A class of equilibrium configurations of Vlasov plasmas carrying a current component along an external magnetic field is presented. The present slab model contains the diamagnetic current jy, and the field-aligned current jz for arbitrary βc (= particle pressure/magnetic pressure of the applied constant field). For fixed βc and field-aligned current, our model admits a family of equilibrium solutions in which the diamagnetic currents range from zero to a maximum value. The amount of diamagnetic current flowing in a machine depends on the width of the machine, the field-aligned current and other plasma parameters. The Helmholtz free energy of the system is calculated under the constraints that the total number of particles and the field-aligned current are conserved. The least unstable equilibrium configuration in a machine is obtained by minimizing the free energy under the stated constraints among all equilibria whose plasma widths do not exceed the width of the machine.

The resonant interaction between three waves, propagating perpendicular to a constant magnetic field in a spatially uniform plasma, is considered. Significant terms, which have been neglected in the coupling coefficients of previous work, are included.

An exact Green'ls function of the Lenard-Bernstein equation for a magnetoplasma subjected to an external d.c. electric field is obtained. The result is applied to the problem of temporal electromagnetic echoes which can be detected at any angle of observation with respect to the external magnetic field B0.

The effects of dynamic shielding of charged-particle interactions on the predicted values of hte transport properties of partially ionized argon are considered. It is found that, at low degreesof ionization, dynamic shielding effects are of little importance, because charged-particle encounters are infrequent, so that the perturbed Lorentzian method may be used to obtain a rapidly convergent solution. At higher degrees of ionization, where the Chapman–Enskog solution converges rapidly, dynamic shielding effects can be taken into account using the expressions presented in the appendix to this paper. It is also found that the static shielding model, assuming complete shielding by electrons only, yields results that are quite close to those obtained when dynamic shielding is considered.

The behaviour of a plasma permeated by a magnetic field, where the field possessess a hyperbolic neutural point, is considered. Results from numerical solutions of the magnetohydrodynamic formulation for such flows are reported. Problems are posed with the solar flare models of Dungey, Sweet & Petschek in mind. No evidence is found to support the idea that compression of the field lines near a hyperbolic null, in the presence of electrical resistance, can radically alter the geometry of those field lines (e.g. the formation of switch-off shocks). These computations do show that, for large values of the magnetic Reynolds number, a rate of annihilation, more rapid than that derived from order-of-magnitude estimates, is possible.

The influence of finite Larmor frequency on the stability of a viscous, finitely conducting liquid in a downward gravitational field under the influence of a uniform magnetic field directed along or normal to gravity, is investigated. The solution in each case is shown to be characterized by a variational principle Based on the variational principle, an approximate solution is obtained for the stability of a layer of fluid of constant kinematic viscosity and an exponentia density distribution. It has been found that finite resistivity and finite Larmor frequency do not introduce any instabifity in a potentially stable configuration. However, for a potentially unstable configuration we find that, for an ideal Hal plasma, the results depend on the orientation of the magnetic field, though the instability persists for all wave-numbers in the presence of non-ideal (finite resistivity and viscosity) effects. For the field aligned with gravity, it is found that a potentially unstable field-free configuration is stabilized if the buoyancy number B ( = gβ/12 V2) is less than unity. For B > 1, the instability arises for wave-numbers exceeding a critical value, which decreases on allowing for Hall terms in the generalized Ohm's law, suggesting a destabilizing influence of finite Larmor frequency. For an ambient horizontal magnetic field, it is found that an ideal plasma is stable, even for B > 0, for perturbations confined to a cone about the magnetic field vector. The angle of the cone of stable propagation, however, decreases on account of finite Larmor frequency.

