FLUVIAL HYDRAULICS

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FLUVIAL HYDRAULICS

S. Lawrence Dingman

2009

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Library of Congress Cataloging-in-Publication Data

Dingman, S.L.Fluvial hydraulics / S. Lawrence Dingmanp.cmIncludes bibliographical references and indexISBN 978-0-19-517286-71. Streamow. 2. Fluid mechanics I. TitleGB1207.D56 2008551.48'3dc22 2008046767

Quotation on p. ix from A Man and His Dog by Thomas Mann, in Death in Venice and SevenOther Stories by Thomas Mann (trans. H.T. Lowe-Porter), a Vintage Book 1930, 1931, 1936 byAlfred A. Knopf, a division of Random House, Inc. Used by permission of Alfred A. Knopf.

9 8 7 6 5 4 3 2 1

Printed in the United States of Americaon acid-free paper

Preface

The overall goal of this book is to develop a sound qualitative and quantitativeunderstanding of the physics of natural river ows for practitioners and students withbackgrounds in the earth sciences and natural resources who are primarily interestedin understanding uvial geomorphology. The treatment assumes an understanding ofbasic calculus and university-level physics.

Civil engineers typically learn about rivers in a course called Open-Channel Flow.There are many excellent books on open-channel ow for engineers [most notablythe classic texts by Chow (1959) and Henderson (1961), and more recent worksby French (1985) and Julien (2002)]. These courses and texts assume a foundationin uid mechanics and differential equations, devote considerable attention to theaspects of ow involved in the design of structures, and generally provide onlylimited discussion of the geomorphic and other more holistic aspects of naturalstreams. By contrast, the usual curricula for earth, environmental, and natural resourcesciences do not provide a thorough systematic introduction to the mechanics ofriver ows, despite its importance as a basis for understanding hydrologic processes,geomorphology, erosion, sediment transport and deposition, water supply and quality,habitat management, and ood hazards.

I believe that it is possible to build a sound understanding of uvial hydraulicson the typical rst-year foundation of calculus and calculus-based physics, and myhope is that this text will bridge the gap between these two approaches. It differs fromtypical engineering treatments of open-channel ow in its greater emphasis on naturalstreams and reduced treatments of hydraulic structures, and from most earth-science-oriented texts in its systematic development of the basic physics of river ows andits greater emphasis on quantitative analysis.

My rst attempt to address this need was Fluvial Hydrology, published in 1984 byW.H. Freeman and Company. Although that book has been out of print for some time,comments from colleagues and students over the years made it clear that the need wasreal and that Fluvial Hydrology was useful in addressing it, and I continued to teacha course based on that text. Student and colleague interest, the publication of newdatabases, a number of theoretical and observational advances in the eld, a growinginterest in estimating discharge by remote sensing, the ready availability of powerfulstatistical-analysis tools, and my own growing discomfort with the Manning equationas the basic constitutive equation for open-channel ow, all led to a resurgence ofmy interest in river hydraulics (Dingman 1989, 2007a, 2007b; Dingman and Sharma1997; Bjerklie et al. 2003, 2005b) and thoughts of revisiting the subject in a newtextbook.

Although my goal remains the same, the present work is far more than a revisionof Fluvial Hydrology. The guiding principles of this new approach are 1) a deeperfoundation in basic uid mechanics and 2) a broader treatment of the characteristics of

vi PREFACE

natural rivers, including extensive use of data on natural river ows. The text itself hasbeen drastically altered, and little of the original remains. However, I have tried tomaintain, and enhance, the emphasis on the development of physical intuitionasense of the relative magnitudes of properties, forces, and other quantities andrelationships that are signicant in a specic situationand to emphasize patternsand connections.

The main features of this new approach include a more systematic review of thehistorical development of uvial hydraulics (chapter 1); an extensive review of themorphology and hydrology of rivers (chapter 2); an expanded discussion of waterproperties, including turbulence (chapter 3); a more systematic development of uidmechanics and the bases of equations used to describe river ows, including statisticaland dimensional analysis (chapter 4); more complete treatment of velocity proles anddistributions, including alternatives to the Prandtl-von Krmn law (chapter 5); a moretheoretically based treatment of ow resistance that provides new insights to thatcentral topic (chapter 6); the use of published databases to quantitatively characterizeactual magnitudes of forces and energies in natural river ows (chapters 7 and 8);more detailed treatment of rapidly varied ow transitions (chapter 10); a more detailedtreatment of waves and an introduction to streamow routing (chapter 11); anda more theoretically based and modern approach to sediment transport (chapter 12).Only the treatment of gradually varied ows (chapter 9) remains largely unchangedfrom Fluvial Hydrology. A basic understanding of dimensions, units, and numericalprecision is still an essential, but often neglected, part of education in the physicalsciences; the treatment of this, which began the former text, has been revised andmoved to an appendix. The number of references cited has been greatly expandedas well as updated and now includes more than 250 items. A diligent attempt hasbeen made to enhance understanding by regularizing the mathematical symbols andassuring that they are dened where used. I have used the center dot symbol formultiplication throughout so that multiletter symbols and functional notation can beread without ambiguity.

A course based on this text will be appropriate for upper level undergraduatesand beginning graduate students in earth sciences and natural resources curriculumsand will likely be taught by an instructor with an active interest in the eld. Underthese conditions, instructors will want to engage students in exploration of questionsthat arise and in discussion of papers from the literature, and to involve them inlaboratory and/or eld experiences. Therefore, I have not included exercises, butinstead provide through the books website (http://www.oup.com/uvialhydraulics)an extensive database of ow measurements, a Synthetic Channel spreadsheet thatcan be used to explore the general nature of important hydraulic relations and the waysin which these relations change with channel characteristics, a simple spreadsheetfor water-surface prole computations, links to other uvial hydraulics and uvialgeomorphological websites that are available through the Internet, and a place forinstructors and students to exchange ideas and questions.

I thank David Severn and Rachel Cogan of the Dimond Library at the University ofNew Hampshire (UNH) and Connie Mutel of the Iowa Institute of Hydraulic Researchat the University of Iowa for assistance with references, permissions, and historicalinformation. Data on world rivers were generously provided by Balazs Fekete of

PREFACE vii

UNHs Institute for the Study of Earth, Oceans, and Space. Cross-section surveydata for New Zealand streams were provided by D.M. Hicks, New Zealand NationalInstitute of Water and Atmospheric Research. I heartily thank Emily Faivre, JohnStamm, David Bjerklie, Rob Ferguson, and Carl Bolster for reviews of variousportions of the text at various stages in its development. Their comments wereextremely helpful, but I of course am solely responsible for any errors and lackof clarity that remain.

This work would not have been possible without the encouragement and supportof my parents in pursuing my undergraduate and graduate education; of the teacherswho most inspired and educated me: John P. Miller at Harvard, Donald R.F. Harlemanof the Massachusetts Institute of Technology, and Richard E. Stoiber at Dartmouth;and of Francis R. Hall and Gordon L. Byers, founders of UNHs Hydrology Program.I owe special thanks to my student Dave Bjerklie, now of the U.S. Geological Surveyin Hartford, Connecticut, whose response to my initial research on the statisticalanalysis of resistance relations and subsequent discussions and research have been amajor impetus for my continuing interest in uvial hydraulics.

The love, support, and guidance of my wife, Jane Van Zandt Dingman, havesustained me in this work as in every aspect of my life.

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Contents

1. Introduction to Fluvial Hydraulics 3

2. Natural Streams: Morphology, Materials, and Flows 20

3. Structure and Properties of Water 94

4. Basic Concepts and Equations 137

5. Velocity Distribution 175

6. Uniform Flow and Flow Resistance 211

7. Forces and Flow Classication 269

8. Energy and Momentum Principles 295

9. Gradually Varied Flow and Water-Surface Proles 323

10. Rapidly Varied Steady Flow 347

11. Unsteady Flow 400

12. Sediment Entrainment and Transport 451

Appendices 514

A. Dimensions, Units, and Numerical Precision 514B. Description of Flow Database Spreadsheet 526C. Description of Synthetic Channel Spreadsheet 527D. Description of Water-Surface Prole Computation

Spreadsheet 530

Notes 531

References 536

Index 549

ix

I am very fond of brooks, as indeed of all water, from the ocean to the smallest weedypool. If in the mountains in the summertime my ear but catch the sound of plashing andprattling from afar, I always go to seek out the source of the liquid sounds, a long way ifI must; to make the acquaintance and to look in the face of that conversable child of thehills, where he hides. Beautiful are the torrents that come tumbling with mild thunderingsdown between evergreens and over stony terraces; that form rocky bathing-pools and thendissolve in white foam to fall perpendicularly to the next level. But I have pleasure in thebrooks of the atland too, whether they be so shallow as hardly to cover the slippery,silver-gleaming pebbles in their bed, or as deep as small rivers between overhanging,guardian willow trees, their current owing swift and strong in the centre, still and gentlyat the edge. Who would not choose to follow the sound of running waters? Its attractionfor the normal man is of a natural, sympathetic sort. For man is waters child, nine-tenthsof our body consists of it, and at a certain stage the foetus possesses gills. For my partI freely admit that the sight of water in whatever form or shape is my most lively andimmediate kind of natural enjoyment; yes, I would even say that only in contemplation ofit do I achieve true self-forgetfulness and feel my own limited individuality merge into theuniversal. The sea, still-brooding or coming in on crashing billows, can put me in a stateof such profound organic dreaminess, such remoteness from myself, that I am lost to time.Boredom is unknown, hours pass like minutes, in the unity of that companionship. Butthen, I can lean on the rail of a little bridge over a brook and contemplate its currents, itswhirlpools, and its steady ow for as long as you like; with no sense or fear of that otherowing within and about me, that swift gliding away of time. Such love of water andunderstanding of it make me value the circumstance that the narrow strip of ground whereI dwell is enclosed on both sides by water.

