Dunes E-Book

Andrew Warren

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: RGS-IBG Book Series
• Sprache: Englisch

Seitenzahl: 394

Veröffentlichungsjahr: 2013

Beschreibung

Dunes is the first book in over a decade to incorporate the latest research in this active and fast-developing field. It discusses the shapes, sizes, patterns, distribution, history and care of wind-blown dunes, and covers all aspects of dunes, terrestrial and in the Solar System.

• The only book to cover all dunes, terrestrial and in the Solar System, in deserts, on coasts, and in the past
• Represents the most current update on the research of dunes for over a decade
• Incorporates the latest research to come out of China where the field is most rapidly expanding
• Discusses the most recent range of skills and technology now focused on the study of dunes
• Brings up-to-date a rapidly expanding field

Leseprobe

Contents

List of Figures

Series Editors’ Preface

Acknowledgements

Introduction

Reference

Part One: <10 m2; <10 years

Chapter One: Wind and Sand

Wind versus Bed

Lift-Off

Thresholds

Sand in Motion

The Transport Rate

References

Chapter Two: Ripples

Subtypes

Models

Pattern

Chapter Three: The Form and Behaviour of Free Dunes

Definitions

Early Stages

The Profile of a Fully Grown Dune

Movement

Size

References

Part Two: 1000 to 10,000 m2; 100 to 1000 years

Chapter Four: Pattern in Free Dunes

Definitions

Wind-Directional Regimes

The Classification of Wind-Directional Regimes

Wind-Directional Regimes and Dune Pattern

Transverse Dunes

Linear Dunes

Sand Sheets

Dunes with Distinctive Sand

References

Chapter Five: Forced Dunes

Dunes Built around Bluff Obstacles

Dunes on Gently Sloping Terrain

Reference

Chapter Six: Dunes and Plants

Wind, Sand and Plants

Dunes among Plants

References

Chapter Seven: Coastal Dunes

Coastal Dunes and Climate

The Beach–Dune System

Exclusively Coastal Dunes

References

Part Three: >0.3 mm; <2,200,000,000 years

Chapter Eight: Sand Seas

Terms

Large Sand Seas

Growth and Development

Sand Seas in Tectonic Basins

Topographically Unconfined Sand Seas

Transfer between Sand Seas

Chapter Nine: A History of Dune Sand

Provenance

Recycling

Maturation

Relationships between Dune Fields and the Sources of Their Sand

References

Chapter Ten: A History of Inland Dunes

Very Ancient Dunes: Siliceous Windblown Sandstones

The Emergence of Familiar Spatial and Dynamic Patterns

Dune Historiography

Quaternary Dune-Building Climates

Dunes in the Early- and Mid-Pleistocene

Late Pleistocene Dunes

Dunes in the Holocene

References

Chapter Eleven: A History of Coastal Dunes

Long Sequences

Sea Level

Other Controls

Calcareous Aeolianite

Reference

Chapter Twelve: Mars, Venus, Titan

Similarities

Differences

Sand

Ripples and Transverse Aeolian Ridges

Dunes

Reference

Part Four: Care

Chapter Thirteen: Local, Short-Term Care (<1000 m2; <10 years)

Dunes in Deserts

Stabilised Dunes in Semi-Arid Areas

Coastal Dunes

References

Chapter Fourteen: Sustainability (>100,000 m2; >10 years)

Constraints

Sustainability

References

Index

RGS-IBG Book Series

Published

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Forthcoming

Smoking Geographies: Space, Place and TobaccoRoss Barnett, Graham Moon, Jamie Pearce, Lee Thompson and Liz TwiggMaterial Politics: Disputes Along the PipelineAndrew BarryPeopling Immigration Control: Geographies of Governing and Activism in the British Asylum SystemNick GillThe Geopolitics of Expertise: Knowledge and Authority in an Integrating EuropeMerje KuusThe Geopolitics of ExpertiseIn the Nature of Landscape: Cultural Geography on the Norfolk BroadsDavid MatlessWorking Lives: Gender, Migration and Employment in Post-war BritainLinda McDowellFashioning Globalisation: New Zealand Design, Working Women and the Cultural EconomyMaureen Molloy and Wendy LarnerOrigination: The Geographies of Brands and BrandingAndy PikeMaking Other Worlds: Agency and Interaction in Environmental ChangeJohn WainwrightEveryday Moral Economies: Food, Politics and Scale in CubaMarisa WilsonAssembling Export Markets: The Making and Unmaking of Global Food Connections in West AfricaStefan OumaRehearsing the State: The Political Practices of the Tibetan Government-in-ExileFiona McConnellArticulations of Capital: Global Production Networks and Regional TransformationsJohn Pickles, Adrian Smith and Robert Begg, with Milan Buček, Rudolf Pástor and Poli Roukova

This edition first published 2013© 2013 John Wiley & Sons, Ltd

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Library of Congress Cataloging-in-Publication Data

Warren, Andrew.Dunes : dynamics, morphology, history / Andrew Warren.pages cmIncludes bibliographical references and index.ISBN 978-1-4443-3969-7 (cloth) – ISBN 978-1-4443-3968-0 (pbk.)

