A class of exact stationary statistical states for the inviscid magnetohydrodynamic equations in two dimensions and in various geometries is found and the corresponding fluctuation spectra are calculated. Some solutions agree with previous computations in the canonical ensemble while other solutions are found. In particular, the Navier—Stokes limit is recovered and maximum cross helicity solutions exist in two dimensions. The difficulty of proving existence and uniqueness of statistical solutions for non-dissipative two-dimensional turbulence is quoted in terms of rugged constants and associated Gibbs measure.

The present paper quantitatively examines how the process of fast reconnexion depends on the magnitude of the local resistivity enhanced in the vicinity of the magnetic neutral point. It is shown that quasi-steady Petschek-type configurations are set up, one for each of the variously imposed local resistivity enhancements. The fundamental structure of the quasi-steady configuration is largely controlled by the initially indented value of locally enhanced resistivity. It is especially remarked that the width of the diffusion region becomes smaller as the locally enhanced resistivity becomes smaller. We find that each of the quasi-steady configurations presents nothing other than the Petschek-type configuration that corresponds to the allowable maximum reconnexion rate for the relevant magnetic Reynolds number. We also see that the magnitude of fast reconnexion rate has a weak dependence on the local resistivity in the diffusion region. All our numerical results are very consistent with previous theoretical work on the fast reconnexion problem, once the problem is reconsidered from another angle. We hence suggest that the process of fast reconnexion should be viewed as a gross instability, inherent to the current sheet system itself, that can be triggered by some local onset of anomalous resistivity.

Equations describing the resistive evolution of a cylindrically symmetric magnetic field through force-free states enable us to study its response to externally applied electric fields. The process is relaxational, and, under certain conditions, the final state is a reversed field configuration. We show, moreover, that the non-standard nature of the equations leads to a non-trivial (adiabatic) evolution in the case of vanishing resistivity.

This is part 2 of a paper concerned with the stability against small perturbations of a certain class of nonlinear wave solutions of the equations that describe a cold unmagnetized plasma. It refers to transverse linearly polarized waves in an electron-positron plasma. A numerical method, based on Floquet's theory of linear differential equations with periodic coefficients, is used to solve the perturbation equations and obtain the instability growth rates. All the three possible types of perturbations are discussed for a typical value of the (large) amplitude of the nonlinear wave: electrically longitudinal slightly unstable modes (with maximum growth rate γ approximately equal to 0·07ω0, where ω0 is the frequency of the nonlinear wave); purely transverse moderately unstable modes (with γ ≃ 0·26ω0); and highly unstable electrically transverse modes (with γ ≃ l·5ω0).

Numerical solutions of the momentum and energy equilibria in a two-temperature diffuse pinch show that it is possible for the electron temperature to have a radially oscillating component with a wavelength of the order of ae(mi/me)1/2, where ae is the electron Larmor radius. This is also the order of magnitude of the optimum wavelength for growth of an electro-thermal instability in a homogeneous plasma (studied in part 3 of this series) and suggests this kind of instability as a possible primary source of a current filamentation in the plasma. It has also been revealed both analytically and computationally in this strictly steady state model that the pressure of one species (ions or electrons) is finite at the wall boundary; the same study carried out on a neo-classical model shows, however, that strict confinement is possible provided some restrictions are imposed upon its free parameters. This is also found with a model employing classical tensor conductivity (studied in part 2).

We study the frequency shift suffered by a wave packet propagating in a homogeneous and stationary plasma, when subjected to the influence of a second wave packet which nonlinearly changes the dielectric properties of the plasma. The opposite cases of an adiabatic frequency shift and a relativistic Doppler shift are discussed in detail. The contribution of the inverse Faraday effect to the adiabatic situation is also taken into account. The results might be relevant in the search for new generators in the sub-millimetre ranges and for the understanding of spectral transfers of energy in a plasma.

Forced magnetoacoustic oscillations in a fully ionized, non-uniform, current carrying plasma column of finite beta are treated theoretically. The results of numerical calculations are given for the specific case of diffuse pinch equilibrium configurations. It is found that, for these configurations, the amplitude of the axial component of the oscillating magnetic field is enhanced and the frequency at which magnetoacoustic resonance occurs is raised. It is also found that the presence of an equilibrium axial current produces a substantial azimuthal component, in addition to the axial component, of the oscillating magnetic field.

Nonlinear Kelvin–Helmholtz instability for an incompressible plasma under the influence of gravity is investigated theoretically, using the method of multiple time-scales. The effect of nonlinearity is to reduce the growth rate for the linear instability and to narrow the unstable wavenumber domain. There are two cases, in one of which the effect is strong and in the other slight. The difference between these cases is studied.

A tensor conductivity model for a fully ionized, two-temperature, cylindrical and diffuse pinch is constructed and its properties analyzed under energy and momentum equilibria conditions. The set of Braginskii's classical two-temperature tensor transport equations are solved for the strict steady state and the radial equilibria profiles of the relevant parameters produced. Unlike the scalar conductivity model (part 1 of this series) tensor coefficients yield profiles with the appropriate ideal boundary conditions, namely zero pressure and temperatures at the plasma radius. Furthermore, spatial oscillations of the electron temperature only occur for extreme values of the free parameters.