Thomas Mann

FLUVIAL HYDRAULICS

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1Introduction to FluvialHydraulics

1.1 Rivers in the Global Context

Although rivers contain only 0.0002% of the water on earth (table 1.1), it is hard tooverstate their importance to the functioning of the earths natural physical, chemical,and biological systems or to the establishment and nutritional, economic, and spiritualsustenance of human societies.

1.1.1 Natural Cycles

The water owing in rivers is the residual of two climatically determined processes,precipitation and evapotranspiration,1 and the general water-balance equation for aregion can be written as

Q = P ET, (1.1)where Q is temporally averaged river ow (river discharge) from the region, P isspatially and temporally averaged precipitation, and ET is spatially and temporallyaveraged evapotranspiration.2 The dimensions of the terms of equation 1.1 may bevolume per unit time [L3 T1] or volume per unit time per unit area [LT1]. (Seeappendix A for a review of dimensions and units.)

At the largest scale, the time-integrated global hydrological cycle can be depicted asin gure 1.1. The worlds oceans receive about 458,000 km3/year in precipitation and

3

4 FLUVIAL HYDRAULICS

Table 1.1 Volume of water in compartments of the global hydrologic cycle.

Area covered Volume Percentage of Percentage ofCompartment (1,000 km2) (km3) total water freshwater

Oceans 361,300 1,338,000,000 96.5 Groundwater 134,800 23,400,000 1.7

Fresh 10,530,000 0.76 30.1Soil water 16,500 0.001 0.05

Glaciers andpermanent snow

16,227 24,064,000 1.74 68.7

Antarctica 13,980 21,600,000 1.56 61.7Greenland 1,802 2,340,000 0.17 6.68Arctic Islands 226 83,500 0.006 0.24Mountains 224 40,600 0.003 0.12

Permafrost 21,000 300,000 0.022 0.86Lakes 2,059 176,400 0.013

Fresh 1,236 91,000 0.007 0.26Saline 822 85,400 0.006

Marshes 2,683 11,470 0.0008 0.03Rivers 148,800 2,120 0.0002 0.006Biomass 510,000 1,120 0.0001 0.003Atmosphere 510,000 12,900 0.001 0.04

Total water 510,000 1,385,984,000 100 Total freshwater 148,800 35,029,000 2.53 100

The global cycle is diagrammed in gure 1.1. From Shiklomanov (1993), with permission of Oxford University Press.

lose 505,000 km3/year in evaporation, while the continents receive 119,000 km3/yearin precipitation and lose 72,000 km3/year via evapotranspiration.

The water owing in riversriver dischargeis the link that balances theglobal cycle, returning about 47,000 km3/year from the continents to theoceans.

Table 1.2 lists the worlds largest rivers in terms of discharge. Note that theAmazon River contributes more than one-eighth of the total discharge to the worldsoceans!

River discharge is also a major link in the global geological cycle, delivering some13.5 109 T/year of particulate material and 3.9 109 T/year of dissolved materialfrom the continents to the oceans (Walling and Webb 1987). Thus, Rivers are boththe means and the routes by which the products of continental weathering are carriedto the oceans of the world (Leopold 1994, p. 2). A portion of the dissolved materialconstitutes the major source of nutrients for the oceanic food web.

River discharge plays a critical role in regulating global climate. Its effects onsea-surface temperatures and salinities, particularly in the North Atlantic Ocean,drive the global thermohaline circulation that transports heat from low to highlatitudes. The freshwater from river inows also maintains the relatively lowsalinity of the Arctic Ocean, which makes possible the freezing of its surface; thereection of the suns energy by this sea ice is an important factor in the earthsenergy balance.

INTRODUCTION 5

RIVERS 2,120

Lakes &Marshes102,000

Atmosphere 12,900

Biomass1,120

Soil water16,500

Oceans1,338,000,000Ground water

10,530,000

Glaciers24,000,000

P =117,000ET = 71,000

Plant uptake = 71,000

Recharge = 46,000

GW = 43,800

Q = 44,700

GW = 2,200

E = 1,000 P= 2,400

2,400

P = 458,000

E = 505,000

Figure 1.1 Schematic diagram of stocks (km3) and annual uxes (km3/year) in the globalhydrological cycle. E, evaporation; ET, evapotranspiration; GW, groundwater discharge;P, precipitation; Q, river discharge. Data on stocks, land and ocean precipitation, oceanevaporation, and river discharge are from Shiklomanov (1993) (see table 1.1); other uxes areadjusted from Shiklomanovs values to give an approximate balance for each stock. Dashedarrows indicate negligible uxes on the global scale

The drainage systems of riversriver networks and their contributing water-shedsare the principal organizing features of the terrestrial landscape. Thesesystems are nested hierarchies at scales ranging from a few square meters to5.9 106 km2 (the Amazon River drainage basin). The worlds largest river systemsin terms of drainage area are listed in table 1.3. At all scales, rivers are the links thatcollect the residual water (precipitation minus evapotranspiration and groundwateroutow) and its chemical and physical constituents and deliver them to the next levelin the hierarchy or to the world ocean.

1.1.2 Human Signicance

As indicated in gure 1.1, the immediate source of most of the water in rivers isgroundwater. Conversely, virtually all groundwater is ultimately destined to becomestreamow. River discharge is the rate at which nature makes water available forhuman use. Thus, at all scales, average river discharge is the metric of the waterresource (Gleick 1993; Vrsmarty et al. 2000b).

Humans have been concerned with rivers as sources of water and food, as routes forcommerce, and as potential hazards at least since the rst civilizations developed along

6 FLUVIAL HYDRAULICS

Table 1.2 Average discharge from the worlds 30 largest terminal drainage basins ranked bydischarge.a

Discharge

% Total dischargeRank River km3/year to oceans m3/s mm/year

1 Amazon 5,992 13.4 190,000 1,0242 Congo 1,325 3.0 42,000 3583 Chang Jiang 1,104 2.5 35,000 6154 Orinoco 915 2.0 29,000 8805 Ganges-Brahmaputra 631 1.4 20,000 3876 Parana 615 1.4 19,500 2317 Yenesei 561 1.3 17,800 2178 Mississippi 558 1.2 17,700 1749 Lena 514 1.1 16,300 213

10 Mekong 501 1.1 15,900 64811 Irrawaddy 399 0.9 12,700 97412 Ob 394 0.9 12,500 15313 Zhujiang (Si Kiang) 363 0.8 11,500 83114 Amur 347 0.8 11,000 11915 Zambezi 333 0.7 10,600 16716 St. Lawrence 328 0.7 10,400 25917 Mackenzie 286 0.6 9,100 16718 Volga 265 0.6 8,400 18119 Shatt-el-Arab (Euphrates) 259 0.6 8,210 26820 Salween 211 0.5 6,690 64921 Indus 202 0.5 6,410 17722 Danube 199 0.4 6,310 25323 Columbia 191 0.4 6,060 26424 Tocantins 168 0.4 5,330 21825 Kolyma 128 0.3 4,060 19226 Nile 96 0.2 3,040 2527 Orange 91 0.2 2,900 9728 Senegal 86 0.2 2,730 10229 Syr-Daya 83 0.2 2,630 7830 So Francisco 82 0.2 2,600 133

aTerminal means the drainage basin is not tributary to another stream.

Data are from web sites and various published sources.

the banks of rivers: the Indus in Pakistan, the Tigris and Euphrates in Mesopotamia,the Huang Ho in China, and the Nile in Egypt.

Water owing in streams is used for a wide range of vital water resourcemanagement purposes, such as

Human and industrial water supply Agricultural irrigation Transport and treatment of human and industrial wastes Hydroelectric power Navigation Food Ecological functions (wildlife habitat)

INTRODUCTION 7

Table 1.3 Topographic data for the worlds 30 largest terminal drainage basins ranked bydrainage area.a

Elevation (m) Averageslope(103)Rank River Area (106 km2) Length (km) Avg. Max. Min.b

1 Amazon 5.854 4,327 430 6,600 0 1.662 Nile 3.826 5,909 690 4,660 0 1.453 Congo (Zaire) 3.699 4,339 740 4,420 0 1.114 Mississippi 3.203 4,185 680 4,330 0 1.665 Amur 2.903 5,061 750 5,040 0 1.806 Parana 2.661 2,748 560 6,310 0 1.597 Yenesei 2.582 4,803 670 3,500 0 1.948 Ob 2.570 3,977 270 4,280 0 1.289 Lena 2.418 4,387 560 2,830 0 1.83

10 Niger 2.240 3,401 410 2,980 0 0.9411 Zambezi 1.989 2,541 1,050 2,970 0 1.6012 Tamanrasettc 1.819 2,777 450 3,740 0 0.8313 Chang Jiang

(Yangtze)1.794 4,734 1,660 7,210 0 3.27

14 Mackenzie 1.713 3,679 590 3,350 0 2.2315 Ganges-

Brahmaputra1.638 2,221 1,620 8,848 0 6.00

16 Chari 1.572 1,733 510 3,400 260 1.1017 Volga 1.463 2,785 1,710 1,600 0 0.5218 St. Lawrence 1.267 3,175 310 1,570 0 1.2219 Indus 1.143 2,382 1,830 8,240 0 5.5020 Syr-Darya 1.070 1,615 650 5,480 0 2.8421 Nelson 1.047 2,045 500 3,440 0 1.0622 Orinoco 1.039 1,970 480 5,290 0 3.0123 Murray 1.032 1,767 260 2,430 0 1.0324 Great Artesian

Basin0.978 1,045 220 1,180 70 0.55

25 Shatt-el-Arab(Euphrates)

0.967 2,200 660 4,080 0 2.84

26 Orange 0.944 1,840 1,230 3,480 0 1.6527 Huang He

(Yellow)0.894 4,168 2,860 6,130 0 2.93

28 Yukon 0.852 2,716 690 6,100 0 2.9329 Senegal 0.847 1,680 250 10,700 0 0.4330 Irharharc 0.842 1,482 500 2,270 0 1.84

aValues were determined by analysis of satellite imagery at the 30-min scale (latitude and longitude) (average pixel is47.4 km on a side). Terminal means the drainage basin is not tributary to another stream.bA minimum elevation of 0 means the basin discharges to the ocean. A nonzero minimum elevation indicates that thebasin discharges internally to the continent, usually to a lake. cRiver system mostly nondischarging under current climate.Source: Data are from Vrsmarty et al. (2000).

