A solution of the one-dimensional time-independent Vlasov–Poisson system, including the condition of no net current, is constructed for a large class of electric potential functions φ(x), including periodic and solitary waves, as well as monotonic transitions. Displaced Maxwell distributions for the free particles, and an unshifted Maxwell distribution for the trapped ions, are used. The condition of positiveness of the unknown distribution function for the trapped electrons (which can be split into two parts, one depending solely on the chosen wave, the other on the given distributions) is examined. It is shown, for a non-linear wave

that this condition plays no rôle when the thickness l of the wave considerably exceeds the Debye length λD(l ≫ λD). If, however, l lies in the neighbourhood of λD(l ≳ λD), there exist limits for the free parameters. The smallest thickness lmin results if the Mach number M lies in the range 1 > M > 2 and the electronion—temperature ratio θ = Te/Ti exceeds 10. This confirms the view that the wave is a steepened ion-acoustic wave. lmin decreases with decreasing amplitude ϕmax of the wave and decreasing number of trapped ions, but does not lie below the Debye length as long as the wave is non-linear. In the linear case, the condition of positiveness imposes no restrictions.

A treatment is given of the problem of constructing normal modes for anarbitrarily bounded system from roots of the linear dispersion relation D( ω, k) = 0 for the corresponding infinite or periodically bounded system. For a system described by continuous macroscopic variables, and of general cylindrical form (uniform along an axis z, say), each transverse eigenmode gives rise to a set of axial normal modes constructed from a pair of dominant roots kαz(ω) of D = 0 satisfying the boundary conditions which are characterized by complex reflexion coefficients for the dominant waves. The implications of the results for the interpretation of experiments on plasma waves and instabilities on finite cylinders are discussed, with particular reference to the effects of end-plate damping and axial current on Q-machines.

The work of Butler & Gribben (1968) on a general formulation ofthe problem of non-linear waves in non-uniform plasmas is extended. The particular case treated in Butler & Gribben, § 6, is discussed again (the distribution functions fs and electrostatic potential depend only on one space co-ordinate, that in the direction of propagation of the wave, the wave is slowly varying only with respect to this co-ordinate and time, the magnetic field vanishes and relativistic effects are negligible). Conditions necessary to avoid secular terms in the solutions of the Vlasov and Poisson equations are rederived, but without resorting to assumed series expansions in e, the perturbation parameter describing the non-uniformity, for the dependent variables, and using a different notation which simplifies and retains the symmetry of the equations. The corresponding general boundary condition, to be satisfied by fs innergy space at the boundary of the trapped particle region, is also derived.

Particular attention is devoted to the Vlasov equation, and the derived general result is used to obtain the appropriate necessary condition from this equation, correct to O(ε). This shows that, unlike the leading-order theory, at the next stage an extra length scale appears in the equations, which would be needed if e.g. the theory were to be used to discuss shock waves. It is argued that this result, taken with the mixing process described in Butler & Gribben (which supplies a mechanism for the formation of shocks), appears to place the theory at least on as satisfactory a level as the Navier-Stokes theory for the discussion of shock wave structures.

Another aspect of the analysis is the comparison with the Whitham (1965a) averaging method for treating the propagation of non-linear waves in other fields. Similar features are pointed out, and it seems likely that the averaging method is equivalent to the leading-order theory. It is thought that the present approach might prove to be useful in other wave problems.

A new method for dealing with a typical quasi-linear problem is presented. The quasi-linear relaxation of the temperature anisotropy of a collisionless plasma is completely described within the framework of the thermodynamics of irreversible processes. This is obtained by means of generalized Onsager relations applied to the weakly unstable system. Free energy of the system is analyzed in detail, to study the net work involved in the process.

The mathematical treatment of the influence of a weak coupling on an assembly of damped oscillators with different eigenfrequencies permits us to exhibit a peculiar property of the solution: namely, the existence of a high frequency solitary, weakly damped mode. We propose to apply these results to the ionic oscillations of a bounded plasma.

A study of the compressive magneto-acoustic waves in a guiding centre plasma shows that the wave-front that emerges from a point disturbance after a finite time is a simple oblate spheroid with the axis of revolution parallel to the field lines. Thus, in a steady three-dimensional supersonic flow of guiding centre plasma a simple analytic expression can be obtained to represent the characteristic surfaces. From a proper linear combination of the governing macroscopic equations, the characteristic equation is obtained. It represents the propagation of disturbances on the characteristic surface. The characteristic theory can be used to study the interaction of the solar wind with the moon and possibly with other planetary bodies.

It is shown that the effect of the cut-off criteria of the electronic partition functions on transport properties varies with the particular coefficient considered. For example, the cut-off has little effect on translational thermal conductivity, but a large effect on viscosity and reactive thermal conductivity.

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.

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.

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.

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.

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.