Multiple–scales perturbation methods are used to derive equations describing wave–wave interactions in two-fluid cosmic-ray hydrodynamics in one Cartesian space dimension. Two problems are considered: (a) the interaction between short-wavelength waves in a non-uniform large-scale background flow, and (b) wave interactions between long-wavelength waves propagating through a uniform background medium. The short-wavelength wave equations describe the interaction between the backward and forward thermal gas sound waves and the contact discontinuity eigenmode via ‘wave mixing’, in which the waves are reflected by gradients in the large scale background flow. The equations also contain quadratic wave interaction terms describing (i) the Burgers self-wave interaction term, (ii) mean–wave-field interaction terms in which a specific wave interacts with the mean field of the other waves, and (iii) three-wave resonant interactions that describe how a sound wave resonantly interacts with the contact discontinuity to generate a reverse sound wave. In the limit of no interaction between the cosmic rays and thermal gas, and for a uniform background state, the equations reduce to two coupled, integro-differential Burgers equations derived by Majda and Rosales for resonantly interacting waves in adiabatic gasdynamics. The short-wavelength equations also contain squeezing instability terms associated with the large-scale cosmic-ray pressure gradient, which were first investigated by Drury and Dorfi. A generalized wave-action equation, or canonical wave-energy equation, variational principles, and WKB analyses of the linearized equations are used to investigate the modification of the cosmic-ray squeezing instability by wave mixing. Coupled Burgers equations are also derived in the long-wavelength regime that describe resonant wave interactions for weak, diffusively smoothed cosmic-ray-modified shocks. In the latter equations, the cosmic-ray-modified sound waves can resonantly interact with either the contact discontinuity or the cosmic-ray pressure-balance mode to generate a reverse sound wave.

The effects of the propagation angle of electromagnetic waves and the pitch angle of non-thermal electrons on the electron cyclotron maser instability and beam–plasma instability are compared. It is pointed out if the direction of propagation of electromagnetic waves is opposite to the direction of the injected electrons, the maser instability will be suppressed, and hence it cannot explain the radiation with narrow bandwidth excited by energetic electrons propagating downwards in solar flares, such as millisecond spikes, blips and the type III bursts with positive-frequency drift. This discrepancy may be solved by the non-resonant wave–particle interaction in the beam–plasma instability, in which the electromagnetic waves propagating in the same direction as the injected electron beam are mainly composed of the ordinary modes (left-circular polarization), while the electromagnetic waves propagating in the opposite direction to the beam are mainly composed of the extraordinary modes (right-circular polarization). This result is compared with a typical model of a magnetic tube in solar flares, in which the non-thermal electrons propagating upwards excite spikes, blips and type III bursts in the metre and decimetre bands, while the electron beams propagating downwards excite these temporal and spectral structures in the microwave bands.

The stopping power S=S1+SB for charged particles in non-ideal degenerate quantum plasmas is calculated with the help of the dielectric formalism in an approximation corresponding to taking account of the Barkas effect (the term SB; the term S1 corresponds to the Bethe formula). It is found that for a high projectile velocity vp>vF (where vF is the Fermi velocity) in non-ideal plasmas, SB∝e3pv−3p (where ep is the charge of the projectile), the well-known law for the Barkas effect, SB∝e3pv−5p being valid only for ideal plasmas. This relation explains a number of experiments on stopping of different charged particles (protons, muons and heavy ions) in metals without the introduction of fitted parameters. The possibility of extending of this theory to gaseous non-ideal plasmas arising in inertial-confinement fusion (ICF) experiments is also briefly discussed.

We describe the existence of an entropy for lattice gas systems of Fermi–Dirac type based on a generalized semi-detailed balance condition. We demonstrate some essential equilibrium and non-equilibrium fluctuation and dissipation properties, including the so-called Onsager's reciprocity relations, as a consequence of this condition. Requirements for the existence of certain statistical properties in discrete lattice gas systems are not directly inferred from those in real continuous systems, but are closely related. Hence understanding in detail the causes and results for lattice gas systems may provide further insights into fundamental microscopic physical processes.

