Mathematical Modeling of Earth's Dynamical Systems

by Slingerland, Rudy; Kump, Lee

• ISBN13: 9780691145143
• ISBN10: 0691145148
• Format: Paperback
• Copyright: 2011-03-28
• Publisher: Princeton Univ Pr

Summary

Mathematical Modeling of Earth's Dynamical Systemsgives Earth scientists the essential skills for translating chemical and physical systems into mathematical and computational models that provide enhanced insight into Earth's processes. Using a step-by-step method, the book identifies the important geological variables of physical-chemical geoscience problems and describes the mechanisms that control these variables. This book is directed toward upper-level undergraduate students, graduate students, researchers, and professionals who want to learn how to abstract complex systems into sets of dynamic equations. It shows students how to recognize domains of interest and key factors, and how to explain assumptions in formal terms. The book reveals what data best tests ideas of how nature works, and cautions against inadequate transport laws, unconstrained coefficients, and unfalsifiable models. Various examples of processes and systems, and ample illustrations, are provided. Students using this text should be familiar with the principles of physics, chemistry, and geology, and have taken a year of differential and integral calculus. Mathematical Modeling of Earth's Dynamical Systemshelps Earth scientists develop a philosophical framework and strong foundations for conceptualizing complex geologic systems. Step-by-step lessons for representing complex Earth systems as dynamical models Explains geologic processes in terms of fundamental laws of physics and chemistry Numerical solutions to differential equations through the finite difference technique A philosophical approach to quantitative problem-solving Various examples of processes and systems, including the evolution of sandy coastlines, the global carbon cycle, and much more

Author Biography

Rudy Slingerland and Lee Kump are professors of geosciences at Pennsylvania State University. Slingerland is the coauthor of Simulating Clastic Sedimentary Basins. Kump is the coauthor of The Earth System.

Table of Contents

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