Table of contents

The Inverse Spectral Problem for a general N × N spectral equation in one space dimension is generalized to include poles of arbitrary orders.

The Eden growth model has been modified to generate dense branched structures in which a unique directed path connects each occupied site to the cluster origin. A measure μ(r) can be assigned to each site on the cluster at position r such that if this site were removed μ(r) other sites would be disconnected from the cluster. By focussing attention on only those sites with large values of μ(r) tenuous ramified structures emerge that resemble a variety of natural tree-like structures. Although these structures superficially resemble those associated with fractal growth models, they can be described in terms of a crossover from a dimensionality of 1 on short length scales to 2 on long length scales. It is suggested that some real networks might exhibit such a crossover geometry and that this should be considered as an alternative to a fractal network with a non-integer dimensionality.

The radius of the bound state of a particle obeying the Dirac equation with a separable interaction is compared to the radius of the bound state of the particle obeying the Schrödinger equation with a phase equivalent separable potential. By appropriate choice of parameters these bound states can be made to simulate the deuteron. Unlike the conjecture based on the results of previous one-dimensional model calculations, a small but significant difference between the relativistic and nonrelativistic radii is found.

Absorption, excitation and time-resolved emission spectra of xenon are measured and analysed. Coherent pulsed VUV light in the wavelength range 1400-1600 Å, generated by nonlinear frequency mixing in Mg vapour, is used as the source of excitation. Vibrational structure of the absorption and excitation spectra of Xe is detected, and two different relaxation channels of electronically excited xenon molecules are investigated.

The equation allowing one to calculate the contribution of various interactions to the semiempirical parameters is derived both for orthogonal and nonorthogonal parameters. This equation is applied to the second order calculation of the configuration mixing 3d⁹ 4d + 3d⁹ 5s and 3d⁹ 4d + 3d⁸ 4s² contribution to the parameters values when fitting the 3d⁹, 4d levels in nickel-like spectra of Ga IV, Ge V, As VI.

It is shown that Raman spectrum of magneto-electric crystals will have additional Stokes and anti-Stokes lines over and above the normal ones on account of violation of time reversal invariance.

Generation of nonlinear plasma oscillations by intense short laser pulses is considered. It is shown that longitudinal plasma waves with relativistic amplitudes and phase velocities close to the speed of light c can be excited in the wake of a strong single laser pulse in a cold plasma. The analytical as well as numerical studies of the relativistic wake-field generation by the laser pulses with realistic shapes are carried out. The nonlinear scheme of the laser wake-field accelerator is suggested. It is found that the maximum amplitude of the excited plasma waves is to be expected at the wave phase velocity v_p < c and not at v_p ≃ c. A second laser pulse sent after the first one is considered and shown that this trailing pulse can both amplify and "absorb" the wake-field created by the first laser pulse. Applications of our results to the proposed Livermore High-Brightness Lasers are discussed.

The dynamics of a nonequilibrium high-frequency discharge under high pressure in a converging cylindrical wave under the condition of intensive gas heating is investigated. The gas dynamic processes are considered within the framework of the isobaric approximation. It is shown that the space-time evolution of the discharge depends essentially on the amplitude of the wave. At small amplitudes, where the field intensity is only slightly above break-down in the centre, at first the electron temperature T_e and density rapidly (explosively) grow up to comparatively high values. The size of the arising plasma formation is much smaller than the wave-length and the skin layer (plasma cord). Later, due to the strong screening by the discharge plasma, the field weakens and n_e, T_e slowly get smaller (in fact it is the regime of dying plasma cord). At large amplitudes and average gas-heating speed the diminishing of the gas density occurs to be an accompanying factor and the dynamics of the discharge is the same as in the case of nonessential Joule heating.

The work considers the possibility of a rigorous problem statement of a plasma electrostatic probe on the basis of an accurate solution to the boundary kinetic problem. For this purpose the effect of screening by the solid plasma has thoroughly been analysed as a physical basis of probe diagnostics. This effect within a one-dimensional kinetic problem for the collisionless plasma has been found to take place only for cylindrical and spherical geometry of a solid. In case of plane geometry there is no effect.

There have been obtained non-equilibrium distribution functions of charged particles describing interaction of cylindrical and spherical probes emitting thermoelectrons with a two-component plasma. Poisson equations and expressions for current density have been stated that describe an electric exchange of the plasma and probes in total. The results obtained have been compared with those of Allen-Bohm's probe theory. Allen-Bohm's theory has turned out to be a particular case of the probe theory developed in the work.