1. Sand dunes–History. 2. Geomorphology. 3. Sand dunes–Environmental aspects.GB631.W36 2013551.3′75–dc23

2012050118

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

Cover image: Sand dune © Isabella Pfenninger / iStockphotoCover design by Workhaus

List of Figures

Figure 1.1

Velocity profile of the wind over a smooth, flat bed in a wind tunnel.

Figure 1.2

Threshold curve for the start of motion (the static threshold) of particles of the density of quartz (observed and modelled data).

Figure 1.3

Static threshold for grain motion (u*tw) related to the gravimetric water content of the sediment at the start and end of an experiment.

Figure 1.4

Modes of grain transport in the wind.

Figure 1.5

Trajectories of near-spherical saltons.

Figure 1.6

Vertical distribution of dimensionless sand flux, for 0.2–0.3 mm particles, at different free-stream velocities.

Figure 1.7

Saturation length or Lsat (the distance taken for the load carried by the wind to adjust it passes from a hard, cohesive to a sandy surface).

Figure 1.8

Comparison of the performance of selected transport formulae for particles 0.25–0.40 mm (medium sand) in size.

Figure 1.9

Variation in the speed and direction of the wind in the field resulting in a very variable rate of transport, labelled ‘intermittent saltation’.

Figure 1.10

Trajectories of platy saltons.

Figure 2.1

Selection from the large range of ripple wavelengths and patterns.

Figure 2.2

Ripple formation.

Figure 3.1

Cross-section of an idealised dune, showing features explained in the text.

Figure 3.2

Proto-dunes driven by a cross-wind on the lee slope of a large transverse dune; lee side of a small artificial dune; bottle slide on the slip-face of a dune; and return flow in the lee of a dune visualised with smoke.

Figure 3.3

Trapping efficiency on dunes of different heights and at different wind speeds.

Figure 3.4

Dune bedding revealed by wind erosion in weakly cemented calcareous aeolianite in the south-eastern Wahiba Sands in Oman.

Figure 3.5

Downwind progress of free dunes.

Figure 3.6

Celerity versus the height of moving dunes.

Figure 3.7

Dune celerity versus wind speed at 10 m above ground on an upwind level surface.

Figure 4.1

Global wind systems.

Figure 4.2

Daily variations of wind speed in a sea-breeze system on the Gulf coast of Texas at Corpus Christi.

Figure 4.3

Classification of directional wind regimes.

Figure 4.4

Plot of the directionality of wind regimes against a generalised measure of the availability of sand for dune-building.

Figure 4.5

Transverse and linear dunes as modelled by a cellular automaton.

Figure 4.6

Cartoon of the three-dimensional pattern of transverse dunes, and associated terminology.

Figure 4.7

Some measures of pattern, illustrated by transverse dunes at 23°03′52″S; 14°32′09″E; 26 km, near Walvis Bay in Namibia.

Figure 4.8

Effect on the shape of barchans of varying a parameter in a model that relates to the grain size of the sand.

Figure 4.9

Network dunes.

Figure 4.10

Reversing dunes in the Kara Kum from Cherednichenko.

Figure 4.11

Wind-tunnel simulation of flow over a model of an actual star dune at ~40°01′28″N; 94°48′22″E; 4 km, in north-central China.

Figure 4.12

Bedding patterns revealed after rainfall on a seif dune in northern Sinai, with author for scale.

Figure 4.13

Cartoon of roll vortices, and their possible relationship to linear dunes.

Figure 4.14

Patterns of advance of large linear dunes.

Figure 4.15

Dune patterns on coversands near Nieuw Bergen in the Netherlands (51°37′N; 6°03′E; 8 km), revealed by laser altimetry.

Figure 5.1

Nomenclature for forced dunes.

Figure 5.2

Echo dune upwind of a large nebkha.

Figure 6.1

Guelph University wind tunnel ready for a study of the effect of one particular arrangement of rigid obstacles on the sharing of drag between the bed and the obstacles.

Figure 6.2

Interaction of winds with plants at increasing cover.

Figure 6.3

Two-parameter plot of outputs of the mathematical model of Yizhaq et al. (2007).

Figure 6.4

Small nebkhas with single lee dunes.

Figure 6.5

Output of a numerical model, which follows the development of a parabolic dune from a blowout.

Figure 7.1

Traces of now-stabilised transgressive dunes in the Landes of south-western France.

Figure 7.2

Magnitude–frequency effects on sand transport by wind on a beach section on Prince Edward Island in Canada.

Figure 7.3

Cartoon of a typical cross-section of coastal dunes, with nomenclature used in the text.

Figure 8.1

Australian sand seas (active; semi-active and stabilised), showing the dominance of sand ridges and the whorl and a half of the driving wind systems (the half whorl is in the West).

Figure 8.2

Stages in the development of the Gran Desierto Sand Sea in north-western Mexico.

