The Hydraulic Geometry of Stream Channels and Some Physiographic Implications

Home , Manning formula, Stream load, William Morris Davis

The Hydraulic Geometrv J J of Stream Channels and

Some Physiographice8 Implications

GEOLOGICAL SURVEY PROFESSIONAL PAPER 252

b UNITED STATES DEPARTMENT OF THE INTERIOR

FRED A. SEATON, Secretary

GEOLOGICAL SURVEY

Thomas B. Nolan, Director

Reprinted 1954, 1959

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington 25, D. nts (paper cover) Price 40 cents The Hvdraulic Geometrv J J of Stream Channels and Some Physiographic Irn plica tions

By LUNA B. LEOPOLD and THOMAS MADDOCK, JR.

GEOLOGICAL SURVEY PROFESSIONAL PAPER 25 2

Quantitative measurement of some of the hydrauZic factors that help to determine the shape of natural stream channels: depth, width, velocity, and suspended load, and how they vary with discharge as simp Ze power functions. Their interrelations are described by the term “hydradicgeometry.”

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1953 PREFACE This report was prepared by the Technical Coordination Branch of the Water Resources Division of the Geological Survey as a part of the investigation of water-utilization problems. The junior author, Mr. Maddock, is assistant chief of the Hydrology Division, Bureau of Reclamation. The authors have benefit,ed greatly from constructive suggestions made by many friends and colleagues. Particular acknowledgment is due J. Hoover Mackin and Walter B. Langbein whose counsel and criticism contributed greatly to whatever of value is contained in this report. Among the persons who read an early draft and whose suggestions are highly valued are A. N. Strahler, John P. Miller, E. W. Lane, L. C. Gottschalk, K. B. Schroeder, C. C. Nikiforoff, Charles E. Stearns, and W. M. Borland. The continuous encouragement of W. W. Rubey is appreciated. L. B. L. m GLOSSARY

Antidune. A transient form of ripple on thc stream bed analogous to a sand clunc.; an antidune progressively moves upstream. Control. The “control” of a stream-gaging station refers to tho nature of the channel downstream from the measuring section which determines the stage-discharge relation. Current meter. An instrument for measuring water speed; consists of cups which are rotated by the flowing water; rate of rotation is a function of water speed. Cumulative frequency, or flow-duration, curve. Graph of rate of daily discharge in relation to cumulative frequency, indicating the percent of time a given discharge is equalled or exceeded. Discharge. The amount of water flowing in a channel expressed as volume per unit of time. 2, Froude number. A measure of tranquility of flow defined as 48__ Gage height. The water-surface elevation referred to some arbitrary datum. Hydraulic gradient. The slope of the energy grade line, or slope of the line reprcsenting the sum of kinetic and potential energy along the channel length. It is equal to the slope of the water surface in steady, uniform flow.

Hydraulic radius. A measure of water depth in a channel defined as cross-sectional area of flowing water divided ~ by the length of wetted perimeter; approximates mean depth in a wide channel. Manning formula. A formula including empiric constants relating open-channel water velocity to hydraulic gradient, hydraulic radius, and channel roughness. Rating curve. The graph of discharge plotted against water-surface elevation for a given cross section of a stream. Sediment station. A river section where samples of suspended load are taken each day, or periodically. Stage, or river stage. Synonymous with gage height. Stream-gaging station. An installation including a mcchanical continuous recorder of water-surface elevation; at the same cross section periodic velocity measurements are made, and are used to construct the rating curve for thc station. These two types of data are used to furnish a continuous record of water flow. Suspended sediment-discharge curve. A graph of water discharge plotted against concomitant measured sus- pcnd ed-setlimen t load.

SYMBOLS

a coefficient representing width at unit discharge k coefficient representing velocity at unit discharge I, exponent in width-discharge relation L suspended-sediment load, weight per unit of time c coefficient representing depth at unit discharge m cxponent in velocity-discharge relation C Chezy coefficient in v=CJRS n Manning roughness factor d mean depth P coefficient representing load at unit discharge D pipe diameter 0 water discharge in cubic feet per second (cfs) .f exponent in depth-discharge relation R hydraulic radius, approximately equal to mean F a function of depth in wide channels 7j sediment-grain diameter S slope of water surface in open channels g acceleration due to gravity 2) velocity j exponent in suspended load-discharge relation W width Iv CONTENTS

Page Page

Abstract,._. .-. . . ~. ~ .. .~ ~ ~.~.~ ~ ~ - ~ . -. . .~.~~. .. 1 The hydraulic geometry of si ream cliannels in rrlation to

.. ~ ~ ~ ~ . .~ ~.~.~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ General st,atemerit,. , . 1 sediment load-Continued

The hydraulic geometry of stream channels ~ . .~ 2 Interrelations of width, depth, vrlocitx, and suq-

Concept, of frequency of dischargc: ~ ~ ~ ~ ~ 2 pended load to a variable discharge 24 Variation of hydraulic characteristics iii a pari icrilar Interrelation5 of nidth, depth, velocily, aiid bt~lload

ect.ion of a river--. . .~ ~ ~ ~ - . ~ .~~ ~ ~. 1 at a given discharge - 28 Variat,ion of hydraulic characteristics in a dowii- (’hannel-shape adjustnient during individual floods 30

stream direction ~... ~ ~ ~.~ ~ ~ ~ ~ ~ ~ - - ~ ~ ~ . . 9 The role of channel roughness and qlopc 111 thr adjrl.;t- Relation of channel shape t,o frrquency of discharge- 10 merit of channel shape to srdlm(~ritload 35 The hydraulic geometry of stream chaririels in relat.ion t,o Sonic physiographic implications- 43

sediment load ~ ~ ~ .. - ~ .. -.- .. -. ~ . . ~ ~.. 19 The stable irrigation canal--an analoa\ to a gt adrd Relations of suspendcd seditrient io tlisrliarge in a river. - __. ._ 13

particular cross sectioii of a rivrr ~ . 19 Sediment and 1 Iir longitudlnal piofilc .- 48 Relations of suspended scdimeiit, to discharge in 11ie Srirriniary and interpretation 51

downstream direct,ioii.. ~ .- -...... ~ .~ ~ 21 References. -. 53 Interrelations of width, dcpt,li, velocity, and 811s- App~iidiu 54

pciided load at a given dischargr-~ ~ . ~~..23 Iiidev 57

ILLUSTRATIONS

FIGURE1. Cumulative frequency, or flow-duration, curve for Powder River at Arvada, Wyo--- .-._~___. ---.-.---~ ---- - 3

2. Cornparison of different rates of discharge at a station and downstream-- ____.__ ___- -.-. -. -. . . . . ~ ~ - - -- - ~ -- - 4 3. Typical relat.ions among width, depth, velocity, discharge, and gage height, (:lieyennc River iic’ar lhgle But,t>c, S. nak__..__.______-.-----.-..--.------.----.--.---.--~~---.-.-5

4. Relation of widtali, depth, and velocity to discharge, Powder River at Arvada, \i-yo., and at locate, >lotit. ~ ~ - - 6 5. Width, depth, and velocity in relation to mean annual discharge as discharge increases dowiistrearii, Powder River and tributaries, Wyoming and Montana------.. . ------.- - - .-. . - - - - - .- - 10 6. Width, depth, and velocit,y in relation to mean annual discharge as discharge incrcascs downstrearii, 13igIiorn

River and tributaries, Wyoming and Montana--. -.. - - - __ ------.- - - - - ~... . - ~ - - .- - - - .. - .. ~ ~ - - 11 7. Width, depth, and velocity in relation to mean annual discharge as discharge incrrases downstrcarn, Arikaree, Republican, Smoky Hill, and Kansas Rivers in t,lie I

of Missouri and lower Mississippi Rivers______- -.- ______._.__ __.- __ -. . . . __ - -.- -.~ ~... . . - -.- . 1 3 9. WVidt,h, depth, and velocity in relation to mean aiinual discharge as discharge increases dowiistreaiii in various river systems 15 10. Width, depth, and velocity in relat,ion to discharges representing two frrqueilcics, R/Iaumcc and Scioto River basins, Oliio.______-_------.------.------~.~------. _...-~-- 17 11. Width, depth, and velocity iii relation to discharge of various frequencies, Mauniee and Scioto Ijivcr basins, Ohio. 18

12. Diagraniiriatic relations of widt,h, dept.h, and velocity t,o discharge at a station and dowiislreairi-. .. ~ - - ~.~ -. . . . 19 13. Relation of suspended-sediment load to discharge, Powder River at Arvada, Wyo .__._.__._-..~. -.--...... --- 20

14. Typical relations in the hydraulic geometry of natural stream channel systems ._____.__ _.___-_--. .~ ~ -. . .--- 22 15. Average relation of suspended-scdiment load t,o ciiannel-shape factors at a colistant, discharge of 500 cfs (froin

river data in appendix) - - - ~.- - ~ - - - __ ------___ ------.. ------.. ~...... ~ ~ ~.~ .~. 23 16. Average relation of suspended-scdimelit, load to channel-shape factors at four discharge rates (frolii river data

in appendix) ______.-.__ __- ______-. ______-.__ -______- -. .. .. _.- ~ ~..-. ... - -.-. - 24 17. 8;xample of width, depth, vt:locit,y iii relation to discharge (coiist,ructed froin fig. 10) .... ~__..- - . ._._. .. _~_. . 25 m 18. ltelation of b, 7’and j (construct,cd froin fig. 16) .______._____._.___.._____.______.___.. ~ ~._ . . ~. ~. 25

19. Average hydraulic geometry of river chaniiels ...____.._.______.______._..~-. .__~ - __... . ._.. - 27

20. Relation of bed load to channel-shape factors at constant discharge, from Gilbert dat,a.... - .. - ~ _..- -. . - -.__ - 29 21. Changes in channel-shape factors, San Juan River near Bluff, Utah, during flood of Sept~errit~er-I~rcc~rribrr1941 - 31 22. Channel cross sections during progress of flood, September-December 1941, San Juan River near Rl~~ff,l;t,ali--- 32 23. Changes in width, depth, velocity, water-surface elevation, and stream-bed elevatioii with discharge during flood

of December 104&.iurle 1041, Colorado River at Grand Caiiyon, Ariz-. _.~___.____.__...... ~ . - - ~..- - 32 v w ILLUSTRATIONS

Page FIQURE24. Channel cross sections during progress of flood, December 1940-June 1941, Colorado River at Grand Canyon, Ariz_~~~~~~~~~__~~~~______~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~34 25. Changes in channel shape factors during progress of flood, May-June 1948, Rio Graiide near Bernalillo, N. &lex_ 35 26. Channel cross sections during progress of flood, May-June 1948, Rio Craride near Bernalillo, N. hIes - - -.- - - 35 27. Relation of T$ idth, depth, velocity, water-surface elevation, and stream-bed elevation to discharge, Colorado River at S-uma, Ariz- ______------37 28. Progressh e changes of Colorado River at Yuma, A\riz., resulting from the construction of engineering works upstream~~_~______.~_.~_____~_~___~~__~~~~~~~~~_~~~~~_~_~_~_38

29. River profiles in reach of Colorado River near Yuma, Ark ______I______--.---39 30. Channel roughness (Manning n) in relation to discharge for various sizes of scdirnent (from flume data) __.__- - - 41 31. Relation of nidth, depth, and velocity to discharge for stable (regime) irrigation canals in India (Lacey data) -. 44 32. Variation of velocity and depth downstream for three assumed longitudinal profiles. ------49 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS AND SOME PHYSIOGRAPHIC IMPLICATIONS

By LUNAB. LEOPOLDand THOMASMADDOCK, JR.

ABSTRACT In a simplified example, if the discharge is constant, the Somc hydraulic characteristics of stream channels-depth, empiric relation is: At a given discharge, an increase in width width, velocity, and suspended load-are measured quantita- at constant velocity is accompanied by a decrease ip suspendcd- tively and vary with discharge as simple power functionq at a sediment load; conversely, at constant discharge and velocity, given river cross section. Similar variations in relation to an increase in width is accompanied by an increase in bed load. discharge exist among the cross sections along the length of a The writers, like many others, postulate that both discharge river under the condition that discharge at all points is equal in and sediment load are factors essentially independent of the frequency of occurrence. The functions derived for a given cross stream channel and depend on the nature of the drainage basin. section and among various cross sections along the river differ Thc typical relations between suspended-sediment load and only in numerical wlues of coefficients and exponent<. These water discharge are shown in quantitative terms. Specifically, functions are: at a given stream cross section, suspended-sediment load per unit of discharge characteristically increases rapidly with in- w=aQb creased discharge. However, suspended load per unit volume d=rQf of water tends to decrease slightly downstream. These characteristics are important determinants of the shape v=kQm of the cross section of a channel and the progressive changes in L = pQ’ its shape downstream. The empiric relation between hydraulic characteristics of the channel and suspended load provides, in where semiquantitative terms, a logical explanation of the observed channel shape. width w= The hydraulic geometry of river channels is presented for d =mean depth several river systems. It is shown that similar equations apply both to rivers and to stable (“regime”) irrigation canals which u=mean velocity neither scour nor aggrade their beds. The analogy demonstrates L=suspended-sediment load, in units of weight per unit time that the average river chabnel-system tends to develop in a way to produce an approximate equilibrium between the channel and &=water discharge, in cubic feet per second (cfs); a, c, k, p, the water and sediment it must transport. This approximate b, f; m,and j are numerical constants. equilibrium appears to exist even in headward ungraded trihu- taries and in a given cross section for all discharges up to the These relationships at a given channel cross section and down- bankful stage. stream when plotted on graphs are greatly similar even for river systems very different in physiographic setting. The relation- GENERAL STATEMENT ships are described by the term “hydraulic geometry.” In the data studied it appeared that when discharges are of Geomorphology, that branch of geology dealing with equal frequency at different points along a river, that is, equalled land forms, their genesis and history, has been classi- or exceeded the same percent of time, the velocity as well as the cally treated almost exclusively in a qualitative manner. width and depth of flow, increases with discharge downstream. William Morris Davis, for pedagogic reasons, invented This increase of velocity downstream results from the fact that the increase in depth overcompensates for the decrease in slope. words and contrived similes which were descriptive and The tendency for velocity to increase downstream exists on most easy to remember, but were supported by intricately streams despite the decreasing particle size downstream. This developed general argument rather than by field data. indicates that the mean velocity in a given reach is not merely a The qualitative approach to geomorphology has indeed function of the size of partirles which must be transported but is been constructive, but it would be desirable to analyze governed by a more complex interaction among several factors. Measurements of suspended-sediment load are available for a some of the concepts quantitatively. number of different river cross sections. The suspended load is There is available a mass of data on streamflow shown to be an index to total load for purposes of explaining the collected over a period of seventy years representing observed average characteristics of natural channel systems. rivers all over the United States. The field procedure An empiric quantitative relation among average measurements used by engineers to measure the rate of water dis- of width, depth, velocity, discharge, and suspended-sediment load is derived from data on natural rivers. This relation shows charge in a river furnishes concurrent measurements of that depth and width, as well as velocity, are functions of the mean velocity,width, shape ond area of the cross sec- load transported in the chsnnel. tion, as well as discharge. Moreover, in the last decade

I 2 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS the systematic collection of data on suspended sediment There are two possible levels at which this mechanism has begun though the number of stations collecting is of adjustment of river-to-load may be discussed. The as yet few. more detailed way would lie in the realm of hydro- The present paper reports on analyses of some of dynamics and would concern the mechanics of sedimcn t these data, keeping in mind possible application to transport. A second and more general approach is geomorphic problems. The available data are not through the considera tion of river hydraulics hut necessarily the best type for an analysis of stream without a detailed treatment of the hydrodynamics. processes. The parameters measured are perhaps not The second approach is the one used in thc present the most meaningful for such analysis. Nevertheless, paper, in part because it leiids itself better to an the available data do provide some generalized rela- understanding of the relation of hydrnulics to geo- tions of interest to the geomorphologist. morphology, and in part, hccause it lies within thr field The plan of the paper is as follows: of experience of thr authors. The characteristics of channel shape in certain river Thc last part of the paper is an attempt to explore in cross sections are examined. At a given cross section, a preliminary way how the relations among the hydrau- the width and depth of the channel and, thcwforc, t,he lic variables brar on some geomorphic problems. velocity change with the amount of water flowing in It is emphasized that the present study should be the cross section. The changes in width, depth, and considered preliminary in character. Data on strearn- velocity in response to changes in discharge are oh- gaging and suspended load discussed here arc clearly served to have certain characteristics which apply to imperfect measures of the variablcs of specific int crest many natural river cross sections, and these similarities to thr geomorphologist. hloivover, the present analysis are described in the first part of the paper. deals primarily with genwal trcnds in thr relations among measured vsriablcs, and much further work is These same characteristics are then examined down- mcessnry to explain the dcvin tions from these trcnds. stream along the length of natural channcls in order And finally, the suspended-load data available are to determine how they change there. Changcs in slope primnrily derived from Wrst ern streams. Comparable and roughness, which help determine the observed varia- data on strtams in humid areas are rriost meager. tions, are reserved for later discussion. These many rrstricttions are recognized by the authors The data used are those collected at regular river who attempt continuously to keep before the mind of measurement stations for purposes of determining the readtr thr limitations of the data, while emphasizing river discharge. Such data are obtained for purposes the gcrieral trcmls rat1ic.r than tlic inniinierablr individ- other than a study of fluvial morphology and are, for ual variations. the writers’ purposes, incomplete. Particularly, data Explanation of t,hese individual variations and on river slope and bed roughness are meagrr but some refinenlent of the gcneral concepts must await further useful ideas may be obtained even from thr incomplete work. operational data. In the next part of the paper, the river-channrl cliar- THE HYDRAULIC GEOMETRY OF STREAM CHANNELS acteristics including slope and roughness arr discussed in relation to observational data on suspended-sediment CONCEPT OF FREQUENCY OF DISCHARGE load. It is recognized at the outset that the unmeas- Before describing the channel characteristics at u ured bed load is of importance and that its omission given river cross section and how those characteristics from the available data is a limitation to the conclu- change tlownstrram, it nppcars desirable to discuss by sions which may be drawn. Yet some interesting rela- way of dcfinition, frequency of discharge and cumulative tions may he obtained by the use of data on suspended frequency, or Mow-duration, curves. The hydrologist load alone. These relations provide some picture of to whom these conceptions are everyday tools will the interaction of the hydraulic characteristics related pardon the inclusion here of a brief discussion aimed at to channel shape and a segment of the stream’s sediment providing the geologist a picture of frequency relations load and therefore seem to be potentially useful tools which is essential to an understanding of what will for the study of A uvial processes. follow. The relations between observed channel character- The procedure used in making stream measurements istics and suspended-sediment load involve the process is well described in the literature (Corbett, 1943; of adjustment betwern channel shape and sediment Linsley, Kohler, and Paulhus, 1949, pp. 182-198). load. Because the mechanism of adjustment between The daily mean discharge is the average of all rates the capacity of a stream to carry a sediment load and for a day, and is expressed in cubic feet per second. available load is important to the understanding of the The mean rate of discharge for each day of the year graded river, any advance in knowledge of this mecha- at measuring stations on many rivers is published by nism leads toward an improved understanding of river the U. S. Geological Survey in the annual Water- grading. Supply Papers. CONCEPT OF FREQUENCY OF DISCHARGE 3

The frcqur~iic~yof a daily mean tlischarge at a b’TIven on a large trunk river and a point on a small tributary rivcr %aging station, is tlrtcrminecl by counting the may have discharge rates of the same cumulative fre- number of occurrenew of this rate throughout a period quency-or frequency, as it is ordinarily expressed; but of rc~~ml111 hpdrologic practice the number of discharge in the tributary may be only a few cubic feet occurrmccbs is iisurilly prcsc~ntmi in the form of a cuinu- per second, and in the trunk river it may be thousands lativci frrqiienty, or flow-duration, curve, an example of cubic fert per second. of which is prrscntcd in figure 1 (aftcr Hembree, Colby, The median discharge is that discharge which is Sn-ciisoii, autl Davis, 1052, lig x)). To construct sut+li cquallcd or exceeded 50 percent of the time; that is, a curvc, A nuinbc~i~of chutcgorirs of rates of disclrargc. half the days have flow greater than the median, and art’ arhi tr:irily chosrn. ‘Flit. nurnher of days on which lialf less. At Arvada the median flow is about 160 cfs. tlic m(wsurrt1 cl2dp mean discharge fell in each category Tlic mean annual discharge, or the arithmetic mean is tictcrriiiiic~lIp tallying the (la tn. The numhcrs in of tlie flow of all individual days at Arvada is about eiic-li (ai1 tcgory :ire then accuniulatc~tifrom tlie largest 430 cfs, or alniost three times the median. hftan an- clisclinrgSr c.ntt.gory to the smallcst, anti each cumulative nual discharge is generally greater than niedian, and figurc is tlivitlrtl by the total numbc~of days of record. tlic mcsan rate of discharge is equallecl or exceetic(i on Thrb rcsultiiii t quotients wpwscnt the percent of timc somewhat less than half the number of days at most pivcn clisclitirgc~sare cqualletl or exceeded at the station. points 011 a river. ‘I’hcse iigiircxs representing pcrccnt of time are then As a rough generalization, it may be stated that the plotted agaiiist tlischargc to provide the flow-duration mean annual ratcs of discharge at all points on a large curve of whi(*Iitigirrc 1 is a sample. On it a discharge numhrr of rivers are equalled or exceeded about the of 1,000 c-fs or II~OI’Eo(~iirs ahout 8 perccnt of the time. sarnc percent of time. This rri(\ans that tlirougli a long pcriod, 8 percent of The values of average annual discharge at measuring the (lags, or 011 tllr averagtl 8 (lays in every 100, ex- stations on many rivers in the United States arc readily perirncwl a tlisc*liwgclof 1,000 cfs or more. available in the Water-Supply Papers of the U. S. Geo- The cumulative frequency, or duration, of a given logical Survey. For example, the average discharge daily discharge is expressed as the percent of time that for the Powder River at Arvada, Wyo., for the pcriod this rate is quallerl or t~sceecled. Discharge rates of the 1916-47 is published with the description of the station same cumiilativt~frequency may differ, depending on in Water-Supply Paper 1086 (1950) on page 237. tlie streams 1)txitig consitlrred. For csample, a point For purposes of definition the conception of a varying discharge at a given river cross section and downstream will now be explained. In figure 2 cross sections A and B represent two points on a river at low discharge; C and D are the same sections at higher discharge. These sections are shown at the right, as A, B, C, and 10.000 D,in their relation to the full length of the river ancl to its watershed at both low and high discharge. Though the highest discharge may flow over bank, this paper

1,000 is not concerned with discharges above bankful. These diagrams illustrate the conception that the -In width and depth of the channel increase at a given 0 I .-c river cross section with increase in discharge. Varia- a; 100 tions in discharge follow a pattern in time that is pe- P culiar to the position of the cross section arid to the ru0 L” river The difierent discharges at a given cross section n vary in frequency. Frequency is a function of the watershcd and its hydrologic and physical characteristics. IJnder the condition of low discharge represented by the diagram in the upper right corner of figure 2, it is postulated that every point along the river is experiencing a discharge that is small for that point; or, if all points along n river system are experiencing a relatively low discharge, the frequency of the rate of discharge at any one point is about the same as the fre- Percent of time flow equalled or exceeded dischorge mdicoted quency of the rate at any other point. Of course the rate FIGVREl.--Comul~tlve frequency, or flow-duration, curve for Powder River at Arvada, Wpo., 1917-50. of discharge, not its frequency, will generally be much 497780 0 - 59 - 2 4 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

