It is shown that complex variable transformations, suitable for obtaining the solution for the field boundary of a system of line currents confined in one cavity by a perfectly conducting uniform plasma, can be used for obtaining the solution to the inverse problem where a perfectly conducting uniform plasma is confined in one cavity by a system of line currents. It is deduced that the minimum number of line currents for confining (not stably) a plasma is two. The equilibrium configurations for several special but simple cases are investigated and discussed.

This paper compares computed results with upstream and downstream thermal pressures, and the downstream ion density and vector flow velocity measured by the Ames plasma probe on the Pioneer 6 spacecraft as earth's bow shock was traversed. The upstream ion density and vector flow velocity measured on Pioneer 6 by this experiment are used as independent variables, together with Pioneer 6 magnetic field measurements. MRD Rankine—Hugoniot equations for an isotropic plasma are used for these computations. Reasonable agreement is obtained between the measured and computed downstream ion density, thermal pressure and convective velocity magnitude and orientation, only when a 1γ change is made to a downstream magnetic field value used in the calculation. There is disagreement between several determinations of shock orientation and that deduced from coplanarity of the reported magnetic fields, and the shock orientation provides the co-ordinate system for solving the Rankine—Hugoniot equations. If a shock orientation determined by velocity coplanarity is used in the calculations, all computed quantities agree well with the experimental results. Other results suggest that a disagreement between computed and measured upstream thermal pressures may not be negligible in comparison with experimental uncertainties in upstream velocity and density. If the computations included anisotropic pressure terms and/or other factors such as plasma turbulence, better agreement might be obtained between computed and measured upstream thermal pressures.

Transverse instabilities in an electron gas with a neutralizing background, in the absence of a permanent magnetic field are discussed, mostly when the equilibrium velocity distribution is slightly spheroidal, i.e. of the form

with α ˜ γ and with the functional form of f0 not given explicitly. It is found that two types of unstable waves exist; one grows at a rate proportional to ε½‖ cos θ‖3 whenever ε > 0, the other at a rate proportional to(ε cos2θ)½ whenever ε cos 2θ > 0, where ε = (γ2/α2) — 1, and θ is the angle the wave vector makes with the axis of symmetry of the velocity spheroid. It is shown that within the approximation we are working the complex phase velocity is pure imaginary, i.e. the unstable waves are standing waves. The effect of a small permanent magnetic field is considered. The phase velocity becomes complex, at least for propagation parallel to the magnetic field and the instability rate decreases by a factor where H0 is the permanent magnetic field and 〈H2〉½ the r.m.s. maximum field that grows before non-linear effects take over. The maximum instability rate is compared with the collision rate and is found to be higher in low density, high temperature plasmas such as the solar wind. An application to collisionless shock theory is suggested.

The dispersion relation for the. Bernstein modes is derived, in the presence of a Fokker—Planck type collision term. Numerical solutions are obtained, and the results give an approximate formula for the collisional damping. The possibility of observing this damping experimentally is discussed.

Non-linear periodic waves propagating across a magnetic field are investigated, taking relativistic effects into account. The governing equations of the phenomena are taken to be the relativistic two-fluid plasma equations for a cold plasma together with Maxwell's equations. The analytical solution in the non-relativistic case was obtained by Adlam and Allen and others, a case where the quasi- neutral approximation is a good one and the inertia of the electrons plays an essential role in the dispersion effect of the plasma.

In contrast, the analysis of the relativistic waves shows that the quasi- neutral approximation becomes invalid and that the charge separation plays an essential role in the dispersion effect while the inertia of the electrons plays minor role only. Therefore the phenomena to be investigated have a length scale of the appropriately defined Debye distance (the ion Larmor radius), while in the non-relativistic case the width of the wave is of the order of the geometric mean of the Larmor radii of the electrons and of the ions. In the present case where the electron mass may be neglected, there exists an exact solution which is a good approximation for a real plasma. The solitary wave solution is found as the limiting case of infinite periodicity.

The quasi-linear theory of plasma waves in a collisionless plasma in the absence of external fields is formulated in a similar way to that given by Bernstein & Engelmann (1966) but is applied to the case when unstable essentially transverse electromagnetic waves with phase velocities of the order of the r.m.s. electron velocity are present. It is deduced that the wave energy reaches a maximum and then decays leaving the plasma stable. A quantitative description of the time development of a plasma with a slightly spheroidal velocity distribution is given in terms of a parameter x characterizing the anisotropy and it is found that this quantity decays exponentially as the waves grow and ultimately decays like t−β with , and correspondingly the space-averaged field energy grows to a maximum and then decays like t−β.

In the course of an experimental investigation of the Manley Rowe oscillations, we have been led to define some of the conditions required for an ideal coupling between three travelling waves. It is shown that, because of homogeneity conditions, the only possible pumping wave is the electron cyclotron wave. Experimental values are given to within an order of magnitude for finite geometry.

Small amplitude waves and collisionless shock waves are investigated within the framework of the first-order Chew—Goldberger—Low equations. For linearized oscillations, two modes are present for propagation along an applied magnetic field. One is an acoustic type which contains no finite Larmor radius effects. The other which contains the ‘fire hose’ instability in its lowest order terms, does possess finite Larmor radius corrections. These corrections, however, do not produce instabilities or dissipation. There are no finite Larmor radius corrections to the single mode present for propagation normal to the applied magnetic field. Normal shock structure is investigated, but it is shown that jump solutions do not exist. An analytic solitary pulse solution is found and is compared with the Adlam—Allen pulse solution.

The low frequency, linear oscifiation of an unbounded, homogeneous, low density plasma in the presence of a strong, uniform magnetic field is examined using a set of three-dimensional hydromagnetic equations derived from the higher-order Chew—Goldberger—Low theory recently obtained by Frieman, Davidson & Langdon (1966). It is found from the higher-order corrections to the dispersion relation that the waves are generally dispersed and the mirror and firehose instabilities are stabilized.

Resonances of a plasma driven into a weakly non-linear régime are investigated by two different models.

(1) A cold plasma driven by an extended wave source is considered. It is found that resonances can occur at fractions as well as multiples of the normal mode resonances. For the fractional resonances the response amplitude grows linearly with time, whereas for the harmonic resonances it grows exponentially with time.

(2) An unmagnetized Vlasov plasma is excited by a two-grid source below the plasma frequency. A resonance is shown to occur when the driving frequency approaches one half the plasma frequency. It is argued that such non-linear resonances may be responsible for the Alouette II observations of resonances at fractions of the plasma frequency.

The statistical mechanics of the weakly turbulent plasma is treated in an original manner by introducing the waves by means of the van Kampen modes. A system of coupled equations is derived which relates the correlations between waves and waves, waves and particles, particles and particles. The quasilinear theory is obtained at the first order, and the coupling wave equation at second order, under suitable assumptions.