Figure 8.3

Model of sand flow through a sand sea.

Figure 8.4

Elongated zones of thick sand in the Nebraska Sand Hills, presumably trailing downwind of a major source of sand.

Figure 8.5

Transfer of sand between sand seas in the Sahara.

Figure 9.1

Sources of dune sand in the Mohave Desert.

Figure 9.2

Relationship between the deposition of alluvium and the growth of dunes on the banks of Cooper Creek in central Australia, over a sequence of wet and dry phases in the last ~100,000 years.

Figure 9.3

Mineral maturity of the sand in the Nebraska Sand Hills compared with the maturity of possible sources.

Figure 9.4

Scanning electron microscope images of ‘aeolised’ sand grains.

Figure 10.1

Outcrop of the Early Permian Coconino Sandstone on the South Rim of the Grand Canyon.

Figure 10.2

Map showing Etjo and Botucatú Sandstones.

Figure 10.3

Climbing structures and slip-face bedding (or cross-strata) in aeolian sandstones.

Figure 10.4

Cross-section through the Etjo Sandstone in Namibia, showing a hierarchy of surfaces numbered in ascending order of their frequency.

Figure 10.5

Maximum global extent of ice sheets and sand deserts at the Last Glacial Maximum.

Figure 10.6

Palaeo-dunes on the High Plains of the USA.

Figure 10.7

European Sand Belt.

Figure 10.8

Stabilised Late Pleistocene parabolic dunes in Poland.

Figure 10.9

Stabilised and active sand seas in China.

Figure 10.10

Extent of stabilised dunes (in Arabic, ‘qoz’) and their patterns in the central Sudan.

Figure 10.11

Fixed and stabilised dunes in the southern Sahara and the West African Sahel, and localities mentioned in this and other chapters.

Figure 10.12

Patterns of palaeodunes in the Azefal/Agneitir/Akchar area on the Saharan boundary in western Mauritania.

Figure 11.1

Displacement of coastal dunes in north-eastern Brazil versus an index of climatic variability.

Figure 12.1

Threshold curves for particle movement on Mars, Earth and Venus.

Figure 12.2

North Polar sand sea on Mars.

Figure 12.3

Transverse Aeolian Ridges on Mars.

Figure 12.4

Selection of Martian dune patterns.

Figure 12.5

Linear dunes on Titan.

Figure 13.1

Potential origins of blowing sand in the vicinity of Tabuk in north-western Saudi Arabia.

Figure 14.1

Sea level, then, now and soon.

Series Editors’ Preface

The RGS-IBG Book Series only publishes work of the highest international standing. Its emphasis is on distinctive new developments in human and physical geography, although it is also open to contributions from cognate disciplines whose interests overlap with those of geographers. The Series places strong emphasis on theoretically-informed and empirically-strong texts. Reflecting the vibrant and diverse theoretical and empirical agendas that characterize the contemporary discipline, contributions are expected to inform, challenge and stimulate the reader. Overall, the RGS-IBG Book Series seeks to promote scholarly publications that leave an intellectual mark and change the way readers think about particular issues, methods or theories.

For details on how to submit a proposal please visit: www.rgsbookseries.com

Neil Coe National University of Singapore

Joanna Bullard Loughborough University, UKRGS-IBG Book Series Editors

Acknowledgements

I am deeply indebted to: Miles Irving at UCL, for his excellent draftsmanship, his attention to detail, his innovation and his forbearance of my many changes of mind; the Department of Geography at UCL, for allowing Miles to work with me and for allowing me access to the library at UCL; Joanna Bullard, whose eagle eye spotted many faults in the first-submitted draft of this book, and who made many valuable comments; my wife, for her constructive indifference; my family, who made a joke of the long gestation of this book; Ian Wilson, who thought big about dunes; my research students Adrian Chappell, Giles Wiggs and Hiroshi Momiji, whose work provided some of the most stimulating material reviewed here and from whose many publications I have extensively borrowed; John Stout in Big Spring, Texas, who has collated the vast and immensely useful Bibliography of Aeolian Research; and my erstwhile co-author, Ian Livingstone (our book was the groundwork for this one); Mary Bourke (Trinity College Dublin) for help with images of Mars; Paul Hesse (University of Wollongong), for the use of a layer of his GIS of Australian dunes; and Peter Bull at Oxford for the photomicrographs of grains of sand; Jack Gillies (Desert Research Institute, Nevada) for his help in getting me a high-enough resolution version of Figure 6.1.

Andrew WarrenLondon, January 2013

Introduction

This book is a search for explanations of the form, dynamics, size, pattern and history of windblown dunes. Any such search, even today, must begin with Bagnold’s (1941) classic, The Physics of Blown Sand and Desert Dunes, a work that has been cited in almost every subsequent publication on dunes.

