Atmospheres and Oceans on Computers: Fundamental Numerical Methods for Geophysical Fluid Dynamics

Preface
	What This Book Is and What It Is Not
	Organization
	Some Historical Notes
	Caveats and Concerns
	References
Acknowledgements
Contents
1 Governing Equations  and Approximations
	1.1 Governing Equations
	1.2 Boundary and Initial Conditions
	1.3 Hydrostatic Approximation
	1.4 Boussinesq Approximation
	1.5 Shallow Water Equations
	1.6 Quasi-geostrophic Equations
	1.7 Summary and Remarks
2 Preliminaries
	2.1 Elliptic Equations
	2.2 Parabolic Equations
	2.3 Hyperbolic Equations
	2.4 Boundary Conditions
	2.5 Taylor Series
	2.6 Finite Difference Approximations
	2.7 Truncation Errors
	2.8 Notation
	2.9 Orthogonal Functions
	2.10 Fourier Series
	2.11 Fourier Transforms
	2.12 Summary and Remarks
	2.13 Exercises
	2.14 Computer Problems
		2.14.1 Truncation Errors in Two Recursion Formulas
3 Time Marching Problems
	3.1 Advection–Diffusion Equation
	3.2 Diffusion
	3.3 Advection
	3.4 Shallow Water Equations
	3.5 Summary and Remarks
4 Diffusion Problem
	4.1 One-Dimensional Diffusion Equation
	4.2 Finite Difference Form
	4.3 Numerical Stability
	4.4 Von Neumann Stability Analysis
	4.5 Stability of the Discrete Diffusion Equation
	4.6 Centered-in-Time Centered-in-Space Scheme
	4.7 Necessary Stability Condition
	4.8 Explicit and Implicit Schemes
	4.9 Convergence and Consistency: DuFort–Frankel
	4.10 Crank–Nicholson
	4.11 A Direct Elliptic Solver
	4.12 Summary and Remarks
	4.13 Exercises
	4.14 Computer Problems
		4.14.1 Vertical Diffusive Mixing
		4.14.2 Vertical Mixing in a Coupled Atmosphere–Ocean Model
		4.14.3 Yoshida's Equatorial Jet Current
5 Advection Problem
	5.1 One-Dimensional Advection Equation
	5.2 Finite Difference Forms
	5.3 Leap-Frog Scheme
	5.4 Stability of the Leap-Frog Scheme: The CFL Condition
	5.5 Numerical Dispersion
	5.6 Initial Value Problem in CTCS Schemes
	5.7 Computational Modes and Unphysical Solutions
	5.8 Getting Rid of the Computational Mode: Asselin Filter
	5.9 Upstream Scheme
	5.10 Diffusive Scheme
	5.11 Lax–Wendroff Scheme
	5.12 Semi-Lagrangian Method
	5.13 Physical Interpretation of the CFL Condition
	5.14 Implicit Scheme
	5.15 Numerical Diffusion
	5.16 Flux Corrective Schemes
	5.17 Summary and Remarks
	5.18 Exercises
	5.19 Computer Problems
		5.19.1 Advection in Atmospheres and Oceans
		5.19.2 Flux Corrective Methods
6 Shallow Water Problem
	6.1 Shallow Water Equations
	6.2 Linear Shallow Water Equations
	6.3 Analytic Considerations
	6.4 Finite Difference Forms: Linear Equations
	6.5 Staggered Grids
	6.6 Finite Difference Forms: Nonlinear Equations
	6.7 Nonlinear Instability
	6.8 Semi-implicit and Time-Splitting Methods
	6.9 Summary and Remarks
	6.10 Exercises
	6.11 Computer Problems
		6.11.1 Geostrophic Adjustment
		6.11.2 Storm Surges
		6.11.3 Quasi-Lagrangian Method Applied to a Nonlinear System
7 Open Boundary Conditions  and Nesting Techniques
	7.1 Open Boundaries
	7.2 Nesting Techniques
	7.3 Radiation Conditions
	7.4 Sponge Condition
	7.5 Flow Relaxation Scheme
	7.6 A Weakly Reflective OBC
	7.7 Summary and Remarks
	7.8 Exercises
	7.9 Computer Problems
		7.9.1 Planetary Waves
8 Generalized Vertical Coordinates
	8.1 Transformation to a General Vertical Coordinate
	8.2 Transformation of the Governing Equations
	8.3 Terrain-Following Coordinates
	8.4 Summary and Remarks
9 Two-Dimensional Problems
	9.1 Diffusion Equation
	9.2 Advection Equation
	9.3 Shallow Water Equations
	9.4 Summary and Remarks
	9.5 Exercises
	9.6 Computer Problems
		9.6.1 Upwelling in the Bay of Guinea
10 Advanced Topics
	10.1 Higher Order Advection Schemes
	10.2 Combined Advection–Diffusion
	10.3 Nonlinear Instabilities
	10.4 Smoothing and Filtering
	10.5 Two-Way Nesting
	10.6 Spectral Method
	10.7 Summary and Remarks
	10.8 Exercises
	10.9 Computer Problems
		10.9.1 Combined Advection–Diffusion
11 Quality Assurance Procedures
	11.1 Sub-grid Scale Parameterizations and Spectral Cutoffs
	11.2 Good Models
	11.3 Quality Assurance Procedures
	11.4 Summary and Remarks
A Appendix A  Introduction to Fortran 2003 via Examples
	A.1  Why Use Fortran?
	A.2  Historical Background
	A.3  Fortran Syntax
		A.3.1  Data Types in Fortran
	A.4  Structure of Fortran
		A.4.1  Declaration of Variables
		A.4.2  Instructions
	A.5  Compiling a Program
	A.6  Sample Programs
		A.6.1  Daynumber Converter
		A.6.2  Temperature Converter
		A.6.3  A More User-Friendly Version of the Converter Program
		A.6.4  Variable Types, Arrays, Loops, and Memory Allocation
		A.6.5  File Input/Output or I/O
		A.6.6  Multidimensional Arrays
		A.6.7  Functions and Subroutines
	A.7  Exercises