The paper presents a paraxial-ray theory of self-defocusing of obliquely propagating Trivelpiece-Gould (TG) modes in a homogeneous plasma. The TG mode with Gaussian distribution of intensity, in a direction perpendicular to its phase velocity, creates a ponderomotive force on electrons. The electrons are redistributed through ambipolar diffusion. The redistributed electrons are responsible for defocusing of the TG mode. The importance of our study is pointed out.

For the simple model of cold electrons streaming against cold ions the complete set of nonlinear stationary waves is expressed in terms of elliptic functions. The conditions for their dynamical connection to a uniform neutral plasma state are taken into account, and the conditions for the neglect of the magnetic field are analysed. The range of existence of stationary waves is found to be confined to the stable regime of the two-stream instability, but covers only part of it. All nonlinear BGK waves that are found within the limits of the model can be shown to bifurcate from the two-stream instability, some of them also exhibiting secondary and further bifurcations. As an exceptional case, all bifurcations can be treated exactly. Close to the linear regime, all nonlinear modes turn out to be unstable. The corresponding instability is caused by a wave decay that transports energy from low to high wavenumbers of the Fourier modes constituting the wave. From the two-stream solutions four- stream solutions with exactly vanishing magnetic field are derived.

I shall shortly be stepping down from the position of Editor of Journal of Plasnia Physics, although continuing for a while more as an Associate Editor.

Magneto-acoustic instability driven by cosmic rays and the waves that they excite was first studied by McKenzie and Webb and by Zank. In the present paper this instability is reconsidered. In general, the cosmic-ray plasma system is unstable. However, depending on how the time scale is chosen, the results and interpretation may be different from those of McKenzie and Webb and of Zank, particularly in the long- and short-wavelength limits. To illustrate this instability in an astrophysical context, we take our Galaxy and supernova remnants as examples. In those examples, length scales shorter than 2.5 kpc and 1.5 pc respectively are susceptible to this instability, and the growth times are of the order of 4 × 108 and 1.5 × 104 yr respectively.

We consider effects of inhomogeneity on the absorption of high-frequency electromagnetic waves, propagating at arbitrary angles relative to the magnetic field, by current-carrying plasmas. An inhomogeneous current is assumed to be immersed in an otherwise homogeneous background, and the absorption of fundamental electron-cyclotron waves is discussed, with emphasis on the dependence of the inhomogeneity effect on wave frequency and angle of propagation.

The radial self-similar expansion of negatively charged impurity or dust particles in a cylindrical geometry is investigated using the multi-fluid theory. Three different types of self-similar solutions are considered. Various asymptotic limits of physical interest in the self-similar space are shown to occur.

We have shown earlier that compressional magnetic fluctuations modify the properties of MHD waves. The modifications can be large even when the fluctuation level is much smaller than the ambient magnetic field. There are two kinds of effects: resonant and non-resonant. The former is a result of the so- called parametric cyclotron resonance, and enables energy transfer from the wave to particle Larmor motion. This transfer causes the damping of waves even when their frequency is so small that ordinary cyclotron damping is negligible. The non-resonant effect does not cause energy exchange, but it does modify wave dispersion and causes mode coupling between the shear Alfvén mode and magnetosonic modes.

It is shown that, although the mathematical analysis of the Alfvén-wave equation does not show any variation at non-zero or zero singular points, the role of surface waves in the physical mechanism of resonant absorption of Alfvén waves is very different at these points. This difference becomes even greater when resistivity is taken into account. At the neutral point the zero-frequency surface waves that are symmetric surface modes of the structured neutral layer couple to the tearing mode instability of the layer. The importance of this study for the energy balance in tearing modes and the association of surface waves with driven magnetic reconnection is also pointed out.

Short-wavelength coherent structures of ion Bernstein modes are studied on the basis of a nonlinear equation taking into account the effect of high-frequencey diamagnetism. Analytical solutions in the form of nonlinear periodic waves and envelope solitons are found, and their properties are discussed. The appearance of such structures may explain some phenomena observed in laboratory and space plasmas.

A generalized non-local nonlinear Schrödinger equation describing the phenomenon of nonlinear Landau damping in a relativistic two-component plasma is deduced using the kinetic-theory approach of Vlasov. Parameters appearing in the equation are evaluated explicitly for the case of a Maxwellian distribution function.

