equilibrium parameters becomes very simple indeed. We find that, whereas the electric field eigenfunctions differ in zero order for the three models, the eigenfrequencies are identical through first order.

This paper is concerned with existence conditions for steady hydromagnetic shock waves propagating in a collisionless plasma along an applied magnetic field. The electrostatic waves are excluded. The conditions are based on the requirement that solutions of the Vlasov-Maxwell equations deviate from a uniform state ahead of a wave. They are given as the conditions on the upstream flow velocity in the wave frame (i.e. in the form of inequalities among the upstream flow velocity and some critical velocities). The conditions crucially depend on the pressure anisotropy, and demonstrate possibilities of exacting collisionless shock waves for high β plasmas.

In this paper, the coupled-mode equations and coupling coefficients for three-wave interaction are derived by a Lagrangian approach for a general medium. A derivation of the Low Lagrangian for a warm plasma is then given, which avoids certain problems associated with the original analysis. An application of the Lagrangian method is made to interaction between collinearly-propagating electrostatic waves, and a coupling coefficient is derived which agrees with a previous result obtained by direct expansion of the non-linear equations. The paper serves primarily to present and demonstrate a conceptually useful and efficient theoretical approach to non-linear wave interactions.

A closed set of guiding centre equations, derived for a rotating plasma in a static magnetic field, is applied to the problem of the stability of a plasma in a sheared field. The rotation is found to have a stabilizing effect in the absence of resistivity.

A pair of coupled, linear differential equations is derived for the rotating plasma in a weakly sheared field. Dispersion relations are obtained by phase integral methods and, in the absence of finite Larmor radius effects and rotation, instability growth rates proportional to η½13 are found which become proportional to when either is included. The inclusion of both finite Larmor radius and rotation gives growing instabilities proportional to η which are stabilized by the rotation when the finite Larmor radius terms predominate.

The stability of anisotropic plasmas to the magnetosonic (or right-hand compressional Alfvén) wave, near the ion cyclotron frequency, propagating almost perpendicular to the magnetic field, is investigated. For this case, and for wavelengths larger than the ion Larmor radius and for large ion plasma frequency (w2p+ ≫ Ωp+) the dispersion relation is obtained in a simple form. It is shown that for T # T' (even T ≫ T) no instabifity occurs. The resonant ters are also included, and it is shown that there is no resonant instabifity, only damping.

An equation for magnetogasdynamic shock polar, given in terms of magnetic induction polar variables (i.e. the magnetic induction polar), is obtained. Evolutionary and entropy non-decrease conditions, given in terms of these polar variables, assume a very simple form, and admit immediate geometrical interpretation. It is found that this magnetic induction polar is far more natural and convenient to use than the shock polar given in terms of conventional velocity polar variables.

Non-linear hydromagnetic waves in a cold collisionless plasma are investigated by numerical and analytic methods. For this problem, Saffman has already implied that there may exist a non-periodic wave different from the well-known solitary wave (or wave train). His analysis is based on the assumption that the quasi-ergodic theorem can be applied to the present problem. However, the propriety of the assumption has not been discussed.

It is seen from the computer calculations that in general the waves do not have periodicity, as Saffman pointed out. Furthermore, some concrete examples show the behaviour of these ergodic waves more clearly. The width of the waves is of the order of ion Larmor radius. These waves are here called quasi-wave trains. In some special cases, the waves reduce to wave trains (with periodicity). Some of them have a length scale of the geometric mean between the Larmor radii of the electron and of the ion.

The system studied is that of a narrow-band electrostatic wave packet in a collision-free plasma. Inhomogeneous effects are represented by a wave-number, which varies linearly with distance. The system is excited by a weak resonant beam, and, to first order in a smallness parameter associated with the weakness of the beam, the resonant-particle distribution function and charge densities are calculated. It is found that second-order resonant particles become stably trapped in the wave, and, after a few trapping periods, make a dominant contribution to the resonant particle charge density. The growth rate due to the resonant beam was found to increase linearly with trapping time, and typically a pulse which traps particles for n trapping periods exhibits a growth rate ˜ n times the linear Landau value. Furthermore, a reactive component of charge density was found that was able to cause a steady change in wave frequency and wave-number. These features of large growth rates and changing frequency should appear in parallel problems involving other wave types. An obvious application is that of VLF emissions in the whistler mode.

Expressing the Vlasov equation in a local system of co-ordinates defined by the magnetic field, the distribution function is expanded in terms of a small parameter e (assumed < 1), the ratio of the ion Larmor radius to a characteristic length of change perpendicular to the magnetic field. Since the finite-Larmorradius (FLR) approximation restricts consideration to weakly unstable systems, the equations apply only to magnetic configurations in which the curvature of the lines of force is weak. In allowing for the effects of plasma pressure (β ˜ I) it is found that the FLR corrections to the stress tensor take their simplest form in the centre-of-mass frame, while the case of low β yields equations best expressed in the guiding-centre frame.

Two alternative approaches to expanding the binary collision integral for particles of very different mass in powers of their small mass ratio m/M are examined in detail. One of them in particular is found to be suitable for a subsequent expansion of the above quantity in series of spherical harmonics. This reduces to a simple algebra the labour of determining the angular-moment form of the Boltzmann equation for Lorentzian mixtures.

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.