The Vlasov equation is solved by a phase-space boundary integration method in order to investigate the nonlinear frequency of an electron plasma mode. Unlike the familiar discrete particle simulation codes in which a limited number of particles may be considered, the present model represents a more realistic plasma case in which a very large number (practically unlimited) of plasma particles evolving in their self-consistent collective field is investigated. The unperturbed collisionless one-dimensional plasma system consists of warm electrons having a water-bag distribution to simulate equilibrium, and of static ions. The initial perturbation is introduced by changing the boundaries of the electron plasma such that υu = υ0 (1 + α.sin kx) and υl = − υ0(1 − α sin kx), υu and υl representing the upper and the lower boundaries, respectively. This corresponds to a standing wave perturbation. The results obtained for different wavelength perturbations λ ≡ 2π/k, and for several perturbation amplitudes α, are presented and discussed.

A study is made of the inelastic collision integral of the Boltzmann equation using scattering probability formalism. The collision operators are expanded in a power series in the square root of the ratio of masses.

Furthermore, a spherical harmonic expansion is made of all the operators so obtained. These developments are valid whatever the shape of the distribution function of the particles.

We have investigated the stability of the screw pinch with the help of the double adiabatic (CGL) equations including the finite Larmor radius effects through the anisotropic pressure tensor. The calculations are approximate, with FLR treated as a first-order correction to the ideal plasma equations. The dispersion relation has been solved for various values of R2 = p∥/p⊥ and α for the rale and imaginary part of the frequency (ω = ωR ± iωI) in three particular cases: (a) μ = 0, the θ-pinch, (b) μ = ∞, the Z-pinch, (c) μ = -α/m, field distubances parallel to the equilibrium field. Here μ is the pitch of the magnetic field in the pressureless plasma surrounding the main column, α is the wave number, m is the azimuthal number, p∥ and p⊥ are plasma pressures along and perpendicular to the magnetic field.

Initial growth rates are calculated for unstable modes of the Taylor type driven by temperature and density gradients in the presence of viscous and thermal diffusion. For a laser-driven plasma it is shown that, even without ablative flow, conduction can sharply reduce unstable growth, and that growth should only occur in the region of low temperature near the ablation front.

The stability of two of the short time scale solutions obtained in the filament model for a stationary field electron ring accelerator is studied by means of a linearized perturbation on the electron and ion distribution functions. The Vlasov equations for the electrons and ions are coupled. The coupling of these equations results in a Fredholm integral equation which can be converted into an infinite system of linear equations. A stability criterion is then obtained from the zeros of the truncated determinants of the coefficients of the infinite system. One of the equilibrium solutions, for which the ion density in phase space is a monotonic non-increasing function of the unperturbed ion Hamiltonian, is entirely stable. The other solution, for which this monotonicity does not hold, may be unstable, but turns out to be stable provided that both the non-monotonicity parameter ∈ and the ratio of the ion period to the electron period are not too large. In the nonmonotonic case, the ion density in physical space also does not drop off to zero monotonically from the centre of the ring. For the unstable cases, the instability is driven by a ‘spike’ in the phase-space ion density located at the maximum value of the unperturbed ion Hamiltonian.

The paper contains an analysis of the linearized plasma-dynamical equations, including the energy conservation equations, and accounting completely for all the dissipative mechanisms, in the absence of external fields. The eigenvalues of the 14 plasmadynamical modes are obtained systematically in terms of the transport coefficients of the plasma.

The kinetic theory of a plasma is developed on the basis of a normal mode analysis-In the absence of any external field, the system approaches a ‘plasmadynamical (PD) state’, described by fourteen amplitudes of the PD modes. The corresponding eigenvalues and eigenvectors are constructed explicitly. When an external time-dependent electric field is switched on, transitions between these modes determine the evolution of the plasma. Assuming that a PD regime is valid, the kinetic equations reduce to a set of equations describing parametric effects and nonlinear mode-mode couplings. A new source term, arising from the interference between collisions and external field, also appears. Some qualitative implications are discussed.

We calculate the electric field fluctuation spectrum in a magnetized plasma considering the influence of a finite amplitude pump wave.

