Physics and Modelling of Wind Erosion, 2nd Edition (Atmospheric and Oceanographic Sciences Library, 37)

Physics and Modelling of Wind Erosion

ATMOSPHERIC AND OCEANOGRAPHIC SCIENCES LIBRARY VOLUME 37

Editors Lawrence A. Mysak, Department of Atmospheric and Oceanographic Sciences, McGill University, Montreal, Canada Kevin Hamilton, International Pacific Research Center, University of Hawaii, Honolulu, HI, U.S.A. Editorial Advisory Board L. Bengtsson A. Berger J.R. Garratt G. Geernaert J. Hansen M. Hantel H. Kelder T.N. Krishnamurti P. Lemke P. Malanotte-Rizzoli D. Randall J.-L. Redelsperger A. Robock S.H. Schneider G.E. Swaters J.C. Wyngaard

Max-Planck-Institut für Meteorologie, Hamburg, Germany Université Catholique, Louvain, Belgium CSIRO, Aspendale, Victoria, Australia DMU-FOLU, Roskilde, Denmark MIT, Cambridge, MA, U.S.A. Universität Wien, Austria KNMI (Royal Netherlands Meteorological Institute), De Bilt, The Netherlands The Florida State University, Tallahassee, FL, U.S.A. Alfred-Wegener-Institute for Polar and Marine Research, Bremerhaven, Germany MIT, Cambridge, MA, U.S.A. Colorado State University, Fort Collins, CO, U.S.A. METEO-FRANCE, Centre National de Recherches Météorologiques, Toulouse, France Rutgers University, New Brunswick, NJ, U.S.A. Stanford University, CA, U.S.A. University of Alberta, Edmonton, Canada Pennsylvania State University, University Park, PA, U.S.A.

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Physics and Modelling of Wind Erosion by

Yaping Shao University of Cologne, Germany

ABC

Dr. Yaping Shao University of Cologne Germany [email protected]

ISBN 978-1-4020-8894-0

e-ISBN 978-1-4020-8895-7

Library of Congress Control Number: 2008932207 All Rights Reserved c 2008 Springer Science + Business Media B.V.
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Preface 1

Wind erosion occurs in many arid, semiarid and agricultural areas of the world. It is an environmental process influenced by geological and climatic variations as well as human activities. In general, wind erosion leads to land degradation in agricultural areas and has a negative impact on air quality. Dust emission generated by wind erosion is the largest source of aerosols which directly or indirectly influence the atmospheric radiation balance and hence global climatic variations. Strong wind-erosion events, such as severe dust storms, may threaten human lives and cause substantial economic damage. The physics of wind erosion is complex, as it involves atmospheric, soil and land-surface processes. The research on wind erosion is multidisciplinary, covering meteorology, fluid dynamics, soil physics, colloidal science, surface soil hydrology, ecology, etc. Several excellent books have already been written about the topic, for instance, by Bagnold (1941, The Physics of Blown Sand and Desert Dunes), Greeley and Iversen (1985, Wind as a Geological Process on Earth, Mars, Venus and Titan), Pye (1987, Aeolian Dust and Dust Deposits), Pye and Tsoar (1990, Aeolian Sand and Sand Dunes). However, considerable progress has been made in wind-erosion research in recent years and there is a need to systematically document this progress in a new book. There are three other reasons which motivated me to write this book. Firstly, in most existing books, there is a general lack of rigor in the description of wind-erosion dynamics; secondly, the emphasis of the existing books appears to be placed primarily on sand-particle motion, while topics related to the modelling of dust entrainment, transport and deposition have not been presented in great detail and thirdly, the results presented in the existing books appear to be mainly experimental and lacking in documentation of the computational modelling effort involved. My intention is to provide a summary of the existing knowledge of wind erosion and recent progress in that research field. The basic contents of the book include the physics of particle entrainment, transport and deposition and the environmental processes that control wind erosion. It is intended to treat the physics of wind erosion as rigorously as possible, from the viewpoint v

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Preface 1

of fluid dynamics and soil physics. A considerable proportion of the book is devoted to the computational modelling of wind erosion. I hope that this book can be used as a reference point for both wind-erosion researchers and postgraduate students. My basic consideration is that wind erosion can only be understood from a multidisciplinary viewpoint and the computational modelling of wind erosion should focus on the development of integrated simulation systems. Such a system should tightly couple dynamic models, such as atmospheric prediction models and wind-erosion schemes, with real data that characterises soil and surface conditions. In the introductory chapter of the book, this basic concept is reiterated, while in Chapter 9 examples of the advocated modelling approach are given. Chapter 2 provides a summary of wind-erosion climatology in the world and selected regions. Chapters 3 and 4 are devoted to the description of atmospheric modelling and land-surface modelling, as these are the prerequisite for the modelling of wind erosion. Chapter 5 is a description of the basic aspects of wind-erosion theory, while Chapters 6, 7 and 8 are dedicated to the entrainment, transport and deposition of sand and dust particles. In Chapter 9, the integrated wind-erosion modelling system and the data requirement are described. The concluding remarks are given in Chapter 12. Cologne, Germany

Yaping Shao November 1999

Preface 2

Since the publication of the first edition of this book in 1999, much progress has been made in the field of wind-erosion research, especially on dust. This is mainly due to the strong interests in understanding the impacts of mineral aerosol on climate change and the role of dust in bio-geochemistry. In this edition, I have updated the contents of the book to reflect the new developments and corrected the mistakes known to me in the first edition. I have also improved the text and the illustrations. Many colleagues have helped with the preparation of this edition. In particular, I wish to thank Drs Masao Mikami, Irina Sokolik, Karl-Heinz Wyrwoll, Qingcun Zeng, Gongbing Peng, Chaohua Dong, Zhaohui Lin, Masaru Chiba, Naoko Seino, Taichu Y. Tanaka, Masahide Ishizuka, Eunjoo Jung and Youngsin Chun for their support. I also wish to thank Ms. Dagmar Jansen for her careful proofreading of the manuscript and Ms. Martina Klose for helping with the manuscript preparation using LaTeX. Cologne, Germany

Yaping Shao March 2008

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Acknowledgements

About 10 years ago, Dr. M. R. Raupach introduced me to the research of wind erosion. I have ever since maintained a strong interest in this field. During these years, I came to know many colleagues, including Professor L. M. Leslie, Dr. J. F. Leys, Dr. G. H. McTainsh, Mr. P. A. Findlater, Professor W. G. Nickling, Dr. D. A. Gillette, Professor H. Nagashima, Dr. B. Marticorena, Dr. G. Bergametti and Dr. I. Tegen among many others, who helped me to develop a understanding of the topics presented in this book. I am grateful to them for the valuable discussions and arguments during the years and to many of them for providing me with their research results for inclusion in this book. In the wind-erosion research community, there prevails truly a collaborative spirit. The development of the integrated wind-erosion modelling system described in Chapter 9 has been a team effort, and I acknowledge explicitly the significant contributions to the project made by my colleagues and friends, especially, Dr. H. Lu, Dr. P. Irannejad, Dr. R. K. Munro, Dr. C. Werner and Mr. R. Morison. The assistance of Dr. P. Irannejad and Mr. H. X. Zhuang in preparing the graphs and the manuscript has been very helpful. The painstaking final corrections by Dr. R. A. Byron-Scott have resulted in improvements to a text which has been written uncomfortably in my second language. Several chapters of the book were drafted during my stay at the Institute for Geophysics and Meteorology, University of Cologne, in 1999 when I was an Alexander von Humboldt Research Fellow. My stay in Germany has been a happy one, and I thank Professor Dr. M. Kerschgens and the Humboldt Foundation for making that possible. My thanks also go to Dr. M. de Jong from Kluwer Academic Publishers for her enthusiastic and patient approach toward publishing this book. Finally, I would like to take this opportunity to express my gratitude to Professor P. Schwerdtfeger, Dr. J. M. Hacker and Dr. T. H. Chen for their continuous encouragements throughout my scientific career.

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Preface 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

Preface 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1

Wind Erosion and Wind-Erosion Research . . . . . . . . . . . . . . . . . 1.1 Wind-Erosion Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Wind-Erosion Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Integrated Wind-Erosion Modelling . . . . . . . . . . . . . . . . . . . . . . . .

1 1 7 9

2

Wind-Erosion Climatology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Climatic Background for Wind Erosion . . . . . . . . . . . . . . . . . . . . . 2.2 Geographic Background for Wind Erosion . . . . . . . . . . . . . . . . . . 2.3 Atmospheric Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Monsoon Winds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Cyclones and Frontal Systems . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Squall Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Global Wind-Erosion Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Major Wind-Erosion Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Dust Weather Records and Satellite Remote Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 North Africa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 The Middle East . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Central Asia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.5 Southwest Asia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.6 Northeast Asia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.7 The United States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.8 Australia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 18 20 21 23 24 26 29 29 30 34 36 37 39 44 45

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Atmospheric Boundary Layer and Atmospheric Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Atmospheric Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Governing Equations for Atmospheric Boundary-Layer Flows . 3.3 Reynolds Averaging and Turbulent Flux . . . . . . . . . . . . . . . . . . . . 3.4 Equations for Mean Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Equations for Turbulent Fluxes and Variances . . . . . . . . . . . . . . . 3.5.1 Turbulent Dust Flux and Dust Concentration Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Turbulent Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Features of Different Atmospheric Boundary Layers . . . . 3.6 Surface Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Flux-Gradient Relationship . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Friction Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Logarithmic Wind Profile and Roughness Length . . . . . . 3.6.4 Stability Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Similarity Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Monin–Obukhov Similarity Theory . . . . . . . . . . . . . . . . . . 3.7.2 Mixed–Layer Similarity Theory . . . . . . . . . . . . . . . . . . . . . 3.8 Turbulent Flow Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Meso-scale, Regional and Global Atmospheric Models . . . . . . . .

49 49 52 56 59 60 60 61 63 67 67 68 71 72 74 75 78 79 85

4

Land-Surface Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.1 General Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.2 Surface Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.3 Soil Moisture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.4 Soil Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.5 Calculation of Surface Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.6 Land-Surface Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.7 Examples of Land-Surface Simulation . . . . . . . . . . . . . . . . . . . . . . 110 4.8 Treatment of Heterogeneous Surfaces . . . . . . . . . . . . . . . . . . . . . . 112

5

Basic Aspects of Wind Erosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.1 Soil-Particle Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.2 Forces on an Airborne Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.3 Particle Terminal Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.4 Modes of Particle Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.5 Threshold Friction Velocity for Sand Particles . . . . . . . . . . . . . . . 134 5.5.1 The Bagnold Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.5.2 The Greeley-Iversen Scheme . . . . . . . . . . . . . . . . . . . . . . . . 138 5.5.3 The Shao–Lu Scheme and the McKenna Neuman Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.6 Threshold Friction Velocity for Dust Particles . . . . . . . . . . . . . . . 142 5.6.1 Relative Importance of Forces . . . . . . . . . . . . . . . . . . . . . . . 142 5.6.2 Stochastic Nature of Threshold Friction Velocity . . . . . . 145

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The Dynamics and Modelling of Saltation . . . . . . . . . . . . . . . . . 149 6.1 Equations of Particle Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 6.2 Uniform Saltation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 6.3 Non-Uniform Saltation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.4 Streamwise Saltation Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 6.5 The Bagnold-Owen Saltation Equation . . . . . . . . . . . . . . . . . . . . . 157 6.5.1 The Bagnold Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.5.2 The Owen Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.6 Other Saltation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 6.7 The Owen Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 6.7.1 The Formulation of Owen . . . . . . . . . . . . . . . . . . . . . . . . . . 165 6.7.2 The Formulation of Raupach . . . . . . . . . . . . . . . . . . . . . . . . 166 6.7.3 Other Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.7.4 Profile of Saltation Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.8 Independent Saltation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 6.9 Supply-Limited Saltation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6.10 Evolution of Streamwise Sand Transport with Distance . . . . . . . 176 6.11 Splash Entrainment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 6.11.1 Wind-Tunnel Observations . . . . . . . . . . . . . . . . . . . . . . . . . 178 6.11.2 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 6.12 Numerical Modelling of Saltation . . . . . . . . . . . . . . . . . . . . . . . . . . 186 6.12.1 Simple Flow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 6.12.2 Large-Eddy Simulation Model . . . . . . . . . . . . . . . . . . . . . . . 187 6.12.3 Particle Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 6.12.4 Aerodynamic Entrainment . . . . . . . . . . . . . . . . . . . . . . . . . . 190 6.12.5 Splash Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 6.13 Understanding of Saltation from Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 6.13.1 Importance of Splash Entrainment . . . . . . . . . . . . . . . . . . . 194 6.13.2 Particle-Momentum Flux, Saltation Flux and Roughness Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 6.14 Saltation in Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 6.14.1 Intermittency of Saltation . . . . . . . . . . . . . . . . . . . . . . . . . . 201 6.14.2 Aeolian Streamers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 6.14.3 Dynamical Similarity of Saltation . . . . . . . . . . . . . . . . . . . 206

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Dust Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 7.1 Dust Flux and Friction Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 7.2 Mechanisms for Dust Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 7.3 Aerodynamic Dust Entrainment . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 7.4 Energy-Based Dust-Emission Scheme . . . . . . . . . . . . . . . . . . . . . . 222 7.5 Volume-Removal-Based Dust-Emission Scheme . . . . . . . . . . . . . . 226 7.5.1 Motion of Ploughing Particle and Volume Removal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 7.5.2 Vertical Dust Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

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7.6 Comparison of Dust Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 7.7 Spectral Dust-Emission Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 7.8 Discussions on Dust Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 8

Dust Transport and Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 8.1 Evidence of Dust Transport and Deposition . . . . . . . . . . . . . . . . . 247 8.2 Lagrangian Dust-Transport Model . . . . . . . . . . . . . . . . . . . . . . . . . 252 8.3 Eulerian Dust-Transport Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 8.4 Vertical Dust Transport by Diffusion . . . . . . . . . . . . . . . . . . . . . . . 261 8.5 Vertical Dust Transport by Convection . . . . . . . . . . . . . . . . . . . . . 273 8.5.1 Convective Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 8.5.2 Cumulus Parameterisation . . . . . . . . . . . . . . . . . . . . . . . . . . 275 8.6 Dry Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 8.6.1 Two-Layer Dry-Deposition Model: Smooth Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 8.6.2 Two-Layer Dry-Deposition Model: Vegetation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 8.6.3 Single-Layer Dry-Deposition Model . . . . . . . . . . . . . . . . . . 286 8.7 Wet Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 8.7.1 The Theory of Slinn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 8.7.2 Scavenging Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 8.7.3 Scavenging Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

9

Integrated Wind-Erosion Modelling . . . . . . . . . . . . . . . . . . . . . . . 303 9.1 System Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 9.2 Wind-Erosion Parameterisation Scheme . . . . . . . . . . . . . . . . . . . . 307 9.3 Threshold Friction Velocity for Natural Surfaces . . . . . . . . . . . . . 308 9.3.1 Drag Partition: Approach I . . . . . . . . . . . . . . . . . . . . . . . . . 310 9.3.2 Drag Partition: Approach II . . . . . . . . . . . . . . . . . . . . . . . . 316 9.3.3 Relationship of λ and z0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 9.3.4 Double Drag Partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 9.3.5 Soil Moisture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 9.3.6 Chemical Binding and Crust . . . . . . . . . . . . . . . . . . . . . . . . 327 9.4 Sand Drift and Dust Emission of Soils with Multiple Particle Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 9.5 Climatic Constraints on Dust Emission . . . . . . . . . . . . . . . . . . . . . 333 9.5.1 Erodibility Derived from Synoptic Data . . . . . . . . . . . . . . 333 9.5.2 Erodibility Derived from Satellite Data . . . . . . . . . . . . . . . 336 9.5.3 Wind-Erosion Hot Spots . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 9.6 Land-Surface Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 9.6.1 Soil Particle-Size Distribution . . . . . . . . . . . . . . . . . . . . . . . 337 9.6.2 Soil-Binding Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 9.6.3 Frontal-Area Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 9.6.4 Soil Moisture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 9.7 Manipulation of GIS Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

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9.8 Examples of Integrated Wind-Erosion Modelling . . . . . . . . . . . . . 350 9.8.1 Wind-Erosion Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . 350 9.8.2 Wind-Erosion Predictions on Global, Regional and Local Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 9.9 Data Assimilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 10 Sand Dunes, Dynamics and Modelling . . . . . . . . . . . . . . . . . . . . . 361 10.1 Classification of Sand Dunes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 10.2 Migration Speed of Transverse Dunes . . . . . . . . . . . . . . . . . . . . . . 370 10.3 Basic Features of Flow over a Sand Dune . . . . . . . . . . . . . . . . . . . 373 10.4 Sand Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 10.5 Computational Fluid Dynamic Simulation . . . . . . . . . . . . . . . . . . 381 10.5.1 Flow-Model Implementation: Non-hydrostatic Model . . . 382 10.5.2 Flow-Model Implementation: Large-Eddy Model . . . . . . . 384 10.5.3 Computation of Erosion and Deposition Rates . . . . . . . . 385 10.6 Discrete Lattice Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 11 Techniques for Wind-Erosion Measurements . . . . . . . . . . . . . . . 391 11.1 Wind-Tunnel Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 11.2 Sand Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 11.2.1 Passive Samplers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 11.2.2 Active Samplers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 11.2.3 Impact Sensors: Sensit, Saltiphone and Safire . . . . . . . . . 397 11.2.4 Sand Particle Counter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 11.3 Dust Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 11.3.1 High- and Low-Volume Air Samplers . . . . . . . . . . . . . . . . . 400 11.3.2 Optical Particle Counter . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 11.4 Deposition Collectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 11.5 Field Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 11.6 Particle-Size Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 11.6.1 Dry Sieving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 11.6.2 Settling Tube and Elutriator . . . . . . . . . . . . . . . . . . . . . . . . 408 11.6.3 Electro-Sensing Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 11.6.4 Laser Granulometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 11.7 Abrasion Emitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 12 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 12.1 Current Research Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 12.2 Dust Cycle in the Earth System . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

1 Wind Erosion and Wind-Erosion Research

1.1 Wind-Erosion Phenomenon Wind erosion is a process of wind-forced movement of soil particles. This process has the distinct phases of particle entrainment, transport and deposition (Fig. 1.1). It is a complex process because it is affected by many factors which include atmospheric conditions (e.g. wind, precipitation and temperature), soil properties (e.g. soil texture, composition and aggregation), land-surface characteristics (e.g. topography, moisture, aerodynamic roughness length, vegetation and non-erodible elements) and land-use practice (e.g. farming, grazing and mining). During a wind-erosion event, these factors interact with each other and, as erosion progresses, the properties of the eroded surface can be significantly modified. In the first instance, wind erosion is a geological and climatic phenomenon which takes place over long periods of time in deserts and arid regions. Most of the time, wind-erosion events proceed unnoticed but sometimes they are most spectacular. Figure 1.2 shows the satellite image of a massive dust storm over the Atlantic on 26 February 2000. During this event, dust from the Sahara Desert was lifted to up to 5,000 m above ground and blown off the African continent by an easterly wind. Dust storms of this magnitude have been observed elsewhere in the world, for instance in the Middle East, China and Australia (Figs. 1.3, 2.4). Between 15 and 19 April 1998, severe dust storms developed over the Gobi Desert in Mongolia and China. In the following days, the duststorm front moved across China and, by April 20, the elongated dusty belt covered a 2,000-km stretch of the east coast of China. The dust clouds were moving across the Pacific on 23 and 24 April and arrived in North America by 27 April (Husar et al. 2001). Wind erosion is the main mechanism for the formation and evolution of sand seas in the world and the long-range transport of sediments from continent to ocean. Recent studies suggest that the global dust emission amounts to 3,000 Mt yr−1 (estimates vary between 1,000 and 10,000 Mt yr−1 ), and a considerable proportion of this dust is deposited in the ocean (Duce et al. 1991). Y. Shao, Physics and Modelling of Wind Erosion, c Springer Science+Business Media B.V. 2008


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Impact on radiation (Optical thickness, backscatter)

Transport by wind & clouds

Wet deposition

Condensation nuclei

Convection

Dry deposition

Wind

Turbulent diffusion Roughness elements Trapped particles

Soil texture & surface crust

Dust emission Saltation

Soil moisture

Fig. 1.1. An illustration of the three phases of wind erosion: entrainment, transport and deposition. Atmospheric conditions, soil properties, land-surface characteristics and land-use practice control the erosion process (Modified from Lu, 1999)

Large quantities of minerals and organic matter are carried with the dust particles and redistributed around the world. The Loess Plateau in China has a soil depth ranging from 30 to 120 m and its formation is believed to be largely due to the deposition of wind-transported particles from the Gobi Desert over many millions of years. On geological time scales, wind erosion contributes greatly to the global mineral and nutrient circulation and to the evolution of surface topography. Particles suspended in the atmosphere, the aerosols, play an important role in the climate system, as they influence the atmospheric radiation balance directly, through scattering and absorbing various radiation components, and indirectly, through modifying the optical properties and lifetime of clouds. Aerosols generated by wind erosion (mineral aerosol) are the most important aerosols in the atmosphere. The global dust emission of 3,000 Mt yr−1 is comparable with the global sea-salt emission, which is estimated to be around 3,300 Mt yr−1 (Penner et al. 2001). Both estimates for the global dust and sea-salt emissions have large uncertainties, probably a factor of two. The radiative forcing of tropospheric aerosols on the atmosphere is currently an active research topic. For climate studies, the key research topics related to wind erosion are the global dust cycle, namely, the emission, transport and deposition of dust, and the atmospheric processes which involve mineral aerosols, such as radiation, cloud formation and precipitation.

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Fig. 1.2. Satellite image of a dust storm over the Atlantic. Dust from the Sahara was blown off the African continent by an easterly wind on 26 February 2000 (NOAA, acknowledgement)

Wind erosion also causes air-quality hazards in populated areas adjacent to major dust sources. In Beijing, for example, the measured near-surface dust concentration during severe dust storms has been reported to be as high as 5–10 mg m−3 . Near dust sources, dust concentration can exceed 20 mg m−3 (Yabuki et al. 2002). The northeast Asian dust storm that occurred between 18 and 24 March 2002 caused severe disruptions of social activities in the northern part of China and Korea (e.g. closure of airports and schools). During the 21–23 October 2002 Australian dust storm, the PM10 concentrations measured in some coastal cities of Australia (e.g. Brisbane) were close to 1 mg m−3 (Chan et al. 2005). Many contaminants that pose risks to human health and the environment are found or associated with dust, including metal, pesticides, dioxins and radionuclide. Thus, to quantify dust sources and to estimate airborne dust concentrations are also important to air-quality studies. In the recent history, human activities have created profound disturbances to the natural environment. Excessive clearance of native vegetation, over grazing and inadequate agricultural practices have resulted in increased frequency and intensity of wind erosion in some parts of the world. Tegen and Fung (1995) estimated that 20–50% of the global dust load is derived from human-disturbed soils. This estimate has been recently repudiated by Prospero et al. (2002), but there is evidence that over human-disturbed surfaces, the rate of wind erosion can be many times that over undisturbed

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Fig. 1.3. Image of dust storms in the Sahara captured by the space shuttle (NASA, acknowledgement)

natural surfaces. During the 1930s, for example, decreased precipitation coupled with intensive agricultural activities lead to severe wind erosion in the Great Plains of the United States, which became known as the dust bowl of the USA. In the Sahel, drought conditions combined with overpopulation also resulted in a considerable increase of wind-erosion events. In China, cultivation on the Loess Plateau may have contributed much to the severe dust storms in northeast Asia. In Australia, some of the recent severe dust storms have originated in the agricultural areas, where the native vegetation has been cleared over the past 200 years. Figure 1.4 shows a dust storm over the Murray River near Mildura (Australia), a farming area claimed from the forests of native Mallee trees. Wind erosion in agricultural areas leads to land degradation. During an erosion event, fine soil particles rich in nutrient and organic matters, are carried away by wind over large distances and this results in the loss of soil nutrients. According to Raupach et al. (1994), the February 1982 Melbourne dust storm generated a loss of 2 million tonnes of topsoil, including 3,400 t of nitrogen and 10 t of phosphorus. The May 1994 dust storms in Australia caused a soil loss between 10 to 20 million tonnes. The preferential removal of fine particles by wind erosion leaves coarser and less fertile material behind. Consequently, eroded soils become less productive and have a smaller water-holding capacity. For land-care purposes, the major tasks of wind-erosion research are

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Fig. 1.4. Dust clouds over the Murray River near Mildura (142◦ E, 32◦ S, Australia). The origin of the dust were the nearby farming areas (J. F. Leys, acknowledgement)

to quantify the risks of wind erosion on different temporal and spatial scales, to identify the responsible factors and to develop wind-erosion prevention measures. Wind erosion involves complex physics that is not yet fully understood. Its study requires the knowledge of a wide range of disciplines including atmospheric sciences (climatology, synoptics, remote sensing, cloud physics and atmospheric boundary layers), fluid dynamics, soil physics, surface hydrology, colloidal sciences, and ecology as well as agricultural sciences and land management. Almost all physical processes related to wind erosion are particle-size dependent. We often use the term ‘sand’ to describe particles in the size range between 60 and 2,000 µm and the term ‘dust’ to describe particles smaller than 60 µm. More precise definitions are given in Chapter 5. Wind erosion is the consequence of two types of forces at work: the aerodynamic forces that tend to remove particles from the surface, and the forces, such as gravity and inter-particle cohesion that resist the removal.

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The former can be quantified by the friction velocity, u∗ , a measure of wind shear at the surface, and the latter by the threshold friction velocity, u∗t , which defines the minimum friction velocity required for wind erosion to occur. While u∗ is related to atmospheric flow conditions and surface aerodynamic properties, u∗t is related to a range of surface properties, such as soil texture, soil moisture and vegetation cover. For dry and bare sandy soil surfaces, u∗t is small, and therefore it is not surprising that wind erosion occurs mainly under such conditions. The balance between u∗ and u∗t is governed by, and is sensitive to, a number of environmental factors, namely, (1) weather (wind, temperature, rainfall, etc.); (2) soil type (soil texture, hydraulic properties, etc.); (3) soil state (wetness, compactness, aggregation, etc.); (4) surface microscopic conditions (aerodynamic roughness length, vegetation coverage, etc.) and (5) surface macroscopic conditions (landforms, windbreaks, etc.). As a consequence, wind erosion is strongly variable in space and intermittent in time. The sporadic nature of wind-erosion events makes the modelling and prediction of wind erosion, even in a qualitative sense, a formidable task. For soil particles to become airborne, lift forces associated with wind shear near the surface or caused by particle impacts must overcome the gravitational and cohesive forces acting upon them. We call the entrainment of particles aerodynamic entrainment if it is dominated by aerodynamic forces, and refer to it as splash or bombardment entrainment if it is mainly caused by the impact of other moving particles. In either situation, the forces involved in the entrainment process vary strongly from case to case, depending on a range of factors, but in particular on particle size. Consequently, the dominant mechanism for particle entrainment also depends on particle size. For sand particles, the entrainment is essentially aerodynamic, while for dust particles the entrainment is primarily due to the impact of saltating sand grains, a phenomenon known as saltation bombardment (Gillette et al. 1982; Shao et al. 1993b). The motion of airborne particles in the atmosphere has two modes, known as saltation and suspension. Saltation refers to the small hopping motion of sand-sized particles in the direction of the wind, while suspension refers to the floating motion of dust-sized particles in the atmosphere. Through saltation, soil particles are transported in the direction of wind during an erosion event. Saltation is the mechanism for the evolution of sand seas on regional scales and the development of sand dunes and fence-line drifts on local scales. It is an interesting dynamic problem which involves the interactions between the fluid phase, the particulate phase and the surface. As particles saltate through the atmospheric surface layer with a strong wind shear, they absorb momentum from the airflow and generate a momentum transport in the vertical direction. During the impact on the surface, the saltating particles may splash more particles into the atmosphere. The deposition of saltating particles is of great significance to the evolution of the land surface. Particles in saltation may be deposited as wind speed reduces due to changes in atmospheric conditions or changes in surface roughness

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(e.g. shrubs) or topography (e.g. hills). Saltation is also associated with a large degree of randomness originating from the lift-off velocities, lift-off angles and turbulent fluctuations in the atmosphere. It is of particular interest in wind-erosion studies to estimate the intensity of horizontal sand drift related to saltation and to statistically describe the stochastic features of this mode of particle motion. In contrast to sand particles, dust particles, once airborne, can remain suspended in the atmosphere for some time and be dispersed beyond the atmospheric surface layer by turbulence and transported over large distances. This process leads to net soil losses from areas of wind erosion. The dispersion of dust particles is a difficult fluid-dynamic problem, known as heavy-particle diffusion. Because dust particles have a density more than 2,000 times larger than the air density, dispersion of dust particles differs from that of neutrallybuoyant fluid parcels. In addition, unless the atmospheric patterns and turbulence properties are adequately pre-specified, it is not possible to predict the transport of dust particles in the atmosphere with a reasonable accuracy. Dust particles suspended in the atmosphere are eventually delivered back to the surface through dry and wet deposition. Dry deposition is the transfer of airborne dust particles to the surface through turbulent and molecular diffusion and gravitational settling, while wet deposition is the transfer of airborne dust particles through precipitation. Both processes are of similar importance. In the first instance, dry deposition is a fluid-dynamic problem dealing with the diffusion of particles through a thin layer immediately adjacent to the deposition surface. It also involves many physical processes that are difficult to quantify, such as the static electrical charge. The difficulty in studying dry deposition also lies in the lack of detailed knowledge of the flow structure in the very thin layer immediately adjacent to the deposition surface. On the other hand, wet deposition involves the process of rain droplets capturing dust particles suspended in air. The study of wet deposition requires detailed understanding of raindrop size distribution, particle concentration and the capture mechanisms.

1.2 Wind-Erosion Research Wind-erosion research has been progressing along several different streams. Wind-Tunnel Experiments Wind-tunnel experiments have been carried out to investigate the physics of wind erosion, both in laboratory and in field. These studies have focused on the estimates of threshold friction velocity for different particle sizes, sand-drift intensity under various wind and surface conditions, dust-emission mechanisms, sand-dune evolution and the impacts of surface roughness elements and vegetation on wind erosion. These studies have contributed greatly to the core knowledge on wind erosion.

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Field Measurements Field measurements of wind erosion have been carried out on different scales in many parts of the world. Sand drift has been measured using saltation traps, and the impacts of land-surface parameters on wind erosion have been studied. Measurements of wind and dust concentration profiles have been made using anemometers and dust samplers mounted on towers. In addition, networks of air samplers and deposition traps have been setup in recent years for measuring dust concentration and deposition over large areas. Such networks are providing valuable data for studying dust movement in the entrainment phase (tower measurements), the transport phase (tower measurements and air samplers) and at the deposition phase (deposition traps). Wind-Erosion Assessment Assessment of wind erosion on continental scales has been performed by considering wind erosivity and wind erodibility. Wind erosivity describes the potential of wind to generate erosion, while erodibility describes the potential of the surface to be eroded. Chepil and Woodruff (1963) proposed to use a wind-erosion index and developed a model for calculating such indices with the data of wind speed, precipitation and evaporation. McTainsh et al. (1990) applied the model of Chepil and Woodruff to determining wind-erosion indices for Australia. With the development of Geographic Information Systems (GIS), more attention has been paid to soil and land-surface factors. Studies of wind-erosion climate based on dust-storm records have been carried out by, for instance, Middleton (1984), Littman (1991), Goudie and Middleton (2001), Qian et al. (2002) and Kurosaki and Mikami (2005) among many others. Satellite Remote Sensing Satellite remote sensing is advantageous in dust-storm monitoring. Sensors on board of satellites detect the radiances of various surfaces of the Earth through different spectral channels. These channels are set in correspondence to the atmospheric radiation windows and water vapour absorption bands. Various satellite-sensed signals are combined (1) to identify and monitor dust storms in real time (Carlson, 1979; Ackerman, 1989); (2) to derive land-surface and atmospheric parameters for dust modelling; (3) to retrieve dust quantities, such as dust load, optical thickness, particle size, etc. (Ackerman, 1997; Zhang et al. 2006); and (4) to derive long-term dust climatology. For example, Prospero et al. (2002) have used the NIMBUS 7 Total Ozone Mapping Spectrometer (TOMS) aerosol index over a 13-year period (1980–1992) to examine the distribution of dust sources on the globe. Empirical Wind-Erosion Modelling Empirical wind-erosion models have been under development for some time. The most widely used is the Wind-Erosion Equation (WEQ) (Woodruff and

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Siddoway, 1965), an empirical model in which the driving parameters are descriptors of soil type, vegetation, roughness, climate and field length. The original WEQ used annual averages of these descriptors to estimate annual average soil loss. For estimates over shorter periods, the WEQ was modified by Bondy et al. (1980) and Cole et al. (1983). More recent revisions have led to the Revised Wind-Erosion Equation (RWEQ) which includes input parameters such as planting date, tillage method and amount of residue from the previous crop; a weather generator is then used to predict future erosion (Comis and Gerrietts, 1994). The empirical nature of the WEQ limits its transferability from the central Great Plains of the USA, for which it was originally developed, to other areas of the world. Also, the complex interactions between the variables controlling wind erosion are not fully accounted for in the WEQ. For this reason, a more process-oriented model called the Wind-Erosion Prediction System (WEPS) has been developed. The WEPS includes submodels for weather generation, crop growth, decomposition, soil, hydrology, tillage and erosion (Hagen, 1991). Large-Scale Field Experiments Several large-scale field experiments have been recently carried out, dedicated to wind erosion, dust storms and dust cycle. The Aeolian Dust Experiment on Climate Impact (ADEC, Mikami et al. 2006) and the Asian Pacific Regional Aerosol Characterization Experiment (ACE-Asia, Huebert et al. 2003; Arimoto et al. 2006) are two examples. Networks have also been constructed to obtain dust observations over large areas. For example, the Aerosol Robotic Network (AERONET) is a federation of ground-based remote sensing aerosol networks. AERONET assesses aerosol optical properties and validates satellite retrievals of these properties. The data include globally distributed observations of spectral aerosol optical depths and precipitable water. The network has been operating since 1993 and has been carrying out routine measurements at around 150 stations distributed all over the world.

1.3 Integrated Wind-Erosion Modelling The approach advocated in this book is integrated wind-erosion modelling. An integrated wind-erosion modelling system enables the simulation and prediction of all aspects of wind erosion, from particle entrainment, transport to deposition. The aim of such a system is to provide quantitative assessment and prediction of wind erosion on scales from local to global. To this end, the integrated system needs to be constructed with six basic components: an atmospheric model, a wind-erosion model, a land-surface scheme, a dust-transport scheme, a data-assimilation scheme and a geographic-information data base. The atmospheric model provides the data required to drive the winderosion scheme, such as friction velocity, u∗ , wind field for dust advection,

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turbulence intensity for dust diffusion and deposition, and precipitation for wet deposition. In addition, the atmospheric model provides the data, such as radiation, required by the land-surface scheme for modelling the environmental variables, such as soil moisture and vegetation cover, which strongly influence wind erosion. Most atmospheric models are coupled with radiation schemes to deal with the impact of mineral aerosols on radiation transfer in the atmosphere. The wind-erosion model enables the quantification of the entrainment, transport and deposition of soil particles of all sizes. For instance, as far as particle entrainment is concerned, it enables the prediction of (1) the threshold friction velocity for wind erosion, (2) the rate of sand transport and (3) the rate of dust emission. Data assimilation is a technique which combines model and data to achieve an optimal simulation or prediction of a problem of concern. This technique has been very successfully applied to atmospheric and oceanic predictions. Because of the lack of dust measurement data, very little has been done so far in applying data assimilation to dust modelling. However, dust measurements are becoming increasingly available. For example, satellites can now provide continuous dust monitoring over large areas and the developments of inverse methods are producing quantitative estimates of dust load in an atmospheric column. Networks of lidar are being established, which provide dust-profile estimates at a number of locations. Further, stations equipped with dust samplers and radiometers are being set up. We expect that in the near future, data assimilation will be an important component of integrated wind-erosion modelling. Reliable land-surface data is important to wind-erosion modelling. They are required, because the properties of the land surface control the erosion threshold friction velocity, the capacity of the soil to release dust and the partitioning of wind-shear stress acting on non-erodible roughness elements and the erodible surface. Three categories of parameters can be distinguished. The first consists of parameters related to soil properties, e.g., soil particlesize distribution and soil-binding strength. The second consists of aerodynamic parameters related to surface roughness and drag partitioning. The third category consists of parameters which specify the soil thermal and hydraulic properties. For the purpose of modelling wind erosion on regional to continental scales, these soil and land-surface parameters can be stored as layers in a geographic information system (GIS). The first attempt of developing an integrated wind-erosion modelling system was probably made by Gillette and Hanson (1989), who used extensive atmospheric and land-surface data to determine the spatial and temporal variations of dust production in the United States. Gillette and Hanson did not use an atmospheric model and did not consider dust transport and deposition. Earlier versions of dust models, more from the atmospheric perspective, have been developed by Westphal et al. (1988) and Tegen and Fung (1994, 1995). In these early dust models, rather crude wind-erosion schemes

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and land-surface data were used. Marticorena and Bergametti (1995), Shao et al. (1996) and Marticorena et al. (1997) developed physics-based winderosion models and applied them to improve the simulations of wind erosion. Shao and Leslie (1997) and Lu (1999) developed an almost fully integrated wind-erosion modelling system which couples a physics-based wind-erosion scheme with an atmospheric model, a land-surface scheme and a geographicinformation database. They have implemented the system for the prediction of dust storms in Australia. Since the late 1990s, a number of dust storm models for global, regional and local dust problems have been developed. Examples of global dust models include those of Zender et al. (2003), Ginoux et al. (2004) and Tanaka and Chiba (2006). Examples of regional dust models include the studies of Nickovic et al. (2001), Liu et al. (2001), Shao et al. (2003) and Uno et al. (2005). Seino et al. (2005) simulated dust storms in the Tarim Basin using a meso-scale dust model. Integrated modelling is a new approach to studying wind erosion. It takes the advantage of the recent rapid expansion in computing power, developments in atmospheric and land-surface modelling, and the increasing availability of land-surface and remote-sensing data. This approach is a major step forward in the quantitative prediction of wind erosion, the comprehensive analysis of wind-erosion processes and the identification of the natural and human factors that affect wind erosion. However, integrated systems are complex. As will become evident in this book, nearly all wind erosion processes are sensitive to parameters which cannot be derived with great certainty. For example, threshold friction velocity is sensitive to soil moisture and vegetation cover. As a consequence, it is difficult to predict wind erosion with great accuracy. Nevertheless, recent studies have demonstrated that integrated wind-erosion modelling systems can produce results (sand-drift intensity, dust emission, dust concentration, etc.) which are comparable in magnitude with observed data, and the uncertainties embedded in the modelling systems are comparable with the uncertainties of observations.

2 Wind-Erosion Climatology

In this chapter, we describe the climatology of wind erosion. We are interested in the spatial patterns and temporal variations of wind erosion and the geographic, climatic and synoptic conditions which determine them. The understanding of wind-erosion climatology is largely based on the various measurements obtained through the techniques of remote sensing, comparison of aerosol samples with soil samples, monitoring air mass trajectories, analysis of dust weather records and numerical modelling. Among these techniques, the analysis of dust weather records and the analysis of remote sensing data are the most popular (Kurosaki and Mikami, 2005; Prospero et al. 2002). Although weather records are relatively scarce for desert areas and the weather-station network is not dense enough in non-desert areas to provide an accurate picture, the basic climatic features of wind erosion on the globe are now quite well known. Earth-observing satellites are now providing a huge amount of data for studying large-scale dust activities. By examining the aerosol optical thickness (Legrand et al. 1994; Moulin et al. 1998) and the absorbing aerosol index (Herman et al. 1997), which can be retrieved from satellite signals, much about large-scale dust activities has been learned.

2.1 Climatic Background for Wind Erosion The global pattern of wind erosion is closely related to the general circulation of the atmosphere. The distributions of solar radiation and albedo over the globe determine that there is a surplus of available energy (net radiation) in the region of low latitudes and a deficit in the region of high latitudes. This distribution of available energy leads to the general circulation of the atmosphere. As the Earth rotates around its axis, an apparent force acts continuously on a moving air stream if it is studied in a coordinate system which follows the Earth’s rotation. This apparent force, known as the Coriolis force, is −2Ω × v for a unit mass of fluid moving with speed v. Ω is the angular velocity vector Y. Shao, Physics and Modelling of Wind Erosion, c Springer Science+Business Media B.V. 2008


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of the Earth’s rotation. The Coriolis force is proportional to the magnitude of v but acts in the direction perpendicular to v (right hand side in the northern hemisphere, and left hand side in the southern hemisphere). In the free atmosphere, the pressure-gradient force and the Coriolis force dominate the behaviour of the flow. The wind at the balance between the two forces is known as the geostrophic wind. In a local coordinate system (with the origin positioned at a specified location on the earth surface), (x, y, z), with x pointing eastward, y northward and z upward, the geostrophic wind is vg = (ug , vg , 0), where ug and vg are given by 1 ∂p ρ ∂y 1 ∂p f vg = ρ ∂x

f ug = −

(2.1) (2.2)

where ρ is air density, p is pressure and f = 2Ω sin φ is the Coriolis parameter with φ being latitude and Ω =| Ω |. Wind in the free atmosphere is quasi-geostrophic, flowing parallel to the isobars. The situation in the atmospheric boundary layer is somewhat different. Here, in addition to the pressure-gradient and Coriolis forces, the airflow is also influenced by friction. As a consequence, wind in the atmospheric boundary layer flows across isobars from the high-pressure region to the low-pressure region. The basic features of the atmospheric general circulation are as illustrated in Fig. 2.1. In the meridional direction, the circulation is characterised by three circulation cells, the Hadley, the Ferrel and the polar cells. Near the equator, warm air rises and flows to the poles in the upper atmosphere. Under the influence of the Coriolis force, the poleward-moving air obtains the westerly momentum and forms westerly flows at around 30◦ N and 30◦ S. At these latitudes, air converges at high levels and subsides, leading to the developments of the sub-tropical highs. The higher pressure in the sub-tropical region leads to flows in the lower atmosphere toward the equator, which complete the Hadley cell. Again, due to the Coriolis force, the airflows moving toward the equator acquire an easterly component and are known as the trade wind. The sub-tropical high also generates poleward flows in the lower atmosphere. In the polar regions, the situation is the opposite, where strong surface cooling causes air to sink and to flow towards the equator in the lower atmosphere. The air moving towards the equator converges at about 50◦ N and 50◦ S with the air originating from the sub-tropical high, forming the polar-front zone. The three meridional circulation cells give rise to three surface wind regimes in each hemisphere: the trade winds of low latitudes, the midlatitude westerly and the polar easterlies. This general circulation pattern has major implications to the distributions of wind, precipitation, temperature and hence wind erosion. The trade winds from the two hemispheres form the inter-tropical convergence zone (ITCZ), which can be clearly identified over the ocean and less so over the land, where it is modified and suppressed by other atmospheric systems. The ITCZ may be broken at several locations by

2.1 Climatic Background for Wind Erosion

15

Fig. 2.1. An illustration of the atmospheric general circulation, showing the meridional circulation in the vertical cross-section of the atmosphere and the pattern of winds over the Earth surface. PF denotes the polar front and Jp and Js the locations of the high-level (about 200 hPa) jet streams (Redrawn from Defant and Defant, 1958)

monsoon flows, but can be traced around the globe. Air in the ITCZ is mostly unstable and convective cells develop frequently, producing heavy rainfall. In the northern-hemisphere summer, the ITCZ is situated around 10◦ N, broken by the monsoon over the Indian Ocean. In the northern-hemisphere winter, the ITCZ is situated around 5◦ N, stretching from the East Pacific, crossing Latin America, the Atlantic Ocean and North Africa. The ITCZ is broken at the east coast of Africa, but reappears at about 10◦ –15◦ S over the Indian Ocean and the West Pacific. The meridional migration of the ITCZ has a significant impact on wind-erosion activities. For example, wind erosion in the Sahara and Sahel region is enhanced as the ITCZ advances to the north, causing convective thunderstorms, but becomes significantly weaker in late summer. In the polar-frontal zone, baroclinic instability allows the developments of cyclones associated with frontal systems which cause systematic rainfall in large areas. The high-pressure system in the sub-tropical latitudes is the most permanent feature of the global atmospheric circulation. The Pacific and Azores anticyclones are large areas of subsiding air. The dominance of the sub-tropical highs near latitudes 20◦ –30◦ N and S results in deficiency in rainfall, because the subsiding air increases the stability of the atmosphere and prevents the formation of clouds. Some of the most arid regions of the world are found in

16

2 Wind-Erosion Climatology

such places, including the Sahara Desert, the Kalahari and Namib Deserts, the Middle East, the Thar Desert, the coastal deserts of northern Chile, Peru, southern California and large areas of western and central Australia. The idealised global circulation pattern as depicted in Fig. 2.1 is significantly modified by the irregular distribution of continents, oceans and mountain ranges which have different thermal and dynamic properties as well as different seasonal variations. The continents have a smaller heat capacity than the ocean and hence undergo stronger annual temperature variations. The thermal contrasts between the continents and the ocean lead to the developments of monsoons which profoundly affect the distributions of wind and precipitation. Monsoons constitute a major component of the global circulation and play a major role in the sub-tropical and tropical regions of Asia and Africa. West, east and middle Africa, north Indian Ocean, India, South China Sea, the southern parts and north-eastern parts of China, Japan, west Pacific Indonesia and the northern part of Australia are areas under the influence of monsoons. The Asian monsoon system is the most prominent in the world and consists of the north-east winter monsoon and the south-west summer monsoon. Monsoons greatly affect the patterns of precipitation in many parts of the world. In India, for instance, there is little precipitation during the winter monsoon between November and March, while rainfall is plentiful during the summer monsoon between June and October. In China, the movement of the precipitation zone from south to north between spring and autumn is closely related to the propagation of the southeast Asian summer monsoon. In April and May, the rain zone is normally situated in south China and, in early summer, in the Changjiang valley before moving further north in late summer. The summer monsoon that produces rainfall, cannot penetrate to the inland areas of China. This results in a diminished precipitation and increased wind erosion in these areas. In addition, the Tibetan Plateau and the high mountains in western China block the moisture from the Indian Ocean in the southwest. As a consequence, north-western and northern China, parts of north-eastern China and Mongolia, suffer deficiencies in rainfall. The Taklimakan Desert situated between the Tian and Kunlun mountain ranges receives no more than 100 mm of rainfall per year, and the Gobi Desert situated along the border of China and Mongolia receives no more than 200 mm of rainfall per year. The Taklimakan and the Gobi Deserts are the largest deserts in the temperate climate zone. In general, wind erosion is more active in dry years. Littman (1991) studied dust-storm frequencies in Asia and found that the occurrence of dust storms not only shows seasonal fluctuations, but also inter-annual variations between 3.6 and 5.5 years. In Australia, severe dust storms mostly occur during drought years, such as 1982 and 1983, 1991, 1994 and 2002. During the summer of 1982–1983, the eastern regions of Australia were transformed into a near desert. Inland of the Dividing Range, there were vast areas of failed cropland and bared grazing land from which topsoil began to erode. Dust

2.1 Climatic Background for Wind Erosion

17

storms started in the spring and were widespread during the summer, with particularly severe dust storms occurring in January and February of 1983 (Garratt, 1997). The underlying mechanism for extensive drought in Australia is the El Ni˜ no which represents the sea surface temperature variations in the equatorial Pacific. During the El Ni˜ no years, warm waters occur in the eastern Pacific along the tropical coast of South America. The El Ni˜ no phenomenon is coupled with the Southern Oscillation in the tropical atmosphere, which represents the variations of the Walker circulation. El Ni˜ no and Southern Oscillation (ENSO) are two closely related processes, the former taking place in the Pacific Ocean and the latter in the tropical atmosphere. The Walker circulation (Fig. 2.2) is a longitudinal circulation in lowlatitude atmosphere, arising from the variations of sea surface temperature across the ocean. Air rises at longitudes of relative heating and sinks at other longitudes of relative cooling. Under the normal state of the ocean and atmosphere across the Pacific Ocean, in the tropical latitudes off the west coast of South America, the south or south-east trade winds generate a surface current of cold water from the south, and this is deflected towards the west by the Coriolis force. The surface water drifting away is replaced by up-welling of cold deep-ocean water. Consequently, there develops a strong gradient in sea surface temperature from this area across the Pacific to the warm waters of the Indonesian archipelago. This temperature contrast drives an eastwest circulation cell in the atmosphere – the Walker circulation. Over the Indonesian region, the relatively warm and humid air rises to produce clouds and rainfall. Over the eastern Pacific, the relatively cool air sinks and thus, the rainfall is scanty by tropical standards. This is accompanied by easterly winds across the Pacific at the surface and westerlies in the upper atmosphere. On occasions, the easterlies of the tropical Pacific weaken and thereby reduce the wind stress, resulting in warm water flowing eastward across the Pacific to displace the cold water off the South American coast. With the appearance of warm surface waters in the central to eastern Pacific, the difference in temperature across the Pacific is reduced. The Walker circulation becomes weak or breaks down altogether, and a more complicated circulation cell structure develops in its place. As a result, cloud formation and rainfall is reduced in the Australian/Indonesian region but increased in the central to eastern Pacific. There is also evidence that these warm waters in the central Pacific can induce a series of atmospheric waves propagating into the Northern Hemisphere. For some El Ni˜ no events these atmospheric disturbances can be the cause of severe weather events over the entire globe. Areas consistently affected by the El Ni˜ no include Australia, Southeast Asia, large areas of China and the United States as well as certain areas in South America and Africa.

18

2 Wind-Erosion Climatology

Fig. 2.2. An illustration of the atmospheric circulations over the tropical Pacific. During El Ni˜ no years, warm surface-ocean waters appear in the tropical East Pacific, accompanied by a weak Walker circulation. El Ni˜ no years are associated with lower rainfall and greater wind-erosion risks in eastern Australia (top). During La Nina years, warm surface-ocean waters appear in the tropical West Pacific, accompanied by a strong Walker circulation. La Nina years are associated with higher rainfall and smaller wind-erosion risks in eastern Australia (bottom)

2.2 Geographic Background for Wind Erosion Wind erosion can only happen in areas where there is supply of sand and dust. However, the formations of sand and dust sources are determined by, apart from aeolian transport, weathering and fluvial (including glaciofluvial) processes. Prospero et al. (2002) pointed out that almost all major present-day dust sources are located in arid topographic depressions where fluvial action is evident.

2.2 Geographic Background for Wind Erosion

19

Fig. 2.3. Implications of fluvial actions to the formation of present-day dust sources

Fluvial actions are of paramount importance to the formation of the present-day dust sources, such as the Bod´el´e Depression (North Africa), the Tarim Basin (China) and the Lake Eyre Basin (Australia). These sources are located in arid regions centred over topographic lows or on lands adjacent to topographic highs. Fluvial processes are efficient in producing fine particles by separating them from the soil matrix and carrying them to deposition basins or alluvial plains (Fig. 2.3). Fluvial action is evident in the dust source regions by the presence of ephemeral rivers and streams, alluvial fans, playas and salt lakes. Alluvial fans form at the base of mountains where water erosion supplies the sediment. The upper part of the alluvial fan is characterized by coarse sediment, and the lower part by fine sediment. The development of alluvial fans over geologic time may extend to large areas to form alluvial basins, or in areas of gentle slopes alluvial plains. Playa, also known as alkali flat, is a flat-bottomed dry lakebed which consists of fine-grained sediments infused with alkali salts. A consistent association of dust sources with playas has been found (Gill, 1996). While playas may be not dust sources themselves, strong dust emission may occur in the alluvial fans that ring the basins in which the playas are found (Reheis et al. 1995). Most prudent-day dust source regions have deep and extensive alluvial deposits resulting from a relatively recent pluvial history. During the pluvial phases, these regions were flooded and thick layers of sediment were deposited and are now exposed to wind erosion. Many of the dust sources were flooded during the Pleistocene (roughly 2 million to 10,000 years ago, e.g., the Lake Chad Basin).

20

2 Wind-Erosion Climatology

Most of the present-day sand and dust sources are confined to regions with annual rainfall below 250 mm. While the global pattern of aridity is determined by the general circulation of the atmosphere, the interferences of landforms on the atmospheric flow field can significantly modify the climate in a region. The most arid places in the world are basins situated in the wake of high mountains. For example, the Great Basin (USA) in the wake of the Sierra Nevada receives less than 120 mm annual rainfall, and the Tarim Basin (China) surrounded by the Tian and Kunlun mountains and the Pamir Plateau receives less than 100 mm annual rainfall. As air approaches the mountain ranges, it is either diverted to flow around them or forced to rise. The adiabatic cooling associated with the upward motion promotes condensation and precipitation and thereby depletes the moisture in the airflow. Further down wind, air descends over the mountains. The adiabatic heating associated with the downward motion suppresses condensation and precipitation and thereby generates hot and dry airflows, known as foehn. In connection to wind erosion, this process has three important consequences. First, an arid shadow develops in the mountain wake. Second, precipitation increases on mountain slopes which leads to fluvial activity and sediment transport from the mountain slopes to the adjacent depressions. Third, complex flow patterns develop in regions around the mountain ranges, which affect dust transport, e.g., dust may be transported along preferred paths or allowed to accumulate in certain areas. Preferred routes of dust transport have been identified on satellite images of dust events. For example, dust is often seen to elongate from south-eastern Iran into the Indus Delta along the southern flanks of the Makran mountains (Prospero et al. 2002). Dust from the Gobi Desert is preferentially transported south-eastwards along the north-western boundary of the Tibetan Plateau and dust from the Thar Desert eastwards along the southern boundary of the Tibetan Plateau (Shao and Dong, 2006). Dust originating from the northeast Tarim Basin is often advected to the southwest of the basin causing high dust concentrations over there (Shao and Wang, 2003).

2.3 Atmospheric Systems Severe dust storms are mostly generated by vigorous weather systems, such as monsoon winds, cyclones and fronts, squall lines and thunderstorms. In these systems, the necessary meteorological conditions for the formation of a substantial dust storm are present, including (i) strong near-surface wind to lift sand and dust particles from the surface; (ii) strong turbulence or convection to disperse particles to high levels; and (iii) strong winds to transport particles over large distances horizontally. Despite these common features, the atmospheric systems which generate dust storms in different parts of the world can be quite different.

2.3 Atmospheric Systems

21

Fig. 2.4. The structure of the atmospheric circulation over North Africa in August. A: Location of the monsoon trough and the low-level easterly jet over the Sahara. To the north of the monsoon trough at about 20◦ N blows the north-easterly Harmattan (hot and dry) and to the south the south-westerly monsoon wind (warm and moist). B: Vertical structure of the atmosphere along the dashed line in A. Strong winds associated with the convections in the monsoon trough may generate dust storms (Redrawn from Barry and Chorley, 2003)

2.3.1 Monsoon Winds In North Africa, Southeast Asia and the Middle East, dust storms are often generated by monsoon winds or monsoon-related convective disturbances. The Harmattan (low-level hot and dry north-easterly) of North Africa and the Shamal (low-level hot and dry northerly) of the Middle East are well-known examples. The patterns of wind and precipitation in North Africa are very

22

2 Wind-Erosion Climatology

Fig. 2.5. Satellite image of the dust storm originating in the Gobi Desert of northwest China on 14 April 1998. The image, from 16 April 1998, shows the dust clouds behind the cold front and near the centre of the storm (SeaWiFS image by Norman Kuring, NASA GSFC, acknowledgment)

much determined by the regular and continuous migration of the monsoon trough between the annual extreme locations of about 2◦ N in January and 25◦ N in August. South of the trough is a shallow south-westerly airflow of c.a. 1,000 m deep, and overriding the south-westerly is an easterly of around 3,000 m deep, known as the midlevel easterly jet. North of the monsoon trough blows the Harmattan from the sub-tropic high pressure center (Fig. 2.4). Along the monsoon trough, shallow easterly waves develop, accompanied by surface cyclones. The cyclones generate strong near-surface winds which entrain large amounts of dust into the atmosphere (Westphal et al. 1988). Isolated storms and a broad zone of disturbances also occur just below the midlevel easterly jet. Another mechanism for the outbreaks of the Saharan dust are the dry convections which mix momentum from the midlevel easterly jet down to the surface where dust is mobilized (Prospero and Carlson, 1981). In the Harmattan affected regions, wide spread dust haze occurs in light wind conditions, known as the Harmattan haze. The Shamal blows almost daily in the Middle East between June and September. It may generate dust storms across Iraq, Kuwait and the Arabian

2.3 Atmospheric Systems

23

Peninsula. The synoptic situation responsible for the Shamal is a zone of convergence between the subtropical ridge extending into the northern Arabian Peninsula and Iraq from the Mediterranean Sea and the Monsoon Trough across southern Iran and the Southern Arabian Peninsula. The northerly flow associated with this synoptic situation is further accelerated due to the channelling effect exerted by the Iranian and the Arabian Plateaus. As the Shamal blows over the Tigris and Euphrates alluvial plains, a tremendous amount of dust may be lifted into the atmosphere to create dust plumes which stretch southward across Kuwait and the northern Persian Gulf. The Shamal dust storms move into an area like a dust wall of 1,000–2,500 m tall and 100–200 km thick. In the dust wall, the visibility drops to near zero. 2.3.2 Cyclones and Frontal Systems Frontal systems associated with cyclones and depressions also generate largescale dust storms. For example, in spring in Northeast Asia, up to 15 Mongolian cyclones (midlatitude cyclones initiating from baroclinic disturbances over the Mongolian Plateau) travel from northwest to southeast along the East Asian trough over the east coast of the Eurasian continent. Mongolian cyclones are accompanied by rigorous cold fronts and post-frontal north-westerly winds reaching up to 20 m s−1 . The cyclones move over the Gobi Desert and generate dust storms in Northeast Asia. Figure 2.5 shows a satellite image of the 16 April 1998 dust storm originating in the Gobi Desert. During this event, dust was transported to as far as the west coast of the United States. Intense wind-erosion events in Australia are mostly associated with Southern Ocean cold fronts propagating from southwest to northeast (Garratt, 1984; Garratt et al. 1989). The 8 February 1983 Melbourne dust storm, for instance, was generated by a dry cold front which represented a sharp demarcation between a hot northwest flow ahead of it and a west-south-west cooler flow behind it (Fig. 2.6). The structure of the front was typical of the summer cool changes in southern Australia, which can be described as propagating gravity currents (horizontal flows of a fluid that is denser than its surroundings). The cold front consists of an advancing wedge of cooler air marked in this case by a wall of dust, which undercuts the warm air mass ahead of the front. The depth of the gravity current at the head was about 300 m. The propagation speed of the gravity current exceeded 20 m s−1 . Figure 2.7 depicts the typical weather pattern that generates dust storms in Australia. A dust episode in Australia lasts two to three days. Initially, a low is positioned over the Southern Oceans and a high to the southwest of Australia. A cold front extends from the low pressure centre to the Great Australia Bight. Strong post-frontal southerly airflows are directed onto the southern coast of the continent, and a strong hot dry northwesterly wind blows over the south-eastern parts of Australia. Then, the low moves rapidly eastward, accompanied by the north-eastward propagation of the cold front. Strong south-westerly wind (12–15 ms−1 ) prevails behind the

24

2 Wind-Erosion Climatology

Height (m)

1000

500

B

20 m/s

A 10 m/s

Cold Air Hot Air

(a)

Dust 0

Wind (m/s)

14 m/s

C

C

B −30

−20

−10

A 0

10

20

20

(b)

10 0

Pres (hPa)

DD (deg)

0 360 240 120 0

(c) 0

1010

(d) 1000

Temp (C)

0 40 30 20

(e) −30

−20

−10 0 Distance (km)

10

20

Fig. 2.6. (a) Transect through a cool change along the axis normal to the front, showing the height of the advancing wedge of cold air and the wind vectors at three sites: ahead of the front (site A), immediately behind the front (site B) and about 30 km behind the front (site C); (b), (c), (d) and (e): respectively, surface wind speed, wind direction, surface pressure and air temperature along the transect

cold front, causing wide-spread dust emission. The dust particles are carried by the south-westerly to converge to the frontal area, resulting in high dust concentration near the front. They are then transported north-westwards and south-eastwards by the diverging flows along the front. A portion of the dust is eventually deposited to the Indian Ocean and the Pacific Ocean. 2.3.3 Squall Lines As thunderstorms develop in an unstable atmosphere, massive warm air may ascend and release latent heat during the formation of clouds. These storms

2.3 Atmospheric Systems

25

Fig. 2.7. Weather patterns associated with the 8–11 February 1996 dust event in Australia. Near surface wind (vectors, ms−1 ) and air temperature (contours, K) are shown

are sometimes accompanied by intense horizontal vortices with violent surface winds, known as squall lines. During summer (May–October) over some deserts in the southern Sahara, the south-western United States and other arid areas of the world, squall lines accompanied by vigorous dust storms are frequently observed. As illustrated in Fig. 2.8, in front of the storm, buoyant warm air rises beyond its condensation level and produces heavy rain. Dry air reaching the storm at high levels from the rear descends to the ground as it is cooled by the evaporation of the precipitation droplets. As the descending current hits the ground, it is deflected forward and moves out of the cloud in large lobes, forming a high wall of dust which rises up to 1,000–2,500 m. Observations show that the flow speed in these lobes can be as large as 50 m s−1 and a flow speed around 20 m s−1 is quite common (Lawson, 1971).

26

2 Wind-Erosion Climatology Tropopause

Cumulonimbus 15

225

228 10

ht A ir

ra ug Co l

d

A ir

D ow nd

ra ug

D ow nd

ht

Height (km)

Freezing Level

258

Co l

d

5

243

Temperature (K)

Motion of Storm

273

Warm Air Inflow Outflow Leading Edge Dust Wall Front

0 0

10

Rain 20

30

40

50

60

Horizontal Distance (km)

Fig. 2.8. Formation of a dust storm in a thunderstorm. The arrows indicate the direction of the air currents; the hachured region represents falling or suspended precipitation; the stippled region depicts a dust storm caused by the down draught of cold air (From Pye and Tsoar, 1990)

The development of thunderstorms and squall lines is closely related to the stability of the atmosphere. Over 20 severe dust storms related to squall lines occur each summer in the Khartoum region of the Sudan when the monsoon trough lies over the area. Strong surface convection is possible with light, relatively moist westerlies overlain aloft by moist easterlies, allowing the formation of cumulonimbus. Similar synoptic situations prevail in the arid south-western United States, where over 10 dust storms occur each year near Phoenix, Arizona. They are initiated by moist tropical air from the Pacific overlain by an upper flow of moist air from the Gulf of Mexico. Dust storms of this kind are usually short-lived local events, lasting only a few hours, and are sometimes followed by heavy rain which settles the dust and stops erosion.

2.4 Global Wind-Erosion Patterns Wind erosion occurs both on natural and disturbed soil surfaces. Natural aeolian surfaces are found mainly in desert areas, while disturbed soil surfaces are found mainly in the semi-arid areas that are under the influence of human activities, such as clearing of native vegetation and cultivation. The aridity of the semi-arid regions, for instance the Saharan/Sahelian boundaries, is strongly influenced by climate change on decadal time scales.

2.4 Global Wind-Erosion Patterns

27

Fig. 2.9. Global distribution of leaf-area index for March 2004 and the locations of the world’s major deserts, including: (1) Great Basin, (2) Sonoran, (3) Chihuahua, (4) Peruvian, (5) Atacama, (6) Monte, (7) Patagonia, (8) Sahara, (9) Somali-Chabli, (10) Namib, (11) Kalahari, (12) Karroo, (13) Arabian, (14) Rub al Khali, (15) Turkestan, (16) Iranian, (17) Thar, (18) Taklimakan, (19) Gobi, (20) Great Sandy, (21) Simpson, (22) Gibson, (23) Great Victoria and (24) Sturt

The locations of the major deserts are shown in Fig. 2.9. Most of the world’s dry lands are concentrated on the major continents in the subtropical region between 20◦ and 30◦ in both hemispheres, except those in East Asia in higher latitudes around 40◦ N. Overall, the dry lands occupy about one third of the Earth’s land surface, with about 4% of them being classified as extremely arid, 15% as arid and 14.6% as semi-arid (Cooke, 1973). The vegetation cover in the world is closely related to the availability of water for plant growth. As an example, Fig. 2.9 shows the distribution of leaf-area index over the globe for March 2004. Leaf-area index is defined as the total one-sided leaf area over a unit ground-surface area and is an indicator of the vegetation cover. Deserts, arid and semi-arid regions correspond well with regions of small leafarea index. Wind erosion on disturbed soil surfaces can be many times stronger than on natural soil surfaces. This is because freshly exposed soils usually contain more erodible particles than natural soils, in which fine material is likely to have been already depleted. Also, disturbed soil surfaces generally have a lower wind-erosion threshold friction velocity, since they are disrupted mechanically by agricultural activities. Tegen and Fung (1995) carried out an analysis of the potential areas of wind erosion over the world. They introduced six surface classes with 1◦ × 1◦ resolution, which are • •

Desert and sparsely vegetated soils (NS) Old cultivated eroded soils (OA)

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2 Wind-Erosion Climatology

Table 2.1. Percentage of potential wind-erosion area per continent (From Tegen and Fung, 1995) Surface Type

Global

Africa

Asia

Europe

North America

South America

Oceania

Land Area (1010 m2 ) NS (%) OA (%) OE (%) RB (%) RC (%) RD (%)

13,014 34 1.2 2.0 0.4 0.5 0.5

2,966 54 1.3 4.7 1.6 0.3 0.5

4,256 26 2.0 1.9 0.0 0.2 0.4

951 10 4.1 0.4 0.0 0.007 0.005

2,191 18 0.6 1.2 0.0 0.7 0.2

1,768 38 0.4 0.9 0.0 2.2 2.0

882 67 0.5 1.4 0.0 0.003 0.05

• • • •

Old uncultivated eroded soils (OE) Saharan/Sahelian boundary shift (RB) Recent cultivated areas (RC); and Recent deforested areas (RD)

Table 2.1 shows the land-area percentage per continent, which is potentially subject to wind erosion. This proportion varies from about 14.5% in Europe to about 69% in Oceania. Between 10% (Europe) and 67% (Oceania) are natural surfaces. Old disturbed surfaces cover 0.4% (South America) to 4.1% (Europe), while recently-disturbed surfaces (RC and RD) cover 0.01% (Europe) to 4.2% (South America). Through modelling dust transport and dust optical thickness in response to individual dust sources, Tegen and Fung (1995) also estimated annual dust emission from natural and disturbed soil surfaces. Their investigation showed that about 30% to 50% of the total dust emission can be attributed to disturbed soil surfaces and the rest to natural soil surfaces. Peterson and Junge (1971) estimated the global dust emission to be approximately 500 Mt yr−1 . This estimate appears to be too conservative. According to D’Almeida (1986), the dust emission from the Sahara and Sahel region alone is 627 and 723 Mt for the years 1981 and 1982, respectively. The estimates of the global dust emission range from 1,000 Mt yr−1 to 5,000 Mt yr−1 (Table 2.2). The modelling work of Tegen and Fung (1994) indicates that the global dust emission is around 3,000 Mt yr−1 , consisting of 390, 1,960 and 650 Mt yr−1 for particle size groups of 0.5 ∼ 1, 1 ∼ 25 and 25 ∼ 50 µm, respectively. The recent model estimates are converging to a value between 1,000 and 2,000 Mt yr−1 . An important improvement in the model estimate lies in that the global atmospheric dust load, retrieved from satellite data, has been used to constrain the behaviour of the model, so that the predicted dust-source regions are consistent with the satellite observations.

2.5 Major Wind-Erosion Regions

29

Table 2.2. Estimates of global dust emission in Mt yr−1 Source Peterson & Junge, 1971 D’Almeida, 1987 Duce et al. 1991 Tegen and Fung, 1994 Werner et al. 2002 Tegen et al. 2002 Luo et al. 2003 Zender et al. 2003 Ginoux et al. 2004 Miller et al. 2004 Tanaka & Chiba, 2006

Africa Asia America Australia Globale 500 1,900 >910 3,000 693 197 52 1,060 1,700 1,114 173 132 1,654 980 415 43 37 1,490 1,430 496 64 61 2,073 517 256 53 148 1,019 1,150 575 46 106 1,877

Comment Note 1 Note 2 Note 3 Model, 0.1–50 µm Model Model, 0.1–10 µm Model Model, < 10 µm Model, 0.1–6 µm Model, Model, 0.2–20 µm

Note 1: Estimates based on average concentration and residence time Note 2: Budget model and sun photometer aerosol-turbidity data for particles smaller than 5 µm Note 3: Deposition of mineral aerosol to ocean

2.5 Major Wind-Erosion Regions 2.5.1 Dust Weather Records and Satellite Remote Sensing Weather records and satellite observations are two major sources of data for studying wind-erosion climatology. Weather stations are relatively densely distributed around the world. At the weather stations, dust events are reported as weather phenomena at regular intervals (e.g. 3 hourly). For some stations, weather records extend continuously over a period of several tens of years or longer. According to the WMO (World Meteorological Organisation) protocol, dust events are classified into four categories according to visibility: • • • •

Dust-in-suspension: widespread dust in suspension, not raised at or near the station at the time of observation; visibility is usually not greater than 10 km Blowing dust: raised dust or sand at the time of observation, reducing visibility to 1–10 km Dust storm: strong winds lift large quantities of dust particles, reducing visibility to between 200 and 1,000 m Severe dust storm: very strong winds lift large quantities of dust particles, reducing visibility to less than 200 m

Although visibility is affected by the presence of dust particles, anthropogenic aerosols and air moisture, it can be assumed that during a dust event, dust particles play a determining role. Thus, dust concentration can be estimated from visibility using empirical relationships derived by fitting visibility to dustconcentration measurements. Dust concentration derived from visibility at

30

2 Wind-Erosion Climatology

individual weather stations can then be used to generate a dust-concentration field through spatial interpolation (Shao and Dong, 2006). Satellite remote sensing is advantageous in monitoring dust events because of its excellent spatial and temporal coverages. Satellite sensors can detect radiation signals from aerosol and various surfaces of the Earth through different spectral channels. The channels are commonly set in correspondence of the atmospheric radiation windows. The main ones are VL (visible light, 0.65– 0.85 µm), NIR (near infrared, 1.6 µm), MIR (middle infrared, 3.7 µm), WV (water vapour, 6.7 µm) and TIR (thermal infrared, 11 and 12 µm). Signals from the various channels can be combined to detect dust events from satellite imagery and to derive variables for quantifying dust load and dust particle size (Ackerman, 1997). The TOMS (Total Ozone Mapping Spectrometer, Nimbus 7 Satellite) data have been shown to be very useful for mapping the distribution of absorbing aerosols (largely comprised of dust and black carbon). The radiative effect of aerosols is measured in terms of an aerosol index defined as
     I340 I340 − AI = −100 log10 I380 ms I380 cl where (I340 /I380 )ms is the measured ratio of the backscattered radiance at 340 nm to that at 380 nm, and (I340 /I380 )cl is the calculated ratio by a radiation transfer model that assumes a pure gaseous atmosphere. The difference between the measured and calculated ratios is attributed to the presence of aerosols. Absorbing aerosols, clouds and non-absorbing aerosols yield positive, near-zero and negative AI values, respectively. The positive AI is the absorbing aerosol index which normally varies between 0 and 30. The TOMS data have been used to identify areas of high dust concentration (e.g. Goudie and Middleton, 2001). 2.5.2 North Africa The Sahara is the largest arid region on Earth. It occupies nearly all of North Africa, extending in length for approximately 5,600 km from the Atlantic coast to the shores of the Red Sea. In the north–south direction, it spans some 2,000 km or 20 degrees of latitude from about 35◦ N in the Atlas region of Algeria to about 15◦ N in the thorn shrub and grassland region, known as Sudan. The semi-arid region around 12◦ N to the south of the Sahara is the Sahel. The Sahel runs from the Atlantic Ocean to the Horn of Africa, changing from semi-arid grassland to thorn savanna. The Sahel countries today include Senegal, Cape Verde, Mauritania, Mali, Burkina Faso, Nigeria, Chad and Sudan. The Sahel receives 150–500 mm of rainfall a year, primarily in the monsoon season. The amount of rainfall is subject to significant variations due to climate change. While it is widely accepted that the Sahara is the largest dust source in the world, the estimates of the source strength vary over a wide range (Table 2.3).