Starting from the Vlasov–Landau kinetic equation, linearized around the global equilibrium state and assuming the finite dimensional approximation to the non-hydrodynamical subspace, we derive the hydrodynamical equations for a weakly coupled, one-component plasma. Explicit expressions are given for the transport coefficients and frequencies of the hydrodynamical modes. These results describe the influence of the plasma oscillations on the hydrodynamical behaviour of the system. Certain differences are obtained compared with the standard phenomenological approach. Also the real part of the frequency for the plasma mode in our model is in agreement with the Bohm–Gross dispersion relation.

A theoretical method is proposed for predicting the properties of the backward propagating cyclotron harmonic waves from the simple dispersion curve for perpendicular propagation. This method is illustrated for the frequency range 1 ω÷ωc ω 2, where ω and ωc, are, respectively, the wave and electron cyclotron frequencies, and is applicable from very low to infinite plasma densities. The behaviour of the backward propagating cyclotron harmonic wave in the whole propagation space and then in the real space can be easily deduced. The method can be extended to any dispersion branch of backward cyclotron harmonic waves.

Using a test particle approach, the potential of a trapped electron is calculated outside the Debye sphere in the electrostatic limit. The potential strongly depends on some characteristic parameters, such as bounce frequencies of the test and background trapped particles, the total number of trapped particles, etc. In some cases, the potential falls off as the inverse of the distance r. The model is limited to the single-wave case; possible generalizations to the many-wave situation in a turbulent plasma are discussed.

A fully relativistic stream superposition model is employed to conduct a linear numerical simulation of a self-consistently confined sheet of collisionless, neutral plasma. This multi-stream model employs a novel variable termed the ‘canonical momentum potential’ (or ‘action function’) to follow the ion and electron dynamics. For the classic, unsheared sheet pinch, growth rates obtained for the tearing instability are in reasonable agreement with previous estimates using an approximate Vlasov approach. Current shear is then introduced into the sheet and growth rates are again measured. Stabilization of the shorter wavelength modes is observed.

The stability of electrostatic waves propagating at large angles with respect to the background magnetic field is studied in collisional, fully ionized plasmas with two types of ion species and carrying a field-aligned current. By considering plasmas with ma/mb ≪ Nb/Na ≪ mb/ma where m and N denote mass and density respectively and subscripts a and b refer to the two ion species, a complicated dispersion relation is reduced to two simple equations for the determination of the real and imaginary parts of the frequency. It is found that, under appropriate conditions, an instability occurs at frequencies slightly above but very close to the ion–ion hybrid frequency. The growth rate scales directly as the electron–ion collisional frequency.

The Sonnerup solution for reconnexion about an X-type magnetic null point is modified to involve only stable shocks. However, the reconnexion process described does not originate at the line breaking point but is engineered by precise boundary conditions in the inflow region. Our solution is also valid in the compressible case. The limit on the reconnexion rate (in Mach number units) is about 1·5. A non-uniqueness of the solution raises the possibility of viewing flare phenomena as abrupt jumps to different branches of the solution.

The two-dimensional stability of Langmuir solitons in investigated taking into account the dynamic ion response and electron nonlinearities. A variational principle for the growth rate is derived which allows one to determine growth rates and regions of instability by standard numerical procedures. The validity of previous methods using trial functions and the variation of action method, as well as the limitations of the corresponding results for the growth rates, neglecting the dynamic ion response, are critically examined.

From hybrid-kinetic theory an eigenvalue equation for electromagnetic perturbations with ω ≈ ωci in collisionless theta-pinches with anisotropic ion energy was recently derived. This equation is presently reduced to two ordinary second-order linear differential equations by an expansion in the thermal ion Larmor radius. The leading-order correction terms contain Cherenkov resonances absent in the homogeneous case, which are reached for speeds ≈ υ| β|-½ and wave-numbers typical for unstable modes in the homogeneous case, indicating the importance of Cherenkov resonance on the stability of modes driven by ion energy anisotropy. These equations are supplemented by appropriate boundary conditions for the case when the plasma is surrounded by a cylindrical, perfectly conducting wall. For weak inhomogeneities a local dispersion equation is obtained. The physical mechanism underlying the influence of Cherenkov resonance parallel to the confining magnetic field as a FLR effect on stability behaviour is illustrated.

Magnetic shear is found to have a strong effect on the propagation characteristics of the lower hybrid parametric daughter waves but no significant effect on the pump wave. The analysis of the OTS instability shows that the convective damping introduced by magnetic shear acts on a distance L⋍ H(me/mi)½, where H is the magnetic shear scale length. There are two regimes for the convective damping, depending on the wavelength of the parametric daughter waves. For small wavelengths the growth rates are linear functions of (kL)-1. For large wavelengths the growth rates are exponentially decreasing functions of (kL)-1.

The beam-whistler and ion cyclotron instabilities are proposed as principal mechanisms for the generation of precursor turbulence in shock waves propagating along the constant magnetic field. The beam is formed from ions which are reflected from a potential barrier whose existence inside the shock layer is assumed. We find that these reflected beam particles drive unstable a spatial continuum of zero group velocity (with respect to the shock front) precursor whistler waves.

The excitation of the slow extraordinary and electron whistler modes with frequencies in the vicinity of the electron plasma frequency is investigated for a plasma which is composed of thermal and suprathermal electrons. Ion dynamics are ignored. The suprathermal electrons are assumed to comprise a long, anisotropic tail parallel to the ambient magnetic field. Instability is found to occur via a relativistic, anomalous gyroresonance with the suprathermal electrons, and to excite waves with frequencies above and below the electron plasma frequency. Landau and cyclotron damping due to the thermal background is included in the treatment.