The equations of resistive magnetohydrodynamics taking into account the Hall effect are solved analytically for the case of a planar plasma flow across the magnetic field. Waves of purely acoustic nature are revealed that propagate in the plasma medium on the background of a steady magnetically accelerated plasma flow of a rather arbitrary form. The magnetic field manifests itself in this process only in that it produces an effective gravity force, the ‘gravitational’ acceleration being proportional to ωeτe. Like acoustic–gravity waves in the atmosphere, such quasi- acoustic–gravity (QAG) waves in a plasma can increase greatly in amplitude as they propagate ‘upwards’, i.e., in this case, counter to the direction of the electric field. In a plasma channel (where the plasma flows between the electrodes), a system of standing QAG waves arises, which has a stationary component. These waves are responsible for a number of distinctive features in the behaviour of Hall plasma flows, in particular for very large gradients and unexpected small-scale fluctuations of physical quantities near the anode, revealed earlier by numerical simulation. In the cases where the plasma-flow parameters provide a high intensity of these fluctuations, the flow appears to be unstable. In another range of parameter values, the QAG waves are strongly coupled with the electric currents that they induce in the electrodes, their amplitudes increase with time, and, again, the flow cannot be steady. The existence of a rather general dimensionless similarity criterion for resistive Hall plasma planar flows is also shown. This criterion can be found directly from the structure of the equations without any restrictions on the plasma parameters or on the form of the flow.

Fluid modelling of an edge plasma is usually performed using a finite-difference scheme on a fixed structured grid. However, both experimental measurement and numerical simulation show the presence of front-like regions characterized by sharp variations of the main plasma parameters such as temperature and radiation power. This is caused in part (i) by strong nonlinearities in the fluid equation coefficients due to abrupt changes of various plasma reaction rates as a function of temperature and (ii) by high anisotropy of the plasma transport along and across magnetic field lines. Manual mesh adoption is usually applied to allow better resolution of the regions with sharp gradients. However, such an approach is very time-consuming and limited. To overcome this problem, we propose to use adaptive unstructured meshes constructed with a new quasi-one-dimensional adaption algorithm. This approach is fast and conservative because we use a new finite-volume scheme. The price of adaptation is high, because numerical algorithms became much more complicated. To avoid unwanted complexity, we suggest an alternative use of a grid-free method, which requires no connectivity of arbitrarily placed vertices. To benchmark the methods and codes in two dimensions, we find analytical and semi-analytical solutions of the nonlinear diffusion–radiation equation, which may have sharp fronts, unconnected boundaries and bifurcated solutions. We use these solutions to study the efficiency of the proposed numerical algorithms.

Both compressive and rarefactive solitons are found to exist in a warm adiabatic dusty plasma with two-temperature isothermal ions. The Korteweg–de Vries (KdV) equation for isothermal ions and KdV-type equations at a critical density of low-temperature ions are derived in order to discuss the properties of dust acoustic solitons. Using a quasipotential analysis, critical Mach numbers M1c and M2c are obtained such that rarefactive dust acoustic solitons exist when 1<M<M1c and compressive dust acoustic solitons exist when 1<M<M2c.

Analytical expressions for the lowest-order distribution functions of a plasma that are valid throughout the plasma are generally not easily obtainable when conditions shift from collision dominance in some regions of the plasma to absence of collisions in other regions. Indeed, the reason for the breakdown of classical approximation methods for distribution functions and the corresponding macroscopic equations in plasma kinetic theory is often the lack of such global lowest-order distribution functions. The main purpose of the present paper is to derive an approximate global distribution function for an outflowing dilute light ion component in a polar-region model of the ionosphere where conditions for the light ions shift from collision dominance in low regions to collisionless conditions high up. For this, a simplified collision integral of a form given by Chandrasekhar will be used. The proposed global solution is given after simpler solutions, as a collisionless solution, an exact collision-dominated solution and a solution corresponding to classical transport theory are derived, for reason of comparison, in the first parts of the paper.

The structure of magnetosonic perturbations around a pair of X lines of a toroidal magnetic configuration is analysed in the framework of ideal MHD, by means of a WKB technique. The conditions for wave trapping are investigated. It is shown that the wave equation for the displacement is separable in the vicinity of the X lines. The relevant eigenvalues and the different types of ray trajectories are discussed.

The nonlinear interaction between four electrostatic waves in a plasma media is investigated analytically in the presence of linear damping or growth and also frequency mismatch, using a nonlinear perturbation method. Depending on the various initial conditions of the wave amplitudes, solutions of different types are discussed.