Recreation Aesthetic enjoyment3

Demand for water for all these purposes is growing with population, and roughlyone-third of the worlds peoples currently live under moderate to high water stress(Vrsmarty et al. 2000b). Water availability at a location on a river is assessed

8 FLUVIAL HYDRAULICS

by analyses of the time distribution of river discharge at that location (discussed insection 2.5.6.2).

On the other hand, water owing in rivers at times of ooding is one of the mostdestructive natural hazards globally. In the United States, ood damages total about$4 billion per year and are increasing rapidly because of the increasing concentrationof people and infrastructure in ood-prone areas (van der Link et al. 2004).Assessmentof this hazard and of the economic, environmental, and social benets and costs ofvarious strategies for reducing future ood damages at a riparian location is basedon frequency analyses of extreme river discharges at that location (discussed insection 2.5.6.3).4

1.2 The Role of Fluvial Hydraulics

The term uvial means of, pertaining to, or inhabiting a river or stream. This bookis about uvial hydraulicsthe internal physics of streams. In the civil engineeringcontext, the subject is usually called open-channel ow; the term uvial is usedhere to emphasize our focus on natural streams rather than design of structures.

An understanding of uvial hydraulics underlies many important scientic elds:

Because the terrestrial landscape is largely the result of uvial processes,an understanding of uvial hydraulics is an essential basis for the study ofgeomorphology.

Fluvial hydraulics governs the movement of water through the stream network,so an understanding of uvial hydraulics is essential to the study of hydrology.

Stream organisms are adapted to particular ranges of ow conditions and bedmaterial, so knowledge of uvial hydraulics is the basis for understanding streamecology.

Knowledge of uvial hydraulics is required for interpretation of ancient uvialdeposits to provide information about geological history.

Knowledge of uvial hydraulics is also the basis for addressing important practicalissues:

Predicting the effects of climate change, land-use change (urbanization, defor-estation, and afforestation), reservoir construction, water extraction, and sea-levelrise on river behavior and dimensions.

Forecasting the development and movement of ood waves through the channelsystem.

Designing dams, levees, bridges, canals, bank protection, and navigation works. Assessing and restoring stream habitats.

One particularly important application of uvial hydraulics principles is in themeasurement of river discharge. Discharge measurement directly provides essentialinformation about water-resource availability and ood hazards.

Because river discharge is concentrated in channels, it can in principle bemeasured with considerably more accuracy and precision than can precipitation,evapotranspiration, or other spatially distributed components of the hydrologicalcycle. Long-term average values of discharge typically have errors of 5% (i.e., the

INTRODUCTION 9

true value is within 5% of the measured value 95% of the time). Errors in precipitationare generally at least twice that (10%) and may be 30% or more depending onclimate and the number and location of precipitation gages (Winter 1981; Rodda 1985;Groisman and Legates 1994).Areal evapotranspiration is virtually unmeasured, and infact is usually estimated by solving equation 1.1 for ET. Thus, measurements of riverdischarge provide the most reliable information about regional water balances. And,because it is the space- and time-integrated residual of two climatically determinedquantities (equation 1.1), river discharge is a sensitive indicator of climate change.Observations of long-term trends in precipitation and streamow consistently showthat changes in river discharge amplify changes in precipitation; for example, a 10%increase in precipitation may induce a 20% increase in discharge (Wigley andJones 1985; Karl and Riebsame 1989; Sankarasubramanian et al. 2001). Dischargemeasurements are also invaluable for validating the hydrological models that arethe only means of forecasting the effects of land use and climate change on waterresources.

Fluvial hydraulics principles have long been incorporated in traditional measure-ment techniques that involve direct contact with the ow (discussed in section 2.5.3.1).New applications combining hydraulic principles, geomorphic principles, and empir-ical analysis are rapidly being developed to enable measurement of ows viaremote-sensing techniques (Bjerklie et al. 2003, 2005a; Dingman and Bjerklie 2005;Bjerklie 2007) (see section 2.5.3.2).

1.3 A Brief History of Fluvial Hydraulics

In order to understand a science, it is important to have an understanding of howit developed. This section provides an overview of the evolution of the science ofuvial hydraulics, emphasizing the signicant discrete contributions of individualsthat combine to form the basis of our current understanding of the eld. As with allscience, each individual contribution is built on earlier observations and reasoning.The material in this section is based largely on Rouse and Ince (1963), and the quotesfrom earlier works are taken from that book. Their text gives a more complete sense ofthe ways in which individual advances are built upon earlier work than is possible inthe present overview. You will nd it fascinating reading, especially after you becomefamiliar with the material in the present text.

As noted above, the rst civilizations were established along major rivers, and itis clear that humans were involved in river engineering that must have been based onlearning by trial and error since prehistoric times. The Chinese were building levees forood protection and the people of Mesopotamia were constructing irrigation systemsas early as 4000 b.c.e. In Egypt, irrigation was also practiced in lands adjacent tothe Nile by 3200 b.c.e., and the earliest known dam was built at Sadd el Kafara(near Cairo) in the period 29502759 b.c.e..

However, science based on observation and reasoning and the written transmis-sion of knowledge rst emerged in Greece around 600 b.c.e. Thales of Miletus(640546 b.c.e.) studied in Egypt. He believed that water is the origin of allthings, and both he and Hippocrates (460380?) two centuries later articulated the

10 FLUVIAL HYDRAULICS

philosophy that nature is best studied by observation. By far the most signicantenduring hydraulic principles discovered by the ancient Greeks were Archimedes(287212 b.c.e.) laws of buoyancy:

Any solid lighter than the uid will, if placed in the uid, be so far immersed that theweight of the solid will be equal to the weight of the uid displaced.

If a solid lighter than the uid be forcibly immersed in it, the solid will be drivenupwards by a force equal to the difference between its weight and the weight of the uiddisplaced.

A solid heavier than a uid will, if placed in it, descend to the bottom of the uid, andthe solid will, when weighed in the uid, be lighter than its true weight by the weightof the uid displaced. (Rouse and Ince 1963, p. 17)

Hero of Alexandria (rst century a.d.) wrote on several aspects of hydraulics,including siphons and pumps, and gave the earliest known expression of the lawof continuity (discussed in section 4.3.2) for computing the ow rate (discharge) ofa spring: In order to know how much water the spring supplies it does not sufce tond the area of the cross section of the ow. It is necessary also to nd the speedof ow (Rouse and Ince 1963, p. 22).

Although the writings of these and other Greek natural philosophers were preservedand transmitted to Europeans by Arabian scientists, there were no further scienticcontributions to the eld for some 1,500 years. The Romans built extensive andelaborate systems of aqueducts, reservoirs, and distribution pipes that are describedin extensive surviving treatises by Vitruvius (rst century b.c.e.) and Frontinus(40103 a.d.). Although aware of the Greek writings on hydraulics, they did notadd to them or even explicitly reect them in their designs and computations. Forexample, although Frontinus understood that the rates of ow entering and leaving apipe should be equal, he computed the ow rate based on area alone and did not seemto clearly understand, as Hero did, that velocity is also involved. Still, as Rouse andInce (1963, p. 32) note, the Roman engineers must have sensed the effects of head,slope, and resistance on ow rates or their systems would not have functioned as wellas they did.

There were no additions to scientic knowledge of hydraulics from the time ofHero until the Renaissance. However, during the Middle Ages, improvements inhydraulic machinery were made in the Islamic world, and a few scholars in Europewere considering the basic aspects of motion, acceleration, and resistance that laidthe groundwork for subsequent advances in physics. During this period,

the writingsand indeed the theories themselveswere numerous and complex, and the background training of few scholars was sound enough to distinguish fallacy fromtruth. Progress was hence exceedingly slow and laborious, and not for centuries did thecumulative effect of many people in different lands clarify these elementary principlesof mechanics on which the science of hydraulics was to be based. (Rouse and Ince1963, p. 42)In contrast to the dominant philosophies of the MiddleAges, the Italian Renaissance

genius Leonardo da Vinci (14521519) wrote, Remember when discoursing on theow of water to adduce rst experience and then reason. Da Vinci rediscoveredthe principle of continuity, stating that a river in each part of its length in an equal

INTRODUCTION 11

time gives passage to an equal quantity of water, whatever the depth, the slope,the roughness, the tortuosity. He also correctly concluded from his observationsof open-channel ows that water has higher speed on the surface than on thebottom. This happens because water on the surface borders on air which is oflittle resistance, and water at the bottom is touching the earth which is ofhigher resistance. From this follows that the part which is more distant fromthe bottom has less resistance than that below and that the water of straightrivers is the swifter the farther away it is from the walls, because of resistance(discussed in sections 3.3, 5.3, and 5.4). From his observations of water waves,he correctly noted that the speed of propagation of (surface) undulations alwaysexceeds considerably that possessed by the water, because the water generallydoes not change position; just as the wheat in a eld, though remaining xedto the ground, assumes under the impulsion of the wind the form of wavestraveling across the countryside (Rouse and Ince 1963, p. 49) (discussed insections 11.311.5).

