Dynamics of the Equatorial Ocean (eBook)

1 An Observational Overview of the Equatorial Ocean 1.1    The Thermocline: the Tropical Ocean as a Two-Layer Model . . . . . .1.2    Equatorial Currents1.3    The Somali Current and the Monsoon1.4    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 Thermocline1.8    Summary  2      Basic Equations and Normal Modes2.1     Model  . 2.2     Boundary conditions2.3     Separation of Variables 2.4    Lamb’s Parameter and all 2.5    Vertical Modes and Layer Models .2.6    Nondimensionalization  3      Kelvin, Yanai, Rossby and Gravity Waves3.1    Latitudinal wave modes: an overview3.2    Latitudinal wave modes  3.3    Dispersion relation3.4    Analytic Approximations to Equatorial Wave Frequencies3.4.1    Explicit formulas 3.4.2    Long wave series3.5    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 Relationships3.12  The Equatorial Beta-Plane as the Thin Limit of the NonlinearShallow Water Equations on the Sphere 4      The “Long Wave” Approximation & Geostrophy4.1    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 WaveApproximation4.7.1    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-Tracing5.1    Introduction5.2    Coastally-Trapped Waves5.3    Ray-Tracing for Coastal Waves5.4    Ray-Tracing on the Equatorial Beta-plane5.5    Coastal & Equatorial Kelvin Waves5.6    Topographic and Rotational Rossby Waves and Potential Vorticity  6      Reflections and Boundaries6.1    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 Phenomena6.7    Quasi-Normal Modes in the Long Wave Approximation: Derivation6.8    Quasi-Normal Modes in the Long Wave Approximation: Discussion6.9    High Frequency Quasi-Free Equatorial Oscillations 6.10  Scattering & Reflection from Islands    7      Response of the Equatorial Ocean to Periodic Forcing7.1    Introduction7.2    A Hierarchy of Models for Time-Periodic Forcing7.3    Description of the Model and the Problem 7.4    Numerical models: Reflections and “Ringing” 7.5    Atlantic versus Pacific7.6    Summary   8      Impulsive Forcing and Spin-up8.1    Introduction8.2    The Reflection of the Switched-On Kelvin Wave8.3    Spin-up of a Zonally-Bounded Ocean: Overview8.4    The Interior (Yoshida) Solution8.5    Inertial-Gravity Waves8.6    Western Boundary Response8.7    Sverdrup Flow on the Equatorial Beta-Plane8.8    Spin-Up: General Considerations8.9    Equatorial Spin-up: Details8.10  Equatorial Spin-up: Summary  9      Yoshida Jet and Theories of the Undercurrent9.1    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 Latitudes9.4.1    Forced eigenoperators: Hermite series 9.4.2    Hutton-Euler Acceleration of Slowly Converging Hermite Series 9.4.3    Regularized Forcing 9.4.4    Bessel Function Explicit Solution for the Yoshida Jet9.4.5    Rational Approximations: Two-Point Pade Approximantsand 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 Currents10.1  Introduction 10.2  Modal Decompositions for Linear, Stratified Flow 10.3  Different Balances of Forces10.3.1  Bjerknes Balance 10.4  Forced Baroclinic Flow10.4.1  Other Balances10.5  The Sensitivity of the Undercurrent to Parameters10.6  Observations of the Tsuchiya Jets 10.7  Alternate Methods for Vertical Structure with Viscosity 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 Ocean11.1  Introduction 11.1.1  Equatorial beams: A Theoretical Inevitability 11.1.2  Slinky Physics and Impedance Mismatch, or How WaterCn 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 Mode11.6  Kelvin Beams: Background11.7  Equatorial Kelvin Beams: Results  12    Stable Waves in Shear12.1  Introduction 12.2 U (y): Pure Latitudinal Shear12.3  Waves in Two-Dimensional Shear 12.4  Vertical Shear and the Method of Multiple Scales 13    Inertial Instability and Deep Equatorial Jets13.1  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 Approximation13.5  Dynamical Equator 13.6  Gamma-plane Instability 13.7  Mixed Kelvin-Inertial Instability 13.8  Summary  14    Kelvin Wave Instability14.1  Proxies and the Optical Theorem14.2  Six Ways to Calculate Kelvin