The nonlinear interaction of two three-wave systems, including the possibility of negative energy waves in the presence of linear damping or growth and frequency mismatch, is investigated in a plasma, where one system of two transverse and one longitudinal wave interacts with a system of three longitudinal waves, and one of the longitudinal waves introduces coupling between the two subsystems. The solutions are analysed under various initial conditions and it is shown that, if one triplet be explosively unstable by itself, the presence of the second triplet can stabilize the solutions, depending on the relative strength of the coupling factor.

The coupling mechanism of boundary sheaths and a wave launcher immersed in a collisionless plasma is experimentally clarified by the addition of an electron emission from the launcher. The wave amplitude increases rapidly with increasing electron emission. The strong increase in a directly coupled signal is a clear observation of ion-acoustic waves which are simultaneously excited at the sheaths.

The problem of linear conversion of an ordinary polarized electromagnetic wave in a magnetized plasma with density gradient parallel to the magnetic field is considered. An expression for the conversion coefficient as a function of angle of incidence, WKB parameter and magnetic field is obtained. The magnetic field leads to a narrowing of the range of angles of incidence leading to linear conversion, compared with the unmagnetized case.

A discussion is given of test particle transport in the presence of specified stochastic magnetic fields, with particular emphasis on the collisional limit. Certain paradoxes and inconsistencies in the literature regarding the form of the scaling laws are resolved by carefully distinguishing a number of physically distinct correlation lengths, and thus identifying several collisional subregimes. The common procedure of averaging the conventional fluid equations over the statistics of a random field is shown to fail in some important cases because of breakdown of the Chapman-Enskog ordering in the presence of a stochastic field component with short autocorrelation length. A modified perturbation theory is introduced which leads to a Kubo-like formula valid in all collisional regimes. The direct-interaction approximation is shown to fail in the interesting limit in which the orbit exponentiation length LK appears explicitly. A higher-order renormalized kinetic theory in which LK appears naturally is discussed and used to rederive more systematically the results of the heuristic scaling arguments.

The MHD equations are used to discuss the formation and structure of shock waves in channel flow of an ionized gas with finite viscosity, thermal and electrical conductivity. The fundamental resulting complicated nonlinear differential equations are solved exactly, and the shock structure is determined completely. A necessary compatibility condition for the evolution of shock waves is obtained in the form:

where Pr and Prm are ordinary and magnetic Prandtl numbers and γ the ratio of specific heats. The flow quantities and parameters are found to depend, in a non-analytical manner, on Pr and Prm, so that special cases of interest are to be treated separately. Numerical results are given and discussed.

Nonlinear Rayleigh-Taylor instability of a heavy, infinitely conducting fluid, supported against gravity by a uniform magnetic field in the vacuum, is studied for three-dimensional disturbances using the method of multiple time-scales. The three-dimensional problem can be reduced to two dimensions as it is found that an instability present for a three-dimensional disturbance of a given wavelength, for a given equilibrium magnetic field, is also present for a two-dimensional disturbance of the same wavelength propagating along an equilibrium magnetic field of lower strength. The instability is studied for both standing and progressive waves. Although in the linear stability problem the instability growth rate for a progressive wave of a given wavelength is equal to that for a stationary wave of the same wavelength, in the nonlinear instability problem studied here these waves are found to have different growth rates. Our results are compared and contrasted with those for the two-dimensional instability problem studied earlier.

We improve our previously derived addition to the BGK collision term, and express it in a simple form. The collision frequency for scattering now depends anisotropically on the velocity vector. We also apply the improved macroscopic equation of momentum flow to the Hall effect, the cold plasma dispersion relation and the cyclotron resonance. The Hall coefficient which is constant in the case of the BGK collision term now depends on the magnetic field. It is also shown that, compared with the almost symmetric classical curves of cyclotron resonance, the new curves are considerably asymmetric and their half-widths are about 3/2 times the classical ones.

The tearing mode of a tokamak plasma without flow may be stabilized by the presence of a conducting wall surrounding the plasma. When the wall has a finite resistivity, its presence does not affect stability, only growth rates. If, however the plasma has a flow relative to the resistive wall, both stability and growth rates may be affected. For the cylindrical, circular cross-section tokamak the problem is formulated as a complex eigenvalue problem, with a complex eigenvalue Δ', which in the limit of vanishing flow becomes identical to the usual ‘delta prime’. The real part of Δ' describes as usual the power absorbed at the singular surface while the imaginary part describes absorption of momentum. It is found that for a plasma with shearless flow, a resistive wall has a stabilizing effect which even for relatively small flow velocities approaches that of a wall of infinite conductivity. Shear flow is found inherently destabilizing, but important only for very large flow velocities.