The hydromagnetic Kelvin–Helmholtz instability is relevant in many complex situations in astrophysical and laboratory plasmas. Many cases of interest are very complicated, since they involve the combined role of velocity shear, of density and magnetic field stratification, and of various geometries in compressible plasmas. In the present work we continue investigating the influence of various physical and geometrical parameters of the plasma on the Kelvin–Helmholtz modes. We use the general dispersion relation for the ideal compressible MHD modes localized near a velocity discontinuity between two uniform plasmas. We study analytically the existence and properties of the modes and their stability, for a velocity jump combined with a density jump, and for any relative orientation of B, u and k (B is continuous). Stability is analysed by means of a general procedure that allows discussion of any configuration and all kinds of perturbations. The boundaries between modes of different kinds are discussed. In contrast to the case of uniform density, for a density jump there are no monotonically unstable modes, only overstabilities. The unstable modes belong to two types. Those with the largest growth rates tend to monotonically unstable modes in the limit of uniform density, and are related to the torsional Alfvén mode. The other overstable modes have no analogue among the purely incompressible modes, and occur in a range of U that is stable in the incompressible limit. We derive bounds for the growth rate of the instability. The present results may serve as a guide to interpret results in more complicated and realistic situations as those occurring in laboratory and natural plasmas.

We investigate the implosion of a dense τ-pinch plasma driven by an annular finite-thickness gas-puff Z-pinch. The imploding Z-pinch traps an axial magnetic field Bz, compressing it to large values in an extremely short time. The temporal variation of Bz then induces an azimuthal τ current on the surface of a fibre placed on the axis, with a rise time an order of magnitude shorter than the rise time of the Z-pinch current. Our numerical results demonstrate that, for a thick gas-puff layer, maximum compression occurs before the current peaks.We also find that at peak compression, fuel densities of the order of 1025 cm-3 and temperatures above 10 keV can be achieved on a time scale of the order of 0.1 ns. Thus a Lawson parameter nτ ≈ 1014 s cm-3 for a DT fibre becomes achievable. The snowplough effect in the Z-pinch exercises a stabilization effect on the growth of sausage and Rayleigh—Taylor instabilities. In the limit of a very thin gas-puff layer, previous results are fully recovered.

Oscillations in inhomogeneous cold plasmas or inhomogeneous magnetofluids are interpreted in terms of the dynamics of their spectra in wavenumber space. By Fourier transforming the basic integro-differential equation of the problem, a generalized wave equation in wavenumber space is derived, thus converting the oscillation and phase-mixing processes in the original χ space into processes of dispersive propagation and scattering of the spectrum in wavenumber space. The Barston singular continuum eigenmodes correspond to stationary scattering states of a monochromatic wave in wavenumber space, whereas the damping phenomena in χ space correspond to transient ‘leaking’ phenomena accompanying scattering and dispersive propagation of a wave packet in wavenumber space.

The nonlinear dynamics of wave envelopes modulated in 2 + 1 dimensions is considered for two systems in plasma physics: (i) Langmuir pulses and (ii) intense (but weakly relativistic) electromagnetic (EM) pulses. Using singular perturbation techniques applied to an envelope approximation, both problems are reduced to the two-dimensional nonlinear Schrödinger (2DNLS) system, which describes the dynamics of two coupled slowly varying potentials. The general 2DNLS system exhibits a rich variety of phenomena, including enhanced (compared with ‘longitudinal’ propagation) modulational stability and (1D) soliton formation; decay of 1D solitons over long time scales; self- focusing regimes (determined by a virial-type condition); as well as integrability and 2D solitons. Applying our recent results on the 2DNLS system, we determine which of these phenomena can actually occur here and compute the parameter regimes. (i) The 2DNLS system for the Zakharov equations is modulationally unstable for all parameter values. It also has an integrable sector and a self-focusing regime. (ii) The 2DNLS system describes coupled ‘longitudinal’ and ‘transverse’ modulations of linearly or circularly polarized EM pulses propagating through a warm unmagnetized two-component neutral plasma with arbitrary masses (i.e. electron—positron or electron—ion). The pulse can accelerate particles to weakly (but not fully) relativistic velocities; relativistic, ponderomotive and harmonic effects all contribute to the nonlinear terms. The resulting 2DNLS system does not admit a self-focusing regime. Parameter values leading to an integrable case (the so-called ‘Davey—Stewartson I’ equations, which admit 2D soliton solutions) are computed; however, the required values would not be attainable in a laboratory or astrophysical setting. None the less, the existence of new nonlinear modulational instabilities associated with the second spatial degree of freedom already represents an important potential limitation on any (1 + 1)-dimensional approach to nonlinear evolution and modulational instability of plasma EM waves.