Instability14.2.1  Power Series for the Eigenvalue14.2.2  Hermite-Padé Approximants 14.2.3  Numerical 14.3  Instability for the 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 Instability15.1  Introduction15.2  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    Nonlinear Equatorial Waves16.1  Introduction16.2  Weakly Nonlinear Multiple Scale Perturbation Theory 16.2.1  Reduction From Three Space Dimensions to One 16.2.2  Three Dimensions & Baroclinic Modes16.3  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 method16.6  Low Order Spectral Truncation (LOST) 16.7  Nonlinear Equatorial Kelvin Waves16.7.1  Physics of the One-Dimensional Advection (ODA) equation16.7.2  Post-Breaking: Overturning, Taylor shock or “soliton clusters” . . . . . . . . . . . . . . . . . . . . . . 16.7.3  Viscous regularization of Kelvin fronts: Burgers’ equationad matched asymptotic pertubation tery 16.8  Kelvin-Gravity Wave Shortwave Resonance: Curving Fronts andUndulations  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 Eq16.13Shear effects on nonlinear equatorial waves16.14Equatorial Modons   16.15A KdV alternative: the Regularized Long Wave (RLW) equation  16.15.1The useful non-uniqueness of perturbation theory16.15.2Eastward-traveling modons and other cryptozoa 16.16Phenomenology of the Korteweg-deVries Equation on anunbounded domain  16.16.1Standard form/group invariance 16.16.2The KdV equation and longitudinal boundaries16.16.3Calculating the Solitons Only16.16.4Elastic soliton collisions16.16.5Periodic BC 16.16.6The KdV cnoidal wave 16.17Soliton Myths and Amazements16.17.1Imbricate series & the Nonlinear Superposition Principle16.17.2The Lemniscate Cnoidal Wave: Strong Overlap of theSoliton and Sine Wave Regimes 16.17.3Solitary waves are not special 16.17.4Why “Solitary Wave” is the most misleading term inoceanography  16.17.5Scotomas and discovery: the Lonely Crowd16.18Weakly 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 Equation17.1  The Nonlinear Schroedinger Equation for Weakly NonlinearWavepackets: Envelope Solitons, FPU Recurrence and SidebandInstability17.2  Linear Wavepackets17.2.1  Perturbation Parameters 17.3   Derivation of the NLS Equation from the KdV Equation17.3.1  NLS Dilation Group Invariance17.3.2  Defocusing17.3.3  Focusing, envelope solitons and resonance17.3.4  Nonlinear plane wave17.3.5  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 andFPU Recurrence17.4  KdV from NLS 17.4.1  The Landau constant: Poles and resonances17.5  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 Resonance17.9.1  Barotropic/baroclinic triads 17.10Long Wave/Short Wave Resonance17.10.1Landau 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 Vorticity17.17Coupled systems of KdV or RLW equations A     Hermite FunctionsA.1   Normalized Hermite Functions: Definitions and Recursion A.2   Raising and Lowering Operators A.3   Integrals of Hermite Polynomials and FunctionsA.4   Integrals of Products of Hermite Functions A.5   Higher Order and Symmetry-Preserving Recurrences A.6   Unnormalized Hermite PolynomialsA.7   Zeros of Hermite Series A.8   Zeros of Hermite FunctionsA.9   Gaussian Quadrature A.9.1   Gaussian WeightedA.9.2   Unweighted Integrand A.10 Pointwise Bound on Normalized Hermite FunctionsA.11 Asymptotic Approximations A.11.1 Interior ApproximationsA.11.2 Airy Approximation Near the Turning PointsA.12 Convergence Theory A.13 Abel-Euler Summability, Moore’s Trick, and TaperingA.14 Alternative Implementation of Euler Acceleration  A.15 Tapering A.16 Hermite Functions on a Finite Interval A.17 Hermite-Galerkin Numerical ModelsA.18 Fourier TransformA.19 Integral Representations B     Expansion of the Wind-Driven Flow in Vertical ModesC     Potential Vorticity and YC.1   Potential VorticityC.2   Potential Vorticity InversionC.3   Mass-Weighted StreamfunctionC.3.1    General Time-Varying Flows C.3.2    Streamfunction for Steadily-Traveling Waves C.4   StreakfunctionC.5   The Streamfunction for Small Amplitude Traveling Waves C.6   Other Nonlinear Conservation Laws Glossary Index References