Mathematical Geoscience

Interdisciplinary Applied Mathematics

Editors

S.S. Antman P. Holmes

L. Sirovich K. Sreenivasan

Series Advisors

C.L. Bris L. Glass

P.S. Krishnaprasad R.V. Kohn

J.D. Muray S.S. Sastry

Problems in engineering, computational science, and the physical and biological sciences

are using increasingly sophisticated mathematical techniques. Thus, the bridge between the

mathematical sciences and other disciplines is heavily traveled. The correspondingly in￾creased dialog between the disciplines has led to the establishment of the series: Interdis￾ciplinary Applied Mathematics.

The purpose of this series is to meet the current and future needs for the interaction between

various science and technology areas on the one hand and mathematics on the other. This is

done, firstly, by encouraging the ways that mathematics may be applied in traditional areas, as

well as point towards new and innovative areas of applications; and, secondly, by encouraging

other scientific disciplines to engage in a dialog with mathematicians outlining their problems

to both access new methods and suggest innovative developments within mathematics itself.

The series will consist of monographs and high-level texts from researchers working on the

interplay between mathematics and other fields of science and technology.

Interdisciplinary Applied Mathematics

For other titles published in this series, go to

www.springer.com/series/1390

Andrew Fowler

Mathematical

Geoscience

Andrew Fowler

MACSI, Department of Mathematics

& Statistics

University of Limerick

Limerick, Ireland

Series Editors

S.S. Antman

Department of Mathematics

and

Institute for Physical Science

and Technology

University of Maryland

College Park, MD 20742, USA

ssa@math.umd.edu

L. Sirovich

Department of Biomathematics

Laboratory of Applied Mathematics

Mt. Sinai School of Medicine

Box 1012

New York, NY 10029, USA

Lawrence.Sirovich@mssm.edu

P. Holmes

Department of Mechanical and Aerospace

Engineering

Princeton University

215 Fine Hall

Princeton, NJ 08544, USA

pholmes@math.princeton.edu

K. Sreenivasan

Department of Physics

New York University

70 Washington Square South

New York City, NY 10012, USA

katepalli.sreenivasan@nyu.edu

ISSN 0939-6047

ISBN 978-0-85729-699-3 e-ISBN 978-0-85729-721-1

DOI 10.1007/978-0-85729-721-1

Springer London Dordrecht Heidelberg New York

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

Library of Congress Control Number: 2011930929

Mathematics Subject Classification (2000): 86.02, 76.02, 35.02, 34.02

© Springer-Verlag London Limited 2011

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Springer is part of Springer Science+Business Media (www.springer.com)

This book is dedicated with affection and

appreciation to Jim Murray and his wife

Sheila

Jim and Sheila Murray in the garden of their home in Connecticut, summer 2010

Preface

The hardest thing to do with this book was to decide what to call it. The original

working title was ‘Mathematics and the environment’, and my aspiration was, and

is, to provide a blueprint for the application of mathematical models to problems in

the environment which involve the use of differential equations.

The environment is becoming fashionable in applied mathematics, but it often

means different things to different people. It may mean oceans and atmospheres,

and numerical modelling; it may mean groundwater flow and related pollution prob￾lems, for example involving remediation of hydrocarbons or dispersal of phosphates

and nitrates in the soil; or it might be the application of statistical methods in the

assessment of risk and uncertainty in, for example, hydrological forecasting.

No doubt these subjects concern the environment, but they are particular topics.

This book is about general scientific problems concerning phenomena in the world

around us. In the sense that ‘mathematical biology’ is the mathematical study of

living things, the logical title for this book would be ‘Mathematical Geology’, the

mathematical study of processes on (or in) the Earth. Unfortunately, Geology is a

subject which tends to carry the narrower meaning of the study of rocks, and it is

partly to get away from this that university departments have increasingly rechris￾tened themselves as departments of Geology and Geophysics, or of Earth Science,

or (most recently) of Earth System Science.

