Comprehensive comments on the theory of stimulated scattering instabilities of high-frequency electromagnetic waves in magnetized plasmas are presented. It is shown that our general dispersion relations are appropriate for deducing valuable information regarding the growth rates of scattering instabilities and the long-term evolution of modulationally unstable waves in space and laboratory plasmas as well as in astrophysical settings.

It is shown that the magnetohydrodynamic (MHD) equilibrium states of an axisymmetric toroidal plasma with finite resistivity and flows parallel to the magnetic field are governed by a second-order partial differential equation for the poloidal magnetic flux function ψ coupled with a Bernoulli-type equation for the plasma density (which are identical in form to the corresponding ideal MHD equilibrium equations) along with the relation Δ*ψ = Vcσ (here Δ* is the Grad–Schlüter–Shafranov operator, σ is the conductivity and Vc is the constant toroidal-loop voltage divided by 2π). In particular, for incompressible flows, the above-mentioned partial differential equation becomes elliptic and decouples from the Bernoulli equation [H. Tasso and G. N. Throumoulopoulos, Phys. Plasma5, 2378 (1998)]. For a conductivity of the form σ = σ(R, ψ) (where R is the distance from the axis of symmetry), several classes of analytic equilibria with incompressible flows can be constructed having qualitatively plausible σ profiles, i.e. profiles with σ taking a maximum value close to the magnetic axis and a minimum value on the plasma surface. For σ = σ(ψ), consideration of the relation Δ*ψ = Vc σ(ψ) in the vicinity of the magnetic axis leads then to a proof of the non-existence of either compressible or incompressible equilibria. This result can be extended to the more general case of non-parallel flows lying within the magnetic surfaces.

A class of Hamiltonian models is considered that both includes many important examples from fluid and plasma theory and admits an elementary and simple abstract formulation. The formalism is illustrated by explicit expressions for both incompressible and compressible ideal magnetohydrodynamics, including a Coriolis force term and, in the incompressible case, allowing for a sharp ideally conducting boundary. The small-amplitude expansion is studied for a general background state, and the abstract setting gives much control of the usually messy algebra. One result is a new general and surprisingly simple symmetric expression for the coupling strength in the resonant three-wave interaction process. It may provide a convenient starting point for wave coupling calculations and seems suitable for the use in conjunction with computer algebra when various concrete models are considered.

A general form of the one-dimensional nonlinear Schrödinger equation is derived; the coefficients depend mainly on the linear dispersion relation, as does the nonlinear term. Envelope solitons may only exist together with some plasma instabilities. An example of solar radio bursts is selected as evidence for envelope solitons in the electromagnetic waveband.

It is well known that ion-beam–plasma interactions can destabilize right- and left-hand polarized electromagnetic waves. Owing to the fact that these instabilities have mostly been studied numerically by solving the hot-plasma dispersion relation, their fluid nature has often gone unnoticed. Choosing the ion background to be the rest frame, it is shown that the right-hand polarized instabilities are the result of a merging of the magnetosonic/electron-cyclotron branch of the dispersion relation with the ion beam. For any given ion-beam density and sufficiently large beam velocity, there are always two right- and two left-hand polarized instabilities leading to forward-propagating electromagnetic waves. It is also shown that all right-hand polarized instabilities are resonant instabilities, satisfying ω−kU+Ωp ≈ 0 around their maximum growth rate (ω and k are the frequency and the wavenumber respectively, U is the beam velocity, and Ωp is the proton gyrofrequency). Likewise, when the two left-hand instabilities are simultaneously present, they are also resonant instabilities satisfying ω ≈ Ωp. The high-frequency right-hand resonant instability (ω [Gt ] Ωp) has a maximum growth rate that depends only on the ratio between the beam density and the total density. The range of the unstable spectrum decreases with increasing beam velocity, leading to highly monochromatic radiation.

A three-dimensional study of the pump depletion and self-focusing (hot spot) effect on stimulated Raman scattering (SRS) reflectivity is presented within the paraxial-ray approximation. The reflectivity of SRS is observed to exhibit a maximum value in the case of optimum SRS gain with pump depletion. The reflectivity with pump depletion is less than without pump depletion and is the usual three-wave interaction case. The effect of plasma temperature on SRS reflectivity is also discussed.