The problem of transport coefficients in statistical mechanics is reconsidered. A general method is given by which the hydrodynamical equations can straightforwardly obtained starting from the kinetic equation for the one-particle distribution function. From the statistical counterparts of the hydrodynamical equations so derived, the statistical expressions for the transport coefficients are immediately identified.

Linearized hydrodynamic modes have recently been the object of very thorough reserach from the viewpoint of irreversible statistical mechanics; in particular, the Brussels school formalism has been used by Résibois to derive the eigenfrequencies of the hydrodynamical modes, whereby operatorial equations for transport coefficients have been obtained (Résibois 1970; see also the instructive book by Balescu (1975) on this subject).

In this paper we propose an exact solution of Vlasov and Maxwell's equations for a bounded hot plasma in order to derive the dispersion relation of the axially-symmetric surface waves propagating along a plasma column. Assuming specular reflexion of plasma particles from the boundary, expressions for the components of the electric displacement vector are obtained on the basis of the Vlasov equation. Their substitution in Maxwell's equations, neglecting the spatial dispersion in the transverse plasma dielectric function, allows us to determine the plasma impedance. The equating of plasma and dielectric impedances gives the wave dispersion relation which, in different limiting cases, coincides with the well-known results.

The statistical theory of the transport coefficients which was formulated in a previous paper (Vasu, 1976) is exemplified here by the case of a two-fluid plasma in a magnetic field. In addition, a general method is given which allows us to calculate frequency dependent transport coefficients; this method is a natural extension of the paper mentioned above. In the static limit, the transport coefficients obtained for the two-fluid plasma in a magnetic field are similar to those reported by Braginsky (1963) although there are some numerical differences. The frequency dependent transport coefficients obtained here include important resonant and time lag effects.

The Fokker-Planck equation is studied for an energetic ion beam injected into a magnetized plasma consisting of Maxwellian ions and electrons with υthi ≪υb≪ υthe. The time evolution of the fast ion distribution is given in terms of an infinite sum of Legendre polynomials for distributions that are axisymmetric about the magnetic field. The effect of charge exchange is included. The resulting ion distribution is somewhat isotropic for velocities much less than the injection velocity, however, the distribution is sharply peaked in both energy and pitch angle for velocities near the injection velocity. Approximate asymptotic expressions are given for the distribution in the vicinity of the injected beam and for velocities greater than the injection velocity. The effect of a weak parallel electric field is also given.

The stability of circularly polarized waves of finite amplitude propagating parallel to the magnetic field is studied. A set of equations for slowly varying waves of arbitrary amplitude is obtained. A discussion of the stability of the waces is based on this set of equations. Earlier results are confirmed; in addition we find that finite amplitude always promotes stability. An amplitude dependent stability condition for long waves, previously obtained by the author, is confirmed.

Cornwall et al. (1970) suggested that if the ring current protons penetrated the plasmapause, intense resonant electromagnetic ion cyclotron turbulence would lead to their rapid precipitation loss. IN this paper we present a computer simulation study of the electromagnetic ion-cyclotron instability in a homogeneous plasma consisting of protons and electrons embedded in a static magnetic field, B0x. The plasma parameters used are and Effects such as sources and losses or space non-uniformities are not considered here. The predictions of the linear theory for the fast build-up of unstable left-hand (proton-like) electromagnetic ion-cyclotron waves are fully confirmed.

There is considerable interest in the use of energetic ion beams to heat magnetically confined plasmas to ignition temperature. An uncertainty in this heating technique is the possible role of instabilities driven by the fast ion beams. For this reason, we have investigated homogeneous plasma modes, including the cross field ion-ion wave and the beam ion-acoustic wave, as well as the ordinary electromagnetic mode, which taps the free energy associated with the beam anisotropy. The growth rates of these modes are calculated and the stability thresholds are compared to parameters for both present injection experiments and the proposed TFTR device.