Beyond registering 34 more citations, and thus paying at least as many ­compliments, my purpose is to make two complements to Blown Sand. The first is a review of the research that it has inspired. The second is to add accounts of dunes formed round living plants, on coasts, in the past and in managed ­environments. Despite their absence from Blown Sand, the explanation of many of these other dunes is almost as dependent on its insights as the dunes it did cover. I have not gone to the extent of including dune-like features on the beds of rivers, estuaries and seas, even: when they share many characteristics of their shape and dynamics; when the word ‘dune’ is now attached to them in the ­scientific literature; and when Blown Sand has lessons for them (as Bagnold later confirmed). I justify this decision, first by the exclusion of these other dunes from the common understanding of the word ‘dune’; second, by noting that the ­contrast in the density of wind and water is responsible for a significant difference between their dunes, such that subaqueous dunes in the same size-range as ­windblown dunes are the dynamic equivalent to windblown mega-dunes, while subaqueous ripples are the dynamic equivalent of windblown dunes (Chapter 3); third, by the small number of places where I have had to invoke subaqueous dunes; and fourth, and finally, most realistically, because the inclusion of ­subaqueous dunes would have significantly enlarged this book.

The surge of publications on windblown dunes that began in the 1950s (Stout et al. 2009) has been maintained in the last two decades by an expansion in the national origins of the research, most evident in the near-dominance of the ­literature in some years by authors in China, India and South America. The surge has also been driven by an equally healthy disciplinary expansion. The range of skills now focused onto dunes includes: mathematical modelling of many kinds; vastly improved instrumentation for the measurement of sand movement in the field and in the wind tunnel; steadily more sophisticated analysis of increasingly better resolved remotely sensed images; dating techniques (especially optically stimulated luminescence, Chapter 10); stratigraphy; pedology; and others. Few can claim to have mastered all these fields, and I have struggled with some, but not to the extent, I hope, that I have not given them their correct weight. My account will show that there are some very active areas of research, such as the dating of palaeodunes, the measurement of flow over dunes, the modelling of dune profiles and patterns, and models of linear dune formation, where my judgements will quickly and surely be overtaken.

The organisation of this book follows the seminal system of the late Stanley Schumm and his colleague, Robert Lichty, which linked the size, age and ­complexity of landforms (Schumm and Lichty, 1965, RL; references in the text marked ‘RL’ are listed in the References at the end of each chapter). With one significant exception, their insight applies very well to dunes. As the size of a dune (ripple or group of dunes) expands, the longer it takes to develop, the more ­complex must be its explanation (as more processes are involved), and its ­primary explanatory variable may change. The study of windblown sand even adds ­dimensional expansion to the Schumm-and-Lichty framework. Thus, Part I of this book is about the smallest relevant space/timescale (<10 m2; <10 years), namely the lifting and carrying of sand by the wind (Chapter 1), ripples (Chapter 2), and the cross-sectional form of a single transverse dune (Chapter 3), in all of which the wind comes from one direction only, and where there is an unlimited stock of sand, in other words a two-dimensional frame, in which the critical ­variables are more or less confined to only wind speed, grain size and moisture content. Part II (1000–10,000 m2; 100–1000 years) must bring in the third spatial dimension if it is to make sense of dune patterns (Chapters 4–7). In this larger frame, two new variables become critical, namely variability in the direction of the wind and the availability of sand. For this change in spatial scale, there must be, following Schumm and Lichty, an expansion in the temporal scale. Part III contains the significant exception to the Schumm-and-Lichty scheme: the space scale must start at 0.3 mm (the approximate size of a grain of dune sand) to allow explanations of the shape and surface texture of single grains as they have evolved over thousands of years of being blown about, while the timescale expands to 2200 ­million years, that being the likely time since the appearance on Earth of the primordial dune. This space-time frame allows explanations of sand seas (Chapter 8); the history of blown sand (Chapter 9); windblown sandstones and continental palaeodunes (Chapter 10); palaeocoastal dunes (Chapter 11); and, finally, dunes on Mars, Venus and Titan (Chapter 12). Part IV (Care) returns to a neater ­space-time frame: first, Local Care in the Short Term (Chapter 13; <1000 m2; <10 years); and second in the long term, in which the aim is Sustainability (Chapter 14; >100,000 m2; >10 years).

The coverage of all this material could be at a number also of explanatory scales, ranging from mere stimulation, to lengthy, rigorous examination of models and the evidence by which they are to be judged. I have taken the broad middle road. It is the path of a geomorphologist, with field experience of aeolian processes, aeolian morphologies and the history of windblown landforms, and delivers a book of manageable size. The target audience is geomorphologists, perhaps not those at school or in the early years at university, but those in the later years and beyond, and for those interested in the geomorphology of related landforms, and of other disciplines, such as ecology, archaeology, geology and engineering, in which dunes play a role.

I have taken the opportunity that Google Earth offers to cite the latitudes and longitudes and ‘eye altitudes’ (in metres or kilometres) of scenes that help the argument. However, a caution about some of the scenes at low eye altitudes: over the years of writing this book, I have noticed that Google Earth frequently updates its images, so that, in some cases, the message I intended to be taken from an image has been lost, as where dunes show unmistakable evidence of a change in wind direction, or where there have been significant changes in the profiles of sand-yielding beaches. Google Mars also helps in understanding Chapter 12. Another website that takes much of the burden of explanation is Wikipedia, where explanations can be found of many of the terms and concepts not fully explained in the text.