Because da Vincis observations were lost for several centuries, they did notcontribute to the growth of science. For example, one of Galileos pupils, BenedettoCastelli (1577?1644?), again formulated the law of continuity more than a centuryafter da Vinci, and it became known as Castellis law. In 1697, another Italian,Domenico Guglielmini (16551710), published a major work on rivers, DellaNatura del Fiumi (On the Nature of Rivers), which included among other thingsa description of uniform (i.e., nonaccelerating) ow very similar to that in thepresent text (see section 6.2.1, gure 6.2). In an extensive treatise on hydrostaticspublished posthumously in 1663, Blaise Pascal (16231662) showed that the pressureis transmitted equally in all directions in a uid at rest (see section 4.2.2.2).

The major scientic advances of the seventeenth century were those of Sir IsaacNewton (16421727), who began the development of calculus, concisely formulatedhis three laws of motion based on previous ideas of Descartes and others, and clearlydened the concepts of mass, momentum, inertia, and force. He also formulated thebasic relation of viscous shear (see equation 3.19), which characterizes Newtonianuids. Newtons German contemporary, Gottfried Wilhelm von Leibniz (16461716),further developed the concepts of calculus and originated the concept of kinetic energyas proportional to the square of velocity (see section 4.5.2).

In the eighteenth century, the elds of theoretical, highly mathematical hydro-dynamics and more practical hydraulics largely diverged. The foundations ofhydrodynamics were formulated by four eighteenth-century mathematicians, DanielBernoulli (Swiss, 17001782), Alexis Claude Clairault (French, 17131765), Jean leRond dAlembert (French, 17171783), and especially Leonhard Euler (Swiss,17081783). Bernoulli formulated the concept of conservation of energy in uids(section 4.5), although the Bernoulli equation (equation 4.42) was actually derivedby Euler. Euler was also the rst to state the microscopic law of conservation ofmass in derivative form (section 4.3.1, equation 4.16). The Frenchmen Joseph LouisLagrange (17361813) and Pierre Simon Laplace (17491827) extended Eulerswork in many areas of hydrodynamics. Although both Euler and Lagrange exploreduid motion by analyzing occurrences at a xed point and by following a uidparticle, the former approach has become known as Eulerian and the latter as

12 FLUVIAL HYDRAULICS

Lagrangian (section 4.1.4). One of Lagranges contributions was the relation for thespeed of propagation of a shallow-water gravity wave (equation 11.51); the PoleFranz Joseph von Gerstner (17561832) derived the corresponding expression fordeep-water waves (equation 11.50).

Many of the advances in hydraulics in the eighteenth century were made possible byadvances in measurement technology: Giovanni Poleni (Italian, 16831761) derivedthe basic equation for ow-measurement weirs (section 10.4.1) in 1717, and Henri dePitot (French, 16951771) invented the Pitot tube in 1732, which uses energy conceptsto measure velocity at a point. One of the most important and ultimately inuentialpractical developments of this time was the work ofAntoine Chzy (17181798), whoreasoned that open-channel ow can usually be treated as uniform ow (section 6.2.1)in which velocity is due to the slope of the channel and to gravity, of whichthe effect is restrained by the resistance of friction against the channel boundaries(Rouse and Ince 1963, pp. 118119). The equation that bears his name, derivedin 1768 essentially as described in section 6.3 of this text, states that velocity(U) is proportional to the square root of the product of depth (Y ) and slope (S),that is,

U = K Y1/2 S1/2, (1.2)where K depends on the nature of the channel. The Chzy equation can be viewed asthe basic equation for one-dimensional open-channel ow. Interestingly, Chzys1768 report was lost (although the manuscript survived), and his work was notpublished until 1897 by the American engineer Clemens Herschel (18421930)(Herschel 1897).

Although Chzys work was generally unknown, others such as the GermanJohannn Albert Eytelwein (17641848) in 1801 proposed similar relations for open-channel ow. Interestingly, Gaspard de Prony (17551839) in 1803 proposed aformula for uniform open-channel ow identical to equation 7.42 of this text, whichis identical to the Chzy relation for conditions usually encountered in rivers. In Italy,Giorgio Bidone (17811839) was the rst to systematically study the hydraulic jump(section 10.1), in 1820, and Giuseppe Venturoli (17681846) made measurementsconrming Eytelweins formula and in 1823 was the rst to derive an equation forwater-surface proles (section 9.4.1).

During this period, James Huttons (English, 17261797) observations of streamsand stream networks led him to conclude that the elements of the landscapeare in a quasi-equilibrium state, implying relatively rapid mutual adjustment tochanging conditions (section 2.6.2). This was a major philosophical advance in theunderstanding of the development of landscapes and the role of uvial processes inthat development.

Other hydraulic advances of the rst half of the nineteenth century included aquantitative understanding of ow over broad-crested weirs (section 10.4.1.2), used inow measurement, published in 1849 by Jean Baptiste Belanger (French, 17891874).Gaspard Gustave de Coriolis (French, 17921843) is best known for formulating theexpression for the apparent force acting on moving bodies due to the earths rotation(the Coriolis force, section 7.3.3.1), and also showed in 1836 the need for a correctionfactor (the Coriolis coefcient; see box 8.1) when using average velocity to calculate

INTRODUCTION 13

the kinetic energy of a ow. John Russell (English, 18081882) made observationsof waves generated by barges in canals (1843), including the rst descriptions ofthe solitary gravity wave (soliton; section 11.4.2). The rst modern textbook onhydraulics (1845) was that of Julius Weisbach (German, 18061871), which includedchapters on ow in canals and rivers and the measurement of water as well as the workon the resistance of uids with which his name is associatedthe Darcy-Weisbachfriction factor (see box 6.2).

As described in sections 3.3.3 and 3.3.4 of this text, there are two states of uidow: laminar (or viscous) and turbulent. Despite the fact that ows in these twostates have very different characteristics, explicit mention of this did not appear until1839, in a paper by Gotthilf Hagen (German, 17971884). In a subsequent study(1854) Hagen clearly described the two states, anticipating by several decades thestudies of Osborne Reynolds (see below), whose name is now associated with thephenomenon. Interest in scale models as an aid to the design of ships grew in thisperiod, and it was in this context that Ferdinand Reech (French, 18051880) in 1852rst formulated the dimensionless ratio that relates velocities in models to those inthe prototype. This ratio became known as the Froude number (sections 6.2.2.2 and7.6.2) after William Froude (English, 18101879), who did extensive ship modelingexperiments for the British government, though in fact he neither formulated nor evenused the ratio.

Advances in the latter half of the nineteenth century, as with many earlier ones,were dominated by scientists and engineers associated with Frances Corps des Pontset Chausses (Bridges and Highways Agency). Notable among these are ArsneDupuit (18041866), Henri Darcy (18031858), Jacques Bresse (18221883), andJean-Claude Barr de Saint-Venant (17971886). Dupuits principal contributions touvial hydraulics were his 1848 analysis of water-surface proles and their relationto uniform ow (section 9.2) and to variations in bed elevation and channel width(section 10.2), and his 1865 written discussion of the capacity of a stream to transportsuspended sediment. Darcy, in addition to discovering Darcys law of groundwaterow, studied ow in pipes and open channels and in 1857 demonstrated that resistancedepended on the roughness of the boundary. Bresse in 1860 correctly analyzedthe hydraulic jump using the momentum equation (section 10.1; equation 10.8).Saint-Venant in 1871 rst formulated the general differential equations of unsteadyow, now called the Saint-Venant equations (section 11.1).

Dupuits interest in sediment transport was followed by the work of MdricLachalas (18201904), which in 1871 discussed various types of sediment movement(gure 12.1), and the analysis of bed-load transport (1879) by Paul du Boys(18471924), which has been the basis for many approaches to the present day(section 12.5.1). Darcys experimental work on ow resistance was carried on by hiscolleague Henri Bazin (18291917), whose measurements, published in 1865 and1898, were analyzed by many later researchers hoping to discover a practical law ofopen-channel ow. Bazins experiments also included measurements of the velocitydistribution in cross sections (section 5.4) and of ow over weirs (section 10.4.1.1).Another Frenchman, Joseph Boussinesq (18421929), though not at the Corps desPonts et Chausses, made signicant contributions in many aspects of hydraulics,including further insight in 1872 into the laminar-turbulent transition identied by

14 FLUVIAL HYDRAULICS

Hagen, the mathematical treatment of turbulence (section 3.3.4.3), and the formulationof the momentum equation (section 8.2.1, box 8.1).

There were also signicant contemporary developments in England. Theseincluded Sir George Airys (18011892) comprehensive treatment of waves and tidesin 1845, including the derivation of the Airy wave equation (equation 11.46), andSir George Stokess (18191903) expansion in 1851 of Saint-Venants equations toturbulent ow and his derivation of Stokess law for the settling velocity of a sphericalparticle (equation 12.19). Combining experiment and analysis, Osborne Reynolds(18421912) made major advances in many areas, including the rst demonstrationof the phenomenon of cavitation (section 12.4.4.3), the seminal treatment in 1894of turbulence as the sum of a mean motion plus uctuations (section 3.3.4.2),and, most famously, the 1883 formulation of the Reynolds number quantifying thelaminar-turbulent transition (section 3.4.2).

