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Quantitative Modeling of Earth Surface Processes
Assuming some knowledge of calculus and
Geomorphology is undergoing a renaissance made
basic programming experience, this quantitative
possible by new techniques in numerical modeling,
textbook is designed for advanced geomorphology
geochronology and remote sensing. Earth surface
courses and as a reference book for professional
processes are complex and richly varied, but analyt-
researchers in Earth and planetary sciences look-
ical and numerical modeling techniques are pow-
ing for a quantitative approach to earth surface
erful tools for interpreting these systems and the
processes. Exercises at the end of each chapter
landforms they create.
begin with simple calculations and then progress
This textbook describes some of the most effec-
to more sophisticated problems that require com-
tive and straightforward quantitative techniques for
puter programming. All the necessary computer
modeling earth surface processes. By emphasizing a
codes are available online at www.cambridge.org/
core set of equations and solution techniques, the
book presents state-of-the-art models currently em-
ployed in earth surface process research, as well
Jon Pelletier was awarded a Ph.D. in geological sci-
as a set of simple but practical tools that can be
ences from Cornell University in 1997. Following
used to tackle unsolved research problems. Detailed
two years at the California Institute of Technology
case studies demonstrate application of the meth-
as the O.K. Earl Prize Postdoctoral Scholar, he was
ods to a wide variety of processes including hill-
made an associate professor of geosciences at the
slope, ¬‚uvial, eolian, glacial, tectonic, and climatic
University of Arizona where he teaches geomorphol-
systems. The computer programming codes used in
ogy. Dr Pelletier™s research involves mathematical
the case studies are also presented in a set of ap-
modeling of a wide range of surface processes on
pendices so that readers can readily utilize these
Earth and other planets, including the evolution
methods in their own work. Additional references
of mountain belts, the transport and deposition of
are also provided for readers who wish to ¬ne-
dust in arid environments, and ¬‚uvial and glacial
tune their models or pursue more sophisticated
processes on Mars.
Quantitative Modeling of
Earth Surface Processes
Jon D. Pelletier
University of Arizona
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo

Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
Information on this title: www.cambridge.org/9780521855976

© J. D. Pelletier 2008

This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2008

ISBN-13 978-0-511-42310-9 eBook (EBL)

ISBN-13 978-0-521-85597-6 hardback

Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.

Preface page ix

Chapter 1 Introduction 1

1.1 A tour of the ¬‚uvial system 1
1.2 A tour of the eolian system 12
1.3 A tour of the glacial system 19
1.4 Conclusions 29

Chapter 2 The diffusion equation 30

2.1 Introduction 30
2.2 Analytic methods and applications 34
2.3 Numerical techniques and applications 57
Exercises 63

Chapter 3 Flow routing 66

3.1 Introduction 66
3.2 Algorithms 66
3.3 ˜˜Cleaning up™™ US Geological Survey DEMs 70
3.4 Application of ¬‚ow-routing algorithms to estimate
¬‚ood hazards 72
3.5 Contaminant transport in channel bed sediments 74
Exercises 85

Chapter 4 The advection/wave equation 87

4.1 Introduction 87
4.2 Analytic methods 88
4.3 Numerical methods 90
4.4 Modeling the ¬‚uvial-geomorphic response of the southern
Sierra Nevada to uplift 93
4.5 The erosional decay of ancient orogens 101
Exercises 107

Chapter 5 Flexural isostasy 109

5.1 Introduction 109
5.2 Methods for 1D problems 111
5.3 Methods for 2D problems 113
5.4 Modeling of foreland basin geometry 116
5.5 Flexural-isostatic response to glacial erosion in the
western US 120
Exercises 123

Chapter 6 Non-Newtonian ¬‚ow equations 125

6.1 Introduction 125
6.2 Modeling non-Newtonian and perfectly plastic ¬‚ows 125
6.3 Modeling ¬‚ows with temperature-dependent viscosity 130
6.4 Modeling of threshold-sliding ice sheets and glaciers over
complex 3D topography 132
6.5 Thrust sheet mechanics 147
6.6 Glacial erosion beneath ice sheets 149
Exercises 160

Chapter 7 Instabilities 161

7.1 Introduction 161
7.2 An introductory example: the Rayleigh--Taylor instability 162
7.3 A simple model for river meandering 164
7.4 Werner™s model for eolian dunes 166
7.5 Oscillations in arid alluvial channels 169
7.6 How are drumlins formed? 174
7.7 Spiral troughs on the Martian polar ice caps 183
Exercise 187

