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
Author(s): Rudy Slingerland, Lee Kump
Publisher: Princeton University Press
Year: 2011
Language: English
Pages: 246
Tags: Науки о Земле;Метеорология и климатология;
Cover......Page 1
Title......Page 4
Copyright......Page 5
Contents......Page 6
Preface......Page 12
1 Modeling and Mathematical Concepts......Page 16
Pros and Cons of Dynamical Models......Page 17
Some Examples......Page 19
Example I: Simulation of Chicxulub Impact and Its Consequences......Page 20
Example II: Storm Surge of Hurricane Ivan in Escambia Bay......Page 22
Steps in Model Building......Page 23
Basic Definitions and Concepts......Page 26
Nondimensionalization......Page 28
A Brief Mathematical Review......Page 29
Summary......Page 37
First Some Matrix Algebra......Page 38
Solution of Linear Systems of Algebraic Equations......Page 40
General Finite Difference Approach......Page 41
Discretization......Page 42
Obtaining Difference Operators byTaylor Series......Page 43
Explicit Schemes......Page 44
Implicit Schemes......Page 45
How Good Is My Finite Difference Scheme?......Page 48
Stability Is Not Accuracy......Page 50
Summary......Page 52
Modeling Exercises......Page 53
3 Box Modeling: Unsteady, Uniform Conservation of Mass......Page 54
Example I: Radiocarbon Content of the Biosphere as a One-Box Model......Page 55
Example II: The Carbon Cycle as a Multibox Model......Page 63
Example III: One-Dimensional Energy Balance Climate Model......Page 68
The Forward Euler Method......Page 72
Predictor–Corrector Methods......Page 74
Stiff Systems......Page 75
Example IV: Rothman Ocean......Page 76
Backward Euler Method......Page 80
Model Enhancements......Page 84
Modeling Exercises......Page 86
4 One-Dimensional Diffusion Problems......Page 89
Example I: Dissolved Species in a Homogeneous Aquifer......Page 90
Example II: Evolution of a Sandy Coastline......Page 95
Example III: Diffusion of Momentum......Page 98
Summary......Page 101
Modeling Exercises......Page 102
5 Multidimensional Diffusion Problems......Page 104
Example I: Landscape Evolution as a 2-D Diffusion Problem......Page 105
Example II: Pollutant Transport in a Confined Aquifer......Page 111
Example III: Thermal Considerations in Radioactive Waste Disposal......Page 114
Finite Difference Solutions to Parabolic PDEs and Elliptic Boundary Value Problems......Page 116
An Explicit Scheme......Page 117
Implicit Schemes......Page 118
Case of Variable Coefficients......Page 122
Summary......Page 123
Modeling Exercises......Page 124
6 Advection-Dominated Problems......Page 126
Example I: A Dissolved Species in a River......Page 127
Example II: Lahars Flowing along Simple Channels......Page 131
Finite Difference Solution Schemes to the Linear Advection Equation......Page 137
Summary......Page 141
Modeling Exercises......Page 143
7 Advection and Diffusion (Transport) Problems......Page 145
Example I: A Generic 1-DCase......Page 146
Example II: Transport of Suspended Sediment in a Stream......Page 149
Example III: Sedimentary Diagenesis:Influence of Burrows......Page 153
Finite Difference Solutions to the Transport Equation......Page 158
QUICK Scheme......Page 159
QUICKEST Scheme......Page 161
Modeling Exercises......Page 162
8 Transport Problems with a Twist: The Transport of Momentum......Page 166
Example I: One-Dimensional Transport of Momentum in a Newtonian Fluid (Burgers’ Equation)......Page 167
An Analytic Solution to Burgers’ Equation......Page 172
Finite Difference Scheme for Burgers’Equation......Page 173
Solution Scheme Accuracy......Page 175
Diffusive Momentum Transport inTurbulent Flows......Page 178
Adding Sources and Sinks of Momentum:The General Law of Motion......Page 180
Summary......Page 181
Modeling Exercises......Page 182
Example I: Gradually Varied Flow in an Open Channel......Page 184
Explicit FTCS Scheme on a Staggered Mesh......Page 190
Four-Point Implicit Scheme......Page 192
The Dam-BreakProblem: An Example......Page 195
Summary......Page 198
Modeling Exercises......Page 200
10 Two-Dimensional Nonlinear Hyperbolic Systems......Page 202
Example I: The Circulation of Lakes, Estuaries, and the Coastal Ocean......Page 203
An Explicit Solution Scheme for 2-DVertically Integrated Geophysical Flows......Page 212
Lake Ontario Wind-DrivenCirculation: An Example......Page 217
Summary......Page 218
Modeling Exercises......Page 221
Closing Remarks......Page 224
References......Page 226
A......Page 232
B......Page 233
C......Page 234
D......Page 235
E......Page 236
F......Page 237
H......Page 238
K......Page 239
M......Page 240
N......Page 241
P......Page 242
S......Page 244
T......Page 245
Y......Page 246