2.5 Major Wind-Erosion Regions

31

Table 2.3. Estimates of annual dust emission from the Sahara Reference Jaenicke, 1979 Sch¨ utz, 1980 D’Almeida, 1986 Prospero, 1996a, b Swap et al. 1996 Marticorena et al. 1997 Callot et al. 2000

Emission (Mt yr−1 ) 260 300 630–710 170 130–460 586–665 760

The exact dust-source areas in North Africa are still being debated and the conclusions reached in the existing studies are not fully consistent (Herrmann et al. 1999). This is not surprising because (1) the geomorphology of the Sahara is complex, comprising rock, sandy and salt deserts, gravely and loamy soils, and a mixture of these different types; (2) the Sahara together with the Sahel is a vast area over which the atmospheric and surface conditions vary considerably in space and time; and (3) the various methods used for dust-source identification, including remote sensing, analysis of dust weather records, analysis of mineral tracer and numerical modelling, have large uncertainties. Four major dust-source areas have been identified based on the analysis of dust weather records, as Fig. 2.10 shows. Source 1 extends from the Spanish Sahara to North Mauritania; source 2 is located in Algeria and Niger in the triangle formed by the Hoggar, Adrar des Iforhas and Air mountains, northeast of Gao (18.0◦ N, 1.0◦ E, Mali); source 3 is situated north to northeast of Dirkou (19.0◦ N, 12.9◦ E, Niger), north of Bilma (18.7◦ N, 12.9◦ E, Niger) off the west side of the Tibesti Massif and source 4 is located in the northern part of the Sudan and the southern part of Egypt. There are other secondary wind-erosion areas, including the areas in the vicinity of In Salah (27.1◦ N, 2.3◦ E, Algeria), Djebock (17.5◦ N, 0.0◦ E, Mali), Mersa Matruh (31.4◦ N, 27.2◦ E, Egypt), or the alluvial land in the regions of Goundam (16.4◦ N, 3.7◦ E, Mali) and Matam (15.6◦ N, 13.3◦ W, Senegal). The area to the south of the Mediterranean Sea is also a noticeable dust source (Goudie, 1983). The meteorological conditions associated with dust storms in this region are depressions in the westerlies, such as the Mediterranean depressions and the Atlantic depressions in the winter months. The dust-source strengths estimated by D’Almeida (1986) and Marticorena et al. (1997) are summarized in Table 2.4. The largest dust sources are sources 1, 2 and 3, each producing a dust emission of over 200 Mt yr−1 . Dust emission from source 4 is relatively weak. Table 2.4 also shows that wind erosion in the Sahara is most active between February and August and the strongest dust emission occurs in March. Wind-erosion activities are relatively weak between September and January.

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2 Wind-Erosion Climatology

Fig. 2.10. Major sources and transport routes of Saharan dust. The dotted areas are the dust sources identified by D’Almeida (1986). Large solid circles mark the stations of the African turbidity network. The locations Bilma and some mountains and mountain chains Tassali N’Ajjer (TN), Tibesti (T), Hoggar (H), Air (AIR), Adrar des Iforhas (AI) and Ennedi (E) have strategic importance for the dust transport. Shaded are the areas of high TOMS absorbing aerosol index identified by Prospero et al. (2002)

Table 2.4. Dust emission from four different sources and the entire Sahara and Sahel region in Mt for each month, the data are averaged for two years. Total A are the results of D’Almeida (1986) for 1981 and 1982 and Total B are the results of Marticorena et al. (1997) for 1991 and 1992 Month Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec.

S1 18.6 26.6 29.5 24.8 21.1 22.7 13.8 12.7 17.7 8.4 8.9 13.1

S2 7.0 22.7 39.2 26.6 17.3 21.6 20.3 12.3 9.1 7.2 9.6 15.0

S3 5.1 23.3 40.4 29.2 28.6 42.9 19.7 24.1 10.5 5.7 2.8 2.8

S4 0.6 0.5 0.5 0.7 0.6 2.3 0.4 2.6 0.6 0.8 2.3 1.0

Total A 31.3 73.1 109.6 81.3 67.6 89.5 54.2 51.7 37.9 22.1 22.3 31.9

Total B 24.3 72.6 132.0 59.7 75.0 52.5 62.9 48.0 27.6 21.4 22.4 26.5

Total

217.9

207.9

235.1

12.9

673.8

625.7

2.5 Major Wind-Erosion Regions

33

The above described dust sources are in general consistent with those identified from the TOMS aerosol index data (Fig. 2.10). The TOMS data show that there are two primary and several secondary dust sources in North Africa. The first primary source is the Bod´el´e Depression between Tibesti and Lake Chad. This region is very dry (annual rainfall at Faya Largeau [18.0◦ N, 19.0◦ E] is less than 17 mm), but is fed with silty alluvium by streams draining from the Tibesti Massif. The second primary source is situated in the western Sahara covering portions of Mauritania, Mali and southern Algeria. This is again a very dry area with annual precipitation ranging from 5 to 100 mm. According to Prospero et al. (2002), the other dust sources in North Africa include: • Tunisia and northeast Algeria: high dust concentration frequently occurs in the region immediately to the south of the Atlas Saharien. The dust source is located between Chott Jerid (Tunisia) and Chott Melrhir (Algeria). • Libyan Desert and Western Desert: the source region in the Libyan Desert extends to the south-western end of the Qattara Depression in the east and is bounded on the west by the Al-Haruj al-Aswad hill range, the Jabal Bin Ghunaymah mountains and the Sarir Tibesti highlands. The source region in the Western Desert in Egypt is bounded by the Nile to the east and the low lands in central Egypt to the west. Dust emission in the eastern Sahara is active during much of the year, but is most intense in May and June. • Nubian Desert and Northern Sudan: the Nubian Desert together with northwest Sudan is a dust-source region. Four areas have been identified: (1) around (18.5◦ N, 25.5–26.0◦ E), to the east of the Mourdi Depression on the northern flanks of the Ennedi Plateau; (2) around (18.0◦ N, 28.0◦ E) at the terminus of Wadi Howar; (3) around (20.0◦ N, 32.0◦ E), between the Nile and the highlands franking the Red Sea; and (4) around (18.0◦ N, 35.0◦ E) in the Nubian Desert. The high dust season occurs in May–July. • Horn of Africa and Djibouti: A dusty area is found to be around (14.0◦ N, 42◦ E), slightly to the east of the Danakil Depression and Kobar Sink in the Rift Valley of northern Ethiopia and Eritrea. Dust activity in this region peaks in June and July, but is weak during the remainder of the year. The Harmattan season lasts up to eight months from October to May. In summer, at the end of it, the monsoon trough migrates to the north and the Sahel is under the influence of the warm and humid south-westerly flow. In March, the monsoon trough advances northward from about 5◦ N and reaches the northern most position of 20◦ –25◦ N in August. In September, the monsoon trough retreats gradually southward and reaches the southern most position of 5◦ S in January. Thus, in the northern part of the Sahara, north-easterly winds prevail almost for the whole year, although the very northern part of the Sahara is subject to the influences of north-westerly flows of frontal systems. Under the influence of the Harmattan, dust storms in the Sahara are quite frequent, for example, on the alluvial plain of Bilma (Niger) and Faya

34

2 Wind-Erosion Climatology

Largeau (Chad). Dust is transported from these regions to the Gulf of Guinea (McTainsh et al. 1997b; Gillies et al. 1996). During the northward migration of the monsoon trough at the end of the Harmattan season, unstable atmospheric conditions develop as warm and humid air flows over the heated land surface, generating intense thunderstorms. These thunderstorms have the potential to entrain large quantities of dust into the atmosphere (Fig. 2.4). As Fig. 2.10 shows, the Saharan dust is transported along four main trajectories: (1) the southward transport towards the Sahel and the Gulf of Guinea is the most important. The annual dust export along this trajectory is approximately 380 Mt, about 55% of the total dust emission from the Sahara. Dust originating from all sources can be carried southward by the Harmattan, and this occurs frequently in winter. A large proportion of the dust originating from the source areas remains on the continent because of the scavenging effect of the ITCZ precipitation zone. Only about 5% (16 Mt of the 380 Mt) of the southward-moving dust reaches 5◦ N; (2) the westward transport to the Atlantic Ocean is about half the strength of the southward transport. The annual dust export along this trajectory is around 190 Mt or 25% of the total dust emission. Dust generated by the westward moving cyclones and convections associated with the midlevel easterly jet is mainly transported along this trajectory. About 75% of the westward-moving dust (143 Mt of the 190 Mt) is deposited in the North Atlantic Ocean; (3) the northward transport to Europe is around 100 Mt yr−1 or 10% of the total dust emission. Eastern Algeria, Tunisia, Libya and Egypt are the main sources of the dust transported over the Mediterranean route in summer. The northward transport is mostly related to the movement of Soudano-Sahelian depressions; and (4) the eastward transport is relatively minor, which occurs predominantly in spring and is commonly associated with the easterward passage of low-pressure systems. Dust transported along this trajectory mostly originates from central Algeria and the Hoggar Massif and Tibesti areas. 2.5.3 The Middle East The Middle East, largely made up of the Arabian Plateau and the TigrisEuphrates Basin, is an area of active wind erosion. The Arabian Plateau slopes down from the southwest high terrains (1,500–3,000 m) bordering the Red Sea towards the northeast flat lands (50–200 m) adjacent to the Persian Gulf. One of the largest sand deserts in the world, the Rubal Khali (or Empty Quarter, 5,82,750 km2 ) occupies much of the southern interior of the Arabian Peninsula. The Rubal Khali is connected to the An Nafud sand sea in the north by the Ad Dahna, a sand corridor 1287 km long. Figure 2.11 shows the patterns of the dust-storm frequency and the blowing-dust frequency in the Middle East. Two dust areas have been identified. The first covers the Tigris-Euphrates alluvial plain in Iraq and Kuwait, the low-lying flat lands in the east of the peninsula along the Persian Gulf

2.5 Major Wind-Erosion Regions

35

Fig. 2.11. Annual frequency of dust storms (visibility less than 1,000 m, left) and annual frequency of dust events (visibility less than 11 km, right) in the Middle East (Middleton, 1986a). Dotted are the areas of high annual TOMS aerosol index

and the Ad Dahna and the Rubal Khali deserts. The alluvial plains have the highest frequency of dust storms in the Middle East (Safar, 1980), with a maximum of 38 dust storms per year in the vicinity of Abadan (30.2◦ N, 48.2◦ E). At Nasirayah, the numbers of dust-storm and blowing-dust days are as high as 33 and 208 per year, respectively (Middleton, 1986a). The second dust area is found off the Oman coast, extending from the coast between 54.0◦ E and 58.0◦ E to 200 km inland (Tindale and Pease, 1999; Prospero et al. 2002). In this region, the number of dust-storm days is rather low, but that of blowing-dust days is as high as 63. The seasonal variation of dust activity in the Middle East is complex and differs for different regions. Over much of the peninsula, dust is active all year long, but is relatively low in the winter months. Dust activity grows strong in March and April, peaks in June and July and weakens in September. The climate in the Middle East is mainly affected by three pressure systems, the Siberian anticyclone in winter over central Asia, the monsoon cyclone in summer over the India Subcontinent and the depressions travelling from northwest Africa across the Middle East in the non-summer seasons. Severe dust storms are summer-time phenomena associated with the Shamal. Much of the dust entrained by the Shamal is deposited in the Persian Gulf and the Arabian Sea. In some areas, e.g., Negev, Jordan, western and northern Iraq and the northern part of Saudi Arabia, the peak dust season occurs in spring and winter. These areas can be thought of as a corridor of the depressions moving eastward from the Mediterranean Sea. These depressions are characteristic of the areas in spring and winter (Katsnelson, 1970). In spring and winter, dust storms are generated by the depressions moving easterward from the Mediterranean. In addition, localized dust storms may be generated by the outflows associated with thunderstorm downdrafts, known as the Haboob.

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2 Wind-Erosion Climatology

2.5.4 Central Asia Wind erosion is active in the Caspian Sea and the Aral Sea region of central Asia, including Turkmenistan, Uzbekistan and Kasakhstan. The Aral Sea is fed by the rivers Amu Darya and Syr Darya. The Amu Darya flows from the northwest of the Pamir between the Karakum Desert and the Kyzylkum Desert and enters the Aral Sea in the south, while the Syr Darya flows from eastern Uzbekistan northwest along the northern boundary of the Kyzylkum Desert in Kazakhstan and enters the Aral Sea in the northeast. The Turan Lowlands to the south and southeast of the Aral Sea, consisting of deserts and dried playas, are prone to wind erosion. As shown in Fig. 2.12, the highest dust-storm frequency (60 yr−1 ) occurs over the Karakum Desert. Some studies suggest that the number of dusty days reaches on average 40–50 yr−1 in Turkmenistan and exceeding 80 yr−1 at some locations. The largest number of dust-event days stands at 108 yr−1 at Takhiyatas (42.8◦ N, 59.4◦ E, Middleton, 1986b). The more recent (1998–2003) synoptic records show that the highest number of dust-event days, exceeding 50 yr−1 , occurs in Kasakhstan to the northeast of the Aral Sea, rather than the sandy deserts between the Caspian and Aral Seas. Dust events are also observed in the Turan Lowlands and in the southeast part of the Turan Plain, nestle against the mountains of Tajikistan (the Gissar Range) and Afghanistan (the Hindu Kush). In the past few decades, human interferences in the Aral water system have resulted in intensified dust activities in the region. The increased irrigation

Fig. 2.12. Annual frequency of dust storms (visibility 0

w’ < 0

w’ > 0

c’ < 0 0

Eddy mixing some air down, and some up.

z

Net upward dust flux

1

c’ < 0

w’ < 0

w’ > 0

2

2

c

0

Net downward dust flux

c’ > 0

c

Fig. 3.2. An illustration of dust transport by turbulence. (a) net upward dust flux during erosion and (b) net downward dust flux during deposition

positively to w c . Thus the average turbulent dust flux is positive for this swirling motion. Similarly, a negative turbulent dust flux would occur if the gradient of dust concentration is positive, as also illustrated in Fig. 3.2b. Turbulence in the atmospheric boundary layer consists of many positive and negative values of instantaneous fluxes, but on average, turbulent fluxes are pointed in the opposite direction of the corresponding gradient (Fig. 3.2). There are distinct contrasts between turbulent fluxes in stable and convective conditions (Fig. 3.3). In convective conditions, intense updraughts (w ˆ > 0) are associated with positive peaks of θˆ and cˆ (where w ˆ = w /σw with σw being the standard deviation of w, θˆ = θ /σθ and σθ being the standard deviation of θ, etc.). In the immediate adjacent larger areas, downdraughts associated with negative values of θˆ and cˆ prevail. Thus, the convective elements represent an effective mechanism for turbulent transfer in the vertical direction as they produce intensive instantaneous fluxes. A situation in which a few strong updraughts are surrounded by a larger area of weaker downdraughts is known as one of penetrative convection. In stable conditions, turbulence is much weaker in general. In addition, the up- and downdraughts are not strongly correlated with the fluctuations of θˆ and cˆ. As a consequence, turbulent transfer is much less pronounced in stable conditions. On the small scales prevalent in the boundary layer, the effect of turbulent diffusion is usually similar to that of molecular diffusion (i.e. it results in a transfer of a quantity in the opposite direction of its gradient and diminishes that gradient). The mean vertical velocity w ¯ in the atmosphere is close to zero throughout the entire boundary layer. Thus, the vertical advective flux

3.4 Equations for Mean Flows

59

Fig. 3.3. Examples of space series of turbulent vertical velocity, w, ˆ potential temperature, θˆ and dust concentration cˆ and the instantaneous correlations w ˆ θˆ and wˆ ˆ c. (a) for stable conditions and (b) for convective conditions

¯ is much smaller than the corresponding turbulent flux. No such (e.g. ρcp w ¯ θ) statement can be made for the horizontal fluxes, where strong mean horizontal winds and strong turbulence can cause fluxes of comparable magnitudes.

3.4 Equations for Mean Flows By splitting the variables into mean and turbulent parts and applying averaging to Equations (3.2), (3.4), (3.6), (3.10), (3.11) and (3.12), we obtain a set of equations for mean boundary-layer flows (for detailed derivations, see Stull, 1988). ∂u ¯j =0 ∂xj ∂ui uj ∂u ¯i u 1 ∂p ¯j ∂u ¯i + + f ij3 u = −δi3 g − ¯j − ∂t ∂xj ρ ∂x ∂xj p¯ = Rd ρT¯ ¯ ¯ ∂Hj ¯j θ ∂cp θ ∂cp u + =− + s¯h ∂t ∂xj ∂xj ¯j q¯ ∂Ej ∂ q¯ ∂ u + =− + s¯q ∂t ∂xj ∂xj

(3.19) (3.20) (3.21) (3.22) (3.23)

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3 Atmospheric Boundary Layer and Atmospheric Modelling

¯j c¯ ∂Fj ∂¯ c ∂u ∂¯ c + =− − wt + sc ∂t ∂xj ∂z ∂xj

(3.24)

where Ej , Fj and Hj are respectively the jth component of turbulent fluxes of moisture, dust and heat, and sq , sc and sh are respectively the sources for moisture, dust and heat; Rj is the jth component of the mean radiation flux. In the above equations, we have neglected the effects of the molecular motion.

3.5 Equations for Turbulent Fluxes and Variances Turbulent fluxes, such as cp uj θ , uj c and ui uj , appear in Equations (3.19)– (3.24). To apply these equations for modelling atmospheric boundary-layer flows, the turbulent fluxes must be determined through either parameterisations or using prognostic equations. As most of the relevant prognostic equations can be found in Stull (1988), for instance, only the prognostic equation for the turbulent dust flux is given here. 3.5.1 Turbulent Dust Flux and Dust Concentration Variance The turbulent dust flux equation can be derived through the following steps: • Substitute c = c + c into Equation (3.12) • Subtract Equation (3.24) from the resulting equation to obtain an equation for c • Similarly, derive an equation for ui • Multiply ui with the c equation and c with the ui equation and combine the two equations to obtain an equation for ui c • Average the ui c equation Finally, the equations for the turbulent dust flux can be written as ∂u c ∂u c ∂ui c +u ¯j i − wt i = ∂t ∂xj ∂z ∂uj ui c ∂u ¯i ∂¯ c c θ g p ∂c −uj c − ui uj − + δi3 ¯ + − 2ui c ∂xj ∂xj ∂xj ρ ∂xi θ

(3.25)

Again, the vertical component of the turbulent dust flux is the most important one. Under the assumption of horizontal homogeneity and zero mean vertical wind, we have ∂w c c c θ g ∂w w c ∂w c 2 ∂¯ − wt = −σw + ¯ − − 2wc ∂t ∂z ∂z ∂z θ

(3.26)

Equation (3.26) implies that the temporal change of turbulent dust flux is mainly influenced by (1) the advection related to settling wt ∂w c /∂z; (2) the

3.5 Equations for Turbulent Fluxes and Variances

61

production related to mean dust concentration profile and turbulence intensity 2 ¯ (3) the turbulent transport −∂w w c /∂z; ∂¯ c/∂z and to buoyancy c θ g/θ; −σw and (4) the molecular dissipation −2wc . The usefulness of Equation (3.26) in wind-erosion studies has not been fully explored. To derive the prognostic equation for dust concentration variance, we multiply the c equation with 2c . By applying the Reynolds averaging to the resulting equation, we obtain ∂uj c
2 ∂σc2 ∂σc2 ∂¯ c ∂σc2   +u ¯j = −2c uj − wt − − 2c ∂t ∂xj ∂z ∂xj ∂xj

(3.27)

where c = kp (∂c /∂xj )2 is the molecular dissipation rate for dustconcentration variance, which is always positive. Equation (3.27) shows that the local change of dust concentration variance is mainly influenced by (1) the advection related to fluid motion and particle settling u ¯j ∂σc2 /∂xj −wt ∂σc2 /∂z; (2) the production related to mean dust concentration gradient and turbuc/∂xj ; (3) turbulent transport −∂uj c
2 /∂xj ; and (4) lent dust flux −2uj c ∂¯ molecular dissipation −2c . As for Equation (3.26), the application of Equation (3.27) in wind-erosion studies requires further exploration. 3.5.2 Turbulent Kinetic Energy By definition turbulent kinetic energy is given by e=

1
2 (u + v
2 + w
2 ) 2

The turbulent kinetic energy equation is one of the most important equations for atmospheric boundary-layer studies. Following a similar procedure for the derivation of the equation of dust-concentration variance, we obtain the equations for the variances of u, v and w. A summation of the three equations leads to the turbulent kinetic energy equation, which can be written as 1 ∂ui p ∂e g ∂ui ∂e ∂e + uj = δj3 uj θ − ui uj − uj − − ∂t ∂xj ∂xj ∂xj ρ ∂xj θ

(3.28)


2 ∂u is the viscous dissipation rate for turbulent kinetic energy. where  = ν ∂xji This term is always positive and causes a decrease in turbulent kinetic energy with time by converting it irreversibly to heat. If we use a coordinate system which is aligned with the mean wind, U , and assume horizontal homogeneity, we obtain a much simplified equation g ∂w e 1 ∂w p τ ∂U ∂e = w θ + − − − ∂t ρ ∂z ∂z ρ ∂z θ

(3.29)

where τ = −ρu w is the vertical component of the momentum flux, which is conventionally taken to be positive when pointing downward. The term gθ¯ w θ

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3 Atmospheric Boundary Layer and Atmospheric Modelling

is the buoyancy production of turbulent kinetic energy. Such a term is related to the sensible-heat flux, which can be either positive or negative. When there is an upward sensible-heat flux (w θ > 0), turbulence will be generated, whereas if there is a downward heat flux (w θ < 0), turbulence will be suppressed. Turbulent kinetic energy in the atmospheric boundary layer shows a clear diurnal variation driven by buoyancy. During the daytime, when the surface is heated up by solar radiation, heat flux is positive leading to strong turbulence. The intensity of turbulence decays rapidly during the afternoon, as the surface begins to cool down and the buoyancy term becomes negative. The (τ /ρ)(∂U/∂z) term represents the production of turbulent kinetic energy by wind shear, also known as mechanical production. This term is always positive, because τ /ρ and ∂U/∂z have opposite signs. In the surface layer, mechanical production is important because wind shear is usually strong there, and this is often balanced by the dissipation rate which is also large in this layer. It is based on this understanding that  surface friction velocity u∗ (defined as u∗ = τ /ρ) is normally used as a scaling velocity for turbulence in the surface layer. In the body of the convective boundary layer, buoyancy production is the dominating term. As a consequence, turbulence is normally strong during the day when the earth surface is heated by solar radiation (Fig. 3.4). Buoyancy-generated turbulence has a cohesive structure with large thermals penetrating the entire mixed layer (Fig. 3.5a), so that the depth of the mixed layer, zi , represents the typical size of the convective eddies. The scaling velocity of convective turbulence is hence defined as

1/3 g w  θ  0 zi w∗ = θ 1.5 0.15 0.3

0.05 1.0 Height (km)

0.15 0.3

0.6 0.9

0.5

0.3 0 12

18

0

6 12 Time (h)

18

0

6

Fig. 3.4. Modelled time and space variation of turbulent kinetic energy in m2 s−2 in an atmospheric boundary layer driven predominantly by buoyancy (Redrawn from Yamada and Mellor, 1975)

3.5 Equations for Turbulent Fluxes and Variances

63

(a)

Wind 1.5 km

Inversion

θ

Thermals Plumes

0

Wind

(b) Inversion

θ Waves 200 m

0

Fig. 3.5. (a): Structure of convective atmospheric boundary layer showing that small plumes rise from the surface and merge into thermals of size similar to the boundary-layer depth and that entrainment of dry and warm air occurs at the capping inversion. Turbulent mixing results in small vertical gradient of mean quantities in the bulk of the boundary layer. (b): Structure of stable atmospheric boundary layer showing turbulence structure, waves, low-level jet and inversion layer (Redrawn from Wyngaard, 1990)

The quantity w e is the vertical turbulent flux of turbulent kinetic energy. At any given height within the atmospheric boundary layer, this term can be either productive or destructive, depending on whether there is a flux convergence or divergence. When integrated over the depth of the boundary layer, the divergence term becomes zero, assuming there is negligible turbulence at the bottom and top of the boundary layer. 3.5.3 Features of Different Atmospheric Boundary Layers Convective Boundary Layers The convective boundary layer occurs when strong surface heating produces convection in the form of thermals and plumes, or when strong radiative

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3 Atmospheric Boundary Layer and Atmospheric Modelling

cooling at the cloud-top generates upside-down convection. The bottom of the convective boundary layer (near the surface) is characterized by a superadiabatic (temperature lapse rate larger than 10◦ C km−1 ) layer where potential temperature decreases with height, while its top is commonly identified with a capping inversion where potential temperature increases with height. The depth of the convective boundary layer is about 1 km, but varies with time during the day. Over the Sahara desert in mid-summer under strong surface heating, the convective boundary layer can be as deep as 5 km. As depicted in Fig. 3.5a, thermal instability within the convective boundary layer results in small plumes which merge into big thermals of rising warm air from the ground and sinking cool air from the cloud top. The thermals effectively transport heat and other quantities (including dust) from the surface through the boundary layer depth to the capping inversion base. Air from above the inversion base is entrained into the convective boundary layer in the regions of sinking motion. Some energetic thermals occasionally penetrate the capping inversion, but most thermals only distort the upper interface of the convective boundary layer, making it convoluted. The strong turbulent motions effectively mix heat, momentum and moisture in the vertical direction. As a consequence, substantial vertical gradients of these quantities cannot be sustained for very long within the bulk of the convective boundary layer. Typical profiles of wind speed, aerosol concentration and potential temperature are sketched in Fig. 3.6a. Near the surface, wind-speed profile is logarithmic and the potential temperature profile is super-adiabatic. In the body of the convective boundary layer, wind speed and potential temperature are almost constant with height. Across the inversion, wind speed and potential temperature increase rapidly with height. Most dust events occur during daytime under convective conditions. Airborne dust particles can be trapped in thermals and rise through the convective boundary layer. As thermals are usually unable to penetrate the inversion, the top of the convective boundary layer marks a sharp decrease in particle concentration. During such events, columns of dust extending from the surface to the top of the convective boundary layer are visually identifiable. Dust particles may remain suspended for hours because of turbulent mixing and only begin to settle in the evening as turbulence decays. Observations show that dust concentration profile in convective boundary layers is similar to those of passive scalars, such as (specific) humidity (Fig. 3.6a). Near the surface, dust concentration decreases sharply with height. In the bulk of the convective boundary layer, dust concentration tends to remain constant or decrease somewhat with height. The decrease is expected because particles settle under the influence of gravity. The rate of the decrease is more obvious for larger particles than for smaller particles. Across the inversion, dust concentration decreases shapely with height (Iwasaka et al. 2003; Yamashita et al. 2005; Yasui et al. 2005).

3.5 Equations for Turbulent Fluxes and Variances

65

(a)

Entrainment Zone

Inversion

z/zi

1 Wind Speed Particle Concentration Potential Temperature 0

z /h

(b)

Low−level Jet Wind Speed Particle Concentration Potential Temperature

Fig. 3.6. (a): Illustration of typical vertical profiles of wind speed, dust concentration and potential temperature in a convective boundary layer; (b) as (a), but for a stable boundary layer

Stable Layer Stable atmospheric boundary layers are characterised by an increase of potential temperature with height. They mostly occur at night, in response to surface cooling by long-wave radiation into space and are commonly associated with a surface inversion. In stable conditions, winds often become light at ground level, but can accelerate to form low-level jets aloft (Fig. 3.5b). The thermally-stable air suppresses turbulence, while the developing nocturnal jet may enhance wind shears which generate turbulence. As a result, turbulence is in general much weaker than in convective boundary layers but sometimes occurs in short and sporadic bursts.

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3 Atmospheric Boundary Layer and Atmospheric Modelling

The depth of the stable boundary layer, h, is not well defined. As turbulence intensity, suppressed by static stability, decreases gradually with height, there is a general agreement that h should be the height where turbulence intensity drops to about 5% of its value in the surface layer. The correspondence of h to the height of the surface inversion and/or that of the wind maximum depends very much on the evolutionary history of the stable layer. The depth of the stable layer is commonly a few hundred metres at most. At night over land, under clear skies and light winds, it may be even smaller, perhaps no more than 50–100 m. Typical wind, dust-concentration and temperature profiles in stable boundary layers are sketched in Fig. 3.6b. The rapid decrease of dust concentration with height in stable boundary layers has been observed (Yasui et al. 2005). The atmospheric boundary layer over land has a well-defined structural evolution, which depends upon the diurnal cycle of surface heating and cooling (Fig. 3.7). The three major components of this structure are the mixed-layer, the residual layer and the stable boundary layer. With the heating up of the surface after sunrise, the surface layer becomes unstable and convective turbulence begins to develop. The depth of the convective layer increases rapidly with time during the morning and the layer becomes fully established at noon. Vigorous convection is maintained in the next few hours by strong surface heating, but decays in late afternoon as the heating weakens. With the cooling down of the surface after sunset, the nocturnal stable boundary 2000 Free Atmosphere

Height (m)

1500 Entrainment Zone 1000

Capping Inversion

Redisual Layer Redisual Layer

500

Convective Mixing Layer Stable Layer Surface Layer

0 Sunrise

Noon

Surface Layer Sunset

Midnight

Sunrise

Local Time Fig. 3.7. Time evolution of atmospheric boundary layer over land under the influence of surface heating and cooling. Three major parts can be identified: a convective mixed layer during the day, a residual layer containing former mixed-layer air, and a nocturnal stable boundary layer

3.6 Surface Layer

67

layer develops over night. Above this stable layer is the residual layer of the convective boundary layer.

3.6 Surface Layer 3.6.1 Flux-Gradient Relationship In the atmospheric surface layer, the turbulent fluxes of momentum, mass and heat are controlled by the strong vertical gradients of the corresponding mean quantities, and hence the mechanism that leads to turbulent fluxes (Fig. 3.2) is analogous to that governing the molecular diffusion in laminar flows. Based on this analogy, we can express the turbulent fluxes of momentum, heat and dust in terms of the gradients in the vertical direction of the corresponding mean variables: τR = Km ρ∂U/∂z ¯ H = −Kh ρcp ∂ θ/∂z c/∂z F = −Kp ρ∂¯

(3.30) (3.31) (3.32)

where Km , Kh and Kp are exchange coefficients for momentum, heat and dust, in dimensions of [L2 T−1 , e.g. m2 s−1 ]; U , θ¯ and c¯ are respectively the mean wind speed, mean potential temperature and mean dust concentration. This formulation of the flux-gradient relationship is known as the K-theory. The K-theory is based on the small-eddy concept: It is assumed that turbulent transport is a result of local mixing by eddies of size much smaller than the characteristic scale in which the corresponding mean quantity varies vertically. Note that τR , in contrast to H and F , is conventionally defined positive when it is pointed downward. The exchange coefficients Km , Kh and Kp are respectively the counterparts of the kinematical molecular viscosity ν, the molecular diffusivity for heat kh and that for dust particles kp , in laminar flows. Therefore, Km is also known as eddy (or turbulent) viscosity and Kh and Kp are known as eddy diffusivity. The magnitudes of Km etc. are typically three orders of magnitude larger than their molecular counterparts. Also, they are not constants but functions of the turbulent properties of the atmosphere. A simple expression for eddy viscosity, Km , can be obtained based on dimensional arguments. As the dimensions of Km are [L2 T−1 ], Km must be a product of a velocity and a length. Because Km represents the capacity of the flow to transfer momentum for a given gradient of mean wind through turbulent mixing, it is intuitive to assume that Km must depend on the intensity and the size of turbulent eddies. This typical size of eddies, l, is also known as the mixing length. Thus, the essence of the mixing-length theory is that Km ∝ u∗ l with u∗ being a velocity scale. Similar arguments can be made for Kh . For neutral surface layers, Kh /Km ≈ 1 but is in general a function of the thermal stability of the surface layer (Section 3.7.1).

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3 Atmospheric Boundary Layer and Atmospheric Modelling

3.6.2 Friction Velocity The transfer of momentum from the atmosphere to the surface is the ultimate driving force for wind erosion (Chapter 5). The momentum flux (i.e. momentum transfer per unit area) is also known as the shear stress or the drag. The dimensions of drag are in [M L−1 T−2 , e.g. N m−2 ]. In the atmospheric surface layer, wind always increases with height and the momentum transfer is always downwards. While the momentum flux is downward, the drag is a force on the surface along the direction of the wind (Fig. 3.8a). Momentum transfer in the flow is realized through both turbulent and molecular motions. The effective (or total) shear stress, τ , is composed of the Reynolds shear stress, τR , and the viscous shear stress, τM , τ = τ R + τM

(3.33)

The relative importance of τR and τM depends on the distance from the surface. In the body of the boundary layer, i.e. above the viscous sub-layer, flows are predominantly turbulent and, in this region, the momentum flux occurs mainly through turbulence, and thus τ is almost identical to τR . Closer to the surface, especially within the viscous layer right next to the surface, the flow is dominated by viscosity, turbulence is weak and τR becomes insignificant. In this region, the momentum flux occurs mainly through molecular motion. The variations of τ , τR and τM with height in the atmospheric boundary layer (a)

(b) z

z

Wind Momentum Flux

τ

Surface Layer τM

τ

u

τR

Viscous Sublayer τ

Fig. 3.8. (a) An illustration of mean wind profile in the surface layer. A downward momentum flux corresponds to shear stress τ in the direction of the wind. (b) Profiles of effective shear stress, τ , Reynolds shear stress, τR , and viscous shear stress, τM . In the surface layer, τ = τR + τM is approximately constant

3.6 Surface Layer

69

are as illustrated in Fig. 3.8b. In the surface layer, τ remains approximately constant with height. The friction velocity, u∗ , is defined as  (3.34) u∗ = τ /ρ Clearly, u∗ is not the speed of the flow but simply another expression for the momentum flux at the surface. As u∗ is a convenient description of the force exerted on the surface by wind shear, it emerges as one of the most important quantities in wind-erosion studies. Equation (3.34) can be rewritten as  u∗ = u w  ∝ σ where σ is the standard deviation of velocity fluctuations. Thus, u∗ is also a descriptor of turbulence intensity in the surface layer and is thus an adequate scaling velocity for turbulent fluctuations there. Depending upon its characteristics, the surface can be considered to be dynamically smooth if the sizes of the surface roughness elements are too small to affect the flow, or otherwise to be dynamically rough. A criterion for determining whether a surface is rough or smooth is the roughness elements Reynolds number, (3.35) Rer = u∗ Dr /ν with Dr being a measure of typical roughness-element size. The surface can be considered to be rough if Rer exceeds about 300. Dr is around 10 mm for a u∗ of 0.4 m s−1 because ν = 1.5 × 10−5 m2 s−1 . If a sufficiently large roughness element protrudes into the flow, it produces a drag on the flow. This drag occurs because of the pressure difference between the windward side and the leeward side of the roughness element. The pressure on the windward side of the roughness element is larger than that on the leeward side. An integration of pressure over the surfaces of all roughness elements on a unit area of the ground surface gives the pressure drag τr = pds The transfer of momentum from the atmosphere to the surface is achieved eventually through the viscous effect and pressure drag on the surface roughness elements. For a smooth surface, the pressure drag is negligible and the flow characteristics in the viscous layer can be understood based on simple physical considerations. The flow speed U in this region is a function of height z, depending on only the viscous-shear stress τ (=τM ), the kinematic molecular viscosity ν, and air density ρ, so that U = U (z, τ, ν, ρ)

(3.36)

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3 Atmospheric Boundary Layer and Atmospheric Modelling

In the viscous layer, τ is nearly constant and can be well approximated by the Newtonian law dU (3.37) τ = νρ dz An integration of Equation (3.37), using the boundary condition U |z=0 = 0, gives τz (3.38) U= νρ or

U u∗ z = u∗ ν

(3.39)

The above equation shows that over a smooth surface, the wind speed varies linearly with height in the viscous layer. This linear distribution is valid up to about u∗ z/ν = 5, which is roughly the upper limit of the viscous layer (Fig. 3.9). Most natural surfaces prone to wind erosion are rough due to the presence of roughness elements such as vegetation, stubble and soil aggregates. The roughness elements protrude out of the viscous layer and cause a wake behind each of them. In this case, the stress is transmitted to the surface not only by the viscous effect, but also by the pressure drop over the surfaces of the roughness elements. The viscous layer is no longer well-defined, as it is convoluted and frequently broken. The flow characteristics are much more complex (Fig. 3.9a) and the functional form such as Equation (3.39) does not fit the data very well.

z

(a)

z

(b) u

u Logarithmic Profile

Logarithmic Profile

z0 Viscous Layer

z0

Roughness Layer

Linear Profile Smooth Surface

Rough Surface

Fig. 3.9. Profile of mean wind in the surface layer (a) over a smooth surface and (b) over a rough surface

3.6 Surface Layer

71

3.6.3 Logarithmic Wind Profile and Roughness Length In the surface layer above either a smooth or a rough surface, where the effective shear stress is approximately constant with height (Fig. 3.8b), the profile of mean wind obeys the logarithmic law. For neutral surface layers Km can be expressed as (3.40) Km = κu∗ z where κ is the von Karman constant. The reported values of κ in most atmospheric boundary-layer studies range between 0.35 to 0.4. Substituting the expressions for τ (i.e. τ = ρu2∗ ) and Km into Equation (3.30), we obtain u∗ ∂U = ∂z κz

z ≥ z0

(3.41)

Note that in Equation (3.41), z cannot be equal to zero but needs to be larger than zero. In other words, Equation (3.41) is valid for z ≥ z0 with z0 being some height above the ground surface. An integration of Equation (3.41) over z from z0 to z yields the logarithmic wind profile   z u∗ ln U (z) = (3.42) κ z0 We call z0 the aerodynamic roughness length of the surface. In winderosion studies, z0 is a widely used parameter and it is important to understand its interpretations. Formally, z0 is a constant of integration introduced in the derivation of the logarithmic wind profile. A z0 would not exist if Equation (3.40) were not used. Equation (3.42) implies that at z = z0 , the mean wind vanishes when extrapolated logarithmically downward. For a fixed mean wind speed at z, larger z0 implies larger u∗ or larger downward momentum flux. Hence, z0 is a description of the capacity of the surface for absorbing momentum. Some experimental studies indicate that z0 is about 1/30 the height of the roughness elements. Such a rule of thumb is useful only for specific cases. In general, the relationship between z0 and the physical size of the roughness elements is not simple. This is because the capacity of the surface in absorbing momentum, which z0 represents, depends on the interferences of the turbulent wakes generated by the roughness elements. We shall examine z0 in more detail in Chapter 6, when we discuss the Owen effect of saltation. The logarithmic wind profile applies both to smooth and rough surfaces, as illustrated in Fig. 3.9. For practical wind-erosion problems, both u∗ and z0 can be determined through fitting the measurements of mean wind with the logarithmic wind profile. If we plot U measured at different z against ln z, we would obtain a straight line with its slope being u∗ /κ and its intercept being −(u∗ /κ) ln z0 . For a smooth surface, the centeroid of momentum absorption occurs at z = 0. A rough surface can be considered to be a smooth surface superposed

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3 Atmospheric Boundary Layer and Atmospheric Modelling

with roughness elements and the centreroid of momentum absorption is no longer at z = 0 but at a distance away from the surface z = zd . This distance, zd , is the displacement height. The logarithmic wind profile can be more generally written as   z − zd u∗ ln (3.43) U (z) = κ z0 Both z0 and zd depend on the geometric features of the roughness elements (e.g. height, width etc.) and their spatial arrangement. For a dense canopy of forest of height zh , zd is about 0.7zh while for a dense urban canopy, zd is close to zh (Macdonald et al. 1998). Equations (3.39) and (3.42) are derived based on simple assumptions on the profiles of u∗ and Km . More generally, we assume that both u∗ and Km are functions of height u∗ = fu (z) Km = u∗ fk (z) The wind profile can now be determined as follows: z fu (z) U (z) = dz d+z0 fk (z)

(3.44)

The linear wind profile given by Equation (3.39) is a special case with fu and fk being constant, and the logarithmic wind profile given by Equation (3.42) is a special case with fu being constant and fk being a linear function of height. Suppose both fu and fk increase linearly with height with fu (z) = a + bz and fk (z) = κz, then we have   zd b a + (z − z0 ) (3.45) U (z) = ln κ z0 κ To have the knowledge of fu and fk is quite useful in studying flow in the saltation layers as we shall discuss further in Chapter 6. 3.6.4 Stability Measures The structure of the atmospheric boundary layer and the behaviour of turbulence both depend on stability. There are a number of stability measures for atmospheric boundary-layer flows, one of which is the static stability given in terms of the vertical gradient of potential temperature, namely, Stable

¯ ∂ θ/∂z >0

Neutral

¯ ∂ θ/∂z =0

Unstable

¯ ∂ θ/∂z 1, buoyancy destruction of turbulent kinetic energy dominates over shear production, thus the flow becomes less turbulent and is dynamically stable. For Rf < 1, shear production dominates over buoyancy destruction, thus the flow becomes more turbulent and is dynamically unstable. Other forms of the Richardson number have the same physical meaning, but different critical values. For example, the gradient Richardson number is given in terms of the gradients of potential temperature and wind speed  −2 g ∂ θ¯ ∂U (3.47) Ri = ¯ θ ∂z ∂z The critical gradient Richardson number is 0.25. Wind shear near the surface can be significantly modified by the stability of the atmospheric boundary layer. Assuming horizontal homogeneity (∂/∂x and ∂/∂y = 0), stationarity (∂/∂t = 0) and that the divergence of turbulent kinetic energy flux is negligible, Equation (3.29) can be simplified to ∂u ¯ g   −=0 w θ + u2∗ ∂z θ¯

(3.48)

If we take  = u3∗ /κz and Km = κzu∗ for the atmospheric surface layer, it follows that  2 ∂u ¯ g w θ u∗ ∂ u ¯ (3.49) = ¯ + ∂z κz ∂z θ κzu∗ In a statically-neutral surface layer, w θ ≡ 0, and an integration of the above equation gives the logarithmic wind profile. For stable and unstable situations, the wind profile is modified as illustrated in Fig. 3.10. For the unstable situation, a stronger wind shear occurs near the surface, as stronger turbulence transfers momentum more efficiently from higher levels to lower levels and

74

3 Atmospheric Boundary Layer and Atmospheric Modelling 10.0

Stable

1.0

z (m)

Neutral

0.1 Unstable

0.01 0

1

2

3

4

5

U (m s-1) Fig. 3.10. Effects of static stability on the profile of mean wind

increases the wind speed in the surface layer. The situation for the stable case is the opposite. The stronger wind shear in the surface layer partially explains why wind erosion occurs mostly during daytime when the boundary layer is unstable.

3.7 Similarity Theories Although atmospheric boundary-layer flows are mostly turbulent, they are not completely random. Observations frequently show consistent and repeatable features and suggest that identifiable physical processes dominate the evolution of certain types of flows. In other words, these flows may be similar in a dynamical sense. Therefore, despite the complexity of atmospheric boundarylayer processes, it is possible to determine the key factors which control the flow behaviour and develop simple laws for the variables of interest. Similarity theories provide a powerful framework for analysis of experimental data, as well as simple parameterisations for representing the complex dynamic processes involved. A similarity theory has three key ingredients. First, problems governed by similar dynamic processes are identified and then characterized with a few dimensionless parameters (e.g. the Reynolds number) which we call similarity parameters. Second, a set of scaling parameters is identified and used to establish non-dimensionalised dependent and independent variables. Third, we derive a set of similarity laws which are universally valid.

3.7 Similarity Theories

75

Similarity theories have different formulations for different regions of the atmospheric boundary layer. The most successful of these are the Monin– Obukhov similarity theory for surface layers and the mixed-layer similarity theory for convective boundary layers. 3.7.1 Monin–Obukhov Similarity Theory We first introduce the Obukhov length LO = −

u3∗ ¯  θ 0 κ(g/θ)w

The meaning of the Obukhov length can be understood from Equation (3.48). Multiplying Equation (3.48) by κz/u3∗ , we obtain −

κz κz g   κz τ ∂U + 3 = 0 wθ + 3 3 ¯ u∗ θ u∗ ρ ∂z u∗

(3.50)

The first term in the above equation is z/LO , and the second and third terms are of order 1. Thus, for small z/LO , the turbulent kinetic energy equation is basically a balance between shear production and dissipation, for large z/LO , the buoyancy production term becomes important. One interpretation of LO is that it is the height above the surface at which buoyancy production of turbulence first dominates the mechanical production of turbulence. In horizontally-homogeneous boundary layers, flow properties vary only in the vertical direction. It is intuitive suggest that the flow properties should only depend on the conditions of the underlying surface. The quantities we need to describe the main features of the flow can only be u∗ , z0 and ¯  θ 0 . This is because u∗ is a descriptor of the shear stress on the sur(g/θ)w face (or friction), z0 is a descriptor of the capacity of the surface for momentum ¯  θ 0 is the buoyancy flux (or heating). absorption, and (g/θ)w The Obukhov length is a combination of u∗ and gθ w θ 0 . The dimensionless number emerging from this group of parameters is z0 /LO , which serves as the similarity parameter. The Monin–Obukhov hypothesis states that all surface layers with identical z0 /LO should behave in a similar way. The scaling parameters can now be defined as follows: Scaling Length

LO

Scaling Velocity

u∗

Scaling Time

LO /u∗

Scaling Temperature

θ∗ = w θ 0 /u∗

Scaling Humidity

q∗ = w q  0 /u∗

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3 Atmospheric Boundary Layer and Atmospheric Modelling

These scaling parameters are then used to form the dimensionless groups of variables under consideration, such as U/u∗ . The Monin–Obukhov hypothesis is that the dimensionless variables are universal functions of the dimensionless independent variable ζ = z/LO and the similarity parameter, z0 /LO , namely, that   z0 U = f ζ, u∗ LO Numerous observations show, however, that the dependence of the similarity functions on z0 /LO is weak and, therefore, it normally does not appear explicitly in the similarity laws, unless for regimes very close to the surface (when z/LO and z0 /LO are comparable). We first consider the similarity laws for wind, temperature and humidity profiles. For instance, the Monin–Obukhov hypothesis for wind is that ∂U/u∗ = g(ζ) ∂z/LO This hypothesis can be rewritten as u∗ ∂U = φm (ζ) ∂z κz

(3.51)

By analogy, we have θ∗ ∂ θ¯ = φh (ζ) ∂z κz q∗ ∂ q¯ = φq (ζ) ∂z κz

(3.52) (3.53)

The similarity functions φm , φh and φq have been determined empirically. Figure 3.11 shows φm and φh derived from observed data. These functions can be described as follows:  for ζ > 0, stable case 1 + βm ζ φm = (3.54) for ζ ≤ 0, unstable case (1 − γm ζ)−1/4 and

 φh = φq =

1 + βh ζ (1 − γh ζ)−1/2

for ζ > 0, stable case for ζ ≤ 0, unstable case

(3.55)

The empirical constants, βm , γm , βh and γh differ slightly, when estimated from different observational data sets, but the frequently used values are βm = βh = 5 and γm = γh = 16. The Monin–Obukhov similarity theory can be used to estimate the turbulent transfer coefficients via the flux-gradient relationship:

3.7 Similarity Theories φm

(a)

-3

φh

(b)

7

-2

-1

7

6

6

5

5

4

4

3

3

2

2

1

1

0 ζ

77

1

2

-3

-2

-1

0

1

2

ζ

Fig. 3.11. Similarity functions φm (a) and φh (b) derived from observed data (Modified from Businger et al. 1971)

τ = ρu2∗ = ρKm

∂U ∂z

∂ θ¯ ∂z ∂ q¯ λl E = −ρλl u∗ q∗ = −ρλl Kq ∂z H = −ρcp u∗ θ∗ = −ρcp Kh

(3.56) (3.57) (3.58)

Using the similarity functions for wind, temperature and humidity gradients, we find that u∗ κz u∗ κz u∗ κz Kh = Kq = (3.59) Km = φm (ζ) φh (ζ) φq (ζ) These are frequently used relationships in land-surface schemes (Chapter 4). The eddy diffusivity for dust particles, Kp , can be derived from a modification of Km (Chapter 8). From the above relationships, we also see that φm Kh = Km φh From Equations (3.54) and (3.55), Kh /Km is order unity for stable conditions, but is more complicated for unstable conditions. The values of U , θ¯ and q¯ at height z can also be determined by integration of the similarity functions over z, giving

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3 Atmospheric Boundary Layer and Atmospheric Modelling

u∗ [Ψm (ζ) − Ψm (ζ1 )] κ ¯ − θ(z ¯ 1 ) = θ∗ [Ψh (ζ) − Ψh (ζ1 )] θ(z) κ q∗ q¯(z) − q¯(z1 ) = [Ψq (ζ) − Ψq (ζ1 )] κ

U (z) − U (z1 ) =

where

Ψi (ζ) =

φi (ζ) dζ ζ

(3.60) (3.61) (3.62)

i = m, h, q

Based on observations, Ψm has been estimated to be  Ψm =

ln( zz0 ) + 4.7ζ ln( zz0 ) − Ψ1

ζ > 0, stable case ζ ≤ 0, unstable case

where
Ψ1 = 2 ln


 (1 + φ−2 (1 + φ−1 π

m ) m ) + ln − 2 tan−1 φ−1 m + 2 2 2

and Ψh to be  Ψh =

0.74 ln( zz0 ) + 4.7ζ 0.74[ln( zz0 ) − Ψ2 ]

ζ > 0, stable case ζ ≤ 0, unstable case

where Ψ2 = ln[(1 + 0.74φ−1 h )/2] Similarity functions can also be determined for turbulence statistics, notably φw = σw /u∗

φθ = σθ /uθ

φ = κz/u3∗

The frequently used expressions of these similarity functions can be found in, for instance, Kaimal and Finnigan (1994). 3.7.2 Mixed–Layer Similarity Theory Mixed-layer similarity theory applies to convective boundary layers. Convective turbulence, driven mainly by buoyancy, penetrates the entire boundary layer. Hence, the typical length scale for the turbulent motion is zi , the height of the capping inversion (Fig. 3.7). In the bulk of the convective boundary layer, strong mixing diminishes vertical wind gradient and therefore the effect of wind shear on the structure of the flow is not important. The quantities ¯  θ 0 and zi . The which define a convective boundary layer are therefore, g/θw mixed-layer scaling parameters can be constructed from these two quantities.

3.8 Turbulent Flow Models

Scaling length

zi

Scaling velocity

w∗ = ( gθ¯ w θ 0 zi )1/3

Scaling time

zi /w∗

Scaling temperature

θ∗ = w θ 0 /w∗

Scaling humidity

θ∗ = w q  0 /w∗

79

The mixed-layer similarity hypothesis is that dimensionless groups formulated with these scaling parameters are universal functions of z/zi . This hypothesis works well for several variables. We first consider the similarity function for the vertical buoyancy flux. The mixed-layer similarity hypothesis implies that the normalized buoyancy flux should be a function of only z/zi . Observations show that this is indeed the case: Buoyancy fluxes in convective boundary layers almost collapse into a single curve if they are normalised with respect to their corresponding surface values, which gives z w θ w θ =   =1−a w∗ θ∗ zi wθ0

(3.63)

where a is a constant around 1.25 (Fig. 3.12). Mixed-layer similarity functions have been derived from the observed data sets, including those for velocity 2 , potential temperature variance σθ2 and turbulent variances σu2 , σv2 and σw kinetic energy dissipation rate  (Fig. 3.12). The similarity functions for σu2 , 2 are σv2 and σw σ2 . σu2 = v2 = 0.35 2 w∗ w∗ 2  2/3  2 z z σw 1 − 0.8 = 1.8 w∗2 zi zi

(3.64) (3.65)

2 atWhile the normalised σu2 and σv2 are almost constant, the normalised σw tains its maximum at about z/zi = 0.5. The normalised  is nearly a constant and the normalised σθ2 reaches a minimum at about z/zi = 0.5.

3.8 Turbulent Flow Models The modelling of an atmospheric boundary-layer flow involves numerically solving a set of governing equations (Section 3.2) subject to initial and boundary conditions. The governing equations and the initial and boundary conditions are discretised in time and discretised in space on a numerical grid which covers the simulation domain. Popular discretisation techniques include finitedifference, finite-volume and pseudo-spectral methods (Fletcher, 1990). The discretisation procedure produces a set of algebraic equations which can be solved numerically to obtain approximate solutions for the flow.

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3 Atmospheric Boundary Layer and Atmospheric Modelling 1.4

(a) 1.2

(b)

u, v

w 1.0

z/zi

0.8 0.6 0.4 0.2 0 0.01

0.1

1

1

10

100

σθ2/θ2

σ2/w*2

*

1.4

(c)

(d)

1.2

z/zi

1.0 0.8 0.6 0.4 0.2 0

0

0.5

w’θ’/w*θ*

1

0.1

1

10

εzi /w*3

2 Fig. 3.12. (a) Normalised velocity variances σu2 , σv2 and σw , (b) potential tempera2   ture variance, σθ , (c) buoyancy flux, w θ , and (d) dissipation rate for turbulent kinetic energy, , in the mixed-layer similarity framework (After Caughey and Palmer, 1979)

Model resolution is determined by the spacing of the grid, ∆. As ∆ is finite, motions on scales smaller than ∆ cannot be represented by the grid, and this implies that a variable to be discretised is inevitably separated into a resolved component and an unresolved component. Observations of atmospheric boundary-layer flows show that there is an energy gap between the mean and turbulent motions and that the energy spectrum of turbulence can be divided

3.8 Turbulent Flow Models

0.1

1 10 Frequency (cycles/hour)

100

Dissipation Subrange

Large Eddy

Energy Gap

Diurnal Variation

Spectral Energy

Synoptic Motion 0.01

Inertial Subrange

Turbulence

Large Scale

0.001

81

1000

Fig. 3.13. Schematic energy spectrum of atmospheric flows, showing the distinct regimes of synoptic scale motion, energy gap, large eddy, inertial subrange and dissipation subrange

into regimes of energy-containing eddies (or large eddies), inertial subrange and dissipation subrange (Fig. 3.13). The Eulerian integral length scale, ΛE , represents the size of the energy containing eddies, and the Kolmogorov inner scale, η, represents the smallest eddy size (Kaimal and Finnigan, 1994). The order of magnitude of ΛE is 10 to 100 m and that of η is 0.001 m. Depending on grid resolution, modelling of atmospheric boundary-layer flows can be further divided into Reynolds-Averaged Simulation (RAS), LargeEddy Simulation (LES) and Direct Numerical Simulation (DNS). The emphasis of RAS is placed on the mean fields and thus the model resolution is relatively coarse with ∆ being similar to or larger than ΛE . In this case, most turbulent fluctuations are not resolved by the numerical grid. In contrast, DNS requires the grid spacing ∆ to be smaller than η and hence the spatial resolution of DNS is so high that all turbulent motions are resolved by the numerical grid. LES is designed to model energy containing eddies, and the grid spacing must be at least many times smaller than ΛE but many times larger than η. LES for atmospheric boundary-layer turbulence often uses a grid spacing less than 10 m. Equations (3.19)–(3.24) are appropriate for the purpose of RAS. The difficulty lies in the treatment of the turbulent fluxes (e.g. uj θ ) from motions which are unresolved by the numerical grid, or sub-grid motions. These fluxes are unknowns introduced through Reynolds averaging. As a result, the system of Equations (3.19)–(3.24) is not closed, since the number of equations is smaller than the number of unknowns. It is possible to derive additional prognostic equations for these new unknowns. For example, the prognostic equation for the vertical component of uj θ in a horizontally homogeneous

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3 Atmospheric Boundary Layer and Atmospheric Modelling

flow can be written as ¯ ∂w w θ σ 2 g p ∂θ ∂w θ 2 ∂θ = σw − + θ¯ + − 2wθ ∂t ∂z ∂z ρ ∂z θ

(3.66)

However, other new unknowns appear in the above equation, such as the triple correlation term w w θ . If a similar procedure is followed to derive additional equations for w w θ , more unknowns will be introduced and the equation system remains open. In fact, the number of unknowns increases faster than the number of equations as we move to higher orders. The basic approach to achieving a closed system of equations is parameterisation. This means that the quantities for which a prognostic equation is missing are expressed in terms of the already-existing prognostic variables and some parameters. Different orders of closure have been used in modelling boundary-layer flows. In first-order closure, equations for Reynolds-averaged variables are retained in the model and second-order moments (e.g. w θ ) ¯ and parameters. In secondare expressed in terms of first-order ones (e.g. θ) order closure, equations for first- and second-order moments are retained in the model and third-order moments (e.g. w w θ ) are expressed by means of second-order moments (e.g. w θ ) and parameters. The parameters are usually empirical constants and/or simple functions of space and time. Parameterisations depend on an understanding of the physical problem involved and thus, different models may employ different parameterisations. K-theory is a simple and widely used first-order closure technique. This parameterisation specifies the relationship between a turbulent flux and the gradient of the corresponding mean quantity. In general, we have that uj ξ  = −Kξ

∂ ξ¯ ∂xj

(3.67)

where the parameter Kξ is the eddy diffusivity for ξ, an arbitrary scalar. Ktheory is best applicable to atmospheric surface layers where the small-eddy concept is adequate. The Monin–Obukhov similarity theory provides a fairly accurate method of estimating the eddy diffusivities for momentum, heat and various passive scalars (Equation 3.59). The e− closure is another widely used technique for modelling boundarylayer flows (Detering and Etling, 1985). The e −  closure is also based on the small-eddy concept, but the eddy viscosity is calculated using the Prandtl– Kolmogorov hypothesis that Km = ce e2 / where ce is a coefficient. For the e −  closure, the prognostic equations for the turbulent kinetic energy, e, and the dissipation rate of turbulent kinetic energy, , are retained in the governing equation system and numerically solved along with the other prognostic equations. The prognostic equation for  can be

3.8 Turbulent Flow Models

83

derived from the equations of motion or from an equation of turbulent vortex intensity (Tennekes and Lumley, 1972), and for the horizontally homogeneous case is
 ∂u w ∂v  w ∂w   g 2 ∂ = c1 − − + c2 ¯ w θ − − c3 ∂t e ∂z ∂z eθ ∂z e

(3.68)

where w  = −

K ∂ c4 ∂z

The constants used in the e −  model are determined empirically by comparing simulated results with observations (such as wind-tunnel observations). The values commonly used are ce = 0.09, c1 = 1.44, c2 = 1.0, c3 = 1.92, c4 = 1.30. Mellor and Yamada (1974, 1982) introduced a hierarchy of models for atmospheric boundary-layer flows, with closure at different levels. The socalled Level-4 model of Mellor and Yamada retains prognostic equations for second moments and requires the solution of 13 partial differential equations. Mellor and Yamada (1974) have separated the velocity covariance prognostic equation into isotropic (i = j) and anisotropic (i = j) parts and performed a scale analysis, identifying the relative importance of each term and systematically neglecting higher-order terms. This scaling analysis reduces the Level-4 model to a Level-3 one. The latter consists of two partial differential equations and eight algebraic equations. A further simplification can be introduced by neglecting the total time derivative and the diffusion term in the Level-3 model. This simplification is based on considerations of computational efficiency. The resulting system is a Level-2 12 closure similar to the e −  closure. The Mellor–Yamada scheme in its various forms is widely used in modelling atmospheric boundary-layer flows. K-theory, e −  closure and Mellor–Yamada schemes are all local closures, in which an unknown quantity at any point in space is parameterized by values and/or gradients of known quantities at the same point. An alternative to local closure is non-local closure, in which an unknown quantity at any point in space is parameterized by values and/or gradients of known quantities at many points. In non-local closure, the advective nature of structured eddies, especially turbulence in convective boundary layers, is recognised in turbulent transport. Non-local closure schemes assume that turbulence is a superposition of eddies, each of which is associated with an advective process taking place on the scale of the eddy size. Non-local closure is much more difficult to implement in numerical models and has therefore not yet been widely used. Since the pioneering work of Deardorff (1970), LES has been applied widely to studying atmospheric turbulent flows (e.g. Moeng, 1984; Schumann, 1993). Unfortunately, we have not yet fully taken the advantage of LES in wind-erosion modelling, although its potential usefulness in studying

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the interactions between turbulent flows and soil erosion is obvious. For problems such as the protection of soil surfaces by windbreaks, deposition of soil particles in the wake region of isolated shrubs, the evolution of sand dunes and particle entrainment by convective turbulence, LES may offer valuable information which is otherwise difficult to obtain. The model structure and numerical procedures for LES and RAS are similar. In fact, LES and RAS can be carried out using the same numerical code with minor modifications. However, we note that LES is designed to model energy-containing eddies and it differs from RAS in several aspects: • •

•

LES requires higher spatial and temporal resolution than RAS, but as the grid size and integral time step for LES increase, the results of LES and RAS will become similar. Instead of Reynolds averaging, averaging over grid size is required. Commonly used techniques include volume averaging (Deardorff, 1970), volume-balance methods (Schumann, 1975) and Gaussian filters (Leonard, 1974). A subgrid model is required to parameterize the effects of small eddies upon the large eddies, rather than those of turbulence upon the mean flow. The Smagorinsky–Lilly (Smagorinsky, 1963) model is widely used.

Figure 3.14 shows an example of LES for flow over a windbreak of height h with a porosity of 0.5. The simulation domain is 12 h long, 4 h wide and 4 h high and is consisted of 72 × 48 × 48 grid points. In this example, LES provides detailed information on turbulence field (Fig. 3.14a) and the mean flow field (Fig. 3.14b) in the vicinity of the windbreak. This type of information is required in studying the effect of windbreaks on the prevention of wind erosion. Typical spatial resolution of Direct Numerical Simulation (DNS) is several millimetres. The requirement of DNS for extremely high spatial resolution implies that it can only be applied to a small spatial domain, as it is too expensive computationally. The usefulness of DNS in wind-erosion studies lies in its ability to provide a tool for investigation of detailed erosion mechanisms. Examples of this are drag partitioning and erosion pattern in the vicinity of individual roughness elements (see Chapter 9). In many ways, DNS provides valuable information for wind-erosion studies, which is difficult to obtain from traditional wind-tunnel experiments. Figure 3.15 shows a simulation of flow speed around roughness elements mounted on the surface and the distribution of local momentum flux in the vicinity of these elements. In this example, five cylinders with a diameter of 5 mm and a height of 10 mm are placed on the surface of a domain 150 mm long, 65 mm wide and 50 mm high. The grid resolution is 1 mm and the input flow speed in 10 m s−1 . The simulation shows the reduced flow speed and the enhanced local momentum flux in the wake region of the obstacles.

3.9 Meso-scale, Regional and Global Atmospheric Models

85

(a)

z/h

3 2 1 0

−1

0

1

2

3

4 x/h

5

6

7

8

9

8

9

(b)

3 z/h

14

2

12 10

1

8

6

4

0

−1

0

1

2

3

4 x/h

5

2

6

7

Fig. 3.14. (a) Simulated instantaneous flow field near a windbreak of height h and porosity 0.5, using a LES model. The windbreak is located at x = 0. (b) As (a), but for mean flow speed in m s−1 (M. Fitzmaurice, with acknowledgment)

3.9 Meso-scale, Regional and Global Atmospheric Models Modelling wind erosion on broad scales depends critically on the modelling of synoptic and sub-synoptic atmospheric systems, such as fronts, thunderstorms and squall lines, which generate dust storms and other wind-erosion events. Sophisticated numerical weather prediction models with high temporal and spatial resolutions are required for this purpose. Weather is influenced not only by the fluid-dynamic processes of the atmosphere but also by a number of physical, geo-chemical and bio-ecological processes. A range of atmospheric models have been developed. Depending on model resolutions and purposes of model application, these models are usually classified into meso-scale, regional, regional-climate and global models. Traditionally, meso-scale models are designed to model atmospheric systems on scales of 10–100 km, regional models (or weather models) to model synoptic systems on the scale of 100–1,000 km and global models to model climatic systems of scale of 1,000–10,000 km.

86

3 Atmospheric Boundary Layer and Atmospheric Modelling (a)

0.06

0.8

0.03

0

0.8

0.8

0.04

0.8

y (m)

0.05 0.8

1.6 0.8

0.8

0.02

0.8

1.6

0

0.8

0.01 0

0

0.06

0.05 (b)

6

0.05

8

106 42

0.04

86 6

4

8

10

4

8

6

0.02

6

4

10

0.01 0

4

2

8

0.03

10

y (m)

0.1

0

0.05

x (m)

0.1

Fig. 3.15. An example for DNS application to wind erosion modelling. Five roughness elements (cylinders with a diameter of 5 mm and height of 10 mm) are mounted on the surface and DNS is applied to determine (a) the momentum fluxes (in N m−2 ) and (b) flow speed (in m s−1 ) in the vicinity of the roughness elements (An Li, with acknowledgment)

Regional-climate models have similar spatial resolutions as regional-weather models, but focus on atmospheric changes on a time-scale much longer than the synoptic time scale, e.g. seasonal, annual or even decadal. However, the traditional boundaries between the models are rapidly diminishing with the ever increasing computing power. For instance, a meso-scale model can be used for large-eddy simulation as well as numerical weather predictions. Most atmospheric models consist of a dynamic framework, a number of modules for physical processes and techniques for data assimilation. The equation system commonly used for numerical weather prediction consists of seven basic equations: three equations for velocity components, the continuity equation, the thermodynamic equation, the moisture equation and the equation of state. If dust transport is also of concern, the dust concentration equation can be added to the equation system. To account for the effect of surface topography, the σ coordinate system (x, y, σ) is often used. The vertical coordinate σ is defined to be p/ps , with p being the atmospheric pressure at a point and ps that at the surface direct beneath it (Simmons and Bengtsson, 1984). In the σ coordinate system, the equations for numerical weather prediction are:

3.9 Meso-scale, Regional and Global Atmospheric Models

∂(ps uu) ∂(ps uv) ∂(ps uσ) ∂ ˙ (ps u) = − − − ∂t ∂x ∂y ∂σ ∂ps ∂φ − RT + Fu + Du +f ps v − ps ∂x ∂x ∂(ps vu) ∂(ps vv) ∂(ps v σ) ∂ ˙ (ps v) = − − − ∂t ∂x ∂y ∂σ ∂ps ∂φ − RT + Fv + Dv −f ps u − ps ∂y ∂y ps ∂φ = ∂σ ρ

87

(3.69)

(3.70) (3.71)

∂(ups ) ∂(vps ) ∂(σp ˙ s) ∂ps =− − − (3.72) ∂t ∂x ∂y ∂σ     ∂T σ˙ d ln ps ∂T ∂T ∂T = −cp u +v − cp σ˙ + RT + cp ∂t ∂x ∂y ∂σ σ dt (3.73) +QT + FT + DT   ∂q ∂q ∂q ∂q =− u +v − σ˙ + Qq + Fq + Dq (3.74) ∂t ∂x ∂y ∂σ In the above equations, σ˙ is the vertical ‘velocity’ in the σ coordinate system and φ = gz is the geopotential. Fu and Fv are the horizontal frictional forces, and FT and Fq are the heating and moisture changes arising from subgrid scale vertical turbulent exchange. QT and Qq are the heat and moisture source/sink terms, and Du , Dv , DT and Dq are the lateral diffusions of momentum, heat and moisture, respectively. The hydrostatic assumption is embedded in the equations of motion and in the continuity equation. This assumption is valid for modelling large-scale atmospheric systems and the spatial resolution needs to be larger than 5 km. For higher resolution, the effects of surface topography and convection may generate strong vertical accelerations which violate the hydrostatic assumption. For modelling sub-synoptic atmospheric systems and intense weather events, such as squall lines, non-hydrostatic models are necessary. Therefore, meso-scale models are mostly non-hydrostatic while regional and global models are mostly hydrostatic. The equation system can be solved numerically using various numerical schemes and computational grids (e.g. Lin et al. 1997). The horizontal resolution of numerical weather-prediction models ranges from several kilometres to several tens of kilometres and in the vertical, about 30 layers are employed with smaller increments in the lower part of the atmosphere. The operation of regional models requires initial and boundary conditions. The preparation of the initial input data is normally based upon objective analysis, which combines observed data from meteorological networks with a model forecast from a previous time step. Objective analyses are produced routinely by meteorological services. Interpolation of the data from the pressure

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3 Atmospheric Boundary Layer and Atmospheric Modelling

surfaces to the σ levels is then carried out using methods such as cubic-spline interpolation. Sometimes, extrapolation is also required using assumed lapse rates for temperature and thermal-wind relationships for the wind components. Dynamical consistency of the initial fields also needs to be preserved. The requirement for boundary conditions is normally fulfilled through nesting the regional model in a global model. Most regional models can also be self-nested, which allows the model to be run over selected areas with increased resolution. Model simulations at higher resolutions are carried out by first running the model with the coarse-resolution data. During these runs, the model outputs atmospheric variables to data files for every nesting time step (e.g. 6 hr). These data files are then processed, i.e. data from the coarse mesh are interpolated to the fine mesh and adjusted to the heights of the new topography, horizontal wind components, temperature and mixing ratio are interpolated to the new model levels using spline functions. Finally, the model is run on the chosen higher-resolution sub-domain. Regional models also require other data sets for modelling the physical processes. These data sets are commonly those for ocean-surface temperature and land-surface properties, including topography, surface roughness lengths, vegetation characteristics and soil properties. It is important that these data sets are also adequately involved in the preparation of model input data and in the nesting procedure. Figure 3.16a shows an example of running a regional model for the Australian region with self-nesting. In this case, the model is run with a hori-

Fig. 3.16. Simulated near-surface wind (arrows, ms−1 ) and temperature (contours, ◦ C) for the 9 February 1996 dust-storm event in central Australia, (a) with a horizontal resolution of 50 km over the Australian region and (b) with a horizontal resolution of 10 km over the framed area in (a) using self-nesting. A cold front can be identified as the region of large temperature gradient

3.9 Meso-scale, Regional and Global Atmospheric Models

89

zontal resolution of 50 km, with the boundary conditions derived from the Limited Area Prediction System of the Australian Bureau of Meteorology, which provides the input data at a horizontal resolution of 75 km. This particular weather event generated intense dust storms in the Simpson Desert of Australia. In order to examine the details of the frontal system, the regional model is run in the self-nested mode for a selected area with the spatial resolution increased from 50 km to 10 km. Figure 3.16b shows the simulated results near the cold front. While the simulation with a coarse grid reveals the general features of the frontal system, the higher resolution model gives a more detailed structure of the frontal system. As wind erosion is strongly variable in space, it is very useful to obtain atmospheric data at high resolution. Since the late 1980s, dust models have been constructed based on regional weather models. These dust models now have considerable skill for the simulation and prediction of dust storms (Uno et al. 2006). Climate models can take many forms ranging from simple one-dimensional energy balance models to complex three-dimensional time-dependent general circulation models (GCMS) of the atmosphere and ocean. From the perspective of wind erosion, climate models are mainly used to study the global dust cycle and the effect of dust on climate change (Zender et al. 2003; Ginoux et al. 2004; Tanaka and Chiba, 2006). For these types of studies, three-dimensional time-dependent GCMS are more applicable, as detailed simulations of atmospheric parameters, such as wind and precipitation, can be obtained. Climate is generally considered to represent the average behaviour of the climate system (including the atmosphere, hydrosphere, lithosphere, cryosphere and biosphere) over some long period of time. It is not associated with the exact sequence of daily weather fluctuations. Because of the inherent unpredictability of atmospheric and ocean flows, climate models are unable to predict the day-to-day sequence of weather events beyond a very few weeks. The true utility of climate models lies in their ability to predict the statistical properties of some future climate state. This statement applies also to long-term wind-erosion modelling. Global climate models use a similar set of governing equations to regional models, formulated mostly also in the σ coordinate system. With the rapid increase of computing power, climate models are being run with increased horizontal and vertical resolution and increased sophistication in the treatment of the complex physical, geo-chemical and bio-ecological processes. As a result, the distinction between climate models and weather models has become less obvious, apart from detailed numerical procedures. However, we note that, as climate models attempt to model climate changes over long-time periods, the fundamental interactions between the atmosphere, the ocean and other components of the climate system become more and more important. Despite this, improvements in climate modelling in recent years have certainly provided the necessary prerequisite for the assessment and prediction of wind-erosion climatology and led to new opportunities in wind-erosion studies.

