The propagation of fast magnetosonic waves in an inhomogeneous medium with planar flow in investigated. The equations describing the rays along which the waves propagate are derived, as well as the equations for the variations of the wave amplitude along the rays. These equations are solved for the case of steady flow that depends only on the radius. In addition, it is shown that a spectrum of eigenmodes may exist if the steady flow contains a shock. For that purpose, the reflection coefficient of a fast magnetosonic wave form a shock is derived, and it is shown that the waves can be localized in a region bounded by a shock and a critical surface. A criterion for stability of the spectrum is derived.

A cylindrically symmetric, electrically driven, dissipative, energy-conserving magnetohydrodynamic equilibrium model is considered. The high-magnetic-field Braginskii electron electrical resistivity η parallel to a constant axial magnetic field B and ion thermal conductivity ĸ perpendicular to B are included in an energy equation and in Ohm's law. The expressions for η and ĸ depend on number density and temperature, which are functions of radius that are obtained as part of the equilibrium solution. The model has plasma-confining solutions, by which are meant solutions characterized by the separation of the plasma into two regions separated by a relatively thin transition region that is an internal boundary layer across which temperature and current density vary rapidly. The inner region has a temperature, pressure and current density that are much larger than in the outer region. The number density and thermal conductivity attain their minimum values in the transition region. The model has an intrinsic value of β, about 6.6%, which must be exceeded in order that a plasma-confining solution exist. The model has an intrinsic length scale, which, for plasma-confining solutions, is a measure of the thickness of the transition region separating the inner and outer regions of plasma. A larger class of transport coefficients is modelled by artificially changing η and ĸ by changing the constant coefficients ηO and ĸO that occur in their expressions. Increasing ĸO transforms a state that does not exhibit confinement into one that does, improves the confinement in a state that already exhibits it, and leads to an increase in ĸ in the confined region of plasma. The improvement in confinement consists in a decrease in the thickness of the transition region. Decreasing ηO subject to certain constraints, also transforms a state that does not exhibit confinement into one that does, improves the confinement in a state that already exhibits it, and leads to a decrease in η in the confined region of plasma. Increasing ηO up to a critical point increases the current, temperature, and volume of the confined region of plasma, and causes the thickness of the transition region to increase. If ηO is increased beyond the critical point, a plasma-confining state cannot exist. In all cases it is found that an increase in ĸ and a decrease in η in the confined region of plasma are associated with an improvement in the confinement properties of the equilibrium state. If the pressure and temperature are given on the cylinder wall, the equilibrium bifurcates when the electric field decreases below a critical value. The equilibrium can bifurcate into a state that exhibits confinement and a state that does not.

A Gould-Trivelpiece mode at significant power levels is excited in a cylindrical beam-plasma system by launching an external signal. The fundamental mode excites a second-harmonic wave in the upper-hybrid mode. When the second-harmonic frequency is close to (where is the upper-hybrid frequency), considerable enhancement in the second-harmonic power level is observed. The results are interpreted using a slab model.

The propagation of magnetovortex waves in an inhomogeneous medium with planar flow and perpendicular magnetic field is investigated. The linearized MHD equations are studied with the aid of the WKB approximation. Within the latter, a set of ordinary differential equations along the streamlines of the basic flow is derived. These equations govern the variations of the transverse as well as the parallel magnetovortex waves along its rays. The amplitude equations are solved explicitly for a particular geometry of the basic flow, and the solutions are investigated for subsonic as well as for supersonic basic flows.

The propagation of an electromagnetic wave through a transparent plasma characterized by a spatial modulation of the density in the direction of propagation is analysed by computing the reflection coefficient of a single plasma slab and then that relevant to a set of many identical or slightly different slabs. The transmission spectra relevant to different density profiles of a single slab are computed to show the effects of (a) the density gradient, (b) a slight non-uniformity among adjacent slabs and (c) plasma modulation on the linear cut-off.

