Dynamics of the Equatorial Ocean
portes grátis
Dynamics of the Equatorial Ocean
Boyd, John P.
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
10/2017
517
Dura
Inglês
9783662554746
15 a 20 dias
Descrição não disponível.
1 An Observational Overview of the Equatorial Ocean 1.1 The Thermocline: the Tropical Ocean as a Two-Layer Model . . . . . . 1.2 Equatorial Currents 1.3 The Somali Current and the Monsoon 1.4 Deep Internal Jets 1.5 The El Nino/Southern Oscillation (ENSO) 1.6 Upwelling in the Gulf of Guinea 1.7 Seasonal Variations of the Thermocline 1.8 Summary 2 Basic Equations and Normal Modes 2.1 Model . 2.2 Boundary conditions 2.3 Separation of Variables 2.4 Lamb's Parameter and all 2.5 Vertical Modes and Layer Models . 2.6 Nondimensionalization 3&nb sp; Kelvin, Yanai, Rossby and Gravity Waves 3.1 Latitudinal wave modes: an overview 3.2 Latitudinal wave modes 3.3 Dispersion relation 3.4 Analytic Approximations to Equatorial Wave Frequencies 3.4.1 Explicit formulas 3.4.2 Long wave series 3.5 Separation of Time Scales 3.6 Forced Waves 3.7 How the Mixed-Rossby Gravity Wave Earned Its Name 3.8 Hough-Hermite Vector Basis 3.8.1 Introduction 3.8.2 Inner Product and Orthogonality 3.8.3 Orthonormal Basis Functions 3.9 Hough-Hermite Applications 3.10 Initialization Through Hough-Hermite Expansion 3.11 Energy Relationships 3.12&nbs p; The Equatorial Beta-Plane as the Thin Limit of the Nonlinear Shallow Water Equations on the Sphere 4 The "Long Wave" Approximation & Geostrophy 4.1 Introduction 4.2 Quasi-Geostrophy 4.3 "Meridional Geostrophy" Approximation 4.4 Boundary Conditions 4.5 Frequency Separation of Slow [Rossby/Kelvin] and Fast [Gravity] Waves 4.6 Long Wave Initial Value Problems 4.7 Reflection From an Eastern Boundary in the Long Wave Approximation 4.7.1 The Method of Images 4.7.2 Dilated Images 4.7.3 Zonal Velocity 4.8 Forced Problems in the Long Wave Approximation 5 Coastally Trapped Waves and Ray- Tracing 5.1 Introduction 5.2 Coastally-Trapped Waves 5.3 Ray-Tracing for Coastal Waves 5.4 Ray-Tracing on the Equatorial Beta-plane 5.5 Coastal & Equatorial Kelvin Waves 5.6 Topographic and Rotational Rossby Waves and Potential Vorticity 6 Reflections and Boundaries 6.1 Introduction 6.2 Reflection of Midlatitude Rossby Waves from a Zonal Boundary 6.3 Reflection of Equatorial Waves from a Western Boundary 6.4 Reflection from an Eastern Boundary 6.5 The Meridional Geostrophy/Long Wave Approximation and Boundaries 6.6 Quasi-Normal Modes: Definition and Other Weakly Non-existent Phenomena 6.7 Quasi-Normal Modes in the Long Wa ve Approximation: Derivation 6.8 Quasi-Normal Modes in the Long Wave Approximation: Discussion 6.9 High Frequency Quasi-Free Equatorial Oscillations 6.10 Scattering & Reflection from Islands 7 Response of the Equatorial Ocean to Periodic Forcing 7.1 Introduction 7.2 A Hierarchy of Models for Time-Periodic Forcing 7.3 Description of the Model and the Problem 7.4 Numerical models: Reflections and "Ringing" 7.5 Atlantic versus Pacific 7.6 Summary 8 Impulsive Forcing and Spin-up 8.1 Introduction 8.2 The Reflection of the Switched-On Kelvin Wave 8.3 Spin-up of a Zonally-Bounded Ocean: Overv iew 8.4 The Interior (Yoshida) Solution 8.5 Inertial-Gravity Waves 8.6 Western Boundary Response 8.7 Sverdrup Flow on the Equatorial Beta-Plane 8.8 Spin-Up: General