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Internal Gravity Waves

Gary A. Glatzmaier

in Introduction to Modeling Convection in Planets and Stars: Magnetic Field, Density Stratification, Rotation

This chapter focuses on internal gravity waves in a stable thermal stratification. When the amplitude of the fluid velocity is small relative to the amplitude of the phase velocity, a linear ... More

This chapter focuses on internal gravity waves in a stable thermal stratification. When the amplitude of the fluid velocity is small relative to the amplitude of the phase velocity, a linear analysis, which neglects advection, provides insight to the relation between the wavelength and frequency of internal gravity waves. Furthermore, when thermal and viscous diffusion play relatively minor roles the system can be further simplified by neglecting diffusion. The chapter first describes the linear dispersion relation before discussing the computer code modifications and simulations. In particular, it explains what modifications would be needed to convert one's thermal convection code to a code that simulates internal gravity waves, including the nonlinear and diffusive terms. Finally, it considers the computer analysis of wave energy.Less

Magnetic Field

Gary A. Glatzmaier

in Introduction to Modeling Convection in Planets and Stars: Magnetic Field, Density Stratification, Rotation

This chapter focuses on magnetoconvection, which refers to thermal convection of an electrically conducting fluid within a background magnetic field maintained by some external mechanism. It first ... More

This chapter focuses on magnetoconvection, which refers to thermal convection of an electrically conducting fluid within a background magnetic field maintained by some external mechanism. It first provides a brief overview of magnetohydrodynamics and the magnetohydrodynamic equations before explaining how to make a 2D model of magnetic field. In this approach, the case of a uniform vertical background field and the case of a uniform horizontal background field are both considered. The chapter then describes how one could simulate a case of a uniform background field that is tilted relative to both the vertical and horizontal axes. It also considers what can be learned about the stability and structure of magnetoconvection and the dispersion relation for magneto-gravity waves from analytical analyses without the nonlinear terms. Finally, it discusses nonlinear simulations of magnetoconvection in a box with impermeable side boundaries, along with magnetoconvection with a horizontal background field and an arbitrary background field.Less

Doodling toward a New “Theory”

in Drawing Theories Apart: The Dispersion of Feynman Diagrams in Postwar Physics

In the spring of 1964, Jerry Finkelstein, a graduate student in Berkeley's physics department, summarized material culled from earlier course notes—courses that had covered several topics in ... More

In the spring of 1964, Jerry Finkelstein, a graduate student in Berkeley's physics department, summarized material culled from earlier course notes—courses that had covered several topics in theoretical particle physics, many of which his adviser, Geoffrey Chew, had first pioneered or championed only a few years earlier. Finkelstein's study notes comprised simple line drawings, though the diagrams looked to be Feynman diagrams to the casual observer. Finkelstein called the examples in his notes by different names such as Landau graphs, polology diagrams, and Cutkosky diagrams. This chapter discusses these examples in detail. Many particle theorists greeted the failure of perturbative techniques for the strong interactions by working with an alternative representation of particles' scatterings, in the form of dispersion relations. Theorists such as Murph Goldberger, Murray Gell-Mann, Francis Low, Geoffrey Chew, and Nambu Yoichiro launched the new phase of dispersion-relations work.Less

Hydrodynamic stability

Stefano Atzeni and JÜrgen Meyer-Ter-Vehn

in The Physics of Inertial Fusion: BeamPlasma Interaction, Hydrodynamics, Hot Dense Matter

This chapter is devoted to hydrodynamic instabilities. Internal confinement fusion (ICF) capsule implosions are inherently unstable. In particular, the Rayleigh-Taylor instability (RTI) developing at ... More

This chapter is devoted to hydrodynamic instabilities. Internal confinement fusion (ICF) capsule implosions are inherently unstable. In particular, the Rayleigh-Taylor instability (RTI) developing at the beam accelerated capsule outer surface tends to destroy the imploding shell, while the deceleration-phase RTI occurring at the inner surface of the stagnating capsule hinders the formation of a central hot spot. Control of this instability is a major challenge facing ICF. Richtmyer-Meshkov (RMI) and Kelvin-Helmholtz (KHI) instabilities also occur in ICF. Starting from basic theory, the instability linear theory is developed in much detail, including the stabilizing effect of ablation on RTI (ablative stabilization). The resulting dispersion relation is then applied to actual ICF implosions, deriving the admissible levels of non-uniformity in capsule make and implosion drive. The nonlinear growth of bubbles and spikes, including turbulent mixing are also described.Less

