• Cover
  • Half-title page
  • Title page
  • Copyright page
  • Contents
  • Preface
  • Acknowledgments
  • Part I Normal Mode Instabilities
    • 1 Preliminaries
      • 1.1 What Is Instability?
      • 1.2 Goals
      • 1.3 Tools
      • 1.4 Numerical Solution of a Boundary Value Problem
      • 1.5 The Equations of Motion
      • 1.6 Further Reading
      • 1.7 Appendix: A Closer Look at Perturbation Theory
    • 2 Convective Instability
      • 2.1 The Perturbation Equations
      • 2.2 Simple Case: Inviscid, Nondiffusive, Unbounded Fluid
      • 2.3 Viscous and Diffusive Effects
      • 2.4 Boundary Effects: the Rayleigh-Benard Problem
      • 2.5 Nonlinear Effects
      • 2.6 Summary
      • 2.7 Appendix: Waves and Convection in a Compressible Fluid
    • 3 Instabilities of a Parallel Shear Flow
      • 3.1 The Perturbation Equations
      • 3.2 Rayleigh’s Equation
      • 3.3 Analytical Example: the Piecewise-Linear Shear Layer
      • 3.4 Solution Types for Rayleigh’s Equation
      • 3.5 Numerical Solution of Rayleigh’s Equation
      • 3.6 Shear Scaling
      • 3.7 Oblique Modes and Squire Transformations
      • 3.8 Rules of Thumb for a General Shear Instability
      • 3.9 Numerical Examples
      • 3.10 Perturbation Energetics
      • 3.11 Necessary Conditions for Instability
      • 3.12 The Wave Resonance Mechanism of Shear Instability
      • 3.13 Quantitative Analysis of Wave Resonance
      • 3.14 Summary
      • 3.15 Appendix: Classical Proof of the Rayleigh and Fjørtoft Theorems
      • 3.16 Further Reading
    • 4 Parallel Shear Flow: the Effects of Stratification
      • 4.1 The Richardson Number
      • 4.2 Equilibria and Perturbations
      • 4.3 Oblique Modes
      • 4.4 The Taylor-Goldstein Equation
      • 4.5 Application to Internal Wave Phenomena
      • 4.6 Analytical Examples of Instability in Stratified Shear Flows
      • 4.7 The Miles-Howard Theorem
      • 4.8 Howard’s Semicircle Theorem
      • 4.9 Energetics
      • 4.10 Summary
      • 4.11 Further Reading
      • 4.12 Appendix: Veering Flows
      • 4.13 Appendix: Spatial Growth
    • 5 Parallel Shear Flow: the Effects of Viscosity
      • 5.1 Conditions for Equilibrium
      • 5.2 Conditions for Quasi-Equilibrium: the Frozen Flow Approximation
      • 5.3 The Orr-Sommerfeld Equation
      • 5.4 Boundary Conditions for Viscous Fluid
      • 5.5 Numerical Solution of the Orr-Sommerfeld Equation
      • 5.6 Oblique Modes
      • 5.7 Shear Scaling and the Reynolds Number
      • 5.8 Numerical Examples
      • 5.9 Perturbation Energetics in Viscous Flow
      • 5.10 Summary
    • 6 Synthesis: Viscous, Diffusive, Inhomogeneous, Parallel Shear Flow
      • 6.1 Expanding the Basic Equations
      • 6.2 Numerical Solution
      • 6.3 2D and Oblique Modes: Squire Transformations
      • 6.4 Shear and Diffusion Scalings
      • 6.5 Application: Instabilities of a Stably Stratified Shear Layer
      • 6.6 Application: Analysis of Observational Data
      • 6.7 Summary
      • 6.8 Further Reading
    • 7 Nonparallel Flow: Instabilities of a Cylindrical Vortex
      • 7.1 Cyclostrophic Equilibrium
      • 7.2 The Perturbation Equations
      • 7.3 Barotropic Modes (m = 0)
      • 7.4 Axisymmetric Modes (l = 0)
      • 7.5 Analytical Example: the Rankine Vortex
      • 7.6 Numerical Example: a Continuous Vortex
      • 7.7 Wave Interactions in Barotropic Vortices
      • 7.8 Mechanisms of Centrifugal and Convective Instabilities
      • 7.9 Swirling Flows
      • 7.10 Summary
      • 7.11 Further Reading
    • 8 Instability in a Rotating Environment
      • 8.1 Frontal Zones
      • 8.2 Geostrophic Equilibrium and the Thermal Wind Balance
      • 8.3 The Perturbation Equations
      • 8.4 Energetics
      • 8.5 The Vertical Vorticity Equation
      • 8.6 Analytical Solution #1: Inertial and Symmetric Instabilities
      • 8.7 Analytical Solution #2: Baroclinic Instability
      • 8.8 Numerical Solution Method
      • 8.9 Instability in the Ageostrophic Regime
      • 8.10 Summary
      • 8.11 Further Reading
    • 9 Convective Instability in Complex Fluids
      • 9.1 Conditional Instability in a Moist Atmosphere or a Freezing Ocean
      • 9.2 Double Diffusive Instabilities
      • 9.3 Bioconvection
      • 9.4 CO[sub(2)] Sequestration
    • 10 Summary
      • 10.1 Equilibrium States
      • 10.2 Instabilities
  • Part II The View Ahead
    • 11 Beyond Normal Modes
      • 11.1 Instability as an Initial Value Problem
      • 11.2 Transient Growth in Simple Linear Systems
      • 11.3 Computing the Optimal Initial Condition
      • 11.4 Optimizing Growth at t = 0[sup(+)]
      • 11.5 Growth at Short and Long Times: a Simple Example
      • 11.6 Example: The Piecewise Shear Layer
      • 11.7 Mechanics of Transient Growth in a Shear Layer
      • 11.8 Generalizing the Inner Product
      • 11.9 Summary
      • 11.10 Appendix: Singular Value Decomposition
      • 11.11 Further Reading
    • 12 Instability and Turbulence
      • 12.1 Secondary Instabilities and the Transition to Turbulence
      • 12.2 Turbulence-Driven Instabilities
      • 12.3 Cyclic Instability
      • 12.4 Further Reading
    • 13 Refining the Numerical Methods
      • 13.1 Higher-Order Finite Differences
      • 13.2 Finite Differences on an Adaptive Grid
      • 13.3 Galerkin Methods
      • 13.4 The Shooting Method
      • 13.5 Generalizations
      • 13.6 Further Reading
  • Appendix A Homework Exercises
  • Appendix B Projects
  • References
  • Index