The names of Americans are conspicuously absent from the history of hydraulicsuntil 1861, when two Army engineers, A. A. Humphreys and H. L. Abbot, publishedtheir Report upon the Physics and Hydraulics of the Mississippi River. In this theyincluded a comprehensive review of previous European work on ow resistance and,nding that previous formulas did not consistently work on the lower Mississippi,attempted to develop their own. Their work prompted others to look for a universalresistance relation for open-channel ow. One signicant contribution, in 1869, wasthat of two Swiss engineers, Emile Ganguillet (18181894) and Wilhelm Kutter(18181888), who accepted the basic form of the Chzy relation and proffered acomplex formula for calculating the resistance as a function of boundary roughness,slope, and depth. Meanwhile, Phillipe Gauckler (18261905, also of the Corps desPonts et Chausses) in 1868 proposed two resistance formulas, one for rivers of lowslope (S < 0.0007),

U = K Y4/3 S, (1.3a)and the other for rivers of high slope (S > 0.0007),

U = K Y2/3 S1/2. (1.3b)Equation 1.3b was of particular signicance because the Irish engineer RobertManning (18161897) reviewed previous data on open-channel ow and stated inan 1889 report (although apparently without knowledge of Gaucklers work) thatequation 1.3b t the data better than others. However, Manning did not recommendthat relation because it is not dimensionally correct (see appendix A), and proposeda modication that included a term for atmospheric pressure. Mannings proposedrelation was never adopted, but ironically, equation 1.3b with K dependent on channelroughness has become the most widely used practical resistance relation and iscalled Mannings equation (section 6.8). As noted by Rouse and Ince (1963, p. 180),What we now call the Manning formula was thus neither recommended nor evendevised in full by Manning himself, whereas his actual recommendation receivedlittle further attention.

The rst half of the twentieth century saw major advances in understandingreal turbulent ows. In 1904, Ludwig Prandtl (German, 18751953) introducedthe concept of the boundary layer (section 3.4.1), and in 1926 that of the mixing

INTRODUCTION 15

length (section 3.3.4.4) which tied Reynoldss statistical concepts of turbulence tophysical phenomena. This laid the groundwork for a very signicant breakthrough: theanalytical derivation of the velocity distribution in turbulent boundary layers, whichwas developed by Prandtl and his student Theodore von Krmn (Hungarian who lateremigrated to the United States, 18811963) and bears their names (section 5.3.1). Thiswork, which grew out of studies of ow over airplane wings, was a major advance inunderstanding and modeling turbulent open-channel ows.

Meanwhile, theAmerican Edgar Buckingham (18671940) introduced the conceptof dimensional analysis (section 4.8.2) to English-speaking engineers in 1915; theseconcepts have guided countless fruitful investigations of ow phenomena.At the sametime (1914) the American geologist Grove Karl Gilbert (18431918) carried out therst ume studies of the transport of gravel. Filip Hjulstrm (Swedish, 19021982) in1935 and Albert Shields (German, 19081974) in 1936 provided analyses of data thatform the basis for most subsequent studies of sediment entrainment (sections 12.4.1and 12.4.2).

An inuential text that appeared during this period was Hunter Rouses(19061996) comprehensive and authoritative Fluid Mechanics for Hydraulic Engi-neers (Rouse 1938), which remains valuable to this day. In 1937, Rouse derivedan expression for the vertical distribution of suspended sediment that is the basisfor most analyses of this phenomenon (section 12.5.2.1), and in 1943 he conciselysummarized experimental data on resistanceReynolds numberroughness relationsfor the full range of ows in pipes in graphical form. A year later, Lewis F. Moody(American, 18801953) published a modied version of this graph (Moody 1944)that has been extended to open-channel ows and become known as the Moodydiagram (see gure 6.8) (Ettema 2006).

The second half of the twentieth century saw signicant advances in characterizingand understanding natural streams. Many of these advances were by Americanswho applied the scientic and engineering insights described above and developednew approaches of analysis and measurement. One of these was the paper byRobert E. Horton (18751945) (Horton 1945), which was pivotal in turning theanalysis of uvial processes and landscapes from the qualitative approaches ofgeographers to a more quantitative scientic basis. A seminal conceptual contributionwas the geologist J. Hoover Mackins (19051968) clear articulation of Huttonsconcept of dynamic equilibrium, the graded stream (Mackin 1948; see section 2.6.2).Building upon these developments, Luna Leopold (19152006) and several of hiscolleagues associated with the U.S. Geological Survey, most notably R. A. Bagnold(English, 18961990), W. B. Langbein (19071982), J. P. Miller (19231961), andM. G. Wolman (1924), in the 1950s began an era of eld research and innovativeanalysis that dened the eld of uvial processes and geomorphology for the rest ofthe century and beyond.

At the same time, V. T. Chow (American, 19191981) (Chow 1959) andFrancis M. Henderson (Australian, 1921) (Henderson 1966) distilled the advancesdescribed above to provide coherent and lucid engineering texts on open-channelhydraulics. These texts made the subject an essential part of civil engineeringcurricula and were a source of insights increasingly adopted and applied by earthscientists.

16 FLUVIAL HYDRAULICS

As the twenty-rst century begins, two major problems of uvial hydraulicsremain far from completely solved: the a priori characterization of open-channelow resistance/conductance (chapter 6) (the K in equation 1.2), and the predictionof sediment transport as a function of ow and channel characteristics (chapter 12).However, the coming years hold promise of major progress in understanding uvialhydraulics and applying it to these and the critical problems described in section 1.2.This promise is largely the result of technological advances such as the ability tovisualize and measure uid and sediment motion, techniques for remote-sensing ofstreams, and advances in computer speed and storage that make possible the modelingof uid ows. The measurements and insights of all the pioneering work described inthe preceding paragraphs and in the remainder of this text will provide a sound basisfor this progress.

1.4 Scope and Approach of This Book

The goal of the science of uvial hydraulics is to understand the behavior ofnatural streams and to provide a basis for predicting their responses to naturaland anthropogenic disturbances. The objective of this book is to develop a soundqualitative and quantitative basis for this understanding for practitioners and studentswith backgrounds in earth sciences and natural resources. This book differs fromtypical engineering treatments of open-channel ow in its greater emphasis onnatural streams and reduced treatments of hydraulic structures. It differs from mostearth-science-oriented texts in its greater emphasis on quantitative analysis basedon the basic physics of river ows and its incorporation of analyses developed forengineering application.

The treatment here draws on your knowledge of basic mechanics (through rst-yearuniversity-level physics) and mathematics (through differential and integral calculus)to develop a physical intuitiona sense of the relative magnitudes of properties,forces, and other quantities and relationships that are signicant in a specic situation.Physical intuition consists not only of a store of factual knowledge, but also of amental inventory of patterns that serve as guides to the parts of that knowledge thatare relevant to the situation (Larkin et al. 1980). Thus, a special attempt is made inthis book to emphasize patterns and connections.

The goal of chapter 2 is to provide a natural context for the analytical approachemphasized in subsequent chapters. It presents an overview of the characteristics ofnatural stream networks and channels and the ways in which geological, topographic,and climatic factors determine those characteristics. It also discusses the measurementand hydrological aspects of the ow within natural channelsits sources and temporalvariability. The chapter concludes with an overview of the spatial and temporalvariability of the variables that characterize stream channels, including the principleof dynamic equilibrium.

Water moves in response to forces acting on it, and its physical properties determinethe qualitative and quantitative relations between those forces and the resultingmotion. Chapter 3 begins with a description of the atomic and molecular structureof water that gives rise to its unique properties, and the nature of water substance

INTRODUCTION 17

in its three phases. The bulk of the chapter uses a series of thought experiments toelucidate the properties of liquid water that are crucial to understanding its behaviorin open-channel ows: density, surface tension, and viscosity. Included here is anintroduction to turbulence, ow states, and boundary layers, concepts that are centralto understanding ows in natural streams.

Chapter 4 completes the presentation of the foundations of the study of open-channel ows by focusing on the physical and mathematical concepts that underliethe basic equations relating uid properties and hydraulic variables. The objective hereis to provide a deeper understanding of the origins, implications, and applicability ofthose equations. The chapter develops fundamental physical equations based on theconcepts of mass, momentum, energy, force, and diffusion in uids. The powerfulanalytical tool of dimensional analysis is described in some detail. Also discussedare approaches to developing equations not derived from fundamental physical laws:empirical and heuristic relations, which must often be employed due to the analyticaland measurement difculties presented by natural streamows. Although most ofthis book is concerned with one-dimensional (cross-section-averaged macroscopic)analysis, this chapter develops many of the equations initially at the more fundamentalthree-dimensional microscopic level.

The central problem of open-channel ow is the relation between cross-section-average velocity and ow resistance. The main objective of chapter 5 is todevelop physically sound quantitative descriptions of the distribution of velocity incross sections. The chapter focuses on the derivation of the Prandtl-von Krmnvertical velocity prole based on the characteristics of turbulence and boundarylayers developed in chapter 3. Understanding the nature of this prole providesa sound basis for scaling up the concepts introduced at the microscopiclevel in chapter 4 and for determining (and measuring) the cross-section-averagedvelocity.

Chapter 6 begins by reviewing the basic geometric features of river reaches andreach boundaries presented in chapter 2. It then adapts the denition of uniform owas applied to a uid element in chapter 4 to apply to a typical river reach and derivesthe Chzy equation, which is the basic equation for macroscopic uniform ows.This derivation allows formulation of a simple denition of resistance. The chapterthen examines the factors that determine ow resistance, which involves applyingthe principles of dimensional analysis developed in chapter 4 and the velocity-prole relations derived in chapter 5. Chapter 6 concludes by exploring resistancein nonuniform ows and practical approaches to determining resistance in naturalchannels.