The time evolution of firehose-unstable Alfvén waves is calculated within the usual weak-turbulence scheme. In § 2 the amplitude equations are established using a third-order solution of Vlasov's equation. From these a spectrum equation is obtained by discarding four-wave correlations; in addition, we derive an equation which characterizes the truncation error ( §3). Both these equations are integrated numerically, together with the velocity-moment equations (§ 4), forward in time. The results as presented in § 5 correspond to a rapid relaxation of the plasma to equilibrium. For large wave-vectors the relaxation time, as well as the equilibrium wave energy, are in good agreement with the quasilinear treatment, and the truncation error is small. But for low wave-numbers the relaxation is much faster, and the wave energy grows higher as predicted by quasilinear theory. This is because the nonlinear particle current leads to a high effective growth rate, especially at small wave-numbers. In this region the truncation error grows appreciably, and may sometimes reach the order of magnitude of the spectrum. But the overall picture as given by quasilinear theory has been confirmed. In § 6 comparison is made with a macroscopic model of Berezin & Sagdeev (1969), where the plasma noise was simulated by ‘computer noise’. For low amplitudes both methods agree qualitatively.

Initially non-conducting gas is ionized by a thin viscous shock wave. Upstream there can be no magnetohydrodynamic interaction because of the zero conductivity, but the conducting downstream region may have a magneticStructure which interacts with the flow variables. A theoretical analysis is made in the zeromagnetic-Prandtl-number (‘non-viscous ’) limit, i.e. ohmic dissipation is the dominant diffusion mechanism. Unlike magnetohydrodynamic shocks in a pre-ionized gas, ionizing shock waves are not necessarily plane-polarized. Thus ‘skew ’ shock structures can exist, in which the upstream and downstream magnetic field vectors and the shock wave normal do not all lie in a single plane. Explicit solutions are given for typical values of the governing parameters, showing how the magnetic field vector rotates about the shock wave normal as its transverse component changes in magnitude through the shock layer. Skew shocks are necessarily sub-Alfvénic downstream. Unlike the pre-ionized case, the range of trans-Alfvénic shock waves is not excluded, since these shocks can absorb Alfvén waves within their structure. With strong magnetic fields it is possible to achieve very high downstream temperatures by Joule heating. Alternatively, in some cases, magnetic energy can be fed into directed kinetic energy, producing an overall expansion shock.

The dispersion curves of mixing mode between electrostatic and electromagnetic waves propagating perpendicularly to the ambient magnetic field are computed for frequency range near the lower hybrid frequency in the case of a finite-β plasma with a Maxwellian velocity distribution. The detailed results apply to a hydrogen plasma. The result indicates that the lower hybrid frequency is neither the resonance frequency of the electromagnetic wave nor the cut-off frequency of the electrostatic wave. In a high-β plasma, the plateau of the dispersion curve disappears and the electrostatic approximation (Bernstein mode) is not valid up to the wavelength of the order of proton Larmor radius. The polarization is left-handed, and a trace of the most dominant intensity ratio of wave magnetic field energy to the electric field energy in each branch is given approximately by the cold-plasma dispersion curve.

A derivation of the quasilinear equations is given, which is sufficiently general to include damped waves. The cause of some difficulties in previous derivations, momentum and energy conservation, and the origin of irreversibility are discussed.

An isotropic plasma occupies an infinite space, the upper half of which (y 0) is a dielectric. A transverse disturbance is incident obliquely on the face of the dielectric. The waves produced in each half of the space are obtained. Possible extensions are discussed. ome particular examples are considered.

Oblique ionizing shock waves are considered as a subclass of the skew shocks discussed in part 1. It is shown explicitly that, for these cases, when the electrical conductivity is a scalar, the magnetic field and velocity vectors lie in a single plane throughout the shock structure. Details of the shock structure are given in the zero-magnetic-Prandtl-number limit for a range of values of the upstream magnetic pressure ratio as the transverse electric field is varied parametrically. Shocks possessing a magnetic structure are necessarily sub-Alfv énic downstream. Thus some structures are trans-Alfv énic (i.e. super-Alfv énic upstream), and others are completely sub-Alfv énic. Unlike the pre-ionized case, the former are stable to rotational Alfv én disturbances because of their ability to form skew shocks. Many features are qualitatively similar to those of skew shocks. For example, very high downstream temperatures may be obtained by Joule heating, and, in some cases, overall expansion shocks may occur. Although magnetic structures can always be found for a range of shock Alfvén numbers from zero up to a value greater than unity, there is an upper limit, above which only the gas shock exists, corresponding to a specific value of the electric field. There is no upper limit on the gas shook speeds. This contrasts with the skew shock case in which the corresponding (saddle-point) solution has an upper Alfvén number limit (slightly above two for the monatomic, infinite-Mach-number case), above which no skew shock solutions exist at all. Finally, the results of previous studies are reviewed in light of the structural requirements of ionizing shock waves.