It is proved that a system of three-wave coupling with weak damping and the same sign of the coupling coefficients can be completely described by the Weierstrass function. On the basis of the bifurcation theory of the Weierstrass function that we have developed, criteria are found for distinguishing three phases of the three-wave coupling, namely the stable phase and the excited explosive and auto-explosive phases. Two kinds of explosion mechanisms in the three-wave coupling are found and described. The mechanism for the excited explosive phase is normal because trigger energy is required in such a mechanism and the threshold λth is proportional to δ2 (δ is the frequency mismatch), whereas the anomalous mechanism for the auto-explosive phase does not need any trigger – only proper changes of internal parameters to make the absolute invariant J less than 27 are required. In addition, the repetition period of the explosive instability is given. Finally, the effects of weak damping on the bifurcation structure of the explosive instability in three-wave interactions are analysed.

The effects of large-scale sweeping velocity on the turbulent cascade to small scales are examined for two problems: the advection of a passive scalar by a multivariate-Gaussian velocity field and incompressible Alfvén-wave turbulence. In both cases, the sweeping produces anisotropy and reduces the strength of cascade. If the direction of the sweep velocity varies with time, a balance is reached between this anisotropy and isotropizing effects associated with the change of direction.

The theory of electron-beam interaction with Bernstein plasma modes is developed for conditions of active space plasma experiments as well as experiments in large vacuum chambers. Translational movement of the beam along a magnetic field and Larmor rotation of the beam are considered as sources of free energy for wave build-up. Part of the energy of translational movement is lost to the excitation of Langmuir plasma oscillations. Bernstein modes then slowly grow owing to a ‘fan’-type instability. In this case the emission spectrum consists of lines near the half-harmonics of the electron gyrofrequency, ≈(n+½)ωc. Larmor gyration of the beam is not destroyed by the excitation of Langmuir turbulence, and thus there is strong hydrodynamic interaction of the rotated beam with the Bernstein modes. Integer harmonics of gyrofrequency nωc are generated in this case, with the spectral maximum near the upper-hybrid plasma frequency.

A two-fluid model is established to study the effect of injection on the structure of cosmic-ray-modified shocks. The model comprises a thermal plasma and cosmic rays. The cosmic rays are considered to be a massless fluid with significant pressure. Its behaviour is governed by the cosmic-ray energy equation, which can be derived from the transport equation. A term in the equation responsible for injection is proportional to the negative divergence of the plasma velocity. The rest of the dependence has to be introduced phenomenologically. It is expected that the injection of cosmic rays increases if the thermal plasma energy density increases. To fix ideas, we assume that the injection is linear in the thermal pressure. By suitable redefinition of some quantities, this model can formally be recast into a model without injection, i.e. the injection is ‘transformed’ away mathematically. Hence the results are similar to those for the modified shocks without injection. Nonetheless, there are differences. The system with injection prefers a smooth shock transition. The reason is that the injection softens the equation of state of the plasma, and induces an extra cosmic-ray partial pressure. The dependence of the effficiency of shock acceleration on the injection is also discussed.

Y. Nomura et al., Stochasticity in the Josephson map [J. Plasma Phys. 56 (3), 493–506 (1996)]

The third author's name should be A. T. Filippov. We apologise for this error.

S. Li et al., Inverse cascades of angular momentum [J. Plasma Phys. 56 (3), 615–639 (1996)]

In Fig. 4 on p. 626, the upper figure shows the modal angular momenta L(0, n) as functions of time, while the lower figure shows the (positive-definite) modal kinetic energies E(m, n) as functions of time. Also, the upper figure should be labelled ‘(a)’ and the lower one ‘(b)’, in order to agree with the figure caption.

High-resolution, direct numerical simulations of three-dimensional incompressible Navier–Stokes equations are carried out to study the energy spectrum in the dissipation range. An energy spectrum of the form A(k/kd)α exp[−βk/kd] is confirmed. The possible values of the parameters α and β, as well as their dependence on Reynolds numbers and length scales, are investigated, showing good agreement with recent theoretical predictions. A ‘bottleneck’-type effect is reported at k/kd≈4, exhibiting a possible transition from near-dissipation to far-dissipation.

A novel Lagrangian integral formalism is applied to pure electron diodes. Analytical expressions are obtained in terms of the injected current and of the potential drop for two DC states. These two states are either completely transmitted flows or flows with partial reflection at a virtual cathode. Splitting rates and other quantities are presented for arbitrary potentials.