The goals of chapter 7 are to develop expressions to evaluate the magnitudes ofthe driving and resisting forces at the macroscopic scale, to examine the relativemagnitudes of the various forces in natural streams, and to show how these forceschange as a function of ow characteristics. Understanding the relative magnitudes offorces provides a helpful perspective for developing quantitative solutions to practicalproblems.

Chapter 8 integrates the momentum and energy principles for a uid element(introduced in chapter 4) across a channel reach to apply to macroscopic one-dimensional steady ows, and compares the theoretical and practical differences

18 FLUVIAL HYDRAULICS

between the energy and momentum principles. These principles are applied to solvepractical problems in subsequent chapters.

Starting with the premise that natural streamows can usually be well approxi-mated as steady uniform ows (chapter 7), chapter 9 applies the energy relations ofchapter 8 with resistance relations of chapter 6 to develop the equations of graduallyvaried ow. These equations allow prediction of the elevation of the water surface overextended distances (water-surface proles), given information about discharge andchannel characteristics. Gradually varied ow computations play an essential role inaddressing several practical problems, including predicting areas subject to inundationby oods, locations of erosion and deposition, and the effects of engineering structureson water-surface elevations, velocity, and depth. Used in an inverse manner, theyprovide a tool for estimating the discharge of a past ood from high-water marks leftby that ood.

Chapter 10 treats steady, rapidly varied ow, which is ow in which the spatialrates of change of velocity and depth are large enough to make the assumptions ofgradually varied ow inapplicable. Such ow occurs at relatively abrupt changesin channel geometry; it is a common local phenomenon in natural streams andat engineered structures such as bridges, culverts, weirs, and umes. Such owsare generally analyzed by considering various typical situations as isolated cases,applying the basic principles of conservation of mass and of momentum and/orenergy as a starting point, and placing heavy reliance on dimensional analysis andempirical relations established in laboratory experiments. The chapter analyzes thethree broad cases of rapidly varied ow that are of primary interest to surface-waterhydrologists: the standing waves known as hydraulic jumps, abrupt transitions inchannel elevation or width, and structures designed for the measurement of discharge(weirs and umes).

The objective of chapter 11 is to provide a basic understanding of unsteady-ow phenomena, that is, ows in which temporal changes in discharge, depth, andvelocity are signicant. This understanding rests on application of the principlesof conservation of mass and conservation of momentum to ows that change inone spatial dimension (the downstream direction) and in time. Temporal changesin velocity always involve concomitant changes in depth and so can be viewed aswave phenomena. Some of the most important applications of the principles of open-channel ow are in the prediction and modeling of the depth and speed of travelof waves such as ood waves produced by watershed-wide increases in streamowdue to rain or snowmelt, waves due to landslides or debris avalanches into lakesor streams, waves generated by the failure of natural or articial dams, and wavesproduced by the operation of engineering structures.

Most natural streams are alluvial; that is, their channels are made of particulatesediment that is subject to entrainment, transport, and deposition by the waterowing in them. The goal of chapter 12 is to develop a basic understanding ofthese processesa subject of immense scientic and practical import. The chapterbegins by dening basic terminology and describes the techniques used to measuresediment in streams. It then explores empirical relations between sediment transportand streamow and how these relations are used to understand some fundamentalaspects of geomorphic processes. The basic physics of the forces that act on sediment

INTRODUCTION 19

particles in suspension and on the stream bed are formulated to provide an essentialfoundation for understanding entrainment and transport processes, and to gain someinsight into factors that dictate the shape of alluvial-channel cross sections. Thetopic of bedrock erosiona topic that is only beginning to be studied in detailisalso introduced. The chapter concludes by addressing the central issues of sedimenttransport: 1) the maximum size of sediment that can be entrained by a given ow(stream competence), and 2) the total amount of sediment that can be carried by aspecic ow (stream capacity).

2Natural StreamsMorphology, Materials, and Flows

2.0 Introduction and Overview

Stream is the general term for any body of water owing with measurable velocityin a channel. Streams range in size from rills to brooks to rivers; there are no strictquantitative boundaries to the application of these terms. A given stream as identiedby a name (e.g., Beaver Brook, Mekong River) is not usually a single entity withuniform channel and ow characteristics over its entire length. In general, the channelmorphology, bed and bank materials, and ow characteristics change signicantlywith streamwise distance; changes may be gradual or, as major tributaries enter or thegeological setting changes, abrupt.Thus, for purposes of describing and understandingnatural streams, we focus on the stream reach:

A stream reach is a stream segment with fairly uniform size and shape,water-surface slope, channel materials, and ow characteristics.

The length of a reach depends on the scale and purposes of a study, but usually rangesfrom several to a few tens of times the stream width. A reach should not includesignicant changes in water-surface slope and does not extend beyond the junctionsof signicant tributaries.

Each stream reach has a unique form and personality determined by the ows ofwater and sediment contributed by its drainage basin; its current and past geological,topographic, and climatic settings; and the ways it has been affected by humans.Thus, natural streams are complex, irregular, dynamic entities, and the characteristicsof a given reach are part of spatial and temporal continuums. The spatial continuum

20

NATURAL STREAMS 21

extends upstream and downstream through the stream network and beyond to includethe entire watershed; the temporal continuum may include the inheritance of formsand materials from the distant past (e.g., glaciations, tectonic movements, sea-levelchanges) as well as from relatively recent oods.

In subsequent chapters, this uniqueness and connection to spatial and temporalcontinuums will not always be apparent because we will simplify the channelgeometry, materials, and ow conditions in order to apply the basic physical principlesthat are the essential starting point for understanding stream behavior. The purposeof this chapter is to present an overview of the characteristics of natural streams andsome indication of the ways in which geological, topographic, and climatic factorsdetermine those characteristics. This will provide a natural context for the analyticalapproach emphasized in subsequent chapters.

2.1 The Watershed and the Stream Network

2.1.1 The Watershed

A watershed (also called drainage basin or catchment) is topographically denedas the area that contributes all the water that passes through a given cross section ofa stream (gure 2.1a). The surface trace of the boundary that delimits a watershed iscalled a divide. The horizontal projection of the area of a watershed is the drainagearea of the stream at (or above) the cross section. The stream cross section that denesthe watershed is at the lowest elevation in the watershed and constitutes the watershedoutlet; its location is determined by the purpose of the analysis. For geomorphologicalanalyses, the watershed outlet is usually where the stream enters a larger stream, alake, or the ocean. Water-resources analyses usually require quantitative analyses ofstreamow data, so for this purpose the watershed outlet is usually at a gaging stationwhere streamow is monitored (see section 2.5.3).

The watershed is of fundamental importance because the water passing throughthe stream cross section at the watershed outlet originates as precipitation on thewatershed, and the characteristics of the watershed control the paths and rates ofmovement of water and the types and amounts of its particulate and dissolvedconstituents as they move through the stream network. Hence, watershed geology,topography, and land cover regulate the magnitude, timing, and sediment load ofstreamow. As William Morris Davis stated, One may fairly extend the riverall over its [watershed], and up to its very divides. Ordinarily treated, the riveris like the veins of a leaf; broadly viewed, it is like the entire leaf (Davis 1899,p. 495).

2.1.2 Stream Networks

The drainage of the earths land surfaces is accomplished by stream networksthe veins of the leaf in Daviss metaphorand it is important to keep in mind thatstream reaches are embedded in those networks. Stream networks evolve in response

(a)

(b)

1st order

2nd order

3rd order

4th order

0

480465

450

435

N420

405390

375360

345330

315300285

270

Weir

255

________________ Stream_ _ _ _ _ _ _ _ _ _ Divide

500 meters

Elevation in meters above mean sea levelContour interval: 15 meters

Figure 2.1 A watershed is topographically dened as the area that contributes all the waterthat passes through a given cross section of a stream. (a) The divide dening the watershed ofGlenn Creek, Fox, Alaska, above a streamow measurement site (weir) is shown as the long-dashed outline, and the divides of two tributaries as shorter-dashed lines. (b) The watershed ofa fourth-order stream showing the Strahler system of stream-order designation.

NATURAL STREAMS 23

to climate change, earth-surface processes, and tectonic processes, and networkcharacteristics affect various dynamic aspects of stream response and geochemicalprocesses. Knighton (1998) provided an excellent review of the evolution of streamnetworks, Dingman (2002) summarized their relation to hydrological processes, andRodriguez-Iturbe and Rinaldo (1997) presented an exhaustive exploration of thesubject.

2.1.2.1 Network Patterns

Network patterns, the types of spatial arrangement of river channels in the landscape,are determined by land slope and geological structure (Twidale 2004). Most drainagenetworks form a dendritic pattern like those of gures 2.1b and 2.2a: there is nopreferred orientation of stream segments, and interstream angles at stream junctionsare less than 90 and point downstream. The dendritic pattern occurs where thereare no strong geological controls that create zones or directions of strongly varyingsusceptibility to chemical or physical erosion. Zones or directions more susceptible toerosion may display parallel, trellis, rectangular, or annular patterns (gure 2.2be).The distributary pattern (gure 2.2f ) usually occurs where streams ow out ofmountains onto atter areas to form alluvial fans, or on deltas that form wherestreams enter lakes or the ocean. Regional geological structures may also causepatterns of any of these shapes to be arranged in radial or centripetal metapatterns(gure 2.2g,h). The presence of these patterns and metapatterns on maps, aerialphotographs, or satellite images can provide useful clues for inferring the underlyinggeology (table 2.1).