So, this book is not just about mathematical geology: it concerns much more than

the study of rocks. Nor is it mathematical geophysics, although it contains a good

deal of this also. It is mathematics and the environment, but where the word ‘envi￾ronment’ is used in a much wider sense than the narrower uses alluded to above.

The two books which are closest to this in theme and subject matter are Andrew

Goudie’s ‘The Nature of the Environment’, and Arthur Holmes’s masterful ‘Prin￾ciples of Physical Geology’. The latter book could almost provide the contents list

for the present one. The difference of course is that my concern here is in providing

mathematical models which can explain some of the physical phenomena which are

described in these two books.

Writing about recent theories for subglacial landforms, Clarke (2005) said that

‘the work has a daunting mathematical level, uncertain relevance, but potentially

vii

viii Preface

interesting implications.’ For an applied mathematician working in seriously inter￾disciplinary subjects, perhaps this slightly barbed comment is as good as it gets.

This book is, I expect, daunting. It is not necessary that hard scientific problems

beget hard mathematical problems when they are done properly, but it ought to be

what you expect. Decent science does not come cheap.

I personally hope that most of this book is relevant, but that is ultimately a matter

for the scientific community. Relevance is promoted by a kind of cultural accep￾tance, and it needs to be argued through, almost religiously.

This book in its earliest form consisted of written course notes for a sixteen

lecture final year undergraduate course at Oxford. I have taught a similar twenty￾four lecture course at masters’ level at Limerick. For such courses, I select four or

five chapters, and selectively teach material from them. For example, the current

Oxford and Limerick courses take material from Chaps. 2, 4, 5 and 10. Of course

the chapters contain much more material than one could cover in four or six lectures;

one could in fact take an entire course from a single chapter. But my purpose here

is to allow a freedom for selection, and also to elaborate the material to the point

where it becomes of research interest. In writing the book, I have been stimulated to

question accepted wisdom, and to explore new ideas, and some of the material has

even been written up in the form of research papers after the fact.

There is a danger in trying to write an encompassing book about mathematical

geoscience, of which I am only too aware. Most obviously, there are many subjects

which have been left out, and for those which are included, there is no space for a

comprehensive exposition. A glance at the reference list will show that I have largely

followed my own personal view of the subject matter. References are given at the

end of each chapter, but do not aim to give a complete review; rather the intention is

to provide pointers for those interested, with the hope that others will engage with

some of the problems. Geoscience is full of extraordinarily interesting problems.

The audience for this book is largely what is called the GFD community, brought

up on fluid mechanics in the oceans and atmosphere, but which has now branched

out into many of the subjects dealt with here. It is my hope that applied mathemati￾cians may chance on the material, and be stimulated to explore some of the models

which are discussed. It is also my hope that geoscientists will find some of the phe￾nomena and ideas interesting, even if some of the technical detail becomes at times

too threatening.

A large number of people have been of considerable assistance and help in the

something like ten years it has taken to finally produce this book. Firstly, I should

thank my publishers at Springer, who have been very patient over the years: Karen

Borthwick, and more recently, Lauren Stoney. I am grateful to Felix Ng, who rapidly

and expertly produced early drafts of some of the figures for Chaps. 2, 4, 7 and 11.

Ian Hewitt produced Fig. 10.14 and Fig. 9.15. Christine Butler unearthed a copy

of Fig. 11.12 from the vaults of the International Glaciological Society. Bill Shilts,

Christian Zdanowicz and Brian Moorman were very helpful concerning the image

in Fig. 10.22; Gary Parker, Norm Smith and Terence McCarthy were equally helpful

concerning Fig. 5.1. Thanks also to Emanuele Schiavi, Stephen O’Brien, Thomas

Vitolo, Dave Cocks, Rachel Zammett, Geoff Evatt, Rob Style, Sarah Mitchell, Chris

Preface ix

Banerji and Sarah McBurnie for their vigilance in spotting errors or providing ad￾vice. Neil Balmforth has been very kind in providing photographs and movies of

roll waves. Duncan Wingham has been a great help sorting out some of the scal￾ing arguments in Chaps. 10 and 11. Eric Wolff was very kind in providing me with

ice core data, and spending time explaining to me how it worked. Torgeir Wiik and

Kjartan Rimstad pointed out errors in Sect. 2.5.7.