Custodian of quantum field theory

JAGDISH MEHRA and KIMBALL A. MILTON

in Climbing the Mountain: The Scientific Biography of Julian Schwinger

Julian Schwinger had now scaled the peak of quantum electrodynamics (QED), not once, but three times, the last time by inventing a new approach to any quantum-mechanical system, the quantum dynamical ... More

Julian Schwinger had now scaled the peak of quantum electrodynamics (QED), not once, but three times, the last time by inventing a new approach to any quantum-mechanical system, the quantum dynamical principle. Now the task of the field theorist, as was already apparent in the 1930s, was to build upon this success of QED and apply the powerful machinery invented to understand the strong and weak nuclear interactions. This chapter describes the story of Schwinger's work in the central period between the quantum field theory revolutions of the late 1940s and the early 1970s, roughly during the period 1957 through 1965. Schwinger's work on the phenomenological field theory, dispersion relations, spin and the TCP theorem, Euclidean field theory, gauge invariance and mass, quantum gravity, and magnetic charge are examined.Less

Introduction to the Mathematics of Rays

John A. Adam

in Rays, Waves, and Scattering: Topics in Classical Mathematical Physics

This chapter provides an overview of the mathematics of rays. It begins with a discussion of the theory of geometrical optics and how it can be formulated by means of the ray equations or the ... More

This chapter provides an overview of the mathematics of rays. It begins with a discussion of the theory of geometrical optics and how it can be formulated by means of the ray equations or the Hamilton-Jacobi equation. The two equations are of seemingly different types, but they are in fact equivalent. The ray equations are the characteristic equations of the Hamilton-Jacobi equation. This remark leads to the geometrical interpretation: the family of rays of geometrical optics is perpendicular to the wavefronts S = constant, if S denotes the appropriate solution of the Hamilton-Jacobi equation. The chapter considers the Hamilton-Jacobi theory in more detail, along with Hamilton's principle, ray differential geometry and the eikonal equation, and dispersion relations. It also presents the general solution of the linear wave equation before concluding with an analysis of the behavior of rays and waves in a slowly varying environment.Less

Propagating surface plasmons

Henri Benisty, Jean-Jacques Greffet, and Philippe Lalanne

in Introduction to Nanophotonics

This chapter deals with surface waves propagating along a metallic interface called surface plasmon polaritons. We first introduce the concept of polariton, a mixed light-matter excitation of solids ... More

This chapter deals with surface waves propagating along a metallic interface called surface plasmon polaritons. We first introduce the concept of polariton, a mixed light-matter excitation of solids which has remarkable properties in order to confine electromagnetic energy. We discuss in detail its dispersion relation. We then discuss gap surface plasmon polaritons propagating between two parallel metallic planes with subwavelength separation. We finally briefly introduce plasmons on graphene, surface phonon polaritons and discuss the contribution of surface polaritons to the local density of states.Less

Functions of a complex variable

Chun Wa Wong

in Introduction to Mathematical Physics: Methods & Concepts

Functions of a complex variable are shown to be more complete and rigid than functions of a real variable. Analytic functions with well defined derivatives satisfy two Cauch–Riemann conditions. ... More

Functions of a complex variable are shown to be more complete and rigid than functions of a real variable. Analytic functions with well defined derivatives satisfy two Cauch–Riemann conditions. Multivalued functions can be made single-valued on a multi-sheet Riemann surface. The values of an analytic function in a region of the complex plane are completely defined by the knowledge of their values on a closed boundary of the region. Important properties and techniques of complex analysis are described. These include Taylor and Laurent expansions, contour integration and residue calculus, Green's functions, Laplace transforms and Bromwich integrals, dispersion relations and asymptotic expansions. Analytic functions are defined by their properties at the locations called singularities (poles and branch cuts) where they cease to be analytic. This feature makes analytic functions of particular interest in the construction of physical theories.Less

Phonons

Ian R. Kenyon

in Quantum 20/20: Fundamentals, Entanglement, Gauge Fields, Condensates and Topology

Phonons are introduced as an example of quasi-particles that can only exist in matter. Debye’s quantum model for heat capacity of solids and comparison with experimentin different temperature ranges ... More

Phonons are introduced as an example of quasi-particles that can only exist in matter. Debye’s quantum model for heat capacity of solids and comparison with experimentin different temperature ranges is presented. The dispersion relations of lattice vibration (phonons) and quantization for chains of atoms presented, revealing the optical and acoustic modes; anharmonic effects are discussed. Crystal lattice structures and Brillouin zones are introduced. Phonon scattering and the Umklapp process described. The variation of the thermal conductivity of dielectrics with temperature is interpreted. X-ray scattering studies of phonon dispersion relations are described. Coupling between phonons with photons in polaritons is explained: Raman scattering studies of GaN used to exhibit the cross-over of their dispersion relations. The Mössbauer effect, a recoilless process, and its dependence on temperature are explained.Less