2.1.2.2 Quantitative Description

Figure 2.1b shows the most common approach to quantitatively describing streamnetworks (Strahler 1952). Streams with no tributaries are designated rst-orderstreams; the conuence of two rst-order streams is the beginning of a second-order stream; the conuence of two second-order streams produces a third-orderstream, and so forth. When a stream of a given order receives a tributary oflower order, its order does not change. The order of a drainage basin is theorder of the stream at the basin outlet. The actual size of the streams desig-nated a particular order depends on the scale of the map or image used,1 theclimate and geology of the region, and the conventions used in designating streamchannels.

Within a given drainage basin, the numbers, average lengths, and average drainageareas of streams of successive orders usually show consistent relations of the formshown in gure 2.3. These relations are called the laws of drainage-networkcomposition and are summarized in table 2.2. Networks that follow these lawsthatis, that have bifurcation ratios, length ratios, and drainage-area ratios in the rangesshowncan be generated by random numbers, so it seems that the evolution ofnatural stream networks is essentially governed by the operation of chance (Leopoldet al. 1964; Leopold 1994). Table 2.3 summarizes the numbers, average lengths, andaverage drainage areas of streams of various orders.

(a) (b)

(c) (d)

DendriticParallel

Trellis

Rectangular

(f)

Distributary

(g)

Radial

(h)

Centripetal

(e)

Annular

Figure 2.2 Drainage-network patterns (see table 2.1). Panels ae are from Morisawa (1985).

NATURAL STREAMS 25

Table 2.1 Stream-network patterns and metapatterns and their relation to geological controls.

Type Description Geological control Figure

Dendritic Treelike, no preferred channelorientation, acute interstreamangles

None 2.2a

Parallel Main channels regularly spacedand subparallel to parallel,very acute interstream angles

Closely spaced faults,monoclines, or isoclinal folds

2.2b

Trellis Channels oriented in twomutually perpendiculardirections, elongated indominant drainage direction,nearly perpendicularinterstream angles

Tilted or folded sedimentaryrocks with alternatingresistant/weak beds

2.2c

Rectangular Channels oriented in twomutually perpendiculardirections, lengths similar inboth directions, nearlyperpendicular interstreamangles

Rectangular joint or faultsystem

2.2d

Annular Main streams in approximatelycircular pattern, nearlyperpendicular interstreamangles

Eroded dome of sedimentaryrocks with alternatingresistant/weak beds

2.2e

Distributary Single channel splits into twoor more channels that do notrejoin

Thick alluvial deposits (alluvialfans, deltas)

2.2f

Radial(metapattern)

Stream networks radiateoutward from central point

Volcanic cone or dome ofintrusive igneous rock

2.2g

Centripetal(metapattern)

Stream networks ow inward toa central basin

Calderas, craters, tectonicbasins

2.2h

After Summereld (1991) and Twidale (2004).

A stream network can also be quantitatively described by designating the junctionsof streams as nodes and the channel segments between nodes as links. Linksconnecting to only one node (i.e., rst-order streams) are called exterior links; theothers are interior links. The magnitude of a drainage-basin network is the totalnumber of exterior links it contains; thus, the network of gure 2.1b is of magnitude 43.Typically, the number of links of a given order is about half the number for the nextlowest order (Kirkby 1993).

The spatial intensity of the drainage network, or degree of dissection of the terrainby streams, is quantitatively characterized by the drainage density, DD, which is thetotal length of streams draining that area, X , divided by the area, AD:

DD XAD . (2.1)

Drainage density thus has dimensions [L1].

26 FLUVIAL HYDRAULICS

100N() = 615exp(1.33)

50

NU

MBE

R O

F ST

REA

MS

10

5

1

STREAM ORDER1 2 3 4 5

100

AD() = 0.18exp(1.48)

50

MEA

N D

RAIN

AG

E A

REA

, km

2

10

5

(a)

(c)

(b)

1

STREAM ORDER1 2 3 4 5

L() = 0.21exp(0.97)

MEA

N S

TREA

M L

ENG

TH, k

m

10

5

1

0.5

STREAM ORDER1 2 3 4 5

Figure 2.3 Plots of (a) numbers, N(), (b) average lengths, L(), and (c) average drainageareas, AD(), versus order, , for a fth-order drainage basin in England, illustrating the lawsof drainage-network composition (table 2.2). After Knighton (1998).

Drainage density values range from less than 2 km1 to more than 100 km1.Drainage density has been found to be related to average precipitation, with lowvalues in arid and humid areas and the largest values in semiarid regions (Knighton1998). In a given climate, an area of similar geology tends to have a characteristicvalue; higher values of DD are generally found on less permeable soils, wherechannel incision by overland ow is more common, and lower values on morepermeable materials. However, it is important to understand that the value of DD

NATURAL STREAMS 27

Table 2.2 The laws of drainage-network composition.a

Average valueand usual

Law of Denition Mathematical form rangeb

Stream numbers(Horton 1945)

RB = N()N(+ 1) N() = N exp(N )N = N(1) RBN = ln(RB)

RB = 3.703 < RB < 5

Stream lengths(Horton 1945)

RL = X(+ 1)X() X() = L exp(L )L = X(1)/RLL = ln(RL)

RL = 2.551.5 < RL < 3.5

Drainage areas(Schumm 1956)

RA = AD(+ 1)AD() AD() = A exp(A )A = AD(1)/RAA = ln(RA)

RA = 4.553 < RA < 6

aRB , bifurcation ratio; RL , length ratio; RA, drainage-area ratio; N(), number of streams of order ; X(), average lengthof streams of order ; AD , average drainage area of streams of order .bGlobal average for orders 36 computed by Vrsmarty et al. (2000a, p. 23), considered to best represent the geomorphiccharacteristics of natural basins.

Table 2.3 Orders, numbers, average lengths, and average areas of the worlds streams.

Ordera Number Average length (km) Average area (km2)

1 14,500,000 0.78 1.62 4,150,000 1.56 7.23 1,190,000 3.13 334 339,000 6.25 1505 96,900 12.5 7006 27,673 25.0 3,2007 4,456 249 18,0008 906 586 82,0009 176 1,300 369,000

10 38 2,645 1,490,00011 2 4,360 4,140,000

aValues for orders 611 taken from Vrsmarty et al. (2000c) assuming that rst-order streams at the scale of their studycorrespond to true sixth-order streams (Wollheim 2005). Values for orders 15 are computed using the global averagebifurcation, length, and area ratios computed by Vrsmarty et al. (2000c): RB = 3.70; RL = 2.55; RA = 4.55.

for a given region will increase as the scale of the map on which measurements aremade increases.

2.1.3 Watershed-Scale Longitudinal Prole

The longitudinal prole of a stream is a plot of the elevation of its channel bed versusstreamwise distance. The prole can be represented as a relation between elevation(Z) and distance (X), or between slope, S0( dZ/dX) and distance. Downstream

28 FLUVIAL HYDRAULICS

distance can be used directly as the independent variable or may be replaced bydrainage area, which increases with downstream distance, or by average or bankfulldischarge, which usually increases with distance.

At the watershed scale, longitudinal proles of streams from highest point to mouthare usually concave-upward, although some approach straight lines, and commonlythere are some segments of the prole that are convex (gure 2.4).

The elevation at the mouth of a stream, usually where it enters a larger stream,a lake, or the ocean, is the streams base level.

This level is an important control of the longitudinal prole because streamsadjust over time by erosion or deposition to provide a smooth transition tobase level.

The relation between channel slope, S0(X), and downstream distance, X, for agiven stream can usually be represented by empirical relations of one of the followingforms:

S0(X) = S0(0) exp(k1 X), (2.2a)or

S0(X) = k2 Xm2 , (2.2b)or by a relation between slope and drainage area, AD,

S0(X) = k3 ADm3 , (2.2c)where the coefcients and exponents vary from stream to stream depending onthe underlying geology and the sediment size, sediment load, and water dischargeprovided by the drainage basin. Increasing values of k1, |m2|, or |m3| representincreasing concavity.

It is generally assumed that the smooth concave profiles modeled by equation 2.2acrepresent the ideal form that evolves over time in the absence of geologicalheterogeneities or disturbances. Deviations from this form that produce convexitiesin the prole are common and are due to 1) local areas of resistant rock formations,2) introduction of coarser sediment or a large sediment deposit by a tributary orlandslide, 3) tectonic uplift, or 4) a drop in base level. Pronounced steepenings dueto these causes are called knickpoints.

Knighton (1998) reviewed many studies of longitudinal proles and concluded,Channel slope is largely determined by 1) the quantity of ow contributed bythe drainage basin and 2) the size of the channel material.

In almost all river systems, bankfull (or average) discharge increases downstreamas a result of increasing drainage area contributing ow; thus, channel slope can beestimated as

S0(X) = k4 Q(X)m4 d(X)m5 , (2.3a)or

S0(X) = k5 AD(X)m6 d(X)m7 , (2.3b)

0500

1000

1500

2000

2500

3000

3500

0 500 1000 1500 2000 2500Distance (km)(c)

Elev

atio

n (m

)

Rio Grande

0

500

1000

1500

2000

2500

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000Distance (km)(a)

Elev

atio

n (m

)

(b)

0

1000

2000

3000

4000

5000

6000

0 500 1000 1500 2000 2500 3000Distance (km)

Elev

atio

n (m

)

Indus

Mississippi

Figure 2.4 Examples of longitudinal proles of large rivers. All examples are basicallyconcave-upward, even those in which discharge does not increase downstream (lower Indus,Murray, Rio Grande), but some have convex reaches, especially pronounced for the Rio Grandeand Indus. Data provided by B. Fekete, Water Systems Analysis Group, University of NewHampshire. (continued)

(d)

(e)

0

50

100

150

200

250

300

350

0 50 100 150 200 250 300 350 400Distance (km)

Elev

atio

n (m

)

Murray

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 1000 2000 3000 4000 5000 6000Distance (km)

Elev

atio

n (m

)

Amazon

0

200

400

600

800

1000

1200

1400

1600

1800

0 1000 2000 3000 4000 5000 6000Distance (km)

Elev

atio

n (m

) Zaire (Congo)

(f)

Figure 2.4 Continued

NATURAL STREAMS 31

where Q is some measure of discharge (e.g., bankfull or average discharge), d issome measure of sediment size (e.g., median sediment diameter), X is downstreamdistance, andAD is drainage area.The values of the empirical exponents m4 throughm7vary from region to region. As discussed in the following section, d tends to decreasedownstream in most stream systems; thus, relations of the form of equation 2.3 predictthat the more rapid the downstream increase in Q or AD or the downstream decreasein d, the more concave the prole.

