A wave-front distortion of strong shock waves in gases for particular Mach numbers M has been repeatedly noted. We propose two approaches according to which there is a threshold Mach number above which the instability occurs; this Mach number is related to the unperturbed gas pressure P1. The first approach is macroscopic, and takes account of radiation; the second is microscopic, and takes account of radiation, electron diffusion to the walls of the shock tube and the non-equilibrium relaxation processes. A mechanism of four reactions is proposed. A nonlinear differential equation system is established. The latter becomes unstable for critical values of the Mach number.

We investigate finite-amplitude magnetoconvection in a rotating fluid in the presence of a vertical magnetic field when the axis of rotation is parallel to a vertical magnetic field. We derive a nonlinear, time-dependent, one-dimensional Landau–Ginzburg equation near the onset of stationary convection at supercritical pitchfork bifurcation when

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and a nonlinear time-dependent second-order ordinary differential equation when Ta=T*a (from below). Ta=T*a corresponds to codimension-two bifurcation (or secondary bifurcation), where the threshold for stationary convection at the pitchfork bifurcation coincides with the threshold for oscillatory convection at the Hopf bifurcation. We obtain steady-state solutions of the one-dimensional Landau–Ginzburg equation, and discuss the solution of the nonlinear time-dependent second-order ordinary differential equation.

Electromagnetic and kinematic boundary conditions for surface waves propagating on the plane interface between a vacuum and a drifting plasma or a stationary plasma are investigated using a specular reflection procedure. The boundary values of specular-reflection solutions are shown to agree with the kinematic relations involving surface charge and surface current. The specular-reflection solutions explicitly show the well-known fact that a cold fluid forms a surface charge on the interface while a warm fluid does not. If the plasma is drifting (beam), the cold fluid forms a surface current in addition, whereas a warm fluid forms neither a surface charge nor a surface current. Consequently, in a drifting cold plasma, the tangential magnetic field, as well as the normal component of the induction vector D is not continuous. We show that the specular-reflection procedure and the conventional matching method yield the same dispersion relations, as they should, and we correct the wrong notion found in literature that the two methods somehow yield different results.

The effect of finite plasma rotation on the equilibrium of an axisymmetric toroidal magnetic trap is investigated. The nonlinear vector equations describing the equilibrium of a highly conducting, current-carrying plasma are reduced to a set of scalar partial differential equations. Based on Shafranov's well-known tokamak model, this set of equations is employed for the description of a kinetic (stationary) plasma equilibrium. Analytical expressions for the Shafranov shift Δ are found for the case of finite plasma rotation, where two regions of possible plasma equilibria are found corresponding to sub- and super-Alfvénic poloidal rotation. The shift Δ itself, however, turns out to depend essentially on the toroidal rotation only. It is shown that in the case of a stationary plasma flow, the solution of the Grad–Shafranov equation is at the same time also the solution of the stationary Strauss equation.

On the basis of a two-fluid, cold-plasma, linear stability calculation with linear friction between electrons and ions, the growth rate of Alfvén waves is derived from the dispersion relation for a uniformly magnetized plasma, in which the plasma resistivity and a uniform electric current carried by an electron beam are both considered. The growth rate is directly proportional to the plasma resistivity, the electric current density and the value of the parameter ωxpe/Ωe (where ωxpe and Ωe are the electron plasma and cyclotron frequency respectively). Moreover, the growth of Alfvén waves is mainly excited in a direction nearly parallel to the ambient magnetic field. The critical value of the velocity of the electron fluid is just equal to the Alfvén velocity. The results of this paper are compared with those for the linear tearing mode.

The nonlinear dynamics of a finite-amplitude Alfvén wave in a dispersive modulational regime is analysed. Use is made of a conservative model to show that turbulence may arise via deterministic chaos. The behaviour of the system is studied as one varies the initial amplitude of the pump Alfvén wave, the dispersion parameter and the plasma parameter β.

The two-dimensional spatial and temporal evolution of a plasma surrounding an electrode whose potential is suddenly decreased is experimentally investigated. The electrode contains a localized convex or a localized concave perturbation. The quasineutral plasma consists of positive ions and various proportions of negative ions and electrons. The results are compared and contrasted with those that are obtained numerically using a particle-in-cell (PIC) simulation and those that had previously been obtained using a fluid-model code.

