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Mathematical Modeling of Earth's Dynamical Systems: A Primer

Rudy Slingerland and Lee Kump
Princeton University Press
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
William J. Satzer
, on

If you’ve ever taught a differential equations course and yearned for fresh examples, this is definitely a book for you. Wouldn’t it be great to give some of those terribly overworked examples a rest? Even some of the more recent examples from chemistry and biology are starting to look a bit fatigued.

This book was written by two earth scientists with the aim of providing what they regard as essential skills for graduate students and advanced undergraduates in their field. Those skills include the ability to translate chemical and physical systems into mathematical and computational models that provide insight into dynamical processes on the earth’s surface and inside. All the models treated here are based on ordinary and partial differential equations.

What kinds of examples do the authors use? One of the first is a model of the radiocarbon content of the biosphere. In simplest form, this leads to exponential decay; with periodic forcing (determined perhaps by the sunspot cycle), more complicated solutions arise. As the book proceeds, the examples get more complex. Other examples include: dissolved species in an aquifer, evolution of a sandy coastline, and pollutant transport in a confined aquifer. One amazing pictorial example from the introduction shows a simulated time sequence of an iron bolide asteroid, one kilometer in diameter, hitting the ocean at a 45 degree angle. Two of the most interesting worked out examples are analysis of complicated circulation patterns in Lake Ontario and modeling a lahar (water and pyroclastic debris flowing down the side of a volcano).

Although the authors concentrate on earth science, pretty much everything is accessible to anyone having some knowledge of basic physics and chemistry. The early chapters of the book pay a lot of attention to the details of developing a model. It is clear that the authors have a good deal of expertise in modeling. Yet they note that their book is intended as a primer, and students are not expected to have any modeling experience. By the nature of their interests, most of the discussion revolves around partial differential equations — nearly all the problems of interest have time and at least one spatial dimension as independent variables. One consequence of this is a secondary focus in the book on finite difference methods for solving partial differential equations.

Besides being a wonderful source of examples, this short book is a pleasing and well-organized introduction to modeling with differential equations.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

Preface xi
Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 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 Diagenes Influence of Burrows 138
Finite Difference Solutions to the Transport Equation 143
QUICK Scheme 144
QUICKEST Scheme 146
Summary 147
Modeling Exercises 147
Chapter 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

Chapter 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

Chapter 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