This special issue marks the 65th birthday of John Dougherty, the founding father of this Journal and its Editor for the first 27 years of its life. Since the Journal is so much John's creation, we thought that it would not be appropriate to let this occasion pass without recognition, and we are pleased that a number of John's former students and a selection of people who have contributed regularly to the Journal or have been Associate Editors have agreed to contribute to this issue.

Most plasma physicists will probably associate John with at least three major contributions to the subject. The first is his classic work with Don Farley on scattering of electromagnetic waves from a plasma, and we are very pleased that Don has written a short note about the genesis of this work and about his collaboration with John. The second is his book on plasma physics written with P. C. Clemmow, which, some 30 years after its publication, is still one of the best available texts on some advanced aspects of the subject. It received the rare accolade of being reprinted in the Addison-Wesley Advanced Book Classics series 21 years after its first publication. Finally, there is his editorship of this Journal, which he continued long after his own interests had shifted from plasma physics to fundamental aspects of statistical mechanics. Under John's stewardship, the Journal published papers on all aspects of plasma physics, but perhaps became specially noted for thoughtful contributions to basic aspects of the subject.

All readers who know John, whether as authors, visitors to Cambridge or in any other capacity, will be aware of his friendliness and willingness to help others. On behalf of the plasma physics community, we would like to pass on to him our appreciation of his contribution, our congratulations on the occasion of his 65th birthday and our best wishes for the future.

What follows are some personal recollections of John that I thought might be of interest to some readers of this special issue of the Journal of Plasma Physics.

Reduced two-fluid equations governing the nonlinear dynamics of drift–Alfvén waves in dusty plasmas with non-zero ion temperature are derived. In the linear limit, we find a dispersion relation that shows the coupling between the ion-drift–Shukla–Varma mode, and electron-drift (magnetostatic) and (inertial or kinetic) Alfvén waves due to the finite collisionless electron skin depth or Larmorradius corrections. In contrast to the case of an electron–ion plasma, when the nonlinear drift–Alfvén vortices are weakly localized, i.e. decrease at infinity as r−1, the presence of the charged dust grains makes exponential localization possible. The physical meaning of such a localization is connected with the fact that charged massive dust granules provide an additional screening that results in stronger localization of the vortex. In several intermediate-β plasmas with 1 [Gt ] β [Gt ] me/mi (me,i is the electron or ion mass), the localization length approaches a minimum value when the vortex velocity is of the order of the ion diamagnetic drift. It then reaches the value ρiδ−1/4d where ρi is the ion Larmor radius and δd is the ratio of the dust to ion densities multiplied by the dust charge number Zd. In the case of very low plasma pressure, β [Lt ] me/mi, the vortex is localized with typical scale (λeρi)1/2δ−1/4d where λe is the electron skin depth. Our investigation can thus predict the velocities of coherent nonlinear structures in space plasmas.

Dispersion in an intrinsically relativistic, one-dimensional, electron–positron pair plasma (a pulsar plasma) is treated exactly, generalizing earlier results that applied in the low-frequency limit and that neglected the cyclotron resonance. The general theory involves two additional relativistic plasma dispersion functions, evaluated at the normal and anomalous Doppler resonances. These two functions are associated with the non-gyrotropic and gyrotropic parts of the response respectively. The functions are evaluated for bell-type and Jüttner distributions. Wave dispersion is discussed for a non-gyrotropic pulsar plasma with a highly relativistic Alfvén speed. Emphasis is placed on crossings of the light line, defined in terms of the parallel phase velocity. Subluminal waves exist only for sufficiently small angles of propagation, and are confined to frequencies below about the mean gyrofrequency of the relativistic particles.

Comprehensive comments on the theory of stimulated scattering instabilities of high-frequency electromagnetic waves in magnetized plasmas are presented. It is shown that our general dispersion relations are appropriate for deducing valuable information regarding the growth rates of scattering instabilities and the long-term evolution of modulationally unstable waves in space and laboratory plasmas as well as in astrophysical settings.