Considerations 8.9 Equatorial Spin-up: Details 8.10 Equatorial Spin-up: Summary 9 Yoshida Jet and Theories of the Undercurrent 9.1 Introduction 9.2 Wind-Driven Circulation in an Unbounded Ocean: f-plane 9.3 The Yoshida Jet 9.4 An Interlude: Solving Inhomogeneous Differential Equations at Low Latitudes 9.4.1 Forced eigenoperators: Hermite series 9.4.2 Hutton-Euler Acceleration of Slowly Converging Hermite Series 9.4.3 Regularized Forci ng 9.4.4 Bessel Function Explicit Solution for the Yoshida Jet 9.4.5 Rational Approximations: Two-Point Pade Approximants and Rational Chebyshev Galerkin Methods 9.5 Unstratified Models of the Undercurrent 9.5.1 Theory of Fofonoff and Montgomery (1955) 9.5.2 Model of Stommel (1960) 9.5.3 Gill(1971) and Hidaka (1961) 10 Stratified Models of Mean Currents 10.1 Introduction 10.2 Modal Decompositions for Linear, Stratified Flow 10.3 Different Balances of Forces 10.3.1 Bjerknes Balance 10.4 Forced Baroclinic Flow 10.4.1 Other Balances 10.5 The Sensitivity of the Undercurrent to Parameters 10.6 Observations of the Tsuchiya Jets 10.7 Alternate Methods for Vertical Structure with Viscosit y 10.8 McPhaden's Model of the EUC and SSCC's: Results 10.9 A Critique of Linear Models of the Continuously-Stratified Ocean 11 Waves and Beams in the Continuously Stratified Ocean 11.1 Introduction 11.1.1 Equatorial beams: A Theoretical Inevitability 11.1.2 Slinky Physics and Impedance Mismatch, or How Water Cn Be As Reflective As Silvered Glass 11.1.3 Shallow Barriers to Downward Beams 11.1.4 Equatorial methodology 11.2 Alternate Form of the Vertical Structure Equation 11.3 The Thermocline as a Mirror 11.4 The Mirror-Thermocline Concept: A Critique 11.5 The Zonal Wavenumber Condition for Strong Excitation of a Mode 11.6 Kelvin Beams: Background 11.7 Equatorial Kelvin Beams: Results 12 Stable Waves in Shear 12.1 Introduction 12.2 U (y): Pure Latitudinal Shear 12.3 Waves in Two-Dimensional Shear 12.4 Vertical Shear and the Method of Multiple Scales 13 Inertial Instability and Deep Equatorial Jets 13.1 Introduction: Stratospheric Pancakes & Equatorial Deep Jets 13.2.1 Linear Inertial Instability 13.3 Centrifugal Instability: Rayleigh's Parcel Argument 13.4 Equatorial Gamma-Plane Approximation 13.5 Dynamical Equator 13.6 Gamma-plane Instability 13.7 Mixed Kelvin-Inertial Instability 13.8 Summary 14 Kelvin Wave Instability 14.1 Proxies and the Optical Theorem 14.2 Six Ways to Calculate Kelvin Instability 14.2.1 Power Series for the Eigenvalue 14.2.2 Hermite-Pade Approximants 14.2.3 Numerical 14.3 Instability for t he Equatorial Kelvin Wave In the Small Wavenumber Limit 14.3.1 Beyond-All-Orders Rossby Wave Instability 14.3.2 Beyond-All-Orders Kelvin Wave Instability in Weak Shear in the Long Wave Approximation 14.4 Kelvin Instability in Shear: the General Case 15 Nonmodal Instability 15.1 Introduction 15.2 Couette and Poiseuille Flow & Subcritical Bifurcation 15.3 The Fundamental Orr 15.4 Interpretation: the "Venetian Blind Effect" 15.5 Refinements to the Orr Solution 15.6 The "Checkerboard" and Bessel Solution 15.6.1 The "Checkerboard" Solution 15.7 The Dandelion Strategy 15.8 Three-Dimensional Transients 15.9 ODE Models & Nonnormal Matrices 15.10Nonmodal Instability in the Tropics 15.11Summary 16 No nlinear Equatorial Waves 16.1 Introduction 16.2 Weakly