It is shown that the classical magnetohydrostatic equations of an infinitely conducting fluid reduce to the integrable potential Heisenberg spin equation subject to a Jacobian condition if the magnitude of the magnetic field is constant along individual magnetic field lines. Any solution of the constrained potential Heisenberg spin equation gives rise to a multiplicity of magnetohydrostatic equilibria which share the magnetic field line geometry. The multiplicity of equilibria is reflected by the local arbitrariness of the total pressure profile. A connection with the classical Da Rios equations is exploited to establish the existence of associated helically and rotationally symmetric equilibria. As an illustration, Palumbo's ‘unique’ toroidal isodynamic equilibrium is retrieved.

The linear stability of drift waves in a poloidally rotating tokamak plasma is considered. The derived dispersion relation features a peaking of the diamagnetic frequency which gives the drift modes an irreducible two-dimensional character. We then show that inverse Landau damping can be suppressed and even stabilized, if the flow's shear is strong. Even though the instability, excited by the Landau resonance, is stronger at a high velocity shear for positive rotation velocities, effects due to the rotation of the plasma can reverse the sign and induce damping of the two-dimensional drift modes. This stabilizing mechanism works only for positive rotation velocities. For negative rotation velocities, we show that only modes with high poloidal mode numbers are unstable.

Energy transformation between fast magnetosonic waves and Alfvén waves is considered. It is shown that the efficiency of this process can be significantly enhanced if the Alfvén wave propagation directions are appropriately chosen. The relevance of our investigation to the non-linear excitation of magnetohydrodynamic waves in space plasmas is discussed.

We have developed a hybrid magnetohydrodynamics (MHD) –kinetic box model valid for standing shear Alfvén waves using the cold plasma MHD equations coupled to a system of kinetic electrons. The guiding centre equations are used for the motion of the electrons and the system is closed via an expression for the field-aligned electric field in terms of the perpendicular electric field and moments of the electron distribution function. The perpendicular electric fields are derived from the ideal MHD approximation. We outline the basic model equations and method of solution. Simulations are then presented comparing the hybrid model results with a cold plasma MHD model. Landau damping is shown to heavily damp the standing shear Alfvén wave in the hybrid simulations when $v_{th} \ge V_{A}$. The damping rate is shown to be in good agreement with the theoretical rate calculated for the model parameters.

We study the dynamics of phase space holes in a non-collisional plasma by numerical integration of the one-dimensional Vlasov equation. The plasma is bounded and connected to vacuum by two interfaces, which gives rise to non-periodic boundary conditions. By choosing different initial conditions, we consider four different situations: development of a sinusoidal wave, two-stream instability, development of an ab initio Bernstein–Greene–Kruskal state and the expansion of a plasma into vacuum. For the latter situation ions move with a realistic mass of 1837 electron masses. The most prominent results of our investigation are: (a) a clockwise revolution of holes in phase space; (b) the appearance of stable oscillations of the total electron momentum, which, despite their striking linearity, cannot be related to electrostatic plasma oscillations. To the best of our knowledge none of these phenomena have been reported in kinetic simulations before: indeed we prove that the non-periodicity of the boundary conditions is an essential requirement for the sustainment of both hole revolution and momentum oscillations.

The non-linear dynamic behaviour of two three-wave systems in plasma with two waves in common has been studied, including the possibility of negative energy waves and also the effect of linear damping or growth and frequency mismatch. Depending on the various initial conditions solutions of different types have been discussed. It has also been shown that one of the triplets can be stabilized by the other one against explosive instability depending on the relative strength of the coupling factor.

It is well known that energetic protons having a ring-type, anisotropic velocity distribution can generate low-frequency electromagnetic waves, propagating nearly transverse to the geomagnetic field lines, in the ring current region of the Earth's magnetosphere by exciting mode 1 and mode 2 non-resonant instabilities and a resonant instability. In this paper, detailed investigations of the resonant and non-resonant instabilities are conducted as a function of plasma parameters such as ring speeds, electron temperatures, magnetic field strength and wave propagation directions. Our analysis is for parameters corresponding to the ring current region of the Earth's magnetosphere.