Most of the references in the text are to the most recent paper to cover the topic in question (not necessarily the pioneer publication). In most cases, there are many predecessors, most of which are listed in the Bibliography of Aeolian Research (BAR) (http://www.csrl.ars.usda.gov/wewc/biblio/bar.htm). The great majority of references in the book are from the BAR, and therefore need no ­listing here. The cross-references ‘earlier’ and ‘later’ in a chapter of what follows refer to other parts of the same chapter. References in the text to the BAR conform to the following system: the letters ‘a’, ‘b’, ‘c’, etc. after the date of a reference refer to the order in which the same family name is listed in the BAR, not necessarily to the author in the reference (for example, a citation of ‘Smith, 2000c’ refers to the third Smith in the BAR list, even if the other Smiths are not cited in this book). Chinese names, where possible, are in the format ‘Xi Jinpeng’, and are listed under the family name (e.g. ‘Xi’) first. The BAR lists single-author papers first, then two-author papers, then papers by three authors or more.

Reference

Schumm, S.A. and Lichty, R.W. (1965) ‘Time space and causality in geomorphology’, American Journal of Science 263 (2): 110–119.

Part One

<10 m2; <10 years

Chapter One

Wind and Sand

Part I of this book (in which this is the first chapter) has a two-dimensional spatial frame. The principal consequence is that the wind is assumed to come from one direction only.

The chapter is about the lifting and carrying of sand by the wind. As to winds at this scale, only their velocity and small-scale patterns of turbulence are ­relevant. As to sand, the primary interest here is in its grain size (shape in a secondary concern). In the most widely accepted taxonomy for the size of sedimentary particles, grains of ‘sand’ have diameters of between 0.625 mm and 2 mm (Wentworth, 1922, RL). A global survey, by its own admission limited, but probably representative, found that most dune sands were in Wentworth’s ‘fine sand’ category (0.10–0.40 mm) (Ahlbrandt, 1979). A quick scan of recent papers confirms this. Another assumption here is that sand is composed only of quartz. Dunes built of smaller and coarser particles, and of other minerals, are described in Chapter 6. Because changes in the shape and surface texture of windblown sand grains have long histories, they are issues for Chapter 4.

Wind versus Bed

The mechanical energy spent when two bodies (such as the wind and the surface) slide past each other is termed ‘shear’. Shear on a surface over which a wind passes is denoted by τ0.

The Law of the Wall

Because, until very recently, the shear of the wind on a surface could not be ­measured, it had to be approached through theory. The first model of the relationship grew out of the work of Ludwig Prandtl and Theodore von Kármán, working first at Göttingen, and came to be known as the ‘Law of the Wall’, where the word ‘wall’ was chosen with wind tunnels in mind. For dunes, ‘bed’ (as in the ‘bed of a stream’) is more appropriate than ‘wall’ and is adopted here. Despite some serious revision (shortly), the Law is still a good introduction to how the wind shears a surface.

The Law is built on two observations (or, as will be seen, simplifying ­assumptions):

The velocity of the wind increases upward from the bed, because friction on the bed retards the wind, and this retardation is transferred, with weakening effect, to the wind at greater heights.

Figure 1.1a

shows the velocity of a wind at successive heights above the bed of a wind tunnel, both on arithmetic scales. In

Figure 1.1b

, the data and the velocity scale are the same as in

Figure 1.1a

, but the height scale is now logarithmic. The Law declares that on a semi-­logarithmic plot, like this, the data fall on a straight line, as they do in this and many other observations in wind tunnels. The slope of the straight line in

Figure 1.1b

is a function of the strength of the wind that it represents: steep slopes represent gentle winds (low velocities even at some height); gentler slopes represent faster, more powerful winds (high velocities near the bed). This relationship comes into the argument again shortly.

Figure 1.1b

also shows that the straight lines depicting winds of different strength reach the same focal point on the vertical (height) axis. Those who built the Law took this to imply that there was a very thin layer of air, just above the bed, at the same height for all winds, where the air was stationary or moving very slowly. The focal point is higher on rougher beds, which is why it is termed the ‘roughness height’ or ‘roughness length’ (shorthand, ‘

z

0

’). Because few measuring instruments, even today, can penetrate this layer,

z

0

has to be estimated. A common estimate is ~1 mm over a smooth, stable sandy surface on Earth. A newer formulation of

z

0

is discussed very shortly.

In order for the Law to apply to fluids of differing density, a parameter, shear velocity (or friction velocity), or u*, was introduced:

where τ0 is the shear force on the bed, and ρa is the density of the fluid; this is the ‘Prandtl–Kármán equation’.

