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This page intentionally left blank **Quantitative Modeling of Earth Surface Processes** Geomorphology is undergoing a renaissance made possible by new techniques in numerical modeling, geochronology and remote sensing. Earth surface processes are complex and richly varied, but analytical and numerical modeling techniques are powerful tools for interpreting these systems and the landforms they create. This textbook describes some of the most effective and straightforward quantitative techniques for modeling earth surface processes. By emphasizing a core set of equations and solution techniques, the book presents state-of-the-art models currently employed in earth surface process research, as well as a set of simple but practical tools that can be used to tackle unsolved research problems. Detailed case studies demonstrate application of the methods to a wide variety of processes including hillslope, fluvial, eolian, glacial, tectonic and climatic systems. The computer programming codes used in the case studies are also presented in a set of appendices so that readers can readily utilize these methods in their own work. Additional references are also provided for readers who wish to refine their models or pursue more sophisticated techniques. Assuming some knowledge of calculus and basic programming experience, this quantitative textbook is designed for advanced geomorphology courses and as a reference book for professional researchers in Earth and planetary sciences looking for a quantitative approach to earth surface processes. Exercises at the end of each chapter begin with simple calculations and then progress to more sophisticated problems that require computer programming. All the necessary computer codes are available online at www.cambridge.org 9780521855976. Jon Pelletier was awarded a Ph.D. in geological sciences from Cornell University in 1997. Following two years at the California Institute of Technology as the O.K. Earl Prize Postdoctoral Scholar, he was made an associate professor of geosciences at the University of Arizona where he teaches geomorphology. Dr Pelletier’s research involves mathematical modeling of a wide range of surface processes on Earth and other planets, including the evolution of mountain belts, the transport and deposition of dust in arid environments, and fluvial and glacial processes on Mars. **Quantitative Modeling of Earth Surface Processes** Jon D. Pelletier University of Arizona CAMBRIDGE UNIVERSITY PRESS 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 www.cambridge.org 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 ISBN-13 978-0-521-85597-6 eBook (EBL) 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. **Contents** Preface ix Chapter 1 Introduction 1 1.1 A tour of the fluvial 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 flow-routing algorithms to estimate flood 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 fluvial-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 vi CONTENTS Chapter 6 Non-Newtonian flow equations 125 6.1 Introduction 125 6.2 Modeling non-Newtonian and perfectly plastic flows 125 6.3 Modeling flows 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 flux 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 flow routing 235 Ключевые слова: ei, s t, cordillera real, efficiencymask, lateral translation, function, vine, equation, alpine environment, predicts, numerical, distance, gradient, alluvial, solution, problem, change, ay, turbulent diffusion, uinta mountains, vegetation band, water, down downtopo, delay feedback, basal, hudson bay, load, min mintopo, w t, geological society, algorithm, elevation, data, bed, condition, river, ice-albedo feedback, delayed feedback, scarp, incompressible viscous, age, diffusion equation, numerical diffusion, lattice size x lattice size y, topography, dust, lattice size y, order, finger lakes, royal society, sediment, area, large, eq ha, coherence resonance, thickness, distribution, stream, iup, i-iup, cubic polynomial, lake, transport, h r, evolution, scale, upwind differencing, basin, min minheight, high-relief terrain, series, terrace, landslide, chemical specie, two-column format, h x, map, rate, yucca mtn, xc, gravitational settling, cx, parameter, ?ow, deposition, velocity, ice sheet, san joaquin, steepest descent, eq, mountain, soil, result, concentration, glacial, plate, pro?les, diffusion, hanaupah canyon, time, allegheny plateau, cosh, wind, ti tf, ax, method, equal, height, width, dem, code, small, perfectly plastic, numerical recipes, great lakes, h t, heightold, boundary, slope, load-accumulation feedback, -widthold iup, direction, fan, drumlin, climate, modi?ed, fortymile wash, uplift, feedback, hb, topologically distinct, modeling, oblique perspective, sheet, minii minjjdown, power, drainage, modi?ed pelletier, journal, bedrock, sand, ice, pixel, bolivian orocline, point, shear, desert pavement, surface, model, southeast corner, fully compacted, process, arid environment, grid, statistical inevitability, cy, topographic, ancient orogens, local directory, ry, step, observed, minii minjjup, lattice size x, threshold, depth, structural geology, increase, vigorous debate, central core, pro?le, tn, illustrates, semi-arid environment, ilattice size, excitable medium, temperature, mar chiquita, initial, pelletier, shear stress, glacier, conveyor belt, wild burro, great deal, normal stress, region, hillslope, load-advance feedback, h b, range, channel, tephra, hemispheric gcm, hortons laws, erosion, ?ux, accelerating, ha, stress, tridag, icealbedo feedback, relief, wa, cayuga lakes, form, relationship, regional classication, ka ka, case, accumulation, dh db, rx, optimal, geological survey, valley, topo