A ------LOW DISCHARGE Constant frequency of discharge at each point LOW DISCHARGE II points experiencing (For exomple,flow equolled or exceeded 90 percent of the time) B

HIGH DISCHARGE Constant frequency of HIGH DISCHARGE dischorge at each point Infrequent condition (All points experiencing (For example,flow equalled or near-flood discharge) exceeded I percent of the time) D

FIGURE2.-Compaiison of different rates or dischnrac nt 3 given river cross aection and at points downstream. greatcr nctu the mouth of a river draining a large area downstream direction are associated with differences in than at some headwater point. Tlic same postulation width, depth, and velocity, as will be shown later. is made for the condition of liigli discharge, as repre- scntctl by thc tliugrum in thc lower right. VARIATION OF EYDRAVLIC CHARACTERISTICS IN A The two diagrams of the watershed are introduced to PARTICULAR CROSS SECTION OF A RIVER emphasize the points, first, that at a given cross section Changes in velocity, depth, and width of flowing different discharges have different frequencies ; and water take place as the discharge increases at a particu- second, that at cross sections situated at various points lar cross section of a river. Because river cross sections along the stream the rates of discharge are usually differ- tend tjo be roughly semielliptical, trapezoidal, or triangu- ent, one from another. Comparison of the various cross lar, increasing discharge results in an increase of each sections along a stream is made in this paper only under of the other three factors. Typical relations of width, the assumed condition that they are experiencing equal depth, and velocity to discharge will be shown for vari- frequency of discharge. ous points on different rivers. These relations will shorn The term “at a station” is used in this paper to mean that these three factors generally increase with increas- at a given cross section, and the term “in a downstream ing discharge as simple power functions. direction” refers to cross sections situated along the In figure 3A a number of variables are each plotted length of a stream. “Change of discharge downstream” against river stage. Stage, or gage height, is the eleva- means different discharges of the same frequency at tion of the water surface above some arbitrary datum, cross sections situated along the length of a stream. and is the quantity recorded by the water-level recorder The diffcrenres in discharge both at a station and in a of a stream gage. The discharge, width. and mean VARIATION OF HYDRAULIC CHARACTERISTICS IN A GIVEN CROSS SECTION 5

velocity are plotted against stage as ordinate. For In stream-gaging practice, the tlrpth and velocity are comparison, width, depth, and velocity are plottccl on mtvisiircd at various distances from the bank. From logarithmic scales against discharge in figure 3R. thrse depth mcnsurements and their spacing, the (TOSS- When velocity is measured by current meter at a gng- sectional area of flowing water is dctcrminetl. ing station for the purpose of computing discharge, the In this report, the depth represents the mean depth width of water surface and the depth and velocity at, :Issociated with a particular rate of clischarge, and was various positions in tlie cross section are recorded. Pig- clctcrniincd by dividing the mcasurcd cross-sectional ure 3 and other similar figures to be discussed were con- :ireti of flowing u ater by the corrcqonding width of stmcted from ciirrent-meter measurements. ~vntersurfacc. This provides, then, ti incan depth cclual The current-meter data which includc measurements to the depth of a rcrtangrilar clitiiinel having the same of width of water surface, depth, and velocity at various surface width as thc actual chnnrirl at the. given stage points across the river measuring-section, are filed in the tind haying the sanic caress-sect ioniil urca. district offices of the U. S. Geological Sur~eyand are Vt’locitj- discussed in this rcpor( is tlie quotient of available to the public for inspection and use. The tliscliarge divided by the arw of the (TOSS section, and original records must be used or copied in the Survey is the mean velocity of the (TOSS scction tis used in offic~,otherwise appropriate arrangements musl he liydraulic practice (see Linsley, Kohlcr, and Paulhus, made to insure against loss of the originals. 1949, p. 197). This mean velocity is not the most The plot of stage (gage height) against discharge is, in meaningful velocity parameter for discussing sediment engineering parlance, the rating curve of a station. transport but it is the only measiirc of velocity for which Rating cwrves of all stream cross sections are similar in :L large volume of data are availnble. Although the that the increase of discharge per unit increase of stage writers recognize its limitations, tlie mean velocity is is greater for high than for low discharges. Thus thc usrtl here in lieu of adequate data on a more meaningful general form of the ciirvc is typical of all open channels. pa ranieter . The width plotted in figure 3 is the width of the water Figure 3A demonstratcs that if width and velocity surface at any given gage height. aro plotted against river sttigc, there is considcrable

c--

Dischorae, in cfs 0 I 2 3 4 velocity, in fps 0 zoo 400 600 800 width, in feet Relation to discharge Relation to river stage 6 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

I00

50 + W W .4- c g 20 x 5 10

LI I I If- I I

". . I" Discharge, in cfs FIGURE4, A.-Relation of width, depth, and velocity to discharge, Powder River at Arvadil. Wyo VARIATION OF HYDRAULIC CHARACTERISTICS IN A GIVEN CROSS SECTION 7

FIGIRE 4, I3 -l

scatter of points and the curve defined is not easily In the following discussions, gaging stations haviw expressed mathematically. If width, depth, and veloc- channel control and cross sections gaged from a cable ity, are plotted against discharge on logarithmic paper, car were chosen when possible. Many stations where their relations to discharge are expressed by nearly measurements were made from bridges were used for straight lines, us can be seen in figure 3.73, and this may lack of more representative data, but these measure- be simply expressed mathematically. ments were utilized only for low stages where width Figure 3R is typical of a large number of gaging was still increasing with gage height; that is, stages stations for which similar curves were drawn. The lower than those where bridge abutments constrict the slopes and intercepts of the lines varied somewhat flow. between stations but straight lines on logarithmic paper Excluding the unnatural effects introduced by con- typified the relations in the data studied. striction such as bridge abutments, the data analyzed Other examples of these curves are presented in indicate that through ranges of discharge up to the figure 4 to indicate that through a considerable range bankful stage, the relations of width, depth, and of discharge this method of plotting provides essen- velocity to discharge in a natural stream section are in tially straight lines on logarithmic paper. There is the form of simple power functions and plot as straight considerable scatter of the points on these graphs. lines on logarithmic paper. When the river overflows Part of the scat,ter apparently results from temporary its banks, of course width increases rapidly and a new scour and fill of the stream bed. At the lower dis- relation to discharge obtains which will not be con- charges measured by wading rather than from a bridge sidered here. or cable, part of the scatter of points may be attributed The artificial effects introduced by atypical locations to the fact that successive measurements are not made of gaging stations do not appear to invalidate the at cxactly the same cross section. For present pur- conclusion just expressed. poses, the general positions of the lines representing It can be stated that the relation of discharge to other mean relations are of interest and further work will be hydraulic factors in natural river cross sections can be required to study the causes and import of the scattering described by the expressions: of the data. In the present study every available current-meter w=aQb (11 discharge measurement was plotted foi stations having d=cQl short records. For stations having long records, of which figures 3 and 4 are typical, the measurements to v=kQm (3) be plotted were chosen at random from theavailalhe data. Enough measurements were obtained to define where Q is discharge, w, d, and v represent water-surface the curve through the whole range of discharge of tjhe width, mean depth, and mean velocity, respectively. particular station. The letters b, f,m, a, c, and k are numerical constants. A gaging station is installed at a river cross section Because width, depth, and velocity are each a func- where the relation of discharge to stage is as stable as tion of discharge as described by the formulas above, possiblc. Specifically, a location is chosen if possible the three equations can immediately be related to one where discharge plots against gage height in a smooth another through the identity curve. For this reason gages are installed either upstream from a “section control,” where critical velocity Q=areaXvelocity, or, .Q= wdv (in the hydraulic sense) is attained over some outcrop or gravel bar, or in a reach of channel where the channel substituting from the above properties determine the st age-discharge relation. The Q=aQbXcQ’X kQ” latter, called a “channel control,” merely means a or relatively straight and uniform reach generally devoid Q=aCkQo+f+m of unusual riffles or bars. In periods of low water, current-meter measurements It follows therefore that are ordinarily made by wading. When wading is impossible, current-meter measurements are often made b +f+m= 1 .O from bridges. and The cross section of a stream at a gaging station aXcXk=l.O above a section control is not representative of an average reach of channel, particularly at high dis- In figures 3B and 4, the constants, b,f, and m repre- charges. More representative sections are found at sent the slopes of the three lines on the graph, and the gaging stations having channel control. These stations sum of these slopes must equal unity. The values of are often equipped with a cable car from which the the slope of the lines are indicated in the efigures and current meter may be operated. their sum is indeed unity. VARIATION OF HYDRAULIC CHARACTERISTICS IN A DOWNSTREAM DIRECTION 9

The constants, a, c, and k represent the intercepts of of width to depth decreases with increasing flow through the lines and are respectively, values of w, d, and at the channel. discharge of unity. Because the values of discharge in Thc relative rates of increase of width and depth are the actual data are generally much larger than unity, functions of channel shape. It can easily be shown that the product of aXcXk is riot computed at unit dischargv a channel of triangular shape is the only channel for but might better be considered in the following light. which the width-to-deptli ratio remains constant with The product of the values of width, depth, and vrlocity changing discharge. Traptzoidal and elliptical chan- at any given discharge must equal that discharge. For nels are characterized by a decrease of the width-to- example, in figure 3B, at 1,000 cfs: w=245 ft, d= 1.7 ft, depth ratio, and in rectangular channels the width-to- and v=2.4 fps; their product wXdXv equals 1,000. depth ratio tlccreases with increasing discharge at a Because it has been stated that as a general rule. ra te even faster than in trapezoidal sections. equations (l), (2), and (3) characterize hydraulic In summary, relations of width, dcpth, and vc1ocit)- relations for representative natural stream channrl to discharge in the natural river cross sections studied cross sections, it is logical to inquire how the sloprs are in the form of simplr power functions. Thr rrlative and intercepts represented by the numerical constants rates of increase of width, dcpth, and velocitj- are vary among streams, and what range of numerical determined by tlic shape of the channel, the slope of values they represent. Width and depth for a given the water surface, and by the roughness of the wettrd discharge vary widely from one cross section to another perimeter. It is emphasized again that the forcgoing and, therefore, the intercept values, a, c, and k,will vary. discussion is merel3- R description of natural river cross Further work is necessary to determine the factors sections. The discussion of how width, depth, velo- which govern these variations and to determine the city, discharge, slope, and roughness interact to pro- extreme limits. However, there is some consistency vide the observed relations is reserved until after the in values of the slopes of the lines for different streams descriptivr presentation is completed. as will now be explained. The aspect of the above discussion which is different) Average values of thr exponents b, .f, and m have from relations discussed in the voluminous literature been computed for 20 river cross sections representing on open-channel hydraulics is thr plotting of width, a large variety of rivers in the Great Plains and the depth, and velocity against discharge on logarithmic Southwest. Though these particular averages are biased paper. A straight-line relation between gage height toward semiarid conditions, they nevertheless provide and discharge on logarithmic paper is widely known some indication of the order of magnitude of typical and is used in man?- offices of the Geological Survey in values. These averages are the construction of rating curves for river srctions having a shifting control. b=0.26 VARIATION OF HYDRAULIC CHARACTERISTICS IN .f=0.40 A DOWNSTREAM DIRECTION m=0.34 Frequency of discharge is especially important in an understanding of the hydraulic geometry of river Tlie appendix includes a summary of some of the systems and how the width, depth, and velocity of data from which the above values were derived. For natural rivers change in a downstream direction. each station, graphs similar to figure 3R were plotted As already emphasized, comparison of various cross froin current-meter measurements, and a straight line sections along the length of a river is valid only under was drawn by eye through the points. No statistical- the condition of constant frequency of discharge at all fitting procedure was used in drawing the line because cross sections. the use to which the data are to be put did not appear The mean annual discharge is equalled or exceeded to justify such accuracy. The slope of each line was about the same percent of time on a large number of measured directly from the graph. rivers. It represents roughly the discharge equalled That the values of the exponents m, b, and .f were or exceeded 1 day in every 4 over a long period. averaged for the 20 river cross sections does not imply Though exactly the mean annual discharge may not any expectation that the values should be closely similar. occur at all points on any given day of record, it is The mean values are presented so that the relative closely approximated for a large number of days each order of magnitude of the three exponents can be visu- year . alized. In a later section a partial and tentative ex- The data forming the basis of the discussion which planation of the variability of these exponents among follows are again current-meter measurements made various rivers will be given. at gaging stations of the Geological Survey. For present purposes it is sufficient to note that at The width, depth, and velocity corresponding to the most river cross sections, depth increases with discharge mean discharge are plotted against mean discharge in somewhat faster than does width (f>b). The ratio figure 5 for all the gaging stations on all tributaries to 10 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

200 EXPLANATION

100 1. Red Forb near Barnun), \\yo.

2. !diddle Fork Powder River abotc I\a!ccc. 1V)o.

3. hliddle Forh Powder River ncar Laycrc, Kfo. 50 4. R'urtli Fork Powder River near Ilar~l~on,1\10.

5. iiorttt Fork Powder niw n~w'kyanrth, \\yo.

6. South Fork I'owdrr River nPor I\a>cw, i\)n,

20 i. Powder Riwr at Suszcu, l\)o,

8. hliddle Fork Crazy \\oman CrrcL nrnr huh, \\yo.

9. North Fork Crazy \\'oman Crrrl, ncnr hffaln, Wyo. 10 IO. Korth Fork Crazy \\oman CrrcL nrar Crriib, K)u. 2 11. Crazy IVomsn Creck near .Arrada, \\)o.

I?. Powder Rirer at Arvada, Kyn. ilr 13. North Fork Clear Creek near Buffalo, \Vyn. I .o 14. Clear Creek near Buffalo, \\yo.

15. South Fork Rock Cree!? near Buffalo, \\yo.

16. Rock Creek near Buffalo, \\yo. 0.5 17. South l'iney Creek at \\illow Par\,, \\lo.

18. Piney Crwk at Krarney, Wyn.

19. Piney Creek at C'cross, li'yo.

2 20. Clear Creek near An-ada. Wyo.

21. Little Pouder Ricer near Broadus, \I lnl.

22. Powder River 3t hlonrhearl, \lank. I .o 23. Powder River near Locate, \!nnt.

0.5 10 50 100 500 1,000 Mean annual discharge, in cfs

Ft(:I.HE b.-Width, depth, and velocity in relation to mean annual discharge as discharge increases downstream, Powder River and tributaries, n')-oniin!! and Mnnrafia the Powder River in Wyoming and Montana for which the rivers plotted, the depth, width, and velocity tend adequate data were available. Each gaging station to increase progressively as power functions of dis- provides one point in each of the plotted diagrams. charge. There is again considerable scat,ter of points Throughout the figure the abscissa value for each gaging in the graphs. There are reaches of the rivers repre- station is its mean annual discharge. sented in figures 5-8 on which either width, or depth, For purposes of comparison, similar graphs represent- or velocity decreases downstream. The present paper, ing the change of width, depth, and velocity with however, is concerned with the general trends and average annual discharge downstream are presented further work will be necessary to explain the details of in figures 6-8. These represent the Bighorn River and deviations from these trends. tributaries in Wyoming and Montana, the Arikaree, The graphs of figures 5-8 are referred to as the Republican, Smoky Hill, and Kansas Rivers in the changes in width, depth, and velocity with increasing Kansas-Nebraska region, and gaging stations along discharge in the downstream direction. It will be the main trunk of the Missouri and lower Mississippi understood that a small tributary having a relatively Rivers. low mean annual discharge may enter the main trunk far Proceeding downstream in a given river, the dis- downstream, but its plotted position on the abscissa charge tends to increase because of the progressively scale will be determined by its mean discharge. increasing drainage area, though of course there are Strictly speaking, the "downstream" graphs in fig- some streams in which discharge decreases downstream, ures 5-8 show how width, depth, and velocity change particularly in arid areas. It can be seen that for all with discharges of equal frequency. Discharges of VARIATION OF HYDRAULIC CHARACTERISTICS IN A DOWNSTREAM DIRECTION 11

FIQURE&--Width, depth, and velocity in relation to mean annual discharge as discharge iirrrrases downstream, Bighorn River and tributaries, Wyoming, and Ilont:ma, and Yellowstone liiver, Monlnna.

EXPLANATION

1. Bighorn River at Manderson, Wyo. R. Wind River at Riverton, Wyo. 15. Medicine Lodge Creek near 113 nttville. IVyo. 2. Bighorn Itiver at Themopolis, Wyo. '3. Wind River near CrOwheHrt. Wso. lfi. (looseberry Creek near Orass Creek, IVyo. 3. Wighoru River at Katie, Wyo. 10. Oreybiill River near HRsin, Wyo. 17. Yellowstone River at ('wain Springs, Mont. 4. Bighorn River St. Xavier, Wyo. 11. Greyhull River at Meeteetsee. \V).o. 14. Tellowstone River at Billings, Mont. 5. Dighofn River near Custer (Hnrdin), Mont. 12. Pop0 Agie River near Riverton, Wyo. 19. Yellowstone Itiver new Ridneq-, \font. 6. Wind River near Duhois, Wyo. 13. North Fork Owl Creek near Anchor, Wyo. 7. Wind River xiear Hurris, Wyo. . 14. Owl ('rerk near l'hermopolis. Wyo.

497780 0 -59 -3 12 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

Mean annual discharge, in cfs

FIGURE?.-Width, depth, and velocity in relation to mean annnel discharge as discharge increases downstream, Arlkaree, Repnbllcan, Smuky IIiII, and Kansds Itivers in tlic Kansas River system, Kansas and Nebraska. EXPLANATION

13. Kansas River a1 \\anlego, Kans. 1. Arikaree Hiver at Haigler, Nelr. 7. Smoky Hill River at Russell, Kans. 14. Kansas Riyer at Topeka, Kans. 2. Republican Kiver at Ci~lbertson,Nebr. 8. Smoky Hill River st Ellsworth, Kans. 15. River 3. Republican River near Rloominglon, Nebr. 9. Smoky Hill River near Langley, Kans. Kansns at Lecofnpton, Knns. if;. Kansas River at hnner Springs, Kans 4. Republican River at Clny Center, Kans. 10. Smoky Ilill River at Linsborg, Kans. 5. Smoky Hill River at Elkader, Kans. 11. Smoky Ilill River at Enterprise, Kans. 0.. Smoky Hill River nt Ellis, Kens. 12. Knusis River at Ogden, Kans. VARIATION OF HYDRAULIC CHARACTERISTICS IN A DOWNSTREAM DIRECTION 13

5,OOC

2,OOG

+ 8 1,003 c ._c

L .- 5CO 3 -

I ..

200

100

1,000 10,000 100,000 I, 000,000 Mean annual discharge, in cfs

FIG~-its8.-V'icll ti, drplh, :~rid\ docity irl relation to mean aI1no;rl (lischarpe as discharge increases downstream, main trunk of Missouri and lower Mississippi Rivers.

EXPLANhTION

1. Missouri River at Bisniarrk, h. Ihk. 4. Missouri River at Kansas City, &Io. 7. Mississippi River at St. Louii, Mo. 2. Missouri lliver at Pierre, S. 1)ak. 5. Missouri River at IIermann, Mo. 8. Mississippi River at Memphis, Tenn. 3. Missouri River at St. Josellli. hfo ti. Mississippi River at Alton, Ill. Y. Mississippi River near Vicksburg, Miss. 14 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS equal frtqucnc*g tcritl to incrcasr. downstream wit11 in- ineiin nrinuril discharge between small tributaries ant1 creasing drainage area. Thc phrase “in the clown- the main stem to allow determination of the slope of strrnm dirvction” is merely usrd as a short designation the lines. A few rivtir systems studied provide lines for this cvnccption. with slopes quite diffcrent from the majority. Figure By me of the graph, tributarics can be compared 9 includes one such anomaly, the Loup River system, directly nith the trunk stream. On figure 5 the gaging in order to demonstrate the range of slopes of lines stations on the main trunk of the Yowlrr Rivrr (as an found in the data studied. In the examplr of the 1,oup example, point 7) and t~ single large lieadwatrr fork River, the ground-wat er contribution from the sand (hlidtlle Fork, point 2) are designated by the solid hills at tlic headwaters is believed to br partly respon- triangles. Stations on a single tributary, Clear Crcek sible for its anomalous behavior. (point 14), and its headwater tributary, North Fork of The streams chosen to be included in figure 9 diff(1r Clear C‘reelc (point 13), nrc designated by solid clircles. marktdly in physiographic setting. The Tombigbee Each of thcsc two srts of points describe a single river River of Alabama is a coastal-plain stream, narrow and stem tint1 arc downstream graphs uncomplicated by deep. At a discharge of 1,000 cfs its depth is about 7.5 tributaries. It can br seen that were lincs to be drawn ft and its width 110 ft. The Republican and Kansas through those two srts of points, separately, there Rivers are typical streams of the Great Plains. The?; would be a difference from the lines on the graphs are relativelq- wide and shallow. At 1,000 cfs the lines representing the T’owdor River basin as a whole. Yet representing thew streams indicate a mean depth of the difference is astonishingly small. 2.3 ft and a width of about 250 ft. In other words, the The general alinrment of points on the downstream intercepts of the graphs on figure 9 differ greatly among graphs indicates that in a given river basin where all rivers but the slopes of the lines are similar in magni- cross sections arc experiencing the same frequency of tude. The rates of increase in depth, width, and dischargr, the corresponding values of depth, width, velocity with discharge downstream map be similar and vclocity at diff went cross sections having thr same among rivers despite marked diff ermces in the width discharge tend to be similar, regardless of where in to depth ratio at any particular discharge. thr wrLtershrt1 or on what tributary the cross sections From the data in figure 9 and other similar data it may be. appears that the velocity tends to increase with mean The graphs presentecl arc examples chosen from a annual discharge downstream in all the rivers studied. largr numbrr drawn for various river systems in the It must be remembered that this generalization is United Stntes. Figure 9, with the plotted points restricted to the situation in which discharge at all through which the mean lines were drawn eliminated, points along the river is of similar frequency. presents similar graphs for selected river basins. Rivers Most geomorphologists are under the impression that, were chosen to represent a diversity of geographic loca- the velocity of a stream is greater in the headwaters tion, and physiographic and geologic types. The basins than in the lower reaches. The appearance of P are also considerably different in size. To indicate this mountain stream, of coursr, gives the impression of diversity of size of drainage area, the stations used to greater kineticity t,han that observed in a large rivrr represent the Tombigbee River in Alabama have mean downstream. The impression of greater velocity up- annual discharges from 700 to 35,000 cfs, whereas the stream stems in part from a consideration of river Belle Fourche basin in Wyoming has mean annual dis- slopes which obviously are steeper in the upper than in charges ranging from 40 to 600 cfs. the lowcr reaches. It will be recalled, however, that The mean lines through t he points representing indi- velocity depends on drpth as well as on slope, as shown vidual gaging stations (figurcas 5-8 are examples) were in the Manning equation fitted by eye. Thcw is considerablr scatter of points aboiit the rcspectivc ni~a~ilincs. h‘loreover, as dis- (4) cussed pr(4oiisl;v, the poiri ts on thr graphs represent vnlors of depth, width, and vclocitp at stream-gaging wherr depth, d, is approximatrly rqual to hydraulic stations situated at positioiis riot uniformly representa- radius for natural rivm sections, v is mean velocity, n is tive of tlic avcragr river cross sections. Despite the channel roughness, and s is slope. It is recognized that po~siblevariations that might hc introduced by thesr slope, s, in equation (4) is the slope of the encrgy grade vonsiderntions, t1iri.e is a tcndcmcy toward parallelism line. For practical purposes throughout most of the of the lincs on the graphs in figure 9. As a rough gen- discussion in the prrscrit paper, slope of the river profile eralization, it may be stated that in a downstream sufficiently approximates the energy gradient for equa- direction the rates of increase in width, depth, and tion (4) to be used as if the slopes were identical. velocity relative to discharge are of the same order of The fact that velocity increases downstream with magni t utle for rivrrs of tliffercrit sized drainage basins mean annual discharge in the rivrrs studied indicates and of widely diff ercnt physiographic settings. that the increase in depth overcompensates for the Some river basins st atlietl have too small a rangc of decreasing river slope. The magnitudc of this rate of VARIATION OF HYDRAULIC CHARACTERISTICS IN A DOWNSTREAM DIRECTION 15

io I00 1,000 10,000 100,000 1,000,000 Mean annual discharge, in cfs

STREAM AND LOCATION 0- - TOMBIGBEE(ALA I 8- REPUBLICAN-KANSAS(KAN) @ FRENCH BROAD (N C ) I&)-+--e- LOUP(NEBR.) @-.-I- BELLE FOURCHE(WY0) 3 MISSISSIPPI, MAIN STEM 9------YELLOWSTONE- BIGHORN (WYO ) @ -'-*- MADRAS IRRIGATION CANALS (INDIA)

