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A technique of fitting a modified hyperbolic tangent to the edge profiles has improved the localization of plasma edge parameters. Non-dimensional edge parameters are broadly consistent with several theories of the L-H transition that use edge gradients in their formulation of a critical threshold parameter. The ion (nabla)B drift direction has only a small effect on the edge plasma conditions measured near the plasma midplane but a large effect on the divertor plasma. The dramatic change of power threshold with the direction of the ion (nabla)B drift implies that phenomena in the divertor region may be critical for the L-H transition.
Systems with competing energy scales are widespread and exhibit rich and subtle behaviour, although their systematic study is a relatively recent activity. This text presents lectures given at a NATO Advanced Study Institute reviewing the current knowledge and understanding of this fascinating subject, particularly with regard to phase transitions and dynamics, at an advanced tutorial level. Both general and specific aspects are considered, with competitions having several origins; differences in intrinsic interactions, interplay between intrinsic and extrinsic effects, such as geometry and disorder; irreversibility and non-equilibration. Among the specific physical application areas are supercooled liquids and glasses, high-temperature superconductors, flux or vortex pinning and motion, charge density waves, domain growth and coarsening, and electron solidification.
Phase transitions in which crystalline solids undergo structural changes present an interesting problem in the interplay between the crystal structure and the ordering process. This text, intended for readers with some prior knowledge of condensed-matter physics, emphasizes the basic physics behind such spontaneous structural changes in crystals. Starting with the relevant thermodynamic principles, the book discusses the nature of order variables and their collective motion in a crystal lattice; in a structural phase transition a singularity in such a collective mode is responsible for the lattice instability, as revealed by soft phonons. This mechanism is analogous to the interplay of a charge-density wave and a periodically deformed lattice in low-dimensional conductors. The text also describes experimental methods for modulated crystal structures and gives examples of structural changes in representative systems. The book is divided into two parts. The first, theoretical, part includes such topics as: the Landau theory of phase transitions; statistics, correlations and the mean-field approximation; pseudospins and their collective modes; soft lattice modes and pseudospin condensates; lattice imperfections and their role in the phase transitions of real crystals. The second part discusses experimental studies of modulated crystals using x-ray diffraction, neutron inelastic scattering, light scattering, dielectric measurements, and magnetic resonance spectroscopy.
Theory of Phase Transitions: Rigorous Results is inspired by lectures on mathematical problems of statistical physics presented in the Mathematical Institute of the Hungarian Academy of Sciences, Budapest. The aim of the book is to expound a series of rigorous results about the theory of phase transitions. The book consists of four chapters, wherein the first chapter discusses the Hamiltonian, its symmetry group, and the limit Gibbs distributions corresponding to a given Hamiltonian. The second chapter studies the phase diagrams of lattice models that are considered at low temperatures. The notions of a ground state of a Hamiltonian and the stability of the set of the ground states of a Hamiltonian are also introduced. Chapter 3 presents the basic theorems about lattice models with continuous symmetry, and Chapter 4 focuses on the second-order phase transitions and on the theory of scaling probability distributions, connected to these phase transitions. Specialists in statistical physics and other related fields will greatly benefit from this publication.
The exact nature of the physics governing the L-H transition seen in tokamak magnetic confinement experiments has eluded fusion researchers for several decades. To date, a first principles model for the transition does not exist. The improved particle and energy confinement realized by the suppression of turbulence in the post-transition H-mode motivates an understanding of the transition and the empirically known conditions necessary for its initiation, generically an input power threshold with key sensitivities to the edge electron density, main ion mass and charge, plasma configuration, divertor conditions, ∇B drift direction, etc. Modern consensus that an increase in the E x B shear at the plasma edge is responsible for the turbulence suppression and formation of a transport barrier invigorates research into possible driving mechanisms. The loss of thermal ions from the imperfectly confining magnetic field of a tokamak manifests as a steady-state radial current in the edge and has long been suspected to play a role in the generation of the E x B shear and hence the L-H transition.The body of this thesis presents the development of a model for the steady-state thermal orbit loss based on the identification of the phase-space loss cone. The presented model boasts several improvements over other loss cone models found in the literature, largely rooted in the careful consideration of local pitch angle scattering on ions within and near the velocity-space boundaries of projections of the phase-space loss cone to observation points in configuration-space. The probability that ions within the loss cone will be lost on a first orbit is estimated by comparing the rates of collisionally scattering out of the loss cone to the periods of orbit loss. The steady-state is determined by the rates of collisional loss cone refueling modified by the statistical chance of first orbit loss. A competition arises between the sufficiently large temperatures necessary for appreciable parts of the distribution to interact with the loss cone and the reduced rate of collisional refueling of high energy ions. The steady-state orbit loss current calculated by the model exhibits several features of the experimentally measured L-H transition power threshold not present in other models. The orbit loss current displays branching behaviors in the edge density, peaking at densities similar to those minimizing the required transition power on ASDEX Upgrade. Additionally, the loss current features the suspected strong ∇B drift direction asymmetry of the orbit loss. The unfavorable drift configuration requires about a factor of two greater input power to produce a similar orbit loss current seen in the favorable drift, again echoing a known behavior of the power threshold. Other explored features that suggest a promising connection between the thermal orbit losses and the transition are the main ion mass and the horizontal position of the X-point. The orbit loss current has been implemented into the edge fluid transport code SOLPS. The first order plasma response to the current is studied over the high-density branch of the loss current. The leading order effect is an increase in the magnitude of the edge Er well and the associated E x B shear. Over the explored parameter space, the input power necessary to reach some threshold Er magnitude lessens on the order of ∼ 10-20% in the presence of the loss current. Thermal ion orbit loss appears capable of influencing the onset of the L-H transition.
The aim of this book is to give, within a single volume, an introduction to the fields of turbulence modelling and transition-to-turbulence prediction, and to provide the physical background for today's modelling approaches in these problem areas as well as giving a flavour of advanced use of prediction methods. Turbulence modelling approaches, ranging from single-point models based on the eddy-viscosity concept and the Reynolds stress transport equations (Chapters 3,4,5), to large-eddy simulation (LES) techniques (Ch. 7), are covered. The foundations of hydrodynamical stability and transition are presented (Ch. 2) along with transition prediction methods based on single-point closures (Ch. 6), LES techniques (Ch. 7) and the parabolized stability equations (Ch. 8). The book addresses engineers and researchers, in industry or academia, who are entering into the fields of turbulence or transition modelling research or need to apply turbulence or transition prediction methods in their work.
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