Chapter 8 Stochastic processes 188

8.1 Introduction 188
8.2 Time series analysis and fractional Gaussian noises 188
8.3 Langevin equations 191
8.4 Random walks 193
8.5 Unsteady erosion and deposition in eolian environments 194
8.6 Stochastic trees and diffusion-limited aggregation 196
8.7 Estimating total ¬‚ux based on a statistical
distribution of events: dust emission from playas 199
8.8 The frequency-size distribution of landslides 205
8.9 Coherence resonance and the timing of ice ages 210
Exercises 221

Appendix 1 Codes for solving the diffusion
equation 222

Appendix 2 Codes for ¬‚ow routing 235

Appendix 3 Codes for solving the advection
equation 242

Appendix 4 Codes for solving the ¬‚exure equation 256

Appendix 5 Codes for modeling non-Newtonian
¬‚ows 263

Appendix 6 Codes for modeling instabilities 267

Appendix 7 Codes for modeling stochastic
processes 274

References 278
Index 290

The colour plates are to be found between pages 108 and 109.

Geomorphology is undergoing a renaissance gists, on the other hand, are adept at reading
made possible by new techniques in numeri- the geomorphic and sedimentary records to ad-
cal modeling, geochronology, and remote sens- dress big questions, but those records often can-
ing. Advances in numerical modeling make it not be fully interpreted using ¬eld observations
possible to model surface processes and their and geochronology alone. A fourth challenge is
feedbacks with climate and tectonics over a geomorphic prediction. In order for applied geo-
wide range of spatial and temporal scales. The morphology to realize its full potential, geomor-
Shuttle Radar Topography Mission (SRTM) has phologists must be able to predict where geo-
mapped most of Earth™s topography at much morphic hazards are most likely to occur, tak-
higher spatial resolution and accuracy than ever ing into account the full complexity of processes,
before. Cosmogenic dating and other geochrono- feedbacks, and the multi-scale heterogeneity of
logic techniques have provided vast new data on Earth™s surface.
surface-process rates and landform ages. Model- Analytical and numerical modeling are pow-
ing, geochronology, and remote sensing are also erful tools for addressing these challenges. First,
revolutionizing natural-hazard assessment and modeling is useful for establishing relationships
mitigation, enabling society to assess the hazards between process and form. Using quantitative
posed by ¬‚oods, landslides, windblown dust, soil models for different processes, modeling allows
erosion, and other geomorphic hazards. us to determine the signatures of those processes
The complexity of geomorphic systems poses in the landscape. In some cases, the histories of
several challenges, however. First, the relation- external forcing mechanisms (e.g. climate and
ship between process and form is often dif¬cult tectonics) can also be inferred. These linkages are
to determine uniquely. Many geomorphic pro- important because there is generally no direct
cesses cannot be readily quanti¬ed, and it is often observation of landforms that enables us to un-
unclear which processes are most important in derstand how they evolved. Modeling has played
controlling a particular geomorphic system, and a signi¬cant role in recent contributions to our
how those processes interact to form the geomor- understanding of many classic landform types.
phic and sedimentary records we see today. Ter- In this book, I will present analytic and numer-
races and sedimentary deposits on alluvial fans, ical models for many different landform types,
for example, are controlled by climate, tectonics, including drumlins, sand dunes, alluvial fans,
and internal drainage adjustments in a way that and bedrock drainage networks, just to name a
geomorphologists have not been able to fully un- few. Modeling is particularly useful for explor-
ravel. Second, surface processes are strongly in¬‚u- ing the feedbacks between different components
enced by ¬‚uid motions, and most classic geomor- of geomorphic systems (i.e. hillslopes and chan-
phic techniques (e.g. ¬eld mapping) are not well nel networks). Channel aggradation and incision,
suited to quantifying ¬‚uid dynamics and their for example, controls the base level of hillslopes,
interactions with the surface. Third, the geomor- and hillslopes supply primary sediment ¬‚ux to
phic community must bridge the gap between channels. Yet many geomorphic textbooks treat
process-based geomorphology and Quaternary ge- these as essentially independent systems. In some
ology. Process-based geomorphologists have made cases, the boundary conditions posed by one as-
great strides in quantifying transport and erosion pect of a geomorphic system can be considered
laws for geomorphic systems, but this approach to be ¬xed. In many of the most interesting un-
has not yet led to major advances in big geologic solved problems, however, they cannot. An un-
questions, such as how quickly the Grand Canyon derstanding of ¬‚uvial-system response to climate
was carved, or how mountain belts respond to change, for example, cannot be achieved with-
glacial erosion, for example. Quaternary geolo- out a quantitative understanding of the coupled