FIGI'RE9.-Width, depth, and vrlocity in relation to mciiii snnual discharge as discharge increases downstreanr in various river systrms change becomes clear by compurison of the cxponents of In summary, the rate of incretlse of depth tlown- depth and slopc. In equation (4)the velocity depends stream tcnds to ovcrcompensate for the decreasing on depth to the power $ and. on slope to the power $. slope and tends to provide a net incwase of vclocity at Increasing depth tends to increase velority in a down- mean annual discharge in the downstream direction of stream direction. A variation of roughness in the a river. downstream direction will also affect the rate of change The straight-line relations on logarithmic paper Show, of velocity and will be discussed later. just as they do for a particular river cross section, that 16 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS width, depth, arid velocity increase downstream with in the downstream direction. Figure 10 presents thc tliscliarge in the form of simple power functions, individual plots of six curves in order to demonstrate the relative goodness of fit. It will be seen that the gaging stations available for the hlaumec and Scioto basins did not represent so large a range in discharges as did some of the rivers discussed earlier, and that the points did not fall in so good alinement as in some other Tlic cocfficicn ts and cxponent>sare ttssignetl the same basins studied. The width-to-discharge relations are lrtttrs as for a given cross section of a river., but the the best; more scatter of points appears on the depth values will be tliffcrcrit for points in a downstream and velocity curves. direction from tliosc for thc given section. Average For the Rilauniec and Scioto basins combined the valiics of the t>sponr>ntsfor river basins studied are curves of discharge in rclation to width, depth, and velocity for each of five frequencies are shown in b=O.5 figure 11. f=0.4 There is a general parallelism within each family of m=0.1 curves, but this is somewhat more consistent in the width-to-discharge than in the velocity-to-discharge To recapitulatc, both at a given river cross section relation. In these data it appears that, with less frc- ant1 in the downstream direction at mean annual rlis- quent discharges (flood flows), the velocity increases charge, width, depth, ant1 velocity increase with dis- somewhat faster with discharge downstream than it charge. The functions representing these relations are does in low flows. A siniilar study made for flood flows similar but involvc~tlifferc>nt expoucnts and numrrieal in some tributaries to the Yellowstorie River in Wyo- constants. ming showed that velocity remained nearly constant downstream despite a rapid decrease in slope. RELATION OF CHANNEL SHAPE TO FREQUENCY It can be seen that there is a pattern of interrelation- OF DISCHARGE ship of stream characteristics which includes relative Thus far the changes in hydraulic characteristics frequency of discharge, and it is now possible to define downstream in a river system have been considwed only important aspects of the hydraulic geometry of stream for a dischargc equal to the mean annual rate at each channels. An understanding of this geometry is basic point. Comparison of the data already presented will to consideration of the adjustment of channel shape to now be made with similar data for infrequent flows. carry the sediment load supplied to the streams. The Computation of the frequcncies of various discharges relation of at-a-station characteristics to the down- at a gaging station has been discussed. The compu- stream characteristics will now be discussed. tation is a somewhat laborious task. Complete cumu- In basinA of figure 12 the line A-C shows the increase lative frequency (flow-duration) curves for all strcam- of width with increase of discharge at an individual gaging stations in a watershed arc availahle for only a gaging station near the headwaters of a stream, and few localities; one of the best collections of such data is B-D shows tlie same relationship for a downstream that by Cross and Bernhagen (1949) for Ohio. station. If the dischargc A is of the same cumulative It was desired to avoid choosing streams regulated frequency (say 50 percent of thc time) at tlic headwater by many reservoirs, and the Maumee and Scioto gage as discliarge B at the downstrearn gage, then the River basins were selccted as being freer of regulation line A-B is the increase of width with discharge down- than most major river systems in Ohio. Duration stream, corresponding to a cumulative frequency of (cumulative frequency) curves were available for 19 50 percent. Also if C arid D have the same cumulative gaging stations in these two watersheds. Though the frequency (I percent of the time), then line C-D is the Maumee River flows northeast into Laltc Erie and the increase of width downstream at a discharge frequency Scioto south into the Ohio River, the two basins have a of 1 percent. Similarly then, K-G is tlie increase of common divide, and study of the data showed no depth with discharge at the upstream gage, and E-F is significant differences in the hydraulic cliaractcristics the increase of depth downstream for a frequency of for the purposcs of this paper. 50 percent. In the example under discussion, the At each gaging station discharges that were equalled curves are drawn so that at any frequency the width- or exceeded 1, 4, 10, 30, and 50 percent of the time, as to-depth ratio remains constant downstream; that is, well as the mean annual discharge, were selected for A--R is parallel to E-F. study. For each of these discharges, the corresponding It was noted earlier that because velocity, depth, and width were determined by analysis Q E wdv of current-meter data, as explained for figure 3. For each discharge frequency, curves were plotted repre- and w=aQb, d=cQf, and v=kQm, senting the change of hydraulic factors with discharge then b+f +m= 1.O RELATION OF CHANNEL SHAPE TO FREQUENCY OF DISCHARGE 17

500

t Q, W ;200 .- - c+ g 100

+ a, 25 ._c JZ t Q W n2

I 100 1,000 10,000 100,000 Discharge, in cfs

I OPERCENT EXPLANATION :- PERCENT OF TIME DISCHARGE EQUALLED OR EXCEEDED :

FIGURElO.--Width, depth, imd velocity in relation to dischnrgos representing two frequenvics, hI:run~eean

If the slope of the width-discharge curve for a given qucncy t,he velocity does not change downstream. The frequency is 0.5 (exponent b=0.5), a typical value for line J-L is the increase of velocity with discharge at the many rivers, and if A-B parallels E-F (as in fig. 12; upstream gage, and K--hl at the downstreamgage. It basin A), then the slope of the velocity-discharge curve is clear that at, the 1 percent frequency, the vclocity is J-K, corresponding to the discharge occurring 50 per- higher at both statiolis than at the more frequent dis- cent of the time,, must be zero (m=O). At that fre- charge but does not change downstream. 18 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

Discharge, in cfs

EXPLANATION PERCENT OF THE TIME FLOW IS EQUALLED OR EXCEEDED : -1.0%; -- 10%; ----- 30%; --- 50% 0-0 MEAN ANNUAL DISCHARGE.

Fi~r.ita1 I .-Width, depth, mil wlocity in relation to discharge ol wrious ln-riiieiwic~s,Maunier tint1 Sriotn Ilivrr hasiiis. Ohio.

In figure- 12, basin B, the width-to-depth ratio in- This is in t~ctwrdwit,]) tlie data which indicate that the creases downstream at any frequency. This increase slope of the width-discharge curve downstrcarn is, is shown by the fact that A’-B’ slopes more steeply between basins, some\vhat more consistent than the than E’- F’. If the slope of A’- 3’ is again 0.5, the othrr rclations. steepening of thcl slope of J‘-K’ makes the points Thc channcl charac*tscristicsof natural rivers tire sceii J’, K’, L’, ant1 X1’ full nearly in a straight line. Basin to ronstitut e, tlmi, an intc.rtleI>rndent system which B is nearly identical in its geometry with tlic compar- can be described by a schrics of grnplis having simple able curves for the Maumee-Scioto basin (fig. ll), and geonir:tric form. The geomctric form of tlic graphs basin A with the Bighorn basin, presented in figure 6. describing these interactions suggests tlic tt.rm ‘Ihy- The slopes of the width-discharge curves in a down- draulic geometry.” Channel characteristics of x par- stream direction were made the samC in both diagrams. ticular river system can be cicscribetl in terms of the SUSPENDED SEDIMENT AND DISCHARGE IN A PARTICULAR CROSS SECTION 19

Basin A Basin B EXPLANATION CHANGE DOWNSTREAM FOR DISCHARGE OF GIVEN FREQUENCY ---- CHANGE AT GAGING STATION FOR DISCHARGES OF DIFFERENT FREQUENCIES

FIGURE12.--l)iagrammatic rclations of width, depth, and velocity to discharge at a stntion and downstream. slopes and intercepts of the lines in the geometric THE HYDRAULIC GEOMETRY OF STREAM CHANNELS patterns discussed in figure 12. Though the listing of IN RELATION TO SEDIMENT LOAD slopes and intercepts of these lines may not provide SUSPENDED SEDIMENT AND DISCHARGE IN A any visual picture of a river basin, comparison of the PARTICULAR CROSS SECTION values of these factors among rivers has useful aspects This second part of the present paper describes the which will be discussed. interrelations of measured sediment wi:h the other Thus far, no consideration has been given to either hydraulic variables. Following a description of the river slope or bed roughness. The importance of these changes in suspendrd-sedirnent load at a station and fachrs is recognized bj~the authors. However, dis- then downstream, hydraulic arid physiographic inter- cussion thus far is, of neccssity, descriptive in nature. pretation of the observed relationships will be explored. Becausr comparable data for slope and roughnrss are Sediment transported by a stream may conveniently not available in so great, a quantity as data 011 width, be divided into suspended load and bed load, but it depth, velocity, and clischarge, the part(iou1ar (lata must be understood that, depending on the flow con- available for analysis nralw it ciesirablc to organize ditions, sediment may at one time be part of the the paper in its present form. suspended load and at another time part of the bed The observed relations of width, dcptli, and velocity load. Moreover, the capacity of a clianncl to carry to discharge in natural rivers have been described. In sediment of one size is at least partially independent thc ncxt part the observed relation of width, depth, of its capacity to carry another size. For this reason velocity, and discharge to measured suspended load Einstein (1950, p. 4) prefers to differentiate between will be presented. Then, the effect of slope and rough- “wash load” consisting of particles finer than those ness will be discussed in an attempt to provide a making up tlie stream bed, and “bed-material load.” partial explanation for tlie relations observed in the No simple and satisfactory method of sampling bed other quantities. load of a natural stream is in usc nt present. Any 497780 0 -59 -4 20 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

samplcr placcd on the stream bed providrs an obstruc- I,000,000 tion to the natural flow and tends to set up turbulent eddics in the immediate vicinit,y w-hich so change the transport near the becl tliat the resulting sample becomes nonrepresentative. The data available for l00,000 - analysis, th eref ore, consist of suspended sediment i sampled by USC of the improved ~lcptli-iiitegratiIig samplers. The sampled load is made up of particles DAEL 10,000 moving outside tlic bed layer. The measured suspended load does not present a fixed percentage of the c total load in motion but varies with a number of factors, as proposed by Maddock and discussed by Lane and Borland (1951). Tlit: student of fluvial processes may at this point seriously question the general applicability of relations derivcd only from data on suspended load. An attempt will now be made to demonstrate tliat relations between sus~-'eiitled-secliniexit load and channel cliaracteristics which can bc derived from tlic available quantitative data justify tlic use made here of the (lata on suspended load Again for purpose of definition, it is desirable to discuss a tool in wide use in hydrologic practice, the graph showing the relation of suspended sediment to water discharge. Tlic rclation of suspended load to discharge (some- I IO 100 1.000 10.000 times culled tlic sediment rating curve) is usually shown Discharge, in cfs as a plot on logarithmic paper of suspended sediment, FIGURE13.-Relation of suspended-sedi~nentload to discharge, Powder River in units of weight per unit of time, against discharge of at Arvnda, Wyo. the watrr-sediment mixture at a particular gaging statioii (see Linslep, Kohler, and Paulhus 1949, p. 331; The basic data on suspended-sediment load used in IIembree, Colby, Swenson, and Davis 1952, p. 69). tlie present study are available to the public, as already A typical relation of suspended load to discharge is explained, in association with current-meter measure- prescntctl in figure 13 (after Hembrec, Colby, Swcnson, ments. Data on suspended sediment will be published and Davis 1952, fig. 33). Tlic data for such a curve arc by the Geological Survey, but as yet there is consider- oht airiccl by daily, weekly, or other periodic sampling able dclay between collection of data and their publi- at tlic scdimeiit station. Such stations arc being cat'ion. installed and maiiitairlecl by Federal and State agencies It will be noted in figure 13 that the curve showing in increasing numbers in the United States. When relation of suspended load to discharge contains a wide tli~ir.recortls encompsss a long enough timc, the growing scatter of points, as is characteristic. This scatter network of stations promises a mass of basic data which indicates that at a given rate of discharge, thc amount will he of importance to the study of river morphology. of suspended sediment may vary considerably from one The scclimcnt is sampled by a depth-integrating day to another. It will be shown later that at a given suspended-load sampler. No fewer than two samples discharge, increased amounts of suspended load supplied are titken at various distances from the stream bank in to a reach tend to promote changes in t,he channel a givcn river cross section. These samples are analysed characteristics leading to increased capacity to trans- in tlie laboratory to determinc concentration of sus- port, particularly through adjustment of the ratio of pcmltd sediment in parts per million by weight. The velocity to depth. Thus, scatter of points above and aamplrs from a given cross section are combined to below the mean line in the graph of suspended load and give a mean conccntrat~ionwliich is then applied to the discharge implies corresponding compensatory arrange- mean discharge for tlic (lay to provide an estimate of ments of points about the mean curves showing relations tons of suspended load passing the station on the of velocity to discharge and depth to discharge. givcn day. This figure is then plotted against the A straight line has been drawn through the points on mean dischnrge for the day as a single point on the the curve in figure 13. The points are so scattered on curve showing relation of suspended load to discharge. some graphs of suspended load and discharge that a Each point on tlie curve of figure 13 was determined smooth curve drawn to represent them has little in this manner. meaning. When the record is sufficiently long, as it SUSPENDED SEDIMENT AND DISCHARGE IN A DOWNSTREAM DIRECTION 21 is in figure 13 and in the data for 20 stations analysed the observed increase of suspended-scdiment concentra- in this report and summarized in the appendix, a tion with increased discharge results from the suspen- straight line on logarithmic paper is a usable approx- sion of the additional material scoured out of the bed imation. Many graphs of suspended load and dis- by concomitant high velocity. charge may be represented by straight lines in the It will he shown later that at certain river measuring middle range of discharge, but by curves or lines of stntions deposition on the bed occurs under the condi- different slopes in the extremely low and high discharge tions of high velocities accompanying the rapidly ranges. increasing suspended loads of flood rises. These It is typical that the line approximating the data on a velocities are higher than those accompanying scour graph of this type slopes steeply, as in figure 13. At of the bed during the decreiising suspended loads of a station suspended load usnnlly incrcascs more rapidly flood recessions. than does water discharge. It will be noted that if This occurrence is sufficiently common to support water and suspended sediment increase at equal rates, the contention that the changes in velocity and depth the sediment concentration is constant. Constant con- during a flood passage are caused by changes in concentration of suspended sediment w ould be represented centration of suspended sediment introduced into the in figure 13 by a line sloping upward to the right at 45'. particular reach. Suspended-sediment concentrntion The fact that the line slopes more steeply indicates may generally be considered an iiidependcnt variable, that suspended-sedimeht concentration increases rapidly with velocity and depth dependent. with discharge at a station. It is observed that the relation of suspended sedi- This usual rapid increase of suspended load with ment to discharge at a given river cross section may be discharge is interpreted by the authors as indicating approximated, through at le,ast a major part of thv that conditions of rainfall and run-off on the watershed range of discharge, by a straight line on logarithmic combine to furnish to the major stream channels a large paper. Such a line is defined by the expression increment of debris for each increment of water. In other words, it is believed that the slope of the curve L=pQ' showing the relation of suspended sediment to dis- in which L is scdiment load in tons pcr day, 11 nnd j are charge is the result of the process of sheet, rill, and gully numerical constants. In the data analyzed, therc is a erosion on the watershed and represents sediment im- tendency for the suspended sediment at a station to posed on the major stream channels which must carry increase faster than dischargc, or the slope, j, of the it away. suspended sediment-discharge curve is greater than No detailed hydraulic theory is available to explain unity. Valuesofjtypicallyliein the range 2.0 to 3.0. why sediment increases at a faster rate than discharge In summary, it is typical at a given cross section of at a given station. However, a number of widely an alluvial stream for the suspended-sedimen t load to known fact,s make this relation seem logical. Large increase with increase in discharge, and at a mor? rapid discharges imply greater rainfalls than small ones. rate. That is, the concentration of suspcnded scdi- Great rains are characterized by a larger percentage of mcnt incrcases with discharge. Considcv-ation of the surface run-off than small rains, for many reasons. physical characteristics of surface run-ofl' and observa- Three reasons are: tions of bed scour lead to the tentative conclusioii that First, the initial rain must fill surface detention and tlie observed increase in sediment concentration results depression storage betore surface run-off can begin. primarily from erosion of the watershcd rather than Infiltration rate decreases rapidly during the initial from scour of the bed of the main stream in the reach period of a rainstorm. where the mcasuremcnt is made. Second, as the soil surface becomes wetted, areas poorly protected by vegetation are subjected to the RELATIONS OF SUSPENDED SEDIMENT TO DISCHARCIE IN A DOWNSTREAM DIRECTION churning effect of raindrop impact (Ellison, 1944) which increases with rainfall intensity. Puddling of Stations where suspended sediment is sampled are the soil surface, if it occurs, probably increases with still too few to provide data for a direct analysis of the duration and intensity of rainfall. change in the concentration of suspended sediment in a Third, moreover, large runoff rates imply greater downstream direction, as was done for the other hydrau- depths and velocities of water flowing in eroding rills. lic factors. These factors tend toward a greater efficiency of erosion In long reaches of a river through more or less liomo- with the increase in duration and intensity of rain. geneous topography the entrance of various tributaries Thus, it might be expected that at a given river cross does not appear to the eye to have any great effect on section sediment production increases faster than dis- the turbidity of the water. Even over great distances charge. t the changes in sediment concentration downstream are It might be presumed that because the bed of an are not sufficient to be distinguished without actual alluvial river tends to be scoured during high discharges, sampling. 22 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

The change in sediment concentration downstream through homogeneous topography and lithology u7ould seem to be small. However, the percentage of area pf lesser surface slope increases wit 1.1 drainage area. Areas of little slope probably contribute less sediment per unit of water run-off than areas of steep slope. Ground- water contributes a larger percentage of flow to n stream draining a large area than to one draining n small area. Inasmuch as the peiwnt of land not contributing t f sediment arid, tlierefore, the percent of sediment-free 0 water tend to increase somewhat with drainage nren, 3 it would be logical to suppose that the average sediment concentratiori slioiild decrease slightly in the downstream direction. Rube>- (1933, p. 505) reached the T same general conclusion. It should be understood, f % however, that individual rivers may differ in this respect n (Mackin, 1948, p. 480). Strong support for this conclusion is provided by the tx observation that the average annual production oi setli- .-t -0 ment per square mile of drainage area decreases with 3 increasing drainage area. In other words, sediment production per unit of area is greater for small than for large drainage areas. This conclusion has been reaclied by the study of erosion of sample upland areas antl valley cross sections in Illinois (Brune, 1950), by the I I analysis of sediment trapped in small reservoirs in I I Wyoming (Hadley, in preparation, and open file), and I I I from similar data on reservoirs in Iowa and Nebraska I I *++ I I t I (Glymph, 1951, p. 6). If sediment contribution per D 012 I o unit of area decreases with increasing area, on the aver- -0 age there should be a tendency toward smaller concen- + 0)C trations of suspended sediment, clownst,ream. .-E U The relation of suspended sediment to discharge both ul I at a station and downstream can now be considered in 'DaJ ; U the hydraulic geometry of stream channels. In figure alC a 14 the curves of width, depth, and velooity in relation ul3 to discharge are similar to those discussed in figure 12. cn In the diagrammatic plotting of suspended-sediment /-EXPLANATION CHANGE OOWNSTREAM FOR load against discharge in figure 14, the dott8ed line I B3 DISCHARGE OF GIVEN FREQUENCY I As-C, is the curve showing the relation of suspended --- CHANGE AT GAGING STATION FOR sediment to discharge for an upstream station, for DISCHARGES OF DIFFERENT FREQUENCIES which the at-a-station width, depth, antl velocit8y Discharge - curves are respectively Ao-Co, AI-C1, and A2--C2. NOTE: ALL SCALES ARE LOGARITHMIC The curves for a downstream station are, respectively, Bo-1301 B1 and Ill, and B,--D,; and curve B,-D,, FI(;~:ILE14.-Tyr)ical wlutions in the hydraulic: Reornetry of natcmrl chanml shows the relation of suspended load to discharge. systems, showing width, clcspth, velocity, arid auslirrided-st,dimrnt load plott,c(l The small number of siispeIidet1-sediment stations against dischurgr huth at a station :rnd downstrr:lm. arranged along the Iengtli of a given river prcdudcs the construction of curves represcnting the change of line sloping upward to the right at 45'. The probable susp~~iicled-sedimentload downstream, but such a cin-vc relation is therefore shown on figure 14 as the solid is a necessary part of the hydraulic geometry. The lines A3-B, and C3--D3 which slope slightly less general argument in the preceding paragraphs appears steeply than 45". to support the hypothesis that concentration of sus- A brief summary will explain the purpose of the next pended sediment usually decreases somewhat down- step in thc analysis. Thc shape of the cross-section stream with discharge of equal frequency. As in the at any point on a river may be clcwxibetl by the observed suspended load-discharge curve of figure 13 equal increase of width, depth, and velocity with increase of concentration downstream would be rearesented bv a discharne. ConsiderinP the rivpr svstcni from head- HYDRAULIC FACTORS AND SUSPENDED LOAD Kr GIVEN DISCHARGE 23 waters dowiistrc~ani,if dischaqy at all points is (Born- For each station, c-ur~x*nt-rnete~.measurc~iient (lata parahlck in percentage of time this dis(-liargc is t~qiialle~l wcrc used to construct ciirvcs relating discharge to or ~xce~(Icd,the width, tlcptli, ant1 velocity are observed width, depth, and velocity, as sliowri in figurcs 3 arid 4. to increasc downstream with increased discliwge. From thcsc data and curvcs showing the gcncral relation Tlicw at-a-station and tlownstrenm relations can be of suspended seclimc.nt and discharge, values of tlirw expressed 1)y simple grtiplis having a gtwnetric pattern. four factors corresponding to a discaharge of 500 cfs This gcoinetric pattern is an expression of the inter- w(’re tabulated. ~v~latioiishipof width, depth, vcloc-ity, rind tlisc.hargc For each station, width wrns plotted against suspended cxprcssetl by the identity loud in figure 15, and at cwli point tlie corrcqonding velocity was entered. Isopleths of velocity were tlien Discharge = widthX dcpthxvelocity drawn; these are the liiics slanting steeply downward to This identity, howcver, must be only one of a nunibcr the right in the figure. of equations which dctcrmine tlie observed relationships. Because the grapli represcnts {lata for a fixed tlis- Therc probably is a relation between thc charscter- charge, every point in the width-vc4ocity field defines istics of the channel shape and the measured suspended- a particular depth inasmuch as sediment load. The purpose of tlie next sections is to define, at least qualitatively, how tlw suspc~ndcdload Q=w d v= const,ant is observed to vary in different rivers having different width, dcpth, and velocity relationships. The analysis Thus isoplet,hs of depth can bc drawn which are the will first be attempted for a given discharge, and then lines slanting gently downward to the right. will be extended to include different discharges. Figure 15, therefore, is a field showing the relation of corresponding values of width, suspended load, velocity, WIDTH, DEPTH, VELOCITY, AND SUSPENDED SEDI- and depth. The graph has many limitations but it is MENT AT A GIVEN DISCHARGE nevertheless a useful t ool for sonir qualitative analyses. Certain aspccts of the effect of suspended sediment Among its limitations are, first, the fact that the on channel shape are best exemplified by consideration velocity field is only roughly defined by the values in of the condition of constant discharge. The following thc available data. Second, only the suspended part discussion coriceriis the analysis of measured sediment of the load is being considered. Third, tlie scatter of load at various gaging stations when the discharge at points on any average curve showing relation of sus- the stations is equal. pended sediment to discharge is large. Data for 20 sediment measuring stations were used Nevertheless, certain general conclusions of apparent in this study. The data consisted of curves showing significance may be drawn from figure 15 and they will relation of suspended sediment to discharge ; these are be trstcd against observed changes in the cliaract eris- summarized in tabular form in the appendix of this tics of river channels. report. Sediment measurements are being made at If a constant discharge is assumed, at constant width many more stations but insufficient length of record or of channcl an increase in velocity is accompanied by absence of simultaneous current-meter dischargc mcas- an increase in suspended sediment. This requires, at urements precluded their use. A fsw records were constant discharge, a decrease in depth. discarded because of unnatural cross sections. Particu- Conversely, if velocity is constant, an increase in larly, data obtained at dams or in sections where tlie width is accompanied by a decrease of suspended load. channcl walls were confined were deemed unsuitable. This is also associated with a decrease in depth.