Nonlinear Multiple Scale Perturbation Theory 16.2.1 Reduction From Three Space Dimensions to One 16.2.2 Three Dimensions & Baroclinic Modes 16.3 Solitary and Cnoidal Waves 16.4 Dispersion and Waves 16.4.1 Derivation of the Group Velocity Through the Method of Multiple Scales 16.5 Integrability, Chaos and the Inverse Scattering method 16.6 Low Order Spectral Truncation (LOST) 16.7 Nonlinear Equatorial Kelvin Waves 16.7.1 Physics of the One-Dimensional Advection (ODA) equation 16.7.2 Post-Breaking: Overturning, Taylor shock or "soliton clusters" . . . . . . . . . . . . . . . . . . . . . . 16.7.3 Viscous regularization of Kelvin fronts: Burgers' equation ad matched asymptotic pertubation tery 16.8 Kelvin-Gravity Wave Shortwave Resonance: Curving Fronts and Undulations 16.9 Kelvin solitary and cnoidal waves 16.10Corner Waves and the Cnoidal-Corner-Breaking Scenario 16.11Rossby Solitary Waves 16.12Antisymmetrc Latitudinal Modes & MKdV Eq 16.13Shear effects on nonlinear equatorial waves 16.14Equatorial Modons 16.15A KdV alternative: the Regularized Long Wave (RLW) equation 16.15.1The useful non-uniqueness of perturbation theory 16.15.2Eastward-traveling modons and other cryptozoa 16.16Phenomenology of the Korteweg-deVries Equation on an unbounded domain 16.16.1Standard form/group invariance 16.16.2The KdV equation and longitudinal boundaries 16.16.3Calculating the Solitons Only 16.16.4Elastic soliton collisions 16.16.5Periodic BC 16.16.6The KdV cnoidal wave 16.17Soliton Myths and Amazements 16.17.1Imbricate series & the Nonlinear Superposition Principle 16.17.2The Lemniscate Cnoidal Wave: Strong Overlap of the Soliton and Sine Wave Regimes 16.17.3Solitary waves are not special 16.17.4Why "Solitary Wave" is the most misleading term in oceanography 16.17.5Scotomas and discovery: the Lonely Crowd 16.18Weakly nonlocal solitary waves . 16.18.1Background 16.18.2Initial Value Experiments 16.18.3Nonlinear Eigenvalue Solutions 16.19Tropical Instability Vortices 16.20The Missing Soliton Problem 17 Nonlinear Wavepackets and Nonlinear Schroedinger Equation 17.1 The Nonlinear Schroedinger Equation for Weakly Nonlinear Wavepackets: Envelope Solitons, FPU Recurrence and Sideband Instability 17.2 Linear Wavepackets 17.2.1 Perturbation Parameters 17.3 Derivation of the NLS Equation from the KdV Equation 17.3.1 NLS Dilation Group Invariance 17.3.2 Defocusing 17.3.3 Focusing, envelope solitons and resonance 17.3.4 Nonlinear plane wave 17.3.5 Envelope solitary wave 17.3.6 NLS cnoidal & dnoidal 17.3.7 N-soliton solutions 17.3.8 Breathers 17.3.9 Modulational ("sideband") instability, self-focusing and FPU Recurrence 17.4 KdV from NLS 17.4.1 The Landau constant: Poles and resonances 17.5 Weakly Dispersive Waves 17.6 Numerical Experiments 17.7 Nonlinear Schroedinger equation (NLS) summary 17.8 Resonances: Triad, Second Harmonic & Long-Wave Short Wave 17.9 Second Harmonic Resonance 17.9.1 Barotropic/baroclinic triads 17.10Long Wave/Short Wave Resonance 17.10.1Landau constant poles 17.11Triad Resonances: The General Case Continued) 17.11.1A Brief Catalog of Triad Concepts 17.11.2Rescalings 17.11.3The general explicit solutions 17.12Linearized Stability Theory 17.12.1Vacillation and Index Cycles 17.12.2Euler Equations and Football 17.12.3Lemniscate Case 17.12.4Instability & the Lemniscate Case 