Because the dimensions of u* are those of velocity (m s–1), it is termed the ‘shear velocity’. The equation is applicable to thin air on the Tibetan plateau, or on Mars; to dense atmospheres at very low temperatures, as near the Poles on Earth, or on Venus; or to any denser fluid, including water. The shorthand, ‘u*’, is used in many places in this and later chapters.

The next step in the building of the Law was to relate u* to measurements of the slope of the velocity/log height plot (Figure 1.1b). This step built on the observation (earlier) that the slopes of the lines are related to the strength of the wind. More precisely,

Figure 1.1 Velocity profile of the wind over a smooth, flat bed in a wind tunnel. In (a), both axes are arithmetic; in (b), the velocity (on the abscissa) is still arithmetic, but the height scale (on the ordinate) is now logarithmic. The data are Bagnold’s (1941, pp. 48–49). The notations are explained in the text. Plots like this are known as ‘Prandtl curves’. Redrawn after Bagnold (1941).

where uz is wind velocity at height z, and κ is Kármán’s constant.

There are various estimates of κ, from theory and experiment; most are ~0.40.

Improving the wind/bed model

For all its simplicity and clarity, the Law is now only the first step in understanding wind shear on a solid surface. Three of its essential components have been found to be oversimplifications.

The velocity profile

Many plots of the velocity of flow against height over a bed do not fit the logarithmic profile. This is shown by attempts to fit straight lines to measurements of the wind at different heights above a beach, which produced a wide range of values of z0 (Bauer et al., 1992). Similar problems have been found with measurements in a wind tunnel (Butterfield, 1999a). The implication is that the way in which u*andz0 in the Law were derived was somewhat arbitrary.

There are many reasons for divergence from the semi-logarithmic curve. Three apply to common situations on dunes.

When sand is in motion, there is a change in the slope in the velocity/height profile near the top of the cloud of bouncing sand. This point has been dubbed ‘Bagnold’s kink’, after its discoverer (Bagnold, 1941, p. 59) (McEwan, 1993).

Above a heated surface, which is the norm in deserts during the day, the air is unstable, and the wind-velocity profile, below about 0.5 m from the bed, is not semi-logarithmic. Errors of up to 15 times the value of

u

*

may occur if the stability condition of the atmosphere is not allowed for (Frank and Kocurek, 1994). The reason for this kind of deviation in the wind-velocity profile is that heat then joins shear as a driver of vertical mixing, and ensures that there is less change in velocity with height than in neutral conditions. The sand-flow rate in these conditions reaches equilibrium more quickly than in neutral ­conditions (Lu Ping and Dong ZhiBao, 2008). In very stable conditions, as during cold nights in some deserts and, more distinctly, in extreme cold at high latitudes or altitudes, the profile also deviates from the logarithmic. Mosaics of surfaces with different responses to heating, also the norm in deserts, create even more complex wind-velocity profiles as the wind repeatedly passes over surfaces of different roughness (Butler

et al

., 2005).

The height distribution of velocity above a sloping or curved bed is less easy to accommodate in estimates of shear. This is an issue in the explanation for flow over dunes (Chapter 3).

There are now many alternatives to the semi-logarithmic model of the relation between wind velocity and height, but most are useful only in wind tunnels, and few have been used by geomorphologists (Bauer et al., 2004).

The roughness height (z0)

This was the second element in the Law to be questioned. As explained earlier, the definition of z0 in the Law is as arbitrary as that of u*.

The most commonly used estimate of z0 in a cloud of bouncing sand is now Owen’s model (1964):

where g, as ever, is the gravitational constant.

Microturbulence

Turbulence is one of the main ways in which energy is transferred from the wind to the bed. In the scale limits of this chapter, it is only small-scale turbulence that is relevant. At this scale, turbulence is structured into ‘burst–sweep’ sequences. A burst is a downward spurt of air that replaces the air that has just been removed by a sweep, which is a slower upward ejection from the bed. On a loose sandy bed, the leading edge of a burst may dislodge sand, which is then taken up by a sweep. Burst–sweep sequences are responsible for most entrainment, even when, as is probably common, they are effective for only about 20% of the time (Sterk et al., 1998). Measurements on a sandy, eroding field in Burkina Faso revealed that the burst–sweep sequence (at that site and on that occasion) had a downwind dimension of 0.25 m (Leenders et al., 2005).

Turbulence at this scale can be measured by the Reynolds stresses, which describe the variation of velocity in three dimensions. Velocity in the windward direction is denoted ‘u’; in the vertical (up or down) ‘w’; and in the lateral (sideways in either direction) ‘v’. u is positive downwind; w is positive upward; v would take the discussion beyond the two-dimensional frame of this chapter; its role in determining the two-dimension form of dunes has anyway hardly been explored. In a burst (towards the bed), u′ is positive (wind-directional flow faster than the mean), and w′ is negative (flow more downward than the mean). In a sweep (movement away from the bed), u′ is negative (wind-directional flow in the sweep being slower than the mean), and w′ is positive (upward flow at a faster rate than the mean). The prime symbol (′) denotes a fluctuating variable.

