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 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.

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 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 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.

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.

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.

A mathematical description of the weakly nonlinear regime of the coherent cyclotron resonance instability in the magnetosphere which has hitherto been neglected in the literature has been developed. Using the concept that the fundamental role played by the inhomogeneity of the Earth's magnetic field is to limit the interaction time between the resonance electrons and whistler mode wave, perturbation expansions of the current equations to third order have been obtained. It is then found that the real component of resonance current can qualitatively account for excess growth found in some magnetospheric transmissions while the imaginary component gives a good quantitative estimate of frequency shifts observed in transmissions of the mains harmonics.