Starting from a set of equations that lead to a linear dispersion relation coupling kinetic Alfvén waves and ion-acoustic waves, three-dimensional KdV equations are derived for these waves. These equations are then used to investigate the three-dimensional stability of solitary kinetic Alfvén waves and ion-acoustic waves by the small-k perturbation expansion method of Rowlands and Infeld. For kinetic Alfvén waves it is found that there is instability if the direction of the plane-wave perturbation lies inside a cone, and the growth rate of the instability attains a maximum when the direction of the perturbation lies in the plane containing the external magnetic field and the direction of propagation of the solitary wave.

The nonlinear stability of a horizontal interface separating two streaming magnetic fluids of finite thickenss is investigated in two dimensions. The fluids are considered to be inviscid and incompressible. The magnetic field is applied along the direction of streaming. The method of multiple scales, in both space and time, is used to examine the stability properties of the system arising from second-harmonic resonance. A pair of partial differential equations for the amplitude of the wave and its second harmonic are derived. These describe the evolution of the wave train up to cubic order, and may be regarded as the counterparts of the single nonlinear Schrödinger equation that occurs in the non-resonant case. The stability condition of this equation is discussed both analytically and numerically, and stability diagrams are obtained. Regions of stability and instability are identified. The nonlinear cut-off wavenumber separating the regions of stability from those of instability is obtained. The equation governing the evolution of the amplitude at the critical point is also obtained, which leads to a nonlinear Klein—Gordon equation.

A lower bound for the total variation Bmax/Bmin of the field strength B on a toroidal vacuum magnetic surface with finite twist (rotational transform) of the field is given as a necessary (not sufficient) condition for the field lines to be geodesics on it. If such a surface existed, it would be omnigenous with respect to the local gradient-B and centrifugal guiding-centre drift approximation, which means that the normal components of these drift velocities would be zero. It is found that the lower bound on Bmax/Bmin depends on the maximum of – (r/B) ∂B/∂r within the surface. Here r is the distance from the origin of the torus. The lower bound is large if this maximum is small, and vice versa. For some more recent stellarator configurations the condition is found to be satisfied at outer magnetic surfaces. It is obtained without the use of expansion techniques, under the assumption that there is a nested system of magnetic surfaces within the omnigenous one.

The transverse spreading of magnetic field lines in a turbulent plasma is investigated analytically in order to obtain a statistical characterization at large spatial scales. We develop a functional-integral method that allows us to calculate in a systematic way statistical averages of physical quantities that depend on the fluctuating field. The known magnetic diffusion coefficient for the shear-free case is corrected with a term arising from the assumption of a finite transverse correlation length. For the case with magnetic shear the functional method provides the appropriate framework for a perturbative approach based on diagram series.

In this work we investigate the nonlinear dynamical interaction of magnetized electrons and ordinarily polarized electromagnetic modes. We show that when either the mode amplitude is sufficiently large or the electronic longitudinal momentum along the external magnetic field is sufficiently small, non-integer harmonics originating from higher-order terms in the expansion of the Hamiltoniari in powers of the electromagnetic field become important, and the relevant action-angle phase space ceases to be well described by standard nonlinear pendulum approximations. We develop appropriate resonance overlap studies in order to determine the conditions to be satisfied at the transition from regular to chaotic orbits, checking the accuracy of the results with numerical integrations of the full dynamical equations. We verify that chaotic orbits occur within a ‘window’ among the possible values of the longitudinal momentum. The width of this window and the degree of overlap between neighbouring resonances within it are both shown to grow with particle action and wave frequency. We also show that once overlap has occurred for a particular value of the action, it persists for larger values of this quantity, a circumstance that leads to a large amount of particle energization.

Dust particles, assumed to be of one size and to exhibit a discrete distribution of electric charges, are treated as heavy ions with a large number of ionization levels. The average of the discrete particle effects on the kinetic equations is approximated by the Lénard–Balescu collision term and by detailed counting to describe transport in velocity space and transitions between the differentionization levels respectively. We estimate analytically and numerically the relaxation times for the dust particles both towards a Maxwellian velocity distribution and towards an equilibrium distribution for the ionization levels. We sum over the ionization levels to obtain a hierarchy of ‘charge-moment’ equations for the single dust density function, and estimate the importance of terms originating from the ionization distribution. Similar terms are also present in the hydrodynamic equations for a dust plasma, and we briefly discuss these.

This paper presents a paraxial theory of relativistic self-focusing of a Gaussian laser beam in plasma for a time-harmonic plane wave at arbitrary beam intensity. Since the relativistic mechanism is instantaneous, the theory is also applicable to self-focusing of laser pulses. The analysis leads to two values for the critical beam power for self-focusing, Pcr1 and Pcr2. When P < Pcr1 < Pcr2, the beam diverges. When Pcr1 <P <pcr2, it first converges, then diverges, and so on. When P > Pcr2 it first diverges, then converges, and so on.

In this paper we consider a set of nonlinear MHD equations that admits in a linear approximation a solution in the form of a slow sausage surface wave travelling along an isolated magnetic slab. For a wave of small but finite amplitude, we investigate how a slowly varying amplitude is modulated by nonlinear self-interactions. A stretching transformation shows that, at the lowest order of an asymptotic expansion, the original set of equations with appropriate boundary conditions (free interfaces) can be reduced to the cubic nonlinear Schrödinger equation, which determines the amplitude modulation. We study analytically and numerically the evolution of impulsively generated waves, showing a transition of the initial states into a train of solitons and periodic waves. The possibility of the existence of solitary waves in the solar atmosphere is also briefly discussed.

The theory of flute-mode stability is developed for a two-energy-component plasma partially terminated by a conducting limiter. The formalism is developed as a preliminary study of the effect of end loss in open-ended mirror machines, where large-Larmor-radius effects are important.

A model of a cylindrically symmetric, force-free magnetic field consisting of a sequence of concentric layers with piecewise-constant α is used to construct models of the surface currents on isolated, force-free magnetic flux tubes. Two boundary conditions are considered: a current-neutralized flux tube (Bφ = 0, Bz φ 0, Bz ≠ O at r > r0), and an isolated current-carrying flux tube (Bφ ≠ 0, Bz = 0 at r > r0). A single-a model that is current-neutralized is a reverse-field pinch, and is unacceptable as a model for a solar flux tube. Examples of two-α models for a current-neutralized flux tube are presented. The models of the surface currents satisfying either boundary condition are shown to simplify considerably when the surface layer is thin. A model consisting of several layers, with piecewise-constant α, may be used to find an approximate solution for a force-free flux tube with an arbitrarily specified current profile.

We present a new theory for the relaxation of two-dimensional collisionless quasi-neutral plasmas by mixing in phase space. In general, mixing in phase space does not lead to equilibrium states in the framework of exact, collisionless theory. However, filamentation on smaller and smaller scales can result in stationary states from a macroscopic (coarse-grained) point of view. In a first step we discuss a general technique to calculate suitably stable equilibria of collisionless plasmas. In a second step we develop a model relaxation that yields a simple transition to a description on macroscopic scales without following the details of the complicated mixing dynamics. Combination of both results allows calculation of final states of collisionless relaxation. We illustrate our approach with the help of a one-dimensional sheet configuration as a simple example.