Introduction to PDEs and waves for the atmosphere and ocean / Andrew Majda.

Format:

Book

Author/Creator:

Majda, Andrew, 1949-

Series:

Courant lecture notes in mathematics 1529-9031 ; 9.

Courant lecture notes in mathematics, 1529-9031 ; 9

Language:

English

Subjects (All):

Waves.

Differential equations, Partial.

Differential equations, Nonlinear.

Oceanography--Mathematical models.

Oceanography.

Atmosphere--Mathematical models.

Atmosphere.

Stratified flow.

Physical Description:

ix, 234 pages : illustrations ; 26 cm.

Place of Publication:

New York : Courant Institute of Mathematical Sciences ; Providence, R.I. : American Mathematical Society, [2003]

Contents:

1.1. Basic Properties of the Equations with Rotation and Stratification 1

1.2. Two-Dimensional Exact Solutions 2

1.3. Buoyancy and Stratification 5

1.4. Jet Flows with Rotation and Stratification 6

1.5. From Vertical Stratification to Shallow Water 6

Chapter 2. Some Remarkable Features of Stratified Flow 9

2.1. Energy Principle 9

2.2. Vorticity in Stratified Fluids and Exact Solutions Motivated by Local Analysis 11

2.3. Use of Theorem 2.4: Exact Two-Dimensional Solutions 16

2.4. Nonlinear Plane Waves in Stratified Flow: Internal Gravity Waves 20

2.5. Exact Solutions with Large-Scale Motion and Nonlinear Plane Waves 24

2.6. More Details for Theorem 2.7 on Special Exact Solutions for the Boussinesq Equations Including Plane Waves 26

Chapter 3. Linear and Nonlinear Instability of Stratified Flows with Strong Stratification 31

3.1. Boussinesq Equations and Vorticity Stream Formulation 33

3.2. Nonlinear Instability of Stratified Flows 37

3.3. Shear Flows 40

Chapter 4. Rotating Shallow Water Theory 49

4.1. Rotating Shallow Water Equations 49

4.2. Conservation of Potential Vorticity 52

4.3. Nonlinear Conservation of Energy 53

4.4. Linear Theory for the Rotating Shallow Water Equations 54

4.5. Nondimensional Form of the Rotating Shallow Water Equations 60

4.6. Derivation of the Quasi-Geostrophic Equations 62

4.7. The Quasi-Geostrophic Equations as a Singular PDE Limit 65

4.8. The Model Rotating Shallow Water Equations 67

4.9. Preliminary Mathematical Considerations 69

4.10. Regorous Convergence of the Model Rotating Shallow Water Equations to the Quasi-Geostrophic Equations 73

4.11. Proof of the Convergence Theorem 75

Chapter 5. Linear and Weakly Nonlinear Theory of Dispersive Waves with Geophysical Examples 79

5.1. Linear Wave Midlatitude Planetary Equations 79

5.2. Dispersive Waves: General Properties 82

5.3. Interpretation of Group Velocity 86

5.4. Distant Propagation from a Localized Source 90

5.5. WKB Methods for Linear Dispersive Waves 93

5.6. Beyond Caustics: Eikonal Equation Revisited 107

5.7. Weakly Nonlinear WKB for Perturbations Around a Constant State 109

5.8. Nonlinear WKB and the Boussinesq Equations 117

Chapter 6. Simplified Equations for the Dynamics of Strongly Stratified Flow 125

6.1. Nondimensionalization of the Boussinesq Equations for Stably Stratified Flow 126

6.2. The Vorticity Stream Formulation and Elementary Properties of the Limit Equations for Strongly Stratified Flow 133

6.3. Solutions of the Limit Dynamics with Strong Stratification as Models for Laboratory Experiments 136

Chapter 7. The Stratified Quasi-Geostrophic Equations as a Singular Limit of the Rotating Boussinesq Equations 147

7.2. The Rotating Boussinesq Equations 148

7.3. The Nondimensional Rotating Boussinesq Equations 151

7.4. Formal Asymptotic Derivation of the Quasi-Geostrophic Equations as a Distinguished Asymptotic Limit of Small Rossby and Froude Numbers 153

7.5. Rigorous Convergence of the Rotating Boussinesq Equations to the Quasi-Geostrophic Equations 157

7.6. Preliminary Mathematical Considerations 160

7.7. Proof of the Convergence Theorem 164

Chapter 8. Introduction to Averaging over Fast Waves for Geophysical Flows 171

8.2. Motivation for Fast-Wave Averaging 171

8.3. A General Framework for Averaging over Fast Waves 173

8.4. Elementary Analytic Models for Comparing Instabilities at Low Froude Numbers with the Low Froude Number Limit Dynamics 177

8.5. The Rapidly Rotating Shallow Water Equations with Unbalanced Initial Data in the Quasi-Geostrophic Limit 182

8.6. The Interaction of Fast Waves and Slow Dynamics in the Rotating Stratified Boussinesq Equations 191

Chapter 9. Waves and PDEs for the Equatorial Atmosphere and Ocean 199

9.1. Introduction to Equatorial Waves for Rotating Shallow Water 199

9.2. The Equatorial Primitive Equations 207

9.3. The Nonlinear Equatorial Long-Wave Equations 220

9.4. A Simple Model for the Steady Circulation of the Equatorial Atmosphere 226.

Notes:

Includes bibliographical references (pages 233-234).

ISBN:

0821829548

OCLC:

51093124