4 Land-Surface Modelling

Wind erosion, a land-surface process itself, is closely related to other landsurface processes, in particular to soil hydrological and surface bio-ecological processes which determine the status of soil moisture and vegetation cover. Wind erosion occurs only if the soil is depleted of moisture and the lack of vegetation cover is serious, as the capacity of the surface to resist wind erosion depends critically on these factors. Hence, land-surface modelling is of critical importance to wind-erosion studies. Land-surface modelling is in itself an important research topic, as vegetation and soil play a major role in the climate system through their exchange of mass, energy and momentum with the atmosphere. On the other hand, atmospheric conditions (wind, temperature and precipitation) also strongly affect the processes of the biosphere and the continental hydrosphere. The interactions between the land surface and the atmosphere constitute an active research area. Sophisticated land-surface models for the simulation of these interactions have been developed in recent years. For atmospheric, hydrological and bio-ecological modelling, a land-surface scheme produces a range of useful outputs, such as (1) soil and vegetation temperatures; (2) surface net radiation, sensible-heat flux, latent-heat flux, ground-heat flux and soil-heat fluxes; (3) soil moisture and (4) flux components related to soil-water balance, including infiltration, soil water fluxes, runoff and drainage. As far as winderosion modelling is concerned, we are most interested in soil moisture in the very top soil layer. In this chapter, we shall concentrate on the simulation of soil moisture using land-surface models. Integrated wind-erosion modelling, as will be described in Chapter 9, and land-surface modelling have much in common. Apart from soil moisture, the parameterisations used in land-surface models for estimating friction velocity and the methodology used for the treatment of heterogeneous surfaces are directly transferable to wind-erosion modelling. Furthermore, wind-erosion and land-surface models share a considerable proportion of input data for soil, vegetation and surface aerodynamic properties. Y. Shao, Physics and Modelling of Wind Erosion, c Springer Science+Business Media B.V. 2008


91

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4 Land-Surface Modelling

4.1 General Aspects The land surface mainly consists of soil (down to the water table) and vegetation. The interest in land-surface modelling originates from the need to provide atmospheric models with better lower-boundary conditions through the specification of the exchanges of momentum, energy and mass between the atmosphere and the land surface. The central task of a land-surface model is to quantify these exchanges which, as shown later, are closely related to surface soil hydrological processes, and hence the modelling of soil moisture becomes one of the critical issues for climate and weather models. The processes which influence these exchanges are very complex, as they depend both on atmospheric conditions and the physical and bio-ecological properties of the land surface. The representation of these processes in a landsurface scheme is much simplified. Figure 4.1 shows the concept for such a land-surface scheme. We are mainly concerned with the energy and water balance of the land surface, including the unsaturated soil layer and plants. The starting point is the soil-temperature equation and the soil-moisture equation of an unsaturated soil layer 1 ∂Gt ∂T =− + sh ∂t Cs ∂z ∂Gw ∂θ =− + sw ∂t ∂z

(4.1) (4.2)

where T is soil temperature, Cs is volumetric soil heat capacity (Cs = ρs cs , ρs is soil density and cs is specific heat capacity of the soil), θ is volumetric soil water content, Gt and Gw are the heat flux and the volumetric water flux through the soil, respectively, sh is a source/sink term for soil heat related to the possible phase change of soil water, and sw is a source/sink term for soil moisture related to processes such as transpiration. Pores of various sizes occur in soil and a proportion of these pores is filled with liquid water. The relative volume of this water in a unit volume of soil is defined as volumetric soil water content, θ, often simply called soil moisture. If all soil pores are filled with water, the soil is saturated and the value of soil moisture is the saturation soil moisture, θs . Under natural conditions, soil cannot be dried completely and a small proportion of water is always trapped in the smallest pores. The minimum value of soil moisture under natural conditions is the air-dry soil moisture, θr . Both θs and θr depend on soil type (see Section 4.6). In Equations (4.1) and (4.2), land-surface properties are assumed to be horizontally homogeneous. The solution of these two equations involves the energy and water fluxes at the interface between the atmosphere and land, as shown in Fig. 4.1. At the interface, Gt is affected by solar, atmospheric and land-surface radiations, turbulent heat transfer, evaporation and heat transfer in the soil, while Gw is affected by precipitation, evapotranspiration and runoff. A land-surface scheme is, in principle, the algorithm required to solve

4.1 General Aspects

93

Rl-upward Precipitation

Rs-reflected

Wind Transpiration Interception & Evaporation Rl-downward Rl-upward

Leaf drip

Rs-reflected

Sublimation Infiltration Flood Flow

Sensible Heat Evaporation Snow

Soil water flux

Ground heat flux

Drainage

Fig. 4.1. An illustration of energy and water balance of the land surface. The energy balance is affected by solar, atmospheric and land-surface radiations, turbulent heat transfer, evaporation and ground heat flux. Rs and Rl denote shortwave and long wave radiations, respectively. The water balance is affected by precipitation, evapotranspiration, runoff and drainage

this equation system for a given land-surface configuration. The processes represented in a land-surface scheme can be broadly divided into three categories: sub-surface thermal and hydraulic processes, bare soil transfer processes and vegetation processes. Let us now consider a unit area of land surface. In the vertical direction, the soil column is represented by a number of soil layers as shown in Fig. 4.1. Simplification of the land-surface configuration in the horizontal direction is also required. Figure 4.2 shows examples for how the complex land surface is represented. The unit area of land surface is further divided into n sub-units, each of which has a different soil and vegetation type (Fig. 4.2a). The energy and soil-water conservation can be considered separately for each sub-unit and

94

4 Land-Surface Modelling (a)

(c)

(b)

σf1= 0

s1

σf2

s2

σfi

si

σfn

sn

σf1= 0

σf2

s1

s2

σf

s=1

Fig. 4.2. An illustration showing how complex land surfaces are simplified in a land-surface scheme. A unit area of the land surface is further divided into n subunits, each of which has a different soil and vegetation type. The fraction of ith soil type is si and the fraction of jth vegetation type is σf j . Figures (a), (b) and (c) represent a successive simplification of the land surface (From Irannejad, 1998)

¯ can be estimated through the (energy or any other) flux for the unit area, X, a weighted average of the Xi for each individual sub-unit, i.e., ¯= X

n

Xi si

i=1

with si being the fraction of land surface for the ith sub-unit. This strategy can be simplified to two soil types and two land types for each unit area (Fig. 4.2b). A further simplification occurs if each unit area contains only one soil type and one land-use type (Fig. 4.2c). Most land-surface schemes use simplest configuration and treat the vegetation as a big leaf covering the fraction σf of the land surface. Of course, the land surface in reality is highly heterogeneous, due to natural and anthropogenically-induced ecosystem diversity, complex morphology, soil variability and atmospheric forcing. Because most of the land-surface processes are highly nonlinear, heterogeneity can profoundly affect the exchanges of momentum, water and energy between the surface and the atmosphere. In more recent land-surface schemes which account for the subgrid-scale variations of surface characteristics and/or atmospheric variables, configurations (a) and (b) are increasingly used.

4.2 Surface Energy Balance At the top of the atmosphere, the solar radiation is 1367 ± 3% Wm−2 . Figure 4.3 shows the averaged energy balance of the Earth’s system in relative terms. About 30% of the incoming solar radiation is reflected by the surface and the atmosphere or scattered by the atmosphere back to space. Of the

4.2 Surface Energy Balance Shortwave Radiation 100

Ref. from clouds 20

Longwave Radiation

Emitted from Scattered from surface atmosphere Ref. from surface 5 7 4

Convective Fluxes

Emitted from atmosphere 64

Emitted from surface 109

5 Abs. by clouds

95

23

6

Abs. by Atmos 17

14 Abs. from clouds

22 Abs. directly

11 Abs. scattered

114 Emitted from surface

96 Abs. from atmosphere

23 6 Latent heat Sensible Heat from surface from surface

Fig. 4.3. Energy balance of the Earth system in percentage (After Bryant, 1997)

effective energy remaining in the system about 70% is transmitted through the atmosphere and absorbed by the Earth’s surface. To balance the absorbed energy, the surface releases energy to the atmosphere through long wave radiation and turbulent transfers of sensible and latent heat. The Earth’s surface provides about 2/3 of the energy input for the atmosphere and is hence the major immediate energy source for atmospheric processes. The surface is the only source of moisture. Evapotranspiration of water from the surface and its condensation in the atmosphere is the link between the energy and water cycles in the Earth’s system. The evolution of soil moisture is closely related to that of soil temperature. This is because surface latent-heat fluxes due to evaporation and transpiration are coupled with the surface sensible-heat flux through the surface energybalance equation Rn (T0 ) − λl Ev (T0 ) − H(T0 ) − Gt0 (T0 ) = 0

(4.3)

where Rn is surface net radiation, λl Ev , is latent-heat flux with Ev being evaporation, H is sensible-heat flux, and Gt0 is ground heat flux. All energy fluxes are functions of the surface (skin) temperature T0 . The net radiation, Rn , is a result of three radiation fluxes Rn = (1 − α)Rs + εRl − εσT04

(4.4)

where σ is the Stefan–Boltzmann constant, Rs is downward shortwave radiation, Rl is downward long wave radiation, α is the surface albedo and ε is the surface emissivity. The emissivity of a substance according to Kirchhoff’s law is equal to its absorptivity for the same wavelength range. Emissivity of the natural substances on the Earth’s surface ranges from 0.90 to 0.99. However, in land-surface schemes a universal emissivity is usually assumed for all surface types. A value of 0.98 can be taken as the representative for both soil and

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4 Land-Surface Modelling

vegetation. Since | Rl − σT04 | is usually small, this assumption does not lead to a large error in calculating the surface energy balance. The soil surface temperature, T0 , is commonly calculated by iterative solution of Equation (4.3) until the energy balance is achieved to within a specified accuracy. All terms in Equation (4.3) are either directly or indirectly dependent on soil moisture. Albedo depends on surface characteristics, solar geometry, and spectral distribution of incident solar radiation. Albedo needs to be parameterised and its parameterisation varies in different land-surface schemes. Splitting the solar spectrum into visible and near-infrared regions for calculating surface albedo has been used, for instance, by Dickinson et al. (1986, 1993). Sellers et al. (1986) and Xue et al. (1991) not only account for the radiation wavelength, but also differentiate between the direct and diffused radiation in calculating surface albedo. However, most land-surface schemes (e.g. Wetzel and Chang, 1988; Noilhan and Planton, 1989) use a single all-spectrum albedo for each soil type, each vegetation type or each surface type (soil and vegetation). Different albedo values are assigned to different vegetation types. For instance, albedo varies between 0.19 (summer) and 0.23 (winter) for range-grassland, between 0.16 (summer) and 0.17 (winter) for deciduous forest and 0.12 for coniferous forest and tropical forest. Almost all land-surface schemes modify the soil surface albedo according to soil moisture. This is done for instance by using the empirical relationship αs = αr +

θ − θr (αs − αr ) θs − θr

(4.5)

where αs and αr are prescribed albedos at saturation soil moisture, θs , and air-dry soil moisture θr , respectively, θ is the soil moisture in the top soil layer. Depending on soil colours, αs varies approximately between 0.13 to 0.26, while the corresponding values of αr are about twice as large.

4.3 Soil Moisture The simplest scheme for modelling the evolution of soil moisture is the singlelayer bucket scheme. The soil layer is considered to be a bucket with no drainage at its lower boundary and to act as a reservoir for precipitation until it is full. The excess water is treated as surface runoff and is not available for evapotranspiration (Fig. 4.4). The depth of the soil layer is commonly chosen as 1 m, based on the fact that soil moisture within this layer shows clear annual variations. Central to the bucket scheme are the concepts of field capacity, θf c , and wilting point, θwp , which reflect the hydraulic properties of different soils. Observations show that the rate of water flow in an unsaturated soil decreases once θ reaches a value close to θs . The soil moisture at which internal flow almost ceases is considered to be a physical property of the soil and is known as field capacity. The wilting point is defined as the soil

4.3 Soil Moisture Pr

Pr

Ev Ro

97

Ev Ro

h1

Soil water flux

h2

hb

(a) Bucket

(b) Force−Restore

Fig. 4.4. (a) A representation of the bucket scheme for soil-moisture simulation. (b) as for (a), but of the force-restore scheme. Pr , Ev and Ro are precipitation, evapotranspiration and runoff, respectively

moisture below which water extraction by plant roots presumably ceases and plants wilt. The available soil moisture, θa , is defined as θa = θ − θwp

(4.6)

The temporal variation of θa is determined by the soil water budget equation: P r − R o − Ev ∂θa = ∂t hB

(4.7)

where hB is bucket depth, Pr is precipitation and Ro is runoff. Runoff is calculated as ⎧ θwp ≤ θ < θf c ⎨0 (4.8) Ro = ⎩ P r − Ev θf c < θ ≤ θ s Evaporation occurs at its potential rate, Ep , when soil moisture is higher than a critical level of 0.75θf c . Otherwise, a moisture availability factor, β, is used to adjust the potential rate of evaporation, according to Ev = βEp

(4.9)

with β = min(1, θa /0.75θf c ). The bucket scheme is simple. In most circumstances, it produces reasonable estimates for soil moisture over a depth of hB (about 1 m). However, as far as wind erosion is concerned, we are mostly interested in the soil moisture of the very top layer, which is probably less than 10 mm deep. For this reason, a direct application of the bucket scheme is inadequate for wind-erosion modelling. One possibility is to divide the soil profile into different zones, each of which behaves as a single layer bucket. In this case, precipitation cascades from the upper to the lower zones, when the upper zones reach the field capacity. The zoned-bucket scheme is a considerable improvement over the single-layer bucket scheme. One other shortcoming of the bucket scheme

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is that it does not simulate well the diurnal variation of soil moisture in the top soil layer, which is also of critical importance to wind erosion. There is observational evidence that wind erosion is usually stronger during daytime than during night time. This is partly due to the convective nature of the atmospheric boundary layer and partly due to the lower soil moisture of the surface during daytime. The diurnal variation of soil moisture in the very top soil layer can be better simulated using the force-restore scheme, which is a two-layer scheme with a thin top layer of about 0.1 m and a bulk layer of about 1 m (Deardorff, 1978). The moisture variation in the top thin layer is forced by rapid changes of the upper boundary conditions, namely precipitation and evaporation, and is restored by moisture diffusion from the deep-soil reservoir. Most force-restore type schemes also include a uniform vegetation canopy. The scheme computes separately the energy balance for the canopy layer and the ground surface. The canopy layer partly shields the ground surface and intercepts some of the solar radiation and precipitation. The intercepted precipitation wets the canopy and evaporates into the atmosphere with a rate of potential evaporation. The equations for predicting soil moisture for vegetated ground are: Pr − Es − rEt − Ro θ2 − θ1 ∂θ1 = C1 + C2 ∂t h1 τ1 P r − Es − E t − R o ∂θ2 = ∂t h2

(4.10) (4.11)

where τ1 denotes a period of one day, θ1 is the volumetric water content of the thin top soil layer of depth h1 , the depth to which diurnal soil-moisture cycle extends, θ2 is the vertically-averaged volumetric water content over the bulk layer, h2 , below which the water flux is negligible, Es is evaporation, Pr is precipitation at the ground surface, Et is transpiration, r = h1 /h2 and C1 and C2 are the force and restore coefficients, respectively. Runoff occurs when θ1 exceeds saturation soil moisture, θs . The evaporation efficiency factor β is now related to θ1 by (4.12) β = min(1, θ1 /θf c ) Apart from the relatively simple treatment of soil moisture discussed above, there are land-surface schemes in which soil moisture is obtained by solving the Richards equation. The volumetric soil-water flux follows Darcy’s law and Gw can be expressed as Gw = −K

∂(ψ + z) ∂z

(4.13)

where K is the hydraulic conductivity and (ψ + z) is the hydraulic head with ψ being the pressure head and z the elevation head. In an unsaturated soil, ψ arises from the capillary suction. This suction can be quantified by ∆P which is the deficit between the pressure within the soil water and the atmospheric

4.3 Soil Moisture

99

pressure. This suction pressure deficit is related to ψ through ψ = ∆P/ρw g. In unsaturated soils, ∆P is negative and hence ψ is negative and is commonly measured in metres. In saturated soils ψ is positive. Substituting Equation (4.13) into Equation (4.2) gives the Richards equation   ∂ ∂ψ ∂θ =− K + 1 + sw (4.14) ∂t ∂z ∂z The Richards equation is highly nonlinear, since K and ψ depend nonlinearly on θ. For given values of Gw or ψ at the boundaries of the simulation domain, numerical solutions of the Richards equation can be achieved using the finiteelement and finite-difference techniques. For finite-difference solutions, the space domain (soil depth) is discretised to parallel layers as desired. The values of θ and ψ of each soil layer are updated at each time step and assigned to the center of each soil layer. The water fluxes are calculated at the interfaces of the model soil layers. A schematic representation of the multi-layer soil scheme is shown in Fig. 4.5. Due to the sharp gradients of ψ, especially near the surface, and nonlinearity of the hydraulic functions, a fine vertical discretisation is required. This in turn demands short time-steps, in order to avoid problems Ev

Tr

Pr Ro0

dz2

dz1

Gwo θ1, ψ1, K1, θ2, ψ2, K2,

Ro1 Gw1 Ro2

dzi

Gw2 θi, ψi, Ki,

Roi

dzi+1

Gwi θi+1, ψi+1, Ki+1,

Roi+1

dz m

Gwi+1 θm, ψm, Km,

Rom

Gm

Fig. 4.5. An illustration of the multi-layer configuration for the numerical simulation of soil moisture. The soil column is divided into a number of layers with the deeper ones having larger thicknesses. The state variables are defined at the middle of each layer, while the fluxes are defined at the boundaries. A similar configuration applies to the numerical simulation of soil temperature

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4 Land-Surface Modelling

of numerical instability. Numerical difficulties can arise sometimes, especially under vertical heterogeneous conditions, in connection with infiltration into initially-dry soils, or for coarse-textured soils, which are characterized by sharp wetting fronts. Solution of the Richards equation requires closure relationships between hydraulic conductivity, soil water potential and soil moisture. Different soilmoisture retention functions, i.e. ψ(θ), and the soil hydraulic conductivity functions, i.e. K(θ), have been proposed. A specific set of ψ(θ) and K(θ) functions is referred to as the soil hydraulic model. One of the widely-used soil hydraulic models is the BC model, proposed by Brooks and Corey (1964) based on the observations of a large number of consolidated rock cores. The soil-moisture retention function of the BC model is  −λ ψ θ − θr = ψ ≤ ψs (4.15) θs − θr ψs where λ is a dimensionless pore-size distribution index, and ψs is the saturation potential. The hydraulic conductivity function of the BC model is  θ − θ (3+2/λ) r (4.16) K(θ) = Ks θs − θr One simplification of the BC model has been suggested by Clapp and Hornberger (1978), in which the air-dry soil moisture, θr , is set to zero and the water retention function is replaced with a parabolic form for the near-saturation situation. A popular hydraulic model used in soil physics is the vG model due to van Genuchten (1980). The soil-moisture retention function of this model is given by m  1 (4.17) Θ= n 1 + (−αv ψ) where Θ is the relative soil water content, defined by Θ = (θ − θr )/(θs − θr ) and αv and m are adjustable parameters. The parameter n is related to m by n=

1 m−1

The hydraulic conductivity function of the vG model is 

m  2 K(θ) = Ks Θ1/2 1 − 1 − Θ1/m

(4.18)

or in terms of potential  K(ψ) = Ks

 −m 2 1 − (−αv ψ)n−1 1 + (−αv ψ)n  m/2 1 + (−αv ψ)n

(4.19)

4.5 Calculation of Surface Fluxes

101

4.4 Soil Temperature The evolution of soil moisture is related to the evolution of soil temperature. The latter obeys Equation (4.1). The soil heat flux Gt can be obtained from the gradient of soil temperature Gt = −κhs

∂T ∂z

(4.20)

where κhs denotes the thermal conductivity. Substituting (4.20) into (4.1) gives ∂T ∂2T = Dh 2 + sh (4.21) ∂t ∂z where Dh (= κChss ) is the soil thermal diffusivity. Neglecting the density and heat capacity of the air in the soil, we obtain Cs = ρs cs = (1 − θs )ρq cq + θρw cw

(4.22)

where subscripts q and w denote the soil solid component and water, respectively. The parameters, Cs , ρs , κhs and Dh , all depend on the soil type and soil moisture. κhs varies by several orders of magnitude in the field and has been determined empirically by McCumber and Pielke (1981). In land-surface schemes, Equation (4.21) is normally solved numerically subject to given initial and boundary conditions and using procedures similar to those employed on the soil-moisture equation. The soil heat flux at the upper boundary, Gt0 , can be estimated from the energy balance Equation (4.3) and, at the lower boundary, Gt can be assumed to be zero, if the depth of the soil layer is sufficiently large. In a manner similar to that employed for soil moisture, the force-restore method proposed by Bhumralkar (1975) and Blackadar (1976) can be applied to modelling soil temperature. Here, the temperature of a thin surface layer is forced by the sum of the surface energy fluxes and restored by heat diffusion from the deep soil. The prognostic equations for the temperatures in the surface and deep soil layers are Gt0 T1 − T2 ∂T1 = −C1 − C2 ∂t Cs h1 τ1 Gt0 ∂T2 =− ∂t Cs h2

(4.23) (4.24)

Again, C1 and C2 are force and restore coefficients, respectively.

4.5 Calculation of Surface Fluxes Surface sensible- and latent-heat fluxes must be calculated so that soilmoisture and temperature equations can be solved. The most popular method used in land-surface schemes for estimating these energy fluxes is the bulk

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4 Land-Surface Modelling

zr

ra rag

zd

rc rd

z0

s2

s1

Fig. 4.6. A resistance network used for the calculation of surface fluxes in a landsurface scheme. In the diagram, z0 , zd and zr are surface aerodynamic roughness length, zero displacement height and reference height, respectively; s1 and s2 are the proportions of bare soil surface and vegetated surface, rag , rd and ra are aerodynamic resistances to the fluxes from z0 to zr , z0 to zd and zd to zr , respectively, and rc is the bulk canopy resistance (From Irannejad, 1998)

aerodynamic method as illustrated in Fig. 4.6. For instance, over a bare soil surface the sensible-heat flux from the surface to a reference level, zr , is determined by the temperature difference between zr and z0 (here z0 is the aerodynamic roughness length of the bare soil surface) and the aerodynamic resistance, rag , which depends on the flow characteristics of the atmospheric surface layer. For more complex situations, resistances in serial can be employed. For evaporation from the soil surface under a canopy, for instance, the resistance rag becomes a superposition of rd and ra , with rd and ra being respectively the resistance for the transfer from the surface to the canopy top (being here the displacement height, zd ) and the resistance from the canopy top to the reference level. The added resistances are necessary to take into consideration the distinctly different flow properties within the canopy and above it. Using the bulk-aerodynamic formulation, the sensible-heat flux from the surface can be written as H = ρcp

T0 − Tr rag

(4.25)

4.5 Calculation of Surface Fluxes

103

with ρ being air density, cp the specific heat capacity of air at constant pressure and (4.26) rag = (Ch U )−1 where Ch is the bulk transfer coefficient for heat, U and Tr are wind speed and temperature at a reference height in the atmosphere (usually the lowest level of the atmospheric model or the observational height). For simplicity, we also assume that the bulk transfer coefficient for any other scalar, such as water vapour, is identical to Ch . Similar treatment can be implemented for calculating the heat flux from the soil surface under the canopy and that from the vegetation surface. The latent-heat flux is somewhat more complicated. The total evaporation from the land surface is the result of evaporation from the soil surface, evaporation from wet fraction of the canopy, and transpiration from its dry fraction. In a manner similar to that for sensible heat, the latent-heat flux from the soil surface can be calculated by using λ l Es = λ l ρ

q0 − q r rag

(4.27)

where Es is the evaporation rate, λl is the latent heat of vaporization (or sublimation) and qr is the specific humidity of the air at the reference height. We must now turn our attention to q0 , the specific humidity at the surface. One possible expression for q0 is q0 = rh q ∗ (T0 )

(4.28)

where q ∗ (T0 ) is the saturated specific humidity at the surface temperature T0 , and rh is the relative humidity of the air at the roughness height. If the surface soil is saturated soil, then rh = 1, q0 = q ∗ (T0 ) and evaporation takes place at its potential rate of Es = Ep = ρ

q ∗ (T0 ) − qr rag

(4.29)

It is appropriate to consider briefly the process of evaporation from a soil surface. When the soil dries, a dry layer forms on the top of the soil with its thickness depending on the evaporation rate and time. In this case, evaporation takes place at the interface between the dry soil layer and the wet soil layer below, known as the evaporation zone. In an unsaturated soil, therefore, q0 depends upon the distance of the drying front from the surface and the rate of vapour diffusion through the pores of the dry soil layer. Based on this, q0 can be calculated from q0 = q ∗ (Ts ) −

rs Es ρ

(4.30)

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4 Land-Surface Modelling

where q ∗ (Ts ) is the saturation specific humidity at the evaporation zone where the temperature is Ts , rs is the resistance to vapour diffusion from the evaporation zone to the surface. It is sufficiently accurate to assume that Ts = T0 . Rearranging Equation (4.27) for q0 , we obtain q0 = qr +

rag Es ρ

(4.31)

Equating (4.30) and (4.31), we can eliminate q0 in calculating soil evaporation by using q ∗ (T0 ) − qr (4.32) Es = ρ rag + rs In Equation (4.32), rs needs to be estimated. The latter equation has been used often to estimate the soil evaporation. Alternatively, Equation (4.27) can be used for the purpose with q0 being approximated by the so-called α and β methods (e.g. Mahfouf and Noilhan, 1991), i.e. ⎧ ∗ ⎨ βq (T0 ) + qr (1 − β), β method (4.33) q0 = ⎩ rh q ∗ (T0 ), α method It thus follows that Es can be estimated by ⎧ q ∗ (T )−q ⎨ ρβ r0ag r , Es = ⎩ rh q∗ (T0 )−qr ρ , rag

β method (4.34) α method

Obviously, both rh and β are functions of soil moisture in the very top soil layer. A very simple choice of β is, for instance, Equation (4.12). A summary of various rh and β expressions can be found in Mahfouf and Noilhan (1991). Transpiration is the flux of water vapour through the exposed plant leaves to the atmosphere. In a manner not unlike the unsaturated-soils case, the transpiration zone is not at the surface, but is inside the ‘valve-like structure’ of the leaves, namely the stomata. Transpiration involves the transfer of water vapour from plant stomata to the reference level through three sequential paths. First, water vapour is transferred by molecular diffusion from within the stomata to the leaf surface at zs , which is surrounded by a quasi-laminar boundary layer. Second, canopy turbulence carries water vapour from the leaf surface to the top of the canopy at zc . Finally, at the scale of the whole canopy, atmospheric turbulence carries the water vapour from zc to the reference height zr . The resistances for the three sequential transfer paths are the stomatal resistance, rst , canopy air resistance, rac and above canopy air resistance, ra . The transpiration rate through this sequential process can be calculated from q ∗ (Tc ) − qr (4.35) Et = ρ rst + rac + ra

4.5 Calculation of Surface Fluxes

105

where q ∗ (Tc ) is the saturation specific humidity at the leaf temperature Tc . As the canopy consists of many individual leaves, it is not practical to consider the resistance for each individual stomata and each individual leaf. Instead, it is much simpler to consider the collection of all leaves to be a big leaf and to use rc , the stomatal resistance of this ‘leaf’ (known also as the bulk stomatal resistance or simply the canopy resistance). The transpiration rate from the entire canopy can then be calculated using Equation (4.35), with rst being replaced with rc . We now briefly describe how the aerodynamic resistance, canopy air resistance and bulk stomatal resistance are estimated in land-surface schemes. In most schemes, the aerodynamic resistance is estimated using the Monin– Obukhov similarity theory. Using this theory, the bulk transfer coefficient Ch is given by κ2 Ch = (4.36) Ψm Ψh where Ψm and Ψh are the similarity functions already given in Chapter 3. It follows that the aerodynamic resistances, rag and ra , can be obtained using Equation (4.26). The calculation of canopy air resistance, rac , is more complicated, as the Monin–Obukhov similarity theory does not apply to flows within the canopy, and rac varies with canopy configurations. One possible method of estimating rac is to specify a profile of eddy diffusivity as shown in Fig. 8.17 and then compute rac by integration of the reciprocal of the eddy diffusivity. The canopy resistance is a quantity which is difficult to measure. In most land-surface schemes, the stomatal resistance of a single leaf, rst , is calculated and the bulk stomatal resistance, rc , is obtained using an Ohm’s-law analogy in which it is assumed that all leaves of the canopy operate in parallel. The common expression for rc is that rc =

rst La

(4.37)

where La is leaf-area index, defined as the one-sided surface of all leaves over a unit area of land surface. Different methods have been used to account for the canopy resistance, ranging from a constant value of rc for different vegetation types to those which calculate rc by a multiple-factorial equation consisting of a minimum resistance and functions of environmental factors which influence the transpiration process. Jarvis (1976) proposed a model to describe the stomatal resistance as a function of radiation, ambient air CO2 concentration, atmospheric vapour-pressure deficit, leaf temperature and leaf-water potential. The model parameters are derived using non-linear regression analysis, assuming that there was no synergistic interaction among the environmental variables. The proposed parameterisation of stomatal resistance is of the form rst = rst,min (F1 F2 F3 F4 )−1

(4.38)

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4 Land-Surface Modelling

where rst,min is the smallest value rst can reach. In Equation (4.38), F1 represents the effect of the photosynthetically active radiation, parameterised by F1 =

f + rst,min /rst,max 1+f

(4.39)

where rst,max is the maximum stomatal resistance, corresponding to a situation of closed stomata, and f is given by f=

1.1 Rs La Rsl

where Rs is solar radiation and Rsl is a limiting value of 30 Wm−2 for forest and of 100 Wm−2 for crops and grassland. The factor F2 measures the effect of water stress on the canopy resistance, as ⎧ θ > θf c ⎨ 1, wp (4.40) F2 = θθ−θ , θ ≤ θ ≤ θf c wp ⎩ f c −θwp 0, θ < θwp Here, θ takes the mean soil moisture in the root zone. F3 represents the effect of the water-vapour deficit, ∆q, in the air surrounding the leaves, which can be written as (4.41) F3 = 1 − 0.6∆q F4 is the factor describing the effect of the air temperature, Ta , on the surface resistance and can be parameterised as F4 = 1 − 0.0016(298 − Ta )2

(4.42)

4.6 Land-Surface Parameters A land-surface scheme requires a set of atmospheric variables as input data, including downward short-wave solar radiation, downward infrared atmospheric radiation, precipitation, air temperature, wind speed, surface pressure and specific humidity. The atmospheric forcing data can be obtained either from observations for given locations or, if the land-surface scheme is coupled with an atmospheric model, from the predictions of that model. The land-surface scheme also requires parameters which specify a range of properties of the land-surface. Numerous studies have shown that the performance of a land-surface scheme is sensitive to the choice of land-surface parameters (Ek and Cuenca, 1994). While different land-surface schemes may use different parameters, they can be divided roughly into the categories for surface vegetation, aerodynamics, radiation, soil hydraulics and thermal properties, as listed in Table 4.1.

4.6 Land-Surface Parameters

107

Table 4.1. Categories of parameters required by land-surface schemes Categories Vegetation

Aerodynamics Radiation properties Hydraulic properties

Thermal properties

Parameters Leaf-area index Fractional vegetation cover Canopy height Root distribution Surface & canopy roughness length Zero-displacement height Surface & canopy albedo for visible light & NIR light Thermal emissivity Field capacity, Wilting point Saturation potential, moisture & hydraulic conductivity Other model dependent parameters Soil-texture & thermal conductivity

It is a challenge to obtain reliable parameters for land-surface modelling with a high resolution over large areas. However, better parameters are gradually becoming available, and there is a sustained effort by the international scientific community to establish a global data set for land-surface schemes. Depending on the soil hydraulic model (e.g. the BC or vG model), the soil-hydraulic parameters have slightly different meanings. With the reported parameter values, the soil-hydraulic curves of various models are quite different. Both the BC and the vG model use five parameters to characterize the soil-hydraulic properties. The physical meanings of θs , θr and Ks in the two models are similar, but their values may differ for a given soil. This is partly due to the subtle differences in the definition of saturation in the two models. For the BC models, θs and Ks are defined at ψ = ψs which is smaller than zero, while for the vG model, they are defined at ψ = 0. ψs in the BC model and αv in the vG model can be loosely labelled as scaling parameters for soilwater potential. Parameters λ in the BC model, and m in the vG model can be loosely labelled as shape parameters. Rawls and Brakensiek (1982) have used 1,323 soil samples and derived a set of parameters for the BC model for the USDA soil-texture classes, as summarized in Table 4.2. Carsel and Parrish (1988) used 5,600 soil samples and derived the parameters for the vG model, as summarized in Table 4.3 for the USDA soil-texture classes. Despite the large data sets used to estimate the soil-hydraulic parameters, the difference between the three common parameters, θs , θr and Ks is considerable. Figure 4.7 shows that, with the reported parameter values, the soil hydraulic curves for various models are very different. The models are more consistent for the coarse- to medium-textured soils and less so for the finer soils. For some regions of the world, land-surface parameters for large areas, including soil types, vegetation and land use can be derived from GIS

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4 Land-Surface Modelling

Table 4.2. Soil-hydraulic parameters for the Brooks and Corey model for 11 USDA soil classes

Sand Loamy sand Sandy loam Loam Silty loam Sandy clay loam Clay loam Silty clay loam Sandy clay Silty clay Clay

θs (m m−3 ) 0.437 0.437 0.453 0.463 0.501 0.398 0.464 0.471 0.430 0.479 0.475

θr (m m−3 ) 0.020 0.035 0.041 0.027 0.015 0.068 0.075 0.040 0.109 0.056 0.090

Ks (m s−1 ) 5.83e-5 1.70e-5 7.20e-6 1.90e-6 3.67e-6 1.20e-6 6.39e-7 4.17e-7 3.33e-7 2.50e-7 1.67e-7

ψs (m) −0.1598 −0.2058 −0.3020 −0.4012 −0.5087 −0.5941 −0.5643 −0.7033 −0.7948 −0.7654 −0.8560

1/λ 1.44 1.81 2.64 3.97 4.27 3.13 4.13 5.65 4.48 6.67 6.06

Table 4.3. Soil-hydraulic parameters for the van Genuchten model for 11 USDA soil classes

Sand Loamy sand Sandy loam Loam Silty loam Sandy clay loam Silty clay loam Clay loam Sandy clay Silty clay Clay

θs (m m−3 ) 0.43 0.43 0.41 0.43 0.45 0.39 0.43 0.41 0.38 0.36 0.38

θr (m m−3 ) 0.045 0.057 0.065 0.078 0.067 0.1 0.089 0.095 0.1 0.07 0.068

Ks (ms−1 ) 8.25e-5 4.05e-5 1.23e-5 2.89e-6 1.25e-6 3.63e-6 1.97e-7 7.18e-7 3.37e-7 5.78e-8 5.56e-7

αv (m) 14.5 12.4 7.5 3.6 2.0 5.9 1.0 1.9 2.7 0.5 0.8

m 0.627 0.561 0.471 0.359 0.291 0.324 0.187 0.237 0.187 0.083 0.083

(Geographic Information System) databases, albeit with subjective interpretations. Such a GIS database is for example currently available for the Australian continent with a nominal spatial resolution of 5 km. In that database, the Australian soils are divided into 31 mapping units and for each unit there is a qualitative description of soil properties and associated land forms. Based on the description, the mapping units can be reclassified into the 11 USDA soil-texture classes, so that soil-hydraulic parameters can be assigned to them based on Tables 4.2 or 4.3. The vegetation data sets provide a range of parameters such as vegetation height, fractional vegetation cover, leaf-area index, minimum vegetation stomatal resistance, vegetation albedo and root distribution. GIS databases for vegetation also exist. Again, for the Australian continent, vegetation is divided into 35 classes according to height, density and number of canopy

4.6 Land-Surface Parameters 103

(a1)

10−6

101

10−14

10−3 103

(b1)

10−6

101

(b2)

10−10

10−1

10−14

10−3 103 (c1)

10−6 K (ms−1)

101 −ψ (m)

(a2)

10−10

10−1

10−1 10−3 103 (d1)

101

(c2)

10−10 10−14 10−6

(d2)

10−10

10−1 10−3 103

10−14 (e1)

101 10−1

109

BC VG

10−3 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Θ

10−6

(e2)

10−10 10−14

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Θ

Fig. 4.7. A comparison of BC and vG soil-water retention curves, −ψ(θ), and soilhydraulic conductivity functions, K(θ), for 5 USDA soil-texture classes: (a) sand; (b) sandy loam; (c) loam; (d) clay loam; and (e) clay. The values of −ψ and K are plotted against Θ = θ/θs

layers. From such a database, an estimate can be made of quantities such as vegetation height, fractional vegetation cover, vegetation albedo and the minimum vegetation stomatal resistance. Some vegetation parameters, such as La and the vegetation height, vary with time. One possibility of obtaining information on these parameters is to use remotely sensed data. The estimate of La for a particular time period can be drawn on remotely-sensed Normalized Difference Vegetation Index (NDVI) data derived from the Advanced Very High Radiometric Resolution satellite records of reflective radiation in the red region (0.55–0.68 µm) and the near infrared region (0.72–1.1 µm) of the electromagnetic spectrum. For certain vegetation types, the NDVI derived from satellite data can be calibrated against independent measurements of La . Aerodynamic parameters, such as surface roughness length and zerodisplacement height can be estimated from leaf-area index and vegetation height (e.g. Raupach, 1994).

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4 Land-Surface Modelling

Land-surface parameters have been prepared for many other regions of the world in a similar fashion to that outlined above. For example, Matthews (1983) produced a global archive of land cover and Wilson and HendersonSellers (1985) produced an archive for land-cover types and a global soil database for soil colour, texture and drainage characteristics. A global ecotype data set has been developed by Olson et al. (1983) in which vegetation types are classified based on carbon density or biomass. The US Geological Survey has derived a global distribution of major ecotypes. This dataset has been aggregated to grid resolution ranging from 0.1◦ for a limited area to those for GCMs by the Max-Planck-Institut f¨ ur Meteorologie (Hagemann et al. 1999). Major international projects, such as the First ISLSCP Field Experiment, have also produced data sets for land-surface modelling. The current parameter data sets have a spatial resolution of around 1◦ × 1◦ . For many selected regions, such as Australia, China, Europe and North America, various groups are compiling land-surface data with better quality and resolution. These data sets mostly have much high resolution down to several kilometres. With the development of remote-sensing techniques, more data are being collected for land-surface modelling. In the near future, data sets of better quality, consistency and resolution will become available. Much of these data are also required for wind-erosion modelling, as will be discussed in Chapter 9.

4.7 Examples of Land-Surface Simulation Despite the complexities and uncertainties involved in land-surface schemes, the performance of many of the latter is encouraging. Here, we present two examples of soil-moisture simulation to illustrate the performance of a landsurface scheme (Irannejad and Shao, 1998) and the usefulness of the model output for wind-erosion studies. The first example is for soil moisture modelling at a given location with pre-specified atmospheric data and land-surface parameters. The atmospheric forcing data and land-surface parameters are obtained from the HAPEXMOBILHY (Hydrologic-Atmospheric Pilot Experiment and Modelisation de Bilan Hydrique) carried out in southern France (Andre et al. 1986). Figure 4.8 compares the simulated annual cycle of soil moisture with the observations for 5 soil layers. The depths of the soil layers are 0.05, 0.15, 0.3, 0.5 and 0.6 m, respectively, and they are centred at 0.025, 0.1, 0.35, 0.75 and 1.3 m. The figure shows that the land-surface scheme correctly predicts the annual cycle of soil moisture. Thus, frequent rainfall and low evaporation keep soil moisture close to the field capacity for the first four months of the year; as precipitation decreases and available energy for evaporation increases, soil water begins to deplete at the beginning of the crop-growing season (early May) and reaches a minimum during August to October. The overall agreement between the simulated and measured soil moistures is excellent. In particular, we note

4.7 Examples of Land-Surface Simulation 0.5

111

(a)

Volumetric Water Content, m3 m−3

0.4 0.3 0.2 0.1 0.0 0.5 0.4 0.3 0.2 0.1 0.0

(b)

0.5 0.4 0.3 0.2 0.1 0.0

(c)

0.5 0.4 0.3 0.2 0.1 0.0 0.5 0.4 0.3 0.2 0.1 0.0

(d)

(f)

Observed Simulated 0

60

120

180

240

300

360

Time, day

Fig. 4.8. Comparison of simulated annual cycle of volumetric water content with measurements in 5 soil layers. The thicknesses of the soil layers are 0.05, 0.15, 0.3, 0.5 and 0.6 m, respectively (From Irannejad and Shao, 1998)

that the simulation fairly accurately reproduces the fluctuations of soil moisture in the first soil layer, which is of critical importance for wind-erosion modelling. In Fig. 4.9, simulated and observed diurnal cycles of surface net radiation and the non-radiative heat fluxes for the Intensive Observational Period of HAPEX-MOBILHY are compared. As can be seen, the land-surface scheme has the capacity to model well the components of the surface energy balance. For wind-erosion modelling on regional to continental scales, soil-moisture simulation over large areas with a high spatial resolution is required. For this purpose, a land-surface scheme can be run coupled with an atmospheric prediction model. Most regional and global atmospheric models now have a land-surface model and can simulate soil moisture to some degree.

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4 Land-Surface Modelling

Rnet, W m−2

800

(a)

600 400 200 0 −200

LH, W m−2

800 600

(b)

400 200 0 −200

SH, W m−2

300

(c)

200 100 0 −100 150

GH, W m−2

(d)

Obs. Sim.

100 50 0 −50 148

153

158

163

168

173

178

183

Time, day

Fig. 4.9. Comparison of the simulated diurnal cycles of net radiation (Rnet), latentheat flux (LH), sensible-heat flux (SH) and ground-heat flux (GH) with observations from HAPEX for the Intensive Observational Period from 28 May to 30 June 1987 (From Irannejad and Shao, 1998)

4.8 Treatment of Heterogeneous Surfaces Most land-surface schemes used in atmospheric models treat a model grid cell as a homogeneous area, with the typical size of such an area being about 100 km for GCMs and 10 km for limited-area models. However, the land surface is rarely homogeneous on these scales and in fact, for any resolution of the atmospheric model, the heterogeneity of the land surface can never be sufficiently well represented (i.e. to the fineness required). This insufficient resolution of the surface has two effects; namely, the aggregation effect (Giorgi and Avissar, 1997) and the dynamical effect (Chen and Avissar, 1994; Avissar and Schmidt, 1998). Both effects are important in atmospheric and landsurface modelling. The aggregation effect occurs because of the non-linear nature of the landsurface processes. The surface fluxes on the grid scale are not simply related to mean atmospheric properties over the grid and the ‘averaged’ land-surface

4.8 Treatment of Heterogeneous Surfaces

113

parameters, and therefore, the surface fluxes on the grid scale can be wrong, if the heterogeneity of the land-surface is poorly represented. Wetzel and Chang (1987) have demonstrated that the spatial soil-moisture variability and the strongly nonlinear connection between evapotranspiration and soil moisture results in a relationship between the regional evapotranspiration and areaaveraged soil moisture which is profoundly different to that which applies to a point. In these situations, the grid-averaged surface fluxes can be substantially different from those calculated using the grid-averaged surface properties. The dynamic effect occurs because land-surface heterogeneity generates subgrid motions which are not resolved by the numerical grid. In general, the subgrid closure for atmospheric fluxes is to parameterise the subgrid fluxes through the gradient of the grid variables. This approach contains no information about the land surface and thus it inevitably has inadequacies. Therefore, for a given resolution of the atmospheric model, we must consider how to represent the unresolved surface properties and their possible impact on both the calculation of the surface fluxes on the grid scale and the effect of subgrid motion generated by subgrid surface contrasts. Different strategies exist for the representation of subgrid land-surface heterogeneity. Averaging Surface Properties If a grid cell is treated as homogeneous, parameters used for land-surface modelling are effective parameters defined either as the most frequently occurring parameter or as a weighted average of the parameters for all surface types within the grid cell. For some parameters, averaging methods which account for local nonlinear processes are available (Lhomme, 1992; Mahrt et al. 1992). For instance, the effective roughness length can be calculated based on the concept of blending height (Wierenga, 1986; Mason, 1988). The advantage of using effective grid-averaged parameters is the reduction of computational costs, but the method nevertheless compromises the adequacy with which the heterogeneity of the surface is represented. PDF Representation This representation of a heterogeneous land surface involves using a probability density function (PDF) to define the spatial distribution of the parameters and integrating relevant processes over that PDF. Johnson et al. (1993) and Li and Avissar (1994) applied this method to study the effect of heterogeneity in stomatal resistance, leaf-area index, surface roughness, albedo and soil wetness on land-surface modelling, and showed that latent- and sensible-heat fluxes are sensitive to the spatial variability of these parameters. Although the continuous PDF captures a wide range of the heterogeneity scales which occur within a grid cell, its application to complicated landsurface schemes encounters a number of difficulties. In practical computation, the PDF needs to be applied over a number of intervals, for each of which the land-surface scheme must be run. Using a small number of intervals tends to

114

4 Land-Surface Modelling

distort the shape of the PDF, while using a large number of intervals is computationally expensive. A land-surface scheme usually incorporates a large number of processes with many input parameters. If the heterogeneity of all processes is to be considered, a multi-dimensional joint PDF needs to be introduced. Suppose that there are N parameters and that the joint PDF can be expressed as a product of N independent PDFs, each divided into I intervals. This operation would require solving the land-surface scheme I N times. This is computationally equivalent to solving the model over a very fine spatial net and changing the geographical spacing to a parameter spacing (Giorgi and Avissar, 1997). To construct a PDF for a single parameter, let alone the range of parameters, requires extensive information about the mean, variance and usually skewness and kurtosis of the distribution of the parameter over the region. Such information is rarely available. Furthermore, the complicated interactions of many processes involved in the natural environment make such a function strongly space- and time-dependent. Explicit Subgrid and Mosaic Methods An alternative is to represent the subgrid-scale heterogeneity in a grid cell by a number of homogeneous subgrids (Avissar and Pielke, 1989). Each subgrid is assumed to have fluxes directly to the atmosphere and independently of other subgrids. Calculations are performed separately for each subgrid and weighted according to the relative surface area to each subgrid in order to provide the grid-averaged state variables and fluxes. This is the explicit subgrid approach. The latter approach can be simplified to the mosaic approach, in which all the areas having the same properties are combined into a tile. Figure 4.10 illustrates the mosaic approach in comparison with the explicit subgrid method for an atmospheric model grid box. The mosaic approach does (a)

5 2 5

1

5 3

6 1

6 8

4 9

4

(b)

5

8

2

6

5

7

1

5 3

6 1

4 9

6 4 1

9

5 2

7 3

6

Introduction of the Subgrid

5

8 2

4 8

7

Rearrangement of Patches

5

6 8

1

6 1

6

5 3

8 4

9

4 8

6 7

Fig. 4.10. An illustration of (a) the mosaic method in contrast to (b) the explicit subgrid method. The different land-surface types are numerated

4.8 Treatment of Heterogeneous Surfaces

115

not consider the spatial variation of atmospheric data within the grid, while the explicit subgrid method retains the possibility of allowing the atmospheric forcing data to vary from subgrid to subgrid. Studies on the impact of the inclusion of subgrid-scale heterogeneity of surface properties and atmospheric data in a land-surface scheme have reached very different conclusions. M¨ older et al. (1996) compared the effect of the mosaic and the explicit-subgrid approaches upon the predicted hydrological variables. They found significantly different partitioning of atmospheric radiation and surface-moisture forcing, using the two different strategies. Their results showed a reduction in the surface evaporation when the area-weighted atmospheric forcing data are used. Ghan et al. (1997) found that subgrid-scale variations in precipitation had the largest impact on the regional evaporation. The variation in evaporation due to neglecting the spatial heterogeneity in precipitation was found to be 15%, while the variations due to neglecting the spatial heterogeneity of vegetation and soil were of the order of 4% and 2%, respectively. On the other hand, Mahrt and Sun (1995) used data from three field experiments and concluded that assuming spatially-constant atmospheric forcing and spatially-varying surface properties provided a close approximation to the area composite fluxes. Spatial Variation of Atmospheric Data Sufficiently large patches of surface heterogeneity lead to the development of local (subgrid) circulations. Studies have been carried out to examine the dynamical effects of surface heterogeneity on mesoscale atmospheric circulations, which are generally unresolved in global circulation models. Studies of the dynamical effects of surface heterogeneity can be found in, for instance, Mahfouf et al. (1987), Segal et al. (1988), Giorgi (1989), Chen and Avissar (1994) and Seth and Giorgi (1996). The dynamic effect of surface heterogeneity leads to substantial spatial variations in atmospheric motion and turbulence characteristics which cannot be represented by the resolved atmospheric motion. If the grid-averaged fluxes are computed from the bulk aerodynamic formulation using grid-averages of atmospheric data and other variables, as is normally done in atmospheric models, the inaccuracy can be substantial.

5 Basic Aspects of Wind Erosion

5.1 Soil-Particle Characteristics The processes of particle entrainment, transport and deposition involve a set of particle-to-flow, particle-to-surface and particle-to-particle interactions. The physical properties of individual particles, such as shape, size and density, play an important role in these interactions. These properties vary greatly and are difficult to measure precisely in practice, but they must be adequately described in wind-erosion models. The shape of a grain includes all aspects of its external morphology, such as its general form (sphericity), its roundness (sharpness of edges and corners) and its surface roughness. Observations show that the shape of soil particles is highly irregular (Gillette and Walker, 1977; Pye, 1994), ranging from spheres to plates in their gross form, from very-angular to well-rounded in their roundness and from rough to smooth in their surface texture. In practice, the size of a particle can be determined by directly measuring its external caliper dimensions. Several techniques are described in Chapter 11, among which sieving is the most-widely used direct-measuring method. A series of progressively finer square-mesh sieves is used to separate particles, principally on the basis of their intermediate axial diameters (Kennedy et al. 1985). However, it is readily understood that the idea of particle size is closely related to particle shape and the external caliper dimensions do not accurately represent the size of a particle unless its shape is well-defined. To overcome the difficulties in precisely measuring particle shape and size, we introduce the concept of equivalent particle size. The aerodynamic behaviour of particles is reasonably well-understood only for spherical particles. It is therefore necessary in theory, and also convenient, to treat soil particles of different shapes as spheres, so that their aerodynamic and/or optical behaviours can be described using mathematical expressions. Nearly all theoretical models for particle entrainment and motion are based on the assumption that the particles are spheres. The equivalent particle size is measured by the diameter of a sphere of which certain aerodynamic or optical Y. Shao, Physics and Modelling of Wind Erosion, c Springer Science+Business Media B.V. 2008


117

118

5 Basic Aspects of Wind Erosion

Table 5.1. Various equivalent particle sizes for particle P, defined by comparing a given property with that of a sphere S (After Allen, 1981) Symbol Name

Definition

Formula

dm

mass d

d of S, same density & mass as P

m = ρp

dv ds dsv

volume d surface d surface-volume d

d of S, same volume as P d of S, same surface as P d of S, same external surface to volume ratio as P

V = 6v s = πd2s dsv = d3v /d2s

dd

drag d

df

da dp dc dA dF

dM

d of S, same resistance to motion as P in a fluid with same viscosity & velocity terminal-velocity d d of S, same density & terminal velocity as P in a fluid of same density & viscosity projected-area d d of S, same projected area as P resting in a stable position projected-area d d of S, same projected area as P in random orientation perimeter d d of S, same perimeter as the projected outline of P sieve d width of the minimum square aperture through which P will pass Feret’s d mean distance between pairs of parallel tangents to projected outline of P Martin’s d mean chord length of projected outline of P

πd3 m 6

πd3

FD =

CD πd2 ρv 2 d 8

A = π4 d2a Mean of all possible da L = πdc

properties are identical to those of the particle under consideration. For example, for a given particle mass, m, the particle mass equivalent diameter, dm , is calculated from 1/3  6m (5.1) dm = πρp where ρp is the particle density. There are many other definitions for equivalent particle size, as summarized in Table 5.1. Methods commonly used in sedimentology for measuring equivalent particle size include electro-optical and settling-velocity methods. For example, in the laser-diffraction method, the volume equivalent diameter is estimated on the basis of the optical properties, while in the settling-tube method (Cui et al. 1983; Malcolm and Raupach, 1991), the terminal-velocity equivalent diameter is estimated. Whether the definitions for equivalent particle diameter, as listed in Table 5.1, are adequate depends on the problems being studied. The usefulness of every definition may be limited to certain types of problems. For example, the terminal-velocity equivalent diameter is advantageous if we are interested in the transport and deposition of particles in the atmosphere, as

5.1 Soil-Particle Characteristics

119

the particle terminal velocity is the key aerodynamic property involved in these processes. But this definition of particle diameter may not be suitable, if our main concern is how dust particles suspended in air might influence the radiation balance of the atmosphere. As far as wind-erosion modelling is concerned, the most useful equivalent particle diameters are mass equivalent diameter, drag equivalent diameter, terminal-velocity equivalent diameter, projected-area equivalent diameter, and sieve diameter. From the view point of the dynamic processes involved in wind erosion, sieve diameter is not the best representation of particle size, but it is widely used because it is relatively easy to measure. The size of soil particles, measured directly or indirectly, ranges from more than 2 m to less than 0.1 µm. Soil particles are crudely divided into four categories referred to as gravel (2,000 µm < d ≤ 2 m), sand (63 < d ≤ 2,000 µm), silt (4 < d ≤ 63 µm) and clay (d < 4 µm). Silt and clay particles are commonly called dust. However, as can be seen from Fig. 5.1, there exist at least three similar but somewhat different standards for particle-size classification. A more detailed classification used in sedimentology is the UddenWentworth grade scale (Udden, 1914; Wentworth, 1922), or a modified version of it (Friedman and Sanders, 1978). In this classification, particles are divided into 20 successive size classes between 2 µm and 2,048 mm, increasing by a factor of 2 from class to class (Table 5.2). To facilitate graphical presentation

Soil Particle Size a. Soil Science Society of America and U.S. Department of Agriculture Soil Particle Size Scale

Clay

Sand

Silt 0.002

Clay

medium

v.fine fine

0.05 0.1 0.25

coarse

0.5

Gravel

v.coarse

2 mm

1

76.2

Particle size relative to a sand grain of 0.15 mm in diameter

Silt 0.15 mm

Sand b. MIT and British Standards Institute

Sand

Silt

Clay

fine

medium

0.002 0.006 0.02

Clay

coarse

0.2

Gravel

coarse

0.6

0.02

2 mm

0.2

Suspension

Gravel

coarse

fine

Stones

2 mm c. International Society of Soil Science

Sand

Silt 0.002

medium

fine

0.06

Saltation

Creep

Long-range transport 0.02mm

0.07mm

0.5mm

Short-range

Fig. 5.1. A comparison of the different systems for particle-size definition (M. Mikami, with acknowledgement)

120

5 Basic Aspects of Wind Erosion

Table 5.2. The Udden–Wentworth particle-size scale commonly used in sedimentology, with class terminology modifications proposed by Friedman and Sanders (1978) (Modified from Pye 1994) Size (mm)

φ

2,048

−11

1,024

−10

512

−9

256

−8

128

−7

64

−6

32

−5

16

−4

8

−3

4

−2

2

−1

1

0

0.5

1

0.25

2

0.125

3

0.063

4

0.031

5

0.016

6

0.008

7

0.004

8

0.002

9

Size class terminology of Wentworth (1922)

Size class terminology of Friedman and Sanders (1978) Very large boulders

Common name

Large boulders Medium boulders Cobbles Small boulders Large cobbles Small cobbles

Gravels

Very coarse pebbles Coarse pebbles Pebbles Medium pebbles Fine pebbles Granules

Very fine pebbles

Very coarse sand

Very coarse sand

Coarse sand

Coarse sand

Medium sand

Medium sand

Fine sand

Fine sand

Very fine sand

Very fine sand

Sand

Very coarse silt Coarse silt Silt

Medium silt

Silt

Fine silt Very fine silt Clay

Clay Clay

5.1 Soil-Particle Characteristics

121

and statistical manipulation, the grade-scale boundaries are logarithmically transformed into φ, using the expression   d (5.2) φ = − log2 d0 with d0 being a normalisation factor, conventionally set to 1 mm. With the recent advances in particle-size analysis techniques, a much higher resolution of particle size can be achieved. For instance, a Coulter Multi-Sizer is capable of identifying 256 size classes. With these new techniques, particle-size measurements can be carried out over smaller φ intervals. The particle-size characteristics of a given soil are best described using the particle-size distribution function. It is first appropriate to give a precise definition of the latter function. Suppose the mass fraction (probability) of particles with a mass smaller than m is P (m). Then the corresponding mass (m) (here δ is used to represent probability density function is p(m) = δPδm differentiation, in order to avoid confusion with particle size d). The mass fraction of particles in a small mass interval [m, m + δm] is p(m)δm or in terms of particle size p(d)δd. Assuming soil particles are spherical, we have m = ρp πd3 /6 and it follows that p(d) =

ρp π 2 d p(m) 2

The quantity p(d), the particle-size distribution density function is commonly used to specify the particle-size characteristics of a soil. We note that p(d) has the dimensions of d−1 , while the particle-size distribution function P (d) is dimensionless, as P (d) is the integral of p(d), i.e.,

d

p(d)δd

P (d) =

(5.3)

0

The mass fraction (or percentage, when multiplied by a factor of 100) for particles in the particle-size range between da and db is now given by db p(d)δd (5.4) P (db ) − P (da ) = da

For instance, the dust mass fraction of a soil sample can be estimated from the above expression by setting da = 0 and db = 63 µm, and the sand mass fraction can be estimated by setting da = 63 µm and db = 2,000 µm. In practical particle-size analyses, we often determine P (d) first and then calculate p(d) via the relationship δP (m) p(d) = δd Soils with different textures can be distinguished by their different particlesize distributions. The soil texture classification used by the U.S. Department

122

5 Basic Aspects of Wind Erosion 100

0 10

90

20

Cla

40

Clay

nta rce Pe 30

60

t

Sandy Clay

40

50

Sil

Silty Clay

of

50

ge

ge

60

nta

of

30

70

tce Pe

y

80

Clay Loam

Sand Clay

Silty Clay Loam

70

Loam

20

80 Loam

10 0 100

Lo

Sand

90

yS

Silt

an

d

80

90

Silty Loam

Sandy Loam

am

70

60

50

40

30

20

10

100 0

Percentage of Sand

Fig. 5.2. Types of soil texture according to the content percentages of sand, silt and clay, as classified by the United States Department of Agriculture

of Agriculture is as shown in Fig. 5.2. In this classification, soils are divided into 12 classes according to the percentages of sand, silt and clay contained in the soil. This classification is now widely used in studies of wind erosion (e.g. Gillette, 1988), surface soil hydrology and atmospheric and land-surface modelling (Chapter 4). A considerable amount of data for particle-size distribution has been collected throughout the world as reported in Gillette et al. (1982), Gomes et al. (1990) and Leys and McTainsh (1996). Figure 5.3 shows examples of particle-size distributions for sand, loam, silty loam and clay soils. The samples are taken in Australia and analysed in the laboratory using the minimally-dispersed and fully-dispersed techniques (McTainsh et al. 1997a). Let us first consider the minimally-dispersed particle size distributions. The sandy soil has a particle-size mode around 200 µm and a high percentage of particles with size between 100 and 300 µm. In contrast, the particular silty-clay soil has a particle-size mode at around 100 µm and has a high percentage of fine particles with size smaller than 10 µm. The situations with the loamy soils are between sandy and silty-clay soils. They also have a particle-size mode around 200 µm, but this is much less prominent than for sandy soil, and they have a proportion of fine particles which is greater than in sandy soil but less than in silty-clay soil. In natural soils, silt and clay particles may exist as individual grains, but more often they exist in association with other particles. In sandy soils, they may be attached to sand grains as individuals or as thin layers coated around them. In soils with a high clay content, dust particles normally exist as soil aggregates. During a minor wind-erosion event, soil aggregates and grains

5.1 Soil-Particle Characteristics

123

(b) Loam

(a) Sand 1.0

Minimally dispersed Fully dispersed

d*p(d)

100

−1

10

0.5 10−2 100

101

102

103

0.0 100

(c) Silty Loam

(d) Silty Clay

1.0

d*p(d)

10−1 10−2 0 10

101

102

103

0.5

0.0

1

10

100

d (µm)

1000 1

10

100

1000

d (µm)

Fig. 5.3. Particle-size distributions plotted in d ∗ p(d) against d for (a) a sandy soil (the insert is the corresponding log-log plot); (b) a loamy soil; (c) a silty-loamy soil and (d) a silty-clay soil. The soil samples are analysed using the minimally- and fully-dispersed techniques (Data from G. H. McTainsh, with acknowledgment)

coated with dust behave in a similar fashion to sand particles. However, as wind erosion intensifies, these aggregates and coats may break up in varying degrees, releasing dust into the air. Therefore, in studying the particlesize characteristics of a particular soil, we distinguish two types of particle-size distributions. The first one is the in-situ particle-size distribution, pm (d), obtained using methods which cause as little as possible undesirable disturbance to the soil samples being analysed. The in-situ particle-size distribution is difficult to obtain, but it can be approximated using the minimally-dispersed particle-size distributions. The second one is the fully-disturbed particle-size distribution, pf (d), which mimics the particle-size distribution during strong wind-erosion events. This particle-size distribution is also difficult to obtain, but can be approximated using the fully-dispersed particle-size distribution which can be measured in laboratory. Hence, the fully-dispersed particle-size analysis provides information for the upper limit of dust emission. As will be described in Chapters 6 and 7, both pm (d) and pf (d) are required for the computation of sediment transport and dust-emission rate. In Fig. 5.3, the particle-size distributions obtained using both the minimally-dispersed and

124

5 Basic Aspects of Wind Erosion

fully-dispersed techniques are shown for four soils samples. As can be seen, the fully-dispersed particle-size distributions can be profoundly different from the minimally-dispersed particle-size distributions, with the amount of fine particles in the fully dispersed analyses being larger than in the minimallydispersed ones. For the silty-clay soil sample (Fig. 5.3d) in particular, a large proportion of particles around 100 µm observed in the minimally-dispersed analysis are aggregates of fine particles. For the sandy soil (Fig. 5.3a, sample collected in the Simpson Desert of Australia), the differences between pm (d) and pf (d) are much smaller. This implies that the sand-sized grains of this soil are indeed sand grains rather than clay aggregates. Hence, the sandy soil has a low content of fine particles and a low capacity for dust emission, as expected.

5.2 Forces on an Airborne Particle The motion of an airborne particle is influenced by several forces, including the gravity force, Fg , aerodynamic drag, Fd , aerodynamic lift, Fl , the Magnus force due to particle rotation, Fm , and the electric force FE . The particle-toair density ratio is defined to be σp = ρp /ρ, where the particle density, ρp , is around 2,600 kg m−3 while the air density, ρ, is approximately 1.2 kg m−3 . Since σp is of order 103 , the action of buoyancy upon an airborne soil particle is negligible and hence, Fg is simply the particle weight. The vertical only component of Fg is −mg with g being the acceleration of gravity. The physical mechanism which leads to aerodynamic drag is as illustrated in Fig. 5.4a. If the particle moves relative to the surrounding fluid, a force in the opposite direction of that relative velocity is exerted by the fluid on the particle. This force is known as the drag that arises from the pressure differences between the frontal region and the wake region of the particle and from the transfer of momentum from the fluid to the particle through molecular motion, namely the viscous effect. There is an equal and opposite force exerted by the particle on the fluid. The force exerted by the flow on the particle is equal to the integral of total stress (or momentum flux) over the surface of the particle. The ith component of the force per unit area on a surface with a unit direction vector n is fi = [−pδij + σij ]nj = −pni + σij nj

(5.5)

where p is pressure, δij is the well-known Kronecker second-order tensor and σij is the viscous stress tensor. The ith component of the drag, Fdi over the surface, S, is given by σij nj dS (5.6) Fdi = − pni dS + S

S

The first term on the right hand side of Equation (5.6) represents the transfer of momentum to the particle by the pressure forces and is independent of the

5.2 Forces on an Airborne Particle

125

(a)

+ + +

Fd

-

(b) --

- F1

+

+ +

+

(c) -A- - Fm

+ B+ + +

Fig. 5.4. An illustration of drag, lift and Magnus forces acting on a spherical particle. (a) Aerodynamic drag due to the viscous effect, flow separation and turbulence in the wake region of the particle. Higher fluid pressure on the sphere is indicated positive and lower pressure negative. (b) Aerodynamic lift due to the Bernoulli effect on a spherical particle in a shear flow. The pressure is higher on the lower side of the particle where the fluid velocity is smaller, while the pressure is lower on the upper side, where the fluid velocity is higher. (c) Magnus force due to a combination of the Bernoulli effect and the viscous effect on a spinning particle. On the upper side of the particle, where the particle spins in the same direction as the fluid motion, the fluid velocity is relatively higher and the pressure is relatively lower than on the lower side of the particle, where the particle spins in the opposite direction to the fluid (Modified from Allen, 1994)

viscosity of the fluid. The second term represents the frictional drag associated with the viscosity of the fluid. The expression for aerodynamic drag given in Equation (5.6) is not useful for practical purposes, as distributions of p and σij over the particle surface are difficulty to measure. Instead, the aerodynamic drag force is often expressed in terms of particle-to-fluid relative velocity, ur , by

126

5 Basic Aspects of Wind Erosion

1 Fd = − Cd ρAur Ur 2

(5.7)

where Cd is the aerodynamic drag coefficient, A is the particle cross-section in the ur direction (equal to πd2 /4 for spherical particles) and Ur is the magnitude of ur . If upi and ui are respectively the ith components of the particle velocity and the velocity of the fluid surrounding that particle, then uri = upi − ui is the ith component of ur , and Ur is given by Ur = (u2r1 + u2r2 + u2r3 )1/2

(5.8)

The magnitude of the aerodynamic drag depends critically on the flow pattern around the particle. This is reflected in the fact that Cd is a function of the particle Reynolds number, Rep = Ur d/ν. The Cd (Rep ) relationship has been investigated in numerous experimental studies and is reasonably well understood, as shown in Fig. 5.5. It has the following interpretation for different regimes of Rep : •

For Rep  1, called the Stokes region, the fluid motion in the vicinity of the particle is primarily influenced by viscous forces and fluid inertia is 104

Rep = 10

,C

es

ok

St

102

d

Rep = 103

= ep

/R

Cd

24

Rep = 106 100

Re p 5 is considered as one of pure saltation.