The best measure of shear on the bed

This is the most important issue raised by all these reservations about the Law. Shear velocity (u*) is, by definition, a description of the mean wind, which is seen in the relationship between u* and the transport rate at progressively smaller averaging intervals (Namikas et al., 2003). Thus, any study of entrainment at a small scale must acknowledge burst–sweep sequences. One alternative for measuring turbulent flow is the ‘Reynolds Shear Stress’, which combines stresses in the ­forward and vertical dimensions: (the overbar denotes the mean). When multiplied by the air density (ρ) this gives a force, .

The value of u* has also been questioned even at larger scales. The solution could be as simple as deriving u* from a wind profile measured down to about 0.05 m of the bed (Bauer et al., 2004), or measuring the free-stream velocity ‘well above’ the bed (Schönfeldt and von Löwis, 2003), both of which solutions are empirical rather than theoretical. Developments in measuring sand transport, including shear or force balances, may deliver more theoretically acceptable measures of shear (Gillies et al., 2000).

Lift-Off

A particle of sand starts to move (is ‘entrained’) when the forces that hold it down are exceeded by those that might rip it away.

Holding down by gravity

The gravitational force is defined thus:

where g is the acceleration owing to gravity; ρp is particle density; ρa is the density of the air (or other fluid); and d is the particle diameter.

In other words, where the densities (ρp and ρa) are constant (when all the sand is of the same mineral, say quartz, and the air density does not change, as in many situations), particles of greater size (d), are held down by a greater gravitational force. The model can accommodate the behaviour of sands that are denser than quartz (say, of magnetite) or less dense (say, of diatomite), and of fluids with different density. Chapter 12 includes a discussion of the effects of the differences in gravity, temperature and air density on the lifting and carrying of sand by the wind on Mars, Venus and Titan.

Holding down by cohesion

Cohesion derives from several of the characteristics of particles. First, finer ­particles pack more closely, which means that they touch each other in more places and are thus more coherent. Second, rough particles touch each other in many more places than do smooth ones and so also cohere better. Third, platy shapes, as of many fine particles (particularly clays) allow much more contact than do rounded shapes, if packing is parallel (as it usually is for clays). Fourth, physicochemical bonds, known as London–van der Waals forces, increase ­cohesion between clay particles of some mineralogies (many clays) but are weaker between particles of some common rock minerals, such as quartz (Cornelis and Gabriels, 2004).

The fifth and sixth forms of cohesion come from water held between particles. The fifth is the meniscus force, which is strongest where a meniscus has a small angle of contact with a particle. This is the case where the voids between particles are small, as they are between fine particles. The strength of this force also depends on the roughness, roundness and surface properties of the particles, and on contaminants in the water.

None of the cohesive forces is as strong as the sixth form of cohesion. It depends on water held (adsorbed) on the surfaces of particles. The amount of water held in this way increases with relative atmospheric humidity, but, contrary to intuition, the static threshold (shortly) peaks at a relative humidity of 35–40%, below and above which value, entrainment is easier (Ravi and D’Orico, 2005). All of these properties are difficult to measure, and their combined effect is a major challenge to modellers and experimentalists (McKenna-Neuman and Sanderson, 2008). The relation between moisture and movement is discussed again later in the section on the dynamic behaviour of moisture in an eroding bed.

In sum, fine particles are more coherent than coarse, other things being equal, which they often are. Cohesion, of all sorts, is a function of (1/d)3, where d is the diameter of the particles. This is the start of an explanation for why dunes are sandy: fine particles cohere too well to be easily moved by the wind (with some exceptions, later).

Raising by lift

Shear moves particles on a loose bed by two kinds of ‘aerodynamic entrainment’. The first, lift, occurs because fast flow is accompanied by low pressure, following Bernoulli’s equation. Flow over a bed of particles is faster than the velocity of the surrounding fluid in two situations: first, where there is a difference in pressure between the slow flow very near the bed, and the faster flow just above it, the slope of the velocity/height curve being steepest there; and second, where the wind is accelerated over a protruding particle.

Lift is more effective on rough than on smooth beds, on moving than static particles, and in sudden changes in velocity, as caused by turbulence. In some circumstances, lift may also be augmented by thermal diffusion from a heated surface, or by electrostatic forces arising from friction between the wind and the sediment (Rasmussen et al., 2009). Lift alone entrains few particles, but it lightens the task of the other forces.

Raising by drag

Drag is usually stronger than lift. ‘Surface drag’ is caused by friction between the wind and the bed. It causes both rolling and sliding. ‘Form drag’ is caused by the difference in pressure between the windward and lee sides of a protruding particle, especially in high turbulence. Because it is greatest on top of the particle, it causes rolling. Both contribute to ‘aerodynamic moment’, which is the force on a particle that is dependent on its projection above the surface: particles that project more (bigger or longer ones) are toppled and therefore entrained more easily. Drag can help to eject particles when they collide or are dragged over projections. The magnitude of both forms of drag is proportional to u*2d2 (d being the particle diameter). Drag, like lift, is effective only very close to the bed, and raises few grains on its own. When the particles are clear of the bed, many acquire spin, which may contribute up to 24% of lift (the Magnus effect) (Huang Ning et al., 2010).