We study the Vlasov–Poisson system of equations in Fourier-transformed velocity space. First we reformulate some results of the linear theory: in the new representation the van Kampen–Case eigenmodes are found to be ordinary functions with convenient continuity properties. We give a transparent derivation of the free-streaming temporal echo in terms of the kinematics of wave packets in Fourier-transformed velocity space. We further extend this analysis to include Coulomb interactions, which allows us to establish a connection between the echo theory, the second-order oscillations of Best and the phenomenon of linear side bands. The calculation of the time evolution of the global second-order electric field is performed in detail in the case of a Maxwellian equilibrium distribution function. We conclude that the phenomenon of linear side bands may be properly explained in terms of the intrinsic features of the equilibrium distribution function.

A new action principle determining the dynamics of the Vlasov–Poisson system is presented (the Vlasov–Maxwell system will be considered in Part 2). The particle distribution function is explicitly a field to be varied in the action principle, in which only fundamentally Eulerian variables and fields appear. The Euler–Lagrange equations contain not only the Vlasov–Poisson system but also equations associated with a Lie perturbation calculation on the Vlasov equation. These equations greatly simplify the extensive algebra in the small-amplitude expansion. As an example, a general, manifestly Manley–Rowesymmetric, expression for resonant three-wave interaction is derived. The new action principle seems ideally suited for the derivation of action principles for reduced dynamics by the use of various averaging transformations (such as guiding-centre, oscillation-centre or gyro-centre transformations). It is also a powerful starting point for the application of field-theoretical methods. For example, the recently found Hermitian structure of the linearized equations is given a very simple and instructive derivation, and so is the well-known Hamiltonian bracket structure of the Vlasov–Poisson system.

Quasi-one dimensional hydrodynamic continuity, momentum and energy equations describing the plasma flow in high-pressure-discharge ablative capillaries are derived. To overcome the formidable difficulties arising in the solution of a fully two-dimensional system of equations, experimental information on the structure (geometry) of the generated plasma is used. Thus the two-dimensional hydrodynamic equations are averaged over the cross-section of the capillary to obtain a quasi-one-dimensional system of equations in which, however, the essential two-dimensional features are present. These include the radial outwards radiative transfer of energy and the radial inwards ablative mass flow. Some particular cases, including their thermodynamical aspects, are discussed. Illustrative analytical and numerical solutions of the equations are also presented.

Sheath potential distributions in electrostatic plasmas including negative ions have been studied using the Bohm equation. Since the mean velocity of the negative ions is not necessarily negligible compared with their thermal velocity, the contribution from non-thermal negative ions has been considered. The Sagdeev potential has been introduced to consider structural instabilities of the Bohm system. Wavy solutions related to ion-acoustic solitons bifurcate from the monotone asymptotic Bohm sheath solutions. The solvability conditions have also been discussed for characteristic parameter such as the concentration of negative ions and the Mach numbers of positive and negative ions.

We present a set of equations modelling a low-pressure plasma column sustained by a travelling electromagnetic wave in the dipolar mode in the presence of a constant external magnetic field. It is shown that, from a practical point of view, only the m = 1 mode (the right-hand-polarized wave) can sustain plasma columns in a wide region of gas-discharge conditions: plasma radius R, wave frequency ωo, magnetic field Bo and low pressures, irrespective of the nature of the gas. We have examined two gas-discharge regimes: freefall/diffusion and recombination respectively. For a given gas-discharge regime the axial column structure and wave-field characteristics are specified by two numerical parameters: σ = ωR/c and ω = ωc/ω, where c is the speed of light and ωc the electron-cyclotron frequency. The main result of our study is that the magnetic field-makes it possible to sustain a plasma column for values of σ smaller than σcr = 0.3726, below which, in the absence of a magnetic field, the dipolar wave cannot produce a plasma. Moreover, at a fixed wave power, the magnetic field – in contrast with the case of plasma columns sustained by azimuthally symmetric waves – increases the plasma density and its axial gradient. The limit of an infinite external magnetic field (Ω → ∞) is also considered. A three-dimensional wave structure is obtained, and it indicates that the wave can be a generalized surface mode, a pure surface or a pseudosurface one.