The radially symmetric self-similar expansion of a dusty plasma is investigated in cylindrical and spherical geometries. The electrons and ions are assumed to be in Boltzmann equilibria, while the dynamics of the dust particles is governed by the fluid equations. The effects of finite dust pressure as well as dust-charge variation are included.

A spectral representation for the isotropic part of the Coulomb collisional operator is given. The particle distribution function is expanded in a series of generalized Laguerre polynomials, and the Coulomb collisional operator is expressed in terms of the spectral amplitudes. When the spectral representation is applied to the Fokker–Planck equation, a system of coupled ordinary differential equations for the spectral amplitudes is obtained. The spectral coefficients related to the Coulomb operator are defined through recurrence relations, which we reduce to simplified form. This makes possible accurate and efficient analytical and numerical evaluations of the interaction matrices. The results presented can be used in analytical investigations of the properties of the Coulomb collisional operator as well as in numerical calculations for plasmas far from thermal equilibrium. The method can also be generalized to include angular dependencies for non-isotropic particle distributions.

The present study consists of two parts. Here in Part 1, a model set of conservationlaws exactly preserving the MHD hyperbolic singularities is investigated to develop the general theory of the nonlinear evolution of MHD shock waves. Great emphasis is placed on shock admissibility conditions. By developing the viscosity admissibility condition, it is shown that the intermediate shocks are necessary to ensure that the planar Riemann problem is well-posed. In contrast, it turns out that the evolutionary condition is inappropriate for determining physically relevant MHD shocks. In the general non-planar case, by studying canonical cases, we show that the solution of the Riemann problem is not necessarily unique – in particular, that it depends not only on reference states but also on the associated internal structure. Finally, the stability of intermediate shocks is discussed, and a theory of their nonlinear evolution is proposed. In Part 2, the theory of nonlinear waves developed for the model is applied to the MHD problem. It is shown that the topology of the MHD Hugoniot and wave curves is identical to that of the model problem.

In Part 1 of this study, a model set exactly preserving the MHD hyperbolic singularities was considered. By developing the viscosity admissibility condition, it was shown that the intermediate shocks are necessary to ensure that the planar Riemann problem is well-posed. Here in Part 2, the MHD Rankine–Hugoniot condition and rarefaction-wave relations are presented in phase space, which allows construction of analytical solutions of the planar MHD Riemann problem. In this process, a viscosity admissibility condition is proposed to determine physically admissible shocks. A complete account of MHD Hugoniot loci is given, leading to a classification of several subproblems in which the solution patterns are qualitatively same. Finally, it is shown that the planar MHD Riemann problem is well-posed using intermediate shocks that have been considered non-evolutionary.

A procedure has previously been developed for the iterative construction of invariants associated with magnetic field-line Hamiltonians that consist of an axisymmetric zeroth-order term plus a non-axisymmetric perturbation. Approximate field-line invariants obtained with this procedure are used to examine the topological properties of magnetic field lines in a parabolic-current MHD equilibrium that was slightly perturbed from axisymmetry in the limit of a periodic cylindrical configuration. Excellent agreement between Poincaré maps and the level curves of the first-order invariant is found for small perturbations. A means of circumventing the ‘small-divisor problem’ in some cases is identified and implemented with outstanding results. These results indicate that this perturbation method can have valuable consequences for future investigations of magnetic field-line topology.

Analytic results for the time dependences of the kinetic and magnetic energies of an incompressible magneto fluid threaded by a strong uniform magnetic field B0 are obtained. The governing equations are the linearised magnetohydrodynamic (MHD) ones, but with the conventional Laplacian dissipation replaced by ion parallel viscous effects. The behaviour is shown to depend on the relative sizes of the Alfvén frequency and the viscous and resistive dissipation rates. For many cases equipartition of the kinetic and magnetic energy holds at the (Fourier) modal level. An important exception to this behaviour occurs for two-dimensional fluctuations, that is when the wavevectors are perpendicular to B0.