Waves and Plasma in the Earth’s High Atmosphere

Gino Segrè and John Stack

in Unearthing Fermi's Geophysics

In this chapter, the chief physics effects in the high atmosphere, the region also known as the ionosphere, are treated. It is assumed to be a dilute two-fluid ionized gas or plasma consisting of ... More

In this chapter, the chief physics effects in the high atmosphere, the region also known as the ionosphere, are treated. It is assumed to be a dilute two-fluid ionized gas or plasma consisting of free electrons and one species of ions. Taking pressure as negligible, Euler-type equations in the presence of electromagnetic forces describe the flow of electrons. Supposing further that velocities and electromagnetic fields can be treated in first order, the equations of motion for the electromagnetic fields are derived and a dispersion relation is then obtained for electromagnetic waves in the presence of the plasma. This relation between frequency and wave number involves the density of electrons, showing how it is that radio waves are reflected in the ionosphere. The situation is studied further for the case when an external magnetic field, such as that of the Earth, is present. If the plasma is propagating parallel to this field, it is shown how right and left circularly polarized waves have different dispersion relations, a phenomenon known as double refraction. Spray electrification and electrical storms are given very brief discussions.Less

Surface Gravity Waves

John A. Adam

in Rays, Waves, and Scattering: Topics in Classical Mathematical Physics

This chapter deals with the underlying mathematics of surface gravity waves, defined as gravity waves observed on an air–sea interface of the ocean. Surface gravity waves, or surface waves, differ ... More

This chapter deals with the underlying mathematics of surface gravity waves, defined as gravity waves observed on an air–sea interface of the ocean. Surface gravity waves, or surface waves, differ from internal waves, gravity waves that occur within the body of the water (such as between parts of different densities). Examples of gravity waves are wind-generated waves on the water surface, as well tsunamis and ocean tides. Wind-generated gravity waves on the free surface of the Earth's seas, oceans, ponds, and lakes have a period of between 0.3 and 30 seconds. The chapter first describes the basic fluid equations before discussing the dispersion relations, with a particular focus on deep water waves, shallow water waves, and wavepackets. It also considers ship waves and how dispersion affects the wave pattern produced by a moving object, along with long and short waves.Less

Analyticity Properties of Feynman Diagrams

Laurent Baulieu, John Iliopoulos, and Roland Sénéor

in From Classical to Quantum Fields

General discussion of the analyticity properties of Feynman diagrams. Cutting rules and the Cutkosky unitarity relations. Unstable particles and second sheet singularities. Dispersion relations and ... More

General discussion of the analyticity properties of Feynman diagrams. Cutting rules and the Cutkosky unitarity relations. Unstable particles and second sheet singularities. Dispersion relations and the Mandelstam representation. The analytic S-matrix theory and the bootstrap hypothesis.Less

Magnons

Olle Eriksson, Anders Bergman, Lars Bergqvist, and Johan Hellsvik

in Atomistic Spin Dynamics: Foundations and Applications

In this chapter we give several examples of how the multiscale approach for atomistic spin-dynamics, as described in Part I and Part II of this book, performs for describing magnon excitations of ... More

In this chapter we give several examples of how the multiscale approach for atomistic spin-dynamics, as described in Part I and Part II of this book, performs for describing magnon excitations of solids. Due to the recent experimental advancements in detecting such excitations for surfaces and multilayers, we focus here primarily on spin wave excitations of two-dimensional systems. The discussion can easily be generalized to bulk magnets, and in fact some examples of bulk properties are given in this chapter as well. Magnons can be categorized as dipolar and exchange magnons, where the latter are in the range of giga Hz frequency, and are the main focus of this chapter.Less

Polaritons

Monique Combescot and Shiue-Yuan Shiau

in Excitons and Cooper Pairs: Two Composite Bosons in Many-Body Physics

Chapter 15 deals with exciton-polaritons. Starting from the polariton creation operator, a many-body formalism analogous to the one for excitons is constructed. In particular, the chapter identifies ... More

