In this paper, the general theory developed by Vladimirov and Moffatt [J. Fluid Mech.283, 125–139 (1995)] and Vladimirov et al. [J. Fluid Mech.329, 187–205 (1996); J. Plasma Phys.57, 89–120 (1997)] is extended to nonlinear (Lyapunov) stability for axisymmetric (invariant under rotations around fixed axis) solutions of the ideal incompressible magnetohydrodynamic flows for a situation of arbitrary flow and a poloidal field. The appropriate norm is the sum of magnetic and kinetic energies and the mean square vector potential of the magnetic field.

In an ion-beam–inhomogeneous-plasma system, when the drift frequency ω* exceeds a harmonic of the ion cyclotron frequency nωci, ion cyclotron harmonic waves are excited via the plasma inhomogeneity. These waves are strongly enhanced by ion-beam injection parallel to the magnetic field. As the plasma inhomogeneity is increased, the number of unstable modes increases as does their amplitude. Ion cyclotron harmonic waves with frequencies up to 4ωci are observed experimentally; at least five modes become unstable, and these unstable waves have different az-imuthal mode numbers l = 1 or 2. The dispersion relations of observed multiple ion cyclotron harmonic waves agree well with the theoretical ones, taking into account the plasma inhomogeneity and the ion beam.

Self-focusing is one of the key issues in laser plasma physics applications. Problems involving a multidimensional beam within an inhomogeneous plasma are diffficult to handle. This paper presents the investigation of two-dimensional self-focusing of a laser beam in a plasma whose density n(r, z) is a function of radial as well as z coordinates. The nonlinear mechanism responsible for modification of the background density and the dielectric function is of ponderomotive type. A variational technique is used here for deriving the equations for the beam width and the longitudinal phase. It is observed numerically that an initially diffracting beam is accompanied by oscillatory self-focusing of the beam with distance of propagation. The effect of inhomogeneity scale lengths is also observed. The increase in Lr (= L∥/L⊥) results in oscillatory self-focusing and defocusing with distance of propagation. Furthermore, critical fields for self-trapping of a laser beam as a function of refraction, diffraction lengths and scale lengths of inhomogeneities are also evaluated. Lastly, whatever parameters are chosen, the phase is always negative.

Based on a database, a number of conclusions are derived concerning the power flow into the divertor region of the ASDEX tokamak. The non-transition to detachment in ASDEX is investigated with respect to the sputtering of the copper divertor plates. Following the definition of the low- and high-recycling regions, the best fittings of the power widths in front of the divertor plates are extracted for Ohmic (OH) or neutral-beam-injection heating in low-confinement mode (NBI, L-mode) shots. On the basis of these fittings, it is concluded that the safety factor q and the electron temperature in the divertor can be considered as the main robust parameters. In contrast, the peak of the power inserted into the divertor does not behave as a robust parameter, thus introducing the problem of colinearities.

General similarity solutions of temporal decay of a power spectrum of Alfvén waves in a magnetized plasma are classified in terms of Lie group parameters. One class of these solutions can be mapped to a solution of the Abel equation. A second class is transformed to a solitary-wave solution where the spectrum at short wavelengths is more rapidly damped than at long wavelengths.

A theory of enhanced plasma and ion lines near the reflection height of a pump wave is presented. It is argued that the high-frequency pressure of the pump wave and the averaged pressure of plasma waves localized in small-scale cavitons contribute to the rapid creation of a cavity in the main Airy maximum. Langmuir waves excited later by parametric decay instability are trapped in such a nonstationary cavity. Analytical expressions for adiabatic eigenmodes of Langmuir waves in a nonstationary cavity are presented. Collisionless attenuation of the eigenmodes of Langmuir waves trapped in the cavity is investigated. It is shown that parametric decay instability in a cavity may account for the excitation of plasma and ion-acoustic waves measured by EISCAT UHF radar.

The transverse evolution of ions in front of two juxtaposed electrodes whose electric potentials are independently applied is calculated. The particle motion is described with nonlinear fluid equations and a self-consistent electric field.

We develop a new asymptotic method of resolution of the two-dimensional equilibrium equation of collisionless plasmas described by the Maxwell–Vlasov equations. This method differs from the classical one proposed by K. Schindler [Earth’s Magnetospheric Processes (ed. B. M. McCormac). Norwood, MA: Reidel, 1972, pp. 200–209.] since we consider free-boundary plasmas. Our method is a generalization of the usual multiscale asymptotic developments. The first-approximation asymptotic solutions are found from the elimination of increasing and singular terms in the next approximation. We apply the method to the mathematical description of nonlinear structures that may form in neutral sheets. Particular solutions describing localized plasmoids (O-point configuration) as well as X-point magnetic configurations are obtained. We also find more general solutions describing a finite number of ‘magnetic islands’ (multiplasmoid solutions) separated by X points.

This paper presents an investigation of the time-dependent setting-up of nonlinearity in a collisionless plasma in the presence of a short-duration laser pulse. Using the paraxial-ray approximation, transient self-focusing of the incident laser pulse and Brillouin back-scattering have also been studied, including pump depletion. The effect of this transient nature on reflectivity has been discussed.

It is known that the wavefront of a shock wave in a shock tube becomes unstable at particular Mach numbers and unperturbed gas pressure. This instability leads to the distortion of the wave front, i.e. particular Mach numbers define a threshold beyond which the wavefront is no longer plane (see Annou and Ferhat 1997, and references therein). In a recent paper (Annou and Ferhat 1997, subsequently referred to as AF), the authors proposed a microscopic model based on an ionization mechanism described by a set of two reaction–diffusion equations in an effort to interpret the wavefront distortion. Using an analytical approach, they argued that their equations have a bifurcation point beyond which the wavefront instability occurs and that the solution thus obtained describes the distortion of the wavefront. In this brief comment, we present analytical and numerical arguments supporting the fact that the proposed model, although it has an instability point, is unable to explain the front behavior.

We respond here to the comment by Houili et al. (2001) on our paper on instabilities of intense shock waves in monatomic gases (Annou and Ferhat 1997).

It has been reported that a strong shock-wave front in a shock tube, ceases to be plane and becomes distorted for particular experimental conditions (Schreffler and Christian 1954). Two different approaches have been developed to explain the instability (Annou 1995; Annou and Ferhat 1997). By means of the macroscopic approach, we can reproduce the experimental data. Nonetheless, many parameters suspected to be at the root of the instability have not been taken into account. Consequently, a microscopic approach has been proposed that copes with the atomic reactions taking place due to the shock wave. An irreducible system of atomic reactions leading to ionization of the gas is proposed. This is an attempt to develop a mathematical tool that has to be improved to account for the front distortion. A system of two nonlinear differential equations is obtained. As a matter of fact, we have been greatly inspired by the Brusselator model. Our achievement has been to link the experimental data to a reaction-diffusion equation system, analogous to the Brusselator. Hence, the analytical treatment of the stability, along with finding the solution beyond the instability, is based on the work of Nicolis and Prigogine (1977). It should be noted, as well, that the paper by Annou and Ferhat (1997) is based essentially on a thesis submitted (1994) for the award of the Magister degree (Annou 1995).One of the authors of the paper (R.A.) had asked Mr Houili to investigate the system numerically, and provided him with the necessary literature. Unfortunately, the comment was submitted for publication without consulting either of the authors of the original paper.