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).

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 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.

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.

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.

Butler & Gribben (1968) showed how to generalize non-linear, uniform, wave type solutions of the Vlasov–Poisson system of equations to allow for slow variations in the properties of the plasma. Here one equation of the derived set (that obtained by ensuring that the appropriate solutions of the Poisson equation are non-secular) is rearranged (i) so as to bring out the physical importance of those particles, trapped and untrapped, travelling near the wave speed, and (ii) so as to yield a form particularly suited to the case when the electric field is small.

Several types of two-dimensional solutions for the equations of magnetohydrodynamics are described. For all these solutions the magnetic field contains at least one hyperbolic neutral point. Two new magnetostatic equilibria are introduced for the ideally conducting case. The magnetic field associated with one of these is used to construct an exact time-dependent solution of the MilD equations where the fluid is necessarily at rest. In the case where the field lines are hyperbolae, it is demonstrated that retention of the energy equation (ordinarily decoupled for incompressible flows) implies that the flow beginning at rest, remains at trest

Hamel & Willis (1966) first showed that anisotropy in temperature could be predicted theoretically in the expansion of a spherical source of gas into a vacuum, thus explaining anisotropies observed to occur in rarefied gas jets. Chou & Talbot (1967) carried out numerical calculations based on similai ideas for a plasma consisting of electrons, ions and neutral particles. These showed that the electron temperature remains isotropic, and little anisotropy occurring among the ions within the (rather extensive) calculations. Here, in the first of two papers, an analytic treatment is applied to the expansion of a fully ionized plasma source, with the particles interacting according to the Coulomb law.

This paper introduces the effects of partial ionization and recombination into the steady supersonic spherical source expansion of a plasma. It presents an analytical solution for a problem previously solved only numerically, using moment equations derived directly from the Boltzmann equation under the assumption of an ellipsoidal distribution function. The recombination is assumed collisionalradiative, as considered by Bates, Kingston & MeWhirter (1962a, b) and an approximate rate constant is derived from their results. The plasma is taken to be optically thick. The results show that, in a highly ionized plasma, the species maintain equal isotropic temperatures for the main part of the flow, with the degree of ionization tending to a finite limit. Recombinations, however, remain sufficiently frequent to keep the temperature above that in an expansion with no recombination. In a weakly ionized plasma the temperature of the heavy particles is almost unaffected by recombinations and falls adiabatically, although the electron temperature remains high. The analytical results are compared with the numerical results of Chou & Talbot (1967).

The system studied is a plasma streaming parallel to a uniform magnetic field with a velocity which varies in a transverse direction. The flow is bounded at y =± a by perfectly conducting planes.

The Poisson-Vlasov equations are used to derive an integro-differential equation for ø the Fourier transform of the electrostatic potential. The kernel of this equation is expanded using a small Larmor radius expansion for ø and for the equilibrium distribution function f0.

Small amplitude electrostatic oscillations in a cold plasma with continuously varying density have been investigated. The problem is the same as that treated by Barston (1964), and in part 1 of this paper, but, instead of starting from a differential equation, we base our treatment on an integro-differential equation for the transverse component of the electric field. This equation is very similar to the Vlasov equation, and we show this to be the cause of the striking similarities between the two theories pointed out in part 1. The normal-mode analysis utilizes the similarity of the integral equation for the eigenfunctions to the equation of anisotropic neutron transport. The continuum eigenfunctions are found in the form of a generalized van Kampen eigenmode, and their calculation is reduced to solving a non-singular integral equation. Equivalence of the eigenfimctions found in this way and of those found by solving the equivalent differential equation is proved, and the question of orthogonality and normalization is discussed.

The propagation and stability of ion and electron drift waves in a non-homogeneous plasma containing impurity ions is considered. It is shown that the presence of impurities in the plasma may significantly affect the stability criteria of these low frequency waves. Furthermore, perturbations at frequencies which are multiples of the impurity ion cyclotron frequency may also be unstable when ½K2u R21 > 1.

The Thomson scattering spectrum, for laser light scattered by a low density plasma, is determined by the Doppler shifts from the moving electrons. It is shown how the full three-dimensional distribution function may be constructed if spectra are observed in several directions. Calculations using a model loss-cone distribution show that four spectra are sufficient for a faithful reconstruction of f(v). Most of the complications that arise at relativistic electron velocities can be avoided by placing a diffraction grating in the scattered laser beam.

The field of a test charge at rest in a plasma is known to be shielded via the collective interaction between the test charge and the particles of the plasma. In a thermal equilibrium plasma, the shielded potential has a spatial variation of the Debye form exp (— r)/r. In this note we will show that, in a collisionless current-carrying plasma, the potential profile along the radial direction parallel to the current changes from the Debye form to a potential well form, and causes a reversal of the ele ctric field of the test charge. The formation of the Debye potential well is attributed to the resonant interaction between the particles and the waves stationary in the test charge frame when the net flux of charged particles is different from zero. This is the same type of resonant interaction which leads to the well-known Landau damping (Landau 1946).

The electrostatic shielding of a test electron embedded in a magnetically confined pure electron gas column is investigated. For rigid-rotor equilibria with nearly uniform density in the column interior, it is shown that the shielding length is the local electron Debye length for the case that the test electron is at rest relative to the mean motion of the column.