Nonlinear electrostatic waves in self-gravitating dusty plasmas are considered in two limiting cases, according to whether the charged-particle dynamics is governed mostly by electrostatic forces or mostly by gravitation. This shows a significant difference between these two plasma media with respect to the envelope dynamics in the nonlinear regime. In the former case, when ω2pα > ω2Jα, the amplitude perturbations are longitudinally unstable only in the short-wave range, and the nonlinear effects can result in the formation of longitudinal dust-acoustic solitary waves. But even weak self-gravitational effects can lead to the existence of a long-wavelength range, where self-gravitation prevents the formation of dust-acoustic solitons, and only transverse solitary structures are possible. In the other limiting case (ω2pα < ω2Jα), there is always a transverse modulational instability, which can lead to transverse solitary waves. In both cases, there is a threshold for solitary-wave formation.

Certain unresolved ambiguities surround pressure determinations for incompressible flows: both Navier–Stokes and magnetohydrodynamic (MHD). For uniform-density fluids with standard Newtonian viscous terms, taking the divergence of the equation of motion leaves a Poisson equation for the pressure to be solved. But Poisson equations require boundary conditions. For the case of rectangular periodic boundary conditions, pressures determined in this way are unambiguous. But in the presence of ‘no-slip’ rigid walls, the equation of motion can be used to infer both Dirichlet and Neumann boundary conditions on the pressure P, and thus amounts to an over-determination. This has occasionally been recognized as a problem, and numerical treatments of wallbounded shear flows usually have built in some relatively ad hoc dynamical recipe for dealing with it – often one that appears to ‘work’ satisfactorily. Here we consider a class of solenoidal velocity fields that vanish at no-slip walls, have all spatial derivatives, but are simple enough that explicit analytical solutions for P can be given. Satisfying the two boundary conditions separately gives two pressures, a ‘Neumann pressure’ and a ‘Dirichlet pressure’, which differ nontrivially at the initial instant, even before any dynamics are implemented. We compare the two pressures, and find that, in particular, they lead to different volume forces near the walls. This suggests a reconsideration of no-slip boundary conditions, in which the vanishing of the tangential velocity at a no-slip wall is replaced by a local wall-friction term in the equation of motion.

The well-known Chirikov–Taylor standard map is studied using the methods of non-equilibrium statistical mechanics. It appears possible to simplify the master equation whenever the stochasticity parameter is small and the initial density profile is sharply localized in phase space: this defines the ‘localized weak-stochasticity’ (LWS) regime. The resulting equation of evolution of the density profile involves a convolution in time. Its most conspicuous feature is the absence of a ‘short’ intrinsic time scale: the memory time is infinite. The master equation can therefore not be Markovianized as in usual kinetic theory (or as in the diffusive regime of the standard map). Moreover, the initial value of the fluctuations influences the rate of change of the density profile at all times. Quite unexpectedly, the contributions of the two terms of the master equation to the density profile are of the same order of magnitude. As a result, they practically cancel each other. It is this curious feature that explains the suppression of the motion by the islands and the KAM barriers that is well known from the study of individual standard map orbits. The LWS approximation is valid for a limited time (which is very long when the initial localization is very sharp).

Nonlinear waves and one-dimensional solitons of the Zakharov–Kuznetsov equation are unstable in two dimensions. Although the wavevector K of a perturbation leading to an instability covers a whole region in (Kx, Ky) parameter space, two classes are of particular interest. One corresponds to the perpendicular, Benjamin–Feir instability (Kx = 0). The second is the wave-length-doubling instability. These two are the only purely growing modes. We concentrate on them. Both analytical and numerical methods for calculating growth rates are employed and results compared. Once a nonlinear wave or soliton breaks up owing to one of these instabilities, an array of cylindrical and/or spherical solitons can emerge. We investigate the interaction of these entities numerically.

