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This chapter explains how to write a postprocessing code, and more specifically how to study the nonlinear simulations using computer graphics and analysis. It first considers how to compute and store results in a file during the computer simulation, assuming the Fourier transforms to x-space are done within the main computational code during the simulation. It then describes the postprocessing code for reading these files and displaying the various fields, along with the use of graphics software packages that provide additional, more sophisticated visualizations of the scalar and vector data. It also discusses the computer analysis of several additional properties of the solution, focusing on measurements of nonlinear convection such as Rayleigh number, Nusselt number, Reynolds number, and kinetic energy spectrum.

This chapter considers fermion zero modes on vortex lines and cosmic strings. The quantum numbers, which characterize the energy spectrum of these fermionic excitations, are the linear (p_z) and angular (Q) momenta along the string. There are two types of fermion zero modes: true and approximate. The approximate fermion zero modes have spectrum which crosses zero as function of Q. Due to discrete nature of angular momentum these modes have a small gap called minigap. This occurs in the core of vortices in conventional s-wave superconductors. The number of the anomalous branches is determined by the winding number of a vortex. This analog of index theorem is obtained using topology in combined (p,r) space. The true fermion zero modes have spectrum which crosses zero as function of p_z. These modes take place inside vortices in triplet superconductors and ³He-B. The singly quantized vortex in ³He-A contains branch with exactly zero energy for all p_z, while half-quantum vortex contains Majorana fermions. Both true and approximate zero modes are obtained using the semi-classical approach which is valid because the core size is much larger than the inverse wavelength of fermions. Analysis is extended to fermions on asymmetric vortices.

This chapter demonstrates the potentialities of the quasiclassical method for selected problems in the theory of stationary superconductivity. The Ginzburg–Landau equations are derived, the upper critical field of dirty superconductors at arbitrary temperatures is calculated, and the gapless regime in superconductors with magnetic impurities is discussed. Effects of impurities on the critical temperature and the density of states in d-wave superconductors are discussed. The energy spectra of excitations in vortex cores of s-wave and d-wave superconductors are calculated.

Chapter 12 presents a detailed analysis of continuous-time signals and systems in the frequency domain, including the theory of Fourier series and Fourier transforms, and key examples relevant for the analysis and synthesis of signals processed in the digital transceiver blocks of a communication system. The amplitude, magnitude, phase, and power spectra are defined and calculated for typical signals. In particular, the Fourier transform of periodic signals is presented, due to its importance in communication systems theory and practice. Using a unique notation that distinguishes energy and power signals, the correlation, power, and energy spectral density functions are inter-related by proving the Wiener–Khintchine theorem. A comprehensive analysis of a linear-time-invariant system, using the concepts of impulse response, system correlation function, and power spectral density, both for power signals and energy signals, is presented. In addition, Parseval’s theorem and the Rayleigh theorem are proven.

Chapter 8 recasts the dynamical properties of turbulence in spectral space. The chapter starts by redeveloping the kinematics of turbulence in Fourier space, introducing the ideas of the spectral tensor, energy spectrum, one-dimensional spectra, and spectral singularities. The Karman–Howarth equation is then rewritten in spectral space and the associated dynamical processes discussed. Two-point spectral closure models are outlined, along with their deficiencies.

Main notions and ideas of wave (weak) turbulence theory are explained with the help of Hamiltonian approach to wave dynamics, and are applied to waves in RSW model. Derivation of kinetic equations under random-phase approximation is explained. Short inertia–gravity waves on the f plane, short equatorial inertia–gravity waves, and Rossby waves on the beta plane are then considered along these lines. In all of these cases, approximate solutions of kinetic equation, annihilating the collision integral, can be obtained by scaling arguments, giving power-law energy spectra. The predictions of turbulence of inertia–gravity waves on the f plane are compared with numerical simulations initialised by ensembles of random waves. Energy spectra much steeper than theoretical are observed. Finite-size effects, which prevent energy transfer from large to short scales, provide a plausible explanation. Long waves thus evolve towards breaking and shock formation, yet the number of shocks is insufficient to produce shock turbulence.