It is shown that some recent solutions representing a dipole field immersed in an infinite plasma, which is perfectly conducting, belong to the vorticity equation in steady inviscid hydrodynamics. Further solutions of this equation, referring to the same problem, are constructed and discussed.

An asymptotic solution is given of hte particle motion in an arbitrary quasimonochromatic wave. Zero stationary magnetic field and untrapped particles are considered. The number of adiabatically trapped particles, the total momentum conservation and the space-averaged distribution function of non-resonant particles are derived for the case of an electrostatic wave with time-dependent amplitude.

Normal ionizing shock waves are considered as a subclass of oblique shocks in which the upstream transverse magnetic field component is zero; i.e. the upstream field is normal to the plane of the shock. Non-trivial (switch-on) normal shocks involve a non-zero downstream transverse field component; magnetically trivial normal shocks are simply gas shocks with an imbedded constant normal magnetic field. As with oblique shocks, switch-on normal ionizing shock waves are plane- polarized, provided the conductivity is a scalar. Ohmic structures are discussed for several values of shock Alfv én number, treating the electric field as a free parameter, as usual. For Alfv én numbers extending from zero to two (for the infinite-Mach-number case), there is always a finite range of E field values. Above two, only the gas shock exists, and this requires a unique electric field value. Because the magnetic field magnitude increases through switch-on shocks, there is no mechanism available for converting magnetic energy into thermal energy, as is the case for oblique or skew shocks. Thus, there is no significant downstream heating above the viscous temperature; and, in some cases, slight downstream cooling may even occur. Expansion shocks are not possible in this geometry. Previous studies are reviewed in the light of structural requirements, and some erroneous results are clarified; in particular, it should be noted that MHD switchon solutions for the pre-ionized case are not imbedded in the family of ionizing switch-on solutions.

It has been proposed by Smith that type III radio bursts may be produced by streams of protons with radius of the order of 7 km, which is of the same order as the characteristic lengths of the nonlinear processes which are supposed to take place in the dynamics of these bursts. In this paper, we consider a method for studying systems with finite transverse dimensions and apply it to a simple model: the scattering of a beam of plasma waves by acoustic turbulence and by the particles of the plasma.

The kinetic equations for infinite homogeneous turbulent plasma in a magnetic field are analyzed using a projection operator which allows the time dependence to be maintained in a more exact and consistent manner than has been possible heretofore. By introducing approximations on the multi-time correlation function rather than the fluctuations, as is conventionally done, a hierarchy of equations is obtained which predicts different behaviour for the system especially in connexion with wave-wave interations. These effects are further highlighted by showing how the present results reduce to those obtained by various authors.

Boundary conditions appropriate to a model with finite ion mass corrections of ideal hydromagnetic theory are reviewed. It is shown that these boundary conditions have been correctly satisfied in earlier theory for the stability of an interface between two semi-infinite regions. This theory predicts instability due to Hall current.

The resonant three-wave interactions in an infinite homogeneous magnetized plasma are re-examined using the improved hierarchy presented in part 1. An equation for the wave energy is obtained which, for a spectrum made up entirely of marginally stable waves, is shown to be a second-order Markoffian differential equation, rather than a first-order equation as obtained by others. The conditions under which a first-order equation is derived are examined and discussed. For a spectrum composed of waves with small growth, rates, it is shown that the equation for the plasmon density always has a stationary solution, thus eliminating the occurrence of an ‘explosive instability’ in such systems.