VELOCITY, IN FEET PER SECOND-

.- - 3 - - 50 .

100 500 1,000, 5,000 10,000 50,000 100,000 500,000 Suspended-sediment load, in tons per day

FIGURE15.-Average relation of suspended scdimrnt-load to channel-shape factors at a constant discharge of 5M) CIS (from river data in appendix). 24 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

In summary, decrmsing width at constant velocitj- WIDTH, DEPTH, VELOCITY, AND SUSPENDED LOAD or increasing velocity at constant width cbacli results in AT A VARIABLE DISCHARGE increased capacity for suspended load at constant dis- Relations similar to those examined at constant discharge. charge will now be discussed for three other discharge These relationships should be rcviewd here in terms rates. Using the same 20 suspended-sediment sampling of the basic data from wliich they were derived. The stations, measurements of depth, width, velocity, and basic data consist of 20 cross sections of different rivers suspended load were plotted for discharges of 250, (see appendix). The rivers include the Loup, Rio 1,000, and 2,000 efs in a manner similar to that used for Grande, Bighorn, Smoky Hill, Powder, Moreau, figure 15. The mean isopleths of velocity for the four Republican-in other words, rivcrs in the Great Plains, discharges were rcplotted in figure 16, with the actual Color ado Plateaus, and Rocky hlountain areas. The velocity values shown on figure 15 eliminated for the data arc thtreforc biased toward Western conditions, sake of clarity. yet the rivers arc rather diverse in general appearance. As in figure 15, when discharge, width, and velocity A station on each river was considered under conditions are specified, depth also is specified. In other words, a when 500 cfs was flowing in the channel. At that par- particular point in the graph of figure 16 specifics ticular dischargc, some of the river sections arc wide, width, depth, and velocitj- for any of the four dis- others narrow, some shallow, others deep. The charges. observed suspended load at that discharge has been Any arbitrary straight line drawn on figure 16 will correlated with the concurrent width, depth, and cross the respective fields of velocity for the four dis- velocity at the particular discharge. charges. Any point on that arbitrary line will repre- The graphical relation shows that a wide river having sent simultaneous values of width and of velocity in one a particular velocity is observed to carry a smaller of the four velocitj- fields. The line may be drawn, suspended load than a narrow river having the same then, to represent any chosen width-to-discharge rela- velocity and discharge. tion and a chosen velocity-to-discharge relation. A It shows also that if two rivers have equal width at corresponding suspended load-discharge relation be- the scme discharge, the one which is flowing at the comes thereby determined. higher velocity is carrying a larger suspended load. An example will help clarify this idea. If one follows These are empiric relations observed in the available the horizontal line in figure 16 representing a width of river data. Nothing has been said regarding why one 100 ft to the 2-fps velocity line representing a discharge river is wide and another narrow, or one is deep and of 250 cfs, he will see that the point of intersection another shallow. Clearly these features depend on represents a suspended load of roughly 1,500 tons per such factors as slope, roughness, and particle size which day. The 2-fps velocity line for 500 cfs intersects the have not yet been considered. same horizontal line at a point equal to a load of 3,000 In the analysis, the rivers were compared at equal tons per day. The 2-fps velocity line for 1,000 cfs discharge. It is now logical to examine the same intersects at 6,500 tons per day, and the same line for a relations at different discharges. discharge of 2,000 cfs at 15,000 tons per day. It will

100 1,000 10,000 100,000 I.o oo ,o o a Suspended-sediment load, in tons per day

EXPLANATION DISCHARGE: 2,ooocfs ; ---- i,ooocfs; ---- 5OOCfS ; ------2socfs

Numbers indiccte the velocity in fps isopleths __ --

FIQUBE16.-Average relation of suspended-sediment load to channelshape factors at four discharge rates (from river data ln appendix). HYDRAULIC FACTORS AND SUSPENDED LOAD AT VARIABLE DISCHARGE 25

be noted that these corresponding values of discharge, where the symbols are as defined previously. In the velocity, and suspended load could be used to define graphs of figure 17 it will be noted that relations between discharge and the separate factors of velocity and suspended load. These relations have b +.f+m =0 + 1.O +O = 1.O been plotted in figure 17. This suspended sediment- discharge curve would represent the mean suspcndcd as is required by the identity sediment-discharge relation for a channel of constant Q = wdv width. In this hypothetical channel, various discharges would have the same mean velocity, 2 fps. By analogy to tlic example plotted in figure 17, it can Becausc for each of the four discharge points, velocity be visualized that for any givcn assuniption of b (the and width are specified, depth also would be specified. rate of incrclasc of width with discharge), the slope j of Therefore the relation of width, depth, and velocity to the curve relating suspended load to discharge will de- discharge can also be plotted, and these plots also are pend on the rate of increase of velocity with discharge shown on figure 17. defined by the exponent m. And because b+m+j=l, 1 Ill I I I I I Ill when b and m are specified, f is also specified. If the ratio of m to f is known and the value of b is known then values of m and f are known. For example, assume arbitrary straight lines on the field of figure 16 and find the slope j of thc suspended load-discharge relation for. various values of b and the ratio m to f. The rcsult,s are plotted in figure 18. Each point in the figure represcnts a given straight line drawn in the graph of figure 16. These lines were drawn in such a way that various values of b were represented. The rtbsulting relation of velocity and load with discharge were plotted just as in figure 17, the slopes of the lines representing values of m and f were measured, and the ratio of rn to f was computed. Values of j and the ratio of m to f were then plotted in figure 18. From figure 18 a number of useful concepts may be inferred. The first concerns the interaction of velocity, depth, and the suspended-sediment load. For any Velocity - o Slope m-0 given rate of increase of width with discharge-that is, I c- i .-c 0 I llll I I11111 % Discharge, in cfs 2.0

FIGURE 17 --Fxample of the relation 01 suspended-sediment load and water dppth to discharge lor specified relations of width to discharee and velocity to discharge (constiuclcd from fig 16). 1.6

In accordance with the proccdure used in the example, any straight line drawn on the width-suspended load- 1.2 velocity field of figure 16 defines the rates of increase of two variables with discharge. The rate of increase of thc othcr two variables with discharge niust be given. 0.8 In thc previous example, the rate of increase of width with clischargc and the rate of increase of velocity with discharge were specified. Then the rates of increase of 0.4 depth and siispcded load with discharge becamc known. The slopcs of these lines can be measured and are the exponcnts b,m,f,andjin the equations 0 0 I 2 3 4 5 w=aQb Slope of the suspended-sedimeni rating curwe(;Awhere L=pd d=cQf I FIGURE18.-Reletion of rates of increase of width, depth, velocity, and suspended v=kQm m load to increasing discharge, expressed by values of b, -,andf (constructed L=pQj from flg. 16). 26 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS for any givcn \ due of b- the diagram indicatrs tlitit Figure 18 appears to apply LO average conditions the vdue of j increases witli an iricretisc in thc m to .f both at a station and downstream. For any particu- ratio. That is, tlic rate of incrvase of suspentlcd-scdi- lar cross section of a given river or for any particular nient loritl with tlisclinrgc is a function of tlic ratio: Aootl hydrograph, it should not be expected that figure 18 would definc thc relations because it will be recalled rate of incrww of vclocity with disc+harge ~~ that figure 15, from which type of data figure 18 was rite of increase of depth witli diwhargc. derived, represents a series of smooth lines drawn Slopes of tlitb lines rtyrrscwting clcpth, width, tint1 through data having considerable scatter in the plot. velocity resptvi ivc.ly, in rchtion to disclinrge arc, ac- Nevertheless, figure 18 is a useful guide in summarizing cording to the implications contained in figure 18, a large volume of data on suspended load of rivers. functions of the slope of the line rcprescnting the rela- It is presumed that as additional data on suspended tion of suspmdctl sediment to discharge. For a given load become available, it will be possible to refine the value of b, the stwpcr the slope of the suspcmlcd sedi- graph of figurc 16 by adjustment of the slope and ment-discliargtx lint., the grcalcr will be the slope of position of the velocity isopleths. From available the velocity-discharge line. data, the relations in figure 16 may be summarized as It! is also possible to observe in figure 18 thc points, follows : When cross sections of different rivers having or zones, which represent the average river conditions. different channel characteristics are compared, the The average values of the rates of increase of width, observed suspended-sediment load will vary directly depth, and velocity with discharge have been listed as a function of the velocity, directly as a function separately, ant1 art' sumniarized her(.. of the depth, and inversely as a function of the width of water surface. The suspended load varies as a large At, a station Downstream power of velocity, and as small powers of depth and b=. 26 b=. 5 width. f=.40 f=.4 The preceding discussion may now be summarized by m=. 34 m=. 1 indicating the relation which the curves of figure 18 bear to the hydraulic geometry of river systems charac- The suspended sediment-discharge relations pro- teriyed by figure 14. To facilitate reading, figure 14 has viding the basic data from which figure 18 was derived, been reproduced in essentially the same form in figure represent a number of different stations. Earlier dis- 19 with the addition of simple block diagrams to allow cussion of typical relations of suspended sediment to easy visualization of the at-a-station and downstream discharge brought out tkic fact that the slope, j, of relations constituting the hydraulic geometry. typical curves is of the order of 2.0 to 3.0. As pre- Figure 19 summarizes the average conditions observed viously indicated, at a typical river cross section, width in the numerous rivers studied. Cross section A repre- increases only slightly with discharge until the bankful sents the conditions at a headwater station during low stage is reached. This rate of increase is described flow. Its position will be noted in the upper physio- by the average value b which is provided by the avail- graphic diagram of a watershed in which low flow able data, b=.26. For the average cross section m m prevails. Cross section C represents the same head- (at a station) -= .85. In figure 18 where --=.85, and f f water station under conditions of high discharge. b=.26, j=2.3. Thus the value read on the curves Similarly, B is a downstream section at low flow, and D of figure 18 compares favorably with the observed aver- the same section at high discharge. age slope of suspended sediment-discharge curves. The slopes of the lines in the graphs of figure 19 It was brought out in tlic discussion of the relations represent the average conditions observed in rivers. of suspended sediment to successive points in the down- In the width-discharge graph, line AO-Bo has a slope stream dirckction that suspended-sediment concentra- upward to the right equal to 0.5, or in the expression tion should decrease slightly downstream. A value of w=aOb, b =0.5, which is representative of the increase j=l.0 means that suspended load is increasing at the of width with discharge downstream. same rate us discharge and represents equal sediment Similarly, in the width-discharge graph, the line concentration. It is to be expected, therefore, that Ao-Co represents the increase of width with disdiarge the observed value for rivers should be somewhat less at a station and has a slope b=0.26. than 1.0. Refcrririg to the mean+valuesof m,f, and b In the depth-discharge graph line, Al-B, has a slope 771 f=0.4, and A,-Cl has a slopeLf=0.4. In the velocity- in tlic downstrcam direction where b= .5, and --= .25, .f discharge graph, line m=O. 1 for the downstream direc- it can he s(wi on figure 18 that j=0.8. Thus figure 18 tion, and m=0.34 at a station. Indicates thaL ronccn tm tion of suspcmlctl sedimcnt Lines in the suspended load-discharge graphs also drcreas~sslightly in a downstream direction for a haw slopes equal to average values for rivers, j=0.8 given frequency of discharge, as the earlier argiinicnt downstream (line A3-B3) and j=2.5 at a station (line indicated. A,- 4'3). HYDRAULIC FACTORS AND SUSPENDED LOAD AT VARIABLE DISCHARGE 27

t A C f .-0 3

t B D f a W D t 21 .-c 0 -0 5

Ut 0 -0 c C .-E U Wv) I U W U C W a v) cn3

EXPLANATION - CHANGE DOWNSTREAM FOR DISCHARGE OF GIVEN FREQUENCY --- CHANGE AT GAGING STATION FOR DISCHARGES OF DIFFERENT FREQUENCIES NOTE. ALL SCALES ARE LOGARITHMIC

FIQWKEIg.--hverage hydraulic gcometry of river channels expressed by relations of width, depth, velocity, suspended-sediment load, roughness, and Slope to discharge at D station and downstream. 25 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

It is n(1w possible to interpret the relation between It may now bc aslcecl how the channels at the Tm-ious m b, and j shown in figure 18 and the diagrams in stations arc sliapcd to promote the equilibrium required 7’ by this comparison. At that given discharge, the figure 19. Figure 18 shows the observed relationship downstream statioii has a larger width and a lowrr I)etween suspended load in rivers and the channcl- velocity, as can be wen in the diagram. This is exactly shape factors. The figure shows that for a given value what figure 15 suggests; that is, at a given discharge, m a smaller suspended-setliInent load requires for equiIib- of b, the Yaluej increases with the ratio -e The river f rium a larger width or lower velocity or both. (lata indicate that for given a rate of increase of width Inasmuch as the diagrammatic relations in figure 19 xith discharge (a given slope of line Ao-Bo), the steeper show the actual average slopes of the various lines the slope of the suspended load-discharge line A3-B3, derived from rivcr data, the comparison at constant the steeper must be the slope of the velocity-discharge discharge again confirms the principles of channel- line A2--B2. This would require a smaller slope of the shape adjustment postulated. cIepth-tlisc.harge line A,--R, in order to preserve the The diagram also includes observed average varia- relatio~iof slopes required by tions of roughness and slope which will be discussed later. 6 if+m = 1.o It is emphasized that tliese relations are those derived Snother way to look at the geometry of figure 19 by comparing a large number of river data. The rela- would be as follows: For a given width, and at a gicen tions arc those existing in natural rivers. Flumes dischnrge, un increase in suspended sediment requires an coiil~lbe constructed in w1iic.h relations of slopc, rough- increase in velocity and a reduction in depth. Esactly ness, and the channel-shape factors are different from this interpretation was drawn from figure 15. thosc existing in rivers. In such fluincs the relations This set of interrelationships provides a visual described above would not necessarily hold. picture of how the channel system of a river is carved It is now desirable to observe what kinds of similar to carry the water and sediment provided by the relations seem to hold for becl load. drainage basin. The at-a-station and downstream relations between discharge and suspended load shown INTERRELATIONS OF WIDTH, DEPTH, VELOCITY, AND BED LOAD AT A GIVEN DISCHARGE bv the geometrical pattern A3B3C3D3may be considered essentially independent of the channel system, but are The experimental flume data of Gilbert (1914)-have fuiictions of the drainage basin. The relations of an advantage, for present purposes, over any other width, depth, and velocity to discharge tend to be comparable experimental data. Gilbert used flumes adj usted to conform to the load-discharge function of various widths under conditions of constant dis- according to the principles just outlined. charge. These data allow an analysis, therefore, of The geometrical relations describe another typical the relation of width, velocity, and load at constant relationship in rivers-that of the upstream and down- discharge. The Gilbert sands were such that the load stream station at a given discharge, represrnted along was carried primarily near the becl, and for present a givcri vertical hein figure 19. As an example, purposes his results may be considered to represent a particular discharge is represented by the dashed primarily the transport of sediment as bed load. vertical line representing the discharge Q’. The A-C The Gilbert data used here were taken from the sum- lines represent changes at the upstreani station and the mary by Johnson (1943, pp. 59-63), in which the data B--l) lines at the downstream station. At the same are presented in centimeter-gram-second units. discharge, Q’, note that the suspended load at For “sand B” of Gilbert, four widths of flume were the downstream station (ordinate where Ba-D, crosses used at a discharge of 15,433 cu cm per sec and five the dashed line) is less than thc suspended load at the widths at a discharge of 5,154 cu cm per sec. upstream station (ordinate where A3-C3 crosses the Figure 20 presents the rclatiou of the bed load dashed line). This is reasonable because a given dis- carried in the full width of the flume to width and charge at the downstream station may represent a observed mean velocity at constant discharge. Thus frequent flow or a low flow condition, whereas to attain the graphs of figure 20 are for bed load and are analogous that same discharge the upstream station must expe- to figure 15 for measured suspended load in natural rience a flood flow. A given discharge is equalled or rivers. Isopleths of velocity are drawn through the exceeded more often at a downstream than at an up- observed velocity values appearing in the graph. stream station, and common experience teaches that It is clear that the isopleths of velocity slope upward the frequent, or low, flow is likely to carry a small to the right in contrast to those in figure 15 which concentration of suspended sediment, and a flood flow slope upward to the left. These graphs provide, then, a large concentration. The given discharge, then, the following tentative interpretation, which, if SUS- seldom occurs in nature simultaneously at the head- tained by further work, may provide important assist- water and downstream station. ance4n the understanding of fluvial morphology. HYDRAULIC FACTORS ANI) BED LOAD AT A GIVEN DISCHARGE 29

At constant discharge, an increased velocity ut constant discharge is assumed. A constant discharge and con- zvidth is associated with an increase of both suspender' load stant velocity are specified in the present discussion of and bed load in transport. At constant velocity and dis- the effect of width on bed load and suspended load. charge, an increase in width is associated with a decrease Further indication of tlie offect of width on bed load of suspended load and an inxrease in bed load in transport. is provided by coriiputations niade using the Einstein It is unfortunate that the lack of data on bed load for (1950) equation. Jn a study for the Bureau of Itecla- rivers dictates reliance on flume observations. How- mation, Lnra and lliller (1951) coiripiitetl by tlic Ein- ever, some general argunients and observation support stein equation tlic bed loncl which would lw transported thc conclusions reached by analysis of flume da ta. in a clianncl of variahle width, under different condi- Lane (1937 p. 138) observes, tions of discharge. The computations involvctl in tlic cquation requirc that a vduc bc nssiirricd for rivrr slope If a channel is supplied with a heavy bed load, in order to bc stablc it must niovc thi.; load along. * * * To be stablc, the channel and bed-mat erinl size. Computations \wre ni:itlc by carrying bed loads, therefore, should have a higher velocity along Lars and hliller for different sized fractions of tlic !oat1 the bed, but the same velocity along the banks, and this could scparately, and combined to provide tlie computctl bcd only occur with a wider, shallower section. load having n size distribution characteristic of Rio This statement by Lane is significant because it Grande sediments. Lura and Miller assumcct slope applies to the interpretation of river channels as they comparable to that of the Rio Grande. are observed to exist. That is, an alluvial river channel The computed values of bed load for a consttint dis- which is relatively wide and shallow probably carries a charge of 500 cfs were plottccl by the writers against relatively large bed load. width nncl mean vclocity provicled by tlic computt~tion, Griffit,h (1927, p. 246) notes that the "* * * large in a manner siniilar to figures 15 and 20. In the grtipli, rivers flowing tlwough the alluvial plains of India, which not included here, isopleths of vclocity sloprd upwird in flood are heavily charged with silt, have in all cases a to the riEht as in figure 20. Thus, hed load coiiiputctl tendency to adopt a broad, shallow section * * *." by the Einstein equntiori appears to increase \\ itli Mackin (1945, p. 454) concludes from the British and increased width at constant dischargc and velocity-at Indian literature that "* * * the broad, shallow least for the assumptions of slope and sediment size ehanncl is the type of cross section best adapted for the made by Lara and Miller. transportation of heavy bed load." In the preceding sect ions sonic relations liavc bccii Though these general observations are of interest in developecl among suspended load, bed load, nntl ('litin- this paper, they imply rather than assert that,a eonstant ne1 characteristivs. By far tlie rnnjority of clatn available on sedirnent load are restricted to measurcments of suspended load only, and further detailed analysis of necessity depends on the nirasurecl frnc tion. Afackin (1948, p. 469) emphasizes that thc size-distribution of the load of a given scgment of a stream dcprnds on '' * * * tlic lithologic characteristics, relief, and erosional processes in its drainage basin, and on processes in operation within the stream itself (as sorting.)" The sizes and quantities of available load determine, in part, whether the stream carries the load as suspended or as bed load. A large part of the total sediment transported by natural rivers is carried as suspended load. This is recognized by some geologists (Mackin, 1948, p. 468), but because bed load nieasurenients are few, the relative proportions of bed and suspended load in natural rivers are not grnerally known. Both suspended and bed load are being measured ip the Niobrara River near Cody, Nebr. (Serr, 1950). The data already published show that for discharges between 200 and 1,000 cfs bed load constitutes an average of about 50 percent of the total load. The percentage of the total carried as bed load varied from 3 IO I00 500 Total load,in grams per second 24 to 65 percent. No significant correlation existed between this percentage and either the water discharge FIGUREZO.-Rclation of bed load to channel-shape factors at constant discharge, showing two discharge$, same sediment material (sand B), from Gilbert data. or the rate of sediment transport (Serr, 1950, p. 15). 30 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

