Dynamics of the Equatorial Ocean

ISBN-13: 9783662554746 / Angielski / Twarda / 2017 / 517 str.

John P. Boyd

Spis treści

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-Padé 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

 

Blurb

This book is the first comprehensive introduction to the theory of equatorially-confined waves and currents in the ocean. Among the topics treated are inertial and shear instabilities, wave generation by coastal reflection, semiannual and annual cycles in the tropic sea, transient equatorial waves, vertically-propagating beams, equatorial Ekman layers, the Yoshida jet model, generation of coastal Kelvin waves from equatorial waves by reflection, Rossby solitary waves, and Kelvin frontogenesis. A series of appendices on midlatitude theories for waves, jets and wave reflections add further material to assist the reader in understanding the differences between the same phenomenon in the equatorial zone versus higher latitudes.