The author's previous work on the Rayleigh–Taylor instability is extended to the Kelvin–Helmholtz instability, and the maximum growth rate of a perturbation and an estimate of its upper bound is obtained for an infinite fluid layer under horizontal rotation where the density, horizontal velocity (shear) and magnetic field are continuously stratified in the direction of gravity. Conclusions are drawn about the possibility of stability for some directions of propagation of the perturbation, even in the case of unstably stratified density. It is also shown that the new terms that appear owing to the interaction of the horizontal shear flow, horizontal rotation and stratified magnetic field increase the range of values that contribute to the estimate of the maximum growth rate compared with previous work. Furthermore, a generalization of the sufficient condition for stability under horizontal rotation alone obtained by Johnson is calculated in the presence of density stratification. A new method is also given to obtain a sufficient condition for stability when a magnetic field is present in addition to rotation and density stratification.

The guiding-centre equations of motion of a classical charged particle in a strong magnetic field and a strongly sheared electric field are derived. They can be used to analyse the dynamics of particles in electromagnetic fields whose spatial profiles are similar to those observed during the H mode in the DIII-D tokamak, for instance. The derivation of the equations of motion is performed up to second order in the drift parameter by applying a Hamiltonian pseudocanonical transformation that removes the gyrophase induced by the magnetic field. The main difference with the standard case of a slowly varying electric field relates to the variation of the new gyrophase and to the expression for the magnetic moment: mv2⊥/2B must be replaced by

formula here

The latter case is also reconsidered – mainly to reveal the consequences of the removal of a hidden divergence for small parallel velocities resulting from the usual averaging transformation.

The electron energy distribution function in a microwave discharge at the electron cyclotron resonance (ECR) condition is studied using a two-dimensional Fokker–Planck equation in the bounce-averaged approach. Our model takes into account the effects of linearized Coulomb collisions, electron cyclotron resonance heating in the quasilinear approximation, the effects of ionization and excitation of atoms, elastic scattering of electrons on atoms, and the self-consistent ambipolar potential. The plasma is considered to be slightly collisional, so that bounce averaging is valid. We perform a numerical investigation of the Fokker–Planck equation and obtain the dependences of the discharge characteristics on the parameters of the model, such as breakdown threshold values of neutral density, and the dependence of the electron density, temperature and ambipolar potential on the parameters of the ECR wave and gas density. Some results are compared with experimental data.

We demonstrate with numerical WKB solutions that stationary density profiles can exist in a crossed-field amplifier. These are density profiles that can exist in equilibrium with a high-frequency (RF) wave propagating in the slow-wave structure. These density profiles are very different from the usual Brillouin profiles. They have a finite and negative density gradient, and extend from the cathode to the anode. They carry a DC current proportional to the RF power in the slow-wave structure. The source of the DC current is a wave–particle resonance that occurs toward the top of the sheath. This resonance drives any Brillouin electron flow unstable. The electrons then redistribute themselves into a new flow – one that can be stationary in the presence of the wave–particle resonance. The new flow has a spoke-like structure, where the spokes carry the current to the anode. Various plots are given, including the DC voltage operating range, DC current flow, phase shifts, and density profiles of the total electron distribution, for various ambient magnetic fields and DC voltages. Numerical values agree reasonably well with the modelled experimental device.

The reaction of an equilibrium tokamak plasma to a sudden localized deposition of heat is investigated by means of numerical simulations of the time-dependent equations of resistive magnetohydrodynamics (MHD) in two spatial dimensions, in order to obtain a better understanding of the structure and dynamics of the hot plasma filaments that are observed in recent tokamak experiments. Simulation results show that the fast heating generates MHD waves and creates a localised hot filament. Pressure perturbations are carried away from the heated area by slow waves, even when the heating is applied on the Alfvén time scale. When a temperature-dependent resistivity is considered, a current peak is formed at the place of the temperature peak, and the order of magnitude of the current is given by the condition that ηJz is constant. Next to the wave dynamics, the simulations show another type of hot filament dynamics caused by j×B forces. These forces drive filaments to the centre of the plasma at a nearly constant velocity, which decreases for smaller resistivities. Two filaments can merge under the influence of this current–current interaction. This process happens on a very long time scale. For realistic tokamak resistivities, the coalescence of filaments may take place on time scales of the order of the experimentally reported lifetime of filaments, and may thus be the mechanism that determines the lifetime of the filamentation process.