Low aspect ratio tokamak (LART) configurations represent a promising approach to thermonuclear fusion. They require auxiliary non-ohmic power for heating, current drive and turbulent transport suppression. This paper pioneers the pre-requisite for the quantitative evaluation of these effects, namely the formulation and solution of the full wave equation for pre-heated (i.e. prior to the addition of any auxiliary heating or current drive) LARTs, the case in which mode-converted Alfvèn waves ($\omega < {\omega}_{ci}$) are used as an additional non-ohmic power source. Arbitrary aspect ratio and magnetic shear tokamaks with non-circular cross-section are properly considered; this general approach includes, as a particular case, that of low aspect ratio (spherical) tokamaks.

The problem is formulated in terms of the vector and scalar potentials (${\bf A}, \Phi$), thus avoiding the numerical pollution occurring in the case of the (${\bf E},{\bf B}$) formulation. Adequate boundary conditions at the vacuum–metallic wall interface and regularity conditions at the magnetic axis are enforced.

For the pre-heated stage, a quite general two-fluid, resistive dielectric tensor operator able to describe the anisotropic plasma response in spherical tokamaks is derived and used; except for its linear character, no geometrical limitations are imposed on it. Consistent equilibrium profiles also including effects of neoclassical conductivity and bootstrap current are used (Wilson 1994).

The wave equations are solved with the aid of a suitable $2\frac{1}{2}{\rm D}$ finite-element algorithm, taking advantage of the up-down symmetry of the problem. Illustrative solutions obtained for Culham's START device are presented and discussed; this includes a sensitivity study to the antenna parameters (geometry, frequency and wavenumbers). Finally, as one of the possible applications of these results, illustrative ensuing ponderomotive forces and the non-inductive current drive in a START-like device are calculated.

A fully nonlinear theory for stationary whistler waves propagating parallel to the ambient magnetic field in a cold plasma has been developed. It is shown that in the wave frame proton dynamics must be included in a self-consistent manner. The complete system of nonlinear equations can be reduced to two coupled differential equations for the transverse electron or proton speed and its phase, and these possess a phase-portrait integral which provides the main features of the dynamics of the system. Exact analytical solutions are found in the approximation of ‘small’ (but nonlinear) amplitudes. A soliton-type solution with a core filled by smaller-scale oscillations (called ‘oscillitons’) is found. The dependence of the soliton amplitude on the Alfvén Mach number, and the critical soliton strength above which smooth soliton solutions cannot be constructed is also found. Another interesting class of solutions consisting of a sequence of wave packets exists and is invoked to explain observations of coherent wave emissions (e.g. ‘lion roars’) in space plasmas. Oscillitons and periodic wave packets propagating obliquely to the magnetic field also exist although in this case the system becomes much more complicated, being described by four coupled differential equations for the amplitudes and phases of the transverse motion of the electrons and protons.

In 1989, it was proved that thunder is not caused by the thermal expansion of the lightning channel. The present paper suggests that it is the result of an air explosion driven by the liberation of chemical bond energy from the N2 and O2 molecules. This has been confirmed by laboratory experiments with air arcs of lightning strength. These arcs are short pieces of lightning strokes and their formation is accompanied by thunder-like sonic booms.

The sawtooth activity in an ohmically heated plasma on the HT-7 tokamak has been investigated experimentally using soft x-ray (SXR) diode arrays, magnetic probes and other diagnostics. Its behaviour and occurrence are correlated closely to the discharge conditions such as the electron density $N_{e}$, the electron temperature $T_{e}$, the safety factor $q_a$ on the plasma boundary and the wall condition, etc. Generally speaking, as $N_{e}$ increases, the sawtooth activity grows, becoming stronger with a longer period $T_{s}$, a larger inversion radius $r_{s}$ and a larger amplitude fluctuation $\Delta I_{sx}/\bar{I}_{sx}$ on the SXR radiation intensity signal, and evolve from a small to a moderate sawtooth, to a large or even a giant sawtooth. Sometimes, a double sawtooth and a monster sawtooth activity emerge without apparent deterioration of the plasma confinement and without major plasma disruption. During the events, abundant magnetohydrodynamic phenomena are observed, including a long partial sawtooth collapse. In this paper, the general sawtooth behaviour and the dependence of the occurrence of the sawtooth instability on the electron density $N_{e}$ and wall condition have been summarized. Experimental properties of two types of double sawtooth and monster sawtooth activities are also presented and discussed. The set up of the HT-7 tokamak and the main diagnostics available are also briefly described.