The effect of finite spectral width on the modulational instability of Langmuir waves has been investigated applying a method developed by Alber to derive a transport equation for the spectral density. The numerical results presented show that the spectrum is stable against modulational perturbation when the spectral width exceeds some critical value. For a Gaussian spectrum, the maximum growth rate is less than that for a monochromatic wave but the domain of modulational instability is extended. For a uniform distribution the shift in the growth rate curve towards the region of shorter wavelength is more pronounced and, for a certain range of spectral width, the maximum growth rate exceeds that for a monochromatic wave.

The point model of Hedrick et al. for the toroidal core plasma in the ELMO Bumpy Torus (with neoclassical non-resonant electrons) is examined in the light of catastrophe theory. Even though the point model equations do not constitute a gradient dynamic system, the equilibrium surfaces are similar to those of the canonical cusp catastrophe. The point model is then extended to incorporate ion cyclotron resonance heating. A detailed parametric study of the equilibria is presented. Further, the nonlinear time evolution of these equilibria is studied, and it is observed that the point model obeys the delay convention (and hence hysteresis) and shows catastrophes at the fold edges of the equilibrium surfaces. Although a detailed analysis of the basin boundaries for the simultaneous point attractors is not made, some simple examples are given which illustrate that the final equilibrium state can be drastically affected, not only by the control parameters (neutral density, ambipolar electrostatic potential, electron and ion cyclotron power densities) but also by the initial conditions of the state vector (plasma density, electron and ion temperatures). Tentative applications are made to experimental results.

The effect of the ponderomotive depression of the plasma density on the power reflexion coefficient is calculated in the first space harmonic approximation. Numerical results demonstrate a considerable growth of the reflexion coefficient when increasing the RF power. Exceptionally, the reflexion coefficient can be almost invariable up to high RF power densities (≃ 10kW cm-2), if the unperturbed plasma density gradient is high (≳ 5 × 1012 cm-4). The relation of the present theory to experiments is discussed.

Scattering of electromagnetic and electrostatic waves by one-dimensional Langmuir solitons in a uniform and isotropic plasma is studied analytically using a perturbation method. Mode conversion via scattering by solitons is also considered. The results are also relevant for the case of ion-acoustic solitons.

A retarded time superposition principle is formulated and proved for the two particle correlation function in a multi-species relativistic, and fully electro-magnetic, plasma. This principle is used to obtain the relativistic collision operator. Starting from the relativistic Klimontovich equation, the statistical (Liouville) average of the Klimontovich equation yields an expression for the collision operator in terms of the two-time two-point correlation function for two particles, G12(1, t12, t2). It is proved that G12(1, t12, t2) can be written in a retarded time superposition form in terms of the self-correlation W11(1, t12, t2) and the discreteness response function P(1, t12, t2). The equation for the pair correlation function G12(1, t12, t2), excluding triplet or higher-order correlations, is thus replaced by the simpler equation for P(1, t12, t2). This is the test particle problem which relates P(1, t12, t2) to the discreteness source term W11(1, t12, t2). The equations for P(1, t12, t2). and W11(1, t12, t2) are solved for stationary, homogeneous plasmas without external fields. With these solutions, the collision operator is expressed in terms of the relativistic dielectric properties of the plasma.

This paper reports a study of the linear Vlasov electromagnetic dispersion equation for density drift instabilities in very low β plasmas. A uniform magnetic field and a weak uniform density gradient are assumed. This paper differs from most other studies of this topic in three important ways: (i) no low-frequency or long-wavelength approximations are made, (ii) no gauge condition is imposed and (iii) a modification of the local approximation is used which is argued to be less arbitrary than the usual local approximation. Numerical solution of the resulting dispersion equation yields stabilization of the universal density drift instability at values of β somewhat higher than those obtained from the local approximation, and near maximum growth shows no evidence of coupling between the universal and Alfvén modes.