Chapter 15 deals with exciton-polaritons. Starting from the polariton creation operator, a many-body formalism analogous to the one for excitons is constructed. In particular, the chapter identifies a novel photon-assisted exchange scattering that has similarity with the exchange interaction scattering existing for Frenkel excitons and for Cooper pairs. The chapter further discusses microscopic derivations of exciton-polaritons; these derivations depend on approximations that are made concerning photon-semiconductor interaction. The chapter also points out the intrinsic many-body nature of a “single” polariton for photons not close to photon-exciton resonance. Microcavity polaritons and the crucial role of photon confinement in the polariton dispersion relation are also discussed.Less

Electrons in solids

Ian R. Kenyon

in Quantum 20/20: Fundamentals, Entanglement, Gauge Fields, Condensates and Topology

Electron energy bands in solids are introduced. Free electron theory for metals is presented: the Fermi gas, Fermi energy and temperature. Electrical and thermal conductivity are interpreted, ... More

Electron energy bands in solids are introduced. Free electron theory for metals is presented: the Fermi gas, Fermi energy and temperature. Electrical and thermal conductivity are interpreted, including the Wiedermann–Franz law. The Hall effect and information it brings about charge carriers is discussed. Plasma oscillations of conduction electrons and the optical properties of metals are examined. Formation of quasi-particles of an electron and its screening cloud are discussed. Electron-electron and electron-phonon scattering and how they affect the mean free path are treated. Then the analysis of crystalline materials using electron Bloch waves is presented. Tight and weak binding cases are examined. Electron band structure is explained including Brillouin zones, electron kinematics and effective mass. Fermi surfaces in crystals are treated. The ARPES technique for exploring dispersion relations is explained.Less

Perfect Crystal Neutron Optics

Helmut Rauch and Samuel A. Werner

in Neutron Interferometry 2nd Edn: Lessons in Experimental Quantum Mechanics, Wave-Particle Duality, and Entanglement

The perfect crystal, LLL-geometry, neutron interferometer is geometrically analogous to the classical Mach–Zehnder interferometer. Its operation depends in exquisite detail on the dynamical theory of ... More

The perfect crystal, LLL-geometry, neutron interferometer is geometrically analogous to the classical Mach–Zehnder interferometer. Its operation depends in exquisite detail on the dynamical theory of diffraction in a perfect crystal. Although understanding the basic ideas of most of the experiments discussed in this book does not depend on these details, actually carrying out experiments does. This chapter is devoted to a dynamical diffraction calculation of the operation of a three-crystal LLL interferometer. A description of the importance of the Pendellösung interference fringes is discussed. The spatial profiles of the beams traversing and exiting the interferometer are calculated and graphically displayed. The original Pendellösung interference experiments of Shull are discussed. The multiple reflection process of neutrons within each crystal plate is discussed and calculated using the Takagi–Taupin equations, showing how the spatial width of the beams increases upon traversing each crystal blade.Less

Guided Waves in Plates and Rods

Dale Chimenti, Stanislav Rokhlin, and Peter Nagy

in Physical Ultrasonics of Composites

In this chapter we consider elastic wave modes which propagate in composites with finite boundaries. There are those waves that exist between the two plane parallel boundaries of a homogeneous ... More

In this chapter we consider elastic wave modes which propagate in composites with finite boundaries. There are those waves that exist between the two plane parallel boundaries of a homogeneous anisotropic solid. We consider that well-known problem, as well as waves in an elastic anisotropic rod, specifically an individual graphite fiber. Composite laminates seen in applications are essentially all multilayered structures, and in many cases can be considered periodically layered. So, we also take up the subject of guided waves in layered plates in later chapters. In a plate geometry, as illustrated in Fig. 5.1, we choose the propagation direction to be parallel to the x1 axis and the x3 axis to be normal to the plate surfaces. This geometry is particularly significant for composite materials since, by design, laminates are often locally planar in nature. While the solutions we find are appropriate for flat plates, with some modifications they describe wave motion in gently curved structures as well. Clear and mathematically straightforward descriptions of the characteristics of plate waves exist for isotropic media. The results obtained for isotropic media are not, however, directly applicable to most composites. We begin by considering the behavior of waves in a uniaxial composite laminate. In later chapters we generalize the calculation to layered orthotropic media, concentrating on the results and physical interpretation rather than the algebraic details. To begin a description of waves in plates, let us consider the possible polarizations of particle motion. Let the plate surfaces lie in the (x1, x2) plane of mirror symmetry with the origin dividing the plate thickness in half, as shown in Fig. 5.1. Then, we will at first assume the wave to be uniform in the x2 direction and propagating in the x1 direction, and (x1, x3) is the plane of symmetry. Particle motion can occur along any axis. Note that in this restricted symmetry, shear partial waves polarized along the x2 axis will have no component of particle motion normal to the plate surfaces. Partial waves are a concept introduced by Rayleigh to acknowledge that a superposition of both shear and longitudinal particle motion is generally needed to produce plate waves polarized in the vertical plane. Less