The generation of nonlinear plasma waves with arbitrary phase velocities by an intense laser pulse is considered in order to determine the dependence of the maximum amplitude of driven waves on the group velocity of the laser pulse (or the phase velocity of excited wakefields). Analytical as well as numerical studies of the problem are carried out within the so-called quasi-static approximation. In the case of square-shaped laser pulses the analytical expression for the maximum amplitude as a function of the pump amplitude and the phase velocity is derived. The numerical analysis in the case of Gaussian pulses confirms the result that follows from the analytical formula: the lower the phase velocity of the wakefleld, the greater is its maximum amplitude for a given amplitude of the driving pulse.

We consider the nonlinear propagation of magnetosonic waves along an unperturbed magnetic field in a plasma of high but finite conductivity when the Hall effect is taken into account, and we give a discussion of the analytical solutions of the nonlinear evolution equation that governs the latter problem. This equation is a generalization of the well-known modified Schrödinger equation and of the modified Burgers equation (it is referred to here as the modified Burgers-Schrödinger equation). Explicit analytical expressions for both a stationary kink-type solution and circularly polarized quasi-periodic waves are obtained and investigated.

The transcendental dispersion equation for electromagnetic waves propagating in the slow mode in sheared non-neutral relativistic cylindrical electron beams in strong applied magnetic fields is solved exactly. Thus, rather than truncated power series for the modified Bessel functions involved, use is made of modern algorithms able to compute such functions up to 18-digit accuracy. Consequently, new and significantly more important branches of the velocity shear instability are found. When the shear-factor and/or the geometrical parameter a/b (pipe-to-beam radius ratio) are increased, the unstable branches join, and the higher-frequency, larger-wavenumber modes are significantly enhanced. Since analytical solutions of the exact dispersion relation cannot be obtained, it is suggested that in all similar cases the methods proposed and demonstrated here should be used to carry out a rigorous stability analysis.

We study nonlinear self-interactions including modulational instability in the case of a plane electromagnetic pulse propagating through a magnetized cold plasma at an arbitrary oblique angle to the external magnetic field. For intended applications to pulsar magnetospheres, the magnetic field is so large that both the electron- and ion-cyclotron frequencies are enormous compared with the plasma frequency or the frequency ω of the wave itself. The plasma is assumed to contain two singly charged species, either electrons and positrons or electrons and ions. (No approximation is made with respect to the mass ratio.) We restrict ourselves to the case eE0/mω ≪ 1 (i.e. the wave amplitude E0 excites the electrons to weakly, but not fully, relativistic velocities). We consider a pulse whose linear polarization is in the plane of the wave vector and the magnetic field. (The orthogonal polarization is purely electromagnetic, and induces no motion along magnetic field lines.) The pulse is assumed to be modulated along the direction of the group velocity vector. We show, using a self-consistent multiple-scales solution, that the envelope obeys the nonlinear Schrödinger equation, and from the coefficients of this equation we derive the conditions for modulational instability. Computation of the nonlinear coefficients requires detailed consideration of ponderomotive, relativistic and harmonic effects, all of which, in the ‘weakly relativistic’ case considered here, enter at the same order in the approximation scheme. Unlike the case of propagation parallel to a strong magnetic field, in oblique propagation we find a wide parameter range for modulational instability and soliton formation on time scales appropriate for pulsar micropulses.

The effect of a large Larmor radius on the stability of an infinitely conducting infinitely extended inhomogeneous plasma with two-dimensional magnetic field has been studied. A dispersion relation is obtained for the homogeneous system, and it is found that it is stable and MHD waves propagate. For an inhomogeneous plasma, a dispersion relation is also obtained and discussed for disturbances propagating transverse to inhomogeneity in (a) a cold plasma and (b) an incompressible plasma. It is found that the inhomogeneous system is unstable in both the cases, in agreement with the results of Lee and Roberts. The values of ωr and ωi are computed numerically, and the variations of ωi>0 and the corresponding ωr with the large-Larmor-radius parameter are shown graphically.