There is abundant experimental, theoretical and computational evidence that certain constrained turbulent fluid systems self-organize into large-scale structures. Examples include two-dimensional (geostrophic) fluids, guiding-centre plasmas and pure-electron plasmas, as well as two- and three-dimensional magnetofluids such as reversed-field pinches and spheromaks. The theoretical understanding of relaxation phenomena is divided into two quite different constructs: selective decay and maximal entropy. Theoretical foundations of both of these principles are largely due to Montgomery and his collaborators. In this paper, selective decay and maximal entropy theories of turbulent relaxation of fluids are reviewed and experimental evidence is presented. Experimental evidence from both 2D fluids and from 3D magnetofluids is consistent with the selective decay hypothesis. However, high-resolution computational evidence strongly suggests that formation of large-scale structures is dictated by maximal-entropy principles rather than selective decay.

Six ansatzes are investigated for their potential to allow three-dimensional (3D) ideal magnetohydrodynamic (MHD) equilibria. The ansatzes are based on a Clebsch representation for the magnetic field, B=∇H × ∇k, and a ‘generalized Clebsch representation’, B=∇×(∇K × ∇k), with ∇k being one of the coordinate directions of a cylindrical coordinate system. Three classes of equilibria, all with a straight magnetic axis, are obtained. Equilibria of the first class have a purely poloidal magnetic field of the Clebsch type with k=z and include the 3D equilibria already known. Equilibria of the other two classes have a purely toroidal (i.e. here longitudinal) magnetic field and pressure surfaces that can be chosen such that poloidal sections are closed. The second class is based on a Clebsch representation with k=θ. Solutions contain a free function of θ that determines the poloidal sections of the pressure surfaces at, say, z=0. The behaviour in the toroidal direction is then fixed but not periodic. For the third class, the generalized Clebsch representation with k=z is used. The equilibria are similar to those of the second class, with two important differences. They contain no free function and field lines are not planar. Finally, 3D vacuum fields, which exhibit 3D magnetic surfaces, are presented. They have the same geometry as the equilibria of the third class, and in fact can be obtained as a certain limit from these equilibria. Possible applications of the equilibria found are mentioned.

The magnetic configuration in the confinement zone of a tokamak admits a magnetic flux-bifurcation for critical values of the edge parameters and given injected power when the initial L-state is so defined as to be associated with a stationary magnetic entropy. Assuming the ITER89-P scaling of the energy confinement time in the L state, it is shown that the power is changing along the bifurcation threshold, for fixed values of the edge parameters, according to the law P=Cn0.77B0.97 (where n is the average density and B is the toroidal magnetic field), where the numerical value of the constant C follows from ITER89-P and is consistent with observations of the L–H transition. It is also shown that the observed independence of the threshold with respect to the current is a consequence of the fact that, at the threshold point, the ratio between the auxiliary power and the Ohmic power PΩ in the confinement region is constant when the current is changing and that the value of PΩ at the threshold is a very slow function of the current.

It is proposed that the differential rotation of the Earth's inner core deduced by Song and Richards is due to a combination of the deceleration of the Earth's rotation and the viscous drag between the Earth's inner and outer cores. If this model is correct then the dynamic viscosity in the outer core of the Earth can be estimated to be μ≈104 poise. Besides providing a novel way of determining the viscosity of the core, this simple model suggests some new tests and shows how astronomical effects can influence geological phenomena.

Bohm sheath criteria and double layers are investigated in general multispecies plasmas containing a number of lighter, thermal species, described by Boltzmann distributions, in the presence of a number of heavier species described by cold fluid theory. A nonlinear description for one-dimensional stationary self-similar electrostatic modes is given using the Sagdeev potential. We generalize the classic Bohm sheath criteria by giving appropriate multispecies definitions of drift and ion-acoustic velocities, leading to a global supersonic condition. We also discuss plasmas with only one Boltzmann species, to show that double layers cannot be supported at all, no matter how many cold fluid species are present, and do so by analytical rather than numerical means, based on the full Sagdeev potential. Series expansions for weak double layers occurring in the literature can only claim to find double layers, erroneously as we indicate, by not confirming whether the necessary consistency relations are obeyed, and also by not checking that ‘neglected’ terms in the expansion overwhelm those retained.

A covariant version of Vlasov theory is used to derive the linear response 4-tensor for a relativistic magnetized thermal plasma, generalizing Trubnikov's 3-tensor form. The charge-continuity and gauge-invariance conditions imply a set of integral identities, and use of these identities allows one to derive a manifestly gauge invariant form for the response 4-tensor. This form implies an alternative version of Trubnikov's 3-tensor. The different versions lead to different expressions in the weakly relativistic small-gyroradius limit. Arguments are given as to how this inconsistency should be resolved.