17.13Resonance Conditions: A Problem in Algebraic Geometry 17.13.1Selection Rules and Qualitative Properties 17.13.2Limitations of Triad Theory 17.14Solitary Waves in Numerical Models 17.15Gerstner Trochoidal Waves and Lagrangian Coordinate Descriptions of Nonlinear Waves 17.16Potential Vorticity Inversion 17.16.1A Proof That the Linearized Kelvin Wave Has Zero Potential Vorticity 17.17Coupled systems of KdV or RLW equations A Hermite Functions A.1 Normalized Hermite Functions: Definitions and Recursion A.2 Raising and Lowering Operators A.3 Integrals of Hermite Polyno mials and Functions A.4 Integrals of Products of Hermite Functions A.5 Higher Order and Symmetry-Preserving Recurrences A.6 Unnormalized Hermite Polynomials A.7 Zeros of Hermite Series A.8 Zeros of Hermite Functions A.9 Gaussian Quadrature A.9.1 Gaussian Weighted A.9.2 Unweighted Integrand A.10 Pointwise Bound on Normalized Hermite Functions A.11 Asymptotic Approximations A.11.1 Interior Approximations A.11.2 Airy Approximation Near the Turning Points A.12 Convergence Theory A.13 Abel-Euler Summability, Moore's Trick, and Tapering A.14 Alternative Implementation of Euler Acceleration A.15 Tapering A.16 Hermite Functions on a Finite Interval A.17 Hermite-Galerkin Numerical Models A.18 Fourier Transform A.19 Integral Representations B&nb sp; Expansion of the Wind-Driven Flow in Vertical Modes C Potential Vorticity and Y C.1 Potential Vorticity C.2 Potential Vorticity Inversion C.3 Mass-Weighted Streamfunction C.3.1 General Time-Varying Flows C.3.2 Streamfunction for Steadily-Traveling Waves C.4 Streakfunction C.5 The Streamfunction for Small Amplitude Traveling Waves C.6 Other Nonlinear Conservation Laws Glossary Index References
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1 An Observational Overview of the Equatorial Ocean 1.1 The Thermocline: the Tropical Ocean as a Two-Layer Model . . . . . . 1.2 Equatorial Currents 1.3 The Somali Current and the Monsoon 1.4 Deep Internal Jets 1.5 The El Nino/Southern Oscillation (ENSO) 1.6 Upwelling in the Gulf of Guinea 1.7 Seasonal Variations of the Thermocline 1.8 Summary 2 Basic Equations and Normal Modes 2.1 Model . 2.2 Boundary conditions 2.3 Separation of Variables 2.4 Lamb's Parameter and all 2.5 Vertical Modes and Layer Models . 2.6 Nondimensionalization 3&nb sp; Kelvin, Yanai, Rossby and Gravity Waves 3.1 Latitudinal wave modes: an overview 3.2 Latitudinal wave modes 3.3 Dispersion relation 3.4 Analytic Approximations to Equatorial Wave Frequencies 3.4.1 Explicit formulas 3.4.2 Long wave series 3.5 Separation of Time Scales 3.6 Forced Waves 3.7 How the Mixed-Rossby Gravity Wave Earned Its Name 3.8 Hough-Hermite Vector Basis 3.8.1 Introduction 3.8.2 Inner Product and Orthogonality 3.8.3 Orthonormal Basis Functions 3.9 Hough-Hermite Applications 3.10 Initialization Through Hough-Hermite Expansion 3.11 Energy Relationships 3.12&nbs p; The Equatorial Beta-Plane as the Thin Limit of the Nonlinear Shallow Water Equations on the Sphere 4 The "Long Wave" Approximation & Geostrophy 4.1 Introduction 4.2 Quasi-Geostrophy 4.3 "Meridional Geostrophy" Approximation 4.4 Boundary Conditions 4.5 Frequency Separation of Slow [Rossby/Kelvin] and Fast [Gravity] Waves 4.6 Long Wave Initial Value Problems 4.7 Reflection From an Eastern Boundary in the Long Wave