2.1.4 Downstream Decrease of Sediment Size

There is a general trend of downstream-decreasing bed-material sediment size invirtually all river systems (gure 2.5a), which is typically modeled as an exponentialdecay:

d(X) = d(0) exp(k6 X), (2.4)where d(0) is the grain size at X = 0 and k6 is an empirical coefcient that varies fromstream to stream (values for various streams are tabulated by Knighton 1998). In manyriver systems, the exponential decay is reset where tributaries contributing coarsematerial enter a main stream (gure 2.5b). Interestingly, the rate of size decrease isespecially pronounced in gravel-bed streams, and an abrupt transition from gravel tosand is often observed.

Two physical processes produce the size decrease: grain breakdown by abrasionand selective transport of ner sizes. Experimental studies have shown that abrasiondoes not produce the downstream-ning rates observed in most rivers (see, e.g.,Ferguson et al. 1996), so selective transport is almost always the dominant processproducing downstream sediment-size decrease.

Hoey and Ferguson (1994) were able to simulate the rates of sediment-size decreaseobserved in a Scottish river using a physically based model. Their results supportedthe strong correlation between downstream rates of slope decrease and of particlesize, as reected in equation 2.3.

2.2 Channel Planform: Major Stream Types

2.2.1 Classication

Channel planform is the trace of a stream reach on a map.

The continuum of channel planforms in natural streams can be initially dividedqualitatively into those with a single thread of ow and those with multiple threads.Channel planforms are further categorized quantitatively by their sinuosity:

The sinuosity, , of a stream reach is dened as the ratio of its channellength, X, to the length of its valley,2 Xv (gure 2.6).

XXv

. (2.5)

25

30

35

40

45

50

55

60

65

70

0

(a)

(b)

2 4 6 8 10 12 14 16 18 20

Mea

n gr

ain

size

, d (

mm

)

d = 69exp(0.042X)

ENTRY OF MAJOR TRIBUTARIES

25

30

35

40

45

50

55

60

65

70

0 2 4 6 8 10 12 14 16 18 20Downstream distance, X (km)

Mea

n gr

ain

size

, d (

mm

)

Figure 2.5 Downstream decrease in sediment size in the River Noe, England. Dots showmeasured values. (a) General trend modeled by exponential decay. (b) Resetting ofexponential decay due to inputs of coarser material by tributaries. From Fluvial Forms andProcesses (Knighton 1998), reproduced with permission of Edward Arnold Ltd.

NATURAL STREAMS 33

Figure 2.6 Sinuosity of a reach of the South Fork Payette River, Idaho. The dashed arrowsrepresent the valley length, Xv, which equals 2.61 km. The channel length, X, is 3.53 km;thus, the reach sinuosity is 1.35. Contour interval is 40 ft. Solid and dashed parallel linesare roads.

BecauseX Xv, it must be true that 1. If the difference in elevation betweenthe upstream and downstream ends of a reach is Z , the channel slope, S0, and valleyslope, Sv, are given by

S0 = ZX

(2.6)and

Sv = ZXv

. (2.7)Therefore,

S0 = Z Xv =

Sv

Sv, (2.8)and we see that, for a given valley slope, channel slope depends on channelplanform.

34 FLUVIAL HYDRAULICS

Figure 2.7 An intensely meandering stream in central Alaska. This stream has migratedextensively and left many abandoned channels. Photo by the author.

The most widely accepted qualitative categories of channel planforms, introducedby Leopold and Wolman (1957), are meandering, braided, and straight:

Meandering reaches contain single-thread ows characterized by highsinuosity ( >1.3) with quasi-regular alternating bends (gure 2.7).

Braided reaches are characterized by ow within permanent banks in twoor more converging and diverging threads around temporary unvegetated orsparsely vegetated islands made of the material being transported by thestream (gure 2.8). At near-bankfull ows, the islands are typically submergedand the ow becomes single thread.

Straight reaches contain single-thread ows that, while not strictly straight,do not exhibit the sinuosity or regularity of curvature of meandering channels.

In many cases the thread of deepest ow (called the thalweg) meanders within thebanks of straight reaches. In nature, straight reaches on gentle slopes are rare, and theiroccurrence often indicates that the stream course has been articially straightened.

A fourth basic category is often added to the three proposed by Leopold andWolman (1957):

Anabranching (also called anastomosing or wandering) reaches containmultithread ows that converge and diverge around permanent, usually

NATURAL STREAMS 35

Figure 2.8 A braided glacial stream in interior Alaska. Photo by the author.

vegetated, islands. Individual threads may be single threads of varyingsinuosity or braided.

These basic categories have been elaborated by Schumm (1981, 1985) to provide theclassication shown in gure 2.9.

2.2.2 Relation to Environmental and Hydraulic Variables

Many empirical and theoretical studies have attempted to relate channel planform tochannel slope, the size of material forming the bed and banks, and the timing andmagnitude of ows of water and sediment provided by the drainage basin (Bridge1993). The pioneering work of Leopold and Wolman (1957) showed that the presenceof these patterns can be approximately predicted by where a given reach plots on agraph of channel slope versus bankfull discharge. They used empirical observationsto dene a discriminant line given by

S0 = 0.012 QBF0.44, (2.9)where S0 is channel slope and QBF is bankfull discharge in m3/s. Braided reachesgenerally plot above the line given by equation 2.9, meandering reaches tend to plotbelow it, and straight reaches may plot on either side.

Figure 2.9 Schumms (1985) classication of channel patterns. The three basic types arestraight, meandering, and braided; anastomosing streams are shown as a special case of braidedstream. The arrows on the left indicate typical associations of stream type with stability, theratio of near-bed sediment transport (bed load) to total sediment transport, total sedimenttransport, and sediment size. From Fluvial Forms and Processes (Knighton 1998), reproducedwith permission of Edward Arnold Ltd.

NATURAL STREAMS 37

0.00001

0.0001

0.001

0.01

0.1

1 10 100 1000 10000 100000

Bankfull Discharge, QBF (m3/s)

Cha

nnel

Slo

pe,

S0

1

510

50

100

500

Figure 2.10 Braiding/meandering discriminant-function lines. Braided reaches plot abovethe lines; meandering reaches, below. Solid line is the discriminant function of Leopold andWolman (1957) (equation 2.9); dashed lines are discriminant-function lines of Henderson(1961) (equation 2.10) labeled with values of d50 (mm).

The approach of Leopold and Wolman (1957) was rened by Henderson (1961),who found that the critical slope separating braided from meandering reaches was alsoa function of bed-material size and that the discriminant line could be expressed as

S0 = 0.000185 d501.15 QBF0.44, (2.10)where d50 is the median diameter (mm) of bed material (measurement and charac-terization of bed material are discussed further in section 2.3.2). The discriminantfunctions given by equations 2.9 and 2.10 are plotted in gure 2.10; note that forHendersons equation, both meandering and straight (< 1.3) channels plot below thelines given by equation 2.10, whereas braided channels plot above them. Henderson(1966) showed that an expression very similar to equation 2.10 could be theoreticallyderived from considerations of channel stability.

More recent studies have pursued similar theoretical approaches. For example,Parker (1976) derived a dimensionless stability parameter P, which is calculated as

P g1/2 S0 YBF1/2 WBF2

QBF, (2.11)

where g is gravitational acceleration and YBF and WBF are bankfull depth andwidth, respectively. When P > 1, a braided pattern develops in which the numberof subchannels in the stream cross section is proportional to P; when P 1, ameandering channel develops. Further theoretical justication of Parkers approachand support of discriminant functions of the form of equation 2.11 is given byDade (2000).

38 FLUVIAL HYDRAULICS

However, the criterion of equation 2.11 has been criticized because it requiresinformation about the channel dimensions (YBF and WBF) and form (S0, whichdepends in part on sinuosity as shown in equation 2.8) and so would be of littlevalue for predicting channel planform. To avoid this problem, van den Berg (1995)developed a theory based on stream power (dened and discussed more fully insection 8.1.3) and proposed that a function relating valley slope, Sv, and bankfulldischarge, QBF , to median bed-material size, d50, can be used to discriminate betweenbraided and single-thread reaches with 1.3. He proposed two discriminantfunctions, one for sand bed streams (d50 < 2 mm),

Sv QBF0.5 = 0.0231 d500.42, (2.12a)and one for gravel-bed streams (d50 > 2 mm),

Sv QBF0.5 = 0.0147 d500.42, (2.12b)where QBF is in m3/s and d50 is in mm. Reaches that plot above the line givenby equation 2.12 are usually braided; those below are usually meandering (i.e.,single thread with 1.3) (gure 2.11). Straight reaches (i.e., single thread with< 1.3) plotted both above and below the discriminant lines, as also found by Leopoldand Wolman (1957). Bledsoe and Watson (2001) rened van den Bergs approach byreplacing the single discriminant equation 2.12 with a set of parallel lines that