In the context of ion-acoustic waves in a magnetized plasma comprising cold ions and non-isothermal electrons, small-amplitude, weakly nonlinear waves have been shown previously by Munro and Parkes to be governed by a modified version of the Zakharov–Kuznetsov equation. In this paper, we consider solitary travelling-wave solutions to this equation that propagate along the magnetic field. We investigate the initial growth rate γ(k) of a small transverse sinusoidal perturbation of wavenumber k. The instability range is shown to be 0 < k < 3. We use the multiple-scale perturbation method developed by Allen and Rowlands to determine a consistent expansion of γ about k = 0 and k = 3. By combining these results in the form of a Padé approximant, an analytical expression for γ is found that is valid for 0 < k < 3. γ is also determined by using the variational method developed by Bettinson and Rowlands. The two results for γ are compared with a numerical determination.

It is shown that sheared plasma flows can generate nonthermal electrostatic waves in a magnetized dusty electron–positron plasma. Linearly excited modes attain large amplitudes and start interacting among themselves. Nonlinearly coupled modes self-organize in the form of coherent vortices comprising a vortex chain and a double vortex. Conditions under which the latter appear are given. The relevance of our investigation to space, astrophysical, and laboratory plasmas is pointed out.

High-frequency electrostatic waves have been observed in a two-electron-temperature plasma. Both bi-Maxwellian and Maxwellian-waterbag models were found to be inadequate in explaining the observed dispersion and damping rates. However, modelling of the hot electron component with a κ-distribution function confirms that the experiments represent observation of the electron-acoustic wave in the laboratory.

The so-called ηi instability is driven by the ion-temperature gradient and affects both particle and energy transport. Using a simple kinetic model, first the onset of the instability is investigated by linear theory. In contrast to most treatments in the literature, a non-local analysis is performed. It turns out that the local approximation is only a very rough approximation, and may drastically overestimate the instability conditions. A numerical solution of the full model confirms the non-local stability analysis. In addition, it allows one to follow the nonlinear development of the instability. Using a numerical scheme based on the Lagrange structure of the equations of motion, the saturation values are obtained. A constriction of the density profile is accompanied by a broadening of the ion-temperature profile. The normal form of the bifurcation is obtained, together with the scaling of the anomalous transport (in terms of the deviation from the marginal condition).

We investigate the properties of turbulent adiabatic shock waves resulting from the self-consistent inclusion of finite Alfvén-wave pressure in the shock balance equations. We demonstrate (i) the absence of a switch-on shock; (ii) the existence of two separated domains where the Mach number is greater than unity that have gas compression, and an associated domain where the gas compression ratio is less than unity (i.e. a rarefaction domain); (iii) the difference between the scattering-centre compression ratio and the gas compression ratio for all upstream wave cross-helicity values except for the isolated case of zero helicity; (iv) the presence of anomalous domains where the cosmic-ray particle spectral index can be negative. All of these results are brought about by including the upstream and downstream turbulent wave energies in the shock balance structure. Without the waves, none of the effects persists.

We derive the approximate form and speed of a solitary-wave solution to a perturbed KdV equation. Using a conventional perturbation expansion, one can derive a first-order correction to the solitary-wave speed, but at the next order, algebraically secular terms appear, which produce divergences that render the solution unphysical. These terms must be treated by a regrouping procedure developed by us previously. In this way, higher-order corrections to the speed are obtained, along with a form of solution that is bounded in space. For this particular perturbed KdV equation, it is found that there is only one possible solitary wave that has a form similar to the unperturbed soliton solution.