Compact ignition experiments can be designed to attain peak densities n₀ ≃ 10²¹ m^-3 reliably and to achieve "low temperature" D-T ignition under conditions where the heating power of the α-particles does not exceed twice the ohmic heating power. An average current density near 1 kA/cm² with a maximum plasma current I_p ≃ 12 MA are the reference design parameters of the representative Ignitor Ult machine. The peak density value is consistent with that of B/R₀, as R₀ ≃ 1.3m and B_Υ ≃ 13 T, when compared to the Alcator, FT, and TFTR machines. The vacuum toroidal field B_Υ is reinforced by the contribution of a large paramagnetic poloidal current (I_θ < 10 MA). The high values of the poloidal field B_p (≃3.9T) produce a strong rate of ohmic heating, while the correspoinding high values of I_p ensure that most of the fusion α-particles can be confined to deposit their energy in the central part of the plasma column. The confinement parameter n₀ τ_E is expected to exceed ≃4 x 10²⁰sec/m³ by a considerable margin on the basis of the confinement properties of plasmas with prevalent ohmic heating. The necessary degree of plasma purity (Z_eff ≲ 1.6) is expected to be ensured by the high particle densities, the high magnetic fields and the relatively low values of the thermal loading on the first wall. Since ignition is achieved through transient conditions rather than, as is frequently assumed, a sequence of steady state conditions, numerical simulations have shown that the ohmic power can remain considerable up to ignition, when programming the rise of the plasma current and the particle density while gradually increasing the cross section of the plasma. The volume where the magnetic parameter q ⩽ 1 and macroscopic m⁰ = 1 plasma modes may be excited, has been shown to maintain quite low values until ignition is attained. The stability margin against these modes is made uniquely favorable by the low values of β-poloidal that are characteristic of Ignitor. An injected heating source of 16MW of ICRF power is included to broaden the scenarios under which ignition is achieved.

The influence of sawtooth oscillations on the alpha-particle heating in an ITER-like reactor is investigated. It is shown that both the radial distribution of the fast alpha energy and the alpha power deposition are strongly dependent on the ratio of the sawtooth oscillation period to the time of Coulomb slowing-down of fast alpha particles. Different values of this ratio are leading to different scenarios of alpha heating in ITER.

A detailed investigation is made of the effect of finite confinement time and radial diffusion on the burn up of fusion produced 1 MeV tritons in hot deuterium plasmas. Explicit expressions are obtained for the concomitant reduction in total triton burn up and applications to JET and TFTR burn up measurements indicate diffusion constants D ≅ 0.1-0.2m²/s for the fast tritons if the reduction is due to radial diffusion. The time dependence and the width of the emission profile of the 14 MeV neutrons from the triton burn up are also analyzed. Recent measurements of 14 MeV emission profiles on JET indicate a much smaller diffusion constant, viz. D ≪ 0.1 m²/s. This implies that the observed finite confinement time of the tritons should not be due to radial diffusion and that the diffusion constant of the fast tritons should be very small.

Standard Fokker-Planck codes calculating the α particle distribution function evolution describe the simple balance between the fusion source term and the collision operator neglecting radial diffusion. In the present paper the effects of radial α-diffusion have been studied by means of a modified kinetic equation containing loss terms due to α-particle and heat transport. Energy dependent transport coefficients are admitted. As a second method, the effect of energy dependence of the spatial α diffusion coefficient is included using a multi group set of α-particle continuity equations. We have determined the sensitivity of the α-evolution to the energy dependence of a diffuson coefficient D_α(E) for 2 cases:

(i) a generic model for an anomalous D_α(E).

(ii) D_α(E) corresponding to stochastic ripple losses.

Close collisions of fusion alpha-particles with thermal plasma ions (both of impurities and of main component) give rise to formation of high energy "halo" in the ion velocity space, extending to the energies of MeV-range even for rather heavy ions, like iron. The implications of this process for the diagnostics of the burning plasmas have been discussed. There seem to be possible two versions of the diagnostics: (i) measurement of Doppler contours of the lines radiated by noncompletely stripped impurity ions; (ii) detection of prompt losses of light impurities. In both versions the spatial resolution can be attained by injecting into the plasma the pellets containing the desired impurities. Another option is to detect the prompt losses of 7MeV alpha-particles formed by collisions between 3.5 MeV alpha-particles.