Probe theory is generally used to find the potential of dust particles immersed in plasma. The orbital motion limited theory (OML) is often used to find the potential at the probe surface, but the assumptions underlying this theory are usually not valid in the case of dust and the more general orbital motion (OM) theory is much harder to calculate. Solutions are given for the OM theory in a range of cases applicable to dust. It is shown that the surface potential the full theory gives reduces to the OML result for small probes. Commonly in dusty plasmas the OML surface potential is used, with the surrounding distribution given by Debye–Hückel, or Yukawa theory. This form, however, neglects ion depletion due to the absorption of particles on the probe surface. In this paper a new analytical solution to the system is given which is applicable to small probes and dust. This new expression is equivalent to Yukawa form, but takes ion absorption into account.

Fluctuations in ambient shear flow exhibit interesting transient phenomena. Shear flow produces not only Kelvin–Helmholtz modes (global exponential instabilities represented by point spectra) but also local algebraic instabilities associated with multiple continuous spectra. Since the generating operator is non-Hermitian, the orthogonality of eigenmodes is broken, and unresolvable mode couplings (resonances) bring about secular behavior (algebraic instability). We analyze electrostatic fluctuations in a magnetized non-neutral (single species) plasma where the electrostatic potential parallels the stream function. This secular behavior is reproduced by solving the initial value problem with a renormalization method.

We investigate the conditions under which ‘high-frequency’ electron acoustic–Langmuir solitons can be constructed in a plasma consisting of protons and two electron populations: one ‘cold’ and the other ‘hot’. Conservation of total momentum can be cast as a structure equation either for the ‘cold’ or ‘hot’ electron flow speed in a stationary wave using the Bernoulli energy equations for each species. The linearized version of the governing equations gives the dispersion equation for the stationary waves of the system, from which follows the necessary – but not sufficient – conditions for the existence of soliton structures; namely that the wave speed must be less than the acoustic speed of the ‘hot’ electron component and greater than the low-frequency compound acoustic speed of the two electron populations. In this wave speed regime linear waves are ‘evanescent’, giving rise to the exponential growth or decay, which readily can give rise to non-linear effects that may balance dispersion and allow soliton formation. In general the ‘hot’ component must be more abundant than the ‘cold’ one and the wave is characterized by a compression of the ‘cold’ component and an expansion in the ‘hot’ component necessitating a potential dip. Both components are driven towards their sonic points; the ‘cold’ from above and the ‘hot’ from below. It is this transonic feature which limits the amplitude of the soliton. If the ‘hot’ component is not sufficiently abundant the window for soliton formation shrinks to a narrow speed regime which is quasi-transonic relative to the ‘hot’ electron acoustic speed, and it is shown that smooth solitons cannot be constructed. In the special case of a very cold electron population (i.e. ‘highly supersonic’) and the other population being very hot (i.e. ‘highly subsonic’) with adiabatic index 2, the structure equation simplifies and can be integrated in terms of elementary transcendental functions that provide the fully non-linear counterpart to the weakly non-linear sech$^{2}$-type solitons. In this case the limiting soliton is comprised of an infinite compression in the cold component, a weak rarefaction in the ‘hot’ electrons and a modest potential dip.

A new model of ball lightning is proposed. The main model assumption is that ball lightning has a core consisting of clouds of electrons and totally ionized ions which oscillate with respect to each other. According to the model, ball lightning emits high energy photons that are sometimes dangerous for human beings, and in a number of situations it can kill humans by electric pulses; the ball lightning energy can be of the order of 106 J and even greater. The electric charges that need to be injected into the atmosphere to create ball lightning and the currents, providing the injection of such charges, are estimated. These estimates predict that ball lightning can be created in the experiments with ordinary lightning or powerful electrical installations.