This chapter discusses various methods that have been used to measure the energy spectrum of electrons in both metals and insulators. For metals a property of paramount importance is the detailed shape of the Fermi surface and the accompanying Fermi velocity, since together they affect transport and other phenomena. Methods to measure these quantities include various magneto-acoustic effects (e.g., the geometric resonance phenomenon), the Gantmakher and Sondheimer size effects, the Azbel–Kaner cyclotron resonance, the anomalous skin effect, the high-field magneto-resistance (as it relates to open versus closed orbits), and various quantum oscillations, the most important of which is the de Hass–van Alphen effect. For determining the energy spectrum at energies below the Fermi surface, the most powerful technique is angle resolved photo-emission spectroscopy (ARPES) which, with the emergence of synchrotron sources, has become increasingly important. Also of growing importance is inverse photo-emission spectroscopy (IPS), a special case of which is termed bremsstrahlung isochromat spectroscopy (BIS). Here the roles of the incoming photon and the final state electron are reversed: electrons of known energy (and angle) impinge on the surface and (in the BIS variant) the energy spectrum of the emitted photons is measured. This technique probes the spectrum for energies above the Fermi energy.

As is usual in the statistical theory of turbulence, the statistical closure problem is formulated in wavenumber space. Equations are derived for the two-time and single-time covariances, in terms of the unknown third-order moment. These equations pose the general closure problem are also the basis of much turbulence phenomenology. They are used to derive the spectral energy balance equation (or Lin equation) and to establish its conservation properties; with and without external stirring forces. Attention is given to both the Edwards and Kraichnan formulations of the triangle condition. In the interests of completeness, the L coefficients, the dimensions of relevant spectral quantities (in both finite and infinite systems), and various useful relationships involving the spectrum are given. The chapter concludes with a treatment of energy conservation in real space. The real-space energy balance (or Karman-Howarth equation) is derived for both freely-decaying and stationary cases. A revisionist treatment of the Karman-Howarth equation which corrects some errors in the turbulence literature is also given.

Chapter 3 discusses the origins and nature of turbulence. The emphasis is on the relationship between non-linearity and chaos and the observable properties of fully developed turbulence. Many elementary ideas are introduced, such as the need for a statistical approach, different methods of taking averages, and Kolmogorov’s theory.

Chapter 12 introduces Graphene, which is a two-dimensional “Dirac-like” material in the sense that its energy spectrum resembles that of a relativistic electron/positron (hole) described by the Dirac equation (having zero mass in this case). Its device-friendly properties of high electron mobility and excellent sensitivity as a sensor have attracted a huge world-wide research effort since its discovery about ten years ago. Here, the associated retarded Graphene Green’s function is treated and the dynamic, non-local dielectric function is discussed in the degenerate limit. The effects of a quantizing magnetic field on the Green’s function of a Graphene sheet and on its energy spectrum are derived in detail: Also the magnetic-field Green’s function and energy spectrum of a Graphene sheet with a quantum dot (modelled by a 2D Dirac delta-function potential) are thoroughly examined. Furthermore, Chapter 12 similarly addresses the problem of a Graphene anti-dot lattice in a magnetic field, discussing the Green’s function for propagation along the lattice axis, with a formulation of the associated eigen-energy dispersion relation. Finally, magnetic Landau quantization effects on the statistical thermodynamics of Graphene, including its Free Energy and magnetic moment, are also treated in Chapter 12 and are seen to exhibit magnetic oscillatory features.