Some applications of Maxwell’s equations in matter

J. Pierrus

in Solved Problems in Classical Electromagnetism: Analytical and Numerical Solutions with Comments

This chapter comprises questions of a miscellaneous nature. They mostly have little in common except that all processes are time-dependent and occur within matter. The first few questions introduce ... More

This chapter comprises questions of a miscellaneous nature. They mostly have little in common except that all processes are time-dependent and occur within matter. The first few questions introduce some important preliminaries. For example, modifying Maxwell’s equations to include the effect of matter. The behaviour of the electromagnetic field at the boundary between two media having different properties is an important topic. The matching conditions (as they are known) are derived from both the integral and differential forms of Maxwell’s equations. Certain specific examples then follow, including some simple applications involving conductors, dielectrics and tenuous electronic plasmas. Along the way, the connection between Maxwell’s electrodynamics and the laws of geometrical optics is demonstrated explicitly.Less

Surface Wind Waves

Eric B. Kraus and Joost A. Businger

in Atmosphere-Ocean Interaction

Rhythmic and monotonously repetitive, but quite unpredictable in its details, the structure of the sea surface is an epitome of the natural world. Surface waves have been studied actively by ... More

Rhythmic and monotonously repetitive, but quite unpredictable in its details, the structure of the sea surface is an epitome of the natural world. Surface waves have been studied actively by mathematicians and physicists since the dawn of modern science. Though the phenomenon seems deceptively simple, it cannot be explained or predicted rigorously by existing theories. Nonlinear interactions between wind, waves, and currents cause theoretical problems as well as make it difficult to obtain comprehensive, interactive data sets. In response to wind and pressure changes at the air-sea interface, the ocean reacts with waves that occupy some nine spectral decades: from capillary waves, which undulate within a fraction of a second over distances smaller than one centimeter, to planetary waves with periods measured in years and wavelengths of thousands of kilometers. The dynamics of all these waves can be related to the set of equations discussed in Section 4.1. For that reason, a consideration of all wave forms could have been combined in the same chapter, but we found it more convenient to divide the subject into two parts. The present chapter deals exclusively with wind-generated waves at the sea surface. They determine the small-scale configuration of the air-sea interface and that affects the turbulent transfers, which are the topic of the following chapter. On the other hand, information and energy transports from the sea surface into the ocean interior by internal and inertial waves, depend upon the state of the upper layers of the ocean. This made it desirable to discuss these wave forms in Chapter 7, after the consideration of planetary boundary layers in Chapter 6. Small-amplitude or linear, harmonic surface waves are considered in Section 4.2. Analysis of these waves has been the classic approach to the topic. Linear waves represent an idealized abstraction, but their analysis does provide basic insights into actual wave dynamics. Linear approximations have to be abandoned when one considers the energy and the momentum of wave fields. This is the topic of Section 4.3. In Section 4.4 we discuss the various sources and sinks of wave energy and momentum. Less

Introduction To Solids And Their Interfaces

Abraham Nitzan

in Chemical Dynamics in Condensed Phases: Relaxation, Transfer and Reactions in Condensed Molecular Systems

The study of dynamics of molecular processes in condensed phases necessarily involves properties of the condensed environment that surrounds the system under consideration. This chapter provides ... More

The study of dynamics of molecular processes in condensed phases necessarily involves properties of the condensed environment that surrounds the system under consideration. This chapter provides some essential background on the properties of solids while the next chapter does the same for liquids. No attempt is made to provide a comprehensive discussion of these subjects. Rather, this chapter only aims to provide enough background as needed in later chapters in order to take into consideration two essential attributes of the solid environment: Its interaction with the molecular system of interest and the relevant timescales associated with this interaction. This would entail the need to have some familiarity with the relevant degrees of freedom, the nature of their interaction with a guest molecule, the corresponding densities of states or modes, and the associated characteristic timescales. Focusing on the solid crystal environment we thus need to have some understanding of its electronic and nuclear dynamics. The geometry of a crystal is defined with respect to a given lattice by picturing the crystal as made of periodically repeating unit cells. The atomic structure within the cell is a property of the particular structure (e.g. each cell can contain one or more molecules, or several atoms arranged within the cell volume in some given way), however, the cells themselves are assigned to lattice points that determine the periodicity. This periodicity is characterized by three lattice vectors, ai, i = 1, 2, 3, that determine the primitive lattice cell—a parallelepiped defined by these three vectors. Less