5.5 Threshold Friction Velocity for Sand Particles Soil particles resting on the surface under the influence of an air stream experience several forces, including the aerodynamic drag, Fd , the aerodynamic lift, Fl , the gravity force, Fg , and the inter-particle cohesive force, Fi (Fig. 5.11). The driving forces for the lift off of sand-sized particles are Fd and Fl , which

5.5 Threshold Friction Velocity for Sand Particles

135

are related to the wind shear near the surface and hence are functions of the friction velocity, u∗ . The threshold friction velocity, u∗t , is the minimum friction velocity required for the aerodynamic forces to overcome the retarding forces, namely Fg and Fi , and to initialise the movement of soil particles. There are well established techniques for determining u∗ by measuring either the profile of the mean wind or measuring directly the momentum flux in the atmospheric surface layer. In wind-tunnel and field studies, u∗t is commonly set to the value of u∗ at which a small percentage of soil particles start to move. Inevitably, the practical estimate of u∗t involves a certain degree of subjectivity in deciding what is a small percentage. Threshold friction velocity is essentially a property of the soil surface, rather than that of a soil particle. It describes the capacity of the surface to resist wind erosion and is affected by a range of factors as will be discussed in later chapters. However, under ideal conditions, u∗t can be expressed as a function of only particle size. To establish the u∗t (d) relationship under ideal conditions is important as it defines the lower limit of the threshold friction velocity in reality. Several theories for u∗t (d) have been derived for soils with uniform and spherical particles spread loosely over a dry and unsheltered surface. The theories developed by Bagnold (1941), Greeley and Iversen (1985) and the improvements due to Shao and Lu (2000) and McKenna Neuman (2003) are described below. 5.5.1 The Bagnold Scheme For a particle of size d, u∗t (d) is determined by the balance of Fd , Fl , Fg and Fi , as shown in Fig. 5.11. At the instant of particle motion the combined retarding effect of Fg and Fi will be overcome by the combined lifting effect of Fd and Fl . The particle will tend to pivot about point P in a downstream direction. The balance of forces at the instant of particle lift-off can be obtained by the summation of moments about the pivot point P , rd Fd + rl (Fl − Fg ) + rm Fm − ri Fi = 0

(5.25)

where rd , rl and ri are moment arm lengths and Fd ≡| Fd |, etc. In the above equation, the moment of the particle is expressed as rm Fm . In general, the moment arm lengths depend complicatedly on the arrangements of particles and are difficult to determine. However, it is plausible to assume that they are all proportional to the particle size, namely, rd = ad d, rl = al d, rm = am d and ri = ai d with ad , al , am and ai being dimensionless coefficients. Equation (5.25) thus becomes ad Fd + al (Fl − Fg ) + am Fm − ai Fi = 0

(5.26)

A simple theory for u∗t (d) can be derived by considering only the balance between the aerodynamic drag and the gravity force

136

5 Basic Aspects of Wind Erosion Fl

Air flow Fd

O rd P Fi

rl ri

Fg

Fig. 5.11. Forces acting on a particle resting on the surface under the influence of an air stream, including the aerodynamic drag, Fd , the aerodynamic lift, Fl , the gravity force, Fg and the cohesive force, Fi ; rd , rl and ri are moment arm lengths associated with the forces. O is the centre of gravity of the particle and P the pivot point for particle motion. If the particle rotates, the moment of the particle is represented as rm Fm in the text to facilitate discussions

ad Fd − al Fg = 0

(5.27)

This consideration is justified for sand-sized particles, for which the other three forces are comparatively small. The magnitude of the drag force, Fd , on a particle protruding into the air stream can be written as Fd =

1 Cds ρAs Us2 2

(5.28)

in analogy to Equation (5.7) for the drag force on a particle in a free air stream. Here, Cds is the aerodynamic drag coefficient for the particle attached to the surface, As is the exposed particle cross section perpendicular to the flow and Us is the mean flow speed at a reference height comparable to the particle diameter. There are difficulties in implementing Equation (5.28) in practice for two reasons: (a) Cds , although analogous to Cd , is not well understood, but is known to depend on the choice of the reference level; and (b) the reference flow speed, Us , is not well defined in the flow very close to the surface, which has a strong shear. A more pragmatic approach is to relate Fd to u∗ by means of: Fd = KD ρd2 u2∗

(5.29)

5.5 Threshold Friction Velocity for Sand Particles

137

where KD is a dimensionless function of the particle-friction-velocity Reynolds number, defined as (5.30) Re∗ = u∗ d/ν Assuming ad = al in Equation (5.27), we obtain  u∗t = AB σp gd

(5.31)

where AB = AB (Re∗t ) is a coefficient depending on Re∗t = u∗t d/ν, the particle-friction-velocity Reynolds number at the threshold friction velocity. AB has been found empirically to be between 0.1 and 0.2 for sufficiently large Re∗t (larger than 3.5), implying that u∗t (d) is proportional to d1/2 (Fig. 5.12c). The coefficient, AB , is referred to as the normalised threshold friction velocity, because it can be written as u∗t AB =  σp gd

(5.32)

For grains larger than approximately 100 µm, the proportionality between u∗t and d1/2 has been confirmed by experimental data. However, observations also show that a minimum u∗t exists around 75–100 µm and, for smaller 100

1.8

10

1.4 1.2 1.0

1 10−3

(b)

1.6 F(Ret)

G(d)

(a)

10−2 10−1 d (mm)

100

0.8 −2 10

10−1

100 Ret

101

102

(c)

u*t (ms−1)

100

10−1 Greeley and Iversen Scheme Bagnold Scheme 10−2 10−3

10−2

10−1 d (mm)

100

101

Fig. 5.12. The Bagnold and the Greeley-Iversen schemes for the prediction of threshold friction velocity for individual particles. The empirical functions G and F used in Equation (5.39) are shown in (a) and (b)

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5 Basic Aspects of Wind Erosion

particles, u∗t increases rapidly with decreasing d. The early interpretation of this behaviour of u∗t is that for Re∗t smaller than about 3.5, the particles lie below the viscous sub-layer and the particles are increasingly less susceptible to aerodynamic drag. In this case, the coefficient AB is no longer a constant but increases rapidly with decreasing particle size and therefore u∗t is no longer proportional to d1/2 . 5.5.2 The Greeley-Iversen Scheme The rapid increase of u∗t with decreasing particle size for small particles is more likely due to inter-particle cohesion, rather than the Reynolds-number effect (Greeley and Iversen, 1985). As particles become smaller, inter-particle cohesion becomes increasingly more important and hence, Fi can no longer be neglected from Equation (5.25). An improvement to the Bagnold model is to consider particle cohesive force and aerodynamic lift in addition to aerodynamic drag and gravity force. The aerodynamic drag, lift and moment forces are all expressed as Fd = KD ρu2∗ d2 Fl = KL ρu2∗ d2 Fm = KM ρu2∗ d2

(5.33) (5.34) (5.35)

where KD , KL and KM (of magnitude around 4, 2 and 1) are dimensionless empirical coefficients associated with aerodynamic drag, aerodynamic lift and moment, respectively. It then follows that: ad Fd + al Fl + am Fm = (ad KD + al KL + am KM )ρu2∗ d2

(5.36)

As detailed information for the coefficients, such as ad and KD , is impossible to obtain, it is sensible to simply set at KT = ad KD + al KL + am KM

(5.37)

Substituting Equations (5.36) and (5.37) into Equation (5.25) and using Equations from (5.32) to (5.35), we eventually obtain A2B =

al π 6 [1

+

6ai Fi πal ρp d3 g ]

at KT

(5.38)

Greeley and Iversen (1985) hypothesised that AB is of the form AB = A1 F (Re∗t )G(d)

(5.39)

where the F (Re∗t ) function accounts for the Reynolds-number dependency of the aerodynamic drag, and the G(d) function accounts for the effects of inter-particle cohesive forces. The constant A1 and the functions F and G

5.5 Threshold Friction Velocity for Sand Particles

139

are determined by fitting Equation (5.38) to observed data. A considerable number of measurements obtained in a series of wind-tunnel experiments with a range of particle sizes, particle densities and wind-tunnel pressures, have been used for the fitting. It has been found that 1

G(d) = [1 + 0.006/(ρgd2.5 )] 2

(5.40)

(with d in mm, g = 9.8 m s−1 and ρ = 1.2 kg m−3 ) and, for different regimes of Re∗t , A1 and F (Re∗t ) attain somewhat different values and expressions, as listed below: Re∗t 0.03 ≤ Re∗t ≤ 0.3 0.3 ≤ Re∗t ≤ 10 Re∗t ≥ 10

A1 0.20 0.13 0.12

F (Re∗t ) (1 + 2.5Re∗t )−1/2 (1.928Re0.092 − 1)−1/2 ∗t 1 − 0.0858 exp[−0.0617(Re∗t − 10)]

The behaviour of Equation (5.39) is depicted in Fig. 5.12. The minimum value of u∗t (d) occurs at d = 75 µm; for particles larger than this, u∗t increases with increasing d (eventually as d1/2 ), due to the increasing dominance of the gravity force. This result is consistent with the theory of Bagnold (1941), as given in Equation (5.31). For smaller particles, u∗t (d) increases rapidly with decreasing d, due to inter-particle cohesive forces. The semi-empirical approximation proposed by Greeley and Iversen (1985) has a considerable credibility as it is well supported by high-quality wind-tunnel measurements. 5.5.3 The Shao–Lu Scheme and the McKenna Neuman Scheme While the Greeley–Iversen scheme for the prediction of threshold friction velocity well describes the wind-tunnel observations reported in Iversen and White (1982), the two empirical functions in Equation (5.39), G(d) and F (Re∗t ), have rather complex expressions which cannot be clearly interpreted on physical grounds. It is possible that these complex expressions are the results of a misfit of G(d). A simpler expression for u∗t , which also has a simple physical explanation, can be achieved through an explicit treatment of the cohesive force, as shown by Shao and Lu (2000). On either a theoretical or experimental basis, it is virtually impossible to determine accurately the cohesive force Fi . The total cohesive force consists of van der Waals forces, capillary forces, chemical binding and electrostatic forces (Mahanty and Ninham, 1976). All these forces are sensitive to soil properties, such as particle shape, surface texture, soil mineralogy, packing arrangement and the presence of soil moisture and soluble salts. For spherical particles free of the influence of moisture and chemical binding, inter-particle cohesion can be attributed mainly to van der Waals and electrostatic forces. For sand particles with a diameter around 100 µm, there is observational evidence (e.g. Corn, 1961) that the cohesive force appears to be linearly proportional to particle size. As will be discussed in Section 5.6.1, for idealised spherical dry

140

5 Basic Aspects of Wind Erosion

particles, it is reasonable to assume in theory that the inter-particle cohesive force is proportional to particle size, so that: Fi = βc d

(5.41)

where βc is a dimensional parameter. For a range of powder particles, Phillips (1980, 1984) has suggested that the order of magnitude of βc is around 10−5 Nm−1 . Equation (5.26) can now be rewritten as π at KT ρu2∗t d2 = al ρp gd3 + ai βc d 6

(5.42)

where the dimensionless coefficient KT should be a function of Re∗t . A rearrangement of the above equation then gives:   γ 2 (5.43) u∗t = f (Re∗t ) σp gd + ρd where f (Re∗t ) = and γ=

π al 1 6 at KT

6 ai βc π al

The function f (Re∗t ) is inversely proportional to KT and hence, it also depends on Re∗t . We assume that f (Re∗t ) can be approximated by a polynomial in Re∗t , e.g. a quadratic function, and estimate the coefficients of it from experimental data. Shao and Lu (2000) fitted Equation (5.43) to the windtunnel observations of Iversen and White (1982) for the particle-size range of 50–1,800 µm, within which the experimental data are probably most reliable. They found that an excellent fitting of the measurements can be achieved simply using f (Re∗t ) = 0.0123 and values of γ varying between 1.65 × 10−4 kg s−2 and 5 × 10−4 kg s−2 . Thus, the Shao–Lu scheme for u∗t is very simple. Figure 5.13 shows a comparison of Equations (5.43) and (5.39), together with some observed data. For the particle-size range 50 ≤ d ≤ 1,800 µm, the predictions using Equation (5.43) and those using Equation (5.39) are in good agreement, but they become increasingly different for smaller particles. For d < 50 µm, there is no reliable experimental data for validation and therefore it is difficult to judge which one of the two expressions performs better. An advantage of Equation (5.43) is that it has a much simpler functional expression than Equation (5.39) and clearer physical interpretations. The Shao–Lu scheme has several interesting features. Equation (5.43) implies that if inter-particle cohesion is considered, u∗t is in general explicitly √ √ proportional to Y1 d + Y2 d−1 rather than d, as Equation (5.31) implies. The d−1 term arises from the effect of inter-particle cohesion. For large particles, the term Y1 d dominates and this implies that the balance between

5.5 Threshold Friction Velocity for Sand Particles 20

2 Greeley & Iversen (1985) Flectcher (1976a,b) Greeley & Iversen (1985) γ = 1.65e−4 = 5e−4 = 3e−4

u*t

1.5

Cleaver & Yates (1973)

15

1

10

0.5

5

0 0 10

141

101

102 d (µm)

103

0 10-1

100 d (µm)

101

Fig. 5.13. Comparison of Equation (5.43) for three different γ values (in kg m−2 ) with Equation (5.39) and the observations of Fletcher (1976a, 1976b), Greeley and Iversen (1985) and Cleaver and Yates (1973) (From Shao and Lu, 2000)

the aerodynamic forces and the gravity force determine the magnitude of the threshold friction velocity. In this particle-size range, the increase of u∗t with d is affected by two factors; namely 1) the increase in the gravity force as represented by the term σp gd and 2) the viscous effect represented by the function f (Re∗t ). The value of f (Re∗t ) does not vary over a wide range but lies between 0.011 and 0.013. This is in correspondence with the AB coefficient in the Bagnold expression, because f (Re∗t ) in Equation (5.43) is essentially A2B which ranges between 0.01 and 0.04. On the other hand, f (Re∗t ) contrasts with the viscous effect described by the expression of Greeley and Iversen (1985), in which the F (Re∗t ) has a peculiar dependency upon Re∗t , as can be seen in Fig. 5.12, which is caused possibly by an under-estimation of the cohesive force. The expression G(d) implies that the cohesive force Fi is proportional to d1/2 rather than proportional to d. The Fi ∝ d1/2 relationship is not supported by the theories described in the following section. As a consequence, the F (Re∗t ) function in Iversen and White (1982) becomes unnecessarily complicated. Equation (5.43) shows that the asymptotic behaviour of the u∗t (d) relationship is u∗t ∝ d1/2 as d → ∞. For small particles, the term Y2 d−1 dominates over Y1 d and thus the threshold friction velocity is determined by the balance between the aerodynamic and cohesive forces. The rapid increase of u∗t with decreasing d shows the strong effect of the cohesive force and the rapidly-diminishing influence of the gravity force. For particle diameters smaller than 50 µm, the cohesive force is at least 100 times larger than the gravity force. The asymptotic behaviour of the u∗t (d) relationship for small particles is u∗t ∝ d−1/2 as d → 0. In nature, the balance of the forces which determines the threshold friction velocity is affected by a number of factors, as we shall discuss in Chapter 9.

142

5 Basic Aspects of Wind Erosion

For example, the aerodynamic force is temperature dependent, because air density ρ increases with decreasing temperature and hence, for a given u∗ , the drag force on a particle is larger if temperature is lower. Further, viscosity ν decreases with decreasing temperature, and KD is viscosity dependent at low Reynolds number. Therefore, u∗t may be temperature dependent. Interparticle cohesion also depends on a number of factors, one of which is soil moisture. McKenna Neuman (2003) considered the effect of soil moisture on the inter-particle cohesive force and suggested that Equation (5.41) can be written as (5.44) Fi = βc d+ | ∆P | Ac where Ac is the contact area between adjacent grains and ∆P is the capillarysuction pressure deficit. Equation (5.43) therefore becomes   γ 2 2 (5.45) u∗t = A2 f (Re∗t ) σp gd + 2 ρd with

6 ai (βc d+ | ∆P | Ac ) π al The coefficient A2 is introduced to account for the temperature effect on u∗t . However, A2 is not substantially different from 1, because the effects due to air-density change and viscosity change arising from temperature change tend to compensate each other. For the four particle sizes, 210, 270, 430 and 610 µm, McKenna Neuman (2003) found A2 to be 1.14, 1.06, 1.06 and 1.25, respectively. Through fitting Equation (5.45) to observations, McKenna Neuman found that γ =

γ  = 10−8 − | ∆P | (1.15 × 10−8 d − 5.8 × 10−5 d2 ) with | ∆P | given in Pa and d in m. The McKenna Neuman scheme, i.e. Equation (5.45), is one method of accounting for the impact of soil moisture on u∗t . Other methods will be discussed in Chapter 9.

5.6 Threshold Friction Velocity for Dust Particles 5.6.1 Relative Importance of Forces The entrainment mechanism for sand and dust particles differs greatly because of the profound differences in the relative importance of the forces acting upon them. In particular, the cohesive force consists of van der Waals forces, electrostatic forces, capillary forces and chemical binding forces, but none of these forces can be predicted precisely. It is nevertheless useful to consider them in some detail, so that their qualitative behaviour and the difficulties involved in determining the threshold friction velocity for dust particles are understood.

5.6 Threshold Friction Velocity for Dust Particles

143

Van der Waals Forces The attraction between uncharged micron-sized particles is due to van der Waals forces, which are short-range forces with the domain of importance under a diameter of a dust particle. Theories originating from the colloidal science exist for the calculation of van der Walls forces in idealised situations, notably the Hamaker and Liftshitz theories as described in Langbein (1974) and Mahanty and Ninham (1976). For a small spherical particle of diameter d with a separation, δ, from a same-sized particle, one approximation for a van der Waals force between the two particles in vacuum is: Fiv =

hw d 32πδ 2

(5.46)

where hw is a coefficient varying between 10−18 J and 10−21 J, depending on the material. The smallest value of δ is conventionally considered to be 0.4 nm, because for regions with separation smaller than 0.4 nm, the interactions between the particles are further complicated, as Verwey and Overbeek repulsion (e.g. Theodoor and Overbeek, 1985) takes place. The above relationship is considered to be valid for δ/d  1. For δ/d > 0.2, the van der Waals attraction becomes negligible, being of the order of the Brownian forces. If particles are surrounded by air, the van der Waals attraction between the two particles may increase due to the interactions between the gas molecules adsorbed on the particles. At room temperature, van der Waals forces between particles can be increased by up to two orders of magnitude with increasing pressure (Xie, 1997). Electrostatic Forces The electrostatic force applicable to dust emission is the non-conductor force. For smooth and ideally-spherical particles, they can be written as Fie =

πBp V 2 d 2δ

(5.47)

where V is the contact potential difference, which generally ranges from 0 to about 0.5 volts, δ is the separation between the two adhering particles and Bp is the permittivity of free space. Capillary Forces Capillary forces are caused by the moisture which condenses from the surrounding air onto the particle surface and then forms liquid bridges between neighbouring particles. The surface tension of these bridges results in cohesion between the particles. The capillary forces can be many times the weight of the particles involved when the surrounding air is close to saturation. It has been shown (McKenna Neuman and Nickling, 1989) that the capillary forces

144

5 Basic Aspects of Wind Erosion

developed at inter-particle contacts surrounded by isolated wedges of water are inversely proportional to moisture tension and directly proportional to the geometric properties of the contacts. An approximation to the capillary forces is 2 πTen (5.48) Fic = Gic | ∆P | where Ten is the surface tension of water, Gic is a dimensionless geometric coefficient describing the shape of the contacts between grains and ∆P is the pressure deficit between the pressure within the water wedge and the atmospheric pressure. Other Forces There are other types of inter-particle forces, such as chemical binding and coulomb forces. Various salts in soils may interact with clay particles and form chemical bonds. The chemical processes involved are complicated, and the magnitude of the binding forces can only be determined through experiments (Chapter 9). Real particles are not ideally spherical, smooth and non-deformable. Small particles often have the shapes of platelets with considerable surface roughness and often show large deformation in the contact region. As a consequence, the theoretical predictions, such as Equations (5.46), are rarely applicable in practice. It is readily appreciated that it is virtually impossible to determine accurately the cohesive force acting on dust particles. Two crude conclusions relevant to estimating the threshold friction velocity for dust particles can be drawn from the above discussions; namely, (1) the cohesive forces are in theory linearly proportional to the size of the dust particles and this simple relationship should be used if no better information is available; and (2) the uncertainties in estimating the cohesive forces are at least several orders of magnitude. Despite the uncertainties, it is interesting to analyse the relative importance of the forces acting on particles of different sizes. We have seen in the previous discussions that the gravity force is proportional to d3 , the aerodynamic forces are proportional to d2 and the total cohesive force is in principal proportional to d. Figure 5.14 shows log-log plots of the gravity force, the total cohesive force, and the total aerodynamic force at u∗ = 0.4 m s−1 , as a function of particle size. As can be seen, all three forces decrease with decreasing particle size, but the gravity force and the total aerodynamic force decrease faster than the cohesive force. In this example, for particles with a diameter less than 10 µm, the cohesive force dominates and the particles cannot be lifted at the given friction velocity. For particles in the size range between 10 and 300 µm, the total aerodynamic force is greater than both the gravity force and the cohesive force and hence the particles in this size range can be entrained by the wind. For particle diameters larger than 300 µm, the gravity force becomes the largest of the three.

5.6 Threshold Friction Velocity for Dust Particles

145

104 Cohesive force Gravitational force Force at threshold Aerodynamic force at u* =0.4 (ms−1)

103 102

F (10−8 N)

101 100 10−1 10−2 10−3 10−4 10−5 10−6

1

10

100

1000

d (µm)

Fig. 5.14. Comparison of the relative importance of the gravity force, the cohesive force and the total aerodynamic force at u∗ = 0.4 m s−1 , as a function of particle size. For fine particles, the cohesive force dominates over the gravity and the aerodynamic forces; for medium sized particles, the aerodynamic force is the largest, and for large particles the gravity force becomes dominant

5.6.2 Stochastic Nature of Threshold Friction Velocity Because the cohesive forces are influenced by a number of extremely complicated factors, it is more rational to consider inter-particle cohesion and hence the threshold friction velocity for dust particles, as a stochastic variable which satisfies certain probabilistic distributions. Indeed, laboratory experiments indicate that a wide range of scatter in the measurements of cohesive forces may occur for nearly-identical macroscopic conditions. This scatter increases with decreasing particle size, and the spread can be as large as several orders of magnitude. Zimon (1982) has suggested various techniques for measuring cohesive forces acting upon small particles and interesting laboratory experiments using powder particles. As shown in Fig. 5.15, a particular detaching force only removes a certain proportion of the fine particles under the same laboratory condition. The proportion of the removed particles increases as the detaching force increases. By gradually increasing the detaching force and measuring the proportion of particles removed, the distribution of the cohesive force can be estimated. Zimon (1982) suggested that the cohesive force can be described using a log-normal distribution

146

5 Basic Aspects of Wind Erosion 1.0

Fraction of particle removal

0.8

5 10 20 40 80

− 10 micron − 20 − 30 − 60 −100

0.6

0.4

0.2

0.0 10-3

10-2

10-1 100 101 Detachment force (10−8N)

102

103

Fig. 5.15. Percentage of particles removed against the detachment force for 5 different particle size groups. Particles are removed if the detachment force is larger than the cohesive force. It is shown that for fine particles, the cohesive force varies over a large range (After Zimon, 1982)

p(Fi ) =

  1 [ln Fi − ln F¯i ]2 √ exp − 2σF2 i Fi 2πσFi

(5.49)

where F¯i is the median value of the cohesive force, σFi is the geometric standard deviation. Figure 5.16 shows p(Fi ) for different-sized spherical glass particles of different diameters resting on a steel surface, using the data of Zimon (1982). It is important to note how p(Fi ) varies with particle size. Figure 5.16 reveals that the geometric standard deviation for the log-normal distribution, σFi , is inversely proportional to the particle size. For particles with a diameter around 100 µm, the distribution function p(Fi ) shows a narrow peak, while for smaller particles p(Fi ) spreads much more widely. Using the data of Zimon (1982), it can be estimated that F¯i and σFi as functions of particle size, for diameters smaller than 100 µm, are F¯i (d) = exp(4.3569 − 0.2138d + 0.0018d2 ) σFi (d) = 4.1095 − 0.04761d

(5.50) (5.51)

where F¯i (d) is in mdyn and d in µm. The phenomenon described above has major implications for the modelling of dust emission, and these have not been carefully considered so far in winderosion studies. It implies that while it is meaningful to define a threshold friction velocity as a single value for sand-sized particles, it is not meaningful to

5.6 Threshold Friction Velocity for Dust Particles

147

15 d d d d d

= = = = =

10 micron 20 50 70 90

p(Fi)

10

5

0 10-6

10-5

10-4

10-3 10-2 −8 Fi (10 N)

10-1

100

101

Fig. 5.16. Probability density functions of cohesive force for spherical glass particles on steel for five different particle-size groups, as determined by Zimon (1982). The probability density functions can be approximated as log-normal distributions which have an increasing geometric standard deviation for smaller particles

do the same for dust particles. The conventional definition and measurements of threshold friction velocity for dust particles represent probably only its lower limit. What is desirable is a distribution of the threshold friction velocity for fine particles, or at least its statistical parameters, such as the mean, median and the standard deviation, so that the distribution of the threshold friction velocity can be constructed. So far, no direct measurements seem to be available. It can be hypothesised, however, that the distribution of the threshold friction velocity should also be close to a log-normal distribution.

6 The Dynamics and Modelling of Saltation

Saltation is one of the three major modes of particle motion during wind erosion, along with suspension and creep. During an erosion event, a large numbers of sand-sized particles hop along the surface in the wind direction, resulting in a horizontal transport of soil mass. This transport and the interactions between the particles, flow, surface roughness elements and topography lead to the evolution of sand dunes, the deformation of surface topography and the development of fence-line drifts in agriculture areas. A saltating particle obtains momentum from the atmosphere and strikes the surface with an increased velocity. The impact of saltating particles is one of the major mechanisms for dust emission, which is known as saltation bombardment (Gillette, 1974; Nickling and Gillies, 1989; Shao et al. 1993b and Alfaro et al. 1997). Because saltation takes place in the atmospheric surface layer, the motion of saltating particles involves the interactions with a flow of strong vertical shear. If the load ratio is high, such as is the case very close to the surface, particle-to-particle collisions may occur. The lift off of particles from the surface and the impact of particles on the surface involve particle-to-surface and fluid-to-surface interactions. The saltation process also contains a large degree of randomness, which originates from lift-off conditions and from turbulent fluctuations in the atmosphere. Sand drift and the statistical behaviour of sand-particle motion are the main topics of saltation dynamics. A range of saltation models has been developed, from simple analytical and semi-empirical ones for idealised situations (Bagnold, 1941; Owen, 1964) to more complicated ones which have to be solved numerically (Anderson and Haff, 1991; Shao and Li, 1999). In this Chapter, we discuss the dynamics of saltation and review the saltation models in the light of the considerable amount of wind-tunnel and field observations.

Y. Shao, Physics and Modelling of Wind Erosion, c Springer Science+Business Media B.V. 2008


149

150

6 The Dynamics and Modelling of Saltation

6.1 Equations of Particle Motion Suppose a particle of mass m moves with velocity up in a flow of velocity u. The acceleration of the particle is determined by the forces acting on the particle, which include the aerodynamic drag, Fd , aerodynamic lift, Fl , gravity force, Fg , Magnus force, Fm , and electric force, Fe . Thus, the equation of particle motion is m

dup = Fd + Fl + Fg + Fm + Fe dt

(6.1)

In Equation (6.1), we have neglected the effect of collision of saltating particles in air. This is sensible because it would otherwise introduce complexities. Using the expressions of Fd , Fl , Fg , Fm and Fe derived in Chapter 5, we obtain m

1 1 1 dup = − Cd ρAur Ur + Cl ρA∇U 2 d + mg + Cm πρd3 (Ω × ur ) + mCe E dt 2 2 8 (6.2)

For a spherical particle, m = πρp d3 /6 and A = πd2 /4, Equation (6.2) becomes 3 Cd 3 Cl 3 Cm dup =− ur Ur + ∇U 2 + g + (Ω × ur ) + Ce E dt 4 σp d 4 σp 4 σp

(6.3)

The spin of the particle is described by the angular velocity Ω. For a spherical particle, the moment of inertia I is known to be I = md2 /10 The airflow itself may be rotating and a quantity for describing the rotation is vorticity, ζ = ∇ × u. The angular velocity of air is simply ζ/2. Thus, the relative angular velocity of the particle with respect to air is Ω − ζ/2. The acceleration of the angular velocity of the particle can now be written as   νπd3 ζ dΩ =− Ω− (6.4) dt I 2 or more simply 60ν dΩ =− dt σ p d2

 Ω−

ζ 2

 (6.5)

The essential features of a saltating particle can be conveniently studied in a 2-dimensional coordinate system with the x-direction aligned with the mean wind and the z-direction pointing upward. Figure 6.1 illustrates the forces acting on a particle moving in a shear flow. The velocity of the flow increases with height. It is assumed that the particle is negatively charged and has a top spin. In the case of top-spin, Ω has only one component, pointing in the y direction (lateral into the page), i.e., Ω = (0, Ω, 0). Equations (6.2) and (6.5) can be written explicitly as

6.1 Equations of Particle Motion

151

z ur

up

Fl

Fm

−u − Fd

Fe Fg

u Fig. 6.1. An illustration of the forces acting on a particle moving in a shear flow. The velocity of the flow increases with height. It is assumed that the particle is negatively charged and has a top spin with the angular velocity Ω pointing in the y direction (lateral into the page)

3 Cd 3 Cm dup =− Ur (up − u) + Ωwp (6.6) dt 4 σp d 4 σp 3 Cd 3 Cl 3 Cm dwp 2 =− Ur wp + (U 2 − Ubot )−g− Ω(up − u) (6.7) dt 4 σp d 4 σp d top 4 σp + Ce E   60ν 1 Utop − Ubot dΩ =− (6.8) Ω− dt σ p d2 2 d where Utop and Ubot are the air speeds at heights corresponding to the top and the bottom of the particle. It is useful to compare the magnitudes of the various forces with that of the gravity force. For sand-sized particles (e.g. d = 100µm), the aerodynamic drag and the gravity forces are of the same order of magnitude during a typical wind erosion event [e.g. O(u∗ ) ≈ 1 m s−1 ], i.e., O(Fd /Fg ) = 1. The magnitude of the aerodynamic lift, Fl , depends on wind shear. Wind shear, and hence Fl , is usually the largest at the surface. Fl quickly decreases with height and at a few centimetres above the ground, O(Fl /Fg ) is 0.1. Particle spin is probably an important feature of saltation. Carefully following the logic behind the derivation of threshold friction velocity u∗t in Chapter 5, we see that u∗t is u∗ at which particle starts to roll, not to lift. Chepil (1945) pointed out that 50% or more of saltating particles spin and another 25% or so have relatively indistinct rotation. Equation (6.8) shows that at the state of equilibrium (dΩ/dt = 0), Ω is given by Ω=

1 Utop − Ubot 2 d

152

6 The Dynamics and Modelling of Saltation

Suppose the wind profile is logarithmic. Then, we have approximately Ω=

1 u∗ 1 2 κ z+d

This implies that in general particles rotate faster and the Magnus force is more important at smaller height. It has been observed that Ω varies over wide range, probably between 200π and 2000π s−1 , and is typically around 600π s−1 (300 revolutions per second, White and Schulz, 1977). At such Ω, O(Fm /Fg ) is 0.1. As has been discussed in Chapter 5, the ratio of Fe /Fg can be as large as 0.3 but is smaller in general. Hence, O(Fe /Fg ) = 0.1 is probably a reasonable assumption. Again, the importance of Fe decreases with height. As a first order of approximation, Fl , Fe and Fm can be neglected. It follows that the equation of particle motion can be simplified to 3 Cd dup =− Ur (up − u) dt 4 σp d 3 Cd dwp =− Ur wp − g dt 4 σp d

(6.9) (6.10)

6.2 Uniform Saltation We first consider the saltation of a single sand particle, or that of many identical sand particles under identical flow and surface conditions. The observations of saltation using photographic techniques by Bagnold (1941) and Rice et al. (1996a) have revealed that particle trajectories normally contain a large degree of randomness arising from irregular lift-off velocities and angles. As turbulence is relatively weak close to the surface and the inertia of sand grains is relatively large, neglecting the influence of turbulent fluctuations on saltation trajectories is sometimes a justifiable approximation. In this case, saltation particles would move through the air with different velocities, reach different heights, travel different distances and impact the surface with different velocities and angles, depending only on the mean-flow characteristics and particle lift-off conditions. If the lift-off conditions for all saltating particles were identical, then their trajectories would be identical. We call this idealised saltation with identical trajectories uniform saltation. The concept of uniform saltation allows the development of simple saltation models and it is justified for the reasons given below. Despite the large variations, saltation trajectories possess some common features as illustrated in Fig. 6.2. Most particles, ejected from the surface by aerodynamic forces or by the impact of other saltating particles, take off with a large angle (about 55◦ ), climb steeply to a maximum height, follow a flatter descending path and finally strike the surface at a small angle (about 13◦ ). Certain aspects of the bulk dynamic behaviour of saltating particles, such as the momentum exchange between particle and flow, can be explained simply and qualitatively by considering the motion of a single particle with a characteristic trajectory.

6.2 Uniform Saltation

153

Outer Layer

zm Saltation Layer θ1

θ2 l

Fig. 6.2. Saltation trajectories (thin lines) and the characteristic saltation trajectory (thick line). l and zm are characteristic saltation length and height, while θ1 and θ2 are particle lift-off and impact angles, respectively

The characteristic trajectory has a lift-off angle, θ1 , impact angle, θ2 , length l and height zm (see Fig. 6.2). This simplification of saltation, first proposed by Bagnold (1941), has been adapted by Owen (1964) for developing a saltation theory. Another reason for studying a single-particle trajectory is that a number of physical quantities related to the ensemble of saltating particles, such as the streamwise transport of sand particles in the direction of the mean wind, can be obtained through the summation of the quantities over all possible trajectories. Equations (6.9) and (6.10) suggest that the particle trajectory is determined if the initial conditions up |t=0 and wp |t=0 are given and the flow speed at all locations along that trajectory is known. An example of a particle trajectory in a logarithmic wind-profile situation with initial conditions up |t=0 = V0 cos θ0 ; wp |t=0 = V0 sin θ0 is as shown in Fig. 6.2 (thick line). The particle lift-off angle is around 55◦ and the lift-off velocity V0 is of the same order of magnitude as u∗ . Several quantities can be determined on the basis of particle trajectories, including particle concentration, streamwise mass flux, momentum flux and energy flux. As particles are assumed to be uniform and to follow identical trajectories, these quantities are dependent upon height only. To calculate them, we divide the maximum trajectory height, zm , into N equal intervals of dz. The particle concentration at height z can be estimated from the probability of finding that particle in the layer z ± 12 dz. This probability is inversely proportional to the vertical velocity of the particle. Suppose that the particle number flux (number of particles lifted per unit area per unit time) is n, then the specific concentration related to uniform particle trajectories is   1 1 nm + (6.11) c˜(z) = ρ | wp+ | | wp− |

154

6 The Dynamics and Modelling of Saltation

where | wp+ | represents the average of | wp | over acceding particles and | wp− | over descending particles in the slice z ± 12 dz. Likewise, the mass flux components in the horizontal and vertical directions are   up up + (6.12) q˜x (z) = nm | wp+ | | wp− |   wp wp q˜z (z) = nm + =0 (6.13) | wp+ | | wp− | The horizontal and vertical components of the horizontal momentum flux τpx and τpz can be determined using   up u p up up + (6.14) τ˜px (z) = nm | wp+ | | wp− |   up wp up wp τ˜pz (z) = nm + = nm(up+ − up− ) (6.15) | wp+ | | wp− | where up+ is the average horizontal velocity component of all acceding particles in the range z ± 12 and up− that of all descending particles. The profiles of c˜, q˜x and τ˜pz associated with a single-particle trajectory are shown in Fig. 6.3a.

6.3 Non-Uniform Saltation In reality, particle trajectories are not identical, as the particles usually lift off from the surface randomly with different velocities and angles, and their motion in the atmosphere is subject to turbulent fluctuations. Figure 6.4a shows a sample of numerically-simulated trajectories for 200 µm particles in a logarithmically-profiled airflow (u∗ = 0.5 m s−1 , z0 = 0.001 m) with turbulent intensities σw /u∗ = 1.2 and σu /u∗ = 2.4 (σu and σw are the standard deviations of wind components u and w, respectively). For comparison, Fig. 6.4b shows a sample of trajectories for 200 µm particles with randomly distributed lift-off velocities in the same turbulent flow. The comparison reveals that for relatively large particles, the randomness in particle motion is almost entirely determined by irregular lift-off conditions while the influence of turbulence is small (the impact of turbulence on the motion of smaller particles is expected to be stronger, as will be discussed in Chapter 7). The lift off of particles as a stochastic variable is completely described by the joint probability density function p(V0 , θ0 ). In general, the quantities considered in Equations (6.11)–(6.15) can be determined through the integration over all possible lift-off velocities and angles. For example, the mass concentration can be expressed as ∞ π c˜(z)p(V0 , θ0 )dθ0 dV0 (6.16) cˆ(z) = 0

0

6.3 Non-Uniform Saltation (b) Trajectory

>

~ q

>

Height Height

~ c

~ τp

>

Trajectory

Height

Height

(a)

155

c

q

τp

Fig. 6.3. (a) An illustration of particle concentration, c˜, horizontal mass flux, q˜x , and the vertical component of momentum flux, τ˜pz , associated with the saltation of a single particle. (b) as (a) but for the saltation of uniform particles with a range of lift-off velocities and angles

The mass, momentum and energy fluxes can be determined in a similar fashion, ∞ π q˜x (z)p(V0 , θ0 )dθ0 dV0 (6.17) qˆx (z) = 0

0

etc. Figure 6.3b illustrates the profiles of cˆ, qˆx and τˆpz associated with saltation of uniform particles with randomly-distributed lift-off velocities and angles. For the saltation of multi-sized particles, a further integration over all particle-size groups is required. For instance, the mass concentration is finally expressed as ∞ cˆ(z)p(d)δd (6.18) c(z) = 0

where p(d) is the particle size distribution as discussed in Chapter 5. The mass, momentum and energy fluxes can be determined in a similar fashion,

156

6 The Dynamics and Modelling of Saltation 0.01 Height (m)

a

Height (m)

0.00 0.00

0.02

0.04

0.04

b

0.02 0.00 0.0

0.2

0.4

Distance (m)

Fig. 6.4. A comparison of randomness in particle trajectories arising from turbulence and from lift-off velocities. (a) A sample of simulated particle trajectories of 200 µm particles in a logarithmic wind (u∗ = 0.5 m s−1 and z0 = 0.001 m) with turbulent intensities σu /u∗ = 2.4 and σw /u∗ = 1.2. The lift-off velocities and angles are 0.63u∗ and 50◦ , respectively. (b) as (a) but particles lift off with a vertical velocity that sat1 exp[−wp /(0.63u∗ )] isfies the exponential probability density function p(wp ) = 0.63u ∗ ◦ and a lift-off angle of 50



∞

qx (z) =

qˆx (z)p(d)δd

(6.19)

0

etc.

6.4 Streamwise Saltation Flux A widely used quantity for specifying the intensity of saltation is the verticallyintegrated streamwise saltation flux, Q. By definition, Q is the vertical integral of the streamwise saltation flux, q(z) (identical to qx (z), but with x-direction aligned with the mean wind, the suffix x can be dropped), ∞ q(z)dz (6.20) Q= 0 −2

−1

The dimensions of q are [M L T ] and the dimensions of Q are [M L−1 T−1 ]. In general, Q is a function of fetch distance. At equilibrium saltation, Q is a constant with respect to fetch distance and hence dQ =0 dx

(6.21)

6.5 The Bagnold-Owen Saltation Equation

157

6.5 The Bagnold-Owen Saltation Equation 6.5.1 The Bagnold Model Most saltating particles are confined to a thin layer close to the surface, which we refer to as the saltation layer. We could define, for instance, the depth of the saltation layer as the height below which 90% of saltating particles are found. For uniform saltation, the depth of the saltation layer is simply the maximum height zm of the saltation trajectory, which is of the order of V02 /2g. The saltation theory of Bagnold (1941) is a uniform-saltation model based on the momentum balance of the saltation layer. In this model, the momentum transfer to the surface arising from the saltation of multiple particles is represented by that arising from the characteristic saltation (Fig. 6.2). The characteristic saltation is assumed to have a lift-off velocity of (up1 , wp1 ), an impact velocity of (up2 , wp2 ), a saltation height of zm and a saltation length of l. It is further assumed that particles lift off vertically and hence, up1 is zero. This latter assumption is based on the early wind-tunnel observations, which we now know are not entirely accurate (typical lift-off angles are around 55◦ ). However, it remains valid to assume that up1  up2

(6.22)

The travel time of a particle over the distance l is approximately 2l/up2 , and the time required for that particle to cover 2zm is approximately 4zm /wp1 . 2 /2g and wp1 is proportional to u∗ , the linear Since zm is proportional to wp1 relationship g up2 = (6.23) l co u∗ can be established, where co is an empirical dimensionless coefficient. A saltating particle obtains momentum from the airflow as it moves through it and loses momentum to the surface at impact. Therefore, the momentum flux in the saltation layer is a consequence of both particle and fluid motions (Fig. 6.7). Within the saltation layer, we have in general that: ρu2∗ = τp (z) + τa (z)

(6.24)

where ρu2∗ represents the total momentum flux transferred to the saltation layer from the airflow above it, τp is the particle-borne momentum flux and τa is the airborne momentum flux, both varying with height. The magnitude of τa at the surface, τa0 , is not exactly known, but Owen (1964) has hypothesised that τa0 should approach ρu2∗t if saltation is at equilibrium. Assuming the correctness of the Owen hypothesis, we have that the particle-borne momentum flux at the surface, τp0 , is τp0 = ρ(u2∗ − u2∗t )

(6.25)

158

6 The Dynamics and Modelling of Saltation

In case of strong saltation, u2∗ u2∗t , Equation (6.25) can be written as τp0 = ρu2∗

(6.26)

Suppose the streamwise sand transport is Q. Then the particle-borne momentum flux can also be estimated by using τp0 =

Q Q (up2 − up1 ) = up2 l l

(6.27)

where we have assumed up1  up2 . Substituting Equations (6.23) and (6.25) into the above equation, we obtain Q=

cρ 3 u g ∗

(6.28)

Equation (6.28) constitutes the Bagnold model for the prediction of streamwise saltation flux and shows that Q is proportional to u3∗ , the cube of the friction velocity. Equation (6.28) best applies to strong saltation when the assumption u2∗ u2∗t is sound. However, by making use of Equation (6.25) instead of Equation (6.26), a straightforward modification to Equation (6.28) can be made   u2∗t co ρ 3 u 1− 2 (6.29) Q= g ∗ u∗ In fact, this prediction of streamwise saltation flux is remarkably close to the later models developed by, for instance, Owen (1964). The observations of Bagnold indicate that the empirical coefficient co depends on particle size and takes a value of 1.5 for 250 µm sand particles. 6.5.2 The Owen Model The Owen (1964) saltation model is a rigorous extension of the Bagnold model and is based on the same concept. The single-particle nature of the Owen model is reflected in the following assumptions which Owen made to facilitate the mathematical analysis; namely, (1) particles are spherical and uniform in size and shape; (2) the particle motion is two-dimensional and the lift-off angle is large; and (3) the entire particle motion, which in reality must be endowed with a certain randomness, is regarded repetitive, in the sense that the trajectories of all particles are identical and independent of time and distance along the surface. The last assumption is justified for saltation at equilibrium, where the streamwise sand transport is independent of distance and the net vertical sand flux at the surface is zero. The Owen model does not deal with issues such as the initialisation of particles and does not account for the entrainment of particles by the impact of other saltating particles.

6.5 The Bagnold-Owen Saltation Equation

159

Owen suggested that the region of flow can be divided into an outer layer and the saltation layer of depth zm and then made the following two hypotheses: • As far as the outer-layer flow is concerned, the saltation layer behaves as roughness elements associated with a roughness length that is proportional to zm . • The concentration of particles within the saltation layer is governed by the condition that the shearing stress borne by the fluid falls, as the surface is approached, to a value just sufficient to ensure that the surface grains are in a mobile state. The first hypothesis arises directly from the early wind-tunnel observations of Bagnold (1941) and Zingg (1953). These observations showed that the profile of wind over a surface of mobile sand and that over a surface with the same aerodynamic roughness length z0 differ (Fig. 6.5). Since moving particles transfer momentum from the airflow to the surface, they behave as momentum sinks randomly distributed in the saltation layer. For the outer layer flow, the net effect of saltating particles is similar to that of stationary roughness elements, i.e. they increase the capacity of the surface to absorb momentum. In other words, saltation leads to an increase in the aerodynamic roughness

ln(z)

u*1

u*2

z0s2 z0s1 z0

U Fig. 6.5. Wind profile corresponding to two friction-velocity values, u∗1 and u∗2 . In case of no saltation, the wind profiles are logarithmic (solid line) and the roughness length is z0 . In case of saltation, the wind profiles are modified (dashed line) and the aerodynamic roughness lengths increased to z0s1 and z0s2

160 z

6 The Dynamics and Modelling of Saltation (c)

(b)

(a)

z0s

z0s z0 U

U

U

Fig. 6.6. An illustration of saltation roughness length. (a) The wind profile over a surface of roughness length z0 with no saltation, (b) over the same surface with saltation and (c) over a vegetation canopy

length from z0 to z0s , which is called the saltation roughness length. Raupach (1991) suggested that the Owen effect can be considered to be analogous to the effect of a vegetation canopy on a flow, as illustrated in Fig. 6.6. The Owen model considers saltation as a self-limiting process governed by aerodynamic particle entrainment, particle-momentum transfer and modification of the wind profile by particle motion. The second hypothesis of Owen is a statement for τa at the surface. According to Equation (6.24), if the total momentum transfer in the saltation layer is assumed to be constant, then the behaviour of τp and τa in the saltation layer of the Owen model is as depicted in Fig. 6.7. Thus, while the total momentum flux τp + τa remains constant with height, τp decreases and τa increases monotonically with height. Owen’s second hypothesis implies that for equilibrium saltation, τa0 = ρu2∗t

(6.30)

This is a plausible hypothesis, but its correctness has not been fully examined or verified with experimental data (Anderson and Haff, 1991; Raupach, 1991). Again, the starting point of the Owen saltation model are Equations (6.9) and (6.10). A general analytical solution of these equations is difficult to obtain because of the non-linear dependency of Cd on Rep . However, Ur is approximately 10u∗ and thus Rep is of the order of 102 , assuming that u∗ ∼ 0.5 ms−1 and d ∼ 200 µm. For this range of Rep , Cd is roughly a constant (Fig. 5.5). Based on this argument, Cd is assumed to be a constant for the entire saltation layer, taking its value at Rep = 10u∗ d/ν. Under the initial conditions of up |t=0 = 0 and wp |t=0 = αu∗ , an analytical solution for both particle velocity and trajectory can be found. The key outcome of the Owen saltation model is again the prediction of streamwise saltation flux, Q, as a function of friction velocity. To achieve this,

6.5 The Bagnold-Owen Saltation Equation

161

z

zm

τp

τa

τ*t

τ

Fig. 6.7. Profiles of particle momentum flux, τp , and air momentum flux, τa , as perceived in the Owen saltation model

we denote the particle-number concentration (of dimensions L−3 ) related to upward-moving particles as o1 and that related to downward-moving particles as o2 . Because of mass continuity, the following relationships can be established within the saltation layer; namely, wp1 (z)o1 (z) = wp1 (0)o1 (0) wp2 (z)o2 (z) = wp2 (0)o2 (0)

(6.31) (6.32)

Because we are considering equilibrium saltation, dQ/dx = 0 and the net vertical mass flux at the surface must be zero, that is: wp1 (0)o1 (0) + wp2 (0)o2 (0) = 0 The streamwise sand transport can now be estimated from zm m(up1 o1 + up2 o2 )dz Q=

(6.33)

(6.34)

0

where m is the mass of a particle. Substituting from Equations (6.31), (6.32) and (6.33) into the above integral and assuming that wp1 (0) = αu∗ , we find that:  zm  up1 up2 dz (6.35) − Q = mαu∗ o1 (0) wp1 wp2 0 The above integral can be evaluated if particle velocity as a function of height z is known. The expressions for up1 , wp1 , up2 and wp2 derived by Owen (1964)

162

6 The Dynamics and Modelling of Saltation

are rather complex and hence are not shown here. Owen’s analysis leads to the simple expression for Q   u2∗t ρ 3 u 1 − (6.36) Q = co ∗ g u2∗ where co = 0.25 + wt /3u∗ is a function of the ratio between the particle terminal velocity wt and the friction velocity u∗ . Equation (6.36) is the core of the Bagnold–Owen saltation model The theory of Owen also gives the profiles of momentum and wind within and above the saltation layer. Since the total momentum flux in the saltation layer is constant with height, it follows from Equation (6.24) that dτa dτp =− dz dz

(6.37)

τp = −m(o1 wp1 up1 + o2 wp2 up2 )

(6.38)

and τp can be estimated from

Note that the downward momentum flux is positive. Applying Equations (6.31), (6.32) and (6.33) to the above equation, we obtain d(up1 − up2 ) dτa = mαu∗ o1 (0) dz dz

(6.39)

By analogy with Equation (3.30), the momentum flux and the wind profile within the saltation layer obey the relationship Ko

τa dU = dz ρ

(6.40)

where the diffusivity Ko ≈ κu∗ zm , with zm acting as the mixing length and u∗ as the scaling velocity for turbulence. Hence, the wind profile in the saltation layer obeys 1 dτa d2 U (6.41) = dz 2 Ko ρ dz Owen (1964) has shown that the solution of the above equation can be simplified to      3/2  z u2∗t 1 z 1 1− 1− 2 + 1− U = 10u∗ 1 − (6.42) 4 zm 6 u∗ zm The description of the wind profile for the outer layer in the Owen model is a restatement of the observations of Bagnold (1941), Chepil (1945) and Zingg (1953), which show that wind profile in the outer layer obeys the logarithmic law

6.6 Other Saltation Equations

163

2

log10 (z/z0s)

1

0

-1

4

8

12

16

20

U/u*

Fig. 6.8. The wind profile outside the saltation layer. Measurements are represented by full circles, and z0s is calculated using Equation (6.44) (Modified from Owen, 1964)

u∗ ln U= κ



z z0s

 (6.43)

The saltation roughness length, z0s , is derived by fitting the above equation to observed data, and it is found that z0s = cz0

u2∗ 2g

(6.44)

The fitting of the logarithmic wind profile in the outer layer to observations is shown in Fig. 6.8. The coefficient cz0 is found to be around 0.02.

6.6 Other Saltation Equations A number of other saltation equations have been proposed. In most of these equations, Q is expressed as a function of friction velocity and threshold friction velocity. Sometimes, Q is expressed as a function of wind speed measured at a reference height in the atmospheric boundary layer. We shall discuss only the first type of saltation equations, since the latter ones are not general: they depend on the height of wind measurements and the roughness length of the surface.

164

6 The Dynamics and Modelling of Saltation Table 6.1. Summary of different saltation equations

Source Bagnold (1937)

Expression d 1/2 ρ 3 co ( D ) g u∗

Constant Value D = 250 µm co = 1.5, uniform sand co = 1.8, natural granted sand co = 2.8, poorly sorted sand d 3/4 ρ 3 ) g u∗ co = 0.83, D = 250 µm co ( D )(1 + uu∗t )2 co = 1.8 to 3.1 co gρ u3∗ (1 − uu∗t ∗ ∗ As Kawamura (1964) co = 2.6

Zingg (1953) Kawamura (1964) White (1979)

u2 ∗t ) u2 ∗ d 1/2 ρ 3 co ( D ) g u∗ (1 − uu∗t ) ∗

co gρ u3∗ (1 −

Owen (1964) Lettau and Lettau (1978)

(a)

co = 0.25 +

wt 3u∗

co = 4.2

(b)

(c)

100

Q (kg m−1 s−1)

10−1

10−2 Bagnold (1937) Zingg (1953) Kawamura (1964) Owen (1964) Lettau & Lettau (1978) White (1979)

10−3 10−4 0.2

0.4

0.6

0.8

u* (ms−1)

1.0 0.2

0.4

0.6

u* (ms−1)

0.8

1.0 0.2

0.4

0.6

0.8

1.0

u* (ms−1)

Fig. 6.9. Streamwise saltation flux predicted using various saltation equations listed in Table (6.1) for 100, 200 and 300 µm particles. The calculation of u∗t uses the Greeley–Iversen model (see Chapter 5) and the Owen coefficient is enlarged by a factor of 4

Various slightly different saltation equations exist (Table 6.1). In the two earlier equations proposed by Bagnold (1937) and Zingg (1953), u∗t is not explicitly used, while the more recent models explicitly contain it. Sarre (1987) has compared the calculations for Q from different saltation equations for the same sand particles and found that these predictions can differ by a factor of 3 or more (Fig. 6.9). This difference is not unexpected. It is rather a reflection of the nature of saltation which contains a large degree of randomness that cannot be described precisely by the simple forms listed in Table 6.1. The common feature of the saltation equations is that they all contain one or more parameters which must be determined empirically. While there is no essential difference in the theory underpinning these equations, slightly

6.7 The Owen Effect

165

different assumptions and the uncertainties in the measurements used to estimate the empirical parameters have resulted in the formal differences between these equations. For example, the Kawamura model (Kawamura, 1964) can be derived by following the same procedure as described in Section 6.5.1, except that in Equation (6.23) wp1 ∝ (u∗ + u∗t ) is assumed instead of wp1 ∝ u∗ . For practical application of the saltation equations, it is necessary to estimate u∗t . In general, the latter depends not only on particle size but also on a range of surface conditions and hence contains large uncertainties. These uncertainties in u∗t can easily outweigh the differences inherent in the functional forms of the saltation equations. Therefore, it is difficult to determine whether one equation is better than another. The recommended saltation equations are those of Kawamura (1964), Owen (1964) and Lettau and Lettau (1978), which are probably equally effective. Figure 6.10 shows an example of using the Owen saltation model to fit the observed saltation transport data (Leys, 1998), which confirms the effectiveness of the Owen model for representing observed data.

6.7 The Owen Effect It is clear from the discussions of the previous section that saltating particles behave like mobile momentum sinks in the saltation layer. For the flow in the saltation layer, the effect of saltation is to reduce the vertical gradient of the flow velocity. For the flow in the outer layer (Fig. 6.2), the effect of saltation is to increase the capacity of the surface in absorbing momentum, in other words, to increase in the aerodynamic roughness length. The effect of saltation on the flow in the atmospheric surface layer is known as the Owen effect. The Owen effect can be studied by analysing four inter-related quantities, including the mean wind, particle-borne momentum flux, airborne momentum flux and saltation roughness length. 6.7.1 The Formulation of Owen According to Owen (1964), the wind profile for the saltation layer is given by Equation (6.42). It follows from Equation (6.40) that   1/2   z u2∗t 1− (6.45) τa = ρu∗ 1 − 1 − 2 u∗ zm From Equation (6.24), we have   1/2 u2 z τp = ρu∗ 1 − ∗t 1 − u2∗ zm The saltation roughness length is given by Equation (6.44).

(6.46)

166

6 The Dynamics and Modelling of Saltation 150

20 Ac

100

15

r 2 = 0.99

An r 2 = 0.99

10 50

5 0

0 0.0 0.2

0.4 0.6 0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

6

0.8 Bc

4

0.6

r 2 = 0.96

Bn r 2 = 0.90

0.4

Q (g/m/s)

2

0.2

0

0.0 0.0 0.2

0.4 0.6 0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0 0.4

0.20 0.15

Cc r 2 = 0.93

0.3

0.10

0.2

0.05

0.1 0.0

0.00 0.0 0.2

0.4 0.6 0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0 0.4

8 6

Cn r 2 = 0.96

Dc r 2 = 0.99

0.3

4

0.2

2

0.1

Dn r 2 = 0.89

0.0

0 0.0 0.2 0.4 0.6 0.8 1.0 1.2

0.0 0.2 0.4 0.6 0.8 1.0 u* (m/s)

Fig. 6.10. Fitting the Owen saltation model with observed data. The measurements were made on four soils with different textures denoted A, B, C and D, which corresponds to the U.S. taxonomy Aridosol (agrid), Aridosol (calic orthidf), Vertisol and Aridisol (haplargid). Two treatments were applied to each soil: bare uncultivated (denoted n) and bare cultivated (denoted c), giving a total of 8 soil-treatment combinations. The parameter r2 gives an indication for the goodness of the fitting with a perfect fit having a value of 1 (From Leys, 1998)

6.7.2 The Formulation of Raupach Owen (1964) and Chamberlain (1983) concluded that saltation roughness length can be described by Equation (6.44). This conclusion is supported by wind-tunnel observations, as shown in Fig. 6.8. However, the observations of Rasmussen et al. (1985) for saltation of beach sand, the measurements made

6.7 The Owen Effect

167

by Gillette et al. (1998) in the Owen’s Lake region and the numerical results of Anderson and Haff (1991) have shown that the saltation roughness length in the natural environment is much larger than observed in wind-tunnel experiments. For example, the data of Rasmussen et al. (1985) indicate that cz0 is about 10 times the value observed in wind-tunnel studies. Raupach (1991) has developed an analytical expression which appears to describe well the behaviour of z0s observed in the natural environment. The difference between the cz0 values obtained in wind tunnel and in natural environment has not yet been fully explained. Raupach (1991) has attributed this difference to the fact that saltation observed in wind tunnels normally does not fully reach equilibrium, 0.63 and thus only transient wind profiles and saltation roughness lengths are observed. They are thus much smaller than those observed in the natural environment, where fullydeveloped saltation can be achieved. It is also likely that wind-tunnel and field measurements are not adequately normalised for purposes of full comparison. This is an area that requires further research. In general, the profile of τa is not known. To avoid this difficulty, Raupach (1991) proposed a simple expression for τa (z) which satisfies several constraints: •

The total momentum flux is constant in the saltation layer and is composed of a particle-borne momentum flux, τp , and an airborne momentum flux, τa , namely, Equation (6.24) must be satisfied. • The particle-borne momentum flux, τp , decreases monotonically with height, while the airborne momentum flux, τa , increases monotonically with height. We require z → ∞, τp → 0 and τa → ρu2∗ . • The characteristic height, Hs , of the τp profile is of the order of the particlejump height wp (0)2 /2g. As the lift-off velocity wp (0) is proportional to u∗ , wp (0) = αu∗ , we obtain (6.47) Hs = br α2 u2∗ /2g •

with br being a coefficient. Owen’s self-limiting hypothesis for equilibrium saltation applies, i.e. τa0 = ρu2∗t .

One functional form for τa (z), which satisfies these constraints, is  1/2 τa = 1 − (1 − r)e−z/Hs ρu2∗

(6.48)

where r = u∗t /u∗ for u∗t ≤ u∗ , and r = 1 for u∗t ≥ u∗ . In a manner similar to Equation (6.40), the wind profile in the saltation layer should obey dU τa = Km (6.49) ρ dz  where the eddy diffusivity Km is chosen as κz τa /ρ. This choice of Km recognises the variations of τa (and hence the intensity of turbulence) in the

168

6 The Dynamics and Modelling of Saltation

vertical direction within the saltation layer and is in contrast with Ko (κu∗ zm ) used in the Owen saltation model and with κu∗ z commonly used in atmospheric boundary layer studies. At the upper boundary of the saltation layer, τa → ρu2∗ , hence Km becomes identical to κu∗ z. Substituting Equation (6.48) into Equation (6.49), we obtain
 u∗ 1 e−z/Hs dU = − (1 − r) (6.50) dz κ z z An integration of Equation (6.50) over z0 and z leads to   
    z z z0 u∗ ln − (1 − r) E −E U (z) = κ z0 Hs Hs

(6.51)

where z0 is the roughness length of the underlying surface and E(x) is the exponential integral defined by ∞ −ξ e dξ E(x) = ξ x The exponential integral has the asymptotic behaviour that E(x) → −η−ln(x) as x → 0 where η = 0.577216 is Euler’s constant; and E(x) → 0 as x → ∞. As we are interested in the wind profile well above the saltation layer, namely z0 z0 z z Hs 1, we have E( Hs ) ≈ 0. Similarly, for Hs  1, we have E( Hs ) ≈ z0 −η − ln( Hs ). If we write Equation (6.51) in the form of Equation (6.43) for Hzs 1, we obtain (6.52) ln(z0s ) = (1 − r) ln(Hs ) + r ln(z0 ) − η(1 − r) A substitution of Equation (6.47) into Equation (6.52) leads to the Raupach model for saltation roughness length  2 1−r u z0r (6.53) z0s = A ∗ 2g where A = br α2 e−η is a constant. A likely value for br is 1 and for α is 0.63. Both are widely cited values in the literature, and it follows that the likely value for A is 0.22. Equation (6.53) shows that z0s is a weighted geometric mean of z0 , the u2 roughness length of the underlying surface, and A 2g∗ which is proportional to the characteristic height of the saltation layer, Hs . In Equation (6.53), z0s has two limiting values. Thus, when there is no saltation, u∗ ≤ u∗t , so that r = 1 and z0s is identical to z0 . On the other hand, for strong saltation, u∗ u∗t , u2 so that r = 0 and z0s = A 2g∗ . The variation of z0s with u∗ is as depicted in Fig. 6.11, for the conditions, z0 = 0.01, 0.1 and 1 mm and A = 0.2 and 0.3. It has been assumed that u∗t = 0.2 ms−1 , which is typical for sand-sized particles.

6.7 The Owen Effect

169

102

z0s (mm)

101

100

z0 =1mm 0.1

10−1

10−2 0.0

0.01

A = 0.2 A = 0.3

0.5

1.0

1.5

2.0

u* (ms−1)

Fig. 6.11. Predictions of z0s using Equation (6.53) over surfaces with undisturbed roughness lengths z0 = 0.01, 0.1 and 1 mm, and with A = 0.2 (solid lines) and 0.3 (broken lines). Also used is u∗t = 0.2 ms−1 (Redrawn from Raupach, 1991)

It is seen from Equation (6.53) that cz0 =

z0s = A1−r 2 u∗ /2g



z0 2 u∗ /2g

r (6.54)

This shows that, in contrast to Equation (6.44), cz0 is not a constant but depends linearly on z0r and more complicatedly on u∗ . The variation of cz0 with u∗ is shown in Fig. 6.12 for the same conditions as in Fig. 6.11. Gillette et al. (1997, 1998) have provided a substantial amount of observational evidence from the Owen Lake experiment supporting the Raupach model. A comparison of the observed and modelled z0s is given in Fig. 6.13, which shows that the measurements of z0s can be well described by Equation (6.53) using A = 0.38. This observed value of A is close enough to the predicted value of 0.22. The key assumption to the Raupach model is Equation (6.48). This assumption is not concerned with how particles move in the atmospheric surface layer and has neglected the possible dependence of τp (z) on the size of saltating particles. Obviously,  1/2 Gr τa = 1 − (1 − r)e−(z/Hs ) (6.55) 2 ρu∗ with Gr > 0 also meets the global constraints proposed by Raupach (1991). In this case, the particle-momentum flux obeys τp (z) = ρ(u∗ − u∗t )[2u∗ − (u∗ − u∗t )e−(z/Hs )

Gr

]e−(z/Hs )

Gr

(6.56)

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6 The Dynamics and Modelling of Saltation

z0=1mm

c0

0.2

0.1 0.1 A = 0.2 A = 0.3

0.01 0.0 0.0

0.5

1.0 u*

1.5

2.0

(ms−1)

Fig. 6.12. Predictions for co using Equation (6.54) for the same conditions and parameter values as Fig. 6.11 (Redrawn from Raupach, 1991)

z0s (mm), modelled

3

2

1

0

0

1

2

3

z0s (mm), observed

Fig. 6.13. Modelled saltation roughness length z0s using Equation (6.53) versus field measurements (Redrawn from Gillette et al. 1998)

Equation (6.55) leads to z0s = (A

u2∗ 1−r r ) z0 2g

(6.57)

6.7 The Owen Effect

171

which is almost identical to Equation (6.53), except that A is now no longer a constant but br α2 exp(−η/Gr ), with Gr being a function of particle size. Thus, that a difference exists between the A value of 0.38 observed by Gillette et al. (1998) and the A value of 0.22 suggested by Raupach (1991) is not unexpected. 6.7.3 Other Formulations Dong et al. (2007) has argued that the profile of wind in the saltation layer can be better described using the power law  U (z) = Uzm

z zm

b (6.58)

where Uzm is the wind speed at the upper boundary of the saltation layer, zm . The exponent b falls in the range between 0.1 and 0.2, increasing with wind speed but decreasing with  the size of saltating particles. Again, suppose the eddy diffusivity Km is κ τa /ρz, then we obtain from Equation (6.58)  2 τa (z) = ρκ2 b2 Uzm

z zm

2b (6.59)

Figure 6.14 is a comparison of the τa profiles in the saltation layer as predicted by using Equations (6.45), (6.48) and (6.59) for particles in the size range of 150–200 µm. The differences in τa , and hence in U , τp and z0s , among the different models are considerable. Equation (6.59) suffers the shortcoming that τa approaches zero as z approaches zero, which is inconsistent with Owen’s second hypothesis. 6.7.4 Profile of Saltation Flux While it is convenient to use the vertically-integrated saltation flux, Q, for quantifying the intensity of saltation, it is important to study the profile of saltation flux, q, both for practical and theoretical purposes. It is sometimes so in practice that q is measured and Q is then estimated via Equation (6.20). Most researchers found empirically that q decays exponentially with height (Williams, 1964; Fryrear and Saleh, 1993), q = q0 exp(−az)

(6.60)

where q0 is the value of q at z = 0 and a is a positive empirical constant. The wind-tunnel observations of Butterfield (1999) show that q is probably somewhat more complicated than Equation (6.60). Shao and Raupach (1992) and Gillette et al. (1997) have suggested that the profile of q can be better approximated by (6.61) q = q0 exp(−a1 z − a2 z 2 )

172

6 The Dynamics and Modelling of Saltation 1

z/zm

0.8 0.6 0.4 0.2

(a)

0 1

z/zm

0.8 0.6 0.4 0.2

(b)

0 1

z/zm

0.8 0.6 0.4 0.2 0

(c) 0

0.2

0.4

0.6

0.8

1

ba; bp

Fig. 6.14. A comparison of ba ≡ τa /τ (solid curve) and bp ≡ τp /τ (dashed curve) profiles predicted by using (a) Equation (6.45), (b) Equation (6.48) and (c) Equation (6.59). Based on the Experiment No. 44 of Dong et al. (2007), u∗t is set to 0.234 m s−1 corresponding to particle size d = 175µm and u∗ to 0.312 m s−1 for the evaluations of Equations (6.45) and (6.48). To evaluate Equation (6.59), b is set to 0.119

Equations (6.60) and (6.61) imply that the maximum of q is q0 which occurs at z = 0 (Fig. 6.15). It must be pointed out that the exponential decay of q with height has not been unequivocally determined immediately adjacent to the surface, because q0 has not been observed directly. The numerical simulations by Anderson and Haff (1988), Zheng et al. (2004) and Shao (2005) indicate that the maximum of q may occur at some small distance above the surface, depending on the distribution of particle lift-off velocity.

6.8 Independent Saltation Saltation in nature is not uniform, because saltating particles do not follow identical trajectories for three reasons: (1) particles differ in size, shape and density; (2) atmospheric boundary layer flows are turbulent; and (3) particles

6.8 Independent Saltation

173

100 u*=0.5 ms−1 u*=0.63 ms−1

z (m)

10−1

10−2

10−3

0

500

1000

1500

2000

2500

q (g m−2s−1)

Fig. 6.15. Observed profiles of streamwise saltation flux q for u∗ = 0.5 m s−1 and u∗ = 0.63 m s−1 from the wind-tunnel experiment by Shao and Raupach (1992)

take off from the surface in a wide range of velocities. Clearly, the saltation theories based on uniform saltation which we have studied so far are only first order approximations. The assumption of uniform saltation is often taken for granted and the Bagnold–Owen or the Kawamura saltation equations are used to fit observations regardless whether saltation is uniform or not. However, this is unsatisfactory in theory. It is for example difficult to interpret the meaning of u∗t in Equation (6.36) in the case of saltation of particles of multiple sizes. A useful simplification to this case is to assume that particles of different size groups saltate independently. Suppose a soil is well mixed and the particlemass size distribution is p(d) and the particle-area size distribution is pA (d). The two size distributions are related by pA (d) =

p(d) 1  d p(d)δ ln d

The vertically-integrated streamwise saltation flux for all particle sizes, Q, can now be approximated by ∞ ˆ Q= (6.62) Q(d)p A (d)δd 0

ˆ where is Q(d) is the vertically-integrated streamwise saltation flux for particles of size d, which can be estimated by for example using Equation (6.29).

174

6 The Dynamics and Modelling of Saltation

Well Mixed Soil Surface

Well Sorted Soil Surface

Fig. 6.16. An illustration of the concept of approximating (a) a well-mixed aeolian surface with (b) a well-sorted one

Similarly, the particle-borne momentum flux at surface, τp0 , can be approximated by ∞ τˆp0 (d)pA (d)δd (6.63) τp0 = 0

where τˆp0 (d) is the particle-borne momentum flux at the surface due to the saltation of particles of size d. The concept of independent saltation is illustrated in Fig. 6.16: An aeolian surface covered by a mixture of particles of different sizes is considered to be identical to a well-sorted surface, and particles of different sizes saltate independently. In reality, saltating particles do interact through mid-air collision and splash (i.e. particles on surface take off due to the impact of saltating particles). The probability for particles to collide mid-air is probably small, because the number density of saltating particles is low. The importance of splash is still being debated, and probably does not play a major role if saltation is at equilibrium. Therefore, the assumption of independent saltation is justifiable, although it has never been verified by experiments. For simplicity, let us sort soil particles into I size bins and suppose that the ith bin has a particle size di and a bin width ∆i and occupies a surface-area fraction Pi di +∆i /2 Pi = pA (d)δd di −∆i /2

which is the fraction of surface covered by particles from the ith size bin. We have I

Pi = 1, Suppose K is the number of erodible bins among the I bins for a given u∗ . Then, the fraction of erodible surface is σ=

K

Pk

6.9 Supply-Limited Saltation

175

Equation (6.62) implies that Q=

K

ˆ k Pk Q

(6.64)

Consider now the balance of momentum at the surface. Suppose the particle-borne momentum flux due to the saltation of particles from the kth bin is τˆpk . Then, the particle-borne momentum flux at z = 0 due to the saltation of all K bin is K

τˆp0k Pk (6.65) τp0 = An extension of the second hypothesis of Owen (1964) to the kth bin is that τˆpk0 must satisfy  0 u∗ ≤ u∗tk (6.66) τˆp0k = τ − τtk u∗ > u∗tk where τtk = ρu2∗tk and u∗tk is the threshold friction velocity for the kth bin. It follows that K

Pk τtk (6.67) τp0 = στ − and the airborne momentum transfer to the surface (at equilibrium saltation) satisfies K

Pk τtk (6.68) τa0 = (1 − σ)τ +

6.9 Supply-Limited Saltation In the previous discussions, we have implicitly assumed that saltation takes place under the condition that the supply of particles for the process is unlimited. In reality, the supply of particles is often limited. The saltation that takes place under unlimited supply of particles can be called potential saltation, and the saltation that takes place under limited supply of particles supply-limited saltation, or source-limited saltation. We denote the verticallyintegrated streamwise flux for supply-limited saltation as Qslm . Obviously, Qslm is smaller than or equal to its potential value Q. Not enough research has been done on supply-limited saltation, but a useful theory can be derived based on the assumption of independent saltation. A special case of supply-limited saltation is that a fraction of the surface, σ, is not erodible while the remaining fraction of the surface, (1 − σ), is made of particles of size d and is erodible. In this case, Equation (6.67) becomes τp0 = σρ(u2∗ − u2∗t )

(6.69)

176

6 The Dynamics and Modelling of Saltation

By substituting Equation (6.69), instead of Equation (6.26), in Equation (6.27), the Bagnold–Owen saltation equation can be written as   u2 cρ (6.70) Qslm = σ u3∗ 1 − ∗t g u2∗ Thus, for supply-limited saltation we have Qslm = σQ In practice, σ is time dependent and typically decays exponentially with time.

6.10 Evolution of Streamwise Sand Transport with Distance The saltation models described above are widely used in practice because of their simplicity, although they are strictly applicable only to equilibrium saltation. We usually do not have sufficient data to decide whether or not saltation is in equilibrium. Nevertheless, it is useful to consider in theory how saltation equilibrium is achieved. The key question here is what role splash entrainment plays in the process of saltation. As saltating particles strike the surface, they may rebound and eject more particles into the air. This process is known as splash entrainment. Bagnold (1941) has vividly described this phenomenon he observed in the wind-tunnel experiments as follows. As a particle moves through the air, it absorbs momentum from the airflow and impacts the surface at a much larger velocity than the lift-off velocity; the particles ejected by the impact lift off with even larger speed and, in turn, impact the surface more violently. Bagnold therefore seems to imply that splash entrainment plays a major role in the dynamics of saltation. However, Owen (1964) has argued explicitly that splash entrainment cannot significantly influence the equilibrium state of saltation, for at equilibration, the ejected particles are unlikely to have sufficient energy. Some later researchers (e.g. Raupach, 1991) have adopted Owen’s argument. Anderson and Haff (1991) suggested that saltation is a self-limiting process controlled by four interacting components: namely, particle lift-off by aerodynamic forces; particle motion; splash entrainment; and modification of the wind profile by saltating particles. This argument is adopted in several other studies (e.g. Werner, 1990; Shao and Li, 1999) and the effect of splash entrainment is considered along with the three other limiting factors included in the Owen model. The models which include splash entrainment are mathematically complicated and the solutions must rely on numerical techniques. Anderson and Haff (1991) have been able to simulate numerically the development of saltation in time, or with fetch distance. Their results suggest that (1) a significant distance is required for saltation to reach equilibrium; and (2) in the initial stage of saltation, before equilibrium is reached, Q increases

6.10 Evolution of Streamwise Sand Transport with Distance

177

Q (gm−1s−1)

100

50

u* = 0.6 (ms−1) Observation Simulation

0 0

10 Distance (m)

20

Fig. 6.17. Observed (solid line) and simulated evolution (dashed line) of streamwise saltation flux Q with distance x for u∗ = 0.6m s−1 . The observation is based on the wind-tunnel experiment of Shao and Raupach (1992) and the simulation is based on the saltation model of Shao and Li (1999)

with distance to a maximum, before decreasing to its eventual equilibrium value. This phenomenon is referred to as the “overshoot” of saltation. Bagnold (1941) has provided some evidence for the dependence of Q on fetch distance, indicating a smooth increase of Q to an equilibrium at about 7 m, without overshoot. A possible explanation for this result is that Bagnold used a relatively short tunnel which might have been insufficiently long for observing the full development of saltation. In fact, wind tunnels used for studies of saltation are usually rather short (with working sections of only several metres) and hence the evolution and equilibration of saltation have probably not been well observed in most of the past wind-tunnel experiments. Shao and Raupach (1992) studied the development of saltation for almost uniform sand grains in a wind tunnel with a working section of 19 metres. An example for the evolution of Q with distance is shown in Fig. 6.17. The qualitative behaviour of Q is similar for different wind velocities, and can be divided into three stages. The first few metres constitute a growth stage, where Q increases with fetch distance x, indicating an avalanche of particle mobilisation and entrainment into the airflow. This is followed by an overshooting stage, where Q maintains high values for several metres. In the equilibration stage over the last few metres, Q gradually decreases with x towards a constant value with insignificant streamwise gradient. The wind-tunnel observations and the numerical results of Anderson and Haff (1991) are in qualitative agreement. However, even the wind tunnel used by Shao and Raupach (1992) was found not long enough for the final equilibrium state to be detected unambiguously. The hypothesis for the evolution of saltation is as follows. In the early stage of saltation, aerodynamic forces are mainly responsible for the entrainment of particles and splash entrainment is insignificant. When these particles are mobilised and lifted a small distance above the surface, they absorb kinetic energy from the airflow. At impact, these particles of higher kinetic energy eject more particles into the air and the splash process dominates over

178

6 The Dynamics and Modelling of Saltation

the aerodynamic entrainment, initiating a growth in the number of airborne particles and hence in Q. As more particles are entrained into the air, the wind profile becomes significantly modified because of the additional momentum transfer from air to the surface by the particles, leading to a reduction in the mean wind speed near the bed. This decreases the particle-ejection rate leading eventually to equilibrium. For saltation in the natural environment, it has also been noted that, for a steady wind over a reasonably homogeneous soil surface, Q comes to an equilibrium within a certain distance (Stout, 1990; Gillette et al. 1996; 1997). This distance is approximately 600 m, much larger than the saltationequilibrium distance known from wind-tunnel experiments. This difference in the saltation-equilibrium distances in wind tunnels and in the natural environment has not yet been fully studied. However, it can be hypothesised that this difference arises mainly from the fact that in the natural environment, the depth of the atmospheric boundary layer is much deeper than in the wind tunnel, and hence the response of the boundary-layer flow to the extraction of momentum by particle motion near the surface is much slower than that of the wind-tunnel flow.