Approximation 4.7.1 The Method of Images 4.7.2 Dilated Images 4.7.3 Zonal Velocity 4.8 Forced Problems in the Long Wave Approximation 5 Coastally Trapped Waves and Ray- Tracing 5.1 Introduction 5.2 Coastally-Trapped Waves 5.3 Ray-Tracing for Coastal Waves 5.4 Ray-Tracing on the Equatorial Beta-plane 5.5 Coastal & Equatorial Kelvin Waves 5.6 Topographic and Rotational Rossby Waves and Potential Vorticity 6 Reflections and Boundaries 6.1 Introduction 6.2 Reflection of Midlatitude Rossby Waves from a Zonal Boundary 6.3 Reflection of Equatorial Waves from a Western Boundary 6.4 Reflection from an Eastern Boundary 6.5 The Meridional Geostrophy/Long Wave Approximation and Boundaries 6.6 Quasi-Normal Modes: Definition and Other Weakly Non-existent Phenomena 6.7 Quasi-Normal Modes in the Long Wa ve Approximation: Derivation 6.8 Quasi-Normal Modes in the Long Wave Approximation: Discussion 6.9 High Frequency Quasi-Free Equatorial Oscillations 6.10 Scattering & Reflection from Islands 7 Response of the Equatorial Ocean to Periodic Forcing 7.1 Introduction 7.2 A Hierarchy of Models for Time-Periodic Forcing 7.3 Description of the Model and the Problem 7.4 Numerical models: Reflections and "Ringing" 7.5 Atlantic versus Pacific 7.6 Summary 8 Impulsive Forcing and Spin-up 8.1 Introduction 8.2 The Reflection of the Switched-On Kelvin Wave 8.3 Spin-up of a Zonally-Bounded Ocean: Overv iew 8.4 The Interior (Yoshida) Solution 8.5 Inertial-Gravity Waves 8.6 Western Boundary Response 8.7 Sverdrup Flow on the Equatorial Beta-Plane 8.8 Spin-Up: General Considerations 8.9 Equatorial Spin-up: Details 8.10 Equatorial Spin-up: Summary 9 Yoshida Jet and Theories of the Undercurrent 9.1 Introduction 9.2 Wind-Driven Circulation in an Unbounded Ocean: f-plane 9.3 The Yoshida Jet 9.4 An Interlude: Solving Inhomogeneous Differential Equations at Low Latitudes 9.4.1 Forced eigenoperators: Hermite series 9.4.2 Hutton-Euler Acceleration of Slowly Converging Hermite Series 9.4.3 Regularized Forci ng 9.4.4 Bessel Function Explicit Solution for the Yoshida Jet 9.4.5 Rational Approximations: Two-Point Pade Approximants and Rational Chebyshev Galerkin Methods 9.5 Unstratified Models of the Undercurrent 9.5.1 Theory of Fofonoff and Montgomery (1955) 9.5.2 Model of Stommel (1960) 9.5.3 Gill(1971) and Hidaka (1961) 10 Stratified Models of Mean Currents 10.1 Introduction 10.2 Modal Decompositions for Linear, Stratified Flow 10.3 Different Balances of Forces 10.3.1 Bjerknes Balance 10.4 Forced Baroclinic Flow 10.4.1 Other Balances 10.5 The Sensitivity of the Undercurrent to Parameters 10.6 Observations of the Tsuchiya Jets 10.7 Alternate Methods for Vertical Structure with Viscosit y 10.8 McPhaden's Model of the EUC and SSCC's: Results 10.9 A Critique of Linear Models of the Continuously-Stratified Ocean 11 Waves and Beams in the Continuously Stratified Ocean 11.1 Introduction 11.1.1 Equatorial beams: A Theoretical Inevitability 11.1.2 Slinky Physics and Impedance Mismatch, or How Water Cn Be As Reflective As Silvered Glass 11.1.3 Shallow Barriers to Downward Beams 11.1.4 Equatorial methodology 11.2 Alternate Form of the Vertical Structure Equation 11.3 The Thermocline as a Mirror 11.4 The Mirror-Thermocline Concept: A Critique 11.5 The Zonal Wavenumber Condition for Strong Excitation of a Mode 11.6 Kelvin Beams: Background 11.7 Equatorial Kelvin Beams: Results 12 Stable Waves in Shear 12.1 Introduction 12.2 U (y): Pure Latitudinal Shear 12.3 Waves in Two-Dimensional Shear 12.4 Vertical Shear and the Method of Multiple Scales 13 Inertial Instability and Deep Equatorial Jets 13.1 Introduction: Stratospheric Pancakes & Equatorial Deep Jets 13.2.1 Linear Inertial Instability 13.3 Centrifugal Instability: Rayleigh's Parcel Argument 13.4 Equatorial Gamma-Plane Approximation 13.5 Dynamical Equator 13.6 Gamma-plane Instability 13.7 Mixed Kelvin-Inertial Instability 13.8 Summary 14 Kelvin Wave Instability 14.1 Proxies and the Optical Theorem 14.2 Six Ways to Calculate Kelvin Instability 14.2.1 Power Series for the Eigenvalue 14.2.2 Hermite-Pade Approximants 14.2.3 Numerical 14.3 Instability for t he Equatorial Kelvin Wave In the Small Wavenumber Limit 14.3.1 Beyond-All-Orders Rossby Wave Instability 14.3.2 Beyond-All-Orders Kelvin Wave Instability in Weak Shear in the Long Wave Approximation 14.4 Kelvin Instability in Shear: the General Case 15 Nonmodal Instability 15.1 Introduction 15.2 Couette and Poiseuille Flow & Subcritical Bifurcation 15.3 The Fundamental Orr 15.4 Interpretation: the "Venetian Blind Effect" 15.5 Refinements to the Orr Solution 15.6 The "Checkerboard" and Bessel Solution 15.6.1 The "Checkerboard" Solution 15.7 The Dandelion Strategy 15.8 Three-Dimensional Transients 15.9 ODE Models & Nonnormal Matrices 15.10Nonmodal Instability in the Tropics 15.11Summary 16 No nlinear Equatorial Waves 16.1 Introduction 16.2 Weakly Nonlinear Multiple Scale Perturbation Theory 16.2.1 Reduction From Three Space Dimensions to One 16.2.2 Three Dimensions & Baroclinic Modes 16.3 Solitary and Cnoidal Waves 16.4 Dispersion and Waves 16.4.1 Derivation of the Group Velocity Through the Method of Multiple Scales 16.5 Integrability, Chaos and the Inverse Scattering method 16.6 Low Order Spectral Truncation (LOST) 16.7 Nonlinear Equatorial Kelvin Waves 16.7.1 Physics of the One-Dimensional Advection (ODA) equation 16.7.2 Post-Breaking: Overturning, Taylor shock or "soliton clusters" . . . . . . . . . . . . . . . . . . . . . . 16.7.3 Viscous regularization of Kelvin fronts: Burgers' equation ad matched asymptotic pertubation tery 16.8 Kelvin-Gravity Wave Shortwave Resonance: Curving Fronts and Undulations 16.9 Kelvin solitary and cnoidal waves 16.10Corner Waves and the Cnoidal-Corner-Breaking Scenario 16.11Rossby Solitary Waves 16.12Antisymmetrc Latitudinal Modes & MKdV Eq 16.13Shear effects on nonlinear equatorial waves 16.14Equatorial Modons 16.15A KdV alternative: the Regularized Long Wave (RLW) equation 16.15.1The useful non-uniqueness of perturbation theory 16.15.2Eastward-traveling modons and other cryptozoa 16.16Phenomenology of the Korteweg-deVries Equation on an unbounded domain 16.16.1Standard form/group invariance 16.16.2The KdV equation and longitudinal boundaries 16.16.3Calculating the Solitons Only 16.16.4Elastic soliton collisions 16.16.5Periodic BC 16.16.6The KdV cnoidal wave 16.17Soliton Myths and Amazements 16.17.1Imbricate series & the Nonlinear Superposition Principle 16.17.2The Lemniscate Cnoidal Wave: Strong Overlap of the Soliton and Sine Wave Regimes 16.17.3Solitary waves are not special 16.17.4Why "Solitary Wave" is the most misleading term in oceanography 16.17.5Scotomas and discovery: the Lonely Crowd 16.18Weakly nonlocal solitary waves . 