evolution and feedbacks between hillslopes and behavior and solution methods for diffusion, the
channels. reader will be poised to understand diffusive phe-
Numerical modeling is also useful in geomor- nomena as they arise in different contexts. Sim-
phology because of the central role that ¬‚uid dy- ilarly, boundary-layer and non-Newtonian ¬‚uid
namics play in landform evolution. Geomorphol- ¬‚ows are also examples of equation sets that
ogy can be roughly de¬ned as the evolution of have broad applicability in Earth surface pro-
landforms by the ¬‚uid ¬‚ow of wind, liquid wa- cesses. Non-Newtonian ¬‚uid ¬‚ows are the basis
ter, and ice above the surface. Yet, ¬‚uid mechan- for understanding and modeling lava ¬‚ows, de-
ics generally plays a minor role in many aspects bris ¬‚ows, and alpine glaciers. In the course of
of geomorphic research. Many landform types developing my quantitative skills and applying
evolve primarily by a dynamic feedback mech- them to a wide range of research problems, I
anism in which the topography in¬‚uences the have been continually amazed at how often a
¬‚uid ¬‚ow (and hence the shear stresses) above the concept or technique from one area can pro-
topography, which, in turn, controls how the to- vide the missing piece required to solve a long-
pography evolves by erosion and deposition. Nu- standing problem in another ¬eld. It is this cross-
merical modeling is one of the most successful fertilization of ideas and methods that I want
ways to quantify ¬‚uid ¬‚ow in a complex environ- to foster and share with readers. By emphasizing
ment, and hence it can and should play a central a core set of equations and solution techniques,
role in nearly all geomorphic research. In this readers will come away with powerful tools that
book, numerical models of ¬‚uid ¬‚ow serve as the can be used to tackle unsolved problems, as well
basis for many of the book™s numerical landform as speci¬c knowledge of state-of-the-art models
evolution models. Despite the power of numeri- currently used in Earth surface process research.
cal models, they cannot be used in a vacuum. All The book is designed for use as a textbook for an
numerical modeling studies must integrate ¬eld advanced geomorphology course and as a ref-
observations, digital geospatial data, geochronol- erence book for professional geomorphologists.
ogy, and small-scale experiments to motivate, Mathematically, I assume that readers have mas-
constrain, and validate the numerical work. tered multivariable calculus and have had some
Earth surface processes are complex and experience with partial differential equations.
richly varied. Most books approach the inher- The exercises at the end of each chapter begin
ent variety of geomorphic systems by serially with simple problems that require only the main
cataloging the processes and landforms charac- concepts and a few calculations, then progress to
teristic of each environment (hillslope, ¬‚uvial, more sophisticated problems that require com-
eolian, glacial, etc.). The disadvantage of this ap- puter programming.
proach is that each geomorphic environment is The purpose of this book is not to provide
presented as being essentially unique, and com- an exhaustive survey of all analytic or numeri-
mon dynamical behaviors are not emphasized. cal methods for a given problem, but rather to
This book follows a different path, taking advan- focus on the most powerful and straightforward
tage of a common mathematical framework to methods. For example, advective equations can
emphasize universal concepts. This framework fo- be solved using the upwind-differencing, Lax--
cuses on linear and nonlinear diffusion, advec- Wendroff method, staggered-leapfrog, pseudo-
tion, and boundary-value problems that quantify spectral, and semi-Lagrangian methods, just to
the stresses in the atmosphere and lithosphere. name a few. Rather than cover all available meth-
The diffusion equation, for example, can be used ods, this book focuses primarily on the simpler
to describe hillslope evolution, channel-bed evo- methods for readers who want to get started
lution, delta progradation, hydrodynamic disper- quickly or who need to solve problems of mod-
sion in groundwater aquifers, turbulent disper- est computational size. In this sense, the book
sion in the atmosphere, and heat conduction will follow the model of Numerical Recipes (Press
in soils and the Earth™s crust. By ¬rst provid- et al., 1992) by providing tools that work for
ing the reader with a solid foundation in the most applications, with additional references for