6.11 Splash Entrainment The current understanding of splash entrainment comes primarily from windtunnel observations and computational models. 6.11.1 Wind-Tunnel Observations The results obtained from various wind-tunnel experiments are consistent. Rice et al. (1995) used high-speed photographic techniques to study the splash of different-sized particles on a sand surface of multiple grain sizes. In their experiment, the sand population was divided into 150–250, 300–350 and 425– 600 µm fractions by sieving. Each particle-size fraction was dyed to a particular colour and the original sand population was reconstructed using the dyed sand grains. Grains were dropped through a tube which extended from the roof of the tunnel to 60 mm above the floor. The basic results reported by Rice et al. (1995) are summarised in Tables 6.2 and 6.3. Table 6.2. Wind-tunnel observations of impact velocity, VI , impact angle, θI , rebound velocity, VR , and rebound angle θR (Modified from Rice et al. 1995) Impactor Coarse Medium Fine

VI (m s−1 ) 2.73 ± 0.56 3.30 ± 0.67 3.79 ± 0.66

θI (◦) 14.62 ± 4.85 11.55 ± 3.29 10.52 ± 2.78

VR (m s−1 ) 1.55 ± 0.49 1.86 ± 0.67 2.09 ± 0.79

θR (◦) 23.04 ± 15.31 30.22 ± 23.66 40.17 ± 30.91

VR /VI 0.57 0.57 0.55

θR /θI 1.79 2.85 4.11

6.11 Splash Entrainment

179

Table 6.3. Wind-tunnel observations of velocity and angle parameters for ejected grains. VS and θS are the velocity and angle of splashed particles at lift off, respectively. nS particles are ejected per impact and PS is the probability of splashing (Modified from Rice et al. 1995) Impactor

Coarse

VS θS Coarse 0.24 ± 0.18 44.34 ± 31.13 Medium 0.24 ± 0.21 46.09 ± 34.57 Fine 0.19 ± 0.10 38.79 ± 23.50 Impactor

Medium nS 1.20 1.13 1.01

Fine

VS θS Coarse 0.24 ± 0.20 50.94 ± 37.87 Medium 0.25 ± 0.22 58.97 ± 42.48 Fine 0.29 ± 0.25 57.31 ± 37.86

PS VS θS 0.29 0.26 ± 0.22 51.35 ± 37.47 0.14 0.28 ± 0.24 54.61 ± 37.93 0.11 0.26 ± 0.22 55.56 ± 36.64

nS 2.94 2.21 1.56

PS 0.82 0.58 0.46

nS 5.60 3.55 2.26

PS 0.92 0.81 0.76

Total nS 3.22 2.40 1.75

PS VS θS 0.74 0.25 ± 0.21 50.69 ± 37.45 0.60 0.27 ± 0.23 56.14 ± 40.23 0.51 0.27 ± 0.23 55.38 ± 36.84

Table 6.2 shows that in similar wind flows, coarse particles travel more slowly than fine particles and impact the surface with a slightly larger angle. The impact angle on average is around 10◦ to 15◦ with a standard deviation between 2◦ and 5◦ . The impact angles of saltating grains are remarkably constant over a range of conditions. At rebound, particles lift off at an angle larger than the impact angle and at a velocity about half of the impact velocity. The ratio of the mean rebound and impact velocities falls between 0.5 to 0.6, independent of particle size. The ratio of the mean rebound and impact angles increases with decreasing particle size. Table 6.3 shows the velocity parameters for ejected grains. As a saltating particle impacts the surface, it has the probability PS of creating a splash, depending both on the size of the impacting particle, dI , and that of the ejected particle, dS . Qualitatively, for similar wind conditions, PS increases with dI , but decreases with dS . The number of splashed particles from the surface varies between 2 and 6, also depending on dI and dS . The lift-off velocity of the splashed particles is much smaller than the saltating particles, around 7–9% of the latter. The lift-off angle of the splashed particles is larger. 6.11.2 Numerical Simulations Computational simulation has been used to generate quantitative relationships for splash entrainment (Anderson and Haff, 1991; Haff and Anderson, 1993). The basic approach technique used is the discrete element method that is reminiscent of the molecular dynamics used in chemistry (Cundall and Strack, 1979; Allen and Tildesley, 1987). In these models, we consider a finite system of particles, calculate the forces acting upon the individual particles

180

6 The Dynamics and Modelling of Saltation

and determine the dynamic evolution of particles by solving the corresponding system of Newtonian equations. The problem is computationally intensive, and hence the simulations available so far are confined to two-dimensional systems which consist of spherical particles of two to three different sizes. In a two-dimensional system confined to the x-y plane, the motion of the ith particle of the system can be completely described using six equations dyi dθi dxi = vxi , = v yi , = ωi dt dt dt dvxi dvyi dωi = Fxi , mi = Fyi , Ii = Toi mi dt dt dt

(6.71) (6.72)

where xi , yi and θi are the coordinates and orientation of the ith particle, vxi , vyi and ωi are its translational and angular velocities, respectively, mi its mass and Ii its moment of inertia about its centroid. Fxi , Fyi and Toi are, respectively, the x and y components of the total force and the total torque exerted upon the particle by neighbouring particles in contact with it. The forces acting on the particle include the gravity force, the aerodynamic forces and the forces which act at points of contact between the particle in question and the surrounding particles. The contact forces are of two types: namely, compressive forces, Fn , which are normal to the tangent line at the contact point, and tangential forces, Ft , which are parallel to this line, as illustrated in Fig. 6.18. The Normal Force: When two particles come into contact, a mutuallyrepulsive normal force, Fn , arises between them due to the stiffness of the particles. For problems in which particles are dispersed from another, the hard-sphere collision model can be used to describe the normal forces. In this model, the details of Fn are suppressed; the particle stiffness is effectively assumed to be infinite, and the main effect of Fn is to partition momentum

Ft2 p1 Ft1

Fn2 Fn1 Fg

p2

Fig. 6.18. Illustration of forces acting on the particle in the centre. Fg is the gravity force, Fn1 and Fn2 are normal forces, while Ft1 and Ft2 are tangential forces exerted by particle p1 and p2 , respectively

6.11 Splash Entrainment

181

k

δij = ri +rj −Rij

Rij

ri

Ri

Rj

rj

Rij = Ri − Rj

Fig. 6.19. Treatment of normal forces in the linear-spring model (Modified from Haff and Anderson, 1993)

between colliding particles according to the principle of momentum conservation. However, when particle contacts are enduring, a finite material stiffness is required in order that Fn may change appropriately to ensure momentum balance in the particle system. Haff and Anderson (1993) proposed a linear-spring model to represent the finite compressibility of contacting particles. In this model, the repulsive force is assumed to be proportional to the amount of overlap of the spheres coming into contact plus a damping term (Fig. 6.19). Suppose the radii of particles i and j are ri and rj and the distance between their centres of mass is Rij , where Rij =| Rij |=| Ri − Rj | with Ri and Rj being the locations of the centroids of particles i and j, respectively. The overlap between the two particles Dij is   ri + rj (6.73) Dij = Rij 1 − Rij and the normal contact force is Fn,ij = −kDij − b

dDij dt

(6.74)

For simplicity, we drop the subscripts and rewrite the above equation for the compressive force exerted along the line of centres on a given grain as Fn = −kD − b

dD dt

(6.75)

182

6 The Dynamics and Modelling of Saltation

or

dD d2 D (6.76) = −kD − b dt2 dt where mr is now the reduced mass. The term −kD, where k is the stiffness constant, means that the normal contact force opposes an increase in D, and the −b dD dt damping term, where b is a damping constant, irreversibly extracts energy from the motion. The minus sign in this term ensures that the damping force always opposes the relative motion between the particles. For two particles undergoing  a collision at relative normal velocity v0 , and b is not too large (i.e. b < 2mr k/mr ), the particle displacement during contact can be found by solving Equation (6.76) to give mr

(6.77) D = Dmax e−bt/2mr sin ωt   where ω = ω02 − (b/2mr )2 , ω0 = k/mr and Dmax = v0 /ω0 . The quantity Dmax is the maximum spring compression, i.e. the point at which all kinetic energy 12 mr v02 has been converted into potential energy 12 kDmax . We expect that the particle deformation during the aeolian particle collision would be small and therefore, Dmax  r This also implies that

v0 r The parameters k and b need to be chosen in the computer simulation. For instance, if large velocities v0 are expected to occur, k must be chosen suitably large in the computer model. Within certain limits, the damping parameter, b, can be chosen to achieve the desired damping rate. The parameters k and b together with mr , determine the duration of the contact in collision k mr

T = π/ω and the damping time scale Tdamp =

2mr b

Computationally, these time scales are the limiting factors applying to the computer simulation of aeolian impact problems, as the numerical time step ∆t must be chosen such that ∆t  min{T, Tdamp }. The Tangential Forces: For spherical particles, surface friction provides a coupling between the translational motion of the centroid of a particle and rotational motion about the centre of mass. We consider first the case in which the surface of one grain is in contact with that of a second and there is mutual slipping at the point of contact; that is, the relative tangential velocity vt,ij between particle j and i at the contact point is non-zero. If ω = ωnω is the particle angular velocity where nω is the unit vector parallel to ω, and ri = ri nr is the vector running from the centre of the

6.11 Splash Entrainment

183

vi ωi

Ft,ij= − ktS,ij

vt,j = ωj. rj (vj .n t,j) n t,j

ri rj

vt,i = ω1. ri (vi . n t,i) n t,i

Sji

Sij

vj

ωj

Fig. 6.20. Treatment of tangential forces in the linear-spring model (Modified from Haff and Anderson, 1993)

particle to the contact point with nr being the unit vector parallel to ri , then the tangential rim velocity of particle i, at the contact point, with vt,i = ωi × ri + (vi · nt )nt

(6.78)

where vi is the velocity of the centroid of particle i and nt = nω ×nr . A similar expression can be derived for vt,j . The relative tangential velocity vt,ij is then of course vt,ij = vt,i − vt,j A simple model for such a slipping contact is the Coulomb friction model, in which Ft is taken to be proportional to | Fn |, through a friction coefficient µt . The direction of Ft is chosen to oppose the slip. For particle i in Fig. 6.20, we therefore have Ft,ij = −µt | Fn,ij | nt,ij

(6.79)

where nt,ij = vt,ij / | vt,ij |. Haff and Anderson (1993) identified two numerical problems in this formulation. First, while the formulation is reasonable as long as particle slip continues, it breaks down if the slip rate becomes too small. Then, in a numerical time step, nt,ij may change sign. If this happens Ft,ij , which may be large, will reverse its direction, leading to the oscillation of nt,ij as the tangential force tries to oppose the instantaneous tangential motion. Second, for a non-slipping contact, the frictional force is not defined. To overcome these problems, a tangential spring with force constant kt can be introduced to each particle at the point of initial contact between two mutually-contacting surfaces. By analogy with Equation (6.76), we obtain Ft,ij = −kt s − bt

ds dt

where s is the tangential-displacement vector defined by

(6.80)

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6 The Dynamics and Modelling of Saltation



t

| vt,ij (t) | dt

s = nt,ij 0

If | Ft,ij |< µt | Fn,ij |, Equation (6.80) is used to supplement the Coulomb friction model. When | Ft,ij |> µt | Fn,ij |, the tangential spring is considered to be broken and the contact is allowed to slip, with the force given by Equation (6.79). This force is used until the slipping becomes small, at which point a new spring is attached at the current contact point. Haff and Anderson (1993) have also proposed that kt and bt should be chosen to be k and 2mr ω0 , respectively. Finally, it is found that

(Fn,ij )x + (Ft,ij )x (6.81) Fxi = j

j

and similar for Fyi . The torque is given by

Toi = Rij × Ft,ij

(6.82)

j

with the summation extending over all contacting particles j. Several model parameters need to be carefully chosen for the model to deliver correct results, including the spring constant, k, the restitution coefficient  [= exp(−πb/2mr ω0 )] and the friction coefficient µt , as well as the numerical time step. Haff and Anderson (1993) performed numerical simulations with a system of 500 quartz particles consisting of 33% 328 µm particles and 67% 231 µm particles, using k = 2 × 103 Nm−1 ,  = 0.7, µt = 0.5 and an integration time step ∆t = 0.1µs. Their sensitivity tests show that the model performance is not too sensitive to the choice of k and µt and the uncertainties arising from  are limited, since its value lies between 0 and 1. The computer model has been applied to study the effect of grain impact velocity and angle upon the impact process. Figure 6.21 shows two examples. In the first example (Fig. 6.21a), a small grain (initial position black) is approaching at 1 m s−1 and ricocheting off the bed, dislodging a few nearsurface grains in the vicinity. In the second example (Fig. 6.21b), a large grain is moving at the same speed along the same trajectory, scattering of the bed and causing a substantial disturbance. Figure 6.22 shows the response of the bed surface to a realistic range of impact velocity (0.25–8 m s−1 ), with the impact angle being held constant at 11.5◦ . As impact velocity increases, both mean rebound speed (particle No. 1) and the mean ejection speed increase approximately linearly. For larger impact velocities, the ejection speed appears to tend towards a constant value. The computational simulations have obtained information on splash entrainment which is consistent with the wind-tunnel observations. The mean ejection velocity from the computer simulation for particles in the 200–300 µm size range is about 10% of the impact velocity, while Rice et al. (1995) have

6.11 Splash Entrainment

(a)

185

(b)

Fig. 6.21. (a) Example of disturbance created when a small particle travelling at 1 ms−1 impacts the surface and rebounds. (b) as (a) but for a large particle travelling at the same speed and along the same trajectory (Redrawn from Haff and Anderson, 1993)

Fig. 6.22. Dependence of vertical ejection velocity on impact speed. Impact angle is fixed at 11.5◦ . The first number represents the rebounding particle and the other numbers represent ejected particles (From Haff and Anderson, 1993)

reported values ranging between 7–9%. The predicted mean ejection angle is around 70◦ , which is slightly larger than the values of 55◦ given by Rice et al. (1995), but well within the scatter of the observed data. The simulated rebound velocity and angle of the incident particle are also in good agreement with wind-tunnel experiments, all suggesting that the rebound velocity is about 56% of the incident velocity, with a rebound angle of around 30◦ to 45◦ . Disagreement exists between the simulated and observed ejection rates, with the former being about thrice those observed experimentally. Haff

186

6 The Dynamics and Modelling of Saltation

and Anderson (1993) have attributed this disagreement to either a possible under-counting of out-of-plane ejecta, as the experiments detect grain motion projected on to a plane containing the incident velocity vector, or to the restriction of the simulation to two dimensions.

6.12 Numerical Modelling of Saltation Compared with the relatively simple saltation models described in the previous sections of this chapter, the objective of the numerical models of saltation is to comprehensively simulate the dynamics of saltation. The basic framework of the existing models is as outlined in Anderson and Haff (1991) and Shao and Li (1999). In these models, saltation is considered as a self-limiting feedback process between air motion, particle motion and particle-air and particle-surface interactions. The numerical models of saltation are in essence a coupling of three basic components: (1) a component for the atmospheric boundary-layer flow in which saltation takes place; (2) a component for particle motion and (3) a component for simulation of particle-and-surface interactions. Although the models are rather complex, they have been shown to be useful in addressing a number of issues which have remained so far inconclusive from experimental studies and conceptual analyses. The treatment of the flow field among the existing models differs considerable. In some models only the mean flow is considered (Ungar and Haff, 1987; and Anderson and Haff, 1991), in others turbulence is considered in addition to the mean flow by means of stochastic modelling (Anderson, 1987 and Shao, 2005) or by large-eddy simulation (Shao and Li, 1999). The employment of large-eddy simulation in saltation modelling is computationally expensive, but is potentially very useful in understanding saltation in turbulent flows. The theory and numerical techniques for boundary-layer flows have been outlined in Chapter 3, but interested readers should refer to more specialised books for details (e.g. Galperin and Orzag, 1993; Lesieur, 1997). 6.12.1 Simple Flow Model The flow component of a saltation model can be made simple or complex, depending on the modelling requirements. Assuming that the air density is not significantly altered by the presence of saltating particles, the governing equation system for boundary-layer flows with saltation can be written as, ∂uj =0 ∂xj 1 ∂p ∂ui ∂ui ∂ 2 ui + uj =− − δi3 g + Km 2 − spi ∂t ∂xj ρ ∂xi ∂xj

(6.83) (6.84)

where Km is again eddy diffusivity and spi is the momentum sink arising from particle motion, which can be expressed as the divergence of particlemomentum flux

6.12 Numerical Modelling of Saltation

spi = −

187

1 ∂τpij ρ ∂xj

For steady-state and horizontally-homogeneous saltation with the x direction aligned with the mean wind, the derivatives with respect to t, x and y in Equation (6.84) vanish and the flow has no components in the y and z directions. The flow speed is maintained by an external force, such as a horizontal pressure gradient ∂p/∂x, which just overcomes the effect of friction at the boundaries. It follows that Equation (6.84) is reduced to a simple balance of force in the x direction ∂τp ∂τa + =0 (6.85) ∂z ∂z Integrating Equation (6.85) from z = ∞ to z, and applying the boundary conditions τp∞ = 0 and τa∞ = ρu2∗ , we obtain τa (z) = ρu2∗ − τp (z)  Since τa = ρKm ∂U/∂z and Km = κ τa /ρz, it follows that  1/2 1 ρu2∗ − τp dU = dz κz ρ

(6.86)

(6.87)

This simple flow model is identical to that used by Owen (1964) and Raupach (1991). Equation (6.87) applies only to steady state and horizontally homogeneous saltation and is not suitable for studying saltation which is evolving with distance and time. Its application to unsteady-state and heterogeneous saltation would imply that an instantaneous adjustment of flow to the change of surface-momentum flux takes place. This is inadequate, of course, because the evolution of flow associated with saltation, similar to the evolution of flow over a surface with changing roughness length in space, requires a considerable distance to reach a balance with the new underlying surface (Bradley, 1968). The other disadvantage of using Equation (6.87) is that it is not suitable for studying the interactions between saltating particles and turbulence (Butterfield, 1991; Stout and Zobeck, 1997). 6.12.2 Large-Eddy Simulation Model More comprehensive simulations of boundary-layer flows can be achieved using the large-eddy simulation (LES) approach. The simulation of turbulent flows with saltation is similar to that without saltation, except that the momentum sink term in Equation (6.84) needs to be estimated from the motion of saltating particles. Suppose J particles are present in a control volume of size ∆3 for the LES model at a given time, sp in Equation (6.84) can be estimated by using J 1 dupj (6.88) mj sp = ρ∆3 j=1 dt

188

6 The Dynamics and Modelling of Saltation

where up is particle velocity. The eddy viscosity Km in Equation (6.84) can be modelled using the Smagorinsky or the e− closure as described in Chapter 3. For modelling saltation, the LES model requires high spatial resolution near the surface, and a stretched numerical mesh with a logarithmic decrease in mesh-size toward the ground should be used. This arrangement ensures a sufficiently-high resolution for the calculation of particle trajectories and other physical quantities, such as saltation flux. The integration time steps for the flow model and the particle trajectory model can be different. For example, the time step for the flow model can be taken as 0.005 s and for the particle trajectory model 0.0001 s (50 particle-trajectory integration steps in every fluid-motion time step). The flow field can be initialised by using a simple logarithmic flow upon which random fluctuations of a given intensity are superposed. The flow model can then be run with a periodic boundary condition such that the flow reaches equilibrium with the surface that has a specified roughness length. Saltation is then introduced and both the fluid motion and the particle motion are allowed to evolve simultaneously. The lower-boundary condition is important, as it affects both the flow field and the aerodynamic entrainment of particles. One possible approach is to divide the flow into an outer region and a saltation layer, as in the Owen saltation model. The depth of the inner region, Hs , is approximately the maximum height which most saltating particles reach. The flow speed on average in the outer region obeys Equation (6.43). The friction velocity, u∗ , can be obtained from the downward momentum flux at Hs . There is no conceptual difficulty in estimating this flux from the flow model which can estimate eddy diffusivity through a sub-grid closure scheme, such as the Smagorinsky or the e −  closure. At the upper boundary of the saltation layer, the momentum flux is entirely determined by the fluid motion, which can thus be calculated by using Equation (6.49). Within the saltation layer, while particle motion results in the momentum flux τp , τp +τa remains constant with height. The behaviour of τa and τp in general should be as illustrated in Fig. 6.7 with τp decreasing with height to zero at height z = Hs and τa increasing with height to ρu2∗ at z = Hs . The particlemomentum flux at the surface, τp0 , can be estimated using Equation (6.92), and the airborne momentum flux at the surface, τa0 , can be determined by using the relationship τa0 = ρu2∗ − τp0 . At the lower boundary, wind should satisfy Equation (6.49). 6.12.3 Particle Motion It is common to study particle motion using a Lagrangian stochastic model by following the trajectories of individual particles. This is more convenient than treating the particulate phase as a continuum in a fashion similar to the fluid phase, with the main advantage being that the particle-and-surface interactions can be treated on clear physical grounds. To facilitate discussion, we confine the description of the model to uniform spherical particles with

6.12 Numerical Modelling of Saltation

x

x

x

x

x

x

x

x

x

x

xp

x

189

x Particle Trajectory

x

x

x

x

Fig. 6.23. Flow speed at the location of the saltating particle, xp , is estimated through an interpolation of flow speed at the grid points marked with crosses

diameter d and mass m. It is straightforward to extend the model formulation to deal with any spectrum of particle sizes. In general, Equations (6.3) and (6.5) can be used to model particle trajectories. If particle spin is neglected, then particle trajectory can be determined using the following equation system 3 Cd dupi =− Ur uri − δi3 g dt 4 σp d dxpi = upi dt

(6.89) (6.90)

where xpi is particle position (i denotes the ith component). Note that the particle-to-fluid relative velocity, uri is given by upi − ui (xp ), where ui (xp ) is determined by interpolating the fluid velocity from Eulerian grid points to the location xp , as illustrated in Fig. 6.23. If the subgrid fluctuations of air motion are also to be considered in the calculation of particle trajectories, uri needs to be expressed as uri = upi − (ui + ui )

(6.91)

where ui is the subgrid fluid-velocity fluctuation. However, this approach requires an additional model for ui and is numerically more complex. Once the particle trajectory is known, relevant physical quantities related to saltation can be estimated, such as particle impact velocity and angle, streamwise saltation flux and particle-borne momentum flux, as discussed in Sections 6.2 and 6.3. One difference, though, is that the quantities need to be calculated for each control column of the flow model. For example, the downward flux of streamwise momentum for a volume of ∆x∆y∆z centred at (x, y, z) can be estimated by using

190

6 The Dynamics and Modelling of Saltation

 K↓  K↑

1 τpz (x, y, z) = mup,k ↓ (x, y, z) − mup,k ↑ (x, y, z) (6.92) ∆x∆y∆t k=1

k=1

where K ↓ denotes the number of downward-moving particles through the area of ∆x∆y in the time interval of ∆t and up,k ↓ denotes the streamwise velocity component of the kth particle, and K ↑ and up ↑ are the corresponding values for the upward-moving particles. Since the vertical flux of the streamwise momentum is of particular importance, τpz is simply referred to as τp . 6.12.4 Aerodynamic Entrainment At the surface, the total upward particle-number flux, nT , consists of the aerodynamic entrainment rate, nA , the rebound rate, nR , and the splash rate, nS , so that (6.93) nT = nA + nR + nS all with the dimensions of [L−2 T−1 ]. The aerodynamic entrainment rate is the upward particle-number flux generated by aerodynamic forces. There have been so far no direct measurements for nA as a function of wind speed and surface conditions, but estimates of its magnitude can be obtained by considering the momentum balance at the surface. Suppose the number flux of incoming particles (of mass m) is nI , which may rebound and eject more particles, and imagine there is no aerodynamic entrainment. In this case, the particle-borne momentum flux at the surface is n  nI nS R

up,k (0) − m up,k (0) + up,k (0) (6.94) τp0 = m k=1

k=1

k=1

where up,k (0) is the horizontal velocity component at the surface of the kth particle. Again, the air-borne momentum flux at the surface is τa0 = ρu2∗ −τp0 . If τa0 > τt , it is reasonable to expect that the aerodynamic entrainment rate will be (6.95) nA = ηA (τa0 − τt ) where ηA is a coefficient with dimensions of [N−1 s−1 ]. Equation (6.95) shows that nA /ηA is a momentum flux (i.e. τa0 − τt ) and it can be expressed as A

nA = mup,k (0) ηA

n

k=1

To estimate the magnitude of ηA , we approximate this momentum flux with mnA αx u∗a0 , because the average of up,k (0) is expected to be proportional to u∗a0 with αx being the proportional constant. It follows that ηA ∝ (mαx u∗a0 )−1

6.12 Numerical Modelling of Saltation

191

For u∗a0 = 0.5 m s−1 and αx = 0.5, ηA ≈ 3 × 109 N−1 s−1 for 100 µm quartz particles (ρp = 2,600 kg m−3 ). Equation (6.95) can now be written as   u2∗t nA = ζA u∗a0 1 − 2 d−3 (6.96) u∗a0 6 where ζA = σp πα is a dimensionless constant around 1.74 × 10−3 for quartz x particles. At the beginning of saltation, τp0 = 0 and u∗a0 = u∗ , therefore, the maximum of nA is   u2∗t (6.97) nA,max = ζA u∗ 1 − 2 d−3 u∗

At the later stages of saltation, nA decreases. If τa0 = ρu2∗t at equilibrium saltation, as hypothesised by Owen (1964), then nA is zero. Equation (6.96) is a hypothesis that is not yet verified by experiment. For numerical simulation, ζA should be considered to be an adjustable parameter which has an order of magnitude around 10−3 until better estimates become available. 6.12.5 Splash Scheme A splash scheme is a probabilistic representation of particle rebound and ejection and the velocity distribution functions of the rebound and ejected particles. Most splash schemes (Anderson and Hallet, 1986; Shao and Li, 1999) are empirical, formulated on the basis of either wind-tunnel observations (Willetts and Rice, 1985, 1986, 1989; Rice et al. 1995, 1996a) or computational simulation, as described in Section 6.11.2. Much of the information used in existing splash schemes is hypothetical. Anderson and Haff (1991) suggested that the rebound probability is approximately (6.98) PR (VI ) = 0.95(1 − e−ζR VI ) where VI is impact velocity. This expression suggests that PR (VI ) monotonically increases from a value of zero at VI = 0 to an asymptotic maximum of 0.95. The coefficient ζR is found empirically to be around 2 s m−1 . If the impact particle-number flux is nI and the probability density function for a given impact velocity is p(VI ), then the rebound particle-number flux is ∞ nR = nI PR (VI )p(VI )dVI (6.99) 0

The probability density function for rebound velocity for a given impact velocity, i.e. p(VR | VI ), cannot yet be described with certainty. In existing studies, simple functional forms, such as the exponential distribution   1 VR (6.100) exp − p(VR | VI ) = αVI αVI

192

6 The Dynamics and Modelling of Saltation

or the gamma distribution k 1 p(VR | VI ) = 2 αVI



VR αVI

k

  VR exp −k αVI

(6.101)

have been used, where αVI is the mean velocity of VR for a given VI and k is the shape factor of the gamma distribution. Numerical and experimental studies show that α falls between 0.5 and 0.6 for typical saltations (Table 6.2). A simple assumption for the probability-density function for the rebound angle θR is that it is truncated Gaussian   1 (θR − θ¯R )2 √ p(θR ) = 0 ≤ θR ≤ π (6.102) exp − 2σθ2R Gn 2πσθR where

Gn = 0

π

  (θR − θ¯R )2 1 √ dθR exp − 2σθ2R 2πσθR

is a constant for normalisation. Again, the statistical parameters, θ¯R and σθR , need to be determined empirically. The observations of Rice et al. (1995), as shown in Table 6.2, give θ¯R = (5.64 − 7.7dI )θI

(6.103)

where dI is the size of the impacting particle in mm and θI is the impact angle. As a particle impacts the surface, it not only rebounds, but also ejects several particles in the process. It is reasonable to assume that the number of ejected particles, NS , at each impact is proportional to the kinetic energy EI of the impacting particles. It seems plausible to suggest that NS (VI ) = ζS

VI2 u2∗t

(6.104)

where ζS is the splash entrainment coefficient, which is dimensionless. If the impact particle-number flux is nI , then the splashed particle-number flux is ∞ nS = nI NS (VI )p(VI )dVI (6.105) 0

Similarly to Equation (6.100), the probability density function for velocity of splashed particles is   1 VS (6.106) exp − p(VS | VI ) = hVI hVI The observations given in Table 6.2 shows that h ≈ 0.08.

6.12 Numerical Modelling of Saltation

193

The kinetic energies available for rebound and splash are not independent of one another and it is useful to consider the energetic constraint on ζS to assure consistency in the model. If the impact velocity is VI , the kinetic energy available for splash and rebounding is mVI2 /2. From Equations (6.98) and (6.100), the energy retained by a rebounding particle is on average   ∞ 2 0.95m VR VR −ζR VI (1−e dVR = 0.95m(αVI )2 (1−e−ζR VI ) ) exp − ER = 2 αV αV I I 0 (6.107) Therefore, the energy available for splashing is ES =

ηS mVI2 − ER 2

(6.108)

where the coefficient, ηS , represents the fraction of energy not converted to heat. For example, for VI = 5 m s−1 and no energy being converted to heat (ηS = 1), about 40% of the total kinetic energy is available for splashing. In case ηS < 1, the proportion of energy available is even smaller. Since the splash rate is given by Equation (6.104) and the splash velocity distribution by Equation (6.106), the energy of splashed particles is   ∞ VS mS 2 dVS VS exp − (6.109) ES = NS 2hVI 0 hVI It follows that ES = 2h2 NS EI

(6.110)

Equations (6.107), (6.108) and (6.110) lead to the theoretical predictions that ηS − 1.9α2 (1 − e−ζR VI ) 2h2

(6.111)

ηS − 1.9α2 (1 − e−ζR VI ) u2∗t 2h2 VI2

(6.112)

NS = and ζS =

For VI = 5 m s−1 , h = 0.08, α = 0.6 and ηS = 1, NS is about 25. For the same case with u∗t,s = 0.3 m s−1 , ζS is about 0.1. The precise value of ηS is difficult to determine. In case NS = 0, ηS attains a value around 0.685. The observations of Rice et al. (1995) show that NS is normally around 2 or 3. This implies that ηS normally has a value around 0.7. An alternative is to estimate ζS from experimental data. Again, using the data given in Table 6.2, ζS is found to be dependent upon both dI and ds , the respective sizes of the impacting and ejected particles. An empirical expression for ζS is then: (6.113) ζS = a1 dI + a2 ds + a3 dI ds + a4 where a1 = 0.1414 × 10−3 , a2 = 0.3627 × 10−3 , a3 = −0.6948 × 10−6 and a4 = −0.0722, with dI and ds both in mm.

194

6 The Dynamics and Modelling of Saltation

6.13 Understanding of Saltation from Numerical Simulations Saltation models can produce quantitative predictions of saltation and are useful tools for improved understanding of saltation dynamics. In a sense, numerical simulations provide complementary information to experimental work and sometimes even replacement for the latter. Saltation models have been developed to study a range of questions including the variation of particle size distribution with height, the entrainment rates for particles in different size bins and the associated profiles of saltation fluxes, particle momentum fluxes and particle concentration (Sorensen, 1985). Anderson and Hallet (1986) developed a model for sediment transport for multiple particles and Anderson (1987) introduced the Lagrangian stochastic technique to sediment transport modelling. Jensen and Sorensen (1986) proposed a model for heterogeneous saltation and estimated the dislodgement rate and the probability distribution of lift-off velocity for each size class by finding the values of these quantities for which their model exactly predicts the transport profile obtained by Williams (1964). Despite considerable uncertainties in the saltation models, useful results have been obtained. 6.13.1 Importance of Splash Entrainment One question that has not been adequately answered is the role of splash entrainment in the process of saltation. There are two different hypotheses. On the one hand, the belief that splash is of fundamental importance to the evolution and equilibrium of saltation has motivated researchers to conduct a series of wind-tunnel experiments (e.g. Rice et al. 1995, 1996b) and to develop dedicated computer models (e.g. Anderson and Haff, 1991). On the other hand, Owen (1964) and Raupach (1991) have suggested that splash does not seriously affect the state of equilibrium saltation. Qualitative answers to this rather fundamental question can be obtained by simple calculations of particle motion in the atmospheric surface layer (Shao and Li, 1999). For these calculations, it is sufficient to assume that the wind profile is logarithmic with specified u∗ and z0 . Particles launched from the surface with different velocities and angles will saltate along the surface and, at impact, a proportion of their kinetic energy will be lost to the surface. Observations suggest that the kinetic energy of the particle at rebound is about half that at impact, i.e. ER = 0.5EI . Figure 6.24 shows the trajectories of 200 µm particles with different liftoff velocities for u∗ = 1 ms−1 . It shows that the saltation of particles with different initial velocities will either decay or enter a steady state. For particles with a small initial lift-off speed (for example, w0 = 0.1u∗ ), the saltation height and length decrease after each collision with the surface, as the energy gain of these particles from the airflow is not sufficient to compensate the energy loss to the ground during the collision. This tendency continues until

6.13 Understanding of Saltation from Numerical Simulations

195

Height (m)

0.0008 w0 = 0.1u* 0.0004

0.0000 0.000

0.002

0.004

Height (m)

0.8 w0 = 0.5u* w0 = 10u* 0.4

0.0

0

10 Distance (m)

20

Fig. 6.24. Particle trajectories with different lift-off velocities and a lift-off angle of 45◦ . The parameters used are u∗ = 1 m s−1 , z0 = 1 mm, d = 200 µm and ER /EI = 0.5. The motion of particles with a small lift-off velocity decays to zero (top), while particles with a large lift-off velocity enter the regime of steady-state motion (bottom)

the motion of the particles virtually becomes a creep and eventually ceases. For particles with a sufficiently-large lift-off velocity, the saltation height and length may increase or decrease (for example, w0 = 10u∗ ) after each collision with the surface. The motion of particles approaches a steady state with all particle trajectories becoming identical. The steady state is determined by the amount of energy which a particle absorbs from the airflow and the amount of energy it loses to the surface at impact. Figure 6.25 shows the relationship between the kinetic energy at impact, EI , and that at rebound, ER , for 100 µm particles under u∗ = 0.5 m s−1 and z0 = 2 mm. Assuming ER = 0.5EI , we would encounter three situations. In the first case (Fig. 6.25a), the particle has a small initial kinetic energy, so that 0 < ER < EC , where EC denotes some critical initial kinetic energy. The energy which the particle gains from the airflow during each hop is less than the energy it loses during the impact. Consequently, saltation becomes weaker after each hop, until the particle motion virtually ceases. In the second case (Fig. 6.25b), the initial kinetic energy of the particle falls between EC and ES (EC < ER < ES ), where ES denotes the kinetic energy of a particle at steady-state saltation. The energy which the particle gains during each hop increases until ER approaches ES . In the third case, the initial kinetic energy of the particle is larger than ES (ER > ES ), and the kinetic energy

196

6 The Dynamics and Modelling of Saltation 0.2

6 (a)

(b) c

EI

4 0.1 a

0.0 0.00

2

EC 0.05 ER

0.10

0 0

b

ES 1

2

3

ER

Fig. 6.25. Relationship between EI and ER for 100 µm particles under u∗ = 0.5 ms−1 and z0 = 2 mm, shown as the thick curves in (a) and (b). The thin dot-dashed diagonal lines represent the ER = 0.5EI relationship. (a) For a particle starting from position a, EI decays to zero; (b) For a particle starting from positions b and c, EI approaches ES . EC is the critical initial kinetic energy

of the particle rapidly decreases to ES . Both Figs. 6.24 and 6.25 show that a critical initial kinetic energy, EC , or a critical initial vertical lift-off velocity, w0c , exists that separates particles into two distinctly different categories of motion. It can be readily seen that w0c decreases with increasing u∗ as shown in Fig. 6.26 for 100, 200 and 300 µm particles. Two conclusions can be made from the above discussions. Firstly, for a given wind, only particles with a lift-off velocity larger than w0c enter steadystate saltation, while particles with a smaller lift-off velocity will stop moving after a few hops. Secondly, within a normal range of lift-off velocities, steadystate saltation occurs only if u∗ is large, while for small u∗ values, particle saltation would normally decay. Of course, only particles moving in the steady state have sufficient kinetic energy to create splash entrainment. The critical question is whether the saltation of the splashed particles by those in the steady-state motion generates sufficiently large lift-off velocity for them to also enter the steady-state motion, as, only if this is the case, can the splashing process play an important role in saltation. The impact velocity of particles in steady-state motion can be estimated from numerical calculation. For example, for 200 µm particles under u∗ = 1 ms−1 and z0 = 1 mm, it is 5.6 m s−1 . The initial velocity of the splashed particles would be 0.45 m s−1 , which is about 0.07 times the impacting velocity, according to the wind-tunnel observations of Rice et al. (1995). This velocity is larger than the critical lift-off velocity (around 0.25 m s−1 ) and ensures that the splashed particles also enter in the steady-state motion. In this case, splash entrainment creates a positive feedback and thus plays an important role in saltation. Figure 6.26 shows the lift-off velocity w0s of the splashed particles against u∗ . While w0c /u∗ decreases with u∗ , w0s /u∗ increases with

6.13 Understanding of Saltation from Numerical Simulations

197

w0c / u* , w0s / u*

1.5 w0c / u*, d = 100 micron 200 300 w0s / u*, d = 100 micron 200 300

1.0

Steady saltation Decay saltation 0.5

0.0 0.0

0.5

1.0

1.5

u* (ms−1)

Fig. 6.26. Critical lift-off velocity (thick lines) normalised by u∗ and the normalized lift-off velocity of the splashed particles (thin lines) as a function of u∗ . The relationships are shown for three particle sizes, d = 100, 200 and 300 µm. Full dots indicate the critical friction velocity, u∗c , for splashing to be effective (Modified from Shao and Li, 1999)

u∗ . The intersection of the two curves defines the critical friction velocity, u∗c , which depends on particle size. The interpretation of u∗c is clear: For u∗ > u∗c , the motion of splashed particles will enter the steady state, leading to increased splashing; for u∗ < u∗c , the motion of splashed particles decays and hence the importance of splashing diminishes. 6.13.2 Particle-Momentum Flux, Saltation Flux and Roughness Length Several other important features of saltation can be examined through numerical modelling, including the particle-momentum flux, the wind profile in the saltation layer, the saltation roughness length and the evolution of the streamwise saltation flux (Shao and Li, 1999). Figure 6.27 shows the simulated profiles of particle momentum flux, τp , for 200 µm particles and three different values of u∗ (u∗ = 0.5, 0.75 and 1 m s−1 ). The numerical simulation shows that τp decreases monotonically with height as expected, but the shape of the τp profile varies considerably with u∗ (and also particle size, not shown). Figure 6.28 shows the simulated evolution of the wind profiles at distances 10, 15 and 20 m from the leading edge of the erodible surface. The simulation shows that the wind profiles are modified near the surface by saltating particles, but are adjusted to a new equilibrium within several sec-

198

6 The Dynamics and Modelling of Saltation 0.3

(a)

(b)

−1

Simulated (u* = 0.75 ms ) Predicted (Gr = 0.7) Predicted (Gr = 1)

Predicted (Gr = 1)

Height (m)

(c)

−1

−1

Simulated (u* = 0.5 ms )

Simulated (u* = 1 ms ) Predicted (Gr = 0.5) Predicted (Gr = 1)

0.2

0.1

0.0 0.0

0.1

0.2

0.0

0.2

0.4

0.6

0.0

0.4

0.8

1.2

Particle momentum flux (Nm−2)

Fig. 6.27. Profiles of particle-momentum fluxes simulated using a numerical saltation model and those estimated using Equation (6.55). (a) For u∗ = 0.5 m s−1 and z0 = 2 mm; (b) as (a) but for u∗ = 0.75 m s−1 ; (c) as (a) but for u∗ = 1 m s−1 (From Shao and Li, 1999)

onds. The numerical simulation agrees with the wind-tunnel measurements of McKenna Neuman and Nickling (1994). As shown in Fig. 6.17, the estimates of streamwise saltation flux using the numerical saltation model compare reasonably well with the wind-tunnel measurements of Shao and Raupach (1992) for u∗ = 0.44, 0.5 and 0.6 m s−1 . This numerical simulation has reproduced a qualitatively similar behaviour of Q with distance x. Both the observed and simulated evolutions of Q show a “growth” stage, where Q increases with x, an “overshooting” stage, where Q maintains high values for several metres, and an “equilibration” stage, where Q gradually decreases with x towards a constant value with little streamwise gradient. The simulated and measured streamwise saltation fluxes showed a reasonable quantitative agreement for u∗ = 0.5 and 0.6 m s−1 , but the simulation under-predicts Q for u∗ = 0.44 m s−1 for x < 10 m. The numerical model also produces information which is otherwise difficult to obtain. For example, it is possible to model the profile of particle-borne momentum flux for given particle size and flow conditions. The simulated profile of τp (z) can be used to fit Equation (6.56) and estimate the Gr parameters for Equation (6.57). Figure 6.29 compares the numerically simulated z0s and its estimates using Equations (6.53) and (6.57) with the observed data of Gillette et al. (1998). The parameters used in the simulation are u∗t = 0.42 m s−1 , z0 = 0.25 mm and d = 90 µm, as reported by the latter authors. For this particular case, the calculations of z0s using Equations (6.53) and (6.57), as well as the numerical results, do not differ substantially. Gillette et al. have also reported on the excellent agreement between Equation (6.53) and the observed data. However, Shao and Li (1999) have argued that this agreement could be accidental, as z0s should have an explicit dependency on the size of

6.14 Saltation in Turbulence

199

2.0

z (m)

x = 10m t = 0s 0.5 5 10

x = 15m

x = 20m

1.0

0.0

1.00

z (m)

0.10

0.01

0.00 0

10

20 0

10 u

20 0

10

20

(ms−1)

Fig. 6.28. Simulated time evolution of wind profile at x=10, 15 and 20 m from the starting edge of the erodible surface. The initial wind profile is logarithmic (From Shao and Li, 1999)

the saltating particles. In general, it would be expected that estimates of z0s using Equations (6.57) and (6.53) might differ.

6.14 Saltation in Turbulence The theories we have discussed so far are mainly concerned with the mean features of saltation. In nature, saltation is a process that fluctuates both in space and time. The variability of saltation is determined by several factors including land-surface conditions, particle-flow interactions and atmospheric turbulence. As atmospheric boundary-layer flows are intrinsically turbulent, so are the entrainment and transport of sand grains. Recent observations suggest that the coherent eddies in the atmospheric boundary layer play an important

200

6 The Dynamics and Modelling of Saltation 0.006 0.005

Model, Raupach (1991) Model, Shao and Li (1999) Filed observation, Gillette et al. (1998) Simulation, Shao and Li (1999)

z0s (m)

0.004

0.003 0.002 0.001 0.000 0.2

0.4

0.6

0.8

1.0

1.2

1.4

u* (ms−1)

Fig. 6.29. Comparison of numerically simulated effective roughness length (full dots) and estimates using Equations (6.53) (dot-dashed line) and (6.57) (solid line) with the observed data of Gillette et al. (1998) (open circle)

Fig. 6.30. An image of a saltation field at Qira in the southern part of the Tarim Basin (Photo by M. Mikami, with acknowledgement)

role in saltation dynamics. Figure 6.30 shows an image of a saltation field, in which patches of intensive saltation and organized patterns of particle motion are identifiable. Stout and Zobeck (1997) found that even during a fairly strong erosion event, the condition of u∗ ≥ u∗t is only satisfied for some fractions of

6.14 Saltation in Turbulence

201

time and hence, saltation in general is an intermittent process, rather than a continuous process. Turbulence also very much affects particle motion. If turbulence is ignored, then the saltation-layer depth is approximately zm ∝

u2∗ 2g

For u∗ = 1 m s−1 , zm is around 5 mm. However, this height is much smaller than the saltation-layer depth of around 0.2–0.3 m often observed in windtunnel and field experiments. Shao (2005) found that the depth of the saltation layer simulated using a saltation model with turbulence is several times that simulated without turbulence. In Fig. 6.30, coherent features in the saltation field can be visually identified, such as the quasi-linear structures of high sand concentration stretching in the direction of wind, which are known as streamers. Baas and Sherman (2005) proposed that streamers are generated by near-surface gusts that originate from large eddies propagating to the ground from higher regions of the boundary layer. These elongated and stretched eddies scrape across the surface and initiate saltation along their path. Existing studies on saltation in turbulence are still preliminary, there is reason to believe that the coherent eddies in the atmospheric boundary layer play a particular role in the dynamics of saltation. These eddies generate gustiness in the surface layer and patchy momentum fluxes to the ground surface (Sterk et al. 1998) and are probably mainly responsible for the intermittency and patchiness of saltation as well as the formation of the coherent structures in saltation fields, such as streamers. 6.14.1 Intermittency of Saltation The intermittency of saltation, γint , can be defined as the fraction of time during which saltation occurs at a given point in a given time period. Figure 6.31a shows a time series of wind speed in the atmospheric surface layer. Saltation is expected to occur only in the time windows when wind speed U exceeds the threshold wind speed Ut (5 m s−1 in the example). Suppose p(U ), the probability density function of U , is known then γint is simply γint = 1 −

Ut

p(U )dU 0

Likewise, if friction velocity is used, then we have u∗t p(u∗ )du∗ γint = 1 − 0

Figure 6.31b is a plot of γint against Ut based on the data of U shown in Fig. 6.31a. In the example, γint is just above 0.4. Stout and Zobeck (1997)

202

6 The Dynamics and Modelling of Saltation 10 (a)

U (ms-1)

Ut 5

0

0

200

400

600

800

1000

Sample

1

γ

(b)

0.5

0

0

2

4 Ut (m s−1)

6

8

Fig. 6.31. (a) An illustration of intermittent saltation. Saltation occurs only in the time windows when U exceeds Ut ; (b) A plot of γint versus Ut . The intermittency of saltation γint rarely exceeds 0.5

used the counts per second of sand impacts on a piezoelectric crystal saltation sensor (SENSIT, see Chapter 11) as a measure of saltation activity and found that γint rarely exceeds 0.5. The intermittency of saltation must be related to the way how momentum is transferred from the atmosphere to the ground surface. As described in Chapter 3, the Reynolds shear stress in a turbulent flow can be understood as the eddy correlation between the streamwise velocity fluctuation (U  ) and the vertical velocity fluctuation w τ = −ρU  w which is related to the mean wind via the K-theory [Equation (3.30)]. The averaging time represented by the overline must be much larger than the typical time scale of the fluctuations. In atmospheric boundary layers, especially under convective conditions, eddies are not completely random but have coherent structures. Several types of coherent eddies, such as micro-bursts, vortex rolls and vorticies, are quite well known in atmospheric boundary-layer

6.14 Saltation in Turbulence

203

studies. These coherent eddies are very efficient in generating exchanges of momentum and other quantities between the atmosphere and the surface. At a given location, coherent eddies only occur in a fraction of time, but the momentum flux achieved in this time fraction may be many times the timeaveraged momentum flux. Likewise, at a given time, coherent eddies only affect a fraction of surface area, but the momentum flux to this fraction of area may be many times the space-averaged momentum flux. In other words, coherent eddies contribute significantly to the spatial and temporal variations of momentum fluxes. The response time of the saltation system has been estimated to be around one second (Sterk et al. 1998). It is thus obvious that under the influences of coherent eddies, saltation must also be intermittent in time and patchy in space. It is thus useful to examine the momentum flux on a time interval comparable with the response time of the saltation system (e.g. one second) and its correlations with the activity of saltation. We refer to this momentum flux as instantaneous momentum flux, τinst , which is by definition −ρU  w . The time series of τinst is characterized by intensive positive contributions superposed on weaker negative contributions. Sterk et al. (1998) reported that the values of τinst can be 10–20 times that of τ , but such large values only occur in limited periods of time. The probability distribution of τinst has a positive skewness, indicating the occurrence of large positive values is less frequent. The features of τ  can be studied conveniently using the quadrant technique. To this end, we categorize τinst into four types of events, namely, outward interaction, ejection, inward interaction and sweep (Fig. 6.32d). An outward interaction is an upward motion of high-velocity air from near the surface (U  > 0, w > 0, Quadrant I), while a sweep is a downward motion of high-velocity air (U  > 0, w < 0, Quadrant IV). An ejection is an upward motion of low-velocity air from near the surface (U  < 0, w > 0, Quadrant II), while an inward interaction is a downward motion of lowvelocity air (U  < 0, w < 0, Quadrant III). Both ejection and sweep contribute to a positive shear stress, while outward and inward interactions contribute to a negative shear stress. On average, shear stress is positive and hence, the contributions by ejection and sweep dominate over those by outward and inward interactions. The high values of instantaneous saltation, qinst , are associated with outward interactions and sweeps, while ejection and inward interactions are not capable of supporting appreciable saltation transport (Fig. 6.32a, b and c). However, a well-defined relationship between qinst and τinst does not seem to exist. Shao and Mikami (2005) found that even the 1-minute averages of saltation flux (measured at 0.3 m) and friction velocity are poorly correlated, although the 20-minute averages of these quantities are well correlated. Sterk et al. (1998) reported that the qinst is better correlated with U  (measured at 3 m above the ground surface) with a correlation coefficient of about 0.6. The poor correlation between qinst and τinst raises two questions. The first question is whether τinst measured at some level above the ground surface is

204

6 The Dynamics and Modelling of Saltation

w’ (m/s)

3

III

II

IV

I

0

(d) w’

(a)

II Ejection

U’(m/s)

−3 5

I Outward Interaction

0

U’ Inward Interaction III

(b) −5 500

IV

(c)

q (cts/s) 0

Sweep

0

1

2

3

Time (min)

Fig. 6.32. Saltation flux in relation to horizontal and vertical turbulent velocity fluctuations during a three-minute period of a wind-erosion event. (a) Vertical velocity fluctuation, (b) horizontal velocity fluctuation, (c) saltation flux in particle counts per second recorded using a saltiphone (modified from Sterk et al. 1998), and (d) a schematic illustration of the four-quadrant technique for studying instantaneous momentum flux, dashed lines represent the joint probability density function of U  and w

the same as, or at least similar to, τinst experienced by the ground surface at the same time. Indeed, this is doubtful. Although the U  fluctuations are know to persist from some height to the surface, the same does not apply to w . For example, a positive w at the 3 m level does not necessarily mean a positive w at surface. We could interpret U  as the average of U over a small time interval, which is by no means a steady-state and horizontallyhomogeneous quantity, and hence there is no reason for τinst to be vertically constant. Sterk et al. (1998) have argued that it is not the streamwise shear stress (represented by U  w ), but the horizontal drag force (represented by U  U  ) which is driving saltation. This argument is interesting and certainly appears to be correct as Fig. 6.32b and c suggest. However, we can interpret their argument as follows: The instantaneous shear stress at the surface is not identical to −ρU  w measured at some level above the ground, but is related to U  by τinst ∝ (U  )2 In other words, it is still the shear stress at the surface which drives saltation, but the instantaneous shear stress at the surface is better related to U 2 rather than U  w measured at some level above the ground.

6.14 Saltation in Turbulence

205

The second question is whether qinst is the best quantity to use for studying the relationship between the fluctuations of saltation and shear stress. The second hypothesis of Owen is not necessarily valid for instantaneous saltations, and there is no reason to expect that qinst must be closely correlated with τinst . Rather, the fluctuations in dislodgement rate, namely nA in Equation (6.95), may be better related to the fluctuations of shear stress. Both questions require further investigations. 6.14.2 Aeolian Streamers Aeolian streamers are elongated regions of intensive saltation in the streamwise direction (Fig. 6.30). Streamers meander laterally, merging and bifurcating as they move downwind. According to Baas and Sherman (2005), streamers can be grouped into family streamers, nested streamers and embedded streamers. Family streamers are individual elongated regions of pronounced saltation, separated by regions of minimal or no saltation. Such streamers occur under relatively low transport conditions. Nested streamers are multiple elongated regions of high transport levels superposed on largescale elongated features of base-level transport. Such streamers occur under medium transport conditions. Embedded streamers are elongated regions of exceptionally high transport embedded in a saltation field under high transport conditions. To quantify the streamers, we introduce the following three parameters: (1) Nstr , streamer number density or number of streamers per unit distance in the lateral direction; (2) Wstr , typical spanwise size of streamers; (3) qstr , the spanwise average of saltation flux. The preliminary results of Baas and Sherman (2005) indicate that Nstr is around 0.9 m−1 and Wstr around 0.2 m. These length scales appear to be stable and independent of general wind forcing, such as shear velocity. Streamers do not appear to be a persistent population of saltating grains that continually mark out the stream feature. Neither do they appear to be governed by the surface micro topographic features or the erodibility of the aeolian surface. Rather, streamers are most likely visual footprints of eddies which propagate from higher levels of the atmospheric boundary layer to the surface layer where they elongate and stretch out (Hunt and Morrison, 2000). It is believed that the mechanisms for the formation of aeolian streamers are similar to those for the development of drift-snow streamers, patches of coherent waving in cereal crops and cat paws on open water. All these phenomena are thought to result from sweeps of gusts originating from coherent eddies that scale with the depth of the boundary layer. The behaviour of streamers again underscores the fundamental importance of coherent eddies to saltation and it appears desirable in future saltation models to explicitly incorporate their contributions to particle entrainment and transport.

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6 The Dynamics and Modelling of Saltation

6.14.3 Dynamical Similarity of Saltation The similarity theories for atmospheric boundary-layer flows have been discussed in Chapter 3. It is likely that saltation is also dynamically similar and universal similarity functions can be derived to represent quantities such as particle concentration, streamwise saltation flux and particle-momentum flux. To illustrate this idea, let us consider the two-dimensional (x and z) saltation in a turbulent flow and assume that the surface is covered with sand particles of size d and the particles lift off with a fixed velocity V0 and a fixed angle θ0 . Then, the saltation trajectories of the particles can be calculated by integrating Equations (6.89) and (6.90). Based on the trajectories, as shown in Section 6.2, we can numerically evaluate c˜ [Equation (6.11)], q˜ [i.e. q˜x , Equation (6.12)] and τ˜p [i.e. τ˜pz , Equation(6.15)] as follows ∆t δ ∆z ∆t up δ q˜(z) = nm ∆z ∆t τ˜p (z) = nm −up wp δ ∆z c˜(z) = nm

(6.114) (6.115) (6.116)

where ∆t is the integration time step for the calculation of particle trajectories, ∆z is a small height interval and δ is 1 if the particle is located in the layer of z ± 1/2∆z, and 0 otherwise. The sum applies to the entire trajectory. If the particles of size d lift off with velocity V0 and angle θ0 which satisfy a joint probability density function p(V0 , θ0 ). Then, we can calculate particle concentration cˆ etc. as follows cˆ(z; d) = Fˆs (d)Jc (z; d) qˆ(z; d) = Fˆs (d)Jq (z; d) τˆp (z; d) = Fˆs (d)Jτ (z; d)

(6.117) (6.118) (6.119)

where Fˆs (d) = n(d)m(d) is the entrainment particle mass flux (or dislodgement rate, Sorensen, 2004). The J-functions are defined as ∆t ∞ π δ · p(V0 , θ0 )dV0 dθ0 (6.120) Jc (z; d) = ∆z 0 0 ∆t ∞ π Jq (z; d) = up δ · p(V0 , θ0 )dV0 dθ0 (6.121) ∆z 0 0 ∆t ∞ π Jτ (z; d) = −up wp δ · p(V0 , θ0 )dV0 dθ0 (6.122) ∆z 0 0 These functions are entirely fluid dynamic functions, if p(V0 , θ0 ) is specified. An analysis of the factors which affect particle motion in an atmospheric

6.14 Saltation in Turbulence

207

surface-layer flow suggests that the J functions must only depend on u∗ , z0 and d, because • u∗ affects wind profile and turbulence intensity. • z0 affects wind profile. • d affects particle response to turbulent flow and particle terminal fall velocity. We expect that the J functions are ‘universal’ functions of z if properly normalized. Several interesting relationships can be established if the J-functions are known. For instance, by making use of Equation (6.119), the dislodgement rate at equilibrium saltation is τ − τt (d) . Fˆs (d) = Jτ (0; d)

(6.123)

The above equation is a manifestation of the second hypothesis of Owen (τp0 = τ − τt ). From Equation (6.118), we also obtain
ˆ Fˆs (d) = Q(d)

−1

∞

Jq (z; d)dz

(6.124)

0

ˆ is the vertically-integrated streamwise saltation flux if the surface is where Q covered with uniform particles of size d. Shao (2005) has estimated Jq using a Lagrangian stochastic model for particle motion in atmospheric surface-layer flows. In the latter study, the profiles of wind and turbulence are specified according to the Monin–Obukhov similarity theory, and p(V0 , θ0 ) is given by p(V0 , θ0 ) = p(V0 )p(θ0 ) with p(V0 ) being an exponential distribution and p(θ0 ) a truncated Gaussian distribution. It has been found that   z Kq (6.125) Jq = Aq exp − Kq Lq where Aq is a dimensionless coefficient, Lq a scaling height and Kq a quantity that varies between 1 and 2. The meaning of Kq is clear. Because the particle response to turbulent fluctuations is particle-size dependent, Jq must also be particle-size dependent. The saltation of large particles is not strongly affected by turbulence because of their large inertial. Hence, Jq is very much determined by the initial conditions and is thus close to an exponential function [because p(V0 ) is exponential], i.e., Kq = 1. The motion of small particles is more strongly affected by turbulence. Hence, Jq is not determined by the initial conditions but the process of turbulent diffusion and is thus close to a

208

6 The Dynamics and Modelling of Saltation

Gaussian function, i.e., Kq = 2. The numerical results of Shao (2005) suggest that Aq , Lq and Kq are functions of u∗ , z0 and d. Therefore, Jq is a function of z with u∗ , z0 and d being parameters. Equation (6.125) is obtained under the assumptions that p(V0 ) is exponential and p(θ0 ) is truncated Gaussian. There are uncertainties associated with these assumptions because there is insufficient data to determine what exactly these distributions should be. Different p(V0 , θ0 ) distributions would result in different J functions. However, the essential (exponential) shape of Jq is consistent with the observations (Section 6.7.4) and thus the assumed distributions are not unreasonable. Suppose now the surface is covered with particles of multiple sizes. Following the discussions on independent saltation in Section 6.8, the entrainment mass flux for the kth particle-size bin Fs (dk ) is simply Pk Fˆs (dk ) and the streamwise saltation flux q(z; dk ) is then q(z; dk ) = Pk Fˆs (dk )Jq (z; dk )

(6.126)

It follows that the streamwise saltation flux for all particles is q(z) =

K

q(z; dk )

k=1

Because Jq is ‘universal’ and Fˆs can be estimated from either Equation (6.123) or (6.124), if Pk is known, then q(z; dk ) and q(z) are known. The saltation similarity theory thus allows the estimation of the profiles of streamwise saltation flux. Further, we have ∞

Q(dk ) =

q(z; dk )δz

(6.127)

Q(dk )

(6.128)

0

and Q=

K

k=1

The saltation similarity theory also allows the estimation of q at any level z. Figure 6.33 shows a comparison of the simulated q using the similarity theory

qmod

0.6

qobs u* 0.5

1

0 104.38

104.38

104.48

104.53

u* (ms−1)

q (gm−2s−1)

2

0.4 104.58

Time (LT, day)

Fig. 6.33. Simulated and observed time series of saltation flux q in g m−2 s−1 , together with friction velocity u∗ for 14 April 2002 at Qira

6.14 Saltation in Turbulence

209

(Shao and Mikami, 2005) and observed q for a site at the southern part of the Tarim Basin (Qira, 80.8◦ E, 36.9◦ N). The q measurements are obtained using the saltation particle counter (Yamada et al. 2002). The comparison shows that the theory well reproduced the main features of the observations. Another useful application of the similarity theory is that it allows the determination of the size distribution of saltation particles at level z. This particle-size distribution can be estimated by p(dk ; z) =

1 q(z; dk ) ∆k q(z)

(6.129)

with ∆k being the width of the kth particle-size bin. A reformulation of Equation (6.126) leads to (6.130) q(z; dk ) = q(zr ; dk )γ(z) where zr is a reference level and γ(z) = Jq (z; dk )/Jq (zr ; dk ) Therefore, it is possible to obtain particle-size distribution at any z, if it is known at zr , because by substituting Equation (6.130) into (6.129), one obtains

γ(z)p(dk ; zr ). (6.131) p(dk ; z) = γ(z)p(dk ; zr )/ At zr = 0, in particular, p(dk ; 0) is the particle-size distribution of saltating particles at the surface, which should be identical to the particle-size distribution of the erodible fraction of the parent soil. 0.2 Surface

qi /q

0.3 m

0.1

0.0 0

200

400

d (µm)

Fig. 6.34. A comparison of particle size distribution at z = 0.3 m with that of the parent soil

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6 The Dynamics and Modelling of Saltation

The predicted variation of particle-size distribution with height is as depicted in Fig. 6.34 which shows that the modal particle size decreases with height. Leys and McTainsh (1996) measured particle-size distributions at various levels between the surface and 2 m and found there is a significant decrease of the modal particle size with height below 0.5 m. The theory of Shao and Mikami (2005) is consistent with the observations of Leys and McTainsh (1996).

7 Dust Emission

Dust-emission rate is the vertical mass flux of dust at the surface. The importance of determining the dust-emission rate is twofold. (1) Dust emission is a key component of the global mineral-dust cycle and the associated nutrient cycle. Dust particles, once airborne, can be diffused by turbulence into the upper levels of the atmosphere and then carried over large distances by winds and eventually deposited back to the surface. This process depletes fine particles from the source regions of wind erosion. In agricultural areas, the loss of dust particles leads to gradual land degradation, because fine particles are mostly rich in nutrients and organic matter. Mineral dust is a major provider of nutrient to the oceans. (2) Dust produced by wind erosion is a major source of atmospheric aerosols which have impacts on air quality, weather and climate. It is not possible to calculate aerosol concentration in the atmosphere with certainty unless the dust-emission rate can be estimated with accuracy. Hence, the need for determining this quantity is obvious both for soil conservation and atmospheric studies. Dust emission over a large source areas can be estimated using diagnostic techniques (e.g. D’Almeida, 1986). This type of study requires reliable threedimensional measurements of dust concentration and the flow field, both of which are usually difficult to obtain for dust source regions. An alternative is to numerically estimate dust emission by means of dust-emission schemes. In this chapter, we examine the mechanisms for dust emission and study some of the existing dust schemes.

7.1 Dust Flux and Friction Velocity We distinguish dust-emission rate, F , from net dust-emission rate, FN . The latter is the difference between F and dust-deposition rate, FD , i.e., FN = F − FD Y. Shao, Physics and Modelling of Wind Erosion, c Springer Science+Business Media B.V. 2008


211

212

7 Dust Emission

We shall study FD in Chapter 8. It appears that F has never been measured directly in practice. Rather, it is inferred from the measurements of the vertical flux of airborne dust in the atmospheric surface layer, Fair , which is constant with height under the assumptions of steady state and horizontal homogeneity. In this respect, Fair is similar to the fluxes of other passive scalars. However, Fair itself consists of a component due to turbulent and molecular diffusion, Fair,d , and a component due to gravitational settling, Fair,s . By analogy to scalar fluxes, the diffusive dust flux can be estimated from the profile of dust concentration by means of Fair,d = −ρKp

∂c ∂z

(7.1)

where ρ is air density and Kp is the eddy diffusivity for dust particles and c is the specific dust concentration. Kp is affected both by atmospheric turbulence and by the particle response to turbulent fluctuations (e.g. the particletrajectory crossing effect and the particle-inertial effect, Chapter 8). Hence, Kp is dependent on dust-particle size for given flow conditions. However, as an approximation, Kp can be assumed to be equal to the eddy diffusivity for scalar and expressed as Kp = Km = κu∗ z for neutral atmospheric boundary-layer flows. For stable and unstable conditions, Km can be correspondingly modified by using the Monin–Obukhov similarity theory as outlined in Chapter 3. Suppose the dust-concentration field is in steady state and is horizontally homogeneous. The dust-concentration equation, Equation (3.24), yields a simple balance of gravitational settling and turbulent diffusion   ∂ ∂c ∂c + Kp =0 (7.2) wt ∂z ∂z ∂z where wt is the settling velocity. An integration of the above equation over z gives ∂c = F0 (7.3) −ρwt c − Kp ρ ∂z where F0 is a constant. Equation (7.3) implies that Fair is constant with height (namely, −ρwt c − Kp ρ∂c/∂z), but Fair,d and Fair,s (namely, −ρwt c) is not. Equation (7.3) can be further integrated to yield the steady state profile of dust concentration   −wt /κu∗  z F0 F0 − (7.4) c = cr + ρwt zr ρwt with cr being the dust concentration at the reference height, zr . Equation (7.4) implies that the dust-concentration profile obeys the power-law with c increasing with decreasing height, in contrast to the profile of a scalar, which commonly obeys the log-law. Suppose F0 = 0, then we have

7.1 Dust Flux and Friction Velocity

 c = cr

z zr

213

−wt /κu∗ (7.5)

Given this dust-concentration profile, we can see from Equation (7.3) that the magnitudes of both Fair,d and Fair,s increase with decreasing height. The conclusion of the above analysis is that F is not identical Fair,d and Fair,d is height dependent. For a steady-state dust concentration field with no dust emission, i.e. F0 = 0, Fair,d also obeys a power law  Fair,d = ρwt cr

z zr

−wt /κu∗ (7.6)

In most field experiments on dust emission reported in the literature, only Fair,d has been measured, and we have implicitly interpreted that F = Fair,d We must bear this interpretation in mind when we use the dust-emission measurements. Gillette (1977) reported probably the most comprehensive set of field measurements of F for 9 different soils and examined the relationship between F and u∗ . In his experiments, dust was collected using a specially-designed membrane filter placed at two different heights, and F was calculated using the discrete version of Equation (7.1). The observed values of F are plotted against u∗ in Fig. 7.1a (for 7 out of the 9 soils), together with the observed verticallyintegrated streamwise saltation flux, Q. Figure 7.1b shows that the measured saltation flux is well described by the Q ∝ u3∗ relationship except for the sandyclay loam case and the clay loam case. In contrast, although F generally increases with u∗ , there is a large scatter among the measurements and it is not possible to describe the data with a definitive F (u∗ ) relationship (Fig. 7.1a). Nickling and Gillies (1989, 1993) examined the relationship between dust emission and u∗ using data sets obtained in Mali (West Africa), Texas (USA) and Yukon (Canada). They reported dust-emission rate measured at 13 experimental sites in southern Arizona by means of a portable wind tunnel. The sites were selected on the basis of soil texture, vegetative cover, land use and anthropogenic disturbance. A considerable scatter also exists in their data if no distinction is made between the soil-surface features. To investigate the scatter in the data, Nickling and Gillies classified the experimental sites based on surface morphology and land use and grouped the data sets accordingly. Five classes were established, as follows: (1) Natural undisturbed desert sites; (2) Sites developed or modified by fluvial processes; (3) Construction sites; (4) Mine Tailings, and (5) Agricultural sites. The regression relationships derived for the first four classes are: • Natural undisturbed desert sites: F ∝ u2.99 ∗ • Sites developed or modified by fluvial processes: F ∝ u3.32 ∗

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7 Dust Emission

10−4 Sand Sand Loamy sand Sand Sand Sandy clay loam Clay loam

F (kg m−2 s−1)

10−5 10−6

(a)

10−7 10−8 10−9 0.1

u* (ms−1)

10−1

10−1

(c)

(b) 10−2 F/ Q m−1 )

Q (kg m−1 s−1)

10−2

1.0

10−3 10−4

10−3 10−4

10−5 10−6 0.1

1.0 u* (ms−1)

10−5 0.1

1.0 u* (ms−1)

Fig. 7.1. (a) Vertical diffusive flux of particles smaller than 20 µm versus friction velocity for nine different soil surfaces; (b) as (a), but for streamwise saltation flux; (c) as (a), but for the ratio of vertical diffusive dust flux and streamwise saltation flux (Redrawn from Gillette, 1977)

• •

Construction sites: F ∝ u4.24 ∗ Mine Tailings: F ∝ u2.93 ∗

A meaningful relationship could not be found for the agricultural sites. The same data set has been partitioned on the basis of the percentage of silt and and clay in the surface sediments. For silty loam, the regression is F ∝ u4.27 ∗ . Similar for sandy soils (silt and clay < 15%), the regression is F ∝ u3.03 ∗ results were obtained in the dust-emission experiment over Mali for five types although a considerable of surface condition. It has been found that F ∝ u4.38 ∗ scatter exists in the data. The above-mentioned measurements show that the vertical dust flux is proportional to un∗ with n varying between 2.9 and 4.4. Gillette and Passi (1988) suggested that the dust-emission rate is proportional to u4∗ in theory, i.e.,

7.1 Dust Flux and Friction Velocity

 F = αg u4∗ 1 −

u∗t u∗

215

 u∗ ≥ u∗t

(7.7)

where u∗t is the saltation threshold friction velocity of the surface and αg is a dimensional coefficient ([αg ] = M L−6 T−5 ). A detailed examination of Fig. 7.1 indicates that this hypothesis does not seem to be strongly supported by the observed data: while Q is linearly related to u3∗ (Fig. 7.1b), F/Q is almost independent of u∗ (Fig. 7.1c). Several other data sets have been published in the literature, as summarized in Table 7.1. These data sets are plotted in Fig. 7.2. As seen, for given Table 7.1. Summary of dust-emission observations. G77: Gillette (1977); N83: Nickling (1983); NG93: Nickling and Gillies (1993); N99: Nickling et al. (1999); G03: Gomes et al. (2003); R03: Rajot et al. (2003) Reference G77 N83 NG93

Site Texas, USA Yukon, Canada Mali, West Africa

N99 G03 R03

Diamantina, Qld, Australia Spain Niger, Sahel

Soil type Sand, loam, sandy loam, clay Sand, silt Dune, bare-crusted surface, shrub Savanna, rice, millet fields Clay pan Silty clay loam Crusted sandy soil

105

104

103 3

u*

F (ug m−2 s−1)

F~ 102

4

u*

F~ 101

5

u F~ *

Yukon (Nickling 1983) Mali (Nickling and Gillies 1993) Australia (Nickling et al. 1999) Texas, Loam (Gillette 1977) Texas, Sandy (Gillette 1977) Texas, Sandy Loam (Gillette 1977) Texas, Clay (Gillette 1977) Spain (Gomes et al. 2003) Niger (Rajot et al. 2003)

100

10−1

10−2 0.1

1.0 u* (ms−1)

Fig. 7.2. Observed dust-emission rates under various conditions, see Table 7.1 for the references

216

7 Dust Emission

u∗ , dust emission varies from 10−1 to 105 µg m−2 s−1 . This indicates that dust emission depends strongly on soil type and soil-surface conditions.

7.2 Mechanisms for Dust Emission It is necessary to examine in detail the dust-emission mechanisms so that better dust-emission schemes can be developed. The mechanisms leading to the entrainment of sand and dust particles differ because the relative importance of the forces acting on them changes with particle size. As shown in Chapter 5, the lift-off of sand particles is determined primarily by the balance between the aerodynamic and gravity forces. For smaller particles, the dominance of the gravity force diminishes and the inter-particle cohesion becomes important. The relative importance of the forces acting on the particles in relation to particle size has been shown in Fig. 5.14. As discussed there, the gravity force is proportional to d3 , the aerodynamic force is proportional to d2 , and although large uncertainties exist in the estimates of cohesive forces, we can assume that the total cohesive force is proportional to d. Figure 5.12 indicates that for particles with d < 20 µm, the cohesive force begins to dominate and hence particles cannot be easily lifted from the surface by aerodynamic forces. As also pointed out in Chapter 5, because particle-to-particle and particle-to-surface cohesions are influenced by a number of complicated factors, it is more rational to consider the total cohesive force, and hence the threshold friction velocity for dust particles, as a stochastic variable which satisfies a probabilistic distribution. Indeed, a wide range of scatter in the measurements of cohesive forces occurs for nearly identical macroscopic conditions. This scatter increases with decreasing particle size, and the spread can be several orders of magnitude (Fig. 5.16). Dust particles under natural conditions exist as dust coats attached to sand grains in sandy soils or as aggregates in clay soils. Figure 7.3 shows an image of sand-sized grains taken by using an electron microscope. Fine particles coated on the grains are clearly visible. During weak wind-erosion events, sand particles coated with dusts and clay aggregates behave as individuals and dust particles may not be released, while during strong wind-erosion events, dust coats and soil aggregates may disintegrate resulting in strong dust emission. Then, what is the amount of dust in a unit soil mass and what is the amount of dust that can be released from a unit soil mass? The maximum amount of dust that can be released from a unit soil mass is defined as the dust-emission potential. It is of paramount importance to know what is the particle-size distribution of the parent soil and how it is related to that of the airborne particles. But what exactly do we mean by particle-size distribution? If a soil sample is collected from the field and its particle-size distribution is analysed in a laboratory, then different laboratory techniques would produce different results because the sizes of soil aggregates can vary with the degree of destructive

7.2 Mechanisms for Dust Emission

217

Fig. 7.3. An image of sand-sized soil particles taken by using an electron microscope. The soil sample is collected from a farmland near Miudura, Australia (M. Ishizuka, with acknowledgement)

forces applied to the soil sample. For example, the particle-size distribution measured by sieving and by laser diffraction following chemical dispersion will be very different. This means particle-size distribution is relative. Likewise, the size distribution of airborne particles ‘seen’ by wind during a wind-erosion event must also depend on the strength of the destructive force exerted on the aeolian particles as they hop along the surface. We distinguish four different particle-size distributions •

ps (d): sediment particle-size distribution: The word “sediment” here refers to airborne particles close to the surface. • pm (d): minimally-disturbed particle-size distribution of the parent soil; The phrase “minimally disturbed” refers to the limiting case in which the disturbance is so weak that the breakup of aggregates does not occur. • pf (d): fully-disturbed particle-size distribution of the parent soil; The phrase “fully disturbed” refers to the limiting case in which aggregates are as much broken up as possible by mechanical forces. • pf c (d): chemically-dispersed particle-size distribution of the parent soil; The phrase “chemically dispersed” refers to the limiting case in which aggregates break up into their basic particle sizes.