I solicited comments on individual chapters from many people, and these have

been of great use. Firstly, my thanks to Garry Clarke and Chris Clark, who provided

images (of Trapridge Glacier and ribbed moraine in Northern Ireland) for the front

cover; sadly they could not be used because it took me so long to finish the book that

in the meantime Springer changed the series design! Bruce Malamud spent a year

in Oxford, and was no end of help in the minutiae of computer technology. I have

received useful critical comments from Tom Witelski, Stephen O’Brien, Eric Wolff,

Richard Alley, Henry Winstanley, Slava Solomatov, Alison Rust, Ian Hewitt, Garry

Clarke, Janet Elliott and Don Drew.

Thanks to Ros Rickaby for discussions on carbon; Andy Ellis and Giles Wiggs

for providing images of dunes; Mark McGuinness for Figs. 5.12 and 5.16; Mike

Vynnycky for discussion on diapirism, and for providing the computations and the

resultant figures in Figs. 8.3, 8.6, 8.10 and 8.11. Thanks also to Sophie Nowicki,

for discussions concerning the grounding line; Rich Katz, for his comments on the

material on ice streams; Ian Hewitt, for discussions about canals and eskers, together

with many other things; my fellow drumliners, Chris Clark, Paul Dunlop, Chris

Stokes and Matteo Spagnolo for much information and insight into the geographic

setting of drumlins; Peter Howell, for comments on viscous beams; Geoff Evatt, for

help in assembling Sect. 11.7.

For a book such as this, it would be remiss not to mention with gratitude the

annual GFD summer school at Woods Hole, where I have variously spent long peri￾ods of time, most recently in 2010, and where I have benefitted from the experience

and wisdom of that excellent community of scholars, in particular Joe Keller, Lou

Howard, George Veronis, and Ed Spiegel. Those who have spent time on the porch

or in the classroom at Walsh Cottage will know what a privilege it is to be there, in

the presence of one of the brightest and wittiest seminar audiences on the planet.

The University of Limerick has supported me through my appointment there as

an adjunct Professor and subsequently, through an award by Science Foundation

Ireland, as Stokes Professor. The funds they have generously provided have enabled

me to maintain a research presence at conferences and workshops, as well as pur￾chasing two of the laptops on which this book was written. They have provided a

pleasant and stimulating working environment, not to mention easy access to the

best countryside in the world.

This book is dedicated to Jim Murray and his wife Sheila. I first met Jim on

a cold, dark December evening in 1970, when I ascended staircase 10 in Corpus

Christi College, Oxford, to be interviewed for a place as an undergraduate. We

peered at each other in the ancient, wood-panelled room by candlelight (these were

the days of miners’ strikes and power cuts). Ever since then, Jim has been the torch￾bearer for my path in applied mathematics, yielding to no man in his quest for the

practical and useful.

x Preface

My view of science, and the act of doing science, is that at best it is like driving

a car on an icy road. You know the car works, the road is flat, but actually, you do

not really know what you are doing. You try out a few things and they more or less

work. You might hit a slippery bit, but if you are lucky you get there somehow. And

if you are not lucky, you end up in the ditch. What you have to avoid is the idea that,

if you end up in the ditch, it is the right place to be. Do not get stuck in the ditch.

Get out of the car and back on the road.

It was Kolumban Hutter who said: you do not finish a book, you abandon it. He

was so right. It is like bringing up a child. You love it, change its nappies, feed it,

nurture it, but by the time it is an adult, it is time to go. Be gone!