The nonlinear saturation of a resistive modified tearing mode in a toroidal plasma is calculated by Simon saturation theory for a weakly linearly unstable (over-stable) regime. The linearly unstable mode is a (1, 1) mode, where the first 1 indicates n, the toroidal mode number about an axisymmetric equilibrium and the second 1 indicates the multiple of the non-zero frequency at which the n = 1 mode oscillates when linearly marginally stable. The saturation amplitude is found to be proportional to (Δ – Δc)½ where Δ, a measure of the driving energy of the instability, is proportional to the difference between the logarithmic derivatives of the radial (i.e. perpendicular to the magnetic surface) perturbation magnetic field in the simplest case, and Δc measures toroidal stabilization due to average magnetic line curvature. To obtain the Simon saturation condition, one must go to third order in the small parameter (Δ –Δc)½. Starting with (1, 1), (1, – 1),( – 1, – 1), and( –1, 1 ) modes in first order in (Δ – Δc)½, the modes (0, 0), (0, 2), (2, 2), (2, 0) are obtained in second order, and these are driven with first-order modes to a third-order (1,1) mode which yields a saturation condition. In the quasi-linear approximation, only the near-zero frequency modes (0, 0) and (2, 0) are considered in the second order. In the present paper, only the axisymmetric perturbation (0,0) is used in second order. This gives a relation between nonlinear saturation amplitude and frequency shift, but does not determine either uniquely because of the undetermined parameter γ in the perturbation solutions. This parameter is determined exactly by the requirement of finiteness of the solutions when the (2, 0) non-axisymmetric near-zero frequency perturbation is taken into account in second order. However, magnetic island width at saturation can still be estimated taking only the (0,0) second order mode because this width depends so insensitively on γ, namely as γ–¼. Under this restriction to (0, 0) in second order, the frequency shift is found to be negative. Its absolute value is proportional to Δ – Δc, while the magnetic island width is proportional to (Δ – Δc)¼ and has a scale determined by t, the thickness of the resistive layer, proportional to η⅓, the cube root of the electrical resistivity.

The negative resistance of double layers has been invoked to explain instability in diode currents, and the flickering aurora phenomenon. In this paper the theory of the negative resistance is extended by a more detailed account of the constraints that are imposed on the electron distributions in the double layer. Attention is concentrated on the energetic trapped electrons which originate at the high potential side of a double layer.

We discuss ensemble inhomogeneous states of general two-dimensional magneto-fluids. A subclass of approximate statistical inhomogeneous states of the model equation is constructed using the principle of maximum entropy. The fields are shown to satisfy generalized nonlinear Poisson equations and some limiting cases are solved analytically. The method is then applied to a pseudo-three-dimensional model of an electromagnetic filamentation instability. The results illustrate the general role of the global constants of the motion on the nature of the statistical profiles for the fields describing the most probable states in magnetofluids.

A purely growing instability characterized by a four-wave interaction has been analysed in a uniform, magnetized plasma. Up-shifted and down-shifted upper-hybrid waves and a non-oscillatory mode can be excited by a pump wave of ordinary rather than extraordinary polarization in the case of ionospheric heating. The differential Ohmic heating force dominates over the ponderomotive force as the wave–wave coupling mechanism. The beating current at zero frequency produces a significant stabilizing effect on the excitation of short-scale modes by counterbalancing the destabilizing effect of the differential Ohmic heating. The effect of ionospheric inhomogeneity is estimated, showing a tendency to raise the thresholds of the instability. When applied to ionospheric heating experiments, the present theory can explain the excitation of field-aligned plasma lines and ionospheric irregularities with a continuous spectrum ranging from metre-scale to hundreds of metre-scale. Further, the proposed mechanism may become a competitive process to the parametric decay instability and be responsible for the overshoot phenomena of the plasma line enhancement at Arecibo.

Forced reconnexion due to tearing modes driven by fast magnetosonic waves in a current sheet with stagnation-point flows is discussed. The current sheet with stagnation-point flows which is weakly unstable against tearing modes can be strongly destabilized by vortex motions due to the ponderomotive force of the fast magnetosonic wave. This forced tearing instability can be driven when the incident fast magnetosonic wave intensity, I, exceeds a critical value given by where is the Alfvén velocity, vg the group velocity of the fast wave, vo the background inflow velocity, l the thickness of the current sheet and k the wavenumber of the forced tearing mode. The growth rate is estimated. Applications to solar flares and magnetopause reconnexion processes are briefly discussed.

It is shown numerically, both for the two-dimensional Navier-Stokes (guidingcentre plasma) equations and for two-dimensional magnetohydrodynamics, that the long-time asymptotic state in a forced inverse-cascade situation is one in which the spectrum is completely dominated by its own fundamental. The growth continues until the fundamental is dissipatively limited by its own dissipation rate.