Mathematical Modeling of Earth's Dynamical Systems
A Primer
1st Edition
Published on 28. March 2011
248 pages
978-1-4008-3911-7 (ISBN)
Description
A concise guide to representing complex Earth systems using simple dynamic models
Mathematical Modeling of Earth's Dynamical Systems gives 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 Systems helps 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
- Professors: A supplementary Instructor's Manual is available for this book. It is restricted to teachers using the text in courses. For information on how to obtain a copy, refer to: http://press.princeton.edu/class_use/solutions.html
Mathematical Modeling of Earth's Dynamical Systems gives 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 Systems helps 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
- Professors: A supplementary Instructor's Manual is available for this book. It is restricted to teachers using the text in courses. For information on how to obtain a copy, refer to: http://press.princeton.edu/class_use/solutions.html
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Persons
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.
Content
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
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