Limerick, Ireland A.C. Fowler

Contents

1 Mathematical Modelling ......................... 1

1.1 Conservation Laws and Constitutive Laws ............. 2

1.2 Non-dimensionalisation . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.2 Approximations . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Qualitative Methods for Differential Equations . . ......... 6

1.3.1 Oscillations ......................... 7

1.3.2 Relaxation Oscillations ................... 8

1.3.3 Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.4 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4 Qualitative Methods for Partial Differential Equations . . . . . . . 18

1.4.1 Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.4.2 Burgers’ Equation . . . . . . . . . . . . . . . . . . . . . . 24

1.4.3 The Fisher Equation . . . . . . . . . . . . . . . . . . . . . 26

1.4.4 Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.4.5 Non-linear Diffusion: Similarity Solutions . . . . . . . . . 29

1.4.6 The Viscous Droplet . . . . . . . . . . . . . . . . . . . . . 31

1.4.7 Advance and Retreat: Waiting Times . . . . . . . . . . . . 33

1.4.8 Blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.4.9 Reaction–Diffusion Equations . . . . . . . . . . . . . . . . 41

1.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . 52

1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2 Climate Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.1 Radiation Budget . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.2 Radiative Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . 66

2.2.1 Local Thermodynamic Equilibrium . . . . . . . . . . . . . 67

2.2.2 Equation of Radiative Heat Transfer . . . . . . . . . . . . 68

2.2.3 Radiation Budget of the Earth . . . . . . . . . . . . . . . . 68

2.2.4 The Schuster–Schwarzschild Approximation . . . . . . . . 72

2.2.5 Radiative Heat Flux . . . . . . . . . . . . . . . . . . . . . 73

xi

xii Contents

2.2.6 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 74

2.2.7 Troposphere and Stratosphere . . . . . . . . . . . . . . . . 75

2.2.8 The Ozone Layer . . . . . . . . . . . . . . . . . . . . . . 77

2.3 Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

2.3.1 The Wet Adiabat . . . . . . . . . . . . . . . . . . . . . . . 81

2.4 Energy Balance Models . . . . . . . . . . . . . . . . . . . . . . . 83

2.4.1 Zonally Averaged Energy-Balance Models . . . . . . . . . 84

2.4.2 Carbon Dioxide and Global Warming . . . . . . . . . . . . 86

2.4.3 The Runaway Greenhouse Effect . . . . . . . . . . . . . . 89

2.5 Ice Ages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

2.5.1 Ice-Albedo Feedback . . . . . . . . . . . . . . . . . . . . 93

2.5.2 The Milankovitch Theory . . . . . . . . . . . . . . . . . . 96

2.5.3 Nonlinear Oscillations . . . . . . . . . . . . . . . . . . . . 97

2.5.4 Heinrich Events . . . . . . . . . . . . . . . . . . . . . . . 98

2.5.5 Dansgaard–Oeschger Events . . . . . . . . . . . . . . . . 100

2.5.6 The 8,200 Year Cooling Event . . . . . . . . . . . . . . . . 102

2.5.7 North Atlantic Salt Oscillator . . . . . . . . . . . . . . . . 104

2.6 Snowball Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

2.6.1 The Carbon Cycle . . . . . . . . . . . . . . . . . . . . . . 109

2.6.2 The Rôle of the Oceans . . . . . . . . . . . . . . . . . . . 114

2.6.3 Ocean Acidity . . . . . . . . . . . . . . . . . . . . . . . . 116

2.7 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . 119

2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

3 Oceans and Atmospheres . . . . . . . . . . . . . . . . . . . . . . . . 139

3.1 Atmospheric and Oceanic Circulation . . . . . . . . . . . . . . . . 139

3.2 The Geostrophic Circulation . . . . . . . . . . . . . . . . . . . . . 141