The acceleration of particles to high energy by relativistic plasma waves has received a great deal of attention lately. Most of the particle-acceleration schemes using relativistic plasma waves rely either on intense terawatt or petawatt lasers or on electron beams as the driver of the acceleration wave. These laboratory experiments have attained accelerating fields as high as 1 GeV cm−1 with the electrons being accelerated to about 100 MeV in millimetre distances. In space and astrophysical plasmas, relativistic plasma waves can also be important for acceleration. A process that is common to both laboratory and space plasmas is the surfatron concept, which operates as a wave acceleration mechanism in a magnetized plasma. In this paper, we present test-particle results for the surfatron process.

The purpose of this work is to understand theoretically what are the possible noise levels in a magnetron or a crossed-field amplifier (CFA), due to parametric three-wave interactions in the electron plasma, at various operating parameters. Our approach is to use the cold-fluid equations and their Fourier decomposition into a background (DC) mode, a pump (RF) mode, and two other noise (RF) modes. The two RF noise modes are assumed to interact parametrically with the large RF pump mode, and to satisfy the standard resonance conditions for the sum of the wave vectors and sum of the frequencies. We use our previous results to determine the background mode and the RF pump mode. Any strong RF electric field propagating in a crossed-field electron vacuum device can drive a Brillouin sheath unstable by means of a Rayleigh instability, whenever a wave–particle resonance can be found inside the sheath. What happens physically is that, at the resonance, the laminar flow of the electrons is strongly disturbed, and a diffusion process ensues, whereby the electrons diffuse away from the resonance region. This upsets the balance in the Brillouin flow, causing the electrons to redistribute into a new average DC density profile – which may be far from the original Brillouin profile, but is a stationary solution of a nonlinear diffusion equation. Using these stationary density profiles, we can then study the propagation of small RF signals on such a DC background, as well as their parametric interactions with the RF pump wave, at various DC voltages and magnetic fields. In addition to being able to predict the operating regime and the DC current flow, these studies demonstrate that parametric interactions probably limit the operating voltage range of a typical magnetron or crossed-field amplifier to about 20% above the Hartree voltage.

A new approach, the propagating-source method, is introduced to solve the time-dependent Boltzmann equation. The method relies on the decomposition of the particle distribution function into scattered and unscattered particles. It is assumed in this paper that the particles are transported in a constant-velocity spherically expanding supersonic flow (such as the solar wind) in the presence of a radial magnetic field. Attention too has been restricted to very fast particles. The present paper addresses only large-angle scattering, which is modelled as a BGK relaxation time operator. A subsequent paper (Part 2) will apply the propagating-source method to a small-angle quasilinear scattering operator. Initially, we consider the simplest form of the BGK Boltzmann equation, which omits both adiabatic deceleration and focusing, to re-derive the well-known telegrapher equation for particle transport. However, the derivation based on the propagating-source method yields an inhomogeneous form of the telegrapher equation; a form for which the well-known problem of coherent pulse solutions is absent. Furthermore, the inhomogeneous telegrapher equation is valid for times t much smaller than the ‘scattering time’ τ, i.e. for times t [Lt ] τ, as well as for t > τ. More complicated forms of the BGK Boltzmann equation that now include focusing and adiabatic deceleration are solved. The basic results to emerge from this new approach to solving the BGK Boltzmann equation are the following. (i) Low-order polynomial expansions can be used to investigate particle propagation and transport at arbitrarily small times in a scattering medium. (ii) The theory of characteristics for linear hyperbolic equations illuminates the role of causality in the expanded integro-differential Fokker–Planck equation. (iii) The propagating-source approach is not restricted to isotropic initial data, but instead arbitrarily anisotropic initial data can be investigated. Examples using different ring-beam distributions are presented. (iv) Finally, the numerical scheme can include both small-angle and large-angle particle scattering operators (Part 2). A detailed discussion of the results for the various Boltzmann-equation models is given. In general, it is found that particle beams that experience scattering by, for example, interplanetary fluctuations are likely to remain highly anisotropic for many scattering times. This makes the use of the diffusion approximation for charged-particle transport particularly dangerous under many reasonable solar-wind conditions, especially in the inner heliosphere.