Introduction to PDEs and waves for the atmosphere and ocean

Written by a leading specialist in the area of atmosphere/ocean science (AOS), this book presents an excellent introduction to this important topic. The goals of these lecture notes, based on courses presented by the author at the Courant Institute of Mathematical Sciences, are to introduce mathematicians to the fascinating and important area of atmosphere/ocean science (AOS) and, conversely, to develop a mathematical viewpoint on basic topics in AOS of interest to the disciplinary AOS community, ranging from graduate students to researchers. The lecture notes emphasize the serendipitous connections between applied mathematics and geophysical flows in the style of modern applied mathematics, where rigorous mathematical analysis as well as asymptotic, qualitative, and numerical modeling all interact to ease the understanding of physical phenomena.Reading these lecture notes does not require a previous course in fluid dynamics, although a serious reader should supplement them with additional information on geophysical flows, as suggested in the preface. The book is intended for graduate students and researchers working in interdisciplinary areas between mathematics and AOS. It is excellent for supplementary course reading or independent study.

Preface	p. ix
Chapter 1. Introduction	p. 1
1.1. Basic Properties of the Equations with Rotation and Stratification	p. 1
1.2. Two-Dimensional Exact Solutions	p. 2
1.3. Buoyancy and Stratification	p. 5
1.4. Jet Flows with Rotation and Stratification	p. 6
1.5. From Vertical Stratification to Shallow Water	p. 6
Chapter 2. Some Remarkable Features of Stratified Flow	p. 9
2.1. Energy Principle	p. 9
2.2. Vorticity in Stratified Fluids and Exact Solutions Motivated by Local Analysis	p. 11
2.3. Use of Theorem 2.4: Exact Two-Dimensional Solutions	p. 16
2.4. Nonlinear Plane Waves in Stratified Flow: Internal Gravity Waves	p. 20
2.5. Exact Solutions with Large-Scale Motion and Nonlinear Plane Waves	p. 24
2.6. More Details for Theorem 2.7 on Special Exact Solutions for the Boussinesq Equations Including Plane Waves	p. 26
Chapter 3. Linear and Nonlinear Instability of Stratified Flows with Strong Stratification	p. 31
3.1. Boussinesq Equations and Vorticity Stream Formulation	p. 33
3.2. Nonlinear Instability of Stratified Flows	p. 37
3.3. Shear Flows	p. 40
3.4. Some Background Facts on ODEs	p. 44
Chapter 4. Rotating Shallow Water Theory	p. 49
4.1. Rotating Shallow Water Equations	p. 49
4.2. Conservation of Potential Vorticity	p. 52
4.3. Nonlinear Conservation of Energy	p. 53
4.4. Linear Theory for the Rotating Shallow Water Equations	p. 54
4.5. Nondimensional Form of the Rotating Shallow Water Equations	p. 60
4.6. Derivation of the Quasi-Geostrophic Equations	p. 62
4.7. The Quasi-Geostrophic Equations as a Singular PDE Limit	p. 65
4.8. The Model Rotating Shallow Water Equations	p. 67
4.9. Preliminary Mathematical Considerations	p. 69
4.10. Regorous Convergence of the Model Rotating Shallow Water Equations to the Quasi-Geostrophic Equations	p. 73
4.11. Proof of the Convergence Theorem	p. 75
Chapter 5. Linear and Weakly Nonlinear Theory of Dispersive Waves with Geophysical Examples	p. 79
5.1. Linear Wave Midlatitude Planetary Equations	p. 79
5.2. Dispersive Waves: General Properties	p. 82
5.3. Interpretation of Group Velocity	p. 86
5.4. Distant Propagation from a Localized Source	p. 90
5.5. WKB Methods for Linear Dispersive Waves	p. 93
5.6. Beyond Caustics: Eikonal Equation Revisited	p. 107
5.7. Weakly Nonlinear WKB for Perturbations Around a Constant State	p. 109
5.8. Nonlinear WKB and the Boussinesq Equations	p. 117
Chapter 6. Simplified Equations for the Dynamics of Strongly Stratified Flow	p. 125
6.1. Nondimensionalization of the Boussinesq Equations for Stably Stratified Flow	p. 126
6.2. The Vorticity Stream Formulation and Elementary Properties of the Limit Equations for Strongly Stratified Flow	p. 133
6.3. Solutions of the Limit Dynamics with Strong Stratification as Models for Laboratory Experiments	p. 136
Chapter 7. The Stratified Quasi-Geostrophic Equations as a Singular Limit of the Rotating Boussinesq Equations	p. 147
7.1. Introduction	p. 147
7.2. The Rotating Boussinesq Equations	p. 148
7.3. The Nondimensional Rotating Boussinesq Equations	p. 151
7.4. Formal Asymptotic Derivation of the Quasi-Geostrophic Equations as a Distinguished Asymptotic Limit of Small Rossby and Froude Numbers	p. 153
7.5. Rigorous Convergence of the Rotating Boussinesq Equations to the Quasi-Geostrophic Equations	p. 157
7.6. Preliminary Mathematical Considerations	p. 160
7.7. Proof of the Convergence Theorem	p. 164
Chapter 8. Introduction to Averaging over Fast Waves for Geophysical Flows	p. 171
8.1. Introduction	p. 171
8.2. Motivation for Fast-Wave Averaging	p. 171
8.3. A General Framework for Averaging over Fast Waves	p. 173
8.4. Elementary Analytic Models for Comparing Instabilities at Low Froude Numbers with the Low Froude Number Limit Dynamics	p. 177
8.5. The Rapidly Rotating Shallow Water Equations with Unbalanced Initial Data in the Quasi-Geostrophic Limit	p. 182
8.6. The Interaction of Fast Waves and Slow Dynamics in the Rotating Stratified Boussinesq Equations	p. 191
Chapter 9. Waves and PDEs for the Equatorial Atmosphere and Ocean	p. 199
9.1. Introduction to Equatorial Waves for Rotating Shallow Water	p. 199
9.2. The Equatorial Primitive Equations	p. 207
9.3. The Nonlinear Equatorial Long-Wave Equations	p. 220
9.4. A Simple Model for the Steady Circulation of the Equatorial Atmosphere	p. 226
Bibliography	p. 233

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