A plasma formed with a low field (1 V cm$^{-1}$) at low temperatures (e.g. $\approx{10}^3$ K) from atomic hydrogen generated at a tungsten filament and strontium which was vaporized by heating the metal. Strong extreme ultraviolet emission was observed, but not when sodium, magnesium or barium replaced strontium or with hydrogen or strontium alone. Characteristic emission was observed which supported a resonant energy-transfer mechanism. Significant Balmer $\alpha$ line broadening corresponding to an average hydrogen atom temperature of 23–45 eV was observed for glow discharges of strontium–hydrogen, helium–hydrogen, argon–hydrogen, strontium–helium–hydrogen and strontium–argon–hydrogen compared with $\approx$3 eV for pure hydrogen, krypton–hydrogen, xenon–hydrogen and magnesium–hydrogen. To achieve the same optically measured light output power, hydrogen–sodium, hydrogen–magnesium and hydrogen–barium mixtures required 4000, 7000 and 6500 times the power of the hydrogen–strontium mixture, respectively. A glow discharge plasma formed for hydrogen–strontium mixtures at an extremely low voltage of about 2 V compared with 250 V for hydrogen alone and sodium–hydrogen mixtures, and 140–150 V for hydrogen–magnesium and hydrogen–barium mixtures.

This paper presents the cross-focusing of two intense laser beams in a collisionless plasma, taking into account the relativistic non-linearity. The non-linearity is not bound to large irradiances and this non-linearity is only a perturbation. It should be noted here that while considering the self-focusing due to relativistic electron mass variation, the electron ponderomotive density depression in the channel may also be important. Therefore, these two non-linearities may simultaneously affect the self-focusing process. In the present paper we have considered the situation when only relativistic non-linearity is important. The non-linearity due to relativistic mass variation depends not only on the intensity of one laser but also on the second laser. Therefore, one laser beam affects the dynamics of the second beam and hence a cross-focusing process takes place. The electric field amplitude of the excited electron plasma wave (EPW) has been calculated and its effect on the cross-focusing process has also been discussed. It is observed that the inclusion of a resonantly excited EPW on cross-focusing is significant and the accelerating electric field of the generated EPW becomes affected. A comparison of the theory with the recent experimental observations has also been presented.

A path integral method is applied to a statistical analysis of electron transport described as a Langevin equation in a disturbed magnetic field line structure; in particular, the transition probability of electrons strongly tied to field lines is considered. When the effect of the Coulomb collisions is interpreted as the Gaussian white noise in configuration space, the radial transport of electrons in the chaotic field line structure is different from Brownian diffusion with the diffusion coefficient of field lines, even if a sufficiently small diffusion coefficient of the collisions is considered.

In this paper the general theory developed by Vladimirov et al. is extended to nonlinear (Lyapunov) stability for axisymmetric (invariant under rotations around a fixed axis) solutions of the ideal incompressible magnetohydrodynamic flows for a particular situation, namely arbitrary field and poloidal flow. The appropriate norm is a sum of magnetic and kinetic energies and the mean square vector potential of the magnetic field.

Lagrangian and Hamiltonian formalisms for ideal fluids and plasmas have, during the last few decades, developed much in theory and applications. The recent formulation of ideal fluid/plasma dynamics in terms of the Euler–Poincaré equations makes a self-contained, but mathematically elementary, form of Lagrange–Hamilton theory possible. The starting point is Hamilton's principle. The main goal is to present Lagrange–Hamilton theory in a way that simplifies its applications within usual fluid and plasma theory so that we can use standard vector analysis and standard Eulerian fluid variables. The formalisms of differential geometry, Lie group theory and dual spaces are avoided and so is the use of Lagrangian fluid variables or Clebsch potentials. In the formal ‘axiomatic’ setting of Lagrange–Hamilton theory the concepts of Lie algebra and Hilbert space are used, but only in an elementary way. The formalism is manifestly non-canonical, but the analogy with usual classical mechanics is striking. The Lie derivative is a most convenient tool when the abstract Lagrange–Hamilton formalism is applied to concrete fluid/plasma models. This directional/dynamical derivative is usually defined within differential geometry. However, following the goals of this paper, we choose to define Lie derivatives within standard vector analysis instead (in terms of the directional field and the div, grad and curl operators). Basic identities for the Lie derivatives, necessary for using them effectively in vector calculus and Lagrange–Hamilton theory, are included. Various dynamical invariants, valid for classes of fluid and plasma models (including both compressible and incompressible ideal magnetohydrodynamics), are given simple and straightforward derivations thanks to the Lie derivative calculus. We also consider non-canonical Poisson brackets and derive, in particular, an explicit result for incompressible and inhomogeneous flows.