This comparison of bcd load anti suspended load in fortuitous, thc foregoing discussion implies that there (he mrasiircments of the Kiobrara River lrd tjlie Bureau might be a definite pattern of channel cliangcs dircctly of Reclamation antl the Grological Survey to begin a related to thc suspendcd-4ediment load providtvi to the joint investigation for the purpose of extending this river by the drainage basin. If this werc triic, tlicsc experiment a1 wad\ (Serr, 195 I ) . Thr hfidtllr 1,0111’ changes slioiiltl be of tlie type requircd by the print iplcs River has a sandy bed and the water appears clear arid derived from a study of figures 15 and 16. frec of suspcndcd load. ‘I’lir engineers of thcw agencies The gaging station on the San Juan River at Bluff, tl the lfidclle Loup River near Dunning as an Utah, might first be considered. This river gencr~nllj~ example of a stream which carried most of its load as has a bed consisting of sand, silt, antl some gravcl, which bed loud. Tlie junior author, at, that tinie head of the cliaractrristically shifts during a flood. At high stages Sedimentation Section, Bureau of Rcclama tion, rrpre- the stream at the Bhiff station is confined by rork walls, sented thiit Bureau in the stiidics which led to the but some changes in width are possible ut low stnges. establishment of the mctLsiirenient progi~~nion the The control at the station is a reach of gravcil antl ,Middle Loup Kiver. boulders downstream from the gaging section, cmtl this The measurements showed, however, tlia t the bed control is subject to shift under flood conditions. Chr- load constitlutetl an average of only 55 per cent of the rcnt-meter measurements are madc from a cnblr car. total load even on the hfiddle 1,oup River (Vice and Daily current-meter data and daily suspended load Serr, 1950, fig. 14). samples are availnble for a few years. These daily Bcforc regulation of the Colorado River by Roultler data are used in figure 21 to show tlic changes that, Dam, meusurcments werc made in the overflow nappe occurred during a river risc between Septcmher 9 and on the spillway of Laguna Dam where presumably the December 9, 1941. A measurement on a partirular total sediment would be caught by a suspended-load day is indicated in figure 21 by a point in each of six sampler. These measurements, when compared with graphs. For example, during the first day of the rising suspended-load data at a normal section, indicated that stage the discharge was 635 cfs, width 171 ft, depth before regulation and, at least at low flow, the Colorado 1.20 ft, velocity 3.08 fps, and suspended load 2,140 River in the lower reaches transported almost all of its tons per day. To simplify the picture, an intermediate sediment load in suspension (Lane 1937, p. 190). rise and fall in discharge between September 16 and 29 The 1948 survey of the sediment deposition in Lake are omitted. Mead showed that 49 per cent by weight (978 million The rising discharge was accompanied by a slight tons) of the total sediment deposited in the reservoir rise in width up to the peak discharge of 60,000 cfs, and consisted of material fine enough to be transported in the falling discharge retraces nearly the samc path in density currents (Gould, 1951, p. 48). the curve showing relation of width to discharge. It might be argued, however, that despite a large sus- Suspended load, in contrast, increased at a very pended load in a river only the bed load is of real signifi- rapid rate as the discharge rose from 600 to 5,000 cfs, cance in fluvial morphology assuming it places t,he greRt- but then load increased at a less rapid rate while the est tax on the energy of the stream. discharge continued up to 60,000 cfs. It is possible that owing to the influence of the engi- Depth rose uniformly with discharge until a discharge neer, one may tend to place too much emphasis on bed of about 5,000 cfs was reached. Then the slope of the load and to underrate the role of suspended load in depth-discharge curve increased as the discharge rose determining the observed characteristics of natural from 5,000 to 60,000 cfs. In the falling stage, a new streams. In the present discussion, an attempt is made slope of the depth-discharge curve was established, LO demonstrate that data on suspended load may pro- which was intermediate between the two slopes in the vide an insight into the mechanisms by which river rising stage. At a discharge of 5,000 cfs, the depth was characteristics are determined. about 3.2 ft in the rising stage and 4.4 in the falling The following detailed consideration of the effect of stage. suspended load on river channel characteristics does not A corresponding adjustment in the velocity-discharge deny the importance of bed load. But if the measure- rurve is apparent. In the rising stage at 5,000 cfs the ments of suspended load allow the development of a velocity was 8.6 fps; at the same discharge in the falling meaningful picture of river channel characteristics, the stage it was about 6.0 fps. importance of the suspended load in fluvial morphology The graph of stream-bed elevation in relation to can hardly be denied. discharge is shown in figure 21. Bed elevation, as plotted, is a relative value which was derived by sub- CHANNEL-SHAPE ADJUSTMENT DURING INDIVIDUAL tracting the depth from the gage height (water-surface FLOODS elevation). This difference is a satisfactory estimate It is known that during the passage of a flood, the of bed elevation in the examples discussed here because channel of an alluvial stream is scoured and filled with width varied but little with discharge. Detailed river considerable rapidity. Though such changes may be cross sections at various intervals during the flood CHANNEL-SHAPE ADJUSTMENT DURING INDIVIDUAL FLOODS 31

EXPLA N AT1 ON DISCHARGE Rising Falling

Width + 0 Depth V v

Velocity x 0 Load 0 . Stage 1 m Bed 0 .

l0,OOO F 3/

,I ct I I Ill I II IIII 0s I i I I I1 50,000 lO0,OOO 1,000 5,000 lO.000 100,000 500 1.000 5,000 10,000 500 50,000 &? Dischorge, in cfs Dischorge, in cfs

FIGURE21.-Chnnges iu width, depth. velocity, water-surface elevation, and stream-bed elevation with discharge during flood of September-Docember 1941, San Juan River ne+ Bluff, Utah. 32 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

Distance. in feet 5,000,000 0

x 730

L 0 a 1,000,000 * c 0 c .-c 500,300 0 -.-0 WC E_ D lnG) I U l00,OOO : W:: rn 50,O 0 0

uct I4 20.000 v 30 FIOI.HE22.-('Iiannd cross srctiom during progress of flood, Seytemhcr- Decrmber 1941, San Juan River iiear BluB, Utah. + W 25 a, .I- have been plotted from original current-meter mcasure- .-C -0- ments. al The bet1 elevation rose during the first part of the .n 20 6 flood rise. Deposition of material on the stream bed e at the measuring section during the early period of the +In c o 15 food can be seen by comparing the cross sections in C ._0 figure 22 for September 9 (635 cfs) and September 1.5 c O (6,560 cfs) . This deposition continued progressively as -W IO long as the slope of the suspended sediment-discharge W curve was steep. At a discharge of about 5,000 cfs 0e thc suspended sediment-discharge curve assumed a 9 -5 -u diffcrent, less steep slope, nnd this smaller slope of the 0C line was maintained as the discharge continued to rise W 0 + to the peak flow of almost 60,000 cfs. mO

L At discharges between roughly 5,000 and 60,000 cfs a, c 0 the suspended-sediment concentration increased less 5 rapidly per increment of discharge than 'in the range -5 below 5,000 cfs. The smaller rate of increase of sus- peiidccl -diment concentration was associated with bed scour. After tlie peak flow, tlie falling discharge -IC was accompanied by a stahlc bed elevation. 5,000 IC.000 50,000 100,000 200,000 Discharge, in cfs This flood is one in which the slopc of the width- discharge curve was about the same throughout the EXPLANATION OF DISCHARGE flood passage, and, thereforc, the iniitiial adjustment Width Depth Velocity Load Stage Bed Rtslng + V X 0 I O among suspended load, velocity, and depth is quite Falling o v 0 0 Qm evident. At 5,000 cfs the suspended load was about 1,000,000 tons per day in the rising stage. In the FIGI'RE23.--Changrs iu M idth, dcpth, vrlocity, wvatcr-surlacr rlcvaliou, tind streani-bed elcvntion nith discharge durins fluod of Ikcemhcr 1!l4?-June l'J41, falling st,tigc at thc same discharge nnd the same width roloriido Hiwr lit (:nnid ('anyon, Ariz. CHANNEL-SHAPE ADJUSTMENT DURING INDIVIDUAL FLOODS 33 tlie suspended load mas onl_v 100,000 tons per day. ccntrations were lower. Tlie San Juan at Bluff ancl The lower suspended load was accompanied by a lower the Colorado at Grand Canyon deposited during a velocity and a correspondingly higher depth. This is part of tlic rising stage, then scoured during the re- indeed in accord with the relation already noted in mainder of the rise antl continued to scour during the figure 15; that is, at constant discharge and width, falling stage. decrease in suspended-sedinimt, load is accompanied Figurc 26 presents three cross sections of the Rio by a decrease in velocity and an increase in depth. Crrantle channd at) the gaging station during the The observed rolntions during this flood are in ac- passage of tlie 1948 spring freshet. The scour of thr cordance with tlie ideas explained in association with bed was irregularly distributed across the section but figure 18. For the same value of b (slope of the width- on the whole, scour progressively occurred up to the discharge line), a decrease in the value of j (slope of peak discharge and deposition occurred during the the suspended load-discharge line) is associaled with u falling stage. decrease in the m to f ratio; that is, the slope of the The foregoing analyses of scour and fill of a river bed velocit y-discharge line (m)decreascd while tlie slope during flood demonstrate that the changes in the bed of the depth-discharge line (f) increased. occurred simultaneously with changes in the rate of A second example of the day-to-day adjustments of change of suspended-sediment concentration. channel shape and suspentlctl loatl is the spring riso of It is the thesis of this paper that the observed changes 1941 on the Colorado River at Grand Canyon (fig. 23). in the stream bed resulted from changes in the sediment In this flood passage, at a discharge of 20,000 cfs load brought into the measuring reach from upstream. flicre is a break in the slope of tlie width-discliarge It is postulated that the hydrodynamic factors involved curve, as well as in the loatl, velocity, and depth tend to promote a mutual adjustment between channel curves. At 20,000 cfs, in the rising stage, the load was shape and the sediment load carried into the reach. about I,OOO,OOO tons per day, mliereas in tlie falling The change in sediment load which results in a change stage the load was about 120,000 tons. This lowered of channel shape involves both bed load and suspended load was accompanied by a definite increase in depth load. However, because only the suspended load is and lowered velocity while the width remained the measured, it is necessary to use the data on the sus- same. pended fraction of the load as an index t9 how the total The bed elevation rose during the first part of the sediment load interacts with the hydraulic variables. flood rise, then lowered as tlie discharge continued to That the suspended fraction is a meaningful index is increase toward the pcak flow, it continued to lower demonstrated by the fact that the relations between during the falling stage. At 20,000 cfs, the bed eleva- the channel-shape factors and suspended load, which are tion was 8.5 ft lower on the falling than on the rising derived from measurements of a number of different stage. The gage height, plotted against discharge, rivers, appear to apply in principle to channel changes showed that the water-surface elevation was the at a given station during an individual flood. Specifi- samr in both rising arid falling stages at each discharge. cally, with no change of channel width, a decrease of At the measuring section, therefore, the difference in suspended load at a given discharge was accompanied rhunncl shape was talcen care of by changes in bed, by an increase in depth by bed scour that resulted in a hit tlie control downstream, a bar of heavy gravcl, dccreasc in velocity. This is the type of change indi- (lid not shift. cated by figure 15. The decrease in velocity provides The reality of deposition on the bet1 during the first the adjustment of capacity for carrying the load of the part of the flood rise can be ohservctl in thc successive particular size-distribution supplied by thc watershed. cross sections in figure 24. The river bet1 was lower In response to a decrease in load, the channel shape on January 12, (5,210 cfs) than on Xfarcli 6 (23,400 cfs). hccaine adjusted through scour to the lower capacity Thcse cross srctions dernonstratc that the bed was required for quasi equilibrium. fillrtl during the first part of thtb flood rise and then In the hypothesis stated it was postulated that thc

I\ 21s scoured as tlic tlischargr ~'os(h LL~JOVC20,000 cfs. dianges in channel shape occurred in response to a A iurtllrr cxamplc whicli illurtratrs another type of change in sediment load brought into the reach. Thls ;ttlj~istmcntto changing suspcntlctl load is providrtl by postulate requires proof that the observed change in t1ic spring rise of 1948 on the liio Grancle at Bernnlillo, suspended-sediment load was not the result of the prtwntctl in figure 25. Thcrc was sonic decrease in observed change of vclocity in tlie reach rather than siislmidcd load at a discharge of 11,000 cfs as the peali tlie cause. This is just the question faced by Mackin flo~ptisstbtl the station. The ciirve of bed elcvation (1948, p. 471) when lie observcd that sliows that t,he river scourcd its bed during the rising As set up in an equation * * * load is a functioll of st:igv hcri thc sediment concentration was high antl velocity. In answer to a query as to which is the cause, and fillotl thf hlduring the falling stage when tlie con- which is the effect, the averrge engineer will assert that velocity 34 ‘THE HYDRAULIC GEOMETRY OF STREAM CHANNELS coiitrole or determiries the load carried by a stream. * * * tration. Undcr such an assumption, increasing veloc- [The equation as a] method of expression is inadequate for ity should be associated with htl scour and dccreasing present purposes because the terms of a11 equation are trans- velocity with bed fill. posable. During the rising flood stage in both thc San Jiiari Tf the high siisl)ericlcd-sccliiiicrit concentrations re- River at Bluff (fig. 21) and in the Colorado River at, sulted from the scouring a(-tion of high vclocities, it is Grnncl Canyon (fig. 2:)) the heck were filling during implied that high velocity in a givcri reach scours the u period of rapidly increasing velocity concomitant channcl in that reach. The increase in sediment iii with a rapidly increasing suspended-sedirneIit loud. transport resulting from the local lwd scour should then The material deposited on the lml obviously came accourit for the observed incwnse in sediment concen- from upstrpam. How far upstream is unknown. It

Distance, in feet 0 IO0 200 300 400 30 I I RIGHT LEFT BANK BANK P -May 19, 1941 - 102,000 cfs- --

F~G~IHE24.--Channrl cross sections dirririK progress of flood, 1)eceiiiht~ly40-Jrine 1041, (‘iilorado River at (trarid Canyon, hriz. EFFECTS OF ROUGHNESS AND SLOPE IN ADJUSTMENT OF CHANNEL 35

are typical of a number of stwanis in western United ... 3 States. The spring floods are derived primarily from snow melt in the mountains. 3Iuch sediment must be carried into the major stream c.hannels by the melt water, but this load is augmcnted by the transport of readily available material tltlpositcd in the channel system during tlic local sumtnw Hoods. The spring flood leaves tlicb river Idof the trunk river at a lower elevation tlian had obtairictl before thc flood, but the siircessiori of summer floods tcnds to aggrade the bed to the level obstnd precding the spring snow melt. The succession of cvmts appcars from available data to be approsimatcly the same each year. It can be said in summary that the scour and fill of the bed of the main stem of an alluvial river during flood appears to be an adjustment of channel shape in response to a varying sedirnmt load. The adjustment takes place rapidly. This should not be interpreted as implying a complete ahsence of a time lag. c aJW c In the long-continued flood passages typical of spring & .c snow-melt contribution to Western streams, the g d $2 progressive adjustment of cliannel to load is easily 7s observed from the data, as has been shown. However, 02 -x-,w% similar graphs for small irregular changes in discharge wc even for the same stations show that the adjustment wzLO c is not so nearly complete as this. A time lag measured s:- 500 1,000 5.000 10,000 50,000 in days and perhaps weeks is required for adjustments Discharge, in cfs to take place, as might be expected. EXPLANATION OF DISCHARGE Width Deplh Velocity Load Slope &d THE ROLE OF CHANNEL ROUGHNESS AND SLOPE Rising + 0 x o I 0 Falling o v e o rn IN THE ADJUSTMENT OF CHANNEL SHAPE TO SEDIMENT LOAD FIGURE2.5.- Changes in width, depth, vrlocity, water-surface elevation, and strearn-ht.tI elevation with discharge during flood of May-Jme 1948, Rio (irsnde new 13rmalillo, N.Mrr. It has been shown earlier that, at constant discharge, suspended-sediment load is related to the shape fac- is presumed that the bulk of the sediment carried past tors-width, velocity, and depth. It was noted that Bluff and Grand Canyon during spring flood was derived at constant width and discharge, increased suspended- originally from erosion in ephemeral gullies, rills, and sediment load would be associated with increased by sheet wash rather than from the channels of the. velocity. But because 9 and w are constant, then larger perennial tributaries and trunk streams. the product rXd must be constant. Any increase in Concerning the seasonal sequences of scour and fill, velocity, therefore, must require a decrease of depth. the three rivers discussed in the prcweding examples As the Manning formula states:

Distance, in feet 0 100 200 300 10

c in which v=velocity W - LEFT d=depth L hydraulic radius ._c s =slope of the energy grade line i ._CP n =roughness paranie t er .caJ W It is clear that an increase in velocity and a decrease E?O W d -; of depth requires an increase in the factor n or, in other terms, increased velocity arid decreased depth would require an increase in channel slope or a decrease of FIGURE26.-Chantiel cross sections during progress of flood, MarJune 1W8. Hio (Jrande near Bernalillo, N. Mex. roughness or both. 36 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

Under the condition of a varying discharge a similar creases. This leads to an explanation of the observed result is obtained. The relations of depth and velocity m2 value of -<- to discharge have been expressed by the equations .f 3 Because the present paper deals with the suspended- d=cQf sediment load of streams, the general effect of sediment v=kQm load on channel roughness will be considered. Vanoni (1941, p. 618-619; 1046, pp. 96 and 127) showed that an increase in susperidcd load tends to decrease channel resistance and thus causes an iiicrease in velocity. He explains this effect as a rcsult of dccreased turbulence. The coefficients c and k are numerical constaiitls for a Buclrley (1922) found similar results in measurements given set of conditions. of the Nile River. Thus The effect of suspended-scdirnent concentration on channel roughness was discussed by Thomas (1946, pp. 43-45) who also concluded that increased concentration was associated with dccreased values of 34,anning Tinder the condition that s and n do not change with roughness factor n. discharge, then It should be expected, then, that when a given river 2 cross section is considered, the large changes in sus- m=- f pended-sediment concentration which occur with 3 (5) changes in discharge should be reflected in important _-_m2 changes in channel roughiicss. In a given cross section f-3 (at a station), an adjustment of water-surface dope where does not appear adcquate to account for the observed sk. m2 -- increases with discharge, ->- variations in the velocity-dcpth relations; so, important n f3 changes in roughness must occur. During the passage of a flood through a given river and when cross section, the slope of the water surface does change. In any given reach slope tends to be steeper during a -Sf decreases with discharge, -<-m2 n f3 flood rise than a recession. The steeper slope on the flood rise is due to the fact that the rise at a given cross The average values of m and f for rivers studied were section precedes the rise at any section downstream. It shown to be will be noted, however, that the normal shape of a flood hydrogpaph tends to make the most marked change At a station Downstream of slope coincide closely with the passage of the flood m= .34 m=.l crest in a given reach. f = .40 f=.4 In the Colorado and San Juan River examples however, it is observed that there was a change in 2=.85 -=.25m f f velocity-depth relations which coincided with a change in suspended load-discharge relations at a discharge It appears, then, that for the rivers studied, the well below the peak. There is no a priori reason that S4 any change of the slope-discharge relation should have factor - incrcascs with discharge at a station and n occurred at that discharge. These facts indicate that dccrcases downstream. In the following discussion such at-a-station adjustments in velocity-depth rela- the separate effects of slope and roughness will be tions result primarily from changes in roughness consitlcretl. It will be shown that slope of the water associated with changes in sediment load, rather than surface tericls to rcmain about constant at a station from changes in slope of the water surface. hloreover, while thc increase in the conccntration of suspended during the passage of a flood lasting several months, load with discharge is associated with a decreasing conditions approach steady flow and therefore changcs Si in water-surface slope as a result of the passage of a roughness. Thus, the at-a-station valuc of - tends 7L flood wave are minimized. to increasc with discliargc, a fact which explains why The velocity-depth changes and their relation to m 2 suspended load observed at a single cross section in the the mean at-a-station value of - exceeds -. f 3 Colorado Riwr, Rio Grandc, and San Juan River are It will further be shown that in the downstream typical of all the cross sections studied for which direction the value of roughness, n, tends to be con- adequate data are available. This appears to support servative, or remain about constant, while slopc de- the idea that such changes are typical of fairly long EFFECTS OF ROUGHNESS AND SLOPE IN ADJUSTMENT OF CHANNEL 37 reaches of channel, measured in tens of miles rather than in tens of yards. Changes in velocity-depth relations typified by the individual floods measured at Grand Canyon, Bluff, and Bernalillo must primarily represent, therefore, changes in roughness rather than in slope. If observed changes in velocity-depth relations can be attributed to changes in roughness, then lower values of roughness are associated with largcr suspended-sediment concentrations, and this is just the effect found by Vanoni, Buckle?, and Thomas. The changes which have occurred in the lower reaches of the Colorado River after the construction of Hoover Dam, appear to confirm these observations, as will now be shown. When the sediment load of the Colorado River was stored in Lake Mead, clear water released from the reservoir caused degradation in the channel reach below Hoover Dam. The main degradation occurred in the reach of about 100 miles between Hoover and Parker Dams. The gaging station at Yuma, Ariz., is about 350 miles below Hoover Dam. A record of observations before and after the construction of Hoover Dam allows a comparison to be made of the channel characteristics at Yuma between a period when the river carried a large suspended load and a period when the load was markedly reduced. The current-meter measurements for that gaging station used in the present analysis are available in the offices of the Geological Survey. In addition to the data on width, depth, velocity, and discharge included in the current-meter data, summary data on river slopes and on suspended-sediment load at Yuma are contained in a detailed report by the Bureau of Reclamation (1949) .l These data provide the basis for the present analysis. The Yuma gaging records are obtained from a cablecar. The river at this point flows in a flat alluvial valley bounded by low scarps probably representing pediment remnants. The channel conditions before I I 1 111 I I IIIIII the changes brought about by the construction of ' 100.000 500 1,000 10,000 Hoover Dam were described by the Geological Survey Discharge, in cfs as follows: FIGURE27.-Relation of width, depth, velocity, water-surlnce elcvatioii, and Channel at all stages composed of sand and silt. Brush arid stream-bed elevation to discbarge, Co~orndoRiver at Yu~ria,.irk. S:inllh! trees growing on both banks. * * * Channel is straight for periods before and after constructiou or Hoover Dam are showu. about 1,400 ft above and a mile below the cable. In figure 27 the width, depth, velocity, gage height, curvcs presented for other river cross sections. The and stream-bed elevation are plotted against discharge relations tit Yunia changed, however, after construction for the Colorado River at Yuma. The figure includes of Hoover Dam. graphs of these relations for the period 1927-34 before Figure 28 presents the progressive cliangc of thc same the construction of Hoover Dam and similar graphs factors with time. The width, depth, velocity, gage for 1945-46 to represent measurements after the con- height, bed elevation, and roughness representing the struction of the dam. It will be noted that the relations average values for each of five discharges are plotted of these factors to discharge are similar to at-a-station against time, in years. Figure 28 includes the annual suspcnded-sedirncnt I The river-slope data usod in the rollowbig paragraphs appeal in figures 5 and 9 in the Bureau of Reclamation report, and the suspended-load data are tabulated in load and annual discharge at Yuma gaging station table 8. D. 33.