Nonlinear electrostatic waves in self-gravitating dusty plasmas are considered in two limiting cases, according to whether the charged-particle dynamics is governed mostly by electrostatic forces or mostly by gravitation. This shows a significant difference between these two plasma media with respect to the envelope dynamics in the nonlinear regime. In the former case, when ω2pα > ω2Jα, the amplitude perturbations are longitudinally unstable only in the short-wave range, and the nonlinear effects can result in the formation of longitudinal dust-acoustic solitary waves. But even weak self-gravitational effects can lead to the existence of a long-wavelength range, where self-gravitation prevents the formation of dust-acoustic solitons, and only transverse solitary structures are possible. In the other limiting case (ω2pα < ω2Jα), there is always a transverse modulational instability, which can lead to transverse solitary waves. In both cases, there is a threshold for solitary-wave formation.

The issue of dynamical anisotropy in helical three-dimensional magnetohydrodynamic turbulence with a mean magnetic field B0 is investigated. Using high-resolution direct numerical simulations, we follow the evolution of various isotropic initial states characterized by their different values of the kinetic helicity. The cross helicity and magnetic helicity of the initial conditions are also varied. In agreement with earlier work, we find that such initial states become anisotropic in of order an eddy-turnover time, with correlation lengths parallel to B0 remaining largely unchanged while finer scales are excited in the perpendicular directions. Moreover, it is found that the development of both the anisotropy and the energy are essentially independent of the initial level of kinetic helicity. The physics associated with this latter feature is discussed.

Plasma screening effects on inelastic Compton scattering of photons by bound atomic electrons of hydrogenic target ions in weakly coupled plasmas are investigated. The particle interaction potential in weakly coupled plasmas is obtained using the Debye–Hückel model. The screened wave functions and energy eigenvalues for the ground and excited states of the target ion are obtained using the Ritz variational method. The expression for the lowest-order transition matrix element is obtained from a two-photon process associated with terms quadratic in the vector potential A. The inelastic Compton scattering cross-section from the 1s ground state to the 2p excited state is obtained as a function of the incident photon energy, including plasma screening effects. It is found that plasma screening effects significantly reduce the inelastic Compton scattering cross-section.

An approximate analytical expression is obtained for the orbits of a charged particle moving in a cusp magnetic field. The particle orbits pass close to or through a region of zero magnetic field before being reflected in regions where the magnetic field is strong. Comparison with numerically evaluated orbits shows that the analytical formula is surprisingly good and captures all the main features of the particle motion. A map describing the long-time behaviour of such orbits is obtained.

The MHD analogue of the Brunt/Väisälä frequency, NB, in a magnetized, ideally conducting plasma is obtained, with the vertical component of the magnetic field, Br, taken into account. The magnetic field vector (Br, Bθ, Bϕ) is assumed to satisfy the condition B·∇B ≈ BrdB/dr, which holds in many cases of interest. The frequency NB happens to depend, generally speaking, on the magnetic field orientation relative to the direction of gravity. However, for an isentropic gas, the convective instability criterion is governed by the magnetic field strength (rather than by the orientation of B). In general, the magnetic field has a stabilizing (destabilizing) effect if B/ρ grows (decreases) along the vertical axis r. This conclusion seems not to depend on the specific magnetic field configuration.

Certain unresolved ambiguities surround pressure determinations for incompressible flows: both Navier–Stokes and magnetohydrodynamic (MHD). For uniform-density fluids with standard Newtonian viscous terms, taking the divergence of the equation of motion leaves a Poisson equation for the pressure to be solved. But Poisson equations require boundary conditions. For the case of rectangular periodic boundary conditions, pressures determined in this way are unambiguous. But in the presence of ‘no-slip’ rigid walls, the equation of motion can be used to infer both Dirichlet and Neumann boundary conditions on the pressure P, and thus amounts to an over-determination. This has occasionally been recognized as a problem, and numerical treatments of wallbounded shear flows usually have built in some relatively ad hoc dynamical recipe for dealing with it – often one that appears to ‘work’ satisfactorily. Here we consider a class of solenoidal velocity fields that vanish at no-slip walls, have all spatial derivatives, but are simple enough that explicit analytical solutions for P can be given. Satisfying the two boundary conditions separately gives two pressures, a ‘Neumann pressure’ and a ‘Dirichlet pressure’, which differ nontrivially at the initial instant, even before any dynamics are implemented. We compare the two pressures, and find that, in particular, they lead to different volume forces near the walls. This suggests a reconsideration of no-slip boundary conditions, in which the vanishing of the tangential velocity at a no-slip wall is replaced by a local wall-friction term in the equation of motion.

