The momentum-space diffusion equation and the kinetic wave equation for resonant wave–wave scattering of electromagnetic and electrostatic waves in a relativistic magnetized plasma are derived from the relativistic Vlasov–Maxwell equations by perturbation theory. The p-dependent diffusion coefficient and the nonlinear wave—wave coupling coefficient are given in terms of third-order tensors which are amenable to analysis. The transport equations describing energy and momentum transfer between waves and particles are obtained by momentum-space integration of the momentum-space diffusion equation, and are expressed in terms of the nonlinear wave—wave coupling coefficient in the kinetic wave equation. The conservation laws for the total energy and momentum densities of waves and particles are verified from the kinetic wave equation and the transport equations. These equations are very useful for the theoretical analysis of transport phenomena or the acceleration and generation of high-energy or relativistic particles caused by quasi-linear and resonant wave—wave scattering processes.

The nonlinear propagation of a superluminal, linearly polarized electromagnetic wave in the presence of a relativistic cold electron beam is investigated. At large amplitudes the wave couples to the electron-beam plasma mode owing to two important nonlinear effects, namely the relativistic variation of the electron mass and the excitation of longitudinal space charge fields by strong v × B forces. The nonlinear propagation equations for the coupled electromagnetic and longitudinal waves are derived within the context of a relativistic cold-fluid model. Nonlinear travelling-wave solutions are sought to describe the saturated state of the coupled system. Using Hamiltonian techniques, a wide variety of solutions are obtained and their characteristics discussed.

We employ the Monte Carlo method for the calculation of anomalous transport developed by Punjabi and Boozer to calculate the particle diffusion coefficient for electrons in the stochastic magnetic fields of the reversed-field pinch (RFP). in the Monte Carlo calculations represented here, the transport mechanism is the loss of magnetic surfaces due to resistive perturbations. The equilibrium magnetic fields are represented by the Bessel function model for the RFP. The diffusion coefficient D is calculated as a function of a, the amplitude of the perturbation. We see three regimes as the amplitude of the tearing modes is increased: the Rechester—Rosenbluth regime where D scales as a2 the anomalous regime where D scales more rapidly than a2 and the Mynick—Krornmes regime where D scales more slowly than a2. Inclusion of the effects of loop voltage on the particle drift orbits in the RFP does not affect the intervals in the amplitude a where these regimes operate.

At a college where I taught, there was a ‘Last Lecture’ series, in which the faculty took turns giving a public lecture that would be the lecture they would give if it were to be their very last one. The invitation to contribute a few nontechnical pages to this issue of Journal of Plasma Physics has a somewhat similar valedictory ring, and inspires mixed feelings.

We extend previous work by Buti to include the effect of ion temperature and investigate how the instability range is modified in the simultaneous presence of collisions and ion temperature. A salient feature of the calculations is that waves that were stable in the collisionless case for certain values of the wavenumber k now become unstable.

The three-dimensional equations of ideal hydrodynamics and ideal MHD are expanded in eigenfunctions of the curl, and the resulting basic interactions of these nonlinear systems are analysed. As the equations are invariant under time and amplitude reversal, a criterion defining the arrow of time is introduced. A new parameter, the center of energy, serves to characterize a basic interaction. In the 3D Euler equations we find four different interactions and their mirror images, two of which can transport energy to smaller wavenumbers. This can lead to the appearance of structures in turbulent flow, and throws doubt on a derivation of Kolmogorov's law based on a cascading of energy to higher wavenumbers.In energy the corresponding two-dimensional equations, which are isomorphic to the guiding centre model in plasma physics, only one interaction exists, with a strong inverse cascade, which can lead to accumulation of energy in the spatially largest accessible modes. In MHD theory it is possible to separate magnetic from kinetic interactions. The former give again four basic interactions, two being regular and two being inverse cascades. One of these is quite strong, and can lead to the MHD dynamo effect. Kinetic energy can be transferred into magnetic energy. The dynamo effect is is accompanied by alignment of velocity and magnetic fields. We show that stationary velocity fields may lead to exponentially growing magnetic fields and we give an explicit criterion for this instability.

Ion-acoustic solitary waves in a weakly relativistic warm plasma at the critical phase velocity are investigated using reductive perturbation theory. The basic set of fluid equations describing the system is reduced to a renormalized warmion modified Korteweg—de Vries (mKdV) equation for the first-order perturbed potential and a renormalized linear inhomogeneous equation for the second-order perturbed potential. The stationary solution of the coupled equations is obtained. The cold-plasma limit is considered in order to rule comparisons with work of Pakira et al.

I review the current status of a problem, relevant to both plasma physics and ordinary fluid mechanics, namely the long-time behaviour of solutions of the perfect fluid equations. In two space dimensions, thanks in particular to the work of D. Montgomery, the situation is now quite clear, since one expects the formation at long times of large vortices in a background of potential flow. In three dimensions, the situation is blurred, although its understanding is a central issue for fully developped turbulence. I present some new estimates for a possible scenario of self-similar blow up of solutions of 3D Euler. That turns out to be a rather subtle question, if one tries to stay consistent with the conservation of circulation and of energy.

The propagation of ion acoustic waves in an inhomogeneous plasma is studied in the presence of finite ion temperature. The standard nonlinear Schrödinger equation is obtained, and the regions of instability are plotted. It is claimed that these extraneous factors tend to lower the value of the critical wavenumber.