plotted~ against time. Also included are grapllv of * Contained in description of tho Yuma gaging station, dated June 3, 1!l38, in the flles of the Oeological Survey, Washington, D. C. the median size of bcd material and river slope. The 38 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

I( Slope interpoloted from

FIGUREB.--Progressive changes of Colorado River at Yuma, Arie., resulting from the construction of cngineering works upstrea~n;seven mean values of dlanll~l chiiriicteristics arc shown for each of five discharges. EFFECTS OF ROUGHNESS AND SLOPE IN ADJUSTMENT OF CHANNEL 39 hi-material size was measured at Bureau of Reclania- tioii sectioii 7-S (1949, fig. 15), situated about 3 miles above Tuniu. River slope was obtained as follows: Successive measurements of particular cross sections WCIT niatle by the Bureau of Reclamation to obscrvth tlic degradation of the channel following building of tlic dam. These measurements pcrmit ted the const ruvt ion of sucwssivc profiles of different reaches of tlicb river. The profiles of the reach from Laguna Dam to tlw AIcsican boundary, published by the Bureau of Rcclaniation (1949, fig. 5), are presented as figure 29 of tlic prcscnt report. The average slope in the 7-niile rctic~li (Rcclaniation sections 7-S to 10-S) which includes Yuma gaging station, is plotted against time in figure 28. The conclusions to bc drawn from figure 28 will now bc presented. Following the construction of Hoover Dani the annual suspended load passing Yunia decreased from an avrrage of nearly 150 million tons to less t,lian FlGlHE ZS.--Hivrr profiles in reach of Colorado River near Yuma, hrik.. COIII- paring conditions in 1902 before the construction of enginwring works opstrrarn 25 million tons. The water discharge fluctuated from x ith conditions in 1940 and 1942 (after Bureau of Hrcldmaliori), one period to another, depending on the releases from the clam and the contribution of tributaries entering and the relative erodibility of bed and banks. The the Colorado below the dam. In the period analyzed new discharge regime lacked the high floods to provide here there was, however, no large, consistent and sus- such widening, and new width-depth-velocity require- tained decrease in water passing Yuma after the dam ments for transportation are balanced by the relative WBS h uil t . erodibility of bed and bank now existing. This great decrease in suspended load supplied to The reduction in width implies a larger capacity for thc reach near Yuma caused the channel at Yuma to suspended load at a given velocity and discharge, and change markedly. Simultaneous with the reduction a smaller capacity for bed load. However, because in suspended load, the channel at Yuma was degraded. velocity has decreased, the new channel can be con- The gage height, or water-surface elevation, decreased sidered to have a lowered capacity for both types of 5 to 6 ft but the elevation of the river bed decreased load at a given discharge. about 9 ft. The difference represents the increase In summary, in this example for which considerable in mean depth, this increase was particularly great at data are available, a decrease in sediment load-which high discharges. Despite a decrease in width from 510 originally, at least, was carried primarily in suspension- to 430 ft accompanying the degradation, the mean was accompanied by an important increase in bed velocity decreased considerably. This decrease of roughness. Slope remained essentially constant. Un- velocity resulted from a considerable increase in rough- doubtedly, a different proportion of the load is carried ness. Thc changes in slope were small and there was as bed load in the present than in the original channel, no progressive change during the period of record. and may be in part responsible for the observed increase The consistency in slope in the reach near Yuma can of roughness. be seen in the river profiles of figure 29. It is possible that changes in slope which have The strong increase in bed roughness was not caused occurred in upstream reaches near the dams may in hy a corresponding, consistent change in the size of time affect the Yuma reach. In the years since the bed material, at least the correlation of the roughness construction upstream began, however, a quasi equi- and the median size of bed material is poor. It appears librium has been established at Yuma by change in more likely that the increase in roughness was an channel characteristics and not by changes iiL slope. effect of the cltw-eased concentration of suspended The example demonstrates that even during a period scdinient . of years adjustment of a channel reach in response to a The explanation for the decrease in channel width change in suspended load may be made primarily by nt Yuma probably lies in the cessation of large floods changes in channel shape through changes in roughness after construction of Hoover Dam. The channel was rather than in channel slope. degraded. The large floods formerly characteristic of The longitudinal profile of most rivers is concave to the reach had resulted in the development of a wide the sky, or the slope tends to decrease downstream. original channel to provide a balance among the re- The change in velocity-depth relations in the down- quirements of width, depth, and velocity for transpor- stream direction, as discussed earlier, must be related, tation of the load which was supplied to the stream, in part, to this progressive change of slopc. The role 40 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS of roughness, as compared with slope, in the mainte- the Manning n is different from and probably lower than m that which applies to the same reach at a different nance of the observed - ratio for the downstream f discharge when the bed is molded into dunes or anti- direction can now be analyzed. dunes. This idea was clearly stated by Rubey (1952, At least three factors affect bed roughness: the p. 131). particle size, the bed configuration, and the sediment In figure 30 the relation of bed roughness measured by load. The effect of sediment load on roughness is in Manning’s n is plotted against discharge for four sizes itself partly determined by particle size and is intri- of sediment in transport. The data were collected by cately related to bed movement, as well as to the sus- the U. S. Waterways Experiment Station and mere sum- pended fraction. The Einstein equation (1950) provides marized by Johnson (1943, pp. 80-84). From Johnson’s a method of determining bed roughness when certain summary, data for figure 30 were taken. The expcri- other factors are given, particularly the slope and the ment concerned the transport of natural river sand by size distribution of bed material. The Einstein equa- water in a flume. tion, however, allows width of the channel to be It wilI be noted in figure 30 that a constant slope was assumed, and therefore does not specify what width- maintained. When sand 1, the coarsest of the four depth ratio will be developed under natural conditions. sizes discussed here was used, an increase of discharge The present analysis is concerned with relations was accompanied by a slightly increasing roughness. among width, depth, and velocity as they are observed At low discharge, the roughness associated with sand 3 to exist in natural rivers and is, furthermore, restricted was only slightly higher than that for sand 1, but wheit in dealing with sediment load to its easily measured the configuration of the bed changed from a “smooth” fraction, or suspended load. condition to one of “general riffles” (terms used by The present discussion of the effect of the three Waterways Experiment Station),the roughness increased factors of size, bed configuration, and suspended load markedly. With further increase of discharge, rougli- on roughness aims at a general qualitative under- ness decreased gradually. standing of how the values of roughness observed in The importance of these relations may be summarized rivers appear to be related to the average characteristics as follows. An increase in bed roughness is one type of of rivers. change that will result in the decrease of velocity with A characteristic of most natural rivers is a decrease respect to depth. Because changes in the velocity- in particle size of the bed material in the downstream depth relation constitute a part of the adjustment of direction, due primarily to abrasion. The authors are channel shape to load, variations in roughness affect in agreement with Mackin’s (1948, p. 482) rejection of capacity for load. the oversimpIif?ed explanation that the downstream Decreasing sediment size results in a decrease of decrease in river slope is caused directly by the de- roughness due to the individual particle. But in hydro- creased caliber of load downstream. dynamically rough channels, the roughness due to bed A change in caliber of load must have an effect on bed configuration may be niuch more important than that roughness and, through roughness, may affect other due to particle size, and sometimes decreased particle stream characteristics. This effect will now be exam- size may result in largcr or more effective bed ripples or ined. waves and thus in increased rouginless. In the present paper a distinction is made between It will be not,ed in figure 30 that the values of rough- roughness resulting from ripples or dunes on the channel ness representing “general wavcs” arc markedly differ- bed and roughness due to the individual particles of bed ent among sizes of sund. Largcr values of roughness are material. A nearly identical distinction was made by associatcd with smaller sand sizes. ‘I’his implies, Einstein and Barbarossa (1951). It is assumed in the therefore, that among the four experimental runs using present discussion that the roughness due to channel different sands, even when the configuration of the bed irregularities includes not merely semipermanent ob- looks similar, for a given discharge and slope thc sedi- structions such as rocks and bars but transitory ment pnrticles of smdler size may arrange thcmsclves conditions of bed ripples and waves which change with in such a manner that effective roughness is greater. discharge. than for larger particles. The Manning roughncss factor in pipes represents a At a station with a specific bed configuration, an more or less permanent characteristic of the boundary increase in discharge may be accompanied by eitlier a surface. This is also true for certain open channels, slight decrease in roughness, as in sands 3, 6, and 7, or drainage canals, and even irrigation canals where the a slight increase. However, with increasing discharge banks and bed are fixed. However, in most sediment- the bed may change successively through conditions laden channels the roughness is not a permanent char- described as smooth, riffles, bars, dunes, arid finally acteristic but must change with the configuration of the antidunes. As indicated by Langbein (1942) , antidunes bed. It is logical to suppose that when the bed of an form when the Froude number approaches a valueof alluvial river is smooth, the roughness as described by 1.0, critical flow. Also, the finer the material, the lower EFFECTS OF ROUGHNESS AND SLOPE IN ADJUSTMENT OF CHANNEL 41

.030 t- - Smaller sediment size - Pm,: r\ - SAND 7 C - mi - 0, .-c v c - rz -s ,020 - m 4- m 0) c - c CT 3 0 - 13L

Larger sediment size -

.OlO 40 60 Q (discharge),in thousands of cubic centimeters per second

EXPLANATION

SLOPE .OOl SANDS I 367 SMOOTH -______--_------3 X V O

LOCAL RIFFLES OR WAVES -e----@ x.b GENERAL RIFFLES OR WAVES - - ---e + v MEAN DIAMETER (MILLIMETERS)--- ,586 ,525 ,347 .310

FIGURE30.-Channel roughness (Mamiing n) in relation to discharge for various sizes of sediment (from flume data). will be the valuc of the Froude number at which the size, a change in bed configuration may violently change transition occurs from a smooth bed to dunes and from bed roughness and its effect may be of the same or tluries to antidunes. greater order than the effect of differences in particle At a given discharge, the change in bed configuration size. may increase the valuc of n more than does a consider- Headwater tributaries flowing on gravel or boulders able increase in discharge during which the bed configu- obviously move the large bed debris only in exceptional ratioii remains the same. flows. The channel roughness in such streams must bc Two general conclusions may be derived from figure 30. First, for the same slope and discharge, the effect determined by the sizes of the individual rocks or cob- of decreasing particle size is a tendency to increase the bles, because the bed material is not ordinarily in roughness even when the condition of the bed is observed movement and is not thrown up into transitory ripples to remain the same. Second, with no change in particle and waves in the sense used in describing the bed of an 42 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

alluvial strt~arri. Thr downstream decrease in the size headwater tributary down to the Powder River, of the gra\-cl or pebblr twd therefore results generally ina Wyoming, listed below. Tlic data apply to the condi- downstrcam decrrase in the value of channel roughness. tions of velocity, depth, and width, prevailing at a This can bc seen in the tlntn tracing thc change from a discharge equal to the mran annual rate.

M(!arl StTation aiinual dis- Slope: charge (cfs) ______

Upstr(JaIii Red Fork near Barniim -. .._ -1 50 0. 0057 0. 058 Gravel. I Middle Fork ah\e Kaycer . -_ -_.I 73 . 0038 . 087 Gravel. 1 Middle Fork near I

Tlic clwrease in vduc~of roughness n is intcrpreted as largely fine sand, there is a progressive diminution of the result of decreasing particle size, but may also be slope but a nearly constant value of channel roughness. influencccl somewhat by a downstream increase in Though the particle size decreases downstream in this suspended-sediment coliwntration in that particular river system, from coarse sand to fine sand and silt, the reach of river. values of roughness are sufficiently large for it to be However, in the Arikarw, Republican, and Kansas assumed that bed configuration rather than size of Rivers in the Kansas River system in which the bed is individual particles controls the roughness.

- -~ ~~ ~~ ~~~ ~ ~ ~ ~ Mean

Station annual dis- Slope ~ Ro'lghness I charge (cfs) I (4

Upstream. Arikaree River at Haigler--- - __... - - - -... .. - ~...... -. - ~..-. .~ 29 0. 0031 0. 040

Republican River at Culbertson.. .. . -. . . . ~...... -. . . . -. -. -. ~ -.-. 205 .0015 . 023

Republican River at Bloomington. -.. . - ~..- - - -.. . ~. ~.- -. -. ~ -.- - - 670 . 0010 . 036 I Republican River at. Clay Center- -.-. . __. .------__.~.- -. - ~.~ - - - - -. 1,000 . 0004 . 023 1 Kansas River at Ogden- ___ -.. -. . . .- - - - -.------.-. -.- -. ~..-. -. 2, 300 . 0004 . 035

Dowristrram Kansas River at LeCompton-_. .- -.. - - ~ - -.___ - - - __ - - -.- - - - -.. -. - - 6, 200 .0004 . 034

~- -- ______~_ ~ ___ -~ - ___ ~ _-

Though the slope decreased downstream , the rough- presumably in part because of the greater suspended- ness showed only a modest decrease that was nearly sediment concentration, as indicated by the lower masked by its variability. ordinate position of line C4-D4. As a generalization, it may be said that the value of The river slope decreases downstream as indicated by channel roughness is controlled by the size of individual the slant downward to the right of line A5-R5. In particles when this size is large; that is, gravel, cobbles, flood flow the river slope is probably very slightly or eve11 coarse sand. When the size of sediment is greater than at low flow, in part, because of the tendency larpc, charinel roughness teiids to decrease downstream for the main thread of water to eliminate some of the as tlir size of the particles drcv-eases. In streams whose minor contortions of the low water channel. Thus the beds are composed primarily of fine material, there is a line AS-CS representing the at-a-station change of river tendency for the bed configuration (ripples, waves or slope is drawn so that it slopes slightly upward to the dunes) to control the clianiicl roughness. In streams right. like thcse roughness tends to change but little in the In summary, the observed relations of velocity to downstream direction when discharge of equal frequency depth in the downstream direction are considered to be obtains along the river length. those required for the transportation of the supplied The average changes of roughncss and slope at a load. It is postulated that these velocity-depth rela- station and downstream are part of the hydraulic tions are provided by the concomitant change down- geometry of the channel system. For alluvial streams stream of roughness and slope. Roughness is governed in which roughness tends to be determined by bed con- by the sediment size, the suspended-sediment concentra- figuration and, therefore, decrpases downstream only tion, and the bed configuration, all of which are inde- slightly, figure 19 indicatcs the average relations with pendent of the channel or nearly so. That part of the discharge. The line A4-B4 representing the down- required relation of velocity to depth not provided by stream change of roughness for a rate of discharge that the downstream change in roughness is provided by occurs frequently or for a low discharge, is nearly change in slope, because slope is the dependent horizontal. For flood flow, the roughness is lower, factor which can be adjusted by stream processes. SOME PHYSIOGRAPHIC IMPLICATIONS 43

The c~haracteristicallyrapid increase of suspended a number of empirical equations that were derived from load with tliscliarge at a station requires for transporta- these data. From these equat ions, wliicli lie considered tion of that load a relatively rapid increase of velocity “basic,” are derivable many other correlalive relation- in comparison u ith depth, or a relatively large value of ships. Included in 11;s basic equations (1 939, p. 24) are -.m For the cross sections studied this was equal to .85. .f I’=-2.67@ 1 sz . .,I liis value, being larger tlim :{, requires that - increase I.‘: I. n with incixnsing disc.li:irp~. This is ac.coniplished pri- where Y=wet t8edperimeter tnari1~-by the decrease of roiighiiess, n, resulting from Q =discharge the incrense of concent ration of suspentled load with V=rnean velocit\- discl1aJge Iz =hydraulic radius mean depth In tlie clownstremi tlirection, however, loud increases F=a sediment factor not quite so fast ab ctiscliarge; this means that tlie concentratiori of suspended setliinent, decreases slightly l’lie eqiiation relating wetted perimeter to discharge tlowiistreani. The transportation of this load requires is itlenticd in form to the relation found in the present that tlie depth increase with tliscliarge rnuch faster than stud>-for river data, inasmuch as the wetted perimeter, m P, is nearly equal to width, w, in wide channels. The does tlie velocity, or tlie - ratio must be small. For f comparable equations are the rivers studied its mean value was 25. 1 s 2 Lacey Present study 7’0 provide that velocity-depth relation, must de- ?1 p=2.67&- --- w=aQb crense tlomiistreum. Hew use of the nearly constant where b=0.5 in downstream direct ion concwitr:Ltion of suspended sediment and the efiect of bet1 cwnfiguration, roughness T! tends to remain about 1,nce.v recognized that liis equation relating wetted m perimeter to discliarge applied to many natural streams, constarit tlownstream. ‘J’hiis the required - ratio is .-f hut having few river data lie could not explore the provided primarily hy the decrease of slope downstream. limits of its applicability. Lacey’s data provided him with only a smiill range SOME PHYSIOGRAPHIC IMPLICATIONS of intercepts, so that his width coeffic-icnts, ranging from 2.50 to 2.80 and averaging 2.6, included a miicli THE STABLE IRRIGATION CANAL-AN ANALOGY smaller range than corresponding coefficients for Amrbr- TO A GRADED RIVER ican rivers. Because of the small range of intercepts An irrigation canal constructed in alluvial material in the canal data and the fact that the few individual tends gradually to scour or fill ibs bed and banks until a measurements on rivers also happened to plot wi tli nearly the same intercept as tlie canal data, Larry (TOSS section is achieved that remains in equilibrium with the water ant1 sediment flowing through it. En- conduded that his width-disc-harge equation was funda- gineers designing canals to carry sediment-laden water mental and that it was indcpcndent of sediment size. have faced many of the same problems that face the The second Lacey equation expresses the relation of geologist attempting to understand the grading of rivers. velocity to depth, A large literature exists, particularly in India, on the Vcc R$ subject of “regime” channels ; that is, channels that, are Becuiisc R, the hydraulic radius, is essentially equal to neither scoured nor filled. mean depth, d, of with rivers this equation is directly Early work by Kennedy (1895) and Lindley (1919) m laid the foundation for the research of Lacey who in comparable to thc -f rclation for rivers discwssecl earlier. two papers (1930, 1939) presented a detailed analysis Equations (2) and (3) of the present study can be of stable canals studied in India. Important addition combined as follows. to this literature have been provided by Imie (1937), Das (1950), Blench (1951a, b), and others. d=cQf In figure 3 1 tlat a on regime canals published by Lacey and (1939, pp. 41,49) have been plotted in the same manner v=kQm as the river data shown in figures 5-8. There is indeed then considerable similarity between the graphs for regime canals and those for rivers. Lacey utilized measurements of channel character- or istics made on reaches of Indian canals that, through m time, had achieved a stable cross section. He proposed v ocd7 44 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

- + 0- W 22 2 .t s -w== hv)-W i--w=i<- T’ c L Ti 00, 0 -Oa I O

EXPLANATION: v----- .PUNJAB REGIME SITES; x--- UPPER BAR1 DOAB; -MADRAS GODAVARI

FIGURE31.--Hclation of width, depth, and velocity to discharge lor stablp (regime) irrigation canals in India (Lacey data).