Using a low-frequency, low-β model of a magnetized plasma, several nonlinear waves are studied. All the waves share the property that they have a self- consistent time-averaged current in the steady state. The first wave is a one- dimensional solitary wave that propagates obliquely to a static magnetic field. The wave magnetic field has a kink structure, which results in the production of the time-averaged current. A two-dimensional wave that propagates parallel to the background magnetic field is also discussed. This is a forced wave, requiring an antenna structure to support it in the steady state. The characteristics of the two-dimensional wave are dependent upon the wave speed, which in turn is determined by the speed of the travelling wave on the external antenna.

Self-consistent nonlinear changes of plasma parameters and characteristics of electromagnetic guided waves are studied with regard to the effect of self- focusing and waveguided channel formation in a cylindrical plasma waveguide. Besides the nonlinearity due to the ponderomotive force action usually involved, ionization nonlinearity specified for different regimes of particle loss mechanisms is considered. An azimuthally symmetric electromagnetic surface wave is the guided mode that forms and sustains the wave channel.The nonlinear dispersion law and the nonlinear changes in the wave field distribution and power flow are obtained, describing the effect of the self-focusing of the waves under conditions of weak nonlinearity.

A mapping method is developed to investigate the problem of determination and control of heat-deposition patterns on the plates of a tokamak divertor. The deposition pattern is largely determined by the magnetic field lines, which are mathematically equivalent to the trajectories of a single-degree-of-freedom time-dependent Hamiltonian system. Maps are natural tools to study the generic features of such systems. The general theory of maps is presented, and methods for incorporating various features of the magnetic field and particle motion in divertor tokamaks are given. Features of the magnetic field include the profile of the rotational transform, single- versus double-null divertor, reverse map, the effects of naturally occurring low M and N, and externally imposed high-M, high-N perturbations. Particle motion includes radial diffusion, pitch angle and energy scattering, and the electric sheath at the plate. The method is illustrated by calculating the stochastic broadening in a single- null divertor tokamak. Maps provide an efficient, economic and elegant method to study the problem of motion of plasma particles in the stochastic scrape-off layer.

The process of phase mixing in inhomogeneous MHD or cold plasmas is interpreted as one of energy propagation in discrete Fourier space. Three instructive scenarios are examined: (i) an isolated inhomogeneity with zero boundary conditions; (ii) a periodic inhomogeneity; and (iii) a monotonic inhomogeneity sandwiched between two semi-infinite uniform regions. In each case the coefficients of the associated wave equation in Fourier space for an appropriately chosen dependent variable are very nearly constant almost everywhere, so the propagation is like that of a free unreflected wave. An exception may arise in the coupling of the lowest modes, which can be highly reflective. It is argued that Fourier space is the simplest and most natural context in which to discuss the development of fine-scale oscillations.

Oblique propagation of ion-acoustic solitons in a magnetized low-β plasma consisting of warm positive and negative ion species along with hot electrons is studied. Using the reductive perturbation method, a KdV equation is derived for the system, which admits an obliquely propagating soliton solution. It is found that if the ions have finite temperatures then there exist two types of modes, namely slow and fast ion-acoustic modes. The parameter determining the nature of soliton (i.e. whether the system will support compressive or rarefactive solitons) is different for slow and fast modes. For the slow mode the parameter is the relative temperature of the two ion species, whereas for the fast mode it is the relative concentraion of the two ion species. For the fast mode it is found that there is a critical value of the negative-ion concentration below which only compressive solitons exist and above which rarefactive solitons exist. To discuss the soliton solution at the critical concentration, a modified KdV equation is derived. It is found that at the critical concentration of negative ions compressive and rarefactive solitons co-exist. The effects of temperature of different ion species, angle of obliqueness and magnetization on the characteristics of the solitons are discussed in detail.

In this paper the characteristics for a single- and a bi-ion plasma in the presence of Alfvén waves are given. In the single-ion case, the analysis is extended to the situation where Alfvén waves saturate and dissipatively heat the plasma. When there is no dissipation, there are three sound waves and one entropy wave in the single-ion plasma. Each sound wave is associated with two Riemann invariants relating the changes in density and wave pressure to changes in the flow. In the case when the Alfvén waves saturate and heat the plasma, there are two sound waves and one modified entropy sound wave. Each wave is associated with two Riemann invariants relating changes in density and entropy to changes in the flow. The analysis for the bi-ion plasma is simplified to very sub-Alfvénic flows. In this case the Alfvén waves behave like another plasma component, and both the electric and Alfvén wave forces have the same structure. The system possesses two entropy waves and four sound waves. Each sound wave is associated with two Riemann invariants relating changes in density and flow velocity along the characteristic curves.