A complex stimulated Raman scattering event against a background of non-thermal ion acoustic waves in an inhomogeneous plasma is described. We obtain analytic forms for the Raman gain due to a five-wave interaction consisting of conventional three-wave Raman scattering followed by the decay of the Raman Langmuir wave into a second Langmuir wave (or a second scattered light wave) and an ion acoustic wave. Very modest levels of ion waves produce a significant effect on Raman convective gain. A combination of plasma inhomogeneity and suprathermal ion fluctuations may offer a means for the control of Raman gain.

A Monte Carlo method is used to simulate ion transport in a radiofrequency parallel-plate glow discharge with a negative-voltage pulse connected to the electrode. The dust generated during the discharge and a self-consistent electric field are taken into account. Charge exchange and elastic collisions between ions and neutral atoms and the collection and Coulomb scattering of ions on the dust particles are examined during the motion of ions in the sheath. It is found that self-consistent field, neutral gas pressure, and dust charge, dust concentration and dust size influence, to varying degrees, the energy distribution and density of ions arriving at the target, and in particular, the dust charge and concentration have significant influences. As they increase, the number of ions arriving at the target is greatly reduced. In summary, although the dust content is very low (of order 10−3) in most plasma processing devices, its influence cannot be neglected, because of the large dust size and charge (in particular, the former). Therefore, in order to produce good results with ion implantation, coating, stripping, and etching, it is necessary to reduce the dust content and control the size of dust particles.

Plasma properties during and immediately after electrical breakdown in a high-pressure gas are investigated. A simple scaling law of electrical discharge at high pressure is obtained. Introducing the normalized net ionization rate ξ, the electron temperature at breakdown is described uniquely in terms of the ionization properties of the gas, the second ionization coefficient γ at the cathode, and the parameter pd, where p is the gas pressure and d is the system dimension. The electron attachment process plays a decisive role in the breakdown phenomenon for a high-pressure gas, whereas it is not important in low-pressure discharges. An analytical expression for the high-pressure plasma density is obtained by making use of the electron rate equation. A simple analytical expression for plasma generation in a high-pressure gas provides important scaling laws in the d.c. electrical discharge system, where the electron attachment process is negligible. It is found that the logarithm of the electron density at breakdown is proportional to the discharge time τb, and is inversely proportional to the pulse risetime and the gap distance d between the two electrodes. The plasma density at breakdown is also an increasing function of the gas ionization energy εi.

The well-known Chirikov–Taylor standard map is studied using the methods of non-equilibrium statistical mechanics. It appears possible to simplify the master equation whenever the stochasticity parameter is small and the initial density profile is sharply localized in phase space: this defines the ‘localized weak-stochasticity’ (LWS) regime. The resulting equation of evolution of the density profile involves a convolution in time. Its most conspicuous feature is the absence of a ‘short’ intrinsic time scale: the memory time is infinite. The master equation can therefore not be Markovianized as in usual kinetic theory (or as in the diffusive regime of the standard map). Moreover, the initial value of the fluctuations influences the rate of change of the density profile at all times. Quite unexpectedly, the contributions of the two terms of the master equation to the density profile are of the same order of magnitude. As a result, they practically cancel each other. It is this curious feature that explains the suppression of the motion by the islands and the KAM barriers that is well known from the study of individual standard map orbits. The LWS approximation is valid for a limited time (which is very long when the initial localization is very sharp).

It is shown that using Padé approximants in the evaluation of the plasma dispersion function Z for a Maxwellian plasma is equivalent to the exact treatment for a plasma described by a ‘simple-pole distribution’, i.e. a distribution function that is a sum of simple poles in the complex velocity plane (v plane). In general, such a distribution function will have several zeros on the real v axis, and negative values in some ranges of v. This is shown to be true for the Padé approximant of Z commonly used in numerical packages such as WHAMP. The realization that an approximation of Z is equivalent to an approximation of f(v) leads the way to the study of more general distribution functions, and we compare the distribution corresponding to the Padé approximant used in WHAMP with a strictly positive and monotonically decreasing approximation of a Maxwellian.

Nonlinear waves and one-dimensional solitons of the Zakharov–Kuznetsov equation are unstable in two dimensions. Although the wavevector K of a perturbation leading to an instability covers a whole region in (Kx, Ky) parameter space, two classes are of particular interest. One corresponds to the perpendicular, Benjamin–Feir instability (Kx = 0). The second is the wave-length-doubling instability. These two are the only purely growing modes. We concentrate on them. Both analytical and numerical methods for calculating growth rates are employed and results compared. Once a nonlinear wave or soliton breaks up owing to one of these instabilities, an array of cylindrical and/or spherical solitons can emerge. We investigate the interaction of these entities numerically.