Preface
Chapter 1. Introduction
1.1. Basic Properties of the Equations with Rotation and Stratification
1.2. Two-Dimensional Exact Solutions
1.3. Buoyancy and Stratification
1.4. Jet Flows with Rotation and Stratification
1.5. From Vertical Stratification to Shallow Water
Chapter 2. Some Remarkable Features of Stratified Flow
2.1. Energy Principle
2.2. Vorticity in Stratified Fluids and Exact Solutions Motivated by Local Analysis
2.3. Use of Theorem 2.4: Exact Two-Dimensional Solutions
2.4. Nonlinear Plane Waves in Stratified Flow: Internal Gravity Waves
2.5. Exact Solutions with Large-Scale Motion and Nonlinear Plane Waves
2.6. More Details for Theorem 2.7 on Special Exact Solutionstr for the Boussinesq Equations Including Plane Waves
Chapter 3. Linear and Nonlinear Instability of Stratified Flows with Strong Stratification
3.1. Boussinesq Equations and Vorticity Stream Formulation
3.2. Nonlinear Instability of Stratified Flows
3.3. Shear Flows
3.4. Some Background Facts on ODEs
Chapter 4. Rotating Shallow Water Theory
4.1. Rotating Shallow Water Equations
4.2. Conservation of Potential Vorticity
4.3. Nonlinear Conservation of Energy
4.4. Linear Theory for the Rotating Shallow Water Equationsoriotv
4.5. Nondimensional Form of the Rotating Shallow Water Equations
4.6. Derivation of the Quasi-Geostrophic Equations
4.7. The Quasi-Geostrophic Equations as a Singular PDE Limituri
4.8. The Model Rotating Shallow Water Equations o ibrgc
4.9. Preliminary Mathematical Considerations
4.10. Rigorous Convergence of the Model Rotating Shallow Water Equations to the Quasi-Geostrophic Equations
4.11. Proof of the Convergence Theorem
Chapter 5. Linear and Weakly Nonlinear Theory of Dispersive Waves with Geophysical Examples
5.1. Linear Wave Midlatitude Planetary Equations
5.2. Dispersive Waves: General Properties
5.3. Interpretation of Group Velocity
5.4. Distant Propagation from a Localized Source
5.5. WKB Methods for Linear Dispersive Waves
5.6. Beyond Caustics: Eikonal Equation Revisited
5.7. Weakly Nonlinear WKB for Perturbations Around a Constant State
5.8. Nonlinear WKB and the Boussinesq Equations
Chapter 6. Simplified Equations for the Dynamics of Strongly Stratified Flow
6.1. Nondimensionalization of the Boussinesq Equations
for Stably Stratified Flow
6.2. The Vorticity Stream Formulation and Elementary Properties of the Limit Equations for Strongly Stratified Flow
6.3. Solutions of the Limit Dynamics with Strong Stratification as Models for Laboratory Experiments
Chapter 7. The Stratified Quasi-Geostrophic Equations as a Singular Limit of the Rotating Boussinesq Equations
7.1. Introduction
7.2. The Rotating Boussinesq Equations
7.3. The Nondimensional Rotating Boussinesq Equations
7.4. Formal Asymptotic Derivation of the Quasi-Geostrophic Equations as a Distinguished Asymptotic Limit of Small Rossby and Froud Numbers
7.5. Rigorous Convergence of the Rotating Boussinesq Equations to the Quasi-Geostrophic Equations
7.6. Preliminary Mathematical Considerations
7.7. Proof of the Convergence Theorem
Chapter 8. Introduction to Averaging over Fast Waves for Geophysical Flows
8.1. Introduction
8.2. Motivation for Fast-Wave Averaging
8.3. A General Framework for Averaging over Fast Waves acribnor
8.4. Elementary Analytic Models for Comparing Instabilities at Low Froude Numbers with the Low Froude Number Limit Dynamics
8.5. The Rapidly Rotating Shallow Water Equationso with Unbalanced Initial Data in the Quasi-Geostrophic Limit
8.6.The Interaction of Fast Waves and Slow Dynamics in the Rotating Stratified Boussinesq Equations
Chapter 9 Waves and PDEs for the Equatorial Atmosphere and Ocean
9.1. Introduction to Equatorial Waves for Rotating Shallow Water
9.2. The Equatorial Primitive Equations
9.3. The Nonlinear Equatorial Long-Wave Equations
9.4. A Simple Model for the