The equilibrium of a resistive axisymmetric plasma with purely toroidal flow surrounded by a conductor is investigated within the framework of nonlinear magnetohydrodynamic theory. It is proved that (a) the poloidal current-density vanishes and (b) apart from an idealized case, the pressure profile should vanish on the plasma boundary. For the cases of isothermal magnetic surfaces, isentropic magnetic surfaces and magnetic surfaces with constant density, the equilibrium states obey an elliptic partial differential equation for the poloidal magnetic flux function, which is identical in form to the corresponding equation governing ideal equilibria. The conductivity, which can be neither uniform nor a surface quantity, results, however, in a restriction of the possible classes of equilibrium solutions; for example for the cases considered, the only possible equilibria with Spitzer conductivity are of cylindrical shape.

An experiment is conducted to confirm the theoretical prediction that a rapidly generated lossy plasma can cause spectral breaking and frequency shift of a high-power microwave pulse. Spectral breaking is the transformation or breaking of a single dominant spectral peak associated with an incident pulse into two spectral peaks. The experiment is conducted by comparing the frequency spectrum of an incident pulse with the spectrum of the pulse transmitted through a self-induced air-breakdown environment. It is shown that as the ionization rate becomes too high, the spectrum of the transmitted pulse breaks up into two peaks: one has an upshifted centre frequency, and the other has a downshifted centre frequency. The results show that the amount of frequency upshift is correlated with the ionization rate, whereas the amount of frequency downshift is correlated with the energy losses from the pulse in the self-generated plasma. These experimental results agree with the theoretical prediction and a numerical simulation, which are also presented.

A stability analysis is performed for solitary ion-acoustic waves in a magnetized plasma in which the electrons are non-isothermal. Including the effect of ion drift velocity and magnetic perturbation, a three-dimensional mKdV equation is derived for ion-acoustic waves. The solitary-wave solution of this equation is found to have a sech4 profile. A stability analysis of this solitary wave is performed using the small-k perturbation expansion method of Rowlands and Infeld. A condition for the onset of instability is obtained. The growth rate of the instability is found to attain a maximum for perturbations in the plane perpendicular to the direction of propagation of the solitary wave.

Stimulated scatterings of large-amplitude electromagnetic waves by Langmuir, dust–ion-acoustic and dust-acoustic waves in unmagnetized dusty plasmas are investigated by employing the standard methods of nonlinear three-wave interactions and by incorporating the effects of grain-charge fluctuations, collisions of electrons and ions with dust grains, the plasma drag on a dust grain (for the case of the dust-acoustic wave) and the dependence of the average dust charge on the dusty plasma parameters. Distinction is made between the charging collisions, when electrons and ions are accumulated onto the grain surface; and Coulomb collisions, when electrons and ions are simply deflected from the grain surface. We investigate the regimes for which Coulomb collisions can be treated under the small-angle-deflection approximation. If the intergrain average spacing is equal to or smaller than the Debye length, the collision frequencies of plasma species with dust grains can be much larger than any collision frequency of the plasma species amongst themselves. In the case of Brillouin stimulated scattering, other important contributions to damping come from Landau and dust-charge fluctuation damping. In the case of dust–Brillouin stimulated scattering, the most important contribution to damping comes from dust-charge fluctuation (if the intergrain average spacing is equal to or smaller than the Debye length) and plasma drag on the dust particles (if the intergrain average spacing is larger than the Debye length). We derive the instability thresholds as a function of the density of the dust grains. Because of the inclusion of the new effects, in both Raman and Brillouin scatterings it is found that the instability threshold powers are drastically increased relative to the dust-free case. In the case of dust–Brillouin scattering, a minimum for the threshold power is found in the transition region between ‘dusty’ and ‘dust-in’ plasma. Growth rates near thresholds are also discussed.

This paper and a forthcoming one develop the spectral theory of ballooning transformations relevant to tokamak physics from first principles in a rigorous and yet intuitively clear manner. The power of the ballooning representation to throw light on the spectral characteristics of the plasma problems to which it is applicable is emphasized, and examples are given to illustrate the general notions. The ballooning representation is shown to be essentially a method to separate variables and reduce two-dimensional partial differential equations with periodic coefficients to infinite sets of soluble ordinary differential equations. This paper is concerned with an elementary approach to the techniques in the context of nearly exactly soluble problems involving the anisotropic diffusion operator in toroidal geometry. Two different perturbation methods are discussed. Applications to plasma instability problems and the subtleties involving the continuous spectra of ballooning operators will be taken up in Part 2.