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Mathematical Modeling of Earth's Dynamical Systems

Book Description

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

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Contents
  5. Preface
  6. 1 Modeling and Mathematical Concepts
    1. Pros and Cons of Dynamical Models
    2. An Important Modeling Assumption
    3. Some Examples
      1. Example I: Simulation of Chicxulub Impact and Its Consequences
      2. Example II: Storm Surge of Hurricane Ivan in Escambia Bay
    4. Steps in Model Building
      1. Basic Definitions and Concepts
      2. Nondimensionalization
      3. A Brief Mathematical Review
    5. Summary
  7. 2 Basics of Numerical Solutions by Finite Difference
    1. First Some Matrix Algebra
      1. Solution of Linear Systems of Algebraic Equations
    2. General Finite Difference Approach
      1. Discretization
      2. Obtaining Difference Operators by Taylor Series
      3. Explicit Schemes
      4. Implicit Schemes
    3. How Good Is My Finite Difference Scheme?
      1. Stability Is Not Accuracy
    4. Summary
    5. Modeling Exercises
  8. 3 Box Modeling: Unsteady, Uniform Conservation of Mass
    1. Translations
      1. Example I: Radiocarbon Content of the Biosphere as a One-Box Model
      2. Example II: The Carbon Cycle as a Multibox Model
      3. Example III: One-Dimensional Energy Balance Climate Model
    2. Finite Difference Solutions of Box Models
      1. The Forward Euler Method
      2. Predictor–Corrector Methods
      3. Stiff Systems
      4. Example IV: Rothman Ocean
      5. Backward Euler Method
      6. Model Enhancements
    3. Summary
    4. Modeling Exercises
  9. 4 One-Dimensional Diffusion Problems
    1. Translations
      1. Example I: Dissolved Species in a Homogeneous Aquifer
      2. Example II: Evolution of a Sandy Coastline
      3. Example III: Diffusion of Momentum
    2. Finite Difference Solutions to 1-D Diffusion Problems
    3. Summary
    4. Modeling Exercises
  10. 5 Multidimensional Diffusion Problems
    1. Translations
      1. Example I: Landscape Evolution as a 2-D Diffusion Problem
      2. Example II: Pollutant Transport in a Confined Aquifer
      3. Example III: Thermal Considerations in Radioactive Waste Disposal
    2. Finite Difference Solutions to Parabolic PDEs and Elliptic Boundary Value Problems
      1. An Explicit Scheme
      2. Implicit Schemes
      3. Case of Variable Coefficients
    3. Summary
    4. Modeling Exercises
  11. 6 Advection-Dominated Problems
    1. Translations
      1. Example I: A Dissolved Species in a River
      2. Example II: Lahars Flowing along Simple Channels
    2. Finite Difference Solution Schemes to the Linear Advection Equation
    3. Summary
    4. Modeling Exercises
  12. 7 Advection and Diffusion (Transport) Problems
    1. Translations
      1. Example I: A Generic 1-D Case
      2. Example II: Transport of Suspended Sediment in a Stream
      3. Example III: Sedimentary Diagenesis: Influence of Burrows
    2. Finite Difference Solutions to the Transport Equation
      1. QUICK Scheme
      2. QUICKEST Scheme
    3. Summary
    4. Modeling Exercises
  13. 8 Transport Problems with a Twist: The Transport of Momentum
    1. Translations
      1. Example I: One-Dimensional Transport of Momentum in a Newtonian Fluid (Burgers’ Equation)
    2. An Analytic Solution to Burgers’ Equation
    3. Finite Difference Scheme for Burgers’ Equation
      1. Solution Scheme Accuracy
    4. Diffusive Momentum Transport in Turbulent Flows
    5. Adding Sources and Sinks of Momentum: The General Law of Motion
    6. Summary
    7. Modeling Exercises
  14. 9 Systems of One-Dimensional Nonlinear Partial Differential Equations
    1. Translations
      1. Example I: Gradually Varied Flow in an Open Channel
    2. Finite Difference Solution Schemes for Equation Sets
      1. Explicit FTCS Scheme on a Staggered Mesh
      2. Four-Point Implicit Scheme
      3. The Dam-Break Problem: An Example
    3. Summary
    4. Modeling Exercises
  15. 10 Two-Dimensional Nonlinear Hyperbolic Systems
    1. Translations
      1. Example I: The Circulation of Lakes, Estuaries, and the Coastal Ocean
    2. An Explicit Solution Scheme for 2-D Vertically Integrated Geophysical Flows
      1. Lake Ontario Wind-Driven Circulation: An Example
    3. Summary
    4. Modeling Exercises
  16. Closing Remarks
  17. References
  18. Index