ALBERT

Note: Contents 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 Niño/Southern Oscillation (ENSO) 1.6 Upwelling in the Gulf of Guinea 1.7 Seasonal Variations of the Thermocline 1.8 Summary References 2 Basic Equations and Normal Modes 2.1 Model 2.2 Boundary Conditions 2.3 Separation of Variables 2.4 Lamb’s Parameter, Equivalent Depths, Kelvin Phase Speeds and All that 2.5 Vertical Modes and Layer Models 2.6 Nondimensionalization References 3 Kelvin, Yanai, Rossby and Gravity Waves 3.1 Latitudinal Wave Modes: An Overview 3.2 Latitudinal Wave Modes: Structure and Spatial Symmetries 3.3 Dispersion Relations: Exact and Approximate Frequencies 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 Applications of the Hough-Hermite Basis: Linear Initial-Value Problems 3.10 Initialization Through Hough-Hermite Expansion 3.11 Energy Relationships 3.12 The Equatorial Beta-Plane as the Thin Limit of the Nonlinear Shallow Water Equations on the Sphere References 4 The “Long Wave” Approximation & Geostrophy 4.1 Introduction 4.2 Quasi-Geostrophy 4.3 The “Meridional Geostrophy”, “Low Frequency” or “Long Wave” Approximation 4.4 Boundary Conditions 4.5 Frequency Separation of Slow [Rossby/Kelvin] and Fast [Gravity] Waves 4.6 Initial Value Problems in an Unbounded Ocean, Linearized About a State of Rest, in the Long Wave Approximation 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 References 5 The Equator as Wall: 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 and Equatorial Kelvin Waves 5.6 Topographic and Rotational Rossby Waves and Potential Vorticity References 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 Wave Approximation: Derivation 6.8 Quasi-normal Modes in the Long Wave Approximation: Discussion 6.9 High Frequency Quasi-free Equatorial Oscillations 6.10 Scattering and Reflection from Islands References 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 References 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: Overview 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 References 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 Forcing 9.4.4 Bessel Function Explicit Solution for the Yoshida Jet 9.4.5 Rational Approximations: Two-Point Padé 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) References 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 in the “Bjerknes” Approximation 10.4.1 Other Balances 10.5 The Sensitivity of the Undercurrent to Parameters 10.6 Observations of Subsurface Countercurrents (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, Wind-Driven Ocean References 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 Can 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 References 12 Stable Linearized Waves in a Shear Flow 12.1 Introduction 12.2 UðyÞ: Pure Latitudinal Shear 12.3 Neutral Waves in Flow Varying with Both Latitude and Height: Numerical Studies 12.4 Vertical Shear and the Method of Multiple Scales References 13 Inertial Instability, Pancakes and Deep Internal Jets 13.1 Introduction: Stratospheric Pancakes and Equatorial Deep Jets 13.2 Particle Argument 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 References 14 Kelvin Wave Instability: Critical Latitudes and Exponentially Small Effects 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-Padé Approximants 14.2.3 Numerical Methods 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 References 15 Nonmodal Instability 15.1 Introduction 15.2 Couette and Poiseuille Flow and Subcritical Bifurcation 15.3 The Fundamental Orr Solution 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 and Nonnormal Matrices 15.10 Nonmodal Instability in the Tropics 15.11 Summary References 16 Nonlinear 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 and 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: ut + cux + buux = 0 16.7.2 Post-Breaking: Overturning, Taylor Shock or “Soliton Clusters”? 16.7.3 Viscous Regularization of Kelvin Fronts: Burgers’ Equation And Matched Asymptotic Perturbation Theory 16.8 Kelvin-Gravity Wave Shortwave Resonance: Curving Fronts and Undulations 16.9 Kelvin Solitary and Cnoidal Waves 16.10 Corner Waves and the Cnoidal-Corner-Breaking Scenario 16.11 Rossby Solitary Waves 16.12 Antisymmetr