A nonlinear emission mechanism of electromagnetic waves at the fundamental plasma frequency has been examined by Chian & Alves. This mechanism is based on the electromagnetic oscillating two-stream instability driven by two oppositely propagating Langmuir waves. The excitation of the electromagnetic oscillating two-stream instability is due to nonlinear wave–wave coupling involving Langmuir waves, low-frequency density waves and electromagnetic waves. In this paper the Chian & Alves model is improved using the generalized Zakharov equations. Attention is directed toward the influence of induced low-frequency and Langmuir waves on the properties of the electromagnetic oscillating two-stream instability. Presumably, the properties derived in the present context may be relevant to both space and laboratory plasmas.

We derive the Alfvén-wave equation for an atmosphere in the presence of a non-uniform vertical magnetic field and the Hall effect, allowing for Alfvén speed and ion gyrofrequency that may vary with altitude; the pair of coupled second-order differential equations for the horizontal wave variables, namely magnetic field or velocity perturbations, is reduced to a single complex, second-order differential equation. The latter is applied to spinning Alfvén waves in a magnetic flux tube, in magnetohydrostatic equilibrium, in an isothermal atmosphere. The exact solution is found in terms of hypergeometric functions, from which it is shown that at ‘high altitude’the magnetic field perturbation tends to grow to a non-small fraction of the background magnetic field. By ‘high-altitude’ is meant far above the critical level, which acts as a reflecting layer for left-polarized waves incident from below, i.e. from the ‘low-altitude’ range. We also obtain the exact solution near the critical level, where the left-polarized wave has a logarithmic singularity, and the right-polarized wave is finite. The latter is plotted in this region of wave frequency comparable to ion gyrofrequency, and it is shown that the Hall effect can cause oscillations of wave amplitude and non-monotonic phases with slope of alternating sign. The latter corresponds to ‘tunnelling’, i.e. waves propagating in opposite directions or trapped in adjoining atmospheric layers; this could explain the appearance of inward- and outward-propagating waves, with almost random phases, in the solar wind beyond the earth, for which the Hall effect on Alfvén waves should be significant.

Phase-sensitive probe diagnostics as well as two-point-correlation analysis of radio-frequency probe signals are applied to investigate small-amplitude lower-hybrid (LH) waves launched into a linear nonuniform plasma column by a slow-wave antenna. The LH wave propagation in the inhomogeneous plasma is described by a model taking into account the wavenumber spectrum of the antenna. Special attention is focused on the interaction of the LH waves with the low-frequency density fluctuations. The LH frequency and wavenumber spectra observed can be explained by scattering of the LH waves from the (incoherent) low-frequency density fluctuations. Moreover, the enhanced damping, which is found to be much higher than expected from collisional and electron Landau dampings, is correlated with the level of low-frequency fluctuations.

The equations of high- and low-beta reduced magnetohydrodynamics (RMHD) are considered anew in order to elucidate the relationship between compressible MHD and RMHD and also to distinguish RMHD from recently developed models of nearly incompressible MHD. Our results, summarized in two theorems, provide the conditions under which RMHD represents a valid reduction of compressible MHD. The equations for low-beta RMHD and high-beta RMHD are shown to be identical. Furthermore, as a direct consequence of our analysis, the conditions under which both two-dimensional incompressible MHD (in terms of the spatial co-ordinates as well as the fluid variables) and 2½ dimensional incompressible MHD (i.e. only two-dimensional in the spatial co-ordinates) represent a valid reduction of three-dimensional compressible MHD are also formulated. It is found that the elimination of all high-frequency and long-wavelength modes from the magneto-fluid reduces the fully compressible MHD equations to either two-dimensional incompressible MHD in the plasma beta (β) limit β ≪ 1, or 2½-dimensional incompressible MHD for β ≈ 1. Our approach clarifies several inconsistencies to be found in previous investigations in that the reduction is exact. Our results and analysis are expected to be of interest for plasma fusion and space and solar physics.