Mathematical Modeling of Earth's Dynamical Systems: A Primer

246 Pages·2011·1.33 MB·English

by  Rudy Slingerland

MATHEMATICAL MODELING

of Earth’s Dynamical Systems
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MATHEMATICAL MODELING

of Earth’s Dynamical Systems
A Primer
Rudy Slingerland
and Lee Kump
Princeton University Press • Princeton And oxford
Copyright © 2011 by Princeton University Press
Published by Princeton University Press, 41 William Street,
Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press, 6 Oxford
Street, Woodstock, Oxfordshire OX20 1TW
press.princeton.edu
Jacket illustration by Scott R. Miller, The Pennsylvania
State University. Comparison of results from a landscape
evolution model (CHILD) simulating the Siwalik Hills
of India and Pakistan with an oblique aerial photo of the
same region. The view is toward the west, and the big
river is the Karnali. There is no vertical exaggeration. For
details see Scott R. Miller and Rudy L. Slingerland (2006),
Topographic advection on fault-bend folds: Inheritance of
valley positions and the formation of wind gaps, Geology,
34(9): 769–772, dpo: 10.1130/G22658.1.
All Rights Reserved
Library of Congress Cataloging-in-Publication Data
Slingerland, Rudy.
Mathematical modeling of earth’s dynamical systems : a
primer / Rudy Slingerland and Lee Kump.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-691-14513-6 (hardcover : alk. paper) —
ISBN 978-0-691-14514-3 (pbk. : alk. paper) 1. Gaia
hypothesis—Mathematical models. I. Kump, Lee R. II. Title.
QH331.S55 2011
550.1’5118—dc22
2010041656
British Library Cataloging- in- Publication Data is available
This book has been composed in Sabon
Printed on acid- free paper. ∞
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
Contents
Preface xi
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
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
vi • contents
Explicit Schemes 29
Implicit Schemes 30
How Good Is My Finite Difference Scheme? 33
Stability Is Not Accuracy 35
Summary 37
Modeling Exercises 38
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
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
contents • vii
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
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
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 Diagenesis:
Influence of Burrows 138
viii • contents
Finite Difference Solutions to the Transport
Equation 143
QUICK Scheme 144
QUICKEST Scheme 146
Summary 147
Modeling Exercises 147
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
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
contents • ix
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