Both pm (d), pf (d) and pf c (d) are idealized particle-size distributions which are soil specific and independent on wind-erosion intensity.

218

7 Dust Emission

Now the potential of dust emission can be estimated by making use of the fully-disturbed particle-size distribution. Suppose the maximum size for dust particles is dd , then the mass fraction of dust, ηf , can be determined as dd pf (d)δd (7.8) ηf = 0

If the total amount of the disturbed soil during a wind erosion event is M , then the maximum amount of dust emission possible would be ηf M . This basic consideration will be used in dust-emission schemes described later in the chapter. In practice, pm and pf can only be approximated. We imagine that pm is obtained by using methods that cause as little disturbance as possible to the soil sample, while pf is obtained by using methods that cause as much mechanical disturbance as possible to the soil sample. To estimate the mass percentage of PM10 (particulate matter with diameter smaller than 10 µm) available for suspension from a soil, Saxton et al. (2000) adapted a simple, single air-burst resuspension procedure. This method measures the amount of dust that is free in the soil sample. The word ‘free’ is used here in the sense that dust particles can be easily blown off by wind. Chandler et al. (2002) developed an instrument which couples a self-abrader emitter with a particulate monitoring device for measuring PM10 and PM2.5 (particulate matter with diameter smaller than 2.5 µm) emission potentials from soils. Their measurements (Table 7.2) confirm that in a soil, a fraction of dust is free (the airburst fraction) and a fraction of dust is aggregated. A proportion of the aggregated dust can be freed through mechanical abrasion (the abrader fraction) while the rest of it can only be freed by means of chemical dispersion (the dispersed fraction). It appears reasonable to suggest that three levels of binding energies exist, namely, ψm , ψf and ψcf , representing respectively the levels of binding energy for Table 7.2. Mass fraction (in %) of PM10 and PM2.5 of eight different soils measured using three different techniques, namely, airburst, abrader and chemical dispersion. For each technique the mean and the standard deviation are shown (Modified from Chandler et al. 2002) Soil

Dq Ds L1 L2 L3 L4 L5 TX

PM10 Airburst ave (std) 0.2 (0.2) 0.5 (0.2) 0.6 (0.1) 0.8 (0.5) 0.6 (0.3) 1.1 (0.7) 0.7 (0.1)

Abrader ave (std) 1.5 (0.8) 2.4 (0.1) 2.1 (0.7) 2.6 (0.6) 3.1 (0.8) 3.9 (1.2) 4.1 (0.1) 0.6 (0.2)

PM2.5 Dispersed ave (std) 18.7 (7.4) 25.9 (3.5) 24.6 (5.7) 28.1 (4.4) 28.4 (3.8) 32.2 (4.5) 37.8 (5.0) 16.0 (3.0)

Airburst ave (std) 0.2 (0.1) 0.4 (0.2) 0.4 (0.1) 0.3 (0.1) 0.3 (0.1) 0.5 (0.3) 0.2 (0.1)

Abrader ave (std) 0.5 (0.3) 1.3 (0.4) 1.0 (0.4) 1.4 (0.5) 1.2 (0.2) 1.9 (0.8) 1.2 (0.3)

Dispersed ave (std) 8.0 (2.2) 11.7 (1.5) 9.6 (1.8) 11.9 (1.6) 11.7 (1.0) 11.8 (1.8) 14.7 (1.2)

7.2 Mechanisms for Dust Emission η

219

(a)

ηf ηm

ψm

Chemical Dispersible

Free

Mechanical Dispersible

p(ψ)

ψf

(b)

ψ

Fig. 7.4. Illustration of dust-emission potential. (a) Dust mass fraction η as a function of binding energy; ηm is the mass fraction of free dust and ηf is that of the mechanically dispersible dust; (b) Probability distribution function of dust binding energy, p(ψ), and the levels of binding energy defining free dust, mechanicallydispersible dust and chemically-dispersible dust

the free dust, mechanically-dispersible dust and chemically-dispersible dust (Fig. 7.4). In practice, pm can be approximated by a particle-size distribution measured using the airburst technique described by Saxton et al. (2000), and pf can be approximated by a particle-size distribution measured using the self-abrader technique described by Chandler et al. (2002). Note that during wind erosion, soil particles do not undergo chemical dispersion. In the light of the above discussions, we propose three dust-emission mechanisms as depicted in Fig. 7.5. 1. Aerodynamic Lift: Dust particles can be lifted from the surface directly by aerodynamic forces (Loosmore and Hunt, 2000). However, as the importance of gravity and aerodynamic forces diminishes for smaller particles and the inter-particle cohesion becomes more important, dust emission arising from direct aerodynamic lift is probably small in general. 2. Saltation Bombardment: As saltating particles (sand grains or aggregates) strike the surface, they cause localized impacts which are strong enough to overcome the binding forces acting upon dust particles, leading to dust emission (Gillette, 1981). This mechanism is also known as sand blasting (Alfaro et al. 1997). The wind-tunnel experiments of Shao et al. (1993b) show that the dust-emission rate caused by this mechanism can be an order of magnitude larger than the aerodynamic entrainment.

220

7 Dust Emission

(a) Aerodynamic Entrainment

U

(b) Saltation Bombardment

Sand Grain

(c) Aggregates Disintegration

Aggregate

Fig. 7.5. Mechanisms for dust emission. (a) Dust emission by aerodynamic lift, (b) by saltation bombardment and (c) through disaggregation

3. Disaggregation: Under natural conditions, dust particles may exist as dustcoats attached to sand grains in sandy soils or as aggregates in soils with a high clay content. During a weak wind erosion event, sand particles coated with dusts and clay aggregates behave as individuals and dust particles may not be released, while during a strong wind-erosion event, dustcoats and soil aggregates may disintegrate resulting in increased dust emission. The importance of this mechanism is probably similar to that of saltation bombardment. This mechanism is also known as auto-abrasion. We can formally express the dust-emission rate arising from these three mechanisms as (7.9) F = F a + Fb + Fc Shao et al. (1993b) examined the emission of loosely packed dust particles caused by saltation bombardment in a wind-tunnel experiment. In that experiment, they placed two beds of material in the wind tunnel: an upstream sand bed of length 1 m, which produced a supply of saltating grains, followed immediately by a dust bed of length 2 m, which was subject to saltation bombardment. They used combinations of four sand- and three dust-particle sizes. The sand was an oven-dried aeolian red soil, sieved into classes of 100– 210, 210–400, 210–530 and 530–1,000 µm with averages around 150, 250, 300

7.2 Mechanisms for Dust Emission 5

F (g m−2 s−1)

4

25

UR = 8.3 ms−1 9.8 11.1 12.9

(a)

3

(b) 20 15

2

10

1

5

0

0

120

240 Time (s)

360

221

480

0

0

120

240

360

480

Time (s)

Fig. 7.6. Wind-tunnel observations of dust fluxes at four different wind speeds for (a) the ‘pure dust’ configuration and (b) the ‘bombardment’ configurations. In the former case, no saltation was introduced, while in the latter case, sand particles ranging between 210 to 530 µm were introduced upstream of the dust bed (From Shao et al. 1993b)

and 600 µm, respectively. The three kinds of dust were kaolin clay powder and two types of talc with mean particle sizes of 3, 11 and 19 µm, respectively. Streamwise fluxes for both sand and dust were measured and dust emission was derived from the latter. Shao et al. (1993b) reported that there was little dust emission even at the maximum flow speed which the tunnel generated (about 20 m s−1 ) if no saltation particles were introduced, while strong dust emission occurred if sand particles were propelled over the dust surface. Figure 7.6 compares the dust-emission rate measured at four different wind speeds with and without saltation being introduced. For the “pure dust” configuration (no saltation), the dust-emission rate decayed rapidly with time, becoming negligible less than 200 s after the onset of the wind. The initial dust fluxes, which were small in comparison with the fluxes induced by saltation bombardment (Fig. 7.6b), were caused by the removal of loose dust particles from the newly prepared bed; once these particles were removed, the dust bed stabilised and was not subject to further erosion. For the bombardment configuration (saltation introduced), the dust flux was substantially larger than even the maximum value produced by aerodynamic forces alone at a corresponding wind speed, and was sustained for as long as there was a supply of both sand and dust. (For the high-wind speed cases, the rapid decreases at the end of each run were caused by exhaustion of the supply of source material, either sand or dust.) Figure 7.7 shows the streamwise saltation flux, Q, and the streamwise dust flux, Qd , (for the particular wind-tunnel configuration, Qd = Ld F , where Ld = 2 m is the length of the dust bed), as a function of wind speed for the configuration of 210–530 µm sands saltating over a bed of 3 µm dusts. The experiments clearly show that F is proportional to Q and confirm that saltation bombardment is a major mechanism responsible for dust emission.

222

7 Dust Emission

ln Q (gm−1s−1)

6

Total Sand Dust

4

2

2.0

2.2

2.4

2.6

ln U (ms−1)

Fig. 7.7. Wind-tunnel observations of streamwise sand flux, streamwise dust flux and total streamwise flux (sand plus dust) plotted against wind speed (From Shao et al. 1993b)

7.3 Aerodynamic Dust Entrainment It is useful to distinguish the dust emission in the absence of saltation from that in the presence of saltation, namely,  no saltation Fa (7.10) F = Fb + Fc with saltation Loosmore and Hunt (2000) carried out wind-tunnel experiments on dust emission with an uncrusted, fine material at relative humidity exceeding 40%. They found that a long-term steady dust flux occurs in the absence of saltation, which fits to the approximate form Fa = 3.6u3∗

(7.11)

where Fa is in µg m−2 s−1 and u∗ is in m s−1 . The above relationship shows that Fa is small in comparison with the dust-emission rates observed during wind-erosion events (Fig. 7.2). The occurrence of a weak dust emission in the absence of saltation is expected because there always exist free dust particles which can be blown off the surface even by weak winds.

7.4 Energy-Based Dust-Emission Scheme As saltation bombardment is a major mechanism for dust emission, the dependence of dust emission upon wind velocity is indirect: It involves the relationship between wind and saltation as an intermediate process. It is

7.4 Energy-Based Dust-Emission Scheme

223

conceptually important that this mechanism is represented in dust-emission schemes. Therefore, a dust-emission scheme should consist of three basic components: namely, (1) a quantitative description of saltation intensity for given wind, surface and soil conditions; (2) a statistical representation of saltation bombardment for given sand-particle size, including particle trajectory, impact-particle velocity and angle; and (3) a description of the binding strength of dust particles and the relationship between dust-mission rate and the intensity of saltation. Answers to the first two questions can be found largely in Chapter 6 and thus the major task of developing a dust-emission scheme is to establish a relationship between the dust-emission rate and saltation intensity, by taking account of the binding strength of dust particles. Only a few dust-emission schemes have been developed so far, but it is useful to classify them into energy-based and volume-removal based schemes, on the basis of how dust emission is related to saltation. The immediate challenge we face is how to quantify the binding strength of dust particles. We have argued in Chapter 5 that using threshold friction velocity as a description of binding strength is no longer adequate, since aerodynamic lift is no longer the primary mechanism for dust emission. Shao et al. (1993b) proposed using binding energy as a descriptor of particle-binding strength. The binding energy, ψ, of a dust particle is equal to the energy required to dislodge the particle from the potential well induced by all binding forces. The concept of binding energy is useful as an alternative to using the concept of cohesive forces, as the latter involves the necessity of determining the directions of these forces. A simple dust-emission scheme can be derived based on the energy balance of a saltating particle during the particle and surface collision. We first consider the case in which a particle of size ds saltates over a surface of dust of size dd . The energy supplied to the surface by the saltator during the collision is ∆E = EI − ER , where EI is the kinetic energy of the particle at impact and ER is the kinetic energy retained by the particle at rebound. (In case the saltator also ejects other particles, ER is the total kinetic energy of all particles ejected from the surface, including the rebounding particle.) If J is the total number of dust particles ejected by the bombardment of the saltator, then J satisfies (7.12) J(ψ + ψk ) = ζe ∆E where ψ and ψk are the binding energy and the initial kinetic energy of a dust particle, respectively; and ζe is a constant of proportionality which is less than 1 because only a fraction of ∆E is used for dislodging dust particles and the rest is lost as heat. If the saltation-impact particle-number flux is nI , then the dust-emission rate is F = md nI J

(7.13)

where md is the mass of a dust particle. The saltation particle-borne momentum flux at the surface associated with nI is τp0 = nI ms (upI − upR )

224

7 Dust Emission

where ms is the mass of a saltator and upI and upR are the streamwise velocity components of a typical saltator at impact and rebound, respectively. Using the Owen hypothesis [Equation (6.25)], we find that nI =

Qg ρu2∗ (1 − u2∗t /u2∗ ) = ms (upI − upR ) co ms u∗ (upI − upR )

(7.14)

where Q is the saltation flux of particles with size ds . To relate ∆E to upI and upR , we approximate ∆E =

ms 2 (u − u2pR ) 2 pI

(7.15)

It follows from Equations (7.12) to (7.15) that F = αe with αe =

md gQ ψ + ψk

(7.16)

ζe upI + upR co 2u∗

The above equation shows that the two main factors which affect dust emission, namely the intensity of saltation and the dust binding strength, are embodied in the proportionality F ∝ md gQ/(ψ + ψk ). F is predicted to be proportional to Q and hence proportional to u3∗ if u∗ is much larger than the threshold friction velocity for saltation. For a given Q, F is inversely proportional to (ψ + ψk ). The above dust-emission scheme is similar to the aeolian-abrasion theory as a weathering process (Dietrich, 1977). The latter authors have concluded that the fundamental parameters which control aeolian abrasion are the kinetic energy of the impacting grain and the bond strength of the abraded material. This has been confirmed by Greeley et al. (1982), who have investigated the susceptibility of surfaces to abrasion, Ar , defined as the mass of material eroded per particle impact. For a given size of impact particle, they have found that Ar is proportional to d3 , where d is the impacting-particle diameter. Hence the combined relationship is that Ar is proportional to the kinetic energy of the impacting particle Ar ∝ d3 v 2 ∝ mv 2 /2

(7.17)

where m is the impacting-particle mass. Hence, the findings are consistent with the above-described dust-emission model. The coefficient αe in Equation (7.16) is the product of two dimensionless ratios, ζe /co and (upI + upR )/2u∗ . In essence, ζe /co represents the proportion of the incoming bombardment energy used for breaking inter-particle bonds, while (upI + upR )/(2u∗ ) is a descriptor of the typical strength of saltation impact. For u∗ u∗t , (upI + upR )/(2u∗ ) is of order 5 and must decrease to

7.4 Energy-Based Dust-Emission Scheme

225

zero as u∗ → u∗t . Therefore, αe in general must depend on u∗ and the size of the saltator, ds . During a strong erosion event, the order of magnitude of αe is five times that of ζe , i.e., O(αe ) = O(5ζe ). It would be close to 1, if we take O(ζe ) = 0.2, O(co ) = 1 and O[(upI + upR )/(2u∗ )] = 5. However, since ζe is poorly understood, αe must be treated as an empirical parameter for practical use, which needs to be calibrated with observational data. The order of magnitude of ψk can be estimated. By definition ψk =

md 2 V 2 dust

where Vdust is the lift-off velocity of dust particles. It is plausible to assume that Vdust ∝ (upI +upR )/2. Therefore, for a 10 µm particle ψk would be around 1.36 × 10−12 J while for a 1 µm particle, 1.36 × 10−15 J. The difficulty of Equation (7.16) is that the binding energy between dust particles, ψ, cannot be estimated with accuracy from measurements or theory. Suppose that the binding energy between individual particles is mainly determined by the van der Waals attraction and the double layer Verwey and Overbeek repulsion (e.g. Theodoor and Overbeek, 1985). Then the attraction potential decays with r−1 (r is distance between the particles) and the repulsive potential decays exponentially with r. Although these theories give us physical understanding, parameter uncertainties make it impossible to use them to estimate ψ. In addition, there are other factors which influence ψ, such as static electricity and the existence of binding agents, such as moisture and salt. The following discussion on the order of magnitude of ψ is speculative. If the magnitude of the combined cohesive force is χi , then ψ can be expressed as rmax χi (r)dr ≈ χ¯i rL (7.18) ψ= rmin

where rmin is the minimum separation allowed between the dust particles below which repulsive forces become important and rmax is the maximum separation within which the inter-particle cohesive forces are effective. Here, χ¯i is the mean value of χi and rL = rmax − rmin . If the separation between particles is larger than rmax , then the cohesive forces are negligible. The magnitude of rmin and rmax are not well known, however. A speculative assumption is that rmax = 0.2 d with d being the particle diameter, and a widely-cited value for rmin in colloidal science is 0.4 nm. As rmax is much larger than rmin , we have that rL ≈ rmax . If we assume χi = βc d [where βc ≈ 10−5 Nm−1 from Equation (5.41)], as discussed in Chapter 5, then ψ = 0.2βc d2 (7.19) The order magnitude of ψ is 2 × 10−16 J for a 10 µm particle and 2 × 10−18 J for a 1 µm particle. Again, we encounter large uncertainties here, because the above estimates put ψ several orders of magnitude smaller than ψk . As pointed

226

7 Dust Emission

out in Chapter 5, even in laboratory tests, ψ may vary over a wide range. The speculative estimates given above probably correspond to the lower limits of ψ values. It would be more realistic to treat ψ as a stochastic variable which possesses a probabilistic distribution. One pragmatic approach is to relate ψ+ψk to the threshold friction velocity of dust particles u∗t,d (Shao et al. 1996), which we treat as an observable quantity. Here we assume ψ + ψk to be the product of the drag force acting over an area of πd2d /4 and a length scale which is proportional to dd , so that ψ + ψk ∝

π 2 ρu d3 . 4 ∗t,d d

(7.20)

Under this assumption, Equation (7.16) can be written as F = βQu−2 ∗t,d

(7.21)

where β = β(dd , ds ) is an empirical function of dd and ds with dimensions of [ms−2 ]. The dependency of β on dd and ds can be understood from Equation (7.16), where we have pointed out that αe must be a function of ds and ψ +ψk must be a function of dd . Wind-tunnel observations confirm this dependency. Shao et al. (1993b) observed that the efficiency of saltation bombardment, namely the ratio F/Q, is approximately independent of wind speed for a specified saltation particle-size class, but increases with saltation-particle size at a given wind speed and decreases with dust-particle size. In order to estimate β from the small data set which is currently available, we suggest that it can be expressed simply as: β(dd , ds ) = β1 (ds )β2 (dd ) Using the wind-tunnel observations, Shao et al. (1996) found that β = 10−5 [1.25 ln(ds ) + 3.28] exp(−140.7dd + 0.37)

(7.22)

with dd and ds being in mm. The behaviour of β is as shown in Fig. 7.8.

7.5 Volume-Removal-Based Dust-Emission Scheme Lu and Shao (1999) proposed a dust-emission scheme which, in contrast to the energy-based scheme, estimates dust emission based on the volume removed by the saltators as they plough into the surface soil. Also in this model, saltation bombardment is considered to be the main mechanism for dust emission. The concept of volume-removal is supported by the wind-tunnel experiments of Rice et al. (1996a, b). In these experiments, Rice et al. used a device to supply sand grains (250–300 µm) which moved over a rough surface and then saltated on a loosely-packed or crusted surfaces of fine particles (d < 53 µm). The high-speed photography they employed shows that sand

7.5 Volume-Removal-Based Dust-Emission Scheme

227

1.5

3 (a)

(b) 1.0

β1

β2

2

0.5

1

0

0

200 400 600 ds (micron)

800

0.0

1

10 dd (micron)

100

3

β (model)

(c) 2

1

0

0

1

2

3

β (obs)

Fig. 7.8. Estimated bombardment parameter, β(dd , ds ), from wind-tunnel data (symbols). (a) Factor β1 × 105 , (b) factor β2 and (c) comparison of observed and fitted values of β × 105 using Equation (7.22)

grains saltating on the loose-packed surface usually rebound after excavating an elongated crater in the bed. The grains from the crater are ejected into the air as a dense cloud which then gradually disperses. The cloud either follows the rebounding grain or emerges in front of it, as shown in Fig. 7.9. After several impacts, the soil surface is as shown in Fig. 7.10. Based on the observations of Rice et al. (1996a, b), we can construct a model which calculates the crater volume and the dust emission generated by one saltation impact. The dust emission generated by a large number of saltating particles can be estimated by a superposition of the individual impacting events. The ploughing process of a particle through the soil surface is illustrated schematically in Fig. 7.11. The particle is assumed to be angular with a massequivalent diameter ds , a mass ms and a moment of inertia I. The origin of the coordinate system is located at the centroid of the impacting particle when it starts to contact the surface. It is assumed that the particle does not break during the impact. The particle impinges upon the soil surface at a velocity VI and an angle-of-attack θI . The particle then ploughs into the surface and pushes soil particles ahead of it. The target soil flows plastically during the ploughing without fracture. The protruding tip of the incident particle covers a trajectory (XT , YT ) into the target soil and forms a crater. The total volume of soil ejected from the crater into the air is equal to the volume of the crater.

228

7 Dust Emission

Fig. 7.9. High-speed film sequences of a single sand particle (250–300 µm) colliding with an unaggregated-soil surface of fine particles d < 53 µm (From Rice et al. 1996b)

In order to derive the equations of motion for the ploughing saltator, we make several plausible simplifications as listed below, 1. The impact particle has no initial rotation and only small rotation during the ploughing process. 2. The ratio of the vertical force to the horizontal force on the particle during ploughing is a constant Kv . 3. A constant plastic pressure (force per unit area) on the particle exists during ploughing and its horizontal component is P. 4. The depth over which the particle contacts the surface is the same as that of the crater, YT (Fig. 7.11). 5. The removed volume is the product of the area swept out by the particle tip and the width of the ploughing face, which is identical to ds , so that tc dXT dt (7.23) YT Ω = ds dt 0 where tc is the time at which ploughing ceases. 6. The resultant vertical and horizontal forces on the particle are located at the center of the surface soil material in contact with the particle. The symmetrical picture of two-dimensional ploughing shown in Fig. 7.11 can be understood as the average situation for grains which are tilted in either

7.5 Volume-Removal-Based Dust-Emission Scheme

229

Fig. 7.10. Plan views (2×3.5 cm) of unaggregated bed under a flow in the direction of the arrow. (a) Before bombardment and (b) after the impacts by several sand grains (From Rice et al. 1996b)

U X

φ Y

dXT

θ

ds/2

XT YT YT

Fig. 7.11. An illustration of the saltation bombardment process. A saltating particle ploughs through the soil, creates a small crater and ejects particles into the air. The horizontal and vertical components of the force exerted on the particle by the target soil are Fh = PY ds and Fv = Kv PY ds , respectively. See text for more details (From Lu and Shao, 1999)

230

7 Dust Emission

direction as they strike the surface. To be consistent with Assumption 2, the projected contact area in the horizontal plane is twice that in the vertical plane. 7.5.1 Motion of Ploughing Particle and Volume Removal The equations of particle motion in the X and Y directions and the equation of angular rotation ω are d2 X + PY ds = 0 dt2 d2 Y ms 2 + Kv PY ds = 0  dt ms

2

I

d ω + Pds Y dt2



ds −Y 2

− 2(Kv PY ds )Y = 0

(7.24) (7.25) (7.26)

where PY ds and Kv PY ds are the horizontal and vertical components of the resistance force acting upon the ploughing particle. The moment of inertia I is assumed to be ms d2s /12. With the initial conditions (X, Y, ω) = (0, 0, 0) and (dX/dt, dY /dt, dω/dt) = (VI cos θI , VI sin θI , 0), the equations can be solved to obtain   VI sin θI VI sin θI t (7.27) sin βv t + VI cos θI − X(t) = βv Kv Kv VI Y (t) = sin θI sin βv t (7.28) βv  3.75VI2 sin2 θI  2(βv t)2 + cos 2βv t − 1 (7.29) ω(t) = 2 2 βv ds 3VI sin θI (sin βv t − βv t) + βv ds 

where βv =

PKv ds ms

The volume removed by the saltator can be calculated through Equation (7.23). The motion of the ploughing particle may encounter two somewhat different cases. In Case 1, the impact particle ploughs into and subsequently leaves the target soil. In this case, we have Y (tc ) = 0 and tc = π/βv , as can be seen from Equation (7.28). Suppose the contribution of ω to XT is neglected, then YT = Y and XT = X, and it is found from Equation (7.23) that   2ms VI2 sin2 θI 1 cot θI − . (7.30) Ω= Kv P Kv

7.5 Volume-Removal-Based Dust-Emission Scheme

231

In other words, EI P Thus, Ω is proportional to the kinetic energy of the impacting saltator and inversely proportional to the plastic pressure of the surface. P strongly influences the relationship between Ω and EI . For fixed EI , Ω is small if P is large (hard surface) and Ω is large if P is small (soft surface). If the contribution of ω to XT is considered then, from Assumption 1, we have YT = Y and XT = X + ωds /2. The evaluation of Equation (7.23), by making use of Equations (7.27)–(7.29), is lengthy but straight forward (Lu and Shao, 1999). It is found that
 7.5πVI sin3 θI VI2 2 (7.31) Ω = 2 (sin 2θI − 4 sin θI )ds + βv βv Ω∝

As θI does not vary over a wide range, we set θI to a fixed value of 15◦ in the above equation in order to reduce the number of independent variables. Further, we set Kv = 2. It follows that    EI ρp 1 + VI . (7.32) Ω = 0.23 P P The above equation shows that Ω is proportional to the kinetic energy of the saltation EI = ms VI2 /2 and inversely proportional to surface plastic pressure. Due to the inclusion of particle rotation,  P also affects the dependency of Ω on VI . For large P (hard surface, VI ρp /P  1), we have

For small P (soft surface, VI



Ω ∝ VI2 ρp /P 1), we have Ω ∝ VI3

In Case 2, the particle stops during its scooping action at some depth, as T its kinetic energy is exhausted, i.e. dX dt |t=tc = 0. For this case, the initial dXT ds dω dX condition dt = dt + 2 dt = 0, together with Equations (7.27) and (7.29), leads to an expression of tc in the form 2 cos βv tc − 2 + cot θI +

7.5VI βv tc sin θI 3.75VI sin θI − sin 2βv tc = 0 (7.33) βv ds βv ds

For a given θI , tc can be determined numerically from the above equation and then used in Equation (7.23) to calculate Ω. Equation (7.33) has no solution for tc when θI ≤ tan−1 (1/4), which implies that it is impossible in this model for particles with an impact angle between 0◦ and 14◦ to be trapped in soil. The observations of Rice et al. (1996a, b) do not contradict this assertion, in that they show that saltating sand particles with θI < 15◦ usually

232

7 Dust Emission

rebound after excavating a crater on the soil surface. However, Equation (7.33) does have a solution for tc where impact angles are large, which implies that particles can be trapped in the soil under these circumstances. In this case, Equation (7.31) shows a Ω ∼ VI3 behaviour. Large impact angles often happen over irregularly-shaped soil surface. Alfaro et al. (1997) also observed that a proportion of sand grains saltating over a loosely-packed clay-particle surface do not rebound. 7.5.2 Vertical Dust Flux We now consider the dust emission induced by saltation bombardment from a soil that contains multi-sized dust particles but uniform sand particles of size ds . If nI is the impact-particle number flux and η is the mass fraction of dust contained in Ω, then the vertical dust flux caused by saltation bombardment is F = cb ηρb ΩnI

(7.34)

where ρb is the bulk density of the soil and cb is a constant of proportionality less than 1, since a proportion of dust particles may remain attached to the aggregates contained in Ω. Substituting Equations (7.14) and (7.32) into Equation (7.34), we obtain    ρb ρp 1 + 10u∗ Q (7.35) F = CΩ gη P P where

cb 0.23 CΩ = co Kv



VI upI − upR



VI u∗

 ≈ cb

In the above calculation, we have assumed that VI /(upI − upR ) and the Owen coefficient and co are both of order of 1 and VI /u∗ is order of 10. Thus, the dust-emission scheme of Lu and Shao (1999) obtained the final form of    ρb ρp 1 + 10u∗ Q (7.36) F = cb gη P P Equation (7.36) reveals two qualitatively important relationships. i • Since Q is proportional to u3∗ , F must be proportional  to u∗ with i = ρp 3 ∼ 4. For sufficiently large P (hard soil), so that 10u∗ p  1, F/Q is independent of u∗ and F ∝ u3∗ ,  ρ while for sufficiently small P (soft soil), so that 10u∗ Pp 1, F/Q increases linearly with u∗ and F ∝ u4∗ .

7.6 Comparison of Dust Schemes

•

233

The F/Q ratio is proportional to the fraction of dust in the soil, η. If the soil does not contain dust (η = 0), then dust emission is not possible. For example, dust emission from a sandy beach will be zero although saltation may be strong. F/Q is inversely proportional to P k1 , with k1 = 1 ∼ 1.5. For a hard surface (large P), saltation bombardment is less efficient in producing dust than for a soft surface (small P). This is because, a saltator with given impact velocity and angle would generate a smaller crater on the hard surface than it would on the soft surface.

For the implementation of the dust-emission scheme of Lu and Shao (1999), methods for estimating P must be developed. P is a property related to the soil tensile strength, or hardness, that is in turn related to the packing density and the inter-particle bond strength. For a simple soil, tensile strength is proportional to d−3 (Smalley, 1970). In general, hard and strongly-cohesive soils have a large P, while soft and weakly-cohesive soils have a small P. However, the soil tensile strength is not a unique property but a measure of the reaction of the soil to the disturbing force imposed. The commonly used method to measure hardness is the static indentation test. As the forces imposed on the surface by saltation bombardment are associated with the impact of saltating particles, it is critical to choose a suitable test indenter with a scale comparable to the size of the saltating particles and a load comparable to the stress imposed by the saltators. This type of tensile strength can be determined by using a small-headed penetrometer (a needle), which measures the penetration pressure experienced by the needle (Rice et al. 1997). The volume-removal based dust-emission scheme has several limitations: (1) Dust emission caused by the disaggregation during saltation is not considered in the model. The combination of these two processes and the random nature of soil aggregates, micro-topography (ripples and ridges that influence the local impact angle) make it difficult to predict the dependency of dustemission rate on the impact velocity; (2) The model for Ω is probably too simplistic; and the ratio of the vertical and horizontal plastic pressure and the contact area between the ploughing particle and surface soil cannot be precisely described; (3) Because the elastic forces have been neglected in the model, it may not be suitable for highly-crusted soils, for which the elastic strains are comparable to plastic stains. The erosion behaviour of crusted soils may not be the same as loosely-packed ones.

7.6 Comparison of Dust Schemes Marticorena and Bergametti (1995) proposed an empirical dust-emission scheme, such that F = 0.01 exp(0.308ηc − 13.82) Q

(7.37)

234

7 Dust Emission 10−1

10−4 Soil 5

Soil 5

10−2

10−5

10−3

10−6

10−4

Observed Shao et al. (1993) Marticorena & Bergametti (1995) Lu & Shao (1999)

Observed Simulated

10−5 10−1

10−8 10−5

Soil 4

Soil 4

10−2

10−6

10−3

10−7

10−4

10−8

10−5 10−1

10−9 10−4 Soil 3

Soil 3

10−2

10−5

10−3

10−6

10−4

10−7

10−5

10−8

10−1

10−4 Soil 2

Soil 2

10−2

10−5

10−3

10−6

10−4

10−7

10−5 10−1

10−8 10−5 Soil 1

Soil 1

10−2

10−6

10−3

10−7

10−4

10−8

10−5 0.1

F (kg m−2 s−1)

Q (kg m−1 s−1)

10−7

1.0

0.2 −1

u* (ms )

0.1

1.0

0.2

10−9

−1

u* (ms )

Fig. 7.12. Comparison between the predictions of three different dust-emission models and the field observations (From Lu, 1999)

where ηc is the percentage of clay and F/Q is in [m−1 ]. This empirical expression is derived by fitting the average ratio of F/Q to the fraction of clay content using the data set of Gillette (1977). Figure 7.12 shows a comparison of the dust-emission schemes of Shao et al. (1993b), Marticorena and Bergametti (1995) and Lu and Shao (1999) with the data of Gillette (1977). In general, all three schemes have produced reasonable agreement with the measurements. Note that the parameters used in the latter

7.7 Spectral Dust-Emission Scheme

235

two models are derived more or less from the data set, while those used in the model of Shao et al. (1993b) are determined independently of the data set. Apart from Soil 3, the model of Shao et al. (1993b) has performed reasonably well.

7.7 Spectral Dust-Emission Scheme In the previous section, we have studied the schemes for the prediction of the bulk (all particle sizes) dust-emission rate. However, dust particles of different sizes have profoundly different optical, aerodynamic and mineralogical characteristics and it is often desirable to estimate the dust-emission rate for a given particle-size range. To this end, we divide dust particles in the size range of (0, dd ) into I size bins with di being the mean particle size and ∆di the bin width for the ith bin. Our task is to develop a spectral dust-emission scheme which enables the prediction of dust-emission rates for all I dust bins. Let us denote the dust-emission rate for the ith dust bin with Fdi . Some schemes are intrinsically non-spectral, e.g., Equations (7.7) and (7.37). Nevertheless, we can use these schemes to compute F first and assume that the airborne dust particle-size distribution, pad (d), is known a priori. Then, Fdi can be calculated as Fdi = pad (di )∆di F This approach is practically useful but ad-hoc, because pad (d) is in general not known. Equation (7.36) is intrinsically spectral. For the computation of Fdi , we can write this equation as    ρb ρp ˆ ˆ 1 + 10u∗ Q(ds ) (7.38) Fdi (ds ) = cbi gηi P P where Fˆdi (ds ) is the emission rate of dust of size di attributed to the bombardment of saltating particles of size ds , which produces a vertically-integrated ˆ if the surface is covered by particles of this size; ηi is the streamwise flux Q fraction of dust of the ith dust-size bin, and cbi is a coefficient which is dustparticle size di dependent. Gomes et al. (1990) obtained samples of airborne particles at the 6 m height for different wind-erosion intensities and carried out particle-size analysis. Their observations suggest that the particle-size distributions of the samples have a bimodal structure with one mode being characteristic of loose soil aggregates around 60 µm and another between 2 and 20 µm resulting from the disaggregation of particles larger than 20 µm caused by saltation bombardment. Under strong wind-erosion conditions, the localized momentum transfer is sufficiently large to break clay platelets coated on the surface of large particles, resulting in the emission of sub-micron particles. In wind-tunnel experiments, Alfaro et al. (1997) used pure quartz sand particles

236

7 Dust Emission 15 measured Modeled Population 2 Population 1

d*p(d)

10

5 u* = 0.40 ms−1

d*p(d)

0 1 10

10

100

5 u* = 0.45 ms−1 0 1 15

10

100

d*p(d)

10

5 u* = 0.53 ms−1 0 1

10 d (µm)

100

Fig. 7.13. Size distributions of the airborne particles produced by the bombardment of sand grains over a clay surface for three different friction velocities (open circles, after Alfaro et al. 1997). The modelled particle-size distributions using Equation (7.39) are shown as solid curves. The particle-size distributions can be considered to be a superposition of two populations, with population 1 being the original kaolin clay and population 2 being generated by saltation bombardment. Selected n for the modelling are γ = e−k(u∗ −u∗t ) with u∗t = 0.27 m s−1 , n = 3 and k = 27.3 (After Lu, 1999)

(mass geometric mean 240 µm and standard deviation 1.25 µm) to bombard a surface of kaolin clay (mass geometric mean 8.57 µm and standard deviation 2.33 µm). The particle-size analysis of the dust samples is presented in Fig. 7.13 for three different wind conditions. At relatively small u∗ (0.4 m s−1 ), the size distribution of the airborne dust is close to that of the original kaolin clay. In this case, the impacts of the sand grains do not seem to be sufficiently energetic to break the clay aggregates. For higher friction velocities (0.45 and 0.53 m s−1 ), the aerosol can be considered to be a mixture of two populations, one made up of fine particles of approximate median diameter 2.8 µm

7.7 Spectral Dust-Emission Scheme

237

and the other made up of aggregates with a diameter similar to that of the original kaolin clay. Lu (1999) proposed to express the particle-size distribution observed by Alfaro et al. (1997) as a weighted average of p1 and p2 which respectively represent the particle-size distributions of the above-mentioned two populations of particles ps (d) = γp1 (d) + (1 − γ)p2 (d),

(7.39)

where γ = e−k(u∗ −u∗t ) and u∗t is threshold friction velocity and k and n are empirical parameters. The comparison of the modelled and the observed ps (d) is shown in Fig. 7.13. The above example confirms that airborne particles sampled during winderosion events of different intensities have different particle-size distributions, due to the breaking up of soil aggregates into finer particles at larger wind speed. More generally, immediately adjacent to the surface, say within the saltation layer, the airborne particle-size distribution ps (d) must be confined to two limiting cases, namely n

ps (d) → pm (d) ps (d) → pf (d)

weak erosion strong erosion.

(7.40) (7.41)

It is therefore possible to represent ps (d) using ps (d) = γpm (d) + (1 − γ)pf (d),

(7.42)

where γ is the weight for pm (d) and (1 − γ) is the weight for pf (d). The choice of γ should satisfy the requirements that γ→1 γ→0

weak erosion, u∗ ∼ u∗t strong erosion, u∗ u∗t ,

(7.43) (7.44)

This technique for the approximation of ps (d) is further illustrated in Fig. 7.14. For small u∗ , ps (d) is similar to pm (d) and for large u∗ , ps (d) approaches pf (d). Several observations can be made in relation to the particle-size distributions: 1. The mass fraction of free dust for a unit soil mass can be estimated from dd pm (d)δd (7.45) ηm = 0

where dd is the upper limit of dust particle size. This fraction of dust has low enough binding energy so that it can be easily lifted from the surface by either aerodynamic forces or mechanical abrasion. Equation (7.45) provides an estimate for the minimum amount of dust that can be emitted from a unit mass disturbed by saltation in a wind erosion event.

238

7 Dust Emission pfc pf pm ps, small u* ps, large u*

p(d) (µm−1)

0.016

0.012

0.008

0.004

0

1

10

100

1000

d (µm)

Fig. 7.14. An illustration of the technique for the approximation of ps (d). For small u∗ , ps (d) is similar to pm (d) while for large u∗ , ps (d) approaches pf (d)

2. Depending on the soil type, there is usually a larger amount of dust which is not free but contained in aggregates and can be released only through the mechanical destructions. The fraction of aggregated dust in a unit soil mass is dd

ηc = ηf − ηm =

pc (d)δd,

(7.46)

0

where pc (d) is given by pc (d) = pf (d) − pm (d). In Equation (7.46), ηf is the sum of the free and the aggregated dust, that is, the total fraction of dust which can be released from a unit soil mass,

dd

ηf =

pf (d)δd.

(7.47)

0

Equation (7.47) provides an upper limit for dust emission that is possible from a unit soil mass disturbed by saltation. The estimates of ηm and ηf are practically important, as they provide the upper and lower limits of dust emission for a given soil type. 3. The fraction of dust (originated from a unit soil mass disturbed by saltation) suspended in the atmosphere can be estimated from ηs =

dd

ps (d)δd.

(7.48)

0

For each dust bin, ηmi , ηf i , ηsi and ηci (corresponding to ηm , ηf , ηs , and ηc ) can be estimated from

7.7 Spectral Dust-Emission Scheme

ηmi =

239

di +∆di /2

di −∆di /2

pm (d)δd,

(7.49)

etc. Suppose a soil consists of dust particles of various sizes and saltators of uniform size ds , and also nI is the impact particle-number flux of the saltators and ηf is the mass fraction of dust contained in a unit soil mass. Then, the dust emission due to saltation bombardment can be estimated from Equation (7.34) which we rewrite here as Fˆb (ds ) = cb ηf ρb ΩnI .

(7.50)

Dust can also be released due to disaggregation. We assume that disaggregation only occurs as the saltators strike the surface. Corresponding to nI , the mass flux of the saltating particles striking the surface is ms nI . The mass fraction of soil aggregates available for breaking is ηc . Hence, the dust emission arising from disaggregation can be expressed as Fˆc (ds ) = cc ηc ms nI ,

(7.51)

where cc , like cb , is a coefficient to be determined. The emission of dust from the ith dust bin generated by the saltation bombardment and disaggregation of particles of size ds is given by Fˆb,di (ds ) = cbi ηf i ρb ΩnI Fˆc,di (ds ) = cci ηci ms nI .

(7.52) (7.53)

Both cbi and cci must be functions of di , probably inversely proportional to the binding energy of dust particles and positively related to the strength of the saltation impact. Again, as discussion in Section 7.4, we use ∆E to represent the strength of the impact. It follows that ∆E ψ(di ) ∆E , cci ∝ ψ(di ) cbi ∝

(7.54) (7.55)

For simplicity, we assume cbi = cci . It follows that Fˆdi (ds ) = cbi nI (ηf i ρb Ω + ηci ms ).

(7.56)

The emission of dust particles of size di associated with the saltation of all sand sized particles (sand plus aggregates) can be estimated as a weighed average d2 (7.57) Fˆdi (d)ps (d)δd. Fdi = d1

240

7 Dust Emission

Finally, the total dust emission is given by F =

I

Fdi

(7.58)

i=1

Recall that cbi in Equation (7.56) is the proportion of ηf i released into the air and (1 − cbi ) is the retained proportion in the saltating and ejected particles. These imply that cbi =

ηsi ps (di ) = (1 − γ) + γσd = ηf i pf (di )

(7.59)

where σd is the ratio between the free dust and the aggregated dust, i.e., σd = pm (di )/pf (di ) Substituting (7.59) and (7.14) into (7.56) leads to Qg Fˆdi (ds ) = cy [(1 − γ) + γσd ] 2 (ηf i σm + ηci ), u∗

(7.60)

where σm is the ratio between mΩ (ρb Ω, mass ejected by bombardment) and ms (mass of impacting particle), i.e., σm = mΩ /ms and cy = 1/7co is of order 0.1. If ηmi  ηf i , then ηci ≈ ηf i , then Equation (7.60) is further simplified to gQ Fˆdi (ds ) = cy ηf i (1 − γ)(1 + σm ) 2 u∗

(7.61)

Thus, for given soil texture and friction velocity, it only requires the estimate of σm to compute Fˆdi (ds ). One possibility of computing σm is to compute Ω via Equation (7.32). There, since mΩ = ρb Ω with ρb being soil bulk density, VI is around 10u∗ and ρp /ρb around 2.6 (assuming ρp = 2,600 kg m−3 and ρb = 1,000 kg m−3 ), we obtain    ρb 2 ρb 1 + 14u∗ (7.62) σm = 12u∗ P P Figure 7.15 shows the dependency of σm on u∗ and P. In summary, the dust emission scheme is given by Equations (7.60) and (7.62) plus an assumption on γ. It states that dust emission is proportional to saltation mass transport, but the proportionality depends on soil texture and soil plastic pressure. We point out the following: •

The dependence of Fdi on soil texture is reflected in ηf i and ηmi . For soils which have no dust, dust emission is not possible.

7.7 Spectral Dust-Emission Scheme

241

102 u* =0.6ms−1 0.8 1.0

P=5000 Pa 10000 50000 100000

σm

101

100

(a) 10−1 0.4

0.6

0.8 u* (ms−1)

1

(b) 1.2 102

103

104

105

106

P (Pa)

Fig. 7.15. (a) For given P, σm as a function of u∗ ; (b) For given u∗ , σm as a function of P

•

The value of γ falls between 0 and 1. This implies that aggregated dust is released only during strong erosion events as u∗ u∗t and γ → 0; for weak erosion events, γ → 1, only the emission of free dust is possible. • Fdi depends on σm which is determined both by saltation impact and soil plastic pressure. The data of Rice et al. (1995) suggest that σm associated with a 275 µm particle and a 4 m s−1 impact velocity varies from 0.076 for crusted soil to 10.5 for unaggregated soil. Equation (7.62) is not an explicit function of ds , although σm does depend on ds through VI as sand particles of different sizes have different impact velocities. However, by assuming VI = 10u∗ , we have neglected this dependency in Equation (7.62). • The scheme requires specifying P which, on the basis of the penetrometer resistance measurements of Rice et al. (1997), varies between 103 Pa for light spray fine soil and 107 Pa for deep wetted soil. Consequently, dust emission is expected to vary significantly with soil surface conditions. This is confirmed by the large scatter among the observed dust emission shown in Fig. 7.2. For P ≥ 3 × 105 Pa, σm becomes negligibly small (< 0.1) under normal wind conditions, implying that saltation bombardment is insignificant in such circumstances and disaggregation is the main mechanism for dust emission. For P ≤ 3 × 103 Pa, σm is much larger than 1, implying saltation bombardment is the dominant mechanism for dust emission while disaggregation is negligible. • The input physical quantities and soil parameters required for the scheme include: (1) friction velocity u∗ and threshold friction velocity u∗t ; (2) minimally- and fully-disturbed particle-size distribution, pm (d) and pf (d); and (3) soil plastic pressure, P.

242

7 Dust Emission

Shao (2004) compared the predictions of dust emission using Equation (7.60) with the observations from the studies summarized in Table 7.1. The purpose of Shao (2004) is to estimate cy and P for practical use. For a given soil, pm and pf are approximated as the sum of several log-normal distributions, e.g.   J 1 wj (ln d − ln Dj )2 √ . (7.63) exp − pm (d) = d j=1 2πσj 2σj2 where J is the number of modes (it is generally sufficient to set J = 4), wj is the weight for the jth mode of the particle-size distribution, Dj and σj are parameters for the log-normal distribution of the jth mode. Nonlinear leastsquares fitting techniques can be used to determine wj , ln(Dj ) and σj from laboratory-analysed particle-size distribution data. For each experiment listed in Table 7.1, a soil type and a u∗t are selected according to the experiment descriptions in the relevant references. The value of cy is initially fixed, P is adjusted until a plot of ln F against ln u∗ shows the desired slope. Then, cy is adjusted to obtain a good fit. The results are summarized in Table 7.3. The values of P estimated by Shao (2004) fall between 1,000 and 50,000 Pa. The range of these values is in general agreement with the mean maximum penetration pressure reported by Rice et al (1997). Loose surfaces have smaller, while hard-crusted surfaces have larger P values. For P ≥ 3e + 5Pa, dust emission becomes independent of P (no saltation bombardment, only aggregates disintegration). The coefficient cy is found to fall between 10−5 and 5 × 10−5 . It has been suggested as a general guidance to set cy ≈ 5 × 10−5 and P ≈ 1,000 to 5,000 Pa for loose sandy soils, and to set cy ≈ 10−5 and P ≈ 30,000 to 50,000 Pa for clay soils. Figure 7.16 shows the comparisons of the predicted Q and F against the observed Q and F for sandy (Gillette soil 1, 2, 4 and 5) and loamy (Gillette Table 7.3. Estimated P and cy for the dust scheme due to Shao (2001, 2004) by comparing the simulated and observed vertical dust fluxes from Gillette (1977), Nickling (1983), Nickling and Gillies (1993), Nickling et al. (1999), Gomes et al. (2003) and Rajot et al. (2003) Soil P (Pa) Sandy (soil 1, 2, 4, 5) 1,500 Loamy (soil 3) 500 Loamy (soil 6) 10,000 Clay (soil 9) 30,000 Yukon Silty clay loam 2,000–5,000 Mali Sandy to clay 20,000–40,000 Diamantina Clay 30,000 Spain Silty clay loam 50,000 Niger Sandy 10,000 Data Texas

cy 10−5 5 5 5 5 30 1 1 1 4

Reference Gillette (1977)

Nickling (1983) Nickling & Gillies (1993) Nickling et al. (1999) Gomes et al. (2003) Gomes et al. (2003)

7.7 Spectral Dust-Emission Scheme 106

104 100 103 10−1

102 Obs Moded, p = 500Pa, cy = 5e−5

Obs, loam Model

10−2 102

101 105 104

101

103 100 102 10−1

101 Obs Moded, p = 1500Pa, cy = 5e−5

Obs, sand Model

10−2 0.2

F (ugm−2 s−1)

105

101

0.4

0.6 0.8 u* (ms−1)

1.0

F (ugm−2 s−1)

Q (gm−1s−1)

102

Q (gm−1s−1)

243

1.2 0.2

0.4

0.6 0.8 u* (ms−1)

1.0

100 1.2

Fig. 7.16. Comparison of model-predicted Q and F with the observed Q and F of Gillette (1977)

soil 3) soils. For both cases, the predictions are in good agreement with the observations. The loamy soil shows an exceptionally small P. Comparisons are also made for Gillette soil 6 and 9, but are not shown due to the very small number of data points. The comparison for the Niger data set is shown in Fig. 7.17. According to Gomes et al. (2003), the observed erosion events appear to be source limited. For such cases, Q is difficult to predict unless u∗t is treated as a function of time. As can be seen from Fig. 7.17a and b, the prediction of Q and consequently that of F are problematic. If observed Q is used in Equations (7.60) to predict F , then the comparison of the predicted F with the observed F is much improved as shown in Fig. 7.17c. There is an important caveat on the choice of cy . The theoretical value of cy in Equation (7.60) is around 0.1, but the calibrated cy mentioned above is several orders of magnitude smaller. A possible explanation to this contradiction is as follows. The pm and pf data used by Shao (2004) were obtained through laboratory particle-size analysis in which chemical-dispersion techniques were applied. Since wind erosion does not involve chemical reactions,

7 Dust Emission 101

Obs Mod,p=10000Pa,cy=4e−5,u*t=0.2ms−1

Q (gm−1 s−1)

Obs Model

103

100

102

10−1

101 (a)

10−2 0.2

0.3

0.4 u* (ms−1)

F (ug m−2 s−1)

244

(b) 0.5 0.2

0.3

0.4 u* (ms−1)

100 0.5

Fobs (ug m−2 s−1)

103

102

101 (c) 100 100

101 102 Fmod (ug m−2s−1)

103

Fig. 7.17. (a) A comparison of the predicted and observed streamwise saltation flux Q; (b) as (a) but for dust emission rate F ; (c) a plot of observed F against the predicted F by using the observed Q. Data used for comparison are the Niger data of Rajot et al. (2003)

pf should be understood as the fully-disturbed particle-size distribution of the parent soil in the limiting case in which aggregates are as much broken up as possible by mechanical abrasion, rather than chemical dispersion. The disaggregation arising from mechanical abrasion is much weaker than that arising from chemical dispersion. It is probably a mistake to approximate pf with a particle-size distribution obtained using a method of chemical-dispersion, and this mistake has probably caused the large discrepancy between the theoretical and the calibrated values of cy . In other words, the calibrated cy values contain the error of approximating pf (d) with pf c (d) [as was done in Shao (2004)]. A recalibration of cy is desirable when better particle-size distribution estimates become available. The techniques for determining pm and pf described by Saxton et al. (2000) and Chandler et al. (2002) appear to be promising.

7.8 Discussions on Dust Schemes

245

7.8 Discussions on Dust Schemes Our current capacity for modelling dust emission is still limited. It is now agreed in general that the main mechanisms for dust emission are saltation bombardment and disaggregation. Therefore, dust emission must be directly related to the intensity of saltation and indirectly related to wind shear at the surface. Also importantly, dust emission must be inversely proportional to the binding strength of dust particles. There are three types of dustemission schemes, i.e., empirical forms, energy-based schemes and volumeremoval based schemes. The energy-based schemes relate dust emission to sand drift, but the derivation of the dust-emission scheme does not state how dust emission precisely takes place. It therefore does not require detailed information about individual saltating particles. The major difficulty of the energy-based schemes is that the binding energy for dust particles is difficult to determine and the kinetic energy converted to heat during the impact is not readily estimated. In contrast, the volume-removal based dust-emission scheme takes into consideration the detailed physical process of dust emission. In this model, the concept of binding is realised through the use of the plastic pressure, P, a descriptor of the binding strength of the soil surface, and through the use of cbi in Equation (7.56), a descriptor of the binding strength of the soil aggregates. The increased understanding of wind-erosion physics has enabled the development of dust-emission schemes on a firmer footing. However, the accuracy of the dust-emission schemes is limited by the lack of capacity in specifying the temporal and spatial variations of model parameters. The difficulty in achieving the accuracy of dust-emission schemes is exacerbated by the fact that dust emission can vary over several orders of magnitude. There is also a serious lack of high quality dust-flux measurements for dust-scheme validation. The few existing dust-emission schemes are listed in Table 7.4. Discussions on the advantages and disadvantages of these schemes are further given in Chapter 9.

Table 7.4. Dust-emission schemes. G77: Gillette (1977); GP88: Gillette and Passi (1988); MB95: Marticorena and Bergametti (1995); SRL96: Shao et al. (1996); LS99: Lu and Shao (1999); S04: Shao (2004) Scheme 1 2 3 4 5

Expression F = αg u4∗ (1 − u∗t /u∗ ) F = 0.01Q exp(0.308ηc − 13.82) F = βQu−2 ∗t,d  F = (cb gηρb /P)(1 + 14u∗ ρb /P)Q ) = cy ηf i (1 − γ)(1 + σm )gQ/u2∗ ; Fˆdi (ds   Fdi Fdi = Fˆdi (d)ps (d)δd; F =

References GP88 MB95 SRL96 LS99

Comments Hypothesis Empirical form Theory Theory

S04

Theory

8 Dust Transport and Deposition

Dust particles, once airborne, can be carried by winds to various distances from the source region and deposited back onto the surface somewhere downstream. Dust transport and deposition are thus key links in the dust cycle. The understanding of these processes has applications in a wide range of research fields. From the perspective of geological studies, dust transport and deposition are important mechanisms for the delivery of sediments and minerals from continent to ocean and for the evolution of surface topography. For example, the formation of the loess plateau in China is believed to be a result of the deposition of dust and silt particles originating from the upstream desert areas. In the atmosphere, the presence of dust particles influences processes such as cloud formation and radiation transfer. High concentration of fine particulates in air is of concern to human health, and PM10 and PM2.5 have been used as key indicators of air quality. Dust particles from certain regions may carry harmful substances, and the fates of such particles are naturally of environmental concern. In this chapter, we focus on the physics of dust transport and deposition and study how these processes can be quantified.

8.1 Evidence of Dust Transport and Deposition The distance dust particles travel in the atmosphere depends on flow conditions and particle size. It is useful to establish an approximate relationship between dust-particle size and travel distance for specific atmospheric conditions. Suppose dust particles are raised by a storm which is strong enough to mix them though an atmospheric layer of depth H. Then the concentration of dust in this layer obeys approximately U

∂F ∂ρc =− ∂x ∂z

(8.1)

where U is wind speed, x is distance, F is vertical dust flux and z is height. An integration of Equation (8.1) over z from the surface to the top of the dust Y. Shao, Physics and Modelling of Wind Erosion, c Springer Science+Business Media B.V. 2008


247

248

8 Dust Transport and Deposition

layer, where F vanishes, gives approximately Um

F0 dρcm = dx H

(8.2)

where the subscript m denotes the averaging over depth H, F0 is the net dust flux at the surface. If dust emission is not active, then F0 represents the dust deposition onto the surface. As will be discussed later in the chapter, if there is no rainfall, dust deposition onto the surface can be expressed as F0 = ρwd cm , where wd is the dry-deposition velocity. The solution of Equation (8.1) is thus   x (8.3) cm (x) = cm (0) exp − xd where xd = Um H/wd . This simple model shows that under the influence of dry deposition, cm decays exponentially with distance. The scaling parameter, xd , represents the typical travel distance [at x = xd , cm (x) = 0.37cm (0)] for given Um , H and particle size. Figure 8.1 shows xd as a function of particle size for H = 1 km and several values of Um . The maximum of xd is around 105 km, which occurs for particles with a diameter of 1 µm. For particles with a diameter around 10 µm, xd falls to between 102 and 103 km. Smaller claysized particles can be transported over continental distances of the order of 103 to 104 km. During strong wind-erosion events, particles can be carried to much larger heights than H = 1 km, as assumed in the example. For instance, the convective boundary layer over the Sahara deserts can be as high as 5 km during the day time. At this height, wind speed is also stronger than assumed in the example, allowing particles to be transported over even larger distances. 105 104

xd (km)

103 102 101 100 10−1 10−2

Um=1 ms−1; u*=0.17 ms−1 2 0.35 4 0.7 8 1.4 10−1

100

101

102

103

d (µm)

Fig. 8.1. Typical dust-travel distance, xd , as a function of particle diameter d in a well-mixed atmospheric boundary layer of constant depth (H = 1 km). The deposition velocity, wd , is estimated using Equation (8.114)

8.1 Evidence of Dust Transport and Deposition

249

Equation (8.3) indicates that, in general, particles travelling over large distances are smaller than 20 µm. This prediction is supported by observations. Particle-size analysis of dust samples collected by aircraft in southwest United States, hundreds of kilometres downstream from the source region, has shown that particles which can be carried over such a distance are mostly smaller than 20 µm, although a fraction of larger particles is also present in the samples (Gillette, 1981). Arimoto et al. (1997) carried out observations over the Pacific and the Atlantic Oceans to investigate the characteristics of dust particles originating from the Asian and African deserts. The samples collected over the Pacific Ocean show mass-mean diameters ranging from 0.36 to 5.4 µm, while the samples collected over the Atlantic Ocean show massmean diameters mostly smaller than 3 µm. In northeast Asia, Chun et al. (2001b) observed that dust particles transported from the source regions in China and Mongolia to South Korea mostly fall into the size range between 1.35 and 10 µm. Ambient dust concentration (Table 8.1) shows large variations, with high values in the areas of dust storms and low values over the ocean. The estimated deposition rate upon Asian desert areas ranges between 14 and 2,100 g m−2 yr−1 (Zhang et al. 1997). The deposition rate into the ocean ranges from less than 0.001 to more than 10 g m−2 yr−1 (Pye, 1987; Tegen and Fung, Table 8.1. Range of maximum dust concentration from various measurements and simulations. LDS stands for local dust storm, LRT for long-range transport, M for measured and L for modelled Study Type LDS LDS LDS LDS LRT LRT

Size (µm) ≤10 ≤10 ≤10 ≤10

LRT

Concentration (µg m− 3) 1m 8×103 –2.5×105 1.5 m 4 × 103 –2.6 × 105 1–10 m 103 –105 1–10 m 26–13,735 Bld. roof 280–5,000 Land 500 surf. Land 3.6–32.3 surf. Land 14.6–62.1 surf. Ocean 2–25 surf. Ocean 0.1–60 surf. 2,000 m 2 × 103

LRT LRT

980 hPa 105 960 hPa 129 (µg kg−1 )

≤10 Clay

LRT LRT LRT LRT

Height

≤2.5

u∗ (m s−1 ) 0.4–0.6 0.3–0.5 0.3–0.6 0.2–1.0

Reference Gillette and Walker (1977), M Gillette and Walker (1977), M Nickling (1978), M Nickling and Gillies (1993), M Gao et al (1992), M Talbot et al (1986), M Perry et al (1997), M Prospero and Nees (1976), M Prospero and Nees (1976), M Pye (1987), M Nickovic and Dobricic (1996), L Westphal et al (1988), L Tegen and Fung (1994), L

250

8 Dust Transport and Deposition

1994). Duce et al. (1991) have given a review of the atmospheric input of aerosols and associated trace elements to the world ocean, using data obtained during cruises and measurements at a number of sites for the North Atlantic Ocean, the North and South Pacific Oceans, the South Atlantic and the Indian Oceans. They estimated dry deposition from dust-particle concentration and dry-deposition velocities and estimated wet deposition from the concentration and precipitation-scavenging ratio. The methods of determining dry- and wet-deposition velocities are described later in this Chapter. The results of Duce et al. (1991) are summarised in Table 8.2 and Fig. 8.2. Table 8.2 shows that both wet and dry deposition in the ocean are several g m−2 yr−1 . On the global scale, the total dry and wet depositions are probably of similar magnitude. Figure 8.2 shows two areas of maximum dust deposition (about 10 g m−2 yr−1 ), one in the North Atlantic to the west of the Sahara region and one in the North Pacific to the east of the Asian continent. Several data sets are available for dust deposition over or near the Asian continent. Gao et al. (1997) have studied on the dust deposition at various sites in China and over the East China Sea and Zhang et al. (1997) have studied dust transport and deposition in the Chinese desert regions. The data from Table 8.2. Dust deposition to the Ocean in g m−2 yr−1 (Duce et al. 1991) Ocean North Pacific South Pacific North Atlantic South Atlantic North Indian South Indian Global total

Dry 1.50 0.13 2.90 0.20 2.00 0.22 1.00

Wet 3.80 0.23 1.10 0.27 5.10 0.60 1.50

Total 5.30 0.35 4.00 0.47 7.10 0.82 2.50

Fig. 8.2. Global fluxes of mineral aerosol into the oceans in mg m−2 yr−1 (From Duce et al. 1991)

8.1 Evidence of Dust Transport and Deposition

251

Table 8.3. Dry and wet depositions of dust in spring at various locations in China, South Korea and over the East China Sea. Mean-deposition fluxes are reported in g m−2 mon−1 , with the range of variations given in parentheses (Data from Gao et al. 1997) Location Xian (38◦ N, 105◦ E) Beijing (40◦ N, 116◦ E) Qingdao (36◦ N, 120◦ E) Xiamen (24◦ N, 118◦ E) E. China Sea(28–32◦ N, 122–130◦ E) Baotou (41◦ N, 110◦ E) Lanzhou (36◦ N, 104◦ E) Kato, Hong Kong (23◦ N, 113◦ E) Kenting, Taiwan (23◦ N, 120◦ E) Cheju, S. Korea (33◦ N, 127◦ E) Mallipo, S. Korea (37◦ N, 128◦ E)

Dry 19 (3.7–33) 15 (5.1–51) 1.9 (0.33–6.9) 1.1 (0.33–2.9) 1.3 (0.36–2.4)

Wet 6.0 (1.2–11) 3.3 (1.2–12) 1.1 (0.19–4.2) 2.5 (0.57–6.9) 1.4 (0.39–2.7)

Total 25 (4.9–44) 18 (6.3–63) 3.0 (0.52–11.1) 3.6 (0.90–9.8) 2.7 (0.75–5.1)

32 35 0.42 (0.061–2.2)

6.90 5.9 0.96 (0.14–5.0)

39 41 1.4 (0.21–7.2)

0.17 (0.005–1.1)

0.38 (0.02–3.8)

0.55 (0.025–4.9)

1.5 (0.44–4.3)

1.7 (0.51–4.9)

3.2 (0.95–9.2)

1.6 (0.39–16)

2.1 (0.93–37)

3.7 (1.3–53)

Table 8.4. Dust deposition in Chinese desert regions. Mean-deposition fluxes (dry deposition plus wet deposition) as well as desert-total depositions are reported, with the range of variations given in parentheses (Data from Zhang et al. 1997) Deserts Taklimakan Gurbantunggut Tsaidam Basin Kumutage Badain Juran Ulan Buh Hobq Mu Us Tengger Small Tengger Keerxing

Area (104 km2 ) 33.76 4.88 3.49 1.95 4.43 1.00 1.61 3.21 4.27 2.33 2.46

Deposition g m−2 mon−1 37.5 (9.2–158.3) 10.8 (3.1–22.5) 19.2 (5.7–40.0) 26.7 (3.3–91.7) 25.8 (8.3–62.5) 55.8 (1.2–175) 35 (6.1–47.5) 31.7 (5.5–108.3) 24.2 (1.3–100)

Desert Total Tg yr−1 150 (37–630) 6.2 (1.8–13) 7.9 (2.4–17) 6.2 (0.8–22) 14 (4.4–33) 6.7 (0.1–21) 6.7 (1.2–9.2) 12 (2.1–42) 12 (0.6–52)

Comments Sand desert Sand desert Sand desert Sand desert Sand desert Sand desert Loess Loess Sand desert Sand land Sand land

these two studies are summarised in Tables 8.3 and 8.4. The observations show that areas close to the dust source have a much larger deposition rate than areas away from the dust origin. The deposition rate in the Chinese desert regions is many times larger than that over the North Pacific, e.g. deposition in the Ulan Buh desert (670 g m−2 yr−1 ) is nearly 70 times that over the

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8 Dust Transport and Deposition

North Pacific (10 g m−2 yr−1 ). Liu et al. (2004) have measured monthly dustdeposition rate at Gaolan (36◦ 13 N, 103◦ 47 E, on the Loess Plateau, Gansu, China) over a two-year time period (May 1998 to April 2000) and found the annual average dust-deposition rate to be around 133 g m−2 yr−1 , similar to the results of Derbyshire et al. (1998). The rate of dust deposition during individual dust storms, reaching up to 11,720 g m−2 yr−1 , can be many times the annual average rate.

8.2 Lagrangian Dust-Transport Model The modelling of dust transport is usually done in conjunction with a host atmospheric model, in either the Lagrangian or the Eulerian framework. In the Lagrangian framework, dust particles are treated as discrete entities in the flow and their trajectories are determined through the integration of the equation of particle motion. Except for very severe dust storms, dust concentration is usually much less than several grams per kilogram of air, and the presence of dust does not significantly alter the density of the air and the dynamic behaviour of the flow. Therefore, it is permissible to determine separately the fluid and the particle motions. Lagrangian models are useful in tracking the paths of particles of different sizes under various atmospheric conditions (Ellis Jr and Merrill, 1995; Kotamarthi and Carmichael, 1993). The concept and numerical techniques for Lagrangian modelling are quite simple, but it is necessary to track a large ensemble of dust particles (about 20,000) in order to obtain statistically-reliable results. This makes Lagrangian models computationally expensive. Turbulence in the atmosphere, especially in the atmospheric boundary layer, adds another dimension of complexity to the problem. The motion of particles in a turbulent flow is affected by a combined effect of turbulent diffusion, particle inertia and the gravity force. The estimate of the necessary parameters to account for the effect of turbulence is a difficult theoretical problem which has not yet been satisfactorily solved. Given the initial position and velocity of a dust particle, its future location can be determined by the integration of Equations (6.89) and (6.90). For longrange dust transport, we are mainly interested in particles whose diameters are smaller than 20 µm. For these particles, the particle-response time Tp (σp d2 /18ν) is around or smaller than 10−2 s, which is small in comparison with the typical time scale of fluid motion. Therefore, Equations (6.89) and (6.90) become upi = ui − δi3 gTp dxi = upi dt

(8.4) (8.5)

The practical task of applying the above model is to estimate the wind field, ui , either by the analysis of observed data from the meteorological

8.2 Lagrangian Dust-Transport Model

253

network or from simulations using numerical weather-prediction or globalcirculation models. In each case, the flow velocity, ui , is available at the points of a grid with a fairly coarse resolution. Current global-circulation models have a typical resolution of around 100 km, and weather-prediction models have a typical resolution of around 10 km. Therefore, spatial interpolation of windfield data is required for the implementation of the dust-transport model. The vertical wind speed, u3 ≡ w, is normally derived in atmospheric models or from observed data, by using the continuity equation ∂u ∂v ∂w + + =0 ∂x ∂y ∂z

(8.6)

Integration of the above equation with respect to z from the surface to a reference level z yields  z ∂u ∂v + dz (8.7) w(z) = − ∂x ∂y 0 However, in the atmosphere, the typical vertical wind speed (0.01 m s−1 ) is about three orders of magnitude smaller than the typical horizontal wind speed (10 m s−1 ). In all atmospheric models and observed data, it is difficult to determine the vertical wind speed with desired accuracy. It is sometimes not meaningful to directly use the vertical wind speed obtained through either data analysis or numerical modelling. Given this general constraint, the calculation of a dust-particle trajectory is usually achieved by calculating the isentropic trajectory for an air parcel and a correction is then made to account for the gravitational settling of the dust particle in the vertical direction. Isentropic trajectories are calculated on surfaces of constant potential temperature, θ, and they are preferred over isobaric trajectories (calculated on surfaces of constant pressure) and isosigma trajectories (calculated on surfaces of constant p/ps ) due to their more realistic representation of the motion of dry air, where dθ/dt is zero (Kuo et al. 1985). Because θ remains constant in an isentropic process, the vertical velocity of the air parcel can be determined via 

∂θ ∂θ ∂θ +v + wθ = − u ∂x ∂y ∂t



∂θ ∂z

−1

Suppose an air parcel and a dust particle are both located at height zft−1 = zpt−1 at time step t − 1 and the air parcel would move to zft following the isentropic surface. Then, the height of the dust particle at time step t can be found by (Fig. 8.3) (8.8) zpt = zft − wt ∆t Figure 8.4 shows an example of the analysis of pressure (hPa) and winds on the 40◦ C isentropic surface for the 14 April 1983 dust-storm event occurred in the Sahara region (Reiff et al. 1986). As can be seen, the pressure on the

254

8 Dust Transport and Deposition θk+1

θk

Air Parcle t

zk t−1 zk

wt ∆t

θk-1

Dust Particle

Surface

Fig. 8.3. An illustration of the isentropic method for the Lagrangian dust-transport model. The isentropic trajectory of an air parcel (open circle) is calculated on surfaces of constant potential temperature, θ, and a correction of that trajectory is applied for the dust particle (full dot)

50

60

30

450

40

470

50 480

420 430 420 370 380 305 330 390 315

470

390 330 420

470

330

316

L

305

20

500

L

40

490

520

700 mb

(440)

540

30

(480)

830

(730)

820 30

820

20 820 800

20

10

0

10

Fig. 8.4. Analysis of pressure (hPa) and wind on the 40◦ C isentropic surface at 1200 UTC 14 April 1983. The thick lines are isobars on the above isentropic surface; the thin lines with arrows are streamlines. The stippled region is the dense dust plume on satellite imagery (From Reiff et al. 1986)

8.3 Eulerian Dust-Transport Model

255

Fig. 8.5. Trajectories calculated backward from a number of points in Europe at 00 UTC 18 April 1983. The outlined area represents the locus of trajectory points that had the southern Sahara and Mauritania as origins. Plotting convention is shown in the key at the lower right: ppp refers to pressure (hPa) and hh/dd to hour and day as the trajectories pass the location. At the cross are given the pressure levels upon arrival of the trajectories (From Reiff et al. 1986)

isentropic surface decreases from south to north, indicating a rise of the isentropic surface to higher levels. An useful application of the Lagrangian method is to identify the origin of dust particles observed at a given location through backward tracking. The backward-tracking technique has been implemented by several researchers. An example is presented in Fig. 8.5, which shows the trajectories calculated backwards from a number of points in Europe. The dust particles have been identified as originating from the southern Sahara and Mauritania.