m along the canal length different from that of the average In the downstream direct,ion for river data --0- .251 f river in the downstream direction. whereas the comparable Lacey exponent is 0.5. The A canal system begins with the largest discharge at difference between the numerical values of the exponent the intake where river water is diverted. Downstream in t,hese equations probably can be attributed to the the flow is gradually divided into successively smaller difference in the downstream change in suspended- lateral canals. A regime canal is by definition one that sediment concentration. It was shown in figure 18 neither scours nor fills its channel. that for any given value of b (the rate of increase of To preserve a condit)ion of neither scour nor fill, as width with discharge), the larger the value of the ratio much sediment must be taken out of the system at the m final points of diversion to the irrigated fields as was - t,he larger is the value of j. In terms of the com- f introduced at the head of the canal system. This is parison of the canal and river data, the regime canals accomplished in practice by dividing, at each point of m diversion, the sediment load in the same proportion as show a larger - value and must experience a rate of f the water. That is, if at a given point of lateral increase of suspended-sediment load with discharge diversion 10 percent of the flow of the main canal is SOME PHYSIOGRAPHIC IMPLICATIONS 45 diverted, then 10 percent of the sediment being carried lies in a consideration of slope. In the design of canals, in the canal is also diverted. Blench (1951b, p. 5) it has usually been found necessary to provide greater implies that this is the practice in handling regime- slope in the small laterals than in the large trunk canals. canal systems when he says: Thus canals in regime have a longitudinal profile convex One [canal] exhibited serious sediment trouble on one branch upward because the flow is reversed as compared with but not on another because engineers did not understand regula- natural rivers-that is, the discharge in a canal system tor designs that could divide sediment suitably. decreases downstream. This is analogous except for Blench makes a similar statement elsewhere (1951a, the direction of flow to the condition in natural rivers, paragraph 2.17): which are steeper in the headwaters where the discharge * * * The Lower Sheluni Canal gave tror;hle [by progre5sive is small than downstream where the flow is great. filling of its bed] because the regulator for the two Branches The rat8e of increase of width with discharge is ap- divided sedimen: unfairly. parently independent of the size of the sediment parti- Equality of division of water and sediment is tanta- cles. Not only is this rate similar far headwater reaches mount to the maintenance of a constant sediment and lower reaches, but is is similar for rivers and canals concentration throughout the regime system. which carry sediment of quite different sizes. Inter- If the suspended-sediment concentration remains cepts represented by a, c, and k, in equations (I) to (3) equal at all discharges, then the slope of the line repre- are probably related to sediment size. Lacey found so senting the relation of suspended load to discharge little variation of a in Indian regime canals that he would equal 1.0. This slope is, by definition, the value considered it to be a constant for all streams. Inspec- j. Now, in figure 18, if in the downstream direction tion of the large differences in intercept of the lines in for both canals and rivers, the exponent b in the width- figure 9 representing natural rivers indicates that the discharge relation equals 0.5 as has been demonstrated, value of a is not a constant. Das (1950, p. 160), re- m analyzing the data on Indian canals, concluded that then the - ratio (corresponding to j=l.O in fig. 18) f this coefficient is “ * * * a function of the nature of m the periphery of the stream, i.e., it represents the nature should apply to regime canals. The -ratio at the f of the soil surface forming the margin of the stream.” value of b=0.5 and j=l.O in figure 18 is seen to be He stated that values of a for canals and rivers in delta about .43, or slightly less than 0.5. Tentativelv this areas where sediment sizes are small are less than for appears to be a satisfactory agreement. canals and rivers where sediment sizes are larger. It m As indicated previously, the smaller value of - ob- will be recognized from common experience that sandy f channels are wide compared with silty or clayey tained from the river data analyzed here indicates that channels. j< 1.0, or that the sediment concentration in the rivers A regime canal is a system in equilibrium. By studied decreases slightly in the downstream direction. definition it, is a canal which over a period of years It can be seen that the general relation shown in neither scours nor fills. It is a system in which slope figure 18 applies to regime canals, as well as to natural and channel characteristics are adjusted over a period rivers. In fact, the Lacey equations for regime canals of time to carry the load supplied to it at the upstream become a particular condition in the general relation m end. In this respect 8 regime canal fits perfectly the involving 6, -,s and j. The particular condition is definition of a graded stream (Mackin, 1948, p. 471). merely that of constant suspended-sediment concen- Mackin’s definition is: tration in the downstream direction. Agraded stream isone in which, over a period of years, slope The comparison of regime canals and natural rivers is delicately adjusted to provide, with available discharge and with prevailing channel characteristics, just the velocity required is valid only in the downstream direction. Canals for the transportation of the load supplied from the drainage carrying water high in sediment concentration would not basin. The graded stream is a system in equilibrium; its dictg- normally be expectrd to maintain the regime condition nostic characteristic is that any change in any of the controlling except at the discharge for which they were designed. factors will cause a displacement of the equilibrium in a direction The lack of a varying discharge at any particular canal that will tend to absorb the effect of the change. cross section precludes a proper comparison with the The discharge is provided to the canal system by a at-a-station relation of rivers. Failure to recognize source independent of the channel itself, just as Mackin this would lead to fallacious reasoning, as in a discussion visualizes a graded river. Load also is supplied by an by Stevens (see Lane 1937, p. 147) in which he compared exterior source. In the regime canal the water and sedi- the width-discharge formula of a particular river gaging ment are diverted into the canal system from a river, station with the Lacey width-discharge formula for whereas in a graded stream, the load and dicharge are canals. supplied from the watershed. In both, however, the Another fact that demonstrates the analogy between source is independent of the stream. regime canals and rivers in the downstream direction The regime canal is in equilibrium. Althougll local 46 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS or tcinporary woiir arid fill mriy occur to II minor ckgree, writers, and in principle is the conclusion reached it is ti tliugnostic characteristic of rcy$ne that a change independently by Rubcy (1952, p. 130). in an! controlling factor cuuscs a displacement of the Because the data in figure 31 represent canal scctions oqiii1it)rium in a direction that will ttdto absorb the. c1iosc1i as exnmplcs of regimc conditions, tlic figure rH’c>ct of tlic change. Wcrc this not true, local and may bc consitlcrcd to represent tlic relations of widtli, tempor;yy fill or sroiir .cvoiild have an acruniulativc. depth, ant1 velocity to discharge in a channcl systcin dftbct and the system would liavc., ovcr a period of yrars, gradcd in the sense used by Macltin. The gracletl c~liangrtlsufficiently for the tcrni “wgirnc” to bv inap- cliannc4 system rc.prescntect in tlic graph is n specific one plicahlc. in that, the graplis apply only to the conditions of It is logicnl to supposc, tlicn, that tlic point in figuw scdirncnt size and constitution of bed and bank esisting 18 wtiicli tlcscrihcs rcginic cmals, rcprcscmts values of in those canals. It has been noted, howcvcr, that tlw m intcrccpts of thcw lincs are the factors which woiilcl b, --,:iii(i j ir1iic.h apply to a gratled clianiid system in .I‘ changc with the type of sediment. thi~s(biisc uscbtl 1)y hIaclrin in his tlefinition of a gratlrtl The relative slopes of th depth-discliargc and rivcr. velocity-discharge lines arc considered to be functions 11 has lwcii tlcitionstratcd 1)y tlict concurrc.nt c1iang:cs of suspcndcd-sedirnent concentration at diflercnt tlis- 111 711, ,f, and j during individual floods t81iatthr gc~ncwil charge. rrlatioiis in figure 18 (Mine, at least in g(m\ral, tlic There might be some logic in postulating, that if inutunl atljustnieIits which owiir at a station mlicw data from a particular river plottetl with ns littlc suspc.ntlcd-scdiri~int (~oiicrntrationcliangcs. Bccausc scatter of points on these graphs as do tlic data on figurc 1 S dcscrihes r(~lations1)oth at a station arid dowii- regime canals, the river could be considcrctl in rcgirnc stretmi for averagc vnlurs of thc factors observcd in aiid thcrefore graded. River data discussed in this rivers, nntl approxiinat cily fits rcgimc. canals, any point report plot in graphs similar to those for regime canals. ni thc pap11 is corisitltmd by tlic authors to reprtwiit and the diff crences in sloptl and intcrcept appear to bv coiiwrixbiit vsliws of m ,.f, antl J rcqiiiretl for cyui1il)riurri. explicable by diff ercnccs in srclirncnt sizci :LI~ iii It IS untlrrstootl, IIO\\’CVtl., that tlic c~xact form ar1tl scdiment concentration. positions of tliv linrs in figiir(1 18 milst, be consitlcretl In practical terms, howrver, a particular gradcd rracli siibjc>ctt to atljustincrit as inor(’ (lata hccornc availablr. of a stream is generally too short to include a larg(1 Iri yracticc, tlie suspciitlui loatl-tlisc1inrg.c rdations enough number of gaging stations to allow a tievelop- viirj- within defiiiitc limits. At-a-station valiics of j ment of graphs adequate to describe in dcfinit ivc manncr probaldy vary bctwcen 2 and 3 antl downstream values tlie slopc and intercepts of the dowristrcam rclatioris I)ctwccn O 5 and I.:{. ’\’alucs of b nppw to rcmaiii Moreover, the slopes of the graphs dcpcntl for clear c~loscto 0.5 tlownstrcani aid 0.1-0.3 at, :I station. definition on a considerable rang(’ in disclinrge in the Thcrcfore, actiial valucs of the varia1)lcs occur in downstream direction. A stream segmcnt may be oiily a liniitcd part of tlic ficltl of valucs shown in graded when there is no increase in discharge tlowii- figure 18. stream. Tlic. conception of adjustment of the cross section The downstream graplis presented iii figures 5-8 do ol a ctiannc.1 in response to cliangrs in tlic load-discharge not necessarily represent river systems which arc rchtionsliips has not 1)cr.n strcsscd in tbe gcologic graded throughout their lcngth. For rxample, if litvrat urc. Only hiac-kin (1948) arid Rubcy (1952) give two gradrcl rc.aches arc scparatetl by a canyon reticti this tlic prorninrncc it t1cscrvc.s. As shown by the within which the river flows in a rock-linctl gorge, thc Yiinin (lata, n chmtge in the gowrning .factors may result c:~nyonsection may be steep and may includc waterfalls in chancgr~in chamP1 chnractrristics without a change in arid would obviously not be called graded. If the riwr slop^. gaging stations availablr for analysis were all located Tlic. prcsc~it, antilysis adtls weiglit, t,o tlic importtatice in thr two gratld rea(*lirs,it is possible that downstream of adjustrnrrit in cliannr4 cross section, arid the writers graphs comparable to figure 5-8 could be plottctl in woiiltl support tli~nltcrriatc wording of the definition wliich therti might be a vt~ygood alinement of points. of a prath~lriver which Alaclrin considrrd but, ap- Figure 6 representing the Bighorn Rivcr system parc3rit ly for t,lic: salw of brevity, tliscarcled. Mackiri includes two canyon sections, the Wind River Ca11yori says (1948, p. 484): tlirough the Owl Cre.c>liMountain, in the reach between The question arises * * * as to whether the foregoing Riverton and Thermopolis, and the Bighorn Canyon definition [of a graded river] which describes the graded stream between St. Xavirr and Custcr. Figure 6 shows a as one “in which stope i:, delicately adjusted,” etc., should not considerable scatter of points about the mean line be revised to read, “111 which slope and channel charactrrzstacs and at least papt of this scatter is due to the riorigraded are delicately adjusted,” etc. conditions of some of the reaches where gaging stations The latter phraseology would be preferred by the are located. SOME PHYSIOGRAPHIC IMPLICATIONS 47

Figure 5 prcscnts tluta for the Powder River systcm. The data presented here lend quantitative support The 1’owdt.r Rivr, at least in the reach from Kaywe, to the idea expressed by Macltin (1948, p. 506) that Wyo., to Locate, Mont., has many of the characteristics “* * * even in the aggrading stream the declivity iistd by hIuclriii (1948, p. 472-474) to support his choicbe of each segment is approximately adjusted to the load of specific rc~~liesof certain streams ns examples of in transit through that segment.” The quantitative graded rcachcs Tcrracc remnants indic.ate tlitit the data show that thc approximate adjustment involves Pawdcr River once flowed at IL high lcvtl. The tcrracc not only slope but! width, depth, and velocity, as surfaces cxliibit smooth, longituditial profiles concnv(’- Macliiii’s sttitcmrnt obviously implies. T1.w quantita- iip~ard,similar to the profile of the prment ilood-plain. tive (lata not only demonstrate the existence of approsi- As in Mach’s Coliinibiii Rivcr example, tlownciitting mate adjustmrnt hut, more importantly, describe it. during a period of planatioii was not prevc.ntcct by Lest the rc.acler conclude from the consisteticsy in the lwdroclc rc&tnncc. Thc undt~lyingmaterial in the hydraulic geometry dcfincd by thc available data bcb- Powtler Rivw Vnl1t.y is unconsoliclntetl alluvium. tween graded reaches and reaches riot linown to be This gradt~lreach is rcprcsentetl in figure Ti by the grntlcd, that there is no difference or that the graded six solid trionglts 1al)c~lctlpoints 2, 3, 7, 12, 22, ant1 23. condition does not exist in nature, it is cmpliasizcd These points liavc oonsidcr*nblc nlinement, and diwiattl her3 that the present study is not aimed at the quanti- from thv mean 1inc.s tlra-wii on the graph vcry littk tative differentiation of graded from nongr:itled rcvdics more than tlic scatter scwi in tliv regime canal data of A graded stream by definition (hIackin, 1948) is one figure 31. But the recognition of thp graded rcstcli whkh is in equilibrium through a period of ycars. The from the alinc.mmt of points would obviously bo hytiraulic geometry as defined liere docs not tlcmnn- impossihlr . strate whether gradual changes art in process, though It appears that thc available data do riot allow when a stream is sufficiently lacking in equilibrium to identification of a gadctl reach from reaches which are be suffering rapid degradation or aggradation, changes not definitely liriown to bc graded. It follows that thc in the values of factors in hydraulic geometry can he gaging-station data talten at points distributtxl over a measured, as was demonstrated for Yuma. drainage systcm without regard to the txisterice of or The present study is an attempt to defiii~in quan- laclt of grade, define n change of width, depth, and titative terms, insofar as available measurrments per- velocity downstream with change in discharge as con- mit, how the factors operate which lead to strcsm sistent as the changes observed in the same type of grading, which maintain the graded condition, and data within a graded reach. The same factors which which cause adjustment when the equilibrium condition tend to maintain equilibrium when a reach becomes is upset. Maclan (1948) discussctl these same factors graded are acting also on reaches not in complete equi- in qualitative terms, and this study is an attempt to lihrium and this action is sufficiently potent to product extend his reasoning by quantitative consideration of consistcnt patterns in the relation among the hytlraulic the hydraulic factors. fnctors of channel shape. In summary, regime irrigation crtr~alsthat’ over a Stations on the upstream reaches and minor trihu- period of years mither scour nor fill are ohscrvcd to tarics, many of which are probably not graded but have have a hydraulic geometry similar to that for natural btwi ctowncut over a period of time, have width-depth- rivers. Differences in the average slopes and intercepts vclocity relations similar to those of the graded reach of the lines representing width, depth, velocity, and of the same river systnm. One of the primary criteria discharge relations in canals and rivers can be qualita- for recognizing a graded reach is the lateral planation tively explained. One important difference lies in the of a stream during a period when the longitudinal profile load-discharge relations. The analogy of regimc-canal maintains stability (as discussed by Mackin, 1948, data to those for rivers leads to the hypothesis that the pp. 472-474). Tlic same factors which operate to m, f, b, and j relations in figure 18 represent reyuire- inaiiitain grade are also acting during the period of ments for equilibrium. tlowncut tirig. Graded reaches of a river are shown to haw width, When a strtwn in the process of tl~wricuttinghas depth, velocit-y, and discharge relations similar to those clewloped a smooth longitudinal profile through a given of reaches not known to be graded. This similarity rc:ich, tlic interaction of thc hydraulic variables provides indicates that the same factors which operatc to main- an approac-11to the condition of grade. That the over- tain equilibrium in a graded river arv also sufficiently till lorigituclinal profilt. of a strcani is in general concave active in nongraded reaches to providc spccsific and con- upnrd and morc or less sniooth, cvcn where only a sistent patterns in the hydraulic variablcs hot11 at a few rcachs are artnallg gradcd, is the result of the interaction of these variables. The tendency toward station arid in a downstream dirwtion. Strcam-gaging the condition of grade providcs a tendency toward data, however, are not available in sufficient quantity, continuity jn thr change of width, depth, velocity, in a sufficiently close geographic network, and over a rouglincm, mcl slopc in the downstream direction. long enough period to allow differentiation between 48 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS graded and nongraded reaches by hydruulic geometry which neither discharge nor load changes appreciably alone, though this might theoretically be possible. downstream. Macliin (1948, p. 482) cites a reach of The hydraulic geometry of rivers is useful in improv- the Greybull River as an example. He states that in ing our understanding of the quantitative values of this reach slope decreases downstream markedly and hydraulic factors which cliaractcrize rivers, and in attributes this ‘‘ * * * largely to a decrease in caliber understanding twttcr tlie nature of the intcraction of of load in transit; pebbles of tlie valley floor and terrace factors that lead to the condition of equilibrium or gravel sheets decrease notably in size in a downvalley grade. direction. ” This observed decrease in particle size can be assumed SEDIMENT AND THE LONGITUDINAL PROFILE to rwult in a decrease in channel roughness (value of Manning n) because the bed material being gravelly or Tlic longitudinal profile of a stream might bc con- pebbly is no doubt so large that the size of the indivi- sidered primarily a product of the change in size of bed dual particles rather than bed configuration clctermincs material in a downstrcam dirclction. The hctl of a the value of n. Consider the Manning equation mountain st ream is usually covcwtl with gravel or boulders and has a steep slope. Some distance downstream the bed may lw sandy, and still (.loser to thc (4) mouth silt or clay niav predominate. At the same time, the slope progressively t1ecrrasr.s downstream. If load and discharge are assumed to remain equal, as One tends to associatc the particle size and slope in a Mackin specifies in his Greybull example, then accord- direct cause-and-eff ect relation. Shulits (1941) de- ing to the relations shown in figure 15 thc velocity- rived an equation for the stream profile from a consider- depth relations required for transportation of the sus- ation of decrease of particle size as a result of abrasion. pended part of the load remain constant. Let it be IIe made the assumption that trhe slope was propor- assumed that this is the condition which prevails tional to the size of tlie bed-load material. through the reach of river cited. Such an assumption To assume that slope is proportional to size of bed rests again on the premise that the suspended load is a material requires the operation of certain other hy- usable index to the poorly understood load conditions draulic factors. It could be argued that a given slope which affect channel shape. One could argue that the is required to provide the particular velocity necessary diminishing size downs treani of the channel gravel in- to carry the particlr size present at each point. This creases the percentage of load carried in suspension, reasoning implies that the decreasing particle size but no data are available to support such a contention downstream requires a decreasing velocity which, in and it is not unreasonable to suppose that suspended turn, is provided by a decreasing slope in the down load remains essentially constant in this reach, as does stream direction. the total load. If constant velocity and depth down- Mach (1948, p. 482) argues correctly that a simi- stream are required to carry the constant suspended larity in curves representing the longitudinal profile load, an observed decrease in slope downstream results and the change in particle size does not prove any direct from a decrease in roughness downstream caused by the cause-effect relationship. He (p. 481) believes the reduction in particle size. relationship to be as follows: In a stream having relatively fine bed material, sand Graded profiles usually decrease in slope in a downvalley di- and silt, the roughness due to particle size usually does rection between tributary junctions, chiefly because of B down- not control. The flow may be hydrodynamically valley decrease in caliber of load due to processes within the stream. The principle involved is that the velocity (and, other “smooth”; with this kind of flow in which the thickness things being equal, the declivity) required for the transportation of its laminar film exceeds the roughness projections of of the coarser fractions of the stream’s load decreases with de- the individual particles, as in some Indian canals crease in grain size, the total amount of the load remaining the (Blench, 1951a, fig. 4.1), or the effective roughness is same. controlled by bed configuration rather than particle This explanation again implies a downstream decrease size. in velocity. The present study allows some further The analysis of figure 30 indicates that in fine-grained explanation of the mechanism implied in the phrase, materials, a decreasing particle size tends to increase “velocity required for the transportation of the coarser bed roughness under conditions of constant slope and fractions of the stream’s load.’’ It has been pointed discharge. Roughness, where it is not controlled by the out that the velocity can be changed by a change in size of individual particles, is affected by the concentra- slope and a change in bed roughness or by either. One tion of suspellded sediment, larger concentrations being might suppose from Mackin’s description that the effect, associated with smaller values of roughness, as has becn of particle size on velocity is offset entirely by slope discussed. Bed load affects roughness but its effect despite his qualification of “other things being equal.” is not distinguishable from that of suspended load in the Examples of natural stream reaches can be found in river data available. SOME PHYSIOGRAPHIC IMPLICATIONS 49

The tendency for particle size to decrease downstream however, that different longitudinal profiles differ in leads to a tendency for roughness to decrease down- their rates of change of velocity and depth in a downstream. As the particles in transit become small stream direction. enough, the configuration of the bed finally tends to For example, in a hypothetical stream in which sedi- govern, and a further decrease in particle size promotes ment is not a factor, it is assumed that the only require- a tendency for increased roughness. This increased ment of the channel system is that it carry a specified roughness is also partly caused by the tendency toward amount of water between two fixed altitudes. More decreased concentration of suspended sediment down- specifically, it is assumed that the watershed is of such stream. shape and hydrologic characteristics that, in the stream Though the relative importance of these three factors being considered, the discharge in the main channel cannot be determined from the river data available, the increases linearly from zero at the headwater divide to net result is a remarkable conservatism in the values of 10 cfs at a point 1,000 ft downstream from the divide. roughness. Particularly in streams handling fine- Three completely arbitrary profiles of the stream grained material, the roughness varies between narrow thalweg having different concavit,y, as shown in figure limits in the downstream direction. 32 lower right, are now assumed. Profile A is a straight The relation between velocity and depth to slope, line. Each profile is drawn between the headwater roughness remaining constant, can now be considered in divide and a point of the channel 10 ft lower and 1,000 the downstream direction. f t distant horizontally. It is obvious that streams having an infinite number It is desired to show the change of velocity and depth of longitudinal channel profiles could carry the water downstream for each of two arbitrarily assumed rela- furnished by the watershed. It will now be shown, tions of width. The two conditions of width for which

Wldth increases Width constant downstream : w=2.75 00.'' downstream 1.010 10 AI Assumed rate of increase of discharge downstream , I.o L

PROFILE A

.05.5 .5 I I I I 1 u 200 600 1000 1.0 10 10 Distance from headwater divide, in feel

---*a- 9 2 2% .s .E 2; =- = L Assumed chonnel profiles 5- 3% PROFILE OZP 0 L 0.1 1.0 10

05.5 .s

la 10 10 , I I 1 CI II

200 600 1000 Distance from headwater divide, in feet

I I 05.5 .5 ' 01 12 Note: Roughness n = 0.025 Oischorge, in cfs IINCREASING DOWNSTRE4M) --C EXPLANATION: -DEPTH; --WIDTH; ------VELOCITY

