Instability in Geophysical Flows

Description

An Open Access overview of physical processes that generate instability in geophysical flows, emphasising numerical methods and simple rules to predict instability. 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