8.3 Eulerian Dust-Transport Model In Eulerian models the particulate phase is assumed to be a continuum like the fluid phase. Thus, dust concentration in the atmosphere obeys the advectiondiffusion type of conservation equation, as other scalars in the atmosphere do. Dust concentration at a given time and location can in principle be determined

256

8 Dust Transport and Deposition

if the sources and sinks of dust are specified and the initial and boundary conditions for the concentration field are given. One major challenge here, though, is that the processes, such as particle diffusion, dry deposition and wet deposition, must be adequately represented in the concentration equation. Since most atmospheric-flow models are formulated in the Eulerian framework, the advection-diffusion type of conservation equation for dust particles can be incorporated into them and solved with similar numerical techniques as already exist in the atmospheric models. Let us rewrite Equation (3.24) as ∂c ∂c ∂c ∂c +u +v + (w − wt ) ∂t ∂x ∂y ∂z ∂ ∂ ∂ ∂c ∂c ∂c Kpx + Kpy + Kpz + S r + Sc = ∂x ∂x ∂y ∂y ∂z ∂z

(8.9)

where Kpx , Kpy and Kpz are the dust-particle eddy diffusivities for the x, y and z directions, Sr is a source (or sink) term arising from wet removal and Sc is a source (or sink) term arising from the dust-flux convergence due to dry and wet convections. The accompanying lower and upper boundary conditions are ∂c = F0 at the surface ρ(w − wt )c − ρKpz ∂z ∂c =0 at the top of the atmosphere Kpz ∂z

(8.10)

where F0 is the dust flux at the surface. In areas of active wind erosion, F0 is determined from the dust-emission schemes described in Chapter 7, and in areas with no active wind erosion, F0 is the flux caused by dry deposition which will be described in Section 8.6 of this chapter. Numerically, the modelling of dust concentration and transport is closely linked to the host atmospheric model. Equations (8.9) and (8.10) together with the conditions for the side boundaries of the simulation domain can be solved for the prediction of dust concentration. The equations and boundary conditions need to be discretised in accordance with the grid system of the host atmospheric model, using either finite differencing or other discretisation schemes. It is more elegant to solve Equation (8.9) simultaneously with other governing equations for the atmosphere, but it is often more convenient to solve it after the host atmospheric model has completed the calculations for each time step and the flow-field data for the new time step have become available. The host atmospheric model also calculates the eddy diffusivities, Kx , Ky and Kz , for passive scalars, from the modification of which the eddy diffusivity for dust particles, Kpx , Kpy and Kpz (Section 8.4) can be derived. Because the particle terminal velocity, wt , and the eddy diffusivities for dust particles, Kpx etc., depend on particle size, it is desirable to separate dust

8.3 Eulerian Dust-Transport Model

257

particles into I dust bins, each with a particle-size range di +∆di . Suppose the concentration of the ith particle-size group is ci , then the total concentration is c=

I

ci

i=1

The transport of each particle-size bin can then be treated independently of the other size bins, using Equation (8.9) and replacing c with ci ∂ci ∂ci ∂ci ∂ci +u +v + (w − wti ) ∂t ∂x ∂y ∂z (8.11) ∂ ∂ ∂ ∂ci ∂ci ∂ci Kpxi + Kpyi + Kpzi + Sri + Sci = ∂x ∂x ∂y ∂y ∂z ∂z A unified set of rules does not seem to exist for the division of dust particles into different size bins (Uno et al. 2006). Westphal et al. (1987) used four particle-size intervals: d ≤ 2 µm, 2 < d ≤ 6 µm, 6 < d ≤ 60 µm and d > 60 µm. Tegen and Fung (1995) also used four, but somewhat different, particle-size groups with d ≤ 2 µm, 2 < d ≤ 20 µm, 20 < d ≤ 50 µm and d > 50 µm. It is useful to introduce a uniform set of dust-size bins to facilitate comparison of future studies. Suppose we use wt as a criterion and assume that wt varies by a factor of 4 within each dust-size bin or wti+1 /wti = 4 As wt is given by Equation (5.20), we have that di+1 /di = 2 From the above considerations, a division of dust particles into different size groups is proposed to be d ≤ 2 µm, 2 < d ≤ 5 µm, 5 < d ≤ 10 µm, 10 < d ≤ 20 µm, 20 < d ≤ 40 µm etc. For numerical reasons, the advection-diffusion equation of the type Equation (8.9) is often solved by splitting the advective from the diffusive terms. The advective term is further split into horizontal and vertical advections. To illustrate the procedure, we can write Equation (8.9) as ∂c∗ + Ah ∂t ∂c∗∗ + Av∗ ∂t ∂c∗∗∗ ∂t ∂c ∂t

=0

(8.12)

=0

(8.13)

= D∗∗

(8.14)

= Sr∗∗∗ + Sc∗∗∗

(8.15)

where Ah , Av and D denote the horizontal advective, vertical advective and diffusive terms, respectively. These equations are solved sequentially. First,

258

8 Dust Transport and Deposition

t

u0 n+1 n n−1 i−2

i−1

i

i+1 x

Fig. 8.6. Illustration of numerical treatment for advection

the horizontal advection will be considered and an interim concentration field c∗ is calculated. The vertical advection will then be considered using c∗ , and another interim concentration field c∗∗ is calculated. This is followed by the treatment of the diffusion term using c∗∗ . Finally, the source terms due to wet removal and convection are considered. The advantage of the splitting is that the numerical treatment required for each step is much simpler than the collective numerical treatment for the full three-dimensional advectiondiffusion equation. The diffusion equation of the form (8.14) can be solved by using the Crank– Nicolson scheme, and the associated tri-diagonal system can be solved using the Thomas algorithm (Press et al. 1989). As far as the horizontal advection is concerned, numerous numerical schemes exist with varying performance in adequacy and numerical stability. If finite-differencing schemes are preferred, one good choice is to use the Heun scheme for temporal differencing and a third-order up-winding scheme for spatial differencing, as illustrated in Fig. 8.6. The Heun scheme is a two-level scheme. The integration of the advection equation ∂c +A=0 ∂t from time step n to time step n + 1 is achieved using the Heun scheme by c∗ = cn − An ∆t   n A∗ A cn+1 = cn − + ∆t 2 2

(8.16) (8.17)

where A∗ is calculated with the interim value c∗ . The discretisation of the advection term, such as u∂c/∂x for the ith point, using the third-order upwinding scheme is  ∂c b1 (ci+1 − ci−1 ) + b2 (ci−2 − 3ci−1 + 3ci − ci+1 ) u ≥ 0 = (8.18) b1 (ci+1 − ci−1 ) + b2 (ci−1 − 3ci + 3ci+1 − ci+2 ) u < 0 ∂x

8.3 Eulerian Dust-Transport Model

259

with b1 = 1/(2δx) and b2 = 1/(6δx). The multi-dimensional wave-propagation slope-limiter approach proposed by LeVeque (1996) is also an excellent scheme for horizontal advection. This scheme is second order accurate in both time and space and it eliminates numerical oscillations and maintains the positivity of the concentration field. For vertical advection, the numerical scheme proposed by Bott (1989) is popular. We rewrite Equation (8.13) as ∂(w − wt )c ∂c =− (8.19) ∂t ∂z With the assumption of constant grid spacing ∆z and time increment ∆t, the finite difference flux form of the above equation reads ∆t n n [f − fk−1/2 ] (8.20) ∆z k+1/2 n where cnk is the value of c at grid point k and time step n and fk+1/2 is the vertical flux [i.e. (w − wt )c] at level k + 1/2 and time step n. Bott’s advection scheme is obtained by a non-linear renormalization of the advective fluxes n n and fk−1/2 . These fluxes are computed using the integrated flux form fk+1/2 of Tremback et al. (1987) and then applying limits to the upper and lower values to make it positive definite. The basic idea is to assume that within grid box k the distribution of c is given by a polynomial of order of l. The fluxes can then be written as  +  − gl,k+1/2 ∆z gl,k+1/2 ck − ck fk+1/2 = ∆t gl,k gl,k+1 cn+1 = cnk − k

with + gl,k+1/2 = max(0, Il+ (ok+1/2 )) − = max(0, Il− (ok+1/2 )) gl,k+1/2 + − + gl,k+1/2 + ) gl,k = max(Il,k , gl,k+1/2

where  is a small number, Il+ (ok+1/2 ), Il− (ok+1/2 ) and Il,k are integrals which can be evaluated as Il+ (ok+1/2 ) =

l

j=0

Il− (ok+1/2 ) =

l

(−1)j ak+1,j j=0

Il,k =

l

j=0

o± k

ak,j j+1 [1 − (1 − 2o+ ] k) (j + 1)2j+1

(j + 1)2j+1

j+1 [1 − (1 − 2o− ] k)

ak,j [(−1)j + 1] (j + 1)2j+1

= ±(onk+1/2 ± | onk+1/2 |)/2

260

8 Dust Transport and Deposition

Table 8.5. Coefficients ak,j for orders up to 2 of polynomials for the Bott advection scheme. Note that l is the order of the polynomial for approximation of c within the grid box k 1 0 1a 1b 2

ak,0 ck ck ck ck

ak,1 − ck+1 − ck ck − ck−1 1 (c − ck−1 ) 2 k+1

ak,2 − − − 1 (c − 2ck + ck−1 ) 2 k+1

Fig. 8.7. Simulated near-surface dust concentration in µg m−3 of 4 particle-size groups for 9 February 1996 over Australia

where onk+1/2 is the Courant number defined by onk+1/2 = (w −wt )nk+1/2 ∆t/∆z and ak,j (j = 1, ..., l) are the coefficients of the polynomial. For a given l (up to 2), which is a reasonable choice for practical purposes, the values of ak,j are listed in Table 8.5. The above is the basis of the Eulerian dust-transport model developed by Lu (1999). She used this model to simulate the long-range transport of Australian dust. As an example, Fig. 8.7 shows the predicted dustconcentration fields of six different particle-size groups for the 9 February 1996 Australian dust storm. The fine-dust particles emitted from the Simp-

8.4 Vertical Dust Transport by Diffusion

261

Fig. 8.8. An east-west cross section of simulated dust concentration in µg m−3 for the 2 to 11 µ particle-size group. The cross section is along 27.5◦ S, right over the source region of the dust-storm event

son Desert in central Australia were transported toward the Indian Ocean in the west and the Pacific Ocean in the east. The dust plume was confined mainly to a narrow region of a cold front. Relatively-coarse particles did not travel a large distance, but were deposited in the vicinity of the source region. Figure 8.8 shows an east-west cross section of dust concentration for the particle-size range between 2 to 11 µm, at latitude 27.5◦ S, just over the Simpson Desert.

8.4 Vertical Dust Transport by Diffusion In addition to the grid-scale vertical advection, two subgrid processes need to be considered in relation to the vertical movement of dust particles in the atmosphere. Turbulent diffusion is the main process in the atmospheric boundary layer for dispersing dust from the surface to the free atmosphere. This mechanism is usually dealt with in dust-transport models by assuming that the diffusive dust flux is proportional to the gradient of dust concentration and introducing the particle eddy-diffusivity coefficient, Kp , so that the dust ∂c . This approach is analogous to the handling of turbulent flux is −ρKpz ∂z

262

8 Dust Transport and Deposition

diffusion of other scalars (Chapter 3). The other major process for the vertical transport of dust includes dry and wet convections, which are capable of moving dust particles rapidly through the entire tropospheric air column. In a sense, this process is an advective process which should be represented by the advective terms in Equation (8.9). However, as the spatial resolution of the atmospheric models is normally insufficiently fine, dry and wet convections with typical sizes of several kilometres are considered to be subgrid phenomena that require extra treatment. We first consider the calculation of particle eddy diffusivity, Kp . Dust particles in the atmosphere are heavy particles, as their density (around 2,600 kg m−3 ) is much larger than that of the air (around 1.2 kg m−3 ). When compared with air parcels, dust particles have different response characteristics to the changes in the flow field. Particles with larger mass (either heavier or larger) would respond more slowly to turbulent fluctuations. This is known as the inertial effect. Also, under the influence of the gravity force, dust particles fall from one turbulent eddy to another. This is known as the gravitational settling effect. Consequently, dust particles and air parcels follow different trajectories during turbulent diffusion. This is referred to as the trajectory-crossing phenomenon (Yudine, 1959). Both the inertial and the gravitational settling effects contribute to trajectory crossing as depicted in Fig. 8.9. Suppose an air parcel and a dust particle are both at location O at time t. After a small time increment dt, the air parcel would move to F and the dust particle to P. The position difference between the air parcel and dust particle at time t + dt is ur dt. It is therefore to be expected that the eddy diffusivity for dust particles, Kp , differs from that for air parcels, K. Csanady (1963) derived a theory which relates Kp to K by taking the trajectory-crossing effect into consideration. The results of Csanady’s theory are that −1/2  T 2 w2 (8.21) Kpx = Kpy = K 1 + 4 L 2 t Lz −1/2  T 2 w2 Kpz = K 1 + L 2 t (8.22) Lz where TL is the fluid Lagrangian integral-time scale, Lz is the integral-length scale of the Eulerian fluid-velocity field. The two scales are related by TL = βLz /σ where β is a dimensionless coefficient and σ is the standard deviation of the turbulent velocity, taken here to be the same for all three velocity components for simplicity. Thus, the above expressions can also be written as 

Kpx Kpz

β 2 w2 = Kpy = K 1 + 4 2 t σ   −1/2 β 2 wt2 = K 1+ σ2

−1/2 (8.23) (8.24)

8.4 Vertical Dust Transport by Diffusion (a) Inertial Effect

263

(b) Gravitational Effect

(c) Trajectory Crossing

F

up

ur d t

u dt

O

dt P

Fig. 8.9. The trajectory-crossing effect of a heavy particle in a turbulent flow, which arises from the inertial and the gravitational settling effects. The location of the air parcel is denoted with an open circle and that of the heavy particle with a full dot

More recent studies on the subject have been carried out by Pismen and Nir (1978), Nir and Pismen (1979), Mei et al. (1991) and Wang and Stock (1993) among many others. Lagrangian stochastic models have been developed by, for instance, Walklate (1987) and Sawford and Guest (1991) amongst others. The derivation of Kp can be more conveniently discussed in the Lagrangian framework, in which turbulent diffusion is considered by examining the ensemble statistics of the random motions of individual air parcels or particles. In the Lagrangian framework, the flow is described by the position of all air parcels X(X0 , τ ) (here τ is time) under the initial condition X(X0 , 0) = X0 . The Lagrangian velocity, defined as Vi (X0 , τ ) =

∂Xi (X0 , τ ) , ∂τ

is related to the Eulerian velocity ui by Vi (X0 , τ ) = ui (X(X0 , τ ))

(8.25)

264

8 Dust Transport and Deposition

The displacement, or the position change of the air parcel relative to its initial position, is given by Yi (τ ) = Xi (X0 , τ ) − X0i If N parcels are released from a source located at X0 , then the ensemble average of the displacement is defined as < Yi (τ ) > =

k=N 1 k Yi (τ ) N k=1

and the deviation of Yi (τ ) from its ensemble average is Yi (τ ) = Yi (τ )− < Yi (τ ) > The displacement tensor, an important quantity in statistical fluid mechanics, is defined as (8.26) Dij (τ ) = < Yi (τ )Yj (τ ) > While < Yi (τ ) > is a description of the mean position of the air parcel ensemble, Dij (τ ) is that of the spread of the ensemble. It is now important to establish the relationships between the displacement tensor and the velocity field. Note that the displacement of an individual fluid element is given by τ

Yi (τ ) =

Vi (X0 , t)dt

(8.27)

0

and the mean displacement can be expressed as τ < Vi (X0 , t) > dt < Yi (τ ) > =

(8.28)

0

From the definition of Yi , we have Yi (τ ) =



τ

Vi (X0 , t)dt

(8.29)

0



and

τ



Dij (τ ) = 0

τ

< Vi (X0 , t1 )Vj (X0 , t2 ) > dt1 dt2

(8.30)

0

If τ is small, the approximation Vi (X0 , τ ) = ui (X0 ) applies and it follows that Yi (τ ) = ui (X0 )τ . In this case, Dij can be written as Dij (τ ) = < ui (X0 )uj (X0 ) > τ 2

(8.31)

where < ui (X0 )uj (X0 ) > is simply the Eulerian velocity covariance at X0 , which is usually written as ui uj . Hence, Dij (τ ) is proportional to τ 2 for small τ .

8.4 Vertical Dust Transport by Diffusion

265

We now briefly review the dispersion theory of Taylor for homogeneous, isotropic and stationary turbulence, which also provides the basis for the calculation of particle eddy diffusivity. Making the formal changes, s = t2 − t1 and t = (t1 + t2 )/2, we can rewrite Equation (8.30) in the following form

τ



τ −s/2

Dij (τ ) =

[Bij (t, s) + Bji (t, s)]dtds 0

(8.32)

s/2

where Bij is the Lagrangian velocity-covariance function Bij (t, s) = < Vi (X0 , t)Vj (X0 , t + s) >

(8.33)

The assumption of stationary, homogeneous and isotropic turbulence implies that Bij (t, s) is dependent on s, but independent of t, and that Bij (s) = Bji (s). It also implies that the velocity variances are independent of time and space and that they can be taken to be identical in all three directions. It follows that τ

Dij (τ ) = 2σ 2

τ −s/2

Rij (s)dtds 0

(8.34)

s/2

where σ is the standard deviation of the Eulerian velocity components and Rij is the Lagrangian velocity-correlation function Rij (s) = Bij (s)/σ 2 Evaluating the time integration in Equation (8.34), we have τ Dij (τ ) = 2σ 2 (τ − s)Rij (s)ds

(8.35)

0

In the above equation, the displacement tensor is expressed in terms of the Lagrangian correlation functions. The diagonal elements of the displacement tensor are of particular importance τ (τ − s)Rii (s)ds no summation over i (8.36) Dii (τ ) = 2σ 2 0

In general, it is acceptedthat the Lagrangian correlation function decays sufτ ficiently rapidly so that 0 sRii (s)ds approaches zero if τ approaches infinity. It follows that for large τ Dii = 2σ 2 TLi τ

no summation over i

with TLi being the Lagrangian integral-time scale given by ∞ TLi = Rii (s)ds no summation over i 0

(8.37)

266

8 Dust Transport and Deposition

This time scale describes the time over which the Lagrangian velocity of an air parcel is significantly correlated while it makes random motions under the influence of turbulence: i.e., for τ < TLi the correlation is significant, while for τ > TLi it is insignificant. Hence, TLi is a property of the turbulence field. The Lagrangian auto-correlation function Rii can be well approximated by an exponential function, so that Rii (τ ) = exp (−τ /TLi )

no summation over i

(8.38)

In the above expression, TLi is the time in which the Lagrangian autocorrelation function drops to 0.37. The basic results of the Taylor dispersion theory can now be stated more clearly: for τ  TLi (called the near field), Dij (τ ) is proportional to τ 2 , while for τ TLi (called the far field), Dij (τ ) is proportional to τ . The eddy-diffusivity tensor is related to the displacement tensor by Kij =

1 dDij | | 2 dt

(8.39)

From Equation (8.35), we have Kij (τ ) = σ 2

τ

Rij (s)ds

(8.40)

0

and for the diagonal terms of Kij τ 2 Kii (τ ) = σ Rii (s)ds

no summation over i

(8.41)

no summation over i

(8.42)

0

For the near field and far field, we have Kii = σ 2 τ Kii = σ 2 TLi

τ  TLi τ TLi

no summation over i

(8.43)

We now return to our main interest, which is the eddy diffusivity for dust particles, Kp , using the same formulation described above for the dispersion of fluid parcels. By analogy with Bij (s), the particle Lagrangian velocitycovariance function Sij (s), and the particle Lagrangian velocity-correlation function, Rpij (s), can be defined as Sij (s) = < upi (X0 , t)upj (X0 , t + s) >

(8.44)

Rpij (s) = Sij (s)/σp2

(8.45)

and respectively, where σp is the standard deviation of the particle-velocity components. In correspondence to Equation (8.40), the eddy diffusivity for dust particles can be expressed as

8.4 Vertical Dust Transport by Diffusion



267

τ

Kpij =

Sij (s)ds

(8.46)

0



or

τ

Kpij = σp2

Rpij (s)ds

(8.47)

0

The link between the eddy diffusivities for dust particle, Kpij , and that for air parcels, Kij , is established by considering the fluid velocity fluctuations ∗ (s) along the dust particle trajectories. By analogy with Bij (s) and Sij (s), Bij is introduced as the covariance function for the fluid velocity along the dustparticle trajectories ∗ (s) = < ui (Xp (t))uj (Xp (t + s)) > Bij

(8.48)

where Xp (t) denotes the dust particle position at time t. Similarly, the correlation function for the fluid velocity along the dust particle trajectories is ∗ ∗ (s) = Bij (s)/σ 2 Rij

It can be shown (Pismen and Nir, 1978) that the relationship between Sij and ∗ is Bij   ∞ 1 |τ −s| ∗ ds (8.49) B (s) exp − Sij (τ ) = 2Tp −∞ ij Tp where Tp is the particle response time. For the limiting case of Tp → 0, ∗ (τ ). It follows from Equations (8.46) and (8.49) that Sij (τ ) → Bij Kpij =

τ

∗ Bij ds +



0

∞

∗ Bij (s) exp



τ

τ −s Tp



∗ ds − Tp Bij (τ )

(8.50)

which has a long-time limit lim Kpij =

t→∞

∞

∗ Bij ds

(8.51)

0

Equation (8.50) shows that the eddy diffusivity for dust particles can be expressed in terms of the covariance function of fluid velocity along the particle trajectory and it depends in general on the particle response time which in turn is a function of particle size. The behaviour of different particles, or more specifically the trajectorycrossing effect, would vary from flow field to flow field. For instance, the diffusion of very heavy particles in a weak turbulence field cannot be expected to be the same as that of not-so-heavy particles in a strong turbulence field. The relative importance of the trajectory-crossing effect is quantified using two non-dimensional parameters; namely, the Stokes parameter and the drift parameter. The Stokes parameter is the ratio between the particle-response time, Tp , and the Eulerian integral-time scale, TE , i.e.,

268

8 Dust Transport and Deposition

St = Tp /TE which is a measure of the relative importance of the inertial effect. The drift parameter is the ratio between the particle’s terminal velocity, wt , and the typical velocity scale of the turbulence field, say u∗ , so that Dr = wt /u∗ which is a measure of the relative importance of the effect of gravity. (Friction velocity, u∗ , is an excellent scaling parameter for turbulence in atmospheric surface layers. For other regions of the atmosphere, other scaling velocities may be required.) As a simple case, suppose the inertial effect can be neglected, namely St  1, as assumed by Csanady (1963). This implies that the trajectorycrossing effect is settling under gravity only. Under this assumption, Tp can ∗ (s) and be set to zero in Equation (8.49) and the relationships Sij (s) = Bij ∗ ∗ ∗ Rpij (s) = Rij (s) apply. Therefore, if Bij (s) or Rij (s) can be estimated, then Kpij can be determined using Equation (8.46). In the following, we shall confine the discussions to the diagonal components of the particle eddy-diffusivity tensor, namely, Kpx ≡ Kp11 , Kpy ≡ Kp22 and Kpz ≡ Kp33 . First, we discuss ∗ ∗ = Bii /σ 2 following the behaviour of the velocity auto-correlation functions Rii the argument of Csanady (1963). Consider a situation in which a stationary, homogeneous and isotropic turbulence field has a mean flow velocity U in the horizontal direction of x and dust particles with St  1 are released into that field. The motion of the particles will be examined in a frame which moves with the mean flow velocity. We assume that the velocity auto-correlation functions of the turbulent flow in the moving Eulerian frame are all exponential functions; namely (with no summation over i),   |τ | (8.52) Rii (τ ) = exp − TLi   |τ | (8.53) Eii (τ ) = exp − TEi   |r| (8.54) fii (r) = exp − Lf i     |r| |r| gii (r) = 1 − exp − (8.55) 2Lf i Lf i where Eii is the temporal fluid Eulerian velocity-correlation function at a given point in space, fii (r) and gii (r) are the longitudinal and transverse spatial fluid velocity-correlation functions at a given time, respectively; TEi is the Eulerian integral-time scale and Lf i is the Eulerian integral-length scale. While Equation (8.52) is widely accepted in the literature and confirmed by observations, the exponential shapes of the Eulerian correlation functions, Equations (8.53) and (8.54), are assumed for the convenience of mathematical

8.4 Vertical Dust Transport by Diffusion

269

manipulation. Equation (8.53) is inconsistent with Kolmogorov’s theory of local isotropy. However, Wang and Stock (1993) have argued that it is the value of the integral-length scale, not the shape of the correlation function, which determines the long-term dispersion of particles and hence Equations (8.53) and (8.54) are still acceptable. Equations (8.54) and (8.55) satisfy the continuity condition for isotropic turbulence, which requires the correlation function in the lateral direction (relative to the fall velocity), g(r), to be related to the correlation function f (r) in the longitudinal direction by r df 2 dr Hence, if f (r) satisfies Equation (8.54), g(r) must satisfy Equation (8.55). The Eulerian integral-length scale is related to the Lagrangian integral-time scale by Lf = σTL /β with β being a constant in the range from 1 to 2. At a small time τ after the release, a dust particle will be somewhere in the vicinity of η = wt τ . There are two limiting cases. The first one is that wt is negligible compared with the velocity scale of turbulence, i.e., Dr  1. In this ∗ is reduced to Rii . The second one is that wt is so large that Dr 1. case, Rii ∗ is reduced to the instantaneous spatial velocity correlation at In this case, Rii two points separated by a distance η, namely either fii (η) in the longitudinal direction or gii (η) in the lateral direction. Csanady (1963) considered these ∗ ) lines are ellipses two limiting cases and hypothesised that constant Rz∗ (≡ R33 satisfying the relationship  2  2 η τ + = const (8.56) TL Lz g(r) = f (r) +

η/Lz

Csanady’s hypothesis for Rz∗ is illustrated in Fig. 8.10. Using Equations (8.52) to (8.56), we obtain the fluid velocity-correlation functions seen by a dust

5

0.2

0

0.5

∗

Rz

.75 =0 τ/TL Fig. 8.10. Hypothetical shape of isolines of the longitudinal (vertical, particle-falling direction) velocity-correlation function Rz∗ (η, τ )

270

8 Dust Transport and Deposition

particle as Rx∗ (τ, η)

=

Ry∗ (τ, η)

   1/2   τ |η| σ2 2 exp − wt + 2 = 1− 2Lz Lz β 

Rz∗ (τ, η)

τ = exp − Lz

 wt2

σ2 + 2 β

(8.57)

1/2  (8.58)

Substituting Equations (8.57) and (8.58) into Equation (8.50), setting Sii = ∗ 2 σ and integrating Equations (8.57) and (8.58) with respect to τ , we obtain Rii the particle eddy diffusivity as given in Equations (8.21) and (8.22). Csanady’s theory shows that Kp can be derived from a modification of K through a correction factor which involves TL , Lz and wt . The above discussions show that the key to modelling Kp lies in the specific ∗ , a quantity which cannot be directly measured. Walklate assumptions for Rii ∗ ; namely, that (1987) proposed an alternative to the Csanady model for Rii ∗ ∗ (τ, η) = R22 (τ, η) = R11 (τ )g11 (η) R11 ∗ R33 (τ, η) = R33 (τ )f33 (η)

(8.59) (8.60)

A combination of the two equations above and Equations (8.52) and (8.53) leads to     |τ | |η| exp − ∗ (8.61) Rx∗ (τ, η) = Ry∗ (τ, η) = 1 − 2Lz TL   |τ | Rz∗ (τ, η) = exp − ∗ (8.62) TL where Lz = σTL /β and TL∗ is given by TL∗ =

σTL σ + βwt

(8.63)

Walklate’s assumption leads to −1  βwt Kpx = Kpy = K 1 + 4 σ −1  βwt Kpz = K 1 + σ

(8.64) (8.65)

The performances of the Walklate and Csanady models differ substantially from one another. In general, the particle-inertia effect also needs to be considered. Using Csanady’s hypothesis (8.56) and substituting Equations (8.57) and (8.58) into Equation (8.49), lengthy expressions for Rp11 (τ ) and Rp33 (τ ) can be derived (Wang and Stock, 1993)

8.4 Vertical Dust Transport by Diffusion

 Rp11 (τ ) = +

σ2 θ 2

σ θ

exp(− Tτp )

Tp TL



T

t 2 1 + ( βw σ ) − 

 t 2 τ exp[− 1 + ( βw σ ) TL ] −1 +

p 2 1 βwt Tp ( TL ) [1+( 2 σ TL θ T

( T p )2 L

βwt σ



1+(

βwt 2 σ ) ]+1

βwt 2 σ )

θ

271

 

+

βwt τ 2σ TL

(8.66) T

Rp33 (τ ) =

σ 2 TLp



where

     τ σ2 βwt 2 τ exp − exp − 1 + ( ) − θ Tp θ σ TL (8.67)   2   2 βwt Tp 1+ −1 θ= TL σ

t 2 1 + ( βw σ )

Equations (8.66) and (8.67) show explicitly that the behaviour of Rpii depends on the Stokes parameter, reflected in Tp /TL which is related to St through Tp /TL = St TE /TL , and the drift parameter, reflected in wt /σ which is related to Dr through wt /σ = Dr u∗ /σ. Figure 8.11 shows an example of the dependency of the Rp11 (τ ) function on wt (normalised with σ) for the given Stokes parameter St = 0.2 and β = 1. Finally, using Equations (8.66) and (8.67), we find that the eddy diffusivities for dust particles are  1 βwt t 2 1 + ( βw σ ) − 2 σ (8.68) Kpx = Kpy = Kx t 2 1 + ( βw σ )

10 0.01

0

0.1

wt /σ

0.4 5

1.0

-0.005

Rp,11

-0.015 0.8 0 0

1.25

2.50 τ/T E

3.75

5.00

τ/TE

Fig. 8.11. Contours of Rp11 in the τ /TE and wt /σ plane. The Stokes number, St , and the parameter, β, are set to 0.1 and 1, respectively (From Wang and Stock, 1993)

272

8 Dust Transport and Deposition 1.0 (a)

(b)

Kp/K

0.8 0.6 0.4 Csanady (1963) Walklate (1987) Wang & Stock (1993)

0.2 0.0 1.0

(c)

(d)

Kp/K

0.8 0.6 0.4 0.2 0.0 0

20

40 60 d (µm)

80

100

0

20

40 60 d (µm)

80

100

Fig. 8.12. Ratio between particle eddy diffusivity, Kp , and scalar eddy diffusivity, K, estimated using the models of Csanady (1963), Walklate (1987) and Wang and Stock (1993). (a) for Kpx /Kx and Kpy /Ky and (b) for Kpz /Kz . For (a) and (b), β = 1 and σ = 0.5 m s−1 are used. (c) As (a) and (d) as (b) but for σ = 0.1 m s−1

Kpz = Kz 

1

(8.69)

t 2 1 + ( βw σ )

Equation (8.69) is identical to Equation (8.24). A comparison of the Csanady, Walklate and the Wang and Stock models is shown in Fig. 8.12. The Csanady model and the Wang and Stock model are identical for Kpz and differ slightly for Kpx and Kpy . The performance of the Walklate model differs considerably from that of the other two models. There is a serious lack of experimental data to test these models, but the Csanady model appears to compare well with the limited data set of Snyder and Lumley (1971). Figure 8.12 reveals that the particle eddy diffusivity and the diffusivity for scalars are not substantially different for particles smaller than 20 µm if turbulence is reasonably strong (e.g. σ = 0.5 m s−1 ). For particles smaller than 20 µm, it is justified to simply use Kp = K for practical purposes in strong turbulent flows. As can also be seen, the difference between Kp and K is increasingly larger for larger particles. For instance, Kp /K becomes less than 0.5 for 100 µm particles for the σ = 0.5 m s−1 case. As the intensity of turbulence decreases (e.g. σ = 0.1 m s−1 ), the modification to Kp is more obvious.

8.5 Vertical Dust Transport by Convection

273

8.5 Vertical Dust Transport by Convection Dust particles can be carried to the upper troposphere by dry and wet convective cells, which penetrate the entire troposphere in some cases. Since the typical (horizontal) size of these convective cells is usually much smaller than the grid size of the atmospheric model (several kilometres compared to several tens of kilometres), dry and wet convection are subgrid processes which cannot be directly represented by the advective terms resolved by the grid. Instead, they need to be parameterised. Furthermore, this mechanism of vertical dust transport differs profoundly from turbulent diffusion in that it is not directly controlled by the gradient of dust concentration, but by the stability of the atmosphere. For these reasons, special treatment for convective dust transport, represented by Sc in Equation (8.9), is necessary. In this section, we describe the convective adjustment method and the cumulus parameterisation method for modelling convective dust transport. 8.5.1 Convective Adjustment Let us consider an air column. If the air is unsaturated (q < qs , where q is specific humidity and qs is saturation specific humidity) and the lapse rate of temperature, Γ (defined as −∂T /∂z, T is temperature), is larger than the dry adiabatic lapse rate, Γd (≡ g/cp , c.a. 9.8◦ C km−1 ), then the atmosphere is unstable and dry convection will occur (Fig. 8.13a). The effect of dry convection is to increase the stability of the atmosphere by reducing the temperature lapse rate from Γ to Γd . The convective process also results in adjustments of humidity and dust concentration profiles. Dry convection does not involve condensation and the release of latent heat. Wet convection occurs if there is sufficient disturbance to force air to rise above the condensation level and the air column is moist unstable, i.e., Γ is larger than Γm , the moist adiabatic lapse rate. Wet convection is often associated with the formation of clouds and the process of precipitation (Fig. 8.13b). A number of parameterisation schemes have been developed since the early 1960s. The choice of the convection schemes depends very much on the resolution of the host atmospheric model. For atmospheric models with a coarse resolution, the three widely-used schemes are the vertical-adjustment scheme (Manabe et al. 1965; Kurihara, 1973), the Kuo scheme (Kuo, 1974 and Anthes, 1977) and the Arakawa-Schubert scheme (Arakawa and Schubert, 1974). It is not intended to give a full account of these schemes herein, but interested readers should refer to Pruppacher and Klett (1997) for details. While the vertical-adjustment scheme may not be the most effective of schemes, it is relatively simple to implement to describe the effect of convection on the dust-concentration profile. In the case of dry and/or wet convection, the adjustment scheme will cause a vertical redistribution of sensible and latent heat. The temperature lapse rate after the adjustment, Γc , is defined to become

274

8 Dust Transport and Deposition T(z)

(a) z

T(z)

(b)

z ta ois M dia ba t

ya Dr

Dry convection Γ > Γd

Wet convection Γ > Γm

dia ba

Condensation level

t

T

T

(d)

(c) z

z After adjustment

k−2 k−1 k k+1

Before adjustment

k+2 c

Fig. 8.13. Schematic illustrations of dry and wet convection. (a) Dry convection occurs if the temperature lapse rate Γ is larger than the dry-adiabatic lapse rate Γd ; (b) Wet convection occurs above the condensation level when the temperature lapse rate Γ is larger than the moist-adiabatic lapse rate Γm ; (c) Layers of an air column in an atmospheric model; and (d) adjustment of dust-concentration profile by convection

 Γc =

Γd Γd

1−rh 1−rhc



+ Γm

rh −rhc 1−rhc

(0 ≤ rh ≤ rhc ) (rhc < r < 1)

(8.70)

where rh is relative humidity and rhc is a critical value set to 0.5 in most cases. If rh < rhc and Γ > Γd , dry-convective adjustment takes place, while if rh > rhc and Γ > Γm , wet-convective adjustment takes place. In a configuration as illustrated in Fig. 8.13c, level k is unstable and adjustment is required if Tk+1 − Tk−1 >

1 [Γc |k+1 (zk − zk+2 ) + Γc |k−1 (zk−2 − zk )] 2

(8.71)

After the adjustment, we have ∗ ∗ Tk+1 − Tk−1 =

1 [Γc |k+1 (zk − zk+2 ) + Γc |k−1 (zk−2 − zk )] 2

(8.72)

Suppose adjustments are to be made for layers k1 to k1 + 2n − 2, then these adjustments are to be done under the constraint that the total sensible and latent heat is conserved, i.e.,

8.5 Vertical Dust Transport by Convection k1 +2n−2

∗ ∗ [cp (Tk−1 − Tk−1 ) + λl (qk−1 − qk−1 )] = 0.

275

(8.73)

k=k1

There are several different ways in which q can be adjusted during a convection process. One possibility is to assume that the relative humidity remains unchanged during the adjustment process. In this case, it then follows that qk∗ = rhk qs (Tk∗ ) where the saturation specific humidity, qs , as a function of Tk∗ is given by the Clausius-Clapeyron equation. While all convection processes modify the vertical profiles of temperature and specific humidity, the extent to which such motions redistribute dust particles through a convective column is unclear. Joussaume (1990) has suggested that convection would lead to a complete vertical mixing through the depth of the convective column and hence a uniform profile of dust concentration (Fig. 8.13d). Alternatively, it appears plausible to assume that the relative pattern of dust-particle redistribution is identical to that of specific humidity, namely, c∗k = ck qk∗ /qk 8.5.2 Cumulus Parameterisation Cumulus parameterisation schemes have been developed for the prediction of convective precipitation in atmospheric models (Arakawa and Schubert, 1974; Fritsch and Chappel, 1980; Tiedtke, 1989). These schemes compute the convective air-mass fluxes associated with clouds. In some studies, several cloud types are considered (Gidel 1983; Dvortsov et al. 1998), while in others one cloud type is used to represent the effect of all cloud types (Gimson, 1997). We use the one-cloud-type model to illustrate the use of cumulus parameterisation in modelling convective dust transport. A convective dust-transport equation is established based on the convective air-mass fluxes associated with clouds. We imagine that an atmospheric model-grid cell is divided into area fractions αu , αd and αe = 1 − αu − αd , occupied respectively by updrafts, downdrafts and the environment (Fig. 8.14). Sc in Equation (8.9) can be written as Sc = −

∂w c |cv . ∂z

(8.74)

The (kinematic) convective dust flux w c |cv can be expressed as w c |cv = αu wpu cu + αd wpd cd + αe wpe ce − w ¯p c¯

(8.75)

where wpu , wpd and wpe are the vertical velocities of particles in the updraft, downdraft and environment, respectively, cu , cd and ce are the corresponding particle concentrations and

276

8 Dust Transport and Deposition

αd

αu

εd

δu wpu we wpd

δd

εu

Fig. 8.14. An illustration of the sub-grid processes of convection. The atmospheric model-grid cell is divided into area fractions αu , αd and αe , occupied respectively by updrafts, downdrafts and environment

c¯ = αu cu + αd cd + αe ce w ¯p = αu wpu + αd wpd + αe wpe

(8.76) (8.77)

Since the particle-response time is usually much smaller than the time scale of convection, it is sufficient here to assume wp = (w − wt ). It follows that wpu = wu − wt ; wpd = wd − wt ; wpe = we − wt . Substituting Equation (8.75) into Equation (8.74), we obtain Sc = −

∂ (αu wpu cu + αd wpd cd + αe wpe ce − w ¯p c¯) ∂z

(8.78)

A cloud model is needed to quantify the mass exchanges between the updrafts, downdrafts and environment. For the cloud updraft, the continuity equation can be written as ∂Mu = Eu − Du ∂z

(8.79)

where Mu [M T −1 ] is the vertical mass transport in the updraft and Eu is the entrainment of mass from the environment into the updraft and Du is the detrainment of mass from the updraft into the environment. In the cumulus parameterisation scheme of Kain and Fritsch (1990), Eu and Du are determined by the buoyancy variation at the interface between the updraft

8.6 Dry Deposition

277

and the environment. The rate of entrainment u and that of detrainment δu , both of dimensions [T−1 ], are simply 1 Eu Au ρ 1 Du δu = Au ρ u =

(8.80) (8.81)

where Au is the cross sectional area of the updraft and ρ is air density. Following the similar procedure, the rate of entrainment and that of detrainment associated with the downdraft of the cloud, d and δd , can be calculated. The governing equations describing the particle transport in the cloud drafts and in the environment are given by ∂wpu cu ∂cu =− + u ce − δu cu ∂t ∂z ∂wpd cd ∂cd =− + d ce − δd cd ∂t ∂z ∂wpe ce (αu u + αd d )ce (αu δu cu + αd δd cd ) ∂ce =− − + ∂t ∂z αe αe

(8.82) (8.83) (8.84)

where u ce is the entrainment rate of dust from the environment into the updraft, δu cu is detrainment rate of dust from the updraft into the environment, etc. Equations (8.82) – (8.84) can be solved numerically, and Sc can be calculated by       ∂wt c¯ ∂cu ∂cd ∂ce + αd + αe − (8.85) Sc = αu ∂t ∂t ∂t ∂z As described in Chapter 2, large-scale dust storms are mostly generated by synoptic systems accompanied by intensive convections, such as the monsoon troughs over the Sahara and the Mongolian cyclones over the Gobi. In such events, convective dust transport can be quite significant. Many of the presentday dust sources are located in basins surrounded by elevated terrains. Jung et al. (2005) suggested that deep convection plays a major role in the export of dust from the Tarim Basin, by carrying dust particles from the near-surface layers to levels well above the surrounding terrains, where they are transported further downstream by the westerlies in the upper troposphere.

8.6 Dry Deposition Dust particles are delivered back to the surface by both dry and wet deposition. Dry deposition is the dust flux from the atmosphere to the surface through molecular and turbulent diffusion and gravitational settling. Wet deposition is the dust flux to the surface through precipitation.

278

8 Dust Transport and Deposition

By analogy with the bulk-transfer formulation of scalar fluxes in the atmosphere, dry-deposition dust flux, Fd , can be expressed as Fd = −ρwd [c(z) − c(0)]

(8.86)

where c(0) and c(z) are dust concentrations at the surface and at the reference level z, respectively, and wd is the dry-deposition velocity. In general, wd is dependent on height, surface characteristics, flow properties and particle size. The dependency of wd on particle size has been emphasised in several studies (e.g. Arimoto et al. 1985; Dulac et al. 1989). So that dry deposition can be estimated with greater accuracy, the entire (mass) size distribution of dust particles is usually divided into I groups and wd is estimated as a weighted average I

fi wdi wd = i=1

where fi and wdi are the mass fraction and the dry-deposition velocity of the ith group. The difficult is, however, that the (mass) size distribution of airborne dust cannot be easily assessed from instantaneous field measurements. In practice, dust samples obtained using devices such as high-volume dust samplers are fitted with a monomodal lognormal particle-size distribution. The fitted particle-size distribution is then represented using up to 100 discrete successive size intervals, with each interval representing 1% of the total mass, that is fi = 0.01. For each given particle-size interval, wdi can be derived from dry-deposition models as discussed in the following sections. For dust particles of 1–5 µm, wd ranges between 0.01 and 0.05 m s−1 . For mineral aerosols over the ocean, wd ranges between 0.003 m s−1 and 0.03 m s−1 (Duce et al. 1991). 8.6.1 Two-Layer Dry-Deposition Model: Smooth Surface In general, the instantaneous vertical dust flux can be expressed as Fd = ρwp c − ρκp

∂c ∂z

(8.87)

where κp is particle molecular diffusivity. It is useful to separate wp into two components, wp = −wt + wp , where wt is particle terminal velocity and wp is the deviation of wp from −wt . Averaging Fd over time gives   ∂¯ c   ¯ (8.88) Fd = ρ −wt c¯ + wp c − κp ∂z where ρwp c is the dust flux due to turbulent mixing, which can be estimated from the K-theory using ∂¯ c (8.89) ρwp c = −ρKp ∂z

8.6 Dry Deposition

279

The vertical dust flux can then be written as (overbar is dropped for simplicity) 
∂c (8.90) Fd = ρ −wt c − (Kp + κp ) ∂z Equation (8.90) shows that dry deposition is comprised of a gravitational settling flux, −ρwt c and a diffusive flux, −ρ(Kp + κp )∂c/∂z which in turn is comprised of a turbulent diffusive flux −ρKp ∂c/∂z and a molecular diffusive flux −ρκp ∂c/∂z. In the atmospheric boundary layer, the physical mechanisms responsible for vertical dust flux vary with height: In the bulk of the atmospheric boundary layer, gravitational settling and turbulent diffusion dominate, while in the laminar layer immediately over the surface, gravitational settling and molecular diffusion dominate. The final stage of deposition from the air to the surface is very complex in detail as it not only depends on flow properties, but also on the properties of the dust particles and the surface. This stage of deposition therefore requires special attention. The situation is somewhat simpler if the surface is smooth and sticky. In this case, the surface can be treated as a perfect sink. In the two-layer dry-deposition model for smooth surfaces, the atmosphere below a certain height (say about 10 m) is divided into two layers, an upper layer of depth δt , where turbulent diffusion dominates over molecular diffusion, and a lower layer of depth δm where molecular diffusion dominates over turbulent diffusion (Fig. 8.15). In the upper layer, the diffusive dust flux can be calculated using

Turbulent Layer

δt

Laminnar Layer δm

Fig. 8.15. An illustration of the two-layer dry-deposition model. The atmospheric boundary layer is divided into an upper layer of depth δt and a lower layer of depth δm . In the upper layer, dry deposition results from settling and turbulent diffusion, while in the lower layer, it results from settling and molecular diffusion

280

8 Dust Transport and Deposition

the bulk-transfer formulation −ρwdt [c(z) − c(δm )] where wdt is the bulk-transfer conductance for dust particles in the upper layer. Note that wdt has the dimensions of a velocity and is also known as the turbulent-diffusion velocity for dust particles. As discussed in Chapter 3 and 4, the bulk transfer conductance for momentum flux, ga , is ga = Cd U with Cd being the drag coefficient and U the mean-flow speed at level z. It is plausible to assume that wdt is linearly related to ga , so that wdt = ξga

(8.91)

where ξ is a proportionality factor which depends upon particle size. The depth of the lower layer, δm , should not be too different from the depth of the laminar layer, δ. We known from Chapter 3, over smooth surface, δ can be approximated by δ ∼ ν/u∗ with ν being the kinematic viscosity. Suppose u∗ = 0.5 m s−1 and ν = 10−5 m s−1 . Then, we have δ ≈ 2 × 10−4 m. The typical time scale for flow in the laminar layer can now be defined as δ/u∗ (i.e. ν/u2∗ ). Hence, the dust-particle Stokes number, which is the ratio between the particle response time and the time scale of the flow, is given by St =

Tp u2∗ ν

(8.92)

By analogy with wdt , a molecular-diffusion velocity, wdm , can be defined. Although wdm requires further consideration, its order of magnitude should be κp −1 = u ∗ Sm wdm ∝ δ where Sm = ν/κp is the Schmidt number. The Stokes–Einstein formula (Fuchs, 1964) can be used to estimate κp as κp =

kB T (1 + 2.5λ/d) 3πρνd

where d is particle diameter, kB is the Boltzmann constant (1.38 × 10−23 J K−1 ), T is temperature and λ is the mean free path of air molecules (c.a. 2 × 10−7 m). For 1 µm dust particles, Sm is of the order of magnitude 106 and wdm is of the order of magnitude 10−6 m s−1 . The dust fluxes for the two layers can now be expressed using the bulk formulation and the particle-diffusion velocities, wdt and wdm

8.6 Dry Deposition

Fd (z) = −ρwt c(z) − ρwdt [c(z) − c(δ)] Fd (δ) = −ρwt c(δ) − ρwdm [c(δ) − c(0)]

281

(8.93) (8.94)

For simplicity, we assume that the dust flux in the upper layer is constant and hence that Fd (z) = Fd (δ). As the surface is assumed to be a perfect dust sink, we have c(0) = 0. By making use of Equation (8.86), wd can be obtained by eliminating c(δ) between Equations (8.93) and (8.94), so that wd = wt +

wdt wdm wt + wdm + wdt

(8.95)

Therefore, the dry-deposition velocity from z to the surface is composed of the gravitational settling velocity and a modification related both to the dry-deposition velocity due to turbulent motion in the upper layer and the dry-deposition velocity due to molecular diffusion in the lower layer. The limit for large particles is wd = wt and that for small particles is wd =

wdt wdm wdm + wdt

(8.96)

Equation (8.95) can also be written as 1 1 wt 1 = + + wd − wt wdt wdm wdt wdm

(8.97)

The advantage of the two-layer approach is that it leaves the possibility open for deriving a model for wdm on the basis of the microscopic physics. If we treat the final stage of deposition purely as a molecular-diffusion problem and ignore the effects of other factors such as static electricity, we can arguably approximate wdm as wdm = f1 (u∗ , St ) + f2 (u∗ , Sm )

(8.98)

The first function, related to the Stokes number, accounts for the particle inertial effect and the second function, related to the Schmidt number, accounts for the capability of the fluid for particle diffusion. According to Slinn and Slinn (1981), wdm can be approximated as wdm =

5 −1/2 ga [Sm + 10−3/St ] 2

(8.99)

Figure 8.16 shows a comparison of the two-layer model results with the observations from water-flume and wind-tunnel experiments. Particles with a diameter between 0.1 to 1 µm have the smallest dry-deposition velocities. For even smaller particles, the deposition velocity increases due to the increased

282

8 Dust Transport and Deposition 100

10−1

u * = 1.17 ms−1 = 0.44

wd (ms−1)

= 0.11 = 0.40

10−2

10−3

10−4 10−5 0.001

0.01

0.1

1

d/2 (µm)

Fig. 8.16. Comparison of modelled dry-deposition velocities (lines) with measurements (symbols) from water-flume and wind-tunnel experiments (Redrawn from Slinn and Slinn, 1981)

efficiency of molecular diffusivity, while for larger particles, the dry-deposition velocity increases as gravitational settling becomes more important. The theory is qualitatively consistent with these important observations. Our understanding of airflow close to the surface is limited and hence large uncertainties exist for the specific forms of f1 (u∗ , St ) and f2 (u∗ , Sm ). Since the lower layer is δm deep, the molecular-diffusion flux in this layer should be −κp ∂c/∂z ∝ κp c/δm . The difficulty with this is that the behaviour of δm is not well understood. If we were to suppose that δm is proportional to δ, then −1 dependency of the form wdm would have a Sm wdm ∝

ν −1 κp ∝ Sm δ δ

However, δm itself may depend on Sm , as shown in Slinn et al. (1978), and −1/3 −2/3 possibly δm ∝ δSm . This would then lead to wdm ∝ (ν/δ)Sm . There are other theories about the behaviour of flow close to the surface. For example, using the surface-renewal theory (e.g. Danckwerts, 1970), we would obtain a −1/2 relationship of the form wdm ∝ Sm . There is a similar uncertainty for the dependency of wdm on the Stokes number. 8.6.2 Two-Layer Dry-Deposition Model: Vegetation Flows in and above a vegetation canopy have been subject to numerous investigations (Gross, 1993; Kaimal and Finnigan, 1994). The description given

8.6 Dry Deposition z

K(z) K~z

283

U(z) Constant shear stress layer

h K ~ constant

α c(z)

Momentun flux divergence balanced by canopy momentum sink

hu Momentum flux divergence balanced by pressure gradient

h1 K~z

z0s

Constant shear stress layer

Fig. 8.17. A typical canopy flow showing wind speed in four different regimes governed by different processes. The approximate profile for eddy diffusivity, K, and the area density of absorbing canopy elements, αc , are also shown

here is an outline of the basic features of canopy flows, which would facilitate the discussions on dry deposition on vegetation. An idealised canopy flow is depicted in Fig. 8.17. In a horizontallyhomogeneous canopy of height h, the velocity of a steady-state flow U is a function of height only and obeys approximately   ∂U 1 ∂p ∂ 2 K =− (8.100) −αc Cd U + ∂z ∂z ρ ∂x The first term in Equation (8.100) represents the canopy-induced momentum sink, where Cd is the drag coefficient and αc is the total vegetation-surface area per unit volume. Approximate solutions of U exist for different canopy layers. The layer above the canopy (z ≥ h) is a constant-flux layer, as αc is zero and the pressure gradient is negligible. Here, K can be approximated by K = κu∗ (z − zd ) with zd being the zero-displacement height. Thus, U in this layer is logarithmic   z − zd u∗ ln (8.101) U= κ z0 where z0 is the canopy roughness length. For the region hu < z < h, vegetation acts as a momentum sink, αc Cd U is approximately constant, ∂p/∂x = 0 and K = κu∗ l with l being the length scale of turbulence. Under these assumptions, U is found to be   z  z  Uh  (1 + δ) exp −γ 1 − + (1 − δ) exp γ 1 − (8.102) U= z h h

284

8 Dust Transport and Deposition

 where γ = h Cd αc /(κl) and δ = hu2∗ /(γKUh ) are known coefficients and Uh is the flow speed at height h. Near the surface (z0l < z < hl ), another layer of constant shear stress (ρu2∗l ) exists and K takes a value of κu∗l z, and thus   z u∗l ln (8.103) U= κ z0l where u∗l and z0l are the friction velocity and roughness length associated with the surface below the canopy, respectively. For hl < z < hu , an intermediate region exists, where the flow obeys 1/2 1 ∂p /(Cd αc ) U= − ρ ∂x


(8.104)

Dry deposition on a vegetation canopy involves the transfer of dust from the atmosphere to a vegetation surface through two sequential pathways. First, turbulent transfer carries dust particles from air above the canopy to air within the canopy, adjacent to individual elements including leaves, stems and ground surface. Second, molecular diffusion carries dust particles through the laminar boundary layers surrounding these elements, which are then absorbed onto them by mechanisms depending on the characteristics of the surface. A complication with dust deposition on vegetation canopy is that the dust-absorbing elements (e.g. leaves, stems, etc.) are spatially distributed. One simplification is to treat the canopy as a big leaf. The surface area of absorbing canopy elements per unit ground area (element-area ndex) Λv is given by h Λv = αc dz 0

In a similar fashion to the two-layer model for smooth surfaces, we use two deposition velocities to represent this process Fdt = −ρwt ca − ρwdt (ca − cc ) Fdm = −ρwt cc − ρwdm Λv (cc − c0 )

(8.105) (8.106)

where Fdt is the dust flux from air above the canopy to air within the canopy, Fdm is the dust flux from the canopy air to the vegetation surface, ca , cc and c0 are the dust concentrations above the canopy at reference height z, in air within the canopy and at the element surface, respectively. Again, as given in Equation (8.91), wdt is proportional to ga which is a known quantity from studies of canopy flows. However, wdm now requires more consideration, because three mechanisms affect the transfer of dust from canopy air to element surfaces, including molecular diffusion, interception and impaction (Fig. 8.18). Interception occurs due to particle trapping by the fine hairs on vegetation elements or forces arising from static electricity. Impaction occurs because some particles moving in the canopy flow may have sufficiently

8.6 Dry Deposition

285

Streamline

Impaction Molecular diffusion Interception

Leaf

Fig. 8.18. Mechanisms of dry deposition on vegetation include impaction, molecular diffusion and interception

large velocity for them to penetrate the laminar flow and impact directly on the surface. These mechanisms can be represented by conductances acting in parallel, so that wdm = wdmb + wdmi + wdmm where wdmb , wdmi and wdmm are deposition velocities associated respectively with molecular diffusion, interception and impaction. The molecular conductance can be estimated by using wdmb =

κ p Sh l 1/2

1/3

where l is the element dimension, Sh = Λv Rel Sm is the Sherwood number, Rel = Uc l/ν is the Reynolds number for the absorbing element and Uc is ambient air velocity within the canopy. The impaction conductance wdbm should be proportional to Uc wdmm = em Uc where em is the impact efficiency which is given empirically by B  St em = St + A

(8.107)

where A and B are empirical constants and St is the Stokes number, defined here as 2Uc Tp /l. Bache (1981) and Peters and Eiden (1992) proposed A = 0.8 and B = 2 for several element shapes. The interception process has been discussed by Slinn (1982), but the expression for wdmi is rather complex (not included here). Interception appears to be important for particles with d around 1 µm and increases wdm . In addition, particle rebound on vegetation elements also affects particle transfer

286

8 Dust Transport and Deposition

from the ambient canopy flow to the element surface. Rebound is significant for particles in the 10–100 µm range and decreases wdm . Discussions on rebound can also be found in Slinn (1982). A similar expression to Equation (8.95) can be found for wd for vegetation canopies by setting Fdt = Fdm and eliminating cc from Equations (8.105) and (8.106), assuming that c0 is known. 8.6.3 Single-Layer Dry-Deposition Model Raupach et al. (1999) proposed a single-layer dry-deposition model which is less demanding on data and parameterisations. In this model, the drydeposition velocity is treated as a bulk single-layer conductance made up of three components acting in parallel wd = wt + gbb + gbm

(8.108)

The molecular conductance, gbb , and the impaction conductance, gbm , can be related to the conductance for momentum, ga . Two processes which contribute to the transfer of momentum from the airflow to the surface, including the pressure drag and the viscous drag, are considered. So we may write ga = gap + gav

(8.109)

where gap is the conductance related to pressure drag and gav is that related to viscous drag. According to Thom (1971), ga , gap and gav are related by gap = fr ga gav = (1 − fr )ga

(8.110) (8.111)

where fr is the ratio of pressure drag to the total drag. Raupach et al. (1999) hypothesised that gbb is proportional to gav and gbm is proportional to gap . For gbb , the relationship is −2/3 gav gbb = av Sm

(8.112)

where av is a factor of order 1, accounting for different effects of inter-element sheltering on the molecular transfer of particles and momentum. For gbm , the relationship is (8.113) gbb = ap em gap where ap is another factor with similar physical meaning as av and em is as given in Equation (8.107). The final form of the single-layer model for wd is −2/3 ] wd = wt + ga [fr ap em + (1 − fr )av Sm

(8.114)

in which ap and av are empirical parameters. Although the theory is a considerable simplification of the multi-layer physics, it includes enough physics to

8.6 Dry Deposition

287

Deposition velocity (m/s)

10 1 0.1

wd wt gbm gbb

0.01 0.001 0.0001 0.00001 0.01

0.1

1 10 Particle diameter (microns)

100

1000

Fig. 8.19. Behaviour of wd , wt , gbb and gbm in the one layer dry-deposition model (From Raupach et al. 1999)

10

Deposition velocity (m/s)

1 0.1 0.01

Terminal velocity u*=0.35 m/s u*=0.7 u*=1.4 SG: u*=0.35 SG: u*=0.7 SG: u*=1.4

0.001 0.0001 0.00001 0.01

0.1

1

10

100

1000

Particle diameter (microns)

Fig. 8.20. Comparison of dry-deposition velocity, wd , calculated using the singlelayer model (lines) with the measurements (symbols) of Chamberlain (1967) for three different friction velocities (From Raupach et al. 1999)

capture the dependence of the three major processes (settling, impaction and molecular diffusion) on particle diameter and wind speed (Fig. 8.19). Its two empirical coefficients are sufficient to permit matching to experimental data and make full use of information about bulk momentum transfer to characterise the aerodynamic properties of the canopy. Figure 8.20 shows a comparison of the single-layer model against the windtunnel data of Chamberlain (1967), for deposition of particles of various sizes

288

8 Dust Transport and Deposition

to artificial sticky short grass of height 0.06 m. The coefficients ap and av have been treated as adjustable parameters, set to ap = 2 and av = 8 in the comparison. The main features of the dependency of wd on particle size are reproduced. Both the predictions and the measurements demonstrate the minimum wd around 1 µm, where none of wt , gbm and gbb is effective, and convergence of wd to wt for large particles. The model satisfactorily reproduces the trend of the data with wind speed. The agreement of the simple model with the observations is comparable with multi-layer models.

8.7 Wet Deposition Wet deposition is the mass flux of dust particles to the surface, which are collected (or scavenged) by precipitation. Figure 8.21 shows an image of the aerosols left behind by a dried raindrop. The image reinforces the view that wet deposition is an important mechanism for the removal of particles from the atmosphere. The scavenging process can be divided into in-cloud scavenging and below-cloud scavenging. In-cloud scavenging refers to the process in which particles are collected by raindrops as they form, for example, particles acting as cloud condensation nuclei (CCN). Below-cloud scavenging refers to the process in which particles are collected by raindrops as they precipitate. In the context of this book, we are mainly interested in the rate at which dust particles are removed for given precipitation rate, type, the size distribution of raindrops and the size distribution of dust particles. In-cloud scavenging and the chemical processes involved in wet deposition are not considered.

Fig. 8.21. A microscope image of dust in a raindrop (Shinjo City, Japan; 18 April 2006; image by O. Abe, with acknowledgement)

8.7 Wet Deposition

289

8.7.1 The Theory of Slinn If the raindrops are uniform spheres of radius R and the raindrop number flux through a cross section of an atmospheric column is mR (dimensions L−2 T−1 ), then 3 pr (8.115) mR = 4πR3 where pr is the rainfall rate (dimensions of L3 L−2 T−1 or LT−1 ). The sky coverage rate (fraction of sky covered by raindrops per unit time) is γs =

3 pr 4R

For instance, if pr is 1 mm hr−1 , R = 0.3 mm, then γs ≈ 1/25 min−1 . This implies that if dust particles were collected by raindrops with a 100% efficiency, they would be completely removed within 25 minutes. The efficiency at which dust particles are collected by raindrops is known as the scavenging rate Λ = es γs

(8.116)

where es is the collection efficiency. Two mechanisms prevent dust particles from being captured with 100% efficiency: (1) small particles tend to follow the streamlines and flow around the raindrop and (2) particles may bounce off during the particle-raindrop collision. To account for these two mechanisms, we denote es = ec er where ec is the collision efficiency and er is the retention efficiency. One interpretation of ec is that ec π(R + r)2 is the collision area as illustrated in Fig. 8.22, where r is dust particle radius. For particles smaller than 1 µm, er is about one, while for particles larger than 10µm, er is less than one (Slinn, 1983). However, the amount of data is insufficient to determine er unequivocally. Therefore, until more data become available, it is rational to assume er = 1. It follows that es and ec are identical. Raindrops and dust particles have different falling velocities as they move through the atmosphere. Suppose a raindrop falls with a terminal velocity wR and a dust particle falls with a terminal velocity wt . Then, the relative speed at which the raindrop approaches the dust particle is (wR − wt ). If the number density (number of particles per unit volume) of dust particles is N , then during the time interval ∆t, the total number of dust particles that can be captured by the raindrop is N (wR − wt )(∆t)π(R + r)2 . The number of particles actually captured by the raindrop is es N π(R + r)2 (wR − wt )∆t where the collection efficiency es is a function of both r and R. In reality, raindrops are not uniform in size but obey a size-distribution density function, nR (R, x, t), that varies in space and time. The decrease of

8 Dust Transport and Deposition

Str ea

ml i

ne

290

Raindrop Effective Cross Section Area

Area Leading to Collision Small Particle

Large Particle

Fig. 8.22. An illustration of the collision efficiency. As dust particles tend to follow the streamline, only those passing the area leading to collision will be captured by the raindrop. While the effective cross section of the raindrop is π(R + r)2 , the area leading to collision is ec π(R + r)2 with ec  1 (Modified from Slinn, 1983)

dust particles of radius r per unit volume due to the collection of all raindrops of all sizes becomes ∞ (wR − wt )π(R + r)2 es nR dR (8.117) ∆N = −N (∆t) 0

where nR (R, x, t)dR is the number density of raindrops in the size range R + dR. The above equation leads to the definition of the scavenging rate, Λ, ∞ 1 dN = (wR − wt )π(R + r)2 es nR dR (8.118) Λ(r; x, t) = − N dt 0 which represents the relative decreasing rate of dust-particle number density. Clearly, we must estimate es in order to evaluate Λ. The main processes which influence the particle collision efficiency, ec (note es = ec if er = 1), include molecular diffusion, impaction and interception. Collection by Molecular Diffusion Consider first the simplest case of particle collection by a stationary raindrop. Using a local coordinate system with its origin located at the centre of the raindrop, the particle-number flux caused by molecular diffusion can be expressed as

8.7 Wet Deposition

291

∂N ∂s where s is the distance from the coordinate centre. As particles are absorbed completely at the surface of the raindrop (s = R), the particle-number concentration at the raindrop surface is N |s=R = 0. At a sufficiently large distance away from the raindrop N |s=∞ = N∞ is the ambient particle concentration. For the steady-state case, fb can be assumed to be constant, the solution for N is   R N = N∞ 1 − s fb = −κp

It follows that

κ p N∞ R where the negative sign implies that the particle-number flux is directed towards the raindrop. In reality, fb may differ from −κp N∞ /R for various reasons. This difference can be quantified using the Sherwood number, which is defined as −1  κ p N∞ Sh = fb − R fb = −

For the ideal situation described above, Sh = 1. For moving raindrops, the gradient of particles near the raindrop is not uniformly distributed around the raindrop. It becomes larger in its front and smaller in its wake. As a consequence, the particle flux caused by molecular diffusion is also stronger on the front side of the raindrop and weaker in the wake. In general, the net effect is an increase of particle number flux over κp N∞ /R. The increase depends on the detailed flow structure around the moving raindrop, which can be characterised by two parameters; namely, the raindrop Reynolds number, ReR = wR R/ν and the Schmidt number, which is the ratio of the kinematic viscosity to the molecular diffusivity for dust particles Sm = ν/κp . The raindrop Reynolds number is a statement on whether or not the flow around the raindrop is turbulent. For ReR  1, flow around the raindrop is laminar, while for ReR 1, a turbulent boundary layer develops (Fig. 8.23a, b). For Sm 1, the velocity boundary layer (∝ ν) would be thicker than an even-thinner particle diffusion layer (∝ κp ), and the particle gradient would be large and the flux to the raindrop would therefore also be large (Fig. 8.23c, d). It has been found empirically that 1/2

1/3 Sh = 1 + 0.4ReR Sm

(8.119)

292

8 Dust Transport and Deposition (b) Re >> 1

(a) Re = 0.1

ν/R

R ν/R Viscous Layer

wt

Viscous Layer wt

(d) Pe > 1

(c) Sc > 1 Diffusion Layer

Viscous Diffusion Layer

Boundary Layer

r/R

ν/R r/R wt

Fig. 8.23. (a) Boundary-layer and flow characteristics around a raindrop for low Reynolds numbers; (b) as (a) but for high Reynolds numbers; (c) relation between the molecular-diffusion layer for dust particles and the viscous boundary layer for Schmidt numbers larger than 1 and (d) the diffusion layer for Peclet numbers larger than 1 (Modified from Slinn, 1983)

The above equation indicates an increase of the Sherwood number with ReR and Sm . Note that the molecular-diffusion collection efficiency is the ratio between the rate of dust collection by the raindrop, 4πR2 Sh fb , and the rate of dust approaching the raindrop π(R + r)2 n∞ (wR − wt ). It follows that the collection efficiency due to molecular diffusion ecb is ecb =

4πR2 Sh | fb | π(R + r)2 N∞ (wR − wt )

(8.120)

Assuming also that r  R and wt  wR , we obtain ecb =

4κp Sh Sh = 4 RwR Pe

(8.121)

where P e = wR R/κp is the Peclet number. Collection by Impaction Impaction is another mechanism which influences the efficiency of collision. The impaction process depends on the response of the particle to the changing

8.7 Wet Deposition

293

flow field near the raindrop. After subtracting the terminal velocity from the particle velocity, the equation of particle motion becomes dupi 1 = − (upi − ui ) dt Tp

(8.122)

Recall that upi is the deviation of particle velocity from its terminal velocity and Tp is the particle response time. Suppose the velocity scale for the flow around the raindrop is Uf , then the time scale of the fluid motion, Tf , is u
˜pi = pi , u ˜i = ui and t˜ = t to the R/Uf . Applying the normalisations u Uf

Uf

Tf

above equation, we obtain the normalized equation of motion 1 d˜ upi = − (˜ upi − u ˜i ) ˜ St dt

(8.123)

with St = Tp /Tf being the Stokes number. As can be seen, particles with a large response time (St → ∞) do not respond to the changes in the flow pattern and will therefore collide with the raindrop. For these particles, the collision efficiency due to impact, ecm = 1. Particles with a small response time (St → 0), will follow the streamline completely and hence not collide with the raindrop. For these particles, the collision efficiency due to impact, ecm = 0. There exists a critical Stokes number, St∗ , so that for St ≤ St∗ , ecm is practically zero. A simple mathematical expression which satisfies the above requirement is   1 (8.124) ecm = exp − St − St∗ It has been suggested that St∗ is around 1/12. Collection by Interception The other mechanism which leads to particle and raindrop collision is interception. The idea behind this is that even if a particle had no mass and followed the streamline exactly, it could still be collected because of its finite size. Fuchs (1964) suggested that the efficiency related to interception is eci =

3r R

(8.125)

Finally, the total collision efficiency due to molecular diffusion, interception and impaction is given by ec = ecb + eci + ecm Different approximations to ecb , eci and ecm are possible, which lead to different expressions of ec . Slinn (1984) proposed a semi-empirical relationship for ec which is essentially the same as the above-described theory but slightly different in formulation

294

8 Dust Transport and Deposition 100 R = 500 µm

Impaction

Collection Efficiency

10-1 10-2

10-3

Total

10-4

10-3

Interception

10-2

10-1

Brownian Diffusion

100

101

r (µm)

Fig. 8.24. Collection efficiency ec , as given by Equation (8.126), plotted as a function of particle radius for R = 500 µm. Assumed is ρp = 1,000 kg m−3 .

ec =

 r  4  r 1/3 1/2 1 + 0.4Re1/2 Sm α + (1 + 2Re1/2 ) +4 + 0.16Re1/2 Sm Pe R R  3/2 St − St∗ + (8.126) St − St∗ + 2/3

where α is the air-to-water kinematic viscosity ratio. In Fig. 8.24, ec predicted using Equation (8.126) is plotted as a function of particle radius for raindrop size R = 0.5 mm. It is seen that particles smaller than 0.1 µm can be effectively collected by the raindrop through Brownian diffusion, while particles larger than 1 µm can be effectively collected through impaction. In the particle radii range between 0.1 and 1 µm, ec shows a characteristic minimum. Particles in this radii range are too large for Brownian diffusion to be effective, but are too small for impaction or interception to be effective. It should be pointed out that the theoretical expressions for ec contain large uncertainties because they do not include or adequately describe the other processes which may be important to collision, such as particle-size growth, electrophoresis and thermophoresis. Some existing data, which may be unreliable themselves, differ from the theory by several orders of magnitude. Dust particles are relatively large particles. Jung (2004) evaluated Equation (8.126) by comparing the predicted ec values with data reported in the literature. She found that Equation (8.126) considerably over predicts ec for the size range of commonly observed mineral aerosols. Based on the studies of Beard and Ochs (1984) and Mason (1975), Jung compiled a set of ec values for particles in the size range of 2–20 µ and raindrops in the radii range between 50 and 3,000 µm (Table 8.6). It is found that for R smaller than about 600 µm, ec increases with R. This increase is due to the dynamic

8.7 Wet Deposition

295

Table 8.6. Calculated collection efficiencies ec (%) using the data from Beard and Ochs (1984) and Mason (1975) R (µm) 50 60 80 100 125 150 200 250 300 400 600 1,000 1,400 1,800 2,400 3,000

r (µm) 2 2.2 2.0 2.3 2.7 3.1 3.4 3.7 3.8 3.9 3.5 3.0 2.0 1.4 1.2 1.0 0.8

3 5.5 5.8 10.0 14.0 17.4 20.2 24.0 26.0 27.2 28.5 28.4 25.7 23.0 21.0 18.5 16.0

4 11.2 13.6 23.8 32.4 38.7 42.3 46.5 49.2 50.4 51.4 52.4 49.4 46.1 43.1 39.1 33.0

5 19.4 26.5 41.1 48.8 54.5 58.2 61.2 63.1 63.9 64.9 65.3 64.6 61.9 58.4 52.7 45.3

6 27.9 38.8 53.0 60.3 64.6 66.9 69.3 70.9 71.8 72.8 73.2 72.6 71.2 68.1 62.1 55.0

8 44.0 57.4 68.5 72.8 76.3 78.4 80.8 81.9 82.5 83.0 83.4 83.3 82.5 80.0 75.4 71.0

10 58.3 67.7 76.6 81.4 84.1 85.6 86.8 87.3 87.8 88.0 88.3 88.7 87.8 86.4 83.7 81.0

12 68.0 74.3 82.2 86.2 88.2 88.9 89.9 90.3 90.7 91.0 91.4 91.3 90.9 90.0 87.9 85.6

15 73.1 80.3 87.7 90.4 91.8 92.4 93.1 93.6 93.9 94.4 94.9 94.7 94.4 93.6 92.0 89.9

20 75.4 87.1 92.5 94.1 95.0 95.4 96.0 96.5 96.8 97.3 97.9 98.1 97.7 97.0 95.9 94.1

effects which allow particles to be captured by the eddies in the wake of the raindrop. For R larger than 600 µm, ec generally decreases with R. This decrease is probably related to the fact that St decreases with increasing R. As St decreases, the tendency for particles to collide with the raindrop is reduced. 8.7.2 Scavenging Rate By assuming r  R, wt  wR and er = 1, Equation (8.118) can be simplified to ∞ ec πR3 wR nR dR (8.127) Λ= R 0 For the evaluation of Λ, it is usually required to specify explicitly the raindropsize distribution function nR in space and time. If all raindrops are identical in size, then ec can be considered to be a constant. It follows that ec ∞ 4 3 πR wR nR dR Λ= (8.128) R 0 3 ∞ Because 0 43 πR3 wR nR dR is simply the rate of rainfall pr , we have Λ = αs ec pr /R

296

8 Dust Transport and Deposition 102

Λ (h−1)

100

10−2

10−4 10−3

pr = 5 mm h−1 R = 100 µm R = 1000 µm 10−2

10−1

100

101

r (µm)

Fig. 8.25. Simulated scavenging coefficient as a function of particle radius for monodisperse raindrops (From Jung, 2004)

where αs is an empirical coefficient. The above expression is almost identical to Equation (8.116). For frontal storms αs = 0.5 and R = 0.3 mm have been suggested. Figure 8.25 shows the predicted Λ for pr = 5 mm h−1 . A comparison of Figs. 8.25 and 8.24 shows that the behaviour of Λ is largely determined by that of ec . Equation (8.118) shows that the computation of Λ in general requires the knowledge of raindrop-size distribution, nR , and the raindrop terminal velocity, wR . Various approximations to nR have been proposed in the literature for given precipitation rate, pr (Table 8.7, Fig. 8.26). Marshall and Palmer (1948) suggested that nR obeys an exponential relationship. Their approximation can be improved by assuming n0 to be a function of pr (Sekhon and Srivastava, 1971). Studies show that the number of raindrops at the smaller end of the size spectrum is generally over predicted using the exponential distributions. To capture this feature, Gamma distributions and log-normal distributions have been proposed (Willis and Tattelman, 1989; Feingold and Levin, 1986). The raindrop terminal velocity, wR , can be estimated using the empirical formula suggested by Beard (1976). The scavenging rates estimated using the four raindrop-size distributions are shown in Fig. 8.27 for pr = 5 mm h−1 . While the qualitative behaviour of the scavenging efficiency is similar in all four cases, quantitative differences can be considerable. Over the entire particle size range, a higher scavenging coefficient is obtained if the exponential distribution is used. It is interesting to know how dust particle-size distribution evolves with time during the process of scavenging. If the scavenging rate Λ is constant, then the scavenging efficiency, Φ, is simply

8.7 Wet Deposition Exponential Two−parameter exponential Gamma Lognormal pr = 5 mm h−1

10−2

n(D) (cm−3 mm−1)

297

10−4

10−6

10−8

0

1

2

3 D (mm)

4

5

6

Fig. 8.26. Comparison of raindrop-size distributions for pr = 5 mm h−1 (From Jung, 2004) Table 8.7. Various approximations to raindrop-size distribution nR Exponential

Parameters n0 = 8 × 103

Reference Marshall and Palmer 1948

λ = 4.1p−2.1 n0 exp(−λD) r Exponential, 2-parameter n0 = 7 × 103 p0.37 r

Sekhon and Srivastava (1971)

n0 exp(−λD)

λ = 3.8 p−0.14 r

Gamma n0 Dα exp(−λD)

n0 = 512.85 × 10−6 M D0 α = 2.16

−(4+α)

λ = 5.5880/D0 D0 = 0.1571 M 0.1681 M = 0.062 R0.913 Lognormal   σ = 1.43 ln2 (D/Dg ) √ n0 exp − n0 = 172 p0.22 r 2 ln2 σ D 2π ln σ Dg = 0.72 p0.21 r

Willis and Tattelman (1989)

Feingold and Levin (1986)

298

8 Dust Transport and Deposition 10−2

Λ (s−1)

10−3

Exponential Two−parameter exponential Gamma Lognormal pr = 5 mm h−1

10−4 10−5 10−6 10−7 −3 10

10

−2

−1 10 r (µm)

10

0

10

1

Fig. 8.27. Same as in Fig. 8.25 except for polydisperse raindrops (From Jung, 2004)

Φ(r, t) = −

N (r, t) − N (r, 0) = 1 − exp(−Λt) N (r, 0)

(8.129)

Aerosol (including dust) particle-size distributions can be conveniently approximated as a sum of I log-normal distributions: 
I

log2 (r/ri ) Ni dN (r) √ = exp − d log r 2 log2 σi 2π log σi i=1

(8.130)

where Ni is the aerosol number concentration of the ith log-normal mode and ri and σi are the mean radius and standard deviation of the ith mode. In Table 8.8, the values of these parameters for several types of tropospheric aerosols suggested by Jaenicke (1993) are listed. The scavenging efficiency in terms of the volume occupied by particles, ΦV , is given by V (t) − V (0) (8.131) ΦV (t) = − V (0)

with V (t) =

4πr3 N (r, t) dr 3

Figure 8.28a shows as an example the evolution of an urban aerosol particle size distribution for pr = 25 mm h−1 . Only a small portion of the particles in the submicron range is removed after many hours of rain. Figure 8.28b shows the urban aerosol particle size distribution after 12 hours rain with pr = 0.5, 5 and 25 mm h−1 . Again, most removal occurs in the coarse particle mode and the particles in the submicron range are hardly removed. For the removal of particles in the submicron range, in-cloud scavenging is much more important than below-cloud scavenging.

8.7 Wet Deposition

299

Table 8.8. Parameters for model aerosol size distributions described as the sum of three log-normal modes Jaenicke (1993) Type

i

Polar

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

Background

Maritime

Remote continental

Desert dust storm

Rural

Urban

Ni (cm−3 ) 21.7 0.186 3.04 × 10−4 129 59.7 63.5 133 66.6 3.06 3200 2900 0.3 726 114 1.78 × 10−1 6650 147 1990 9.93 × 104 1.11 × 103 3.64 × 104

ri (µm) 0.0689 0.375 4.29 0.0036 0.127 0.259 0.0039 0.133 0.29 0.01 0.058 0.9 0.001 0.0188 10.8 0.00739 0.0269 0.0419 0.00651 0.00714 0.0248

log σi 0.245 0.300 0.291 0.645 0.253 0.425 0.657 0.210 0.396 0.161 0.217 0.380 0.247 0.770 0.438 0.225 0.557 0.266 0.245 0.666 0.337

8.7.3 Scavenging Ratio Supposing that raindrops and dust particles are distributed evenly in horizontal direction and that the dust particle concentration in the atmosphere is c(z), then Λc is the mass of particles taken out of a unit volume of air per unit time. The wet deposition at the surface is thus ∞ ρΛ(z)c(z)dz (8.132) Fw = 0

It can also be determined from observations of rainfall rate at the surface pr0 , and the concentration of particles in the rainwater, cR0 , in the form Fw = ρw pr0 cR0

(8.133)

where ρw denotes water density. We therefore have that Fw = ρw pr0 c0

cR0 c0

(8.134)

The ratio cR /c is called the scavenging ratio and s0 ≡ cR0 /c0 is the scavenging ratio at the surface. By using Equations (8.132) and (8.134) we have

300

8 Dust Transport and Deposition

dV/dlog r (µm−3 cm−3)

50 Initial t = 10 min t = 30 min t = 60 min t = 6 hrs t = 12 hrs

40 30 20 10 (a) 0

10−2

100 r (µm)

dV/dlog r (µm−3 cm−3)

50 Initial pr = 0.5 mm h−1 pr = 5 mm h−1 pr = 25 mm h−1

40 30 20 10 (b) 0

10

−2

10

0

r ( µm)

Fig. 8.28. (a) Evolution of urban aerosol particle-size distribution under the condition of rainfall of 25 mm h−1 ; and (b) the urban aerosol particle-size distribution after 12 hours of rainfall with rates of 0.5, 5 and 25 mm h−1 (From Jung, 2004)

1 s0 = ρw pr0 c0



∞

ρΛcdz

(8.135)

0

which shows that s0 is related to the scavenging rate in the atmosphere, λ. The values of s0 can be reasonably well estimated from the measurements of cR0 and c0 . In practice, the removal of dust particles via wet deposition Fw is often determined by using the scavenging ratios, as Fw can be now expressed as Fw = ρw pr0 s0 c0

(8.136)

Hence wet deposition can be estimated from precipitation, the scavenging ratio and the airborne dust concentration, all measured at the surface. The scavenging ratio, s0 , is a function of many parameters such as particle size,

8.7 Wet Deposition

301

particle shape, the vertical distribution of dust concentration in the atmosphere, the vertical extent of the rain and the rain cloud, etc. It is therefore difficult to predict its value accurately. In the literature, it has been reported that s0 for mineral aerosols ranges from 100 to 2,000 (Duce et al. 1991). For wet depositions over seas, the suggested values for s0 are around 1,000 for mineral aerosols, varying between 500 to 2,000 (Uematsu et al. 1985).

9 Integrated Wind-Erosion Modelling

The purpose of developing an integrated wind-erosion modelling system (IWEMS) is to produce quantitative predictions of wind erosion on scales from local to global. The system should have the capacity of modelling the whole wind-erosion process, from particle entrainment through transport to deposition. This is a formidable task because, as depicted in Fig. 1.1, wind erosion is governed by a range of factors involving the atmosphere and the land surface. Thus, an IWEMS must couple models for the atmospheric, landsurface and aeolian processes and databases for land-surface parameters, so that the dynamics of wind erosion and the environmental control factors are adequately represented. The first attempt of combining atmospheric and land-surface data for large-scale wind-erosion assessment was made by Gillette and Hanson (1989) in their investigation on the dust production in the United States. Since the later 1980s, global, regional and local dust models have been under development in the atmospheric research community. Earlier attempts on dust modelling were made for instance by Westphal et al. (1988) and Joussaume (1990). More recent examples of global dust models include the studies of Zender et al. (2003), Ginoux et al. (2004) and Tanaka and Chiba (2006) among others. Examples of regional dust models include the studies of Shao and Leslie (1997), Nickovic et al. (2001), Liu et al. (2001) and Gong et al. (2003). Seino et al. (2005) and Uno et al. (2005) simulated dust storms in the Tarim Basin using a meso-scale dust model. Shao et al. (2003) applied an integrated system to the real-time predictions of Asian dust storms with considerable success. The approach of integrated wind-erosion modelling can be put into perspective with several other methods used for wind-erosion assessment. The simplest one is to estimate the climatic distribution of wind strength and frequency for a given area. Such a distribution gives an indication of the potential for wind to generate erosion, but not the actual wind-erosion rate and pattern. Another method is to compute a wind-erosion index (Chepil et al. 1963; McTainsh et al. 1998) as a function of the environmental factors which Y. Shao, Physics and Modelling of Wind Erosion, c Springer Science+Business Media B.V. 2008


303

304

9 Integrated Wind-Erosion Modelling

affect wind erosion, such as wind speed, precipitation and evaporation. The Wind-erosion Equation (WEQ, Woodruff and Siddoway, 1965) is an empirical scheme in which the driving parameters are descriptors of soil type, vegetation, roughness, climate and field length. Originally, WEQ uses annual averages of these descriptors to estimate the annual average soil loss. For estimates over shorter periods, WEQ was modified by Bondy et al. (1980) and Cole et al. (1983). More recent revisions to WEQ have led to the Revised Wind-erosion Equation (RWEQ) which includes input parameters such as planting date, tillage method and amount of residue from the previous crop; a weather generator is then used to predict future erosion (Comis and Gerrietts, 1994). The empirical nature of WEQ has limited its transferability to other areas of the world from the Central Great Plains of the USA, for which it was originally developed. Also, the complex interactions between the variables controlling wind erosion are not fully accounted for in the empirical WEQ. For this reason, a new, more process-oriented model called the Wind-erosion Prediction System (WEPS) has also been under development in the USA. This model includes sub-models for stochastic weather generation, crop growth, decomposition, soil, hydrology, tillage and erosion (Hagen, 1991). By contrast, an IWEMS comprehensively integrates the dynamical and physical models for a range of atmospheric, surface hydrological and aerolian processes with data sets of land-surface parameters and wind-erosion measurements. Such a system takes the advantages of the progresses in atmospheric modelling, data assimilation, remote sensing and GIS (Geographic Information System), and thereby significantly expands the frontiers of aeolian research. Wind-erosion, together with dust modelling, has now become an important component of Earth system modelling. An IWEMS is powerful in that it provides quantitative estimates of wind-erosion related quantities on a wide range of scales. Of course, such a system is more difficult to implement and demands for more data. In this chapter, we discuss IWEMS structure and functioning, the practical problems one encounters in their implementations and the possible solutions.

9.1 System Structure A possible framework for an IWEMS is as illustrated in Fig. 9.1, which consists of the following four major components: 1. 2. 3. 4.

Modelling Monitoring Database Data-assimilation.

The modelling component comprises an atmospheric model and modules for aeolian, land-surface thermal and hydrological processes. The atmospheric model, either global, regional or meso-scale, serves as the host for the other modules. Most atmospheric models today have advanced numerics for

9.1 System Structure

305

Fig. 9.1. The structure of an integrated wind-erosion modelling system consisting of the components for modelling, monitoring, database and data-assimilation

atmospheric dynamics and sophisticated treatments for atmospheric physical processes, such as radiation, clouds, convection, turbulent diffusion, etc. The atmospheric model is normally coupled with a land-surface scheme designed to parameterise the energy, momentum and mass exchanges between the atmosphere and the land surface, as discussed in Chapter 4. For wind-erosion modelling, the land-surface scheme produces friction velocity and soil moisture as outputs. These outputs, together with other land-surface parameters, are then used in the wind-erosion scheme for the predictions of quantities such as saltation flux and dust-emission rate. The transport and deposition models obtain wind, turbulence and precipitation data from the atmospheric model, and dust-emission rate and particle-size information from the dust-emission model to predict dust concentration and deposition. Parameters are required for the dynamic models to characterise the aerodynamic, radiative, thermal, hydraulic and aerolian properties of the land surface. For wind-erosion modelling, for instance, data are required for specifying soil texture, vegetation coverage, vegetation leaf area, roughness frontal area etc. Some of these parameters, for example, soil texture and vegetation leaf-area index, are common for both wind-erosion and surface hydrological modelling. It is advantageous to use a GIS to efficiently manage, manipulate and visualize the large amount of land-surface data. Another component of IWEMS is wind-erosion monitoring. Data are necessary both for model validation and data assimilation. Dust-related observations are becoming increasingly available through the following channels:

306

9 Integrated Wind-Erosion Modelling

1. Dust concentration can be derived from visibility measurements using empirical relationships, which are made routinely at weather stations. The visibility-derived dust-concentration data are quantitatively rather inaccurate, but provide an excellent spatial and temporal coverage of dust activities. 2. Dust concentration measurements are also available from high- and lowvolume samplers. These measurements are probably the most accurate, but are poor in spatial and temporal resolutions, because the measurements are made at a small number of locations and a measurement may take days even weeks to complete. 3. A network of lidars is now functioning which produces aerosol-profile observations at a number of locations in dust affected areas (Sugimoto et al. 2003; Shimizu et al. 2004). Lidar data are nevertheless indirect and are limited to selected areas. During severe dust events, lidar only provides information for the lower part of the atmosphere, as lidar signals often cannot penetrate the dense dust layers. 4. Satellite remote sensing provides excellent spatial and temporal coverage of wind-erosion and dust-storm events, although the difficulties of converting satellite signals to physical quantities, such as dust load and dust size distribution are not yet fully overcome. 5. Dust concentration can also be derived through radiation measurements of, for example, sun photometers. Radiation measurements can be used to determine dust optical thickness which is directly comparable with that derived from satellite data. Wind-erosion models are only approximations to a set of complex processes. Even if these processes were accurately represented in the models, model simulations may still diverge from reality due to inaccuracies in initial conditions and errors in model parameters and forcing data. This occurs often because the models are non-linear and sensitive to minute changes in initial conditions and parameters. It is a considerable advantage to use observations to constrain the model simulations close to the reality. For this purpose, methods must be derived to optimally combine observations and model simulations. The procedure of combining observed data with model simulations to produce an optimal prediction of an evolving system is known as data assimilation, optimal in a sense that the prediction error is minimized. The technique of data assimilation has been widely used in numerical weather, ocean and climate predictions (e.g. Kalnay, 2003). It is also desirable to develop the data assimilation technique as a component of IWEMS. Some progress has been made in this area although there exist insufficient homogenized winderosion data for the purpose. We expect that with the rapid progress in remote sensing technology and other data retrieving techniques, the assimilation of dust measurements into IWEMS simulation will soon become feasible.

9.2 Wind-Erosion Parameterisation Scheme

307

9.2 Wind-Erosion Parameterisation Scheme The wind-erosion (parameterisation) scheme for the calculation of streamwise saltation flux, Q, and dust-emission rate, F , is a key component of IWEMS. As discussed in Chapters 6 and 7, the predictions of Q and F are achieved through modelling the capacity of wind to generate erosion, represented by the friction velocity, u∗ , and through modelling the resistance of the surface against wind erosion, represented by the threshold friction velocity, u∗t . A wind-erosion scheme must quantitatively predict Q and F using a small number of parameters and adequately represent the influences of relevant environmental factors. Figure 9.2 illustrates the wind-erosion scheme proposed by Shao (2001, 2004). The quantity which drives the wind-erosion scheme is u∗ , and the main land-surface parameters required by the scheme are listed in Table 9.1. The parameters required by the atmospheric model and the land-surface model are not listed. In addition, the scheme requires the specification of constants such as particle density ρp and air density ρa and a several empirical constants, including the Owen coefficient co , the dust emission coefficient cy and A and

Fig. 9.2. The structure of the wind-erosion scheme proposed by Shao (2001, 2004) (Diagram by I. Sokolik, with acknowledgement)

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9 Integrated Wind-Erosion Modelling

Table 9.1. A list of the parameters required by the wind-erosion scheme proposed by Shao (2001, 2004), denoted S04, and the scheme proposed by Marticorena and Bergametti (1995), denoted MB95. Parameters required by the atmospheric model and land-surface model are not included S04 pm (d) pf (d) ρp P θ θr λ σ MB95 ηc pad (d) θ θr z0 z00 δx σ

Meaning Minimally-disturbed PSD Fully-disturbed PSD Particle density Soil plastic pressure Soil moisture Air-dry soil moisture Frontal-area index Fraction of erodible area Meaning Clay content Airborne dust PSD Soil moisture Air-dry soil moisture Surface roughness length Underlying roughness length Roughness elements separation Fraction of erodible area

Dimension m−1 m−1 kg m−3 Nm−2 mm−3 mm−3 Dimension % m−1 mm−3 mm−3 m m m -

Usage Saltation flux & dust emission Saltation flux & dust emission Saltation flux & dust emission Volume-removal dust-emission Threshold friction velocity Threshold friction velocity Drag partition Saltation flux & dust emission Usage Dust emission Dust emission Threshold friction velocity Threshold friction velocity Drag partition Drag partition Drag partition Saltation flux & dust emission

b used for the calculation of fθ . These empirical coefficients have considerable uncertainties and are soil type dependent. Marticorena and Bergametti (1995), Alfaro and Gomes (2001) and Zender et al. (2003) have also proposed wind-erosion schemes. There are no fundamental differences among the schemes, although aspects of the formulations do differ. The scheme due to Marticorena and Bergametti (1995) is illustrated in Fig. 9.3. A comparison of Figs. 9.2 and 9.3 shows that the main differences between the two schemes lie in the computations of threshold friction velocity and dust emission. As already pointed out in Chapter 7, the dust-emission scheme employed by Marticorena and Bergametti (1995) is an empirical scheme derived by fitting the ratio of F/Q to the fraction of clay content using the data of Gillette (1977). It is not a spectral scheme for dust emission and to use it in a spectral sense requires the specification of airborne dust distribution in the source region, pad (d). The input parameters required by the Marticorena and Bergametti (1995) scheme are also listed in Table 9.1.

9.3 Threshold Friction Velocity for Natural Surfaces A key variable to be determined in a wind-erosion scheme is the threshold friction velocity for a soil surface, u∗t , in contrast to that for a soil particle. Several surface and soil-related factors strongly affect the magnitude of u∗t ,

9.3 Threshold Friction Velocity for Natural Surfaces

309

Fig. 9.3. The structure of the wind-erosion scheme proposed by Marticorena and Bergametti (1995). Descriptions which are similar to those in Fig. 9.3 are omitted

including soil texture, soil moisture, salt concentration, surface crust and the presence of roughness elements on the surface, such as vegetation and pebbles. These factors are usually heterogeneous in space and vary slowly in time. Some of them may be modified during a wind-erosion event. For example, the particle-size distribution of the topsoil may become coarser as small particles are transported away from the source and the aerodynamic roughness length may increase as large soil aggregates emerge from the surface. Consequently, u∗t may also change during the erosion process. A pragmatic approach to account for the effects of soil and surface factors on u∗t is to express u∗t in the following form u∗t (ds ; λ, θ, sl , cr , ...) = u∗t (ds )fλ (λ)fw (θ)fsc (sl )fcr (cr )...

(9.1)

where u∗t (ds ) is the threshold friction velocity for sand particles of size ds in the idealised situation when soil is dry, bare and free of crust and salt. It is a function of ds only, which can be determined from wind-tunnel experiments using loose sand. In Equation (9.1), λ is the frontal-area index describing the characteristics of surface roughness elements, θ is the volumetric soil moisture, sl is the salt content of the soil and cr is a descriptor of surface crustiness. The multiplicators fλ , fw , fsc and fcr are the correction functions for surface-roughness elements, soil moisture, salt concentration and surface crust, respectively. All these multiplication functions are larger than or equal to 1 and hence u∗t (ds ) is the minimum value of u∗t (ds ; λ, θ, sl , cr , ...). More

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9 Integrated Wind-Erosion Modelling

multiplication functions can be introduced if more surface and soil properties need to be considered. In the following, we examine these functions in detail. 9.3.1 Drag Partition: Approach I Aeolian surfaces are always composed of roughness elements, such as large soil aggregates, pebbles, stubble and vegetation. For such surfaces, the total drag (or shear stress or momentum flux) can be split into a pressure drag on the roughness elements, τr , a skin drag on the ground surface, τs , and a skin drag on the roughness-element surface, τc , so that τ = τ r + τs + τc

(9.2)

The pressure drag τr arises from the pressure differences between the front and the back sides of the roughness elements, which lead to a momentum reduction of the mean flow by production of turbulence. The ground-surface drag τs arises from the friction of the ground surface, and the roughness-elementsurface drag τc arises from that on the surface of the roughness elements. If the roughness-element density (number of roughness elements per unit groundsurface area) is low, then τc can be neglected and we obtain τ = τr + τ s

(9.3)

Suppose over an area S, there exist n uniform roughness elements of simple geometry (e.g. rectangular blocks) and the breadth, height and basal area of the roughness elements are respectively b, h and ab . Then, the two important parameters used to describe the bulk geometric features of the roughness elements, namely, the frontal-area index λ and the basal-area index η can be defined as nbh S nab η= S

λ=

(9.4) (9.5)

It is clear from the definition that λ is the area projected into the flow by the roughness elements per unit ground area, also known as the roughness density. In contrast, η is the area projected into the ground surface by the roughness elements per unit ground area, also known as the fraction of cover. In Equation (9.3), τs is the area-averaged shear stress on a unit ground area. The area-averaged shear stress on a unit exposed ground area is τs = τs (1 − η)−1

(9.6)

It is τs which drives wind erosion. Thus, it is important to determine the ratios τs /τ and τr /τ and their variation with the density and configuration of the roughness elements, in order to estimate τs for given τ . The

9.3 Threshold Friction Velocity for Natural Surfaces

311

problem of drag partition has been studied in theory by many researchers (Lettau, 1969; Wooding et al. 1973; Arya, 1975) and by means of experiments (Marshall, 1971; Musick and Gillette, 1990; Wolfe and Nickling, 1996; Wyatt and Nickling, 1997). The concept of effective shelter area Ash and effective shelter volume Vsh introduced by Raupach (1992) is useful for the simplification of the problem (Fig. 9.4). Ash describes the reduction of ground shear stress τs in the element wakes and is the area integral of the normalized ground-stress deficit 
τs (x, y) 1− dxdy (9.7) Ash = τs0 S where τs (x, y) is the actual ground stress at location (x, y) and τs0 is the ground stress far away from the roughness elements. The meaning of Ash is that it is the area within which the stress on the ground must be set to zero (a) Coordinates

z b = Breadth h = Height

Wind h

y

b x

(b) Effective Shelter Area (Plan) τs = τs0 τs = τs0 Low τs Actual τs Countours

tan−1(u* /Uh )

b τs = 0 in area A

(c) Effective Shelter Volume

Actual φ Countours

φ = 0 in volume V

Fig. 9.4. Effective sheltering area Ash and effective sheltering volume Vsh (From Raupach, 1992)

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9 Integrated Wind-Erosion Modelling

to produce the same integrated stress deficit as that induced by the sheltering element, i.e., τs (x, y)dxdy = S

τs0 dxdy S−Ash

Vsh describes the effect of a given roughness element upon the pressure drag on other elements in its vicinity. The drag force per unit volume, φ, on an obstacle of frontal area ao is φ ∼ ao U 2 , where U is flow speed. Vsh is the volume integral of the normalised deficit in φ which is induced by placing a roughness element on the surface, so that 
φ(x, y, z) 1− dxdydz (9.8) Vsh = φ0 Γ where φ0 is the undisturbed value of φ far from the sheltering element and Γ is the air volume over surface S. Vsh is the volume within which the drag force on the obstacle must be set to zero, to produce the same integrated force deficit as that induced by the sheltering element, that is φdxdydz = φ0 dxdydz Γ

Γ −Vsh

Raupach (1992) made two hypotheses. One specifies Ash and Vsh for a single roughness element, and the other specifies the interactions between the roughness elements. They are: • Hypothesis I: Ash and Vsh can be expressed as Ash = cbhUh /u∗ Vsh = hAsh

(9.9) (9.10)

where c is a coefficient and Uh is the mean wind speed at h. • Hypothesis II: For roughness elements distributed either uniformly or randomly across a ground surface, the combined effective shelter area or volume can be calculated by randomly superimposing individual shelter areas or volumes. The validities of the two hypotheses are limited to the situations of low roughness densities. As roughness density increases, the interactions between the wakes associated with individual roughness elements become stronger and their collective effect cannot be simply described by superimposition. Ground Stress We consider now the attenuation of the ground stress as the roughness density increases. Suppose n roughness elements with an effective shelter area Ash are placed on a ground area S. Then it follows from Hypothesis II that

9.3 Threshold Friction Velocity for Natural Surfaces

τs (n) = τs (0)

 1−

Ash S

n

 =

1−

λAsh nbh

313

n (9.11)

For a constant λ, taking n → ∞ and substituting for Ash from Equation (9.9), it follows that     λAsh Uh τs (λ) = exp − = exp −c λ (9.12) τs (0) bh u∗ The above equation describes how the stress on the ground is reduced as λ increases while Uh is held constant. Further, introducing an un-obstructed surface drag coefficient Cs for the ground without obstacles, τs (λ = 0) = ρCs Uh2

(9.13)

so that Equation (9.12) becomes   Uh τs (λ) = ρCs Uh2 exp −c λ u∗

(9.14)

This shows that at constant Uh , the ground stress follows an exponential decay with increasing roughness density. Stress on Roughness Elements The drag force on an isolated roughness element φ can be written as φ = ρCr bhUh2

(9.15)

where Cr is the pressure drag coefficient for an isolated roughness element. If n roughness elements are placed on a ground area of S, the force per unit ground area acting on these roughness elements is  n Vsh nφ 1− (9.16) τr (n) = S Sh by invoking Hypothesis II. It follows that, by making use of Equation (9.4)  n λVsh 2 (9.17) τr (λ) = λρCr Uh 1 − nbh2 Substituting for Vsh using Equation (9.10) and allowing n → ∞ with λ held constant, we obtain   Uh (9.18) τr (λ) = λρCr Uh2 exp −c λ u∗ This implies that τr increases linearly with λ if λ is small. But as λ further increases, mutual sheltering progressively attenuates τr which then follows an exponential decay with λ.

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Total Stress From Equations (9.14) and (9.18), the total stress is   Uh 2 τ = τs + τr = ρUh (Cs + λCr ) exp −c λ u∗

(9.19)

This is an implicit equation in τ (note that τ = ρu2∗ ). Stress Partition It follows from Equations (9.14), (9.18) and (9.19) that 1 τs = τ 1 + βr λ βr λ τr = τ 1 + βr λ

(9.20) (9.21)

where βr = Cr /Cs is the ratio of the pressure-drag coefficient to friction-drag coefficient. The theory of Raupach shows that the stress partition is controlled entirely by βr . The agreement of the theoretic predictions with the windtunnel observations of Marshall (1971) is shown in Fig. 9.5, where (τr /τ )1/2 is plotted against log(λ). Both theory and data agree in showing that stress partition becomes insignificant (τr /τ is close to 1 and τs /τ is small) when λ exceeds a value in the range of 0.003–0.1. 1.2

1

(τr /τ)1/2

0.8

0.6

Raupach (1992) Wooding et al. (1973) Arya (1975) b/h = 0.5 (Cylinder) b/h = 1 (Cylinder) b/h = 2, 3 (Cylinder) b/h = 5 (Cylinder) b/h = 2 (Hemisphere)

0.4

0.2

0 −4

−3

−2

−1

0

log(λ)

Fig. 9.5. Predictions of stress partition for Cr = 0.3 and Cs = 0.0018 compared with the data of Marshall (1971). Also shown are the predictions of Wooding et al. (1973) and Arya (1975) (From Raupach, 1992)

9.3 Threshold Friction Velocity for Natural Surfaces

315

The drag-partitioning theory described above offers a correction to u∗t for the effect of roughness elements. Using Equations (9.6) and (9.20), the roughness correction function fλ = u∗t (λ)/u∗t (λ = 0) can be expressed as fλ = (1 − mr σr λ)1/2 (1 + mr βr λ)1/2

(9.22)

where σr is the basal area to frontal area ratio (σr = η/λ) and mr is a tuning parameter less than 1, which accounts for non-uniformity in the surface stress. Raupach et al. (1993) recommended βr ≈ 90, mr = 0.5 and σr ≈ 1 as typical values. This prediction represents well the wind-tunnel observations of fλ by Marshall (1971), Lyles and Allison (1976), and Gillette and Stockton (1989), as well as the field data of Musick and Gillette (1990). In the analysis of Raupach (1992), isolated simple roughness elements (e.g. cylinders and cubes) are considered in a turbulent layer which is much deeper than the height of the roughness elements. Embedded in this is the assumption that the effect of roughness elements on the structure of the turbulent layer is negligible. It appears therefore that the theory is best applicable to describing the effect of large soil aggregates and pebbles on wind erosion. In a natural landscape, the other major type of roughness elements is vegetation, which is rarely solid and exhibits varying degrees of porosity to wind flow. Wolfe and Nickling (1996) and Wyatt and Nickling (1997) have examined stress partitioning over several different surfaces with sparse desert vegetation (creosote bush dominated rangeland, mixed creosote bush and bursage rangeland, abandoned farmland and playa), while Lancaster and Baas (1998) have studied the drag partition over a surface covered partially with salt grass. These studies show that although the field data can be represented by the theory of Raupach (1992) and Raupach et al (1993), for certain types of surface roughness elements, the values of βr , σr and mr differ from those suggested by Raupach et al. (1993). For example, the values of βr , σr and mr best describing the data of Wyatt and Nickling (1997) are 202, 1.45 and 0.16, respectively, in contrast to 90, 1 and 0.5 as suggested by Raupach et al (1993). In particular, the value of mr found by Wyatt and Nickling (1997) is much lower than 0.5 (whereas the theoretic value of mr is unity). The relationship between correction function fλ and frontal-area index λ based on Wyatt and Nickling (1997) is shown in Fig. 9.6. The lower value of mr may stem from the nature of the turbulence developed in the wake of solid and porous objects. Compared with solid roughness elements, porous elements produce wakes which extend farther behind the objects because of the significant through-flow and little flow acceleration around the element edges. However, it is difficult to assess the validity of the mr values in general, because mr does not represent a defined physical property, but accounts for the non-uniformity of surface stress or the difference between maximum bed-average shear stress, an abstraction which essentially cannot be known without a better understanding of the distribution of surface shear stress, the nature of element-generated wakes and the dissipation of momentum. Otherwise, mr acts as a tuning coefficient which conceals processes excluded in the model.

316

9 Integrated Wind-Erosion Modelling 1.0

0.6

f

−1 λ

0.8

0.4

0.2 −3

−2

−1

0

log λ

Fig. 9.6. Relationship between correction function fλ and frontal-area index λ (fλ−1 is plotted). Measurements from several experiments (full dots) are shown together with the prediction (solid line) using Equation (9.22) with βr , σr and mr equal to 202, 1.45 and 0.16, respectively (Modified from Wyatt and Nickling, 1997)

The drag coefficient, Cr , which is directly associated with the βr parameter, also deserves attention. For vertical-axis cylinders, Cr is roughly a constant of 0.25 to 0.3 in the range of (roughness-element) Reynolds number between 103 and 105 (Taylor, 1988). The direct measurements of drag on vegetation conducted by Grant and Nickling (1998) suggest that Cr is larger for vegetation at all porosities, ranging between 0.3 and 0.9. 9.3.2 Drag Partition: Approach II An alternative approach to drag partition is to represent the effect of roughness elements in terms of aerodynamic roughness length, which is a more integrative parameter. Suppose the overall (roughness elements plus underlying surface) roughness length is z0 , then the profile of wind in the atmospheric boundary layer is approximately logarithmic   z u∗ ln (9.23) U (z) = κ z0 If the roughness elements are not too closely spaced (i.e. λ < 0.05), an internal boundary layer would grow behind each individual roughness element. The modified wind profile in the internal boundary layer also follows the logarithmic law. Supposing the roughness length of the underlying surface is z00 , we then have

9.3 Threshold Friction Velocity for Natural Surfaces

U (z) =

u∗s ln κ



z z00

317

 z 0, which is not fully consistent with the physics considerations behind the scheme. Scheme-IV: A binding-energy based scheme has been proposed by Shao et al. (1993b, 1996). The emission of dust of size dd due to the saltation of sand grains of size ds is given by F (dd , ds ) = β1 (ds )β2 (dd )Q(ds )u−2 ∗t (dd )

(9.68)

β1 (ds ) = 10−5 [1.25 ln(ds ) + 3.28] β2 (dd ) = exp(−140.7dd + 0.37)

(9.69)

where (9.70)

with dd and ds being in mm. The rate of dust emission for the jth dust particle-size bin Fj is determined by dj2 d2 β2 (d)u−2 (d)p(d)δd · β1 (d)Q(d)p(d)δd (9.71) Fj = ∗t dj1

d1

where dj1 and dj2 are the lower and upper limit of the jth dust particle-size bin and d1 and d2 are the lower and upper size limit of saltating particles. The total dust emission rate is F =

J

j=1

Fj

(9.72)

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9 Integrated Wind-Erosion Modelling

with J being the number of dust particle-size bins. An assumption which lies in the core of Scheme-IV is that the process of dust emission is a linear combination of saltation bombardment of various sand particle sizes. A specific problem with Equation (9.68) is that β1 and β2 are derived based a small set of idealized wind-tunnel experiments. Scheme-V: Taking into consideration the three dust emission mechanisms (aerodynamic entrainment, saltation bombardment and aggregates disintegration), Lu and Shao (1999) and Shao (2001, 2004) suggested that gQ F˜ (dj , ds ) = cy ηfj [(1 − γ) + γσp ](1 + σm ) 2 u∗

(9.73)

where F˜ (dj , ds ) is the dust emission rate for the jth particle-size bin generated by the saltation of particles of size ds , cy is a dimensionless coefficient and γ is a weighting function which satisfies  1 u∗ → u∗t (9.74) γ= 0 u∗ → ∞ Q is the streamwise saltation flux of ds ; g is acceleration due to gravity; σp = pm (di )/pf (di ) with pm (di ) and pf (di ) being the minimally- and fullydisturbed particle-size distribution. The bombardment efficiency σm is estimated by    ρb 2 ρp 1 + 14u∗ (9.75) σm = 12u∗ P P where ρb is soil bulk density and P is soil plastic pressure. This is a spectral model, because the rate of dust emission for any particle size bin is determined by d2 (9.76) F˜ (dj , d)δd Fj = d1

The total dust emission rate is F =

J

Fj

(9.77)

j=1

with J being the number of dust size bins. The values of cy fall between 10−5 and 5 × 10−5 and P falls between 1,000 and 50,000 Pa. In Scheme-V, the micro-physics of dust emission has been taken into consideration. It is more complex than the other schemes, but is still very simple. The input parameters required by Scheme-V have physical interpretations. However, the applications of Scheme-V are humped by the lack of soil and land-surface data that are not yet readily available, in particular the soil plastic pressure and the minimally- and fully-disturbed parent soil particle-size distributions.

9.5 Climatic Constraints on Dust Emission

333

In general, dust emission mechanisms require further investigation. The difficulty in achieving the accuracy of dust emission schemes is exacerbated because dust emission can vary by several orders of magnitude. A major problem facing the dust modelling community is the lack of direct dust flux data for model validation.

9.5 Climatic Constraints on Dust Emission Because of the large uncertainties in regional and global dust models, it is desirable to apply climatic constraints to the estimates of dust emission, such that the simulated dust-source pattern and intensity are not inconsistent with climatic observations. To this end, we introduce a erodibility index, Se , and define Se = 0 as the non-source region and Se > 0 as the source region. Two particular useful sources of data for establishing the climatic constraints on erodibilty are surface weather data and satellite data. 9.5.1 Erodibility Derived from Synoptic Data The synoptic records from weather stations are the best data available for analysing dust climatology. According to the WMO (World Meteorological Organisation) protocol, dust events are classified according to visibility into the categories of: • Dust in Suspension: widespread dust in suspension, not raised at or near the station at the time of observation; visibility is usually not greater than 10 km • Blowing Dust: raised dust or sand at the time of observation, reducing visibility to 1–10 km • Dust Storm: strong winds lift large quantities of dust particles, reducing visibility to between 200 and 1000 m • Severe Dust Storm: very strong winds lift large quantities of dust particles, reducing visibility to less than 200 m Synoptic records are collected at a large number of stations at 3-hourly intervals over the past few decades. These records are valuable for studying dust-storm distribution and frequency, near-surface dust concentration, climatic background and synoptic systems for dust-storm generations. For a given weather station, the frequencies of the four dust-event categories, respectively denoted as fDIS , fBD , fDS and fSDS , can be estimated as fDIS = ADIS /Aobs etc. where ADIS is the number of dust-in-suspension records and Aobs is that of total synoptic records. The frequency of all dust events is then fDE = fDIS + fBD + fDS + fSDS

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9 Integrated Wind-Erosion Modelling

Visibility is recorded along with the dust-event report. Although visibility is affected both by aerosols and water vapour, it is reasonable to assume that during a dust event, dust particles are the determining factor. Thus, dust concentration can be estimated from visibility using empirical relationships derived by fitting dust-concentration measurements to visibility. Several such relationships have been proposed by Chepil and Woodruff (1957), Equation (9.78), Patterson and Gillette (1977), Equation (9.79), Tews (1996), Equation (9.80), and Shao et al. (2003), Equation (9.81) C = (7078/Vis )1.25 C = (10507/Vis )1.07 C = (2032/Vis )0.877  3802.29Vis−0.84 Vis < 3.5 C= exp(−0.11Vis + 7.62) Vis ≥ 3.5

(9.78) (9.79) (9.80) (9.81)

where Vis is visibility in [km] and C is dust concentration in [µg m−3 ]. It needs to be pointed out, however, that dust concentration estimated in this way has large uncertainties for the following reasons: (1) the data sets used for deriving the relationships have been small; (2) the dependency of visibility on dust concentration is affected by dust particle size and air humidity; and (3) visibility measurements are subjective and often inaccurate. For a given value of Vis , the estimates of C using the above four relationships may greatly differ. Albeit with uncertainties, indicative dust concentrations can be estimated empirically at weather stations from the visibility records. These concentration estimates can then be used to generate a dust-concentration field by spatial interpolation. The erodibility index can be derived by combining the dust-concentration field with soil, vegetation and topography data. Suppose a region of concern is divided into I × J grid cells, then it is appropriate to calculate Se (i, j) as follows Se (i, j) = C(i, j)δ(i, j) we can set δ(i, j) = 0 if any one of the following criteria is satisfied: • •

Surface of cell (i, j) is a water surface or snow-covered surface. Surface of cell (i, j) is covered by vegetation or roughness elements (e.g. pebbles) with frontal-area index exceeding λmin (e.g. 0.1). • Average dust concentration for cell (i, j), C(i, j), is smaller than Cmin (e.g. 10 µg m−3 ). • Topographic elevation above sea surface at cell (i, j) is higher than hmin . As an example, Se (i, j) derived using the above described technique using the synoptic data of 27 May 1998 – 26 May 2003 for the region of (30 150◦ E, 5 60◦ N) is shown in Fig. 9.14. The distribution of the synoptic stations and the visibility-derived dust concentration, averaged over the 6-year period, is shown Fig. 9.14a. The distribution of Se (i, j) derived without topography

9.5 Climatic Constraints on Dust Emission

335

Fig. 9.14. Erodibility index over Asia, derived by combining visibility data and GIS data. (a) Dust concentration estimated from visibility data for individual weather stations; (b) Erodibility index after taking into consideration water, snow and vegetation surfaces; (c) Erodibility index after taking topography into consideration

data is shown in Fig. 9.14b, and that with topography correction is shown in Fig. 9.14c. In the study domain, the Gobi Desert region (A), the Tarim Basin (B), the Thar Desert (C), the Iran-Afghanistan-Pakistan border region (D), the Caspian-Aral Sea region (E), the Arabian Peninsula (F) and the Northeast

336

9 Integrated Wind-Erosion Modelling

Africa region (G) emerge as climatic dust source regions, a result consistent with those presented in Chapter 2. In IWEMS, Se (i, j) defines the potential regions of dust emission. Of course, Se (i, j) is not a quantitatively-useful measure for dust-emission rate, because higher dust concentration does not always mean stronger dust emission. The actual dust-emission rate for an individual dust event will still have to be estimated through the use of the dust-emission parameterisation scheme. 9.5.2 Erodibility Derived from Satellite Data Satellite observations over a sufficiently long period of time can also be used to estimate erodibility index. As described in Chapter 2, the TOMS aerosol index, averaged over the 13-year period of 1980–1992, provided evidence that most climatologically-strong dust sources coincide with topographic depressions where alluvial sediments have accumulated on geological time scales (Prospero et al. 2002). Based on this understanding, a method of computing Se as a function of topographic elevation has been proposed by Ginoux et al. (2001). In Zender et al. (2003), Se is used to define the areas where sediments may have accumulated through the transport by surface water runoff. Such areas can be estimated from digital elevation maps. They have reported that the use of a spatially distributed Se dramatically improves the spatial correlation between the simulated dust emissions with the TOMS aerosol index. 9.5.3 Wind-Erosion Hot Spots Satellite imagery shows that in some areas, dust events exhibit remarkably repetitive patterns and strong dust emissions are often confined to well-defined areas with certain geomorphologic and hydrological features. These areas are simply called hot spots. Figure 9.15(a) and (b) compare the images of dust storms in the Bodele Depression, Chad, on 2 February 2004 and 6 April 2004. The dust patterns for the two events, and indeed for many more events, are remarkably similar. From the dust clouds, some of the hot spots can be clearly identified. Similar observations can be made in Fig. 9.15(c) and (d) for the dust storms in Afghanistan. Efforts have been made to establish a high-resolution (e.g. 250 m) database of dust sources by identifying the hot spots from satellite imagery for regions of specific interests, such as Afghanistan and Iraq. Such databases are particularly useful for high-resolution regional dust modelling.

9.6 Land-Surface Parameters Land-surface parameters are required to quantify the land-surface properties which affect wind erosion, such as the capacity of soil to release dust and the threshold friction velocity for erosion to take place. In the context of

9.6 Land-Surface Parameters

337

Fig. 9.15. (a) Dust storm image in the Bodele Depression, Chad, 12:25UTC 2 Feb 2004, observed by satellite, Aqua; (b) as (a), but for 6 Apr 2004; (c) Dust storm image in Afghanistan, 06:15UTC 23 Sep 2003, observed by satellite, Terra; (d) as (c) but for 17 Aug 2004

wind-erosion modelling, land-surface parameters can be divided into three categories. These are • • •

Category 1: parameters for specifying soil properties, such as soil texture, soil-salt content, soil-binding strength (either binding energy or soil plastic pressure depending on dust scheme), etc. Category 2: parameters for specifying surface aerodynamic properties, such as frontal-area index, erodible fraction, roughness length, etc. Category 3: parameters related to soil thermal and hydraulic properties required by land-surface modelling

For wind-erosion modelling on regional to global scales, land-surface parameters can be stored as layers in a geographic information database. 9.6.1 Soil Particle-Size Distribution For broad-scale wind-erosion modelling, we often encounter the difficulty that measured particle-size distributions are not available for certain locations. One possible approach to overcoming this difficulty is to divide soils into

338

9 Integrated Wind-Erosion Modelling

different soil-texture classes and assign a representative pm (d) and pf (d) to each of these classes. The representative pm (d) and pf (d) would have to be established on the basis of soil samples collected elsewhere but for the same soil texture. Soils can be roughly divided into soil-texture classes using the USDA soiltexture triangle (Fig. 5.2). The choice of this classification has two advantages. Firstly, soil classification for wind-erosion modelling should be consistent with those used in related studies, notably atmospheric, land-surface and hydrological modelling as well as soil physics. The USDA soil-texture classification has been widely used in these research fields. Secondly, some data has already been collected throughout the world and these data sets are mostly organised according to the USDA soil classification. One useful way of organising particle-size data is to consider a soil particlesize distribution to be a superposition of N simple distributions. Log-normal distributions are popular choices (Gomes et al. 1990; Chatenet et al. 1996). In this case, we have   N

(ln d − ln Dj )2 wj dP (d) √ = exp − d ln(d) j=1 2πσj 2σj2 where P (d) is the probability distribution function, wj is the weight for the jth mode of the particle-size distribution and Dj and σj are parameters for the log-normal distribution of the jth mode. The above equation can also be written as   N

(ln d − ln Dj )2 wj √ (9.82) exp − d × p(d) = 2σj2 2πσj j=1 The advantage of representing p(d) using Equation (9.82) is that it allows extrapolation of particle-size information obtained from a limited number of soil samples to the broad region. The particle-size distribution for each soil texture is fitted with several log-normal distributions and, in doing so, each particle-size distribution is represented by a small set of parameters, usually 9 or 12. Fitting observed data by means of non-linear least-squares methods is suitable for determining the parameters wj , ln(Dj ) and σj in Equation (9.82). This is in essence an optimisation problem, as the non-linear least-squares method requires the squared error between the fitted and observed particle-size distributions (i.e., the optimisation function) to be minimized (or optimized) by choosing a particular combination of parameters (wj , ln(Dj ) and σj ). For the non-linear least-squares fitting, the optimisation function, fopt , is defined as fopt =

K

[dk pobs (dk ) − dk pf it (dk )]

2

k=1

where pobs and pf it are respectively the measured and modelled values of particle-size distribution for particle-size class dk , with k referring to the kth

9.6 Land-Surface Parameters

339

Table 9.3. Estimated log-normal size-distribution parameters for four Australian soils for minimally-dispersed and fully-dispersed particle-size distributions Samples

Clay Silt

Sand USDA

(%) (%)

(%)

Simpson (m) 0.03 0.7

mode 1 w1

mode 2

ln(D1 ) σ1

w2

mode 3

ln(D2 ) σ2

w3

ln(D3 ) σ3

99.27

S

1.00 5.05

Simpson (f) 0.02 1.27 98.71

S

1.00 4.98

0.34

Betoota (m) 0.08 17.91 82.01

S

0.76 5.11

0.55 0.16 4.20

0.32 0.08 3.31

1.20 40.20 58.60

LS

0.37 5.10

0.51 0.33 4.17

0.25 0.30 3.40

0.99

Manilla (m) 0.31 33.24 66.45

SL

0.08 5.06

0.22 0.88 4.50

0.95 0.04 2.40

0.92

Betoota (f) Manilla (f)

0.34 0.65

1.80 45.28 52.92

SL

0.87 4.45

0.94 0.13 2.49

0.94

Cooper (m) 0.06 31.98 67.96

SL

0.77 4.50

0.39 0.11 4.19

0.39 0.12 3.46

0.46

Cooper (f)

SiL

0.10 4.99

0.89 0.56 3.92

0.62 0.34 2.28

0.91

4.25 67.87 27.88

size class and K referring to the total number of particle-size classes. The nonlinear optimisation problem has many local optimal points and the global optimal point (where fopt obtains the absolute minimum value) is usually difficult to find. Hence a satisfactory fitting requires a number of try-anderror tests. Practically, the initial values of the parameters need to be altered several times until optimisation is achieved. There is a need to apply the above-described procedure to establishing a global (and regional) database of both minimally- and fully-disturbed particlesize distributions for the purposes of wind-erosion modelling. A number of samples have been collected and analysed for Australian soils. As an example, the fitted parameters for four Australian soils are listed in Table 9.3. The four soil samples are collected in the Simpson Desert and at Betooma, Manilla and Cooper Floodplain. Except for the Simpson Desert sample, the minimally- and fully-dispersed analyses show considerable differences in particle-size distributions. For instance, the Cooper Floodplain sample is a sandy loam according to minimally-dispersed analysis but a silty loam according to fully-dispersed analysis. Clearly, the Cooper Floodplain soil has a much larger capacity for dust emission than the Simpson Desert soil. Numerical tests show consistently that only up to three or four log-normal distributions (trial functions) are required to achieve a good representation of observed particle-size distributions. Using four or more trial functions produces almost the same results, with one or two weights degenerating close to zero. In some cases, more trial functions may even lead to worse results, due to numerical errors. This consistency implies that particle-size distributions indeed have inherent physical modes which are identifiable through mathematical fitting. Table 9.3 shows the Australian soils in general consist of three basic modes. The first mode, which is also most prominent, is related to sandsized particles with a ln(D) close to 5, corresponding to the particle size of medium sand around 150 µm. The second mode has a ln(D) around 4.1 to 4.5, corresponding to the particle size of fine sand between 60 and 90 µm and

340

9 Integrated Wind-Erosion Modelling

Fig. 9.16. (a) Observed and modelled particle-size distributions for sand (Simpson), (b) for loamy sand (Betoota), (c) sandy loam (Manilla) and (d) silty loam, in terms of d × p(d) versus d. Both minimally-dispersed particle-size distribution (a1, b1, c1 and d1) and fully-dispersed particle-size distribution (a2, b2, c2 and d2) are shown

the third mode has a ln(D) around 2.8 to 3.5, corresponding to the particle size of silt between 16 and 33 µm. The weight for the third mode increases as soil texture changes from sand to clay. Figure 9.16 shows the fitted results for minimally-dispersed and fullydispersed particle-size distributions for sand (Simpson), loamy sand (Betoota), sandy loam (Manilla) and silty loam (Cooper). For the sandy soil, the difference between the minimally-dispersed and fully-dispersed particle-size distributions is small. The differences are larger for the loamy sand and sandy loam soils. The minimally-dispersed particle-size distribution for the silty loam soil shows a prominent mode around 100 µm. In the corresponding

9.6 Land-Surface Parameters

341

fully-dispersed case, there is a shift of weight towards finer particles. This comparison reveals that in the silty loam soil, most grains around 100 µm are aggregates of silt or clay. Particle-size distributions of soil samples collected from various parts of the world have been analysed using the technique of dry sieving. Soil samples are usually oven dried and then sieved into 10 size classes ( 11.0 7.0 - 11.0 4.7 3.3 2.1 1.1

-

Stage 1 Stage 2

7.0 4.7 3.3 2.1

0.65 - 1.1 0.43 - 0.65 < 0.43

Filter Plate

Stage 8 Back Up

Airflow

Fig. 11.7. Structure of an Andersen sampler

402

11 Techniques for Wind-Erosion Measurements

so that only small particles are collected (e.g. diameter less than 11 µm). Teflon-coated glass filter, polytetrafluoroethylene (PTEE) filter and quartz filter are commonly used. The choice of filter type depends on the chemical analyses to be conducted on the sample. Some types of filter have high blank contents of certain metal elements and are thus unsuitable for collecting dust for detecting those metal elements. Laboratory analyses are important for the understanding of dust composition, mineralogy and morphology. In the context of wind-erosion physics, the following three types of analysis deserve particular attention (Yabuki et al. 2005). (1) The knowledge of mass concentrations of water-soluble ions and water-insoluble elements is useful for modelling dust emission. Water-soluble + + − −2 + + − ions mainly include NH+ 4 , Na , Mg2 , Ca2 , K , Cl , NO3 and SO4 . The level of concentration of certain water-soluble ions gives an indication of the −2 likely origins of the aerosol. For example, NO− 3 and SO4 are typically related to secondary air pollution particulates ammonium sulphate [(NH4 )2 SO4 ] and ammonium nitrate (NH4 NO3 ), while Na+ and Cl− are more related to sea salt. Water-soluble ions usually makes up 10% to 20% (in mass) of a dust sample. Water-insoluble elements mainly include Na, Mg, Al, K, Ca, Fe, Ti and Mn. The ratios of Na/Al, Mg/Al, K/Al and Fe/Al are often estimated. By comparing these ratios with those of desert soils, it is possible to identify the origin of the dust. (2) Analysis of dust mineralogy can be done by using techniques such as X-ray powder diffraction. Dominant soil constituting minerals are quartz (SiO2 ), feldspar (KAISi3 O8 ) and various evaporites, including calcite (CaCO3 ), gypsum (CaSO4 ), halite (NaCl), thenardite (Na2 SO4 ) etc. Other minerals include biotite, chlorite, dolomite, anatase etc. (3) Particle morphology, including particle size, shape and structure can be analysed using scanning electron microscopes. 11.3.2 Optical Particle Counter An Optical Particle Counter (OPC) measures aerosol concentration and size distribution (Mikami et al. 2005b). In general, an OPC contains a laser illuminated optical system which uses a photodetector to collect the scattered light from a single particle as it passes through the laser beam. The photodetector signals (pulses) are amplified, and the pulse height corresponds to a particle-size category. Each scattered pulse represents a particle count and this is incremented in the appropriate size category to obtain the number of particles in a given size interval. An OPC consists of three major components: (1) the airflow system; (2) the optical system; and (3) the electronics system (Fig. 11.8). Through the airflow system, the sample air is focused and confined to the boundaries of the laser beam. This is achieved by isokinetically merging the sampled airflow with a filtered sheath airflow, prior to its entering the optical sampling chamber. An OPC is a very low-volume air sampler with a flow rate of around 100 mL min−1 . The range of particle-size measurement is

11.4 Deposition Collectors

403

Particle Inlet

Detector

SHEATH FLOW

Sample Cavity

Laser

Filter Pump

Sample outlet Filter Sheath flow Flowmeter

Sample flow Flowmeter

Fig. 11.8. Structure of an optical dust particle counter

typically between 0.3 and 5 µm, but some OPC have the eight particle-size groups: 0.3–0.5, 0.5–0.8, 0.8–1.35, 1.35–2.23, 2.23–3.67, 3.67–6.06, 6.06–10.0 and 10.0–25.0 µm. OPC operates by counting single particles. The singleness is ensured by adjusting the sampling and the sheath flow rates. However, if the particle number concentration at the inlet exceeds a certain limit (known as the saturation particle concentration), the adjustment of flow rates does not ensure that only a single particle enters the optical viewing volume at a given instant. If multiple particles are present in the optical viewing volume at the same instant, the signal will be biased both in particle size and counts. The saturation particle concentration for an OPC is 107 cm−3 . This limit is sufficiently large for wind-erosion studies, because dust concentration during natural dust events is generally lower.

11.4 Deposition Collectors Dust deposition is affected by the properties of the ground surface, among other factors such as wind conditions, ambient dust concentration and particle size. A useful way of studying dust deposition on natural surfaces is to employ surrogate surfaces which imitate the natural ones. In wind-tunnel experiments,

404

11 Techniques for Wind-Erosion Measurements

surrogate surfaces are easy to install and to manage. A variety of surrogate surfaces have been tested, including water or antifreeze, glass marbles, moist filter paper, sticky surfaces, grass, moss, paper, snow, glass, plastic, metal, artificial grass, etc. (Sow et al. 2006). However, surrogate surfaces are difficult to use in the field for long-term dust-deposition monitoring. Instead, dust-deposition collectors are often used. Many types of collector have been tried, ranging from simple household buckets to special devices, but none of collectors can be considered to be the professional standard. Two of the more widely used types are the Marble Dust Collector (Goossens, 2006) and the Inverted Frisbee Collector (Hall et al. 1994). A Marble Dust Collector consists of a rectangular plastic tray (c.a. 50 × 30 × 10 cm3 ) and a sieve container on top of the tray. The sieve container is filled with two layers of marbles (c.a. 15 mm in diameter) which form a marble filter. Dust settles on and between the marbles and is washed by rain (or using water) into the plastic tray and collected via an outlet underneath the tray. The reason of using marbles is because they have very low microroughness and can prevent the splash of the dust from the collector by raindrops. The marble filter also acts as a dust trap which protects the dust that has settled into it from resuspension. The Inverted Frisbee Collector consists of a circular stainless steel bowl about 30 cm in diameter and 4 cm deep and an aerodynamically-shaped aluminium deflector ring around the bowl. The inner diameter of the ring is 38.4 cm and the outer diameter is 64 cm. A variation of the Inverted Frisbee Collector is to fill the bowl with marbles. Goossens (2006) examined the efficiency of the two types of collectors using water surface as reference and concluded that the efficiencies of these collectors are usually 50% lower, often much lower, than that of the water surface.

11.5 Field Measurements Field measurements are essential for understanding wind-erosion processes and for verifying wind-erosion models. A problem in the past with field measurements is that they are not sufficiently cohesive and reliable for testing wind-erosion models, especially dust models. While it is difficult to define in general what a cohesive data set should be, a tentative list of quantities that should be measured is as proposed in Table 11.1. The measurements of saltation flux can be made using an array of saltation traps, such as the Fryrear sand traps (Figs. 11.3 and 11.9). The traps are usually mounted on a tower to measure the fluxes of sand particles at various levels over a height of about 2 m. The main advantage of the Fryrear traps is that they are robust and easy to maintain for long-term wind erosion monitoring. For measuring the saltation of different particle-size groups in high frequency, SPC (Figs. 11.6 and 11.9) appears to be the most suitable sensor developed so far. Figure 11.9 shows an example setup of a field wind-erosion

11.5 Field Measurements

405

Table 11.1. A list of parameters to be measured in field experiments Measurements Purposes Saltation flux & particle size Sand drift & saltation models Dust concentration & particle size Dust concentration, emission & deposition Wind speed Friction velocity, roughness length, land-surface model Wind direction Weather Air temp., humidity & pressure Weather, land-surface model Solar radiation Weather, land-surface model Precipitation Weather, land-surface model, crust Soil moisture Threshold friction velocity, crust Frontal-area index Threshold friction velocity, roughness length Fraction of cover Erodible area, saltation & dust models Soil particle-size distribution Threshold friction velocity, saltation & dust models

Fig. 11.9. An example for the set up of a wind-erosion monitoring system in field. Dust concentrations and size-distributions are measured using Optical Particle Counters (1, OPC at 1, 2 and 3.5 m); TSP concentrations are measured using high-volume air samplers (2, TSP at 1, 2, 3.5 and 5 m) and PM10 concentrations are measured using a Dust Track (3, at 2 m); Saltation flux is measured using an array of Fryrear sand traps (4, at 0.1, 0.2, 0.5, 1 and 2 m), Sand Particle Counters (5, SPC at 0.05, 0.1 and 0.3 m) and a Sensit (6, at 0.1 m). It also consists of a micro-meteorological station for recording wind profiles and other meteorological data (7, AWS) (Experiment set up by M. Mikami, J. F. Leys and M. Ishizuka; photo by M. Mikami, with acknowledgment)

monitoring station. In this example, Fryrear sand traps were mounted at five different levels (0.1, 0.2, 0.5, 1 and 2 m). The traps were unevenly spaced in the vertical, as sand transport occurs mostly near the surface. Three SPCs

406

11 Techniques for Wind-Erosion Measurements

were used for measuring turbulent saltation fluxes of 32 particle size groups at 0.05, 0.1 and 0.3 m above the ground surface. It is also important to measure the atmospheric and land-surface quantities which influence wind erosion. The atmospheric quantities, in particular, wind speed, air temperature and air humidity at various levels, as well as radiation and precipitation, can be measured using a micro-meteorological station. In the example shown in Fig. 11.9, an array of anemometers is mounted on a mast for measuring the profile of wind speed (at heights 0.5 and 2.16 m). From the wind measurements, surface roughness length and friction velocity can be derived, assuming a logarithmic profile of the mean wind. Land-surface quantities, such as soil texture, soil moisture, soil crust and surface cover should be measured. TSP measurements can be obtained using dust samplers mounted on a tower (Fig. 11.9). Such a system can operate automatically and the dustsampling equipment can be triggered by predetermined wind speeds to run for predetermined periods. The dust samplers actively draw air through fine fibreglass filters, which can be weighted subsequently and analyzed in laboratory. In addition to TSP measurements, Optical Particle Counters can be used to measure dust concentration and size distribution on various levels. The Andersen sampler can be used as a complementary instrument to the OPC, which can be used to collect samples for more detailed laboratory chemical, mineralogical and morphological analyses. In addition to field measurements at selected sites, a large amount of dustrelated measurements have been collected through monitoring networks and remote sensing. In relation to air-quality monitoring, networks of air samplers for measuring aerosol concentration have been operating and expanding. The measurements are quite accurate, but are not dedicated to dust measurements and are low in spatial and temporal resolution. There are very few air samplers in desert areas, and a measurement takes days even weeks to make. In relation to weather monitoring, networks of ground-based solar radiation measurements using instruments such as sun photometers have been operating for decades. Ground-based radiation measurements are useful for determining quantities, such as dust optical thickness. Remote sensing has been offering new opportunities for wind-erosion research. Numerous specialized books have been written on remote sensing. In the context of wind erosion, lidar network and satellite remote sensing deserve particular attention. In northeast Asia, a lidar network has been functioning since the early 2000s (Sugimoto et al. 2003; Shimizu et al. 2004). From lidar signals, vertical aerosol profiles can be retrieved. The great advantage of the lidar network is that it allows continuous and three-dimensional observations of dust storms. However, lidar observations are still limited to a small number of locations and in case of severe dust storms, lidar only provides information for the lower part of the troposphere, as lidar signals often cannot penetrate the dense dust layer. Satellite data have been used for monitoring dust storms, identifying dust-emission hot spots and examining global

11.6 Particle-Size Analysis

407

dust climate. Satellite data are also essential for deriving land-surface parameters, such as vegetation cover and aerodynamic roughness length, which are necessary for wind-erosion and land-surface modelling. However, technical difficulties in converting satellite signals to physical quantities, such as dust load and dust particle size, are not yet fully solved. There are obvious difficulties for satellites to detect dust below clouds.

11.6 Particle-Size Analysis Particle-size distribution is a key parameter determining the entire process of wind erosion, from entrainment through transport to deposition. It can be estimated from either a geometric or the dynamic point of view. From the geometric perspective, particle-size distributions can be determined using one of the three methods: (a) dry or wet sieving; (b) electro-optical techniques, including Coulter Counter analysis and laser granulometry, and (c) computerised image analysis. From the dynamic perspective, the distribution of the particle terminal velocity can be measured using a settling tube or an elutriator. The choice of the most appropriate method depends largely on the amount of fine material present in the soil sample and the intended applications of the data set. Samples which contain only small amounts of fine material can be analysed through dry-sieving or settling-tube analysis, whereas Coulter-Counter analysis or laser granulometry are more adequate if the sample contains a significant amount of dust particles. Image analysis can be employed if both size and shape information are needed. It is important to recognize that dust particles may exist as soil aggregates or as coats attached to sand grains. They are released mainly through saltation bombardment and aggregates disintegration (self abrasion). For modelling purposes, it is desirable to determine the minimally-disturbed particle-size distribution, through gentle sieving or settling-tube analysis of soil samples and, in addition the total amount of dust particles that can be released during wind-erosion events. McTainsh et al. (1997a) proposed a composite method for particle-size analysis, by using sieving to determine the particle-size distribution for the particle-size range d > 75 µm, Coulter Multisizer for the particle-size range between 2 to 75 µm, and a Pipette for the particle-size range smaller than 2 µm. 11.6.1 Dry Sieving Dry sieving is undertaken using a stack of successively-finer sieves which are mounted on an electrically-powered shaker (Pye and Tsoar, 1990). The shakers have simple vibrating, rotating and tilting actions or have a hammer action. Each sieve consists of a stainless-steel, brass, phosphor-bronze or nylon mesh. Nests of sieves with aperture dimensions at quarter-phi (Equation 5.2) or half-phi intervals are commonly used in practice. The optimum size used

408

11 Techniques for Wind-Erosion Measurements

Table 11.2. Recommended sieve aperture and maximum permissible sieve loading Mesh (mm) Load (kg) Mesh (mm) Load (kg)

20 2.0 1.18 0.1

14 1.5 0.6 0.075

10 1.0 0.425 0.075

6.3 0.75 0.3 0.05

5 0.5 .212 0.05

3.35 0.3 0.15 0.04

2.0 0.2 0.063 0.025

for dry sieving depends on the number of sieves and the dimensions of the mesh aperture. Standard permissible sieve-loading according to the British Standards Institution is given in Table 11.2. 11.6.2 Settling Tube and Elutriator The settling tube and elutriator are instruments used for determining the distribution of particle terminal velocity, wt , which is simply related to the distribution of particle size (Fig. 5.7). While the physical principles for the functioning of a settling tube and an elutriator are the same, the design and the analysis procedure for the two instruments are different. For a settling tube, a soil sample is dropped in still air, and the times required by the particles to travel through the settling tube are measured (Malcolm and Raupach, 1991). The particle terminal velocities are derived from these times, because particles with larger wt travel faster through the tube than those with smaller wt . For an elutriator, an air stream is blown upward through a soil sample contained in a cup, which is thereby fluidised; grains with wt less than the vertical flow velocity are entrained and removed (Chepil, 1951). The distribution of wt is determined by gradually increasing the flow speed through the cup. A possible configuration of a settling tube is as illustrated in Fig. 11.10. A settling tube is a vertical tube, approximately 10 m long and is fitted with a releasing device at the top. Soil samples of several grams are released on computer command. At the bottom of the tube, a collection, weighing and recording apparatus is fitted. Particles travelling through the tube are collected and weighed by using a load cell that measures the accumulated mass as a function of time. The load cell can be connected to a digital logger. In still air, the vertical-velocity component of particle motion, wp , is given by 3 dwp =− Cd (Rep )wp | wp | −g (11.2) dt 4σp d The variables used in the above equation are as defined in Chapter 5. The particle terminal velocity, wt , derived from the above equation satisfies Equation (5.18). The particle trajectory z(t) is the distance fallen in time t, following release at z = 0 and t = 0, and is therefore given by t wp (t)dt (11.3) z(t) = 0

11.6 Particle-Size Analysis

409

Release Mechanism Solenoid

Tube

~ 10 m

Power Supply Control Box Strain Gauge

Datalogger

Computer

Fig. 11.10. Configuration of a settling tube. The vertical tube is approximately 10 m long fitted with a releasing device at the top. Soil samples of several grams are released on computer command. At the bottom of the tube, a collection, weighing and recording apparatus is fitted

For particle sizes in the Stokes region, the solution of Equation (11.2) and the subsequent integration give the particle speed and trajectory as wp (t) = −wt (1 − e−t/Tpt ) w2 z(t) = −wt t + t (1 − e−t/Tpt ) g

(11.4) (11.5)

where /Tpt is defined by Equation (5.19). The above equations represent a correct trajectory z(t) in the Stokes region (Rep  1) where a linearised drag term (i.e., Cd = 24/Rep ) is physically correct. At higher particle Reynolds numbers, the above equations are only an approximation to the true trajectory, worsening as Rep increases. However, this approximation turns out to be useful for Rep well beyond the true Stokes regime (Malcolm and Raupach, 1991). In a settling-tube analysis, the time th for the particle to travel through the settling tube of length h is measured. Replacing z with −h in Equation (11.5), we obtain an estimate of wt from th by means of
  th g h wt2 1 − exp − (11.6) wt = + th gth wt The equation is a transcendental one for wt , but can easily be solved by iteration. The mass accumulation is measured as a function of time, and this

410

11 Techniques for Wind-Erosion Measurements

75 mm

~3 m

Particle trap with filter

Filter

200 mm

Fluidising cup

Air flow and flow speed device

Fig. 11.11. Illustration of an elutriator

accumulation can be converted to mass accumulation as a function of wt through Equation (11.6) or as a function of equivalent particle size. An elutriator consists of a fluidising cup in which a soil sample to be analysed is placed, a tube of several metres length, a particle trap with a filter and a flow-speed control device (Fig. 11.11). An elutriator functions on the same principle as a settling tube. When an airflow of velocity w passes through the fluidising cup, a proportion of the particles will be lifted. When these particles move through a sufficiently long tube, their motion reaches an equilibrium and their travel speed relative to the air flow is −wt . Particles with terminal velocities smaller than w would move through the tube, while particles with terminal velocities larger than w would fall back to the cup. The procedure of particle-size analysis is therefore to start the analysis at a low flow speed, to separate fine particles from the soil sample, and then gradually increase the flow speed in order to separate the larger particles. The cup holding the sample is usually corn-shaped, so that the fluidisation of the sample bed can be achieved even at low flow speed through the tube, which is necessary for separating fine particles from the sample. 11.6.3 Electro-Sensing Methods One of the instruments used for particle-size analysis is the Coulter Multisizer. The instrument is best suited for handling small samples with a narrow

411

Electrodes

11.6 Particle-Size Analysis

Particles

Sensing Zone

Fig. 11.12. Illustration of the electrical sensing zone, showing an aperture tube immersed in an electrolyte with particles passing through the aperture (Redrawn from McTainsh et al. 1997a)

particle-size range. For such samples, the time required for the analysis is short, the resolution is very high, and the reproducibility is good. The Multisizer is less well-suited to samples with a broad particle-size range. The Coulter Multisizer is based on the Coulter principle (Fig. 11.12). The number and size of particles are measured by suspending the sample in a conductive liquid and measuring the electrical current between two electrodes on either side of a small aperture, through which particles are sucked. As each particle passes through the sensing zone and aperture it changes the impedance of the current between the two electrodes, producing a pulse with a magnitude proportional to the particle volume. These current pulses are scaled, counted and accumulated in 256 size-related channels from which a particle-size distribution is produced. The Coulter Multisizer produces size distributions in terms of volume, number and particle surface area. The nature of sample pre-treatment has a significant effect on the results of the analyses. As the sample must be analysed in a liquid electrolyte [3% tri-sodium orthophosphate (N a3 P O4 12H2 O) plus 50% glycerol], the Coulter Multisizer cannot perform undispersed particle-size analyses, i.e., the analyses are always more or less dispersed. If the analysis is done without intentionally dispersing the sample through additional chemical or physical treatments, it is called minimally-dispersed and the resultant particle-size distribution is referred to as the minimally-dispersed particle-size distribution. For most soils, this is probably not too different from the in-situ particle-size distribution or the best approximation to it currently available. For practical purposes of wind-erosion modelling, the minimally-dispersed particle-size distribution is considered to be identical to the in-situ particle-size distribution. For the

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fully-dispersed analysis, the soil sample receives chemical and vigorous physical dispersions to reduce it to its fundamental particle-size constituents. A typical chemical treatment is to place the sample in a soil dispersant, such as 3% tri-sodium orthophosphate and 1M sodium hydroxide (NaOH). The particle-size analysis of the soil sample after the chemical and physical treatments gives the fully-dispersed particle-size distribution. This particle-size distribution is the best approximation available for that of the sediment during a very strong wind-erosion event. In wind-erosion modelling, the fully-dispersed particle-size analysis is used to estimate the mass fraction of dust in a given soil. One of the advantages of the Coulter Multisizer is its capacity to analyze samples in very small quantities down to about 0.1 g, while other particle-size analysis techniques, such as hydrometer and sieve analyses, require samples up to 30 g. This feature of the Coulter Multisizer allows the particle-size analysis of sediment samples collected using high-volume samplers. The Coulter Multisizer also produces a relatively-high resolution for particle-size analysis, since particles are sized into 256 classes. The multisizer can measure particles over a size range from 0.45 to 1200 microns, but there are practical difficulties with analyses at the coarse end (>150 µm), and the Multisizer appears to underestimate the clay fraction (