The scenario of heavy-ion reactions around the Fermi energy is explored. The quantum BUU equation is solved numerically with and without nonlocal corrections and the effect of nonlocal corrections on experimental values is calculated. A practical recipe is presented which allows reproducing the correct asymptotes of scattering by acting on the point of closest approach. The better description of dynamical correlations by the nonlocal kinetic equation is demonstrated by an enhancement of the high-energy part of the particle spectra and the enhancement of mid-rapidity charge distributions. The time-resolved solution shows the enhancement of neck formation. It is shown that the dissipated energy increases due to the nonlocal collision scenario which is responsible for the observed effects and not due to the enhancement of collisions. As final result, a method is presented how to incorporate the effective mass and quasiparticle renormalisation with the help of the nonlocal simulation scenario.

Light energy penetrating the sea is diminished almost exponentially with depth with an accompanying drastic change in the energy spectrum as the result of absorption by various components in the seawater. Such a change in the light environment will affect phytoplankton life directly. Accordingly, in the study of light in the sea, much attention has been drawn toward the contribution of phytoplankton to the light field and also how much energy or which parts of the light spectrum are utilized at various depths by phytoplankton. Spectral distribution of underwater irradiance is determined by the processes of absorption and scattering from various components of the seawater. Since absorption plays a much more important role in spectral variation than scattering (Preisendorfer, 1961), the spectral absorption of each component should be studied in order to adequately interpret the variation of spectral irradiance in the sea. The materials absorbing light are phytoplankton, other particles, dissolved organic substances, and the water itself. Spectral characteristics of the light environment in the sea are determined by the variable ratios of these components. Several authors have attempted to measure directly the spectral absorption of individual components in seawater (Kirk, 1980; Okami et al., 1982; Kishino et al., 1984; Carder and Steward, 1985; Weidemann and Bannister, 1986). However, the determination of the absorption coefficient of natural phytoplankton is quite difficult, because no suitable technique is available for the direct measurement of absorption. Accordingly, there is still considerable uncertainty about light absorption by phytoplankton under natural conditions. Phytoplankton photosynthetic efficiency is important for the algae as well as the other organisms in the same ecosystem. Photosynthetic efficiency can be estimated fundamentally from quantum yield, which is obtained by measuring three parameters: photosynthetic rate, spectral downward irradiance, and the spectral absorption coefficient of phytoplankton. The optical system of a recently designed spectral irradiance meter is shown in Fig. 4-1. The meter has two independent cosine collectors that receive downward and upward spectral irradiance, respectively, by rotation of the mirror placed behind the collectors. After collimation, light reflected by the mirror is separated by a beam splitter.

Serious contradictions to the existence of electrons in nuclei impinged in one way or another on the theory of beta decay and became acute when Charles Ellis and William Wooster proved, in an experimental tour de force in 1927, that beta particles are emitted from a radioactive nucleus with a continuous distribution of energies. Bohr concluded that energy is not conserved in the nucleus, an idea that Wolfgang Pauli vigorously opposed. Another puzzle arose in alpha-particle experiments. Walther Bothe and his co-workers used his coincidence method in 1928–30 and concluded that energetic gamma rays are produced when polonium alpha particles bombard beryllium and other light nuclei. That stimulated Frédéric Joliot and Irène Curie to carry out related experiments. These experimental results were thoroughly discussed at a conference that Enrico Fermi organized in Rome in October 1931, whose proceedings included the first publication of Pauli’s neutrino hypothesis.

By using quasi-geostrophic modons constructed in Chapter 6 as initial conditions, rotating-shallow-water modons are obtained through the process of ageostrophic adjustment, both in one- and in two-layer configurations. Scatter plots show that they are solutions of the rotating shallow-water equations. A special class of modons with an internal front (shock) is shown to exist. A panorama of collision processes of the modons, leading to formation of tripoles, nonlinear modons, or elastic scattering is presented. The modon solutions are then used for initialisations of numerical simulations of decaying rotating shallow-water turbulence. The results are analysed and compared to those obtained with standard in 2D turbulence initializations, and differences are detected, showing non-universality of decaying 2D turbulence. The obtained energy spectra are steeper than theoretical predictions for ‘pure’ 2D turbulence, and pronounced cyclone–anticyclone asymmetry and dynamical separation of waves and vortices are observed.