16.18.1Background 16.18.2Initial Value Experiments 16.18.3Nonlinear Eigenvalue Solutions 16.19Tropical Instability Vortices 16.20The Missing Soliton Problem 17 Nonlinear Wavepackets and Nonlinear Schroedinger Equation 17.1 The Nonlinear Schroedinger Equation for Weakly Nonlinear Wavepackets: Envelope Solitons, FPU Recurrence and Sideband Instability 17.2 Linear Wavepackets 17.2.1 Perturbation Parameters 17.3 Derivation of the NLS Equation from the KdV Equation 17.3.1 NLS Dilation Group Invariance 17.3.2 Defocusing 17.3.3 Focusing, envelope solitons and resonance 17.3.4 Nonlinear plane wave 17.3.5 Envelope solitary wave 17.3.6 NLS cnoidal & dnoidal 17.3.7 N-soliton solutions 17.3.8 Breathers 17.3.9 Modulational ("sideband") instability, self-focusing and FPU Recurrence 17.4 KdV from NLS 17.4.1 The Landau constant: Poles and resonances 17.5 Weakly Dispersive Waves 17.6 Numerical Experiments 17.7 Nonlinear Schroedinger equation (NLS) summary 17.8 Resonances: Triad, Second Harmonic & Long-Wave Short Wave 17.9 Second Harmonic Resonance 17.9.1 Barotropic/baroclinic triads 17.10Long Wave/Short Wave Resonance 17.10.1Landau constant poles 17.11Triad Resonances: The General Case Continued) 17.11.1A Brief Catalog of Triad Concepts 17.11.2Rescalings 17.11.3The general explicit solutions 17.12Linearized Stability Theory 17.12.1Vacillation and Index Cycles 17.12.2Euler Equations and Football 17.12.3Lemniscate Case 17.12.4Instability & the Lemniscate Case 17.13Resonance Conditions: A Problem in Algebraic Geometry 17.13.1Selection Rules and Qualitative Properties 17.13.2Limitations of Triad Theory 17.14Solitary Waves in Numerical Models 17.15Gerstner Trochoidal Waves and Lagrangian Coordinate Descriptions of Nonlinear Waves 17.16Potential Vorticity Inversion 17.16.1A Proof That the Linearized Kelvin Wave Has Zero Potential Vorticity 17.17Coupled systems of KdV or RLW equations A Hermite Functions A.1 Normalized Hermite Functions: Definitions and Recursion A.2 Raising and Lowering Operators A.3 Integrals of Hermite Polyno mials and Functions A.4 Integrals of Products of Hermite Functions A.5 Higher Order and Symmetry-Preserving Recurrences A.6 Unnormalized Hermite Polynomials A.7 Zeros of Hermite Series A.8 Zeros of Hermite Functions A.9 Gaussian Quadrature A.9.1 Gaussian Weighted A.9.2 Unweighted Integrand A.10 Pointwise Bound on Normalized Hermite Functions A.11 Asymptotic Approximations A.11.1 Interior Approximations A.11.2 Airy Approximation Near the Turning Points A.12 Convergence Theory A.13 Abel-Euler Summability, Moore's Trick, and Tapering A.14 Alternative Implementation of Euler Acceleration A.15 Tapering A.16 Hermite Functions on a Finite Interval A.17 Hermite-Galerkin Numerical Models A.18 Fourier Transform A.19 Integral Representations B&nb sp; Expansion of the Wind-Driven Flow in Vertical Modes C Potential Vorticity and Y C.1 Potential Vorticity C.2 Potential Vorticity Inversion C.3 Mass-Weighted Streamfunction C.3.1 General Time-Varying Flows C.3.2 Streamfunction for Steadily-Traveling Waves C.4 Streakfunction C.5 The Streamfunction for Small Amplitude Traveling Waves C.6 Other Nonlinear Conservation Laws Glossary Index References
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