3.2.1 Eddy Viscosity . . . . . . . . . . . . . . . . . . . . . . . . 141

3.2.2 Energy Transport . . . . . . . . . . . . . . . . . . . . . . 142

3.2.3 Global Energy Balance . . . . . . . . . . . . . . . . . . . 147

3.2.4 Choosing Coordinates . . . . . . . . . . . . . . . . . . . . 148

3.2.5 Non-dimensionalisation . . . . . . . . . . . . . . . . . . . 150

3.2.6 Day and Night, Land and Ocean . . . . . . . . . . . . . . . 153

3.2.7 Parameter Estimates . . . . . . . . . . . . . . . . . . . . . 154

3.2.8 Basic Reference State . . . . . . . . . . . . . . . . . . . . 155

3.2.9 A Reduced Model . . . . . . . . . . . . . . . . . . . . . . 156

3.2.10 Geostrophic Balance . . . . . . . . . . . . . . . . . . . . . 158

3.3 The Planetary Boundary Layer . . . . . . . . . . . . . . . . . . . 159

3.4 Poincaré and Kelvin Waves . . . . . . . . . . . . . . . . . . . . . 160

3.5 The Quasi-geostrophic Approximation . . . . . . . . . . . . . . . 164

3.5.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . 168

3.5.2 The Day After Tomorrow . . . . . . . . . . . . . . . . . . 171

3.6 Rossby Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

3.6.1 Baroclinic Instability . . . . . . . . . . . . . . . . . . . . 176

3.6.2 The Eady Model . . . . . . . . . . . . . . . . . . . . . . . 176

Contents xiii

3.7 Frontogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

3.7.1 Depressions and Hurricanes . . . . . . . . . . . . . . . . . 180

3.8 The Mixed Layer and the Wind-Driven Oceanic Circulation . . . . 182

3.9 Western Boundary Currents: The Gulf Stream . . . . . . . . . . . 188

3.9.1 Effects of Basal Drag . . . . . . . . . . . . . . . . . . . . 189

3.9.2 Effects of Lateral Drag . . . . . . . . . . . . . . . . . . . 191

3.10 Global Thermohaline Circulation . . . . . . . . . . . . . . . . . . 192

3.11 Tides and Tsunamis . . . . . . . . . . . . . . . . . . . . . . . . . 193

3.11.1 The Tidal Equations . . . . . . . . . . . . . . . . . . . . . 194

3.11.2 Ocean Tides . . . . . . . . . . . . . . . . . . . . . . . . . 198

3.11.3 Seiches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

3.11.4 Amphidromic Points . . . . . . . . . . . . . . . . . . . . . 201

3.11.5 Tsunamis . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

3.12 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . 211

3.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

4 River Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

4.1 The Hydrological Cycle . . . . . . . . . . . . . . . . . . . . . . . 223

4.2 Chézy’s and Manning’s Laws . . . . . . . . . . . . . . . . . . . . 225

4.3 The Flood Hydrograph . . . . . . . . . . . . . . . . . . . . . . . . 226

4.4 St. Venant Equations . . . . . . . . . . . . . . . . . . . . . . . . . 230

4.4.1 Non-dimensionalisation . . . . . . . . . . . . . . . . . . . 231

4.4.2 Long Wave and Short Wave Approximation . . . . . . . . 231

4.4.3 The Monoclinal Flood Wave . . . . . . . . . . . . . . . . 232

4.4.4 Waves and Instability . . . . . . . . . . . . . . . . . . . . 235

4.5 Nonlinear Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

4.5.1 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 238

4.5.2 Roll Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 238

4.5.3 Tidal Bores . . . . . . . . . . . . . . . . . . . . . . . . . . 248

4.6 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . 256

4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

5 Dunes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

5.1 Patterns in Rivers . . . . . . . . . . . . . . . . . . . . . . . . . . 267

5.2 Dunes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

5.2.1 Sediment Transport . . . . . . . . . . . . . . . . . . . . . 273