FIOURE32.-Variation of velocity nnd depth downstream for three assumed longitudinal profiles; increase of discharge downstream is speciEed, nnd two rates Of change of width downstream nre compared; channel roughness constant. 50 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS comput,ations have Thc longitiitlinal profilcks postiilatctl iir tlic c~xiimplv discharge relation is arc hut samples of an infinite niiin1)t~,cnch of wliiclr would be associated with particular clianncl c1i:mctci.- istics necessary for fulfillment of the 1iydrtLiiIic lnnrs and (2) Any of thcsc iiifiiiite number of combinations woiild l~, quite capa1)lc of carrying the tliscliargc of wnter posttu- Diagram A1 hns an Intcd. HOW~VW,it map he prcsiimctl thnt only oiw of ortlcr of iiingnitude of tlic 0.5 ~~lr~~lpS~~OWI~for 11at1ira1 the possiblc combinations would ol)tnin in iint,rirc Tftc rivers. In diagram A2 tlic width is assumcd constant equilibrium projile di-fers from other possibla pri)jYus iit tlownstrcairi. Bccriusc discliargt. is assumctl to be the relatire rates of increase of idocity anti depth it! ilic clircctly proportional to lcngtli along thc cliaiiiicl, downstream rlireciion. diagram A2 lias a witltli-discharge rela tiori The longiturliiial p.ofil~of :I strcvim ropwsc.!its tl1t5 simultaneous solution of eight cquat ions involving cight w= cs" unlcnowiis. No attmipt is iiiatlt. to intlicatcl tlw nil t UIY of the rtlntions or tho liytlrodyi~ninies lwliiiid tlwni. It has been assumcd in tlic example that t,hc rougli- Rut, according to the rcitsonillg in this papc~tlit v ti? ness, usiug Manning n as a nieasure, is constant tlirougli- in which the variables are tl(~pcntlentmay- ho suinmw- out, and has the raliie 0.025. At any position along tlic ized :is follows: profile, the slope can be measured from thc assumed The first equation reprcscnts continuity: profile curve. By assuming a depth, velocity ca,n be computed. The tiiscliarge is given for each position Q=wdv (The flow equation.) (A) by reading from tlie discharge curve. Width, being a function of dischnrgc, is specified. Tbcwfore, for The sccond is indopendent of thc strcani chnnncl itsclf each position along tlie strpam length, two independtiit aiid represents the independent cliartictcrist ics of tliv equatioris relating velocity to depth arc nvailahlc: drainage basin : L= F(Q) (The load-discliarge rc.latioii wsult iiig from tlit: charactcristics of the chinage area.) (B1 in which s and n me known. Also The nest cquations reprcscnt relations iii wliicli tlw Q=wdv values of the factors and the spcicifica relations n11101ig them dcperitl on processcs or action in the stream; in which (3 and 20 are known. The equations can be that is, processcs riot indtIwn(1cnt of the stivnm itstilt': solved simultaneously to compute velocity and depth. - Velocity, depth, and width have been plotted against g = F(Q) (The particle size chringcs downstiwin.) (C') discharge to show the changes in these factors in the w=F(cZ) (A width-dcpth rehti~iithat iiivolvcs downstream direction. thci relative erodibility of bctl mtl hik.) (D Inspection of the six sets of curves shows that, regard- 1 less of the Inminer in which width changes downstream, increase in concavity of the stream profile decreases the rate at which velocity increases downstrtiam, and (Where. .(I is t,hc sizv of the st~liinctnt increases the rate of deepening downstream. The n=F(d, 5, Q, s) (11') particles ; this relates sediment size to rougliiwss ti .) 171 profiles differ, therefore, primarily in the - ratio. f A channel that does riot increase in width downstmum obviously milst accommodate the increase in (liscliarge charge diagram corresponding to a particular rntc by incresst. in velocity or depth. of cliangt of sejimerit coriccritlatioii.) ((; 1 It will be noted that in diagram AI, for which the profile is a straight line but in which tlic width incwases Thc final equation is purely liytli~nulicand r(~piw(~iit~ downstream at approximately the natural rate, veloc- thc cqunditp bctwccn energy lossos in frictioii iiiid tliv ity increases downstrcam at a rate greater than for dccrtvlst. of c'nc'rgy potcnt ia 1, or h(':L( 1: normal rivers In diagram BI of a profile of moderate concavity, thc vclocitp tlcc*rcases downstream. 11 concavity intermdtite between profiles A1 and B 1 would providc a combination of depth and velocity curves that would approximate conditions in a natural stream having other characteristics similar to those assumed in the cxamplc. SUMMARY AND INTERPRETATION 51 l’he primary determinant of the slope of a stream is rainfall washes the loosened waste down the initial thr past physiographic history which determines the slopes to incipient streams. difierenccs in altitude and the distance between the “The machinery of the destructive processes is thus mouth and tlic watershed divide. The problem dis- put in motion,” Davis says, ‘(and the destructive de- cussed here concerns the form of the longitudinal profile vclopment of the region is begun.” which will be cut between these two points. It has Larger rivers quickly deepen their valleys; the rela- been demonstrated thnt conditions of approximate tively brief period of youth is characterized by rapidly equilibrium tend to br estahlislird in a reach of the increasing relief. The mature stage that follows is the stroam as soon as a more or less smooth longitudinal period of maximum relief and greatest variety of form. profile has been established in that reach even though After a transition period of slowly decreasing relief is downcutting may continur. Within the limits of the an indefinitely long old age of faint relief in which conditions detcrminrtl by thc physiograpliic history, furthcr changes are exceedingly slow. the intrractions which tlcterriiinc the form of the longi- Throughout the cycle rivcrs are both the routes over tudinal profile can be surnmnrizrd : which the erosion products are carried to the sea, and Channels of diffcrent shnpcs have charucteristic the rncchanism for effecting the transportation. distributions of velocity in the cross section. With Davis (1902) not only defined stages in the develop- thc same discliarge ti very shallow and wide channd ment of land forms through the cycle, but described a t.encts to haw a grcnter rtvoragc shear in the vertical complete cycle of river lifc consisting of successive plane than in the horizontal. Conversely a very deep stages of youth, adolescence, maturity, and old age. ;iritl 1~:~i~i~~~vchannt4 trnds to havc grmter average At the stage of maturity, a river attains the condi- shcar on thc bank than on the bed. ‘l’he relative tion of gradc which connotes tlic existenco of equilib- r~odililityof 1)ed and bank allows the channel sliapc rium in the river system. Mackin (1948, pp. 471, to tlcbvrlop a vrlocity distribution which is in approxi- 484) says: mate rqnilitrium with thc particdm erodihility charac- A graded stremi is one in which, over a period of years, slope teristics. 111 this mariner the increase of width with [and channel characteristics are] delicately adjusted to provide, discharge (value of b) prohably is determined. Why with available discharge * * * just the velocity required this exponent b should so consistently be nearly equal for the transportation of the load supplied from the drainage basin. to 0.5 in the downstream tlirection for widely different rivws is not known and constitutes an important Equilibrium, or grade, is attained by the larger unsolved problcin . streams relatively early in the geomorphic cycle, and Different concavities of longitudinal profile are by those of medium and smaller size at progressively nssochtcd with different relations of velocity and depth later stages. But even in thc youthful topography of to discharge, as required by the hydraulics of open the headlands where streams are gradually downcutting channels. However, a particular rate of increase of and are therefore not graded, tlic stream-channel char- 1)otli velocity and depth downstream is necessary for acteristics and slope tend to be adjusted to the discharge maintenance of approximate equilibrium in a channel, and sediment load. Even the ungraded parts of a inasmuch as the drainage area produces sediment and channel system exhibit a neat pattern characterized by water in a characteristic manner. a tendency toward a regular increase of width and , depth downstream with increasing discharge and with Roughness varies with sediment load and with a concomitant decrease particle size but the interaction of opposing tendencies in slope. The present paper describes in quantitative terms . rtxsults in a small range of values of roughness down- strrnni in most rivcrs. Thc suspended load and its the manner in which width, depth, and vclocity change change downstream characteristic of natural rivers with discharge downstream and at a given river cross require a particular rate of increase of velocity and section. It is demonstrated that these factors tend depth downstrcam. ITnder the conditions of a nearly to change progressively downstream in all natural stream channels studied in so nearly the Sam(’ way as constant roughnrss, to provide the rrquired velocity- graded reaches change that quantitative data available tlrbptli rclations slopc must generally decrease down- from the existing network of stream gaging stations strcarn; it is for this reason that the longitudinal profile of nrarly all natural strcnrns carrying sediment is are not adequate in volume or geographic coverage to differentiate graded from ungraded reaches. This dem- conc>tvr to the sky. onstrates the strong tendency toward establishment of SUMMARY AND INTERPRETATION a quasi equilibrium operating in channels carrying water and sediment. Rivers play a iiiajor rolv in the geographic cycle. As The mechanism of the interaction which promotes explained by Davis (1899), the successive stages of the development of at least quasi equilibrium may lie geomorphic tlevtlopmcnt of a region begin ideally with summarized as follows: uplift. Surfacc roclis are at t ncktd by weathering, and Load and discharge are characteristics of the hydrol- 52 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS ogy, geology, and physiognomy of the drainage basin. Thus for a given width and discharge, the total load They are furnished to the trunk stream channels and depends on the ratio of velocity to depth. are essentially independent of the trunk channel system. The velocity-depth relatioils required for transporta- Flood flows are characterized by large sediment loads, tion of the supplied load are provided in part by adjust- and in terms of the suspended-load data available for ment’of bed roughness which is a function of the bed analysis, are characterized by relatively large suspended- material size and the suspended-sediment concentra- sediment concentrations. Lou7 flows are characterized tion. With the roughness so determined, slope tends by low concentrations of suspended load. to be adjusted to provide the velocity-depth relations Slope is the one hydraulic factor which can be ad- necessary to carry the load in quasi equilibrium. justed over a period of time by processes within the During the passage of a flood through a given cross stream and may be considered the factor which makes section of a river channel, a larger velocity is necessary the final adjustment, as may be required for quasi to carry in quasi equilibrium the relatively great equilibrium, after the interaction of a11 the other factors suspended-sediment concentration of the high discharge has rwulted in establishment of mutually adjusted than the small concentrations of low flow. This in- values of those factors. crease in velocity at a cross section is not provided by The factors which are semidependent -that is, a marked change in slope because there is insufficient neither independent as are load and discharge, nor dc- time for such an adjustment to be made, and because pendent as is dope-are width, depth, velocity, bed there must be some continuity of slope downstream in roughness, and size of sediment particles. The nature order that any cross section may generally be as close of the interdependence of these factors appears to be to equilibrium as any other. as follows: The changes in velocity-depth relations required for The shape of the river cross section, roughly de- quasi equilibrium with load are brought about primarily scribed by its width-to-depth ratio, determines thc by changes in bed roughness associated with the distribution of velocity and shear in the cross section. changes in suspended-sediment concentration. The mean velocity in an open channel usually is equalled In the downstream direction, however, suspended- at a position 0.6 of the distance from the surface to the sediment concentration decreases only slightly with bed. In two channels having the same mean velocity discharge. The average decrease in the particle size and discharge, the deeper one will have a smaller average downstream tends to decrease bed roughness, but this shear between the bed and the position in the vertical is counteracted by the effect of decreasing sediment where a velocity equal to the mean value exists. This concentration and of changing configuration of the bed. may partly explain, at least qualitatively, why for the The result is that roughness changes but little down- same discharge and mean velocity, a greater bed load stream. The particular rates of increase of velocity characterizes a wide shallow section than a deep narrow and depth with increase of discharge downstream re- one. Even less well understood is the observed fact that quired for transport of the load are, therefore, main- a wide, shallow stream carries less suspended load than tained by the downstream decrease of slope. a deep narrow one of the same velocity and discharge. The interactions just summarized operate in both It may be that more of the total load will be carried as ungraded and graded reaches. They account for the bed load in a wide shallow reach than in a deep narrow maintenance of grade once the condition of equilibrium reach of a stream if the velocity and discharge are equal. is achieved, but their influence is sufficiently potent, In reality, however, changes in cross section in different even in reaches not yet graded. to cause the drainage reaches of the same stream are usually accompanied by system to develop in a highly coordinated and sgste- some change in velocity even though the discharge may matic manner. Because of these interactions rivers be equal. tend to increase in width and depth downstream With The increased bed shear in a wide shallow reach hav- increase of discharge in a remarkably consistent manner. ing a given velocity promotes a tendency to scour. A Each branch in the channel system of a river tends narrow deep reach of the same velocity is apparently to occupy a valley proportioned to its size. This inter- characterized by less shear on the bed but greater shear relationship is so similar in rivers widely different in on the bank. Thus there is a tendency to adjust the physiographic setting that its importance to the under- cross-sectional shape in accordance with the relative standing of land forms was recognized more than a erodibility of bed and bank. century ago. Geomorphology cannot move ahead, For a given width and discharge however, the total however, if we remain content to describe processes of load depends on the velocity, and any change in velocity land sculpture only in qualitative terms. When viewed requires a concomitant change in depth to satisfy quantitatively, the interactions of the various hydraulic continuity expressed by the equation factors. are found to be more complex than the qualitative analyses have led us to believe, but the unravelkg &= wdv of their complexities constitutes advance of knowledge. REFERENCES 53

REFERENCES KENNEDY,R. G., 1895, Prevention of silting in irrigation canals: Inst. Civil Eng. Proc., vol. 119, pp. 281-290. BLENCH,THOMAS, 195la, Hydraulics of sediment-bearing canals LACEY,GERALD, 1930, Stable channels in alluviuni: Inst. Civil and rivers, Evans Industries Lts., Vancouver, B. C., Canada. Eng. Proc., vol. 229, pt. 1, pp. 259-384. 1951b, Regime theory for self-formed. sediment-bearing 1939, Regime flow in incoherent alluvinm: Central channels: Am. SOC.Civil Eng. Proc., vol. 77, separate 70, Board Irr. (India) pub. 20, Simla. pp. 1-18. LANE, E. W., 1937, Stable channels in erodible materials: RRUNE,G. M., 1950, Dynamic concept of sediment sources: Am. SOC.Civil Eiig. Trans., no. 102, pp. 123-194. .4m. Geophys. Union Trans., vol. 31, no. 4, pp. 587-594. LANE,E. W., and BnRLAND, w.M., 1951, Estimating bed-load: 13TiCXLEY, 11. B., 1922-23, The influence of silt on the velocit'y Of Am. Geophys. Union Trans., vol. 32, no. 1, pp. 121-123. flowing wat,er in open channels: Inst. Civil Eng. hoc., LANGBEIN,W. R., 1!)42, Hydraulic criteria for sand waves, Am. v01. 216, pp. 183-211. Geophys. Union Trans., pt. 2, pp. 615-618. CORBXTT,D. AI., and others, 19.43, St,ream-gaging procedure: LARA,J. M., and MILLER,C. It, 1951, Corlveya~icechannel- U. 8. Geol. Survey Water-Supply Paper 888. widening study, Middle Rio Grande project: U. S. Bur. C>ROSS, W. P., and BERNHAGEN,R. H., 1949, Ohio st,reani-flow Reclamation (open-file report). characteristics, part 1, flow duration: Ohio Dept. of Nat- LINDLEY,E. S., 1919, Reginir channels, Punjab Eng. Congress ural Resources Bull. 10. Proc., vol. 7. DAS,ISHAR, 1950, Theory of flow of water and universal hg- LINSLEY,R. K., KOHLICR,M. A., PAT;I,HVS,J. L. €I., 1949. draulic diagrams: Jour. Central Board of Irr. (India), Applied hydrology, New York, McGraw-Hill Book Go., VO~.7, 110. 2, pp. 151-162. 689 pp. DAVIS, W. M., 1899, The geographical cycle, Geog. Jour. 11, MAChIN, J. Hoover, 1948, Concept of thr graded river: Geol. pp. 481-504. SOC.Am. Bull. 59, pp. 463-512. 1902, Base-level, grade and peneplain, Jour. Geol. 10, R~JBEY,W. W., 1933, Equilibrium conditions in debris-laden pp. 77-111. streams: Am. Geophys. Union Trans. pp. 497-505. EINSTEIN,H. A,, 1950, The bed-load function for sediment trans- ___ 1952, The gcology and mineral resources of Hardin and portation in open channel flows: U. S. Dept,. Agr. Tech. Brussels quadrangles, Illinois, U. S. Geol. Survey, Prof. Bull. 1026. Paper 218. EINSTEIN,H. A,, and BARBAROSSA,N. L., 1951, River channel SERR3d, E. I?., 1950, Progress report, investigatiolis of fluvial roughness: Am. Soc. Civil Eng. Proc., vol. 77, 12 pp. sediments of the Niobrara River near Cody, Nebr., U. S. ELLISON,W. D., 1945, Some effects of raindrops and surface-flow Geol. Survey Circ. 67. on soil erosion and infiltration: Am. Geophys. Union Trans., 1951, Measurement of bedload sediment, Am. Geophys. vol. 26, no. 3, pp. 415-430. Union Trans., vol. 32, pp. 123-126. GILBERT,G. K., 1914, Transportation of debris by running water: SHULITS,S., 1941, Rational equation of river-bed profile: Ani. U. S. Geol. Survey Prof. Paper 86. Geophys. Union Trans. pt. 3, pp. 622-629. GLYMPH,L. M., 1951, Relation of sedimentation to accelerated STEVENS,J. C., 1937, Discussion of LANE,E. W., Stable channels erosion in the Missouri River Basin, U. S. Dept,. Agr. Soil in erodible material: Am. SOC.,Civil Eng. Trans., no. 102, Cons. Service, SCS-TP-120, 20 pp. pp. 145-149. COULD,HOWARD R., 1951, Some quantit,ative aspects of Lake THOMAS,A. R., 1946, Slope formulas for rivers and canals: Mead turbidity currents: SOC. Econ. Paleontologists and Jour. Central Board of Irr. (India), vol. 3, no. 1, pp. 40-49. Mineralogists, special pub., no. 2, pp. 34-52. ___ 1949, Analysis of hydraulic data of some boulder rivers: GRIFFITH,W. M., 1927, A theory of silt and scour, Inst. Civil Jour. Central Board of Irr. (India), vol. 6, no. 1. pp. 67-71. Eng. Proc., vol. 223. VANONI,V. A,, 1941, Some experimcnts on the transportation of HADLEY,R. F., The usefulness of small reservoirs in erosional suspended load: Am. Geophys. Union Trans., pt. 3, pp. studies of minor drainage basin. U. S. Geol. Survey, 608-620. (open-file report,). _- 1946, Transportation of suspended sediment by water: .___Sources of sediment and sedimentation rates in small Am. SO~.Civil Eng. Trans., no. 111, pp. 67-133. reservoirs, Cheyenne River basin above Angost,ura Dam: U. S. Geol. Survey (in preparation). VICE, R. B., and SERR,E. F., 1950, Progress report, investiga- HEMBREE,C. H., COLBY,B. R., SWENSON,I€. A,, and DAVIS, tions of fluvial sediments, Middle Loup River near Dunning J. R., 1952, Progress report, sedimentation and chemical and Milburn, Nebr., U. S. Geol. Survey, Water Resources quality of water in the Powder River drainage basin, Wyo. Division, (open-file report). and Mont.: U. S. Geol. Survey (in preparation). U. S. Army, Corps of Engineers, 1935, Studies of river bed HORTON,R. E.. 1945, Erosional development of streams and their materials and their movement, with special references to drainage basins-Hydro-physical approach to quant,itative the lower Mississippi River: U. S. Waterways Exper. Sta. morphology: Geol. SOC. Am. Bull., vol. 56, no. 3, pp. Paper 17, 161 pp., Vicksburg, Miss. 2 7 5-3 70. U. S. Bureau of Reclamation, 1949, Report of river control work JOHNSON,J. W., 1943, Laboratory investigations on bed-load and investigations, Lower Colorado River Basin, calendar transportattion arid bed roughness: U. S. Dept'. Agr. Soil years 1948 and 1949: Boulder City, Nev., 124 pp. (open- Cons. Service, Sedimentation Sect., SCS-TP-50. file report). 54 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

APPENDIX APPENDIXA.-Channel-shape characteristics of rivers at stage corresponding to mean annual dischurge [Data on width, nrm, mean velocity, and mew depth represent approximations to the average and vary somewhat, depending on the particular srction where measurernentr are niade] ~- rea of cross Drainage Yrnrs of ean annua lean velor- dean depth dischargr ‘idth (frei section ity (fps) (feet) rea (square record (cis) ;quare feet) milrs) ___- Mobile River Basin

Tombigbee Rivqr at Abc.rdeen, miss^...... 19 2, 8V3 142 1,633 1. a 11.5 2,310 1.973 0. 5 13. n 823 Ruttahatchee River near Calcdonia. miss^ .. ~ .~ ~ ~ ~ .. I1 1. ow1 152 4. Tombiebee River at Columbus, Miss- ...... 30 5, 791 270 3,220 1.8 11.9 490 Sipsey River near Elrod, Ala...... 11 715 93 650 1.1 7.0 51 5 in, 2411 2. 2 22.3 15,500 Tombigbee River near Coatopa, Ala . ~ ~~~.~. ~~~~~ ~ ~ 19 21,450 458 Tombigbee River near Leroy, Ala..-...... 19 26. 250 613 13,360 2. 0 26.0 19,100 Scioto River Ijasiii Scioto Rivrr at LaHm?. Ohio 1R 198 90 342 .6 3.8 2.55 Ginto Hivrr near IJrosi)ect. Ohio . 1s 452 125 300 1. 5 2. 4 571 drioto Riwr nwr 1)uhIin. Ohio 1. 20 743 moo RIB .9 4.3 988 Bcioto Riwr at Columbus, Ohio ... 26 1,308 mo 1,093 1. 2 3.9 1, 624 Scioto liivrr nrar (‘irdeville. Ohio ... n 1,815 280 3, 0311 .6 10.8 2.635 Scioto River at Chillicothe, Ohio ...... 26 3,289 m 1,840 1.8 9. 2 3.84i Tennessee Rivrr Basin 67. 9 French Rroad River at Rosman. N. C ...... 12 229 77 151 1. 5 2. n Frrnrh Rroad Rivrrat Calvcrt. N. C ...... 23 335 92 168 2.0 1.8 103 4 34 25. 22. 7 15 .9 11.7 26 125 55 88. n I. 4 1. 6 40. 1 13 1.589 296 828 I. 9 2. n 676 13 2,511 190 1.115 2. 2 5. 9 I. 56i French Broad River near Newport Trnn 26 2, 7F4 346 1, 833 1. 5 5.3 1,858 fi 154 64 115 1.3 1.8 90. 8 Watauga River near Sugar Grove, k C 13.8 Noland Creek near Rryson City, I\’ C 2 12 42. 8 30 57. 2 .R 1.9 Tennessee Rivrr at Knoxville, Tenn 48 12.820 933 13.450 1.0 14.4 R, 934

St. Lawrence River Basin 266 1. 1 2. 8 369 St. Joseph River near Blakeslee, Ohio ~~ 6 282 95 Maumee Riverat Antwerp, Ohio ...... 22 1.547 245 1,078 1.4 4.4 2,049 Tiffin River at Stryker Ohio~-...... 14 292 72 34R :8 4.8 444 Tiflin River near Rrunkrsburg, Ohio ...... 7 448 85 332 1.4 3.9 766 1. 0 2. 33.1 Auglaize River near Fort Jennings, Ohio ~ ~ .~.~ ~ ~ . .~~ 21 27R 100 2.85 8 Ottawa River at Allentown, Ohio...... 16 119 61 97 1.2 1. E lfi8 Ottawa River at Kalida, Ohio ...... 5 141 Ro 128 1.1 1.6 315 Blanchard River near Findlay. Ohio...... 18 219 65 114 1.9 1. E 313 643 Rlanchard River at Olendorf, Ohio ...... 7 54R 80 304 1. Y 3. 8 Rlanchard River nrar Dupont, Ohio ...... 7 452 120 64fi .7 5.4 749 Auglaize River near Defiance, Ohio ...... 27 1,598 285 1. n,w 1. 5 3. 7 2. 328 5. 530 Maumee River near Ohio ...... 19 3.612 440 1.826 2.0 4. 2 Defiance, 6.314 Maunwe River at n’aterville, Ohio.~...... 22 4.269 730 1. 644 2. 6 2. 2 Yellowstone River Basin 37 2,969 242 813 3. 6 3.4 2. 630 16 6,331 2.53 1,343 4.8 6.2 11,180 3 177 t8 9i 1.8 1. 4 233 3 8% 134 335 2. 5 2. f 1. 221) 3 1.315 155 372 3. 5 2.4 1. 920 34 1, in6 159 318 3. 5 2. [ 2.320 6 683 123 406 1. I 3. : 2. 010 42 1, 908 218 543 3. 5 2. ! 8. 080 5 10. : 17 9. : 1. 7 .( 58. 2 10 45. ! 26 24. 1 I. 9 .L 484 4 20. I 1Y 10. ! 1.9 0. I 155 G 1, Y50 258 650 3.0 2. : 11,900 5 38 24 21 1.8 .< 86 27 364 u7 122 3.0 1.2 690 17 200 77 100 2. 0 1. : 1.130 19 2.383 175 627 3. R 3. 15,900 13 3,676 264 1,296 2.8 4. < ...... 14 4,535 371 1,855 2. 4 5. ( 20,700 5 154 41 68 2.3 1. f 189 5 202 57 86 2. 4 1. ! 429 8 294 96 116 2. 5 1. ; 1, 190 4 4.390 420 1, a50 2.4 4. 1 13 11,860 308 3,750 3. 2 11. ! ...... I 50 25 26 1.9 1. I 142 74 41 56 1.3 1.1 450 157 84 77 2.0 1. ‘ 980 13 16 15 .9 1 25 39 22 26 1. 5 1. : 69 48 57 27 1.8 1, 150 136 84 75 1.8 I 3,om 20 19 15 1.3 t 83 21 18 19 1. 1 l.! 52 1 22 24 18 1.2 174 1 63 33 38 1. 7 1. I 956 31 434 112 138 3. 2 1.: 6.050 1 16 23 19 .8 I 29 3 70 42 39 1.8 .I 120 1 17 ,u) 19 1.0 44 ; 37 19 23 1.6 1. : 60 3 47 33 33 1.4 1. I 29 E 101 45 69 1. 5 1. ! 106 fi 55 63 1.2 1. 267 1: 222 84 124 1.8 1.: 1,110 56 24 29 1. Q 1’ ...... 1; 528 124 211 2. E 1. 8,030 1: 763 16 273 2. E 1. 12. cion See footnotes at end of table APPENDIX 55

APPENDIXA.-Channel-shape characteristics of rivers at stage corresponding to mean annual dischurge--ontinued [D:%taon width. area. mean velocity. and mean depth represent approximations to the average and vary somewhat, depending on the particular section where mensurcmcnts are made]

Iean annual irra 01 cross Drainsgr Years of section Mean dcpth discharge Vidth (fret) (feet) ma (square record (cfs) square feet) miles)

Belle Fourche River Basin

Belle Fourche River below Moorcroft, Wyo...... 7 79 39 47 1. 7 1.2 1.730 Belle Fourche River at Hulett. Wyo ...... 12 fig. 5 .50 