Expressions for the field intensities in a warm ionized plasma for two type of modes (electromagnetic and electroacoustic) pertaining to a microstrip matched coaxial termination are obtained using linearized hydrodynamic theory coupled with a vector function technique. The radiation power and directivity for the respective modes are evaluated for different ratios of plasma to source frequencies. It is observed that electromagnetic fields are similar to other microstrip antenna structures, but more directive in a particular direction. The electroacoustic mode exhibits a complete absence of multilobed structure, which was found for other antenna structures. The efficiency of this structure in a plasma is poor.

Fully nonlinear kinetic Alfvén solitary waves are investigated, taking account of the drifting effect of electrons in the direction of the external magnetic field in a low-β plasma (where β = 8πn0T/c2B02 is the ratio of the kinetic pressure to the magnetic pressure). Both compressive and rarefactive Alfvén solitons are found to exist in different ranges of kz (direction of wave propagation), depending on the values of M (the Mach number) and ve (the drift velocity, occurring implicitly in an effective parameter A). The existence domain of kinetic Alfvén solitons is found to be concentrated in the upper regime of β (≪ 1) and lower regime ofM ( < 1), but the values of M in the case of rarefactive solitons are observed to be higher than those of compressive solitons. The range of existence of kinetic Alfvén solitons is substantially narrowed when the initial drift velocity v′e of the electrons is taken into account.

We study the quasimode decay of a lower-hybrid wave and a damped ion cyclotron wave in a plasma having two kinds of electrons. This decay channel is also investigated for a cylindrical plasma. The behaviour of the threshold and growth rate with variations in Tn/Tc and non/noc are studied, and a comparison is made with previous results. Our results show that the growth rate and the threshold for the onset of parametric decay are influenced by the presence of the second electron species.

The entropy associated with absolute equilibrium ensemble theories of ideal, homogeneous, fluid and magnetofluid turbulence is discussed, and the three-dimensional fluid case is examined in detail. A σ function is defined, whose minimum value with respect to global parameters is the entropy. A comparison is made between the use of global functions σ and phase functions H (associated with the development of various H theorems of ideal turbulence). It is shown that the two approaches are complementary, though conceptually different: H theorems show that an isolated system tends to equilibrium, while σ functions allow the demonstration that entropy never decreases when two previously isolated systems are combined. This provides a more complete picture of entropy in the statistical mechanics of ideal fluids.

Montgomery and co-workers have developed a framework to describe the steady state of a turbulent magnetofluid, without the usual recourse to linearization. Thus the magnetic and velocity fields emerge in their fully nonlinear form as a consequence of the selective decays of the invariants of the system. Using this statistical theory of magnetohydrodynamic turbulence, the pressure, magnetic and flow fields of a solar coronal loop have been determined. The spatial and time profiles of the loop pressure are derived. A comparison with the observed properties of the loops is made, whenever possible.

The plasma transport equations for a weakly collisional plasma have previously been derived for four different time scales. This paper is devoted to the derivation of the plasma transport equations for the two other complementary regimes: the intermediately collisional regime (ICR) (i.e. for the case where the transit time w1 is of the same order as the collision time is of the same order as the collision time ), and the strongly collisional regime (SCR) (i.e. for the case of ) for different time scales. It is shown that the lowest-order gyromotion is unperturbed by collisions. On the Alfvén time scale, one merely obtains for both the intermediately collisional case and the strongly collisional case the single-fluid ideal MHD equations, if certain additional requirements are satisfied. On the MHD-collision time scale, one arrives at the full set of transport equations, where in both cases, contrary to the weakly collisional case, no turbulent terms are found. On the resistive diffusion timescale, one ends up with the known transport equations, with the addition of turbulent contributions.

The stability of a finite-amplitude standing Alfvén wave of circular polarization in a low-β plasma is studied using a set of nonlinearly coupled MHD wave equations. In the presence of a standing Alfvén pump, two distinct gratings associated with the density fluctuations are excited: those due to the ponderomotive beating of the pump magnetic field, and those due the induced magnetic fluctuations. The roles played by the two gratings in the mode coupling are analysed. Both convective and purely growing regimes of the MHD parametric instabilities can be produced by a standing Alfvén wave. In both regimes, the maximum growth rate increases as the pump amplitude increases, and decreases as increases. Tn the presence of the second grating, a new unstable convective regime appears that widens the overall instability bandwidth.

Nonlinear wave structures of large-amplitude ion-acoustic waves in a plasma with a relativistic electron beam are studied using the pseudopotential method. The region of existence of large-amplitude ion-acoustic waves is examined, and it is shown that the condition for their existence depends sensitively on parameters such as the relativistic effect of the electron beam, the ion temperature, the electrostatic potential and the electron beam density. It turns out that the region of existence spreads as the relativistic effect (Mach number) increases and the ion temperature decreases. New properties of large-amplitude ion-acoustic waves in a plasma with a relativistic electron beam are predicted.

The basic set of equations describing nonlinear ion-acoustic waves in a cold collisionless plasma, in the limit of long wavelengths, is reconsidered. First, a travelling-wave solution is found up to third order by means of a straightforward perturbation approach based on the smallness of the wavenumber. As a result, a positive dressed solitary wave shows up, which is larger, taller and faster than the KdV soliton, the first-order result. Furthermore, the accuracy of this approach is tested and compared with previous result. Secondly, the reductive perturbation techique to study higher-order corrections is revised and adapted to the present problem.