A reduced MHD fluid model for the unstable toroidicity-induced shear Alfvén eigenmode (TAE) is described. This consists of four coupled time evolution equations for the poloidal magnetic flux, toroidal component of vorticity, energetic particle density and parallel flow velocity, which are solved numerically using the three-dimensional initial value code FAR in toroidal geometry. The TAE mode is readily excited and exhibits similar scalings as have been predicted analytically.

The trapped electron collisional damping of global toroidicity-induced Alfvén eigenmodes (TAE-modes) is investigated. The collisions of trapped electrons with passing ones and ions lead to the essentially enhanced damping rate of TAE-modes in comparison with usually assumed damping due to the magnetic curvature drift of the electrons.

Effects of nonlocal coupling between the fusion alpha particle current density and the fast magnetosonic wave electric field on cyclotron absorption of these waves by alphas are investigated. Such a nonlocality being due to the alpha particle motion along magnetic field lines is very strong because the alpha particle collisions with electrons and bulk ions are very rare and the collisional decorrelation of the wave-particle phase is weak. An alpha particle performs dozens of revolutions around the torus before the dephasing occurs. The expressions for the alpha particle current density and RF power absorbed by alphas in a tokamak are obtained and analysed. They account for the collisional decorrelation, finite Larmor radius and the magnetic drift of a phase.

The effect of a poloidal magnetic field null (X-point region) on the orbits of energetic fusion products is analysed. It is found that the resulting poloidal distribution of lost particles is strongly peaked in the vicinity of the X-point, for an axisymmetric magnetic field distribution. This effect should be considered, together with the effect of field ripple, in the analysis of energetic particle diagnostics and of loading of wall structures.

Microwave scattering by the fluctuations of fusion plasmas is one of the most promising α-diagnostic techniques. Previous investigations have concentrated on the fluctuations near the slow wave branch in the lower hybrid range of frequencies. The small signal and the lack of sensitivity to the contribution of α-particles to the total cross-section near the slow branch severely limits the effectiveness of this technique. In this paper, we report results of investigations of scattering by fluctuations in the lower hybrid range of frequencies near the fast branch. Surprisingly, when both fast and slow branches exist, the scattering amplitudes are comparable. More important, the α-contribution is larger for the fast branch and the fast branch has a larger parameter space where it exists. Specifically, the slow branch exists only above the lower hybrid frequency, while the fast branch can exist at all frequencies up to the electron cyclotron range of frequencies. We find numerically that the scattering amplitudes near the fast branch below the lower hybrid frequency are several orders of magnitude larger than those near the slow branch above that frequency where it can exist. This may make microwave scattering by fast waves a more attractive α-diagnostic technique.

The high energy proton component generated by the high power (~ 10 MW) NBI with beam ions energy ~ 1 MeV into the helium plasma is investigated. The resulting proton component is shown to simulate the alphas in a thermonuclear reactor quite well. The protons driven collective effects are reported. It is shown that in a tokamak with high energy NBI the experimental investigation of hot protons driven TAE modes and the determination of the threshold value of the proton beta of TAE excitation will be possible in a wide plasma parameters range.

The dynamics of plasma temperature profiles depends on the simultaneous processes of diffusion and creation, e.g. heating by alpha particles, as well as losses, e.g. bremstrahlung, and furthermore on the effects of boundaries, which play an essential role in the formation of equilibria. The temperature evolution is governed by a partial differential equation of the reaction-diffusion type. The heating by alpha particles is represented by a source term which depends on temperature to a certain power. The present contribution summarizes the concepts and recent results of a new technique of analysis based on a central expansion in the spatial variable. The problem of studying the nonlinear partial differential equation governing the evolution of a temperature profile is transformed to a description in terms of three characteristic dynamic variables (amplitude, width and shape) which depend only on time and which are governed by three coupled first order differential equations. Such equations allow for dynamic description of selforganisation in open as well as bounded systems. The technique simplifies the descripion of the dynamic properties of the temperature evolution of a burning fusion plasma. It provides physical insight in the dynamics of the system by analysis of the interplay between the various simultaneous processes. In particular, the problems of existence, accessibility and approach to possible equilibria can be studied by means of the new technique.

Results are presented accounting for a selection of extensive numerical calculations of the evolution in space and time for a dynamic system, governed by a nonlinear diffusion equation with source and loss terms (which could also be nonlinear). The system is assumed to be circularly symmetric and limited by an external boundary where the value of the depending variable (e.g. temperature) is supposed to be zero.