Preface ....................................................... xi
1 Modeling and Mathematical Concepts .......................... 1
Pros and Cons of Dynamical Models ........................... 2
An Important Modeling Assumption ............................ 4
Some Examples ............................................... 4
Example I: Simulation of Chicxulub Impact and Its
Consequences .......................................... 5
Example II: Storm Surge of Hurricane Ivan in Escambia
Bay ................................................... 7
Steps in Model Building ..................................... 8
Basic Definitions and Concepts .......................... 11
Nondimensionalization ................................... 13
A Brief Mathematical Review ............................. 14
Summary .................................................... 22
2 Basics of Numerical Solutions by Finite Difference ......... 23
First Some Matrix Algebra ............................... 23
Solution of Linear Systems of Algebraic Equations ....... 25
General Finite Difference Approach ......................... 26
Discretization .......................................... 27
Obtaining Difference Operators by Taylor Series ......... 28
Explicit Schemes ........................................ 29
Implicit Schemes ........................................ 30
How Good Is My Finite Difference Scheme? ................... 33
Stability Is Not Accuracy .................................. 35
Summary .................................................... 37
Modeling Exercises ......................................... 38
3 Box Modeling: Unsteady, Uniform Conservation of Mass ....... 39
Translations ............................................... 40
Example I: Radiocarbon Content of the Biosphere as
a One-Box Model ...................................... 40
Example II: The Carbon Cycle as a Multibox Model ........ 48
Example III: One-Dimensional Energy Balance Climate
Model 53 Finite Difference Solutions of Box Models ... 57
The Forward Euler Method ................................ 57
Predictor-Corrector Methods ............................. 59
Stiff Systems ........................................... 60
Example IV: Rothman Ocean ............................... 61
Backward Euler Method ................................... 65
Model Enhancements ...................................... 69
Summary .................................................... 71
Modeling Exercises ......................................... 71
4 One-Dimensional Diffusion Problems ......................... 74
Translations ............................................... 75
Example I: Dissolved Species in a Homogeneous Aquifer ... 75
Example II: Evolution of a Sandy Coastline .............. 80
Example III: Diffusion of Momentum ...................... 83
Finite Difference Solutions to 1-D Diffusion Problems ...... 86
Summary .................................................... 86
Modeling Exercises ......................................... 87
5 Multidimensional Diffusion Problems ........................ 89
Translations ............................................... 90
Example I: Landscape Evolution as a 2-D Diffusion
Problem .............................................. 90
Example II: Pollutant Transport in a Confined Aquifer ... 96
Example III: Thermal Considerations in Radioactive
Waste Disposal ....................................... 99
Finite Difference Solutions to Parabolic PDEs and
Elliptic Boundary Value Problems ....................... 101
An Explicit Scheme ..................................... 102
Implicit Schemes ....................................... 103
Case of Variable Coefficients .......................... 107
Summary ................................................... 108
Modeling Exercises ........................................ 109
6 Advection-Dominated Problems .............................. 111
Translations .............................................. 112
Example I: A Dissolved Species in a River .............. 112
Example II: Lahars Flowing along Simple Channels ....... 116
Finite Difference Solution Schemes to the Linear
Advection Equation ........................................ 122
Summary ................................................... 126
Modeling Exercises ........................................ 128
7 Advection and Diffusion (Transport) Problems .............. 130
Translations .............................................. 131
Example I: A Generic 1-D Case .......................... 131
Example II: Transport of Suspended Sediment in
a Stream ............................................ 134
Example III: Sedimentary Diagenesis: Influence of
Burrows ............................................. 138
Finite Difference Solutions to the Transport Equation ..... 143
QUICK Scheme ........................................... 144
QUICKEST Scheme ........................................ 146
Summary ................................................... 147
Modeling Exercises ........................................ 147
8 Transport Problems with a Twist: The Transport of
Momentum .................................................. 151
Translations .............................................. 152
Example I: One-Dimensional Transport of Momentum in
a Newtonian Fluid (Burgers' Equation) ............... 152
An Analytic Solution to Burgers' Equation ................. 157
Finite Difference Scheme for Burgers' Equation ............ 158
Solution Scheme Accuracy ............................... 160
Diffusive Momentum Transport in Turbulent Flows ........... 163
Adding Sources and Sinks of Momentum: The General Law of
Motion ................................................. 165
Summary ................................................... 166
Modeling Exercises ........................................ 167
9 Systems of One-Dimensional Nonlinear Partial
Differential Equations .................................... 169
Translations .............................................. 169
Example I: Gradually Varied Flow in an Open Channel .... 169
Finite Difference Solution Schemes ........................ for
Equation Sets .......................................... 175
Explicit FTCS Scheme on a Staggered Mesh ............... 175
Four-Point Implicit Scheme ............................. 177
The Dam-Break Problem: An Example ...................... 180
Summary ................................................... 183
Modeling Exercises ........................................ 185
10 Two-Dimensional Nonlinear Hyperbolic Systems .............. 187
Translations .............................................. 188
Example I: The Circulation of Lakes, Estuaries, and
the Coastal Ocean ................................... 188
An Explicit Solution Scheme for 2-D Vertically
Integrated Geophysical Flows ........................... 197
Lake Ontario Wind-Driven Circulation: An Example ....... 202
Summary ................................................... 203
Modeling Exercises ........................................ 206
Closing Remarks .............................................. 209
References ................................................... 211
Index ........................................................ 217
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