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.

Higher-order nonlinear and dispersive effects on ion-acoustic solitons in a plasma consisting of two types of cold positive ions and two-temperature non-isothermal electrons are studied using a reductive perturbation method. An integrated form of the basic sets of fluid equations consisting of two types of positive ions and two types of non-isothermal electrons is derived in terms of a pseudo-potential. A perturbation solution of this equation is obtained by the Bogoliubov–Mitropolsky method. At the lowest order, we obtain a modified Korteweg–de Vries solitary-wave solution, which has a sech4 profile. Calculations are carried out at the next two higher orders. The secularity-removing condition at each order gives a correction to the velocity of the solitary wave at the corresponding order. The variations of Mach number and width of the ion-acoustic soliton with amplitude at each order are shown graphically for plasmas with H+ and He+ or with He+ and Ar+ ions, with two-temperature non-isothermal electrons.

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.