5.2.2 Bedload . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

5.2.3 Suspended Sediment . . . . . . . . . . . . . . . . . . . . . 274

5.3 The Potential Model . . . . . . . . . . . . . . . . . . . . . . . . . 275

5.4 St. Venant Type Models . . . . . . . . . . . . . . . . . . . . . . . 279

5.5 A Suspended Sediment Model . . . . . . . . . . . . . . . . . . . . 282

5.6 Eddy Viscosity Model . . . . . . . . . . . . . . . . . . . . . . . . 285

5.6.1 Orr–Sommerfeld Equation . . . . . . . . . . . . . . . . . . 285

5.6.2 Orr–Sommerfeld–Exner Model . . . . . . . . . . . . . . . 289

5.6.3 Well-posedness . . . . . . . . . . . . . . . . . . . . . . . 290

xiv Contents

5.7 Mixing-Length Model for Aeolian Dunes . . . . . . . . . . . . . . 292

5.7.1 Mixing-Length Theory . . . . . . . . . . . . . . . . . . . 293

5.7.2 Turbulent Flow Model . . . . . . . . . . . . . . . . . . . . 295

5.7.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . 295

5.7.4 Eddy Viscosity . . . . . . . . . . . . . . . . . . . . . . . . 296

5.7.5 Surface Roughness Layer . . . . . . . . . . . . . . . . . . 296

5.7.6 Outer Solution . . . . . . . . . . . . . . . . . . . . . . . . 298

5.7.7 Determination of p10 . . . . . . . . . . . . . . . . . . . . 300

5.7.8 Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

5.7.9 Shear Layer . . . . . . . . . . . . . . . . . . . . . . . . . 302

5.7.10 Linear Stability . . . . . . . . . . . . . . . . . . . . . . . 305

5.8 Separation at the Wave Crest . . . . . . . . . . . . . . . . . . . . 308

5.8.1 Formulation of Hilbert Problem . . . . . . . . . . . . . . . 311

5.8.2 Calculation of the Free Boundary . . . . . . . . . . . . . . 314

5.9 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . 317

5.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

6 Landscape Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

6.1 Weathering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

6.2 The Erosional Cycle . . . . . . . . . . . . . . . . . . . . . . . . . 332

6.3 River Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

6.4 Denudation Models . . . . . . . . . . . . . . . . . . . . . . . . . 334

6.4.1 Sediment Transport . . . . . . . . . . . . . . . . . . . . . 335

6.4.2 Non-dimensionalisation . . . . . . . . . . . . . . . . . . . 336

6.4.3 The Issue of Time Scale . . . . . . . . . . . . . . . . . . . 339

6.5 Channel-Forming Instability . . . . . . . . . . . . . . . . . . . . . 339

6.5.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . 340

6.5.2 Steady State Solution . . . . . . . . . . . . . . . . . . . . 341

6.5.3 Uplift and Denudation . . . . . . . . . . . . . . . . . . . . 343

6.5.4 Geomorphically Concave Slopes are Unstable . . . . . . . 344

6.5.5 WKB Approximation at High Wave Number . . . . . . . . 347

6.5.6 Turning Point Analysis . . . . . . . . . . . . . . . . . . . 348

6.5.7 Rivulet Theory: δ 1 . . . . . . . . . . . . . . . . . . . . 354

6.6 Channel Formation . . . . . . . . . . . . . . . . . . . . . . . . . . 358

6.6.1 Channel Solutions . . . . . . . . . . . . . . . . . . . . . . 361

6.6.2 Bank Migration, Stability and Blow-up . . . . . . . . . . . 361

6.7 Channels and Hillslope Evolution . . . . . . . . . . . . . . . . . . 363

6.7.1 Hillslope Evolution . . . . . . . . . . . . . . . . . . . . . 366

6.7.2 Detachment Limited Erosion . . . . . . . . . . . . . . . . 367

6.7.3 Headward Erosion . . . . . . . . . . . . . . . . . . . . . . 370

6.7.4 Side-Branching . . . . . . . . . . . . . . . . . . . . . . . 371

6.8 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . 371

6.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

7 Groundwater Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

7.1 Darcy’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388