readers who want to ¬ne-tune their models. The 1D and when we model the evolution of surfaces
appendices provide the reader with computer we will refer to the model as 2D. This conven-
code to illustrate technique application in real- tion may seem strange at ¬rst, but it makes the
world research problems. Hence, many of the ap- book more consistent overall. For example, if we
plications I cover in this book necessarily come model heat ¬‚ow in a thin layer using the diffu-
from my own research. In focusing on my own sion equation, everyone would agree that we are
work, I don™t mean to imply that my work is solving a 2D problem for T (x, y, t). If we use the
the only or best approach to a given problem. diffusion equation to model the evolution of a
The book is not intended to be a complete survey hillslope described by h(x, y, t), that, too, is math-
of geomorphology, and I knowingly have left out ematically a 2D problem even though it represents
many important contributions in favor of a more a 3D landform.
focused, case-study approach. I wish to acknowledge my colleagues at the
Modelers are not always consistent in the way University of Arizona for taking a chance on
that they use the terms one- (1D), two- (2D), an unconventional geomorphologist and for cre-
and three-dimensional (3D) when referring to a ating such a collegial working environment. I
model. Usually when physicists and mathemati- also wish to thank my graduate students Leslie
cians use the term 1D they are referring to a Hsu, Jason Barnes, James Morrison, Steve De-
model that has one independent spatial variable. Long, Michael Cline, Joe Cook, Maria Banks, Joan
This can be confusing when applied to geomor- Blainey, Jennifer Boerner, and Amy Rice for help-
phic problems, however, because a model for the ful conversations and collaborations. I hope they
evolution of a topographic pro¬le, h(x, t), for ex- have learned as much from me as I have from
ample, would be called 1D even though it repre- them. Funding for my research has come from
sents a 2D pro¬le. Similarly, a model for the evo- the National Science Foundation, the Army Re-
lution of a topographic surface h(x, y, t) would be search Of¬ce, the US Geological Survey, the NASA
described as 2D even though the surface itself is Of¬ce of Space Science, the Department of En-
three-dimensional. In this book we will use the ergy, and the State of Arizona™s Water Sustain-
convention that the dimensionality of the prob- ability Program. I gratefully acknowledge support
lem refers to the number of independent spa- from these agencies and institutions. Finally, I
tial variables. Therefore, when we model 2D to- wish to thank my wife Pamela for her patience
pographic pro¬les we will classify the model as and support.
Chapter 1


and Jones, 1999). The structural style of exten-
1.1 A tour of the ¬‚uvial system sional faulting varies greatly from place to place
in the western US, but Figure 1.2 illustrates one
simple model of extension that can help us un-
The ¬‚uvial system is classically divided into
derstand the topography of the modern Basin
erosional, transportational, and depositional
and Range. The weight of the high topography
regimes (Schumm, 1977). In the Basin and Range
in the region created a vertical compressive force
province of the western US, Cenozoic tectonic
on the lower crust. That compressive force was
extension has produced a semi-periodic topog-
accompanied by an extensional force in the hori-
raphy with high ranges (dominated by erosion)
zontal direction. Mechanically, all rocks respond
and low valleys (dominated by deposition). In this
to a compressive force in one direction with an
region, all three ¬‚uvial-system regimes can be
extensional force in the other direction in or-
found within distances of 10--20 km. As an in-
der to preserve volume. This combination of com-
troduction to the scienti¬c questions addressed
pressive vertical forces and extensional horizon-
in this book, we start with a tour of the pro-
tal forces created faults at angles of ≈ 5--30—¦ to
cess zones of the ¬‚uvial system, using Hanau-
the horizontal (with smaller angles at greater
pah Canyon, Death Valley, California, as a type
example (Figure 1.1).
Crustal blocks ¬‚oat on the mantle, with the
1.1.1 Large-scale topography of the basin weight of the crust block supported by the buoy-
and range province ancy force that results from the bottom of the
crustal block displacing higher-density mantle
The large-scale geomorphology of the Basin and
rock. The shape of each crustal block partially
Range is a consequence of the geometry of faults
determines how high above Earth™s surface the
that develop during extension and the subse-
block will stand. Normal faulting creates a series
quent isostatic adjustment of each crustal block.
of trapezoidal crustal blocks that, while mechan-
In the late Cretaceous, the Basin and Range was
ically coupled by faults, can also rise and fall ac-
an extensive high-elevation plateau, broadly simi-
cording to their shape. A crustal block with a
lar to the Altiplano-Eastern Cordillera of the cen-

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