Incompressible turbulent flows are investigated in the framework of ideal magnetohydrodynamics. All the field quantities vary with only two spatial dimensions. Equilibrium canonical distributions are determined in a phase space whose co-ordinates are the real and imaginary parts of the Fourier coefficients for the field variables. In the geometry considered, the magnetic field and fluid velocity have variable x and y components, and all field quantities are independent of z. Three constants of the motion are found (one of them new) which survive the truncation in Fourier space and permit the construction of canonical distributions with three independent temperatures. Spectral densities are calculated. One of the more novel physical effects is the appearance of macroscopic structures involving long wavelength, self-generated, magnetic fields (‘magnetic islands’) for a wide range of initial parameters. Current filaments show a tendency toward consolidation in much the same way that vorticity filaments do in the guiding-centre plasma case. In the presence of finite dissipation, energy cascades to higher wavenumbers can be accompanied by vector potential cascades to lower wavenumbers, in much the same way as, in the fluid dynamic (Navier-Stokes) case, energy cascades to lower wavenumbers accompany enstrophy cascades to higher wavenumbers. It is suggested that the techniques may be relevant to theories of the magnetic dynamo problem and to the generation of megagauss magnetic fields when pellets are irradiated by lasers.

The stability of electromagnetic drift waves in high-β anti-loss-cone plasmas is investigated. These waves with propagation transverse to the external magnetic field are found to be unstable provided ρ > ρc, where ρ is the parameter which characterizes the strength of the anti-loss-cone. The growth rates associated with this instability (which may be called the anti-loss-cone drift instability) are ˜ 0·1Ωe and they increase with the strength of the anti-loss-cone.

Wave-wave coupling is investigated for spatially growing waves in a cold beam-cold plasma system. Numerical solutions of the wave coupling equations are obtained for various combinations of three waves as well as several interacting triplets of waves that are of experimental interest. It is found that most three wave interactions in this system are explosive if they involve the most unstable mode. The results are compared, where applicable, with analytic criteria for the unboundedness of the solutions.

Because of the technical difficulties associated with the production of ultrashort beam pulses, the efforts to release energy from thermonuclear microexplosions are mainly directed at present towards confinement in a high-Z material solid wall, as in the original proposal by the author for relativistic electron beam induced fusion. The most common approach today is by the implosion of a solid high-Z material shell, for example, with an intense relativistic electron beam. We will show that better utilization of the beam energy may be possible with beam bombardment of a curved concave wall, setting off a shock wave which can be focused precisely onto the thermonuclear target. In addition, by a proper variation of the atomic weight and density in the layer of the wall material to be ablated by the energy deposition of the beam, the pulse length of the shock wave can be made substantially shorter than the pulse length of the beam.

A general formalism for describing two-time fluctuations in magnetized plasma is presented. Two-time expectations of one-body operators (phase functions) are written in terms of the phase space density autocorrelation function

where δN is the fluctuation in the singular Klimontovich microdensity. It is shown that is the first member of a set of two-time quantities

which collectively obeys the linearized BBGKY cumulant hierarchy in the (Xi, t) variables, with initial conditions successively smaller in the plasma parameter . We study in detail the case of fluctuations in thermal equilibrium, although the general formalism holds also for the non-equilibrium case. To lowest order in εP, Γ obeys the linearized Vlasov equation. From this are recovered all of Rostoker's results for fluctuations excited by Cherenkov emission and absorbed by Landau damping, as well as a constructive proof of the test particle superposition principle. To first order, Γ obeys (in the Markovian approximation) the linearized Balescu-Guernsey-Lenard equation. For frequencies and wavenumbers in the hydrodynamic regime, the velocity moments of Γ obey linearized fluid equations with classical transport coefficients (i.e. essentially those computed by Braginskii in the 3-D case). It has been found that the classical theory is in disagreement with certain computer and laboratory experiments performed in strong magnetic fields. This defect is attributed to the absence in the classical theory of contributions to the collision operator, hence transport coefficients, of fluctuations long-lived on the Vlasov scale. Analogous difficulties arise in the theory of hydrodynamics in neutral fluids. To improve the plasma theory, a renormalization of the two-time hierarchy is proposed which sums selected terms from all orders in εP and thus treats the hydrodynamic fluctuations self-consistently. The resulting theory retains appropriate fluid conservation laws, thereby avoiding erroneous results encountered in certain diffusing orbit theories, when the fluid viscosity is indiscriminantly replaced by the test particle diffusion coefficient. In order to explain the results of the computer simulations, the theory is applied in part 2 to the problem of anomalous hydrodynamic contributions to the transport coefficients.