Raising by bombardment

When particles are lifted into the wind, they pick up momentum from the wind in their trajectory above the surface and take some back to the bed on their return. This is ‘bombardment’, and, when sand is in movement, it is more powerful than any of the other sand-raising mechanisms. It both dislodges loose grains and breaks up aggregates (pellets) and crusts (both later). Once sand is lifted in sufficient amounts, further entrainment is almost wholly by bombardment.

Thresholds

This section adds to, but does not yet complete, the explanation for why dunes are built of sand. The wind can lift grains if it has sufficient power, which is to say (following the line of reasoning earlier), it is fast enough. The critical condition, when sand begins to blow, is the threshold velocity (ut), or, more generally, the threshold shear velocity (u*t).

In Bagnold’s (1941, pp. 85–90) terms, the static (or fluid) threshold is the wind velocity at which grains start to move under the influence of lift and drag alone; and the dynamic (or impact) threshold is passed when particles are bombarding the bed. The dynamic threshold occurs at a lower velocity than the static threshold, because of bombardment. A much later model shows that in most places on Earth, the velocity at the dynamic threshold is ~0.96 (the ratio is different on Mars, Venus and Titan, as described in Chapter 12; Almeida et al., 2008a).

Bagnold (1941, pp. 85–90) built the first mathematical model of the static threshold, which described the balance between the lifting and retaining forces on a particle. Subsequent theorising was reviewed by Cornelis and colleagues (2004b; Figure 1.2), who developed a model of their own, which is simpler and more testable than some earlier versions.

Figure 1.2 Threshold curve for the start of motion (the static threshold) of particles of the density of quartz (observed and modelled data) (Cornelis and Gabriels, 2004). The curve for the dynamic threshold is added and very approximate. Reprinted with permission from John Wiley & Sons.

Thresholds have been found to be much more complicated than this, in theory, in wind tunnels, and in the field. Even in dry sand (moist sand is discussed shortly), surface conditions, such as roughness, the grain-size mix, and other factors, each produce their own thresholds, and these may vary in time and space (Baas, 2007). The following list, therefore, is far from exhaustive. Different thresholds occur when: (1) saltation reaches the intensity at which it can move sand in reptation (shortly); (2) bouncing grains are powerful enough to disperse clods, pellets, and crusts, of varying cohesiveness (Hu ChunXiang et al., 2002) (also shortly); (3) ripples appear (Chapter 2); and (4) ripples move from a ­‘subcritical’ shape (with gentle lee slopes), to a ‘supercritical’ shape, in which there are small slip faces (Hoyle and Woods, 1997).

Even within one of these groups of threshold, there is a range of behaviour. In a wind tunnel, the static threshold occurs at a spread of velocities, from that at which particles begin to rock back and forth; start rolling; or are lifted from the bed; or, at a larger scale, between the point at which there are only a few flurries of movement in response to bursting turbulence and the point at which the whole bed is mobile. If the wind is accelerated in the wake of even quite small roughness elements, local entrainment occurs at velocities lower than the ambient threshold (Sutton and McKenna Neuman, 2008). The difference between early, sporadic movement and the movement of the whole bed produces static thresholds with a wide spectrum of values. It would be better to choose a probability density of values, although this is seldom done (Williams et al., 1994).

In the field, thresholds are yet more elusive. Measurement is more complex (both of the wind and of blowing sand): winds are gustier; sediments have more sizes and densities; controls like moisture or crusting can limit the supply of loose particles (later); there is far greater spatial variability in all these controls; and features like sand streamers (shortly) complicate measurements of sediment movement. In one field experiment, sand was seen to move at a velocity below a theoretical threshold (calculated from grain size and wind speed), almost certainly because of high instantaneous wind velocities (Rasmussen and Sørensen, 1999). In another experiment, now on a wet beach, it was found that each size and size-mixture of sand, and each set of environmental variables, had its own threshold (Wiggs et al., 2004a). Many of the problems involved in measuring thresholds in the field may be overcome by using terrestrial laser scanning, which can quickly and non-invasively measure surface topography, moisture content of the surface, and perhaps even the sizes of grains in saltation (shortly; Nield and Wiggs, 2011). A recent study, based on measurements in the field, discovered yet another cause of variability: the way in which the threshold is calculated from data (Barchyn and Hugenholtz, 2011).

Grain size

This section continues, but does not, even yet, complete, the explanation for why most dunes are sandy. The relationship between the static threshold and the grain size of sand is shown by a simple experiment: trays, containing grains of a succession of sizes of particle, each with only one size, are exposed, one by one, to increasing wind speeds in a wind tunnel, and the threshold velocity (ut) or threshold shear velocity (u*t) at which each size of particle begins to move is plotted against its size. The results of such an experiment are shown in Figure 1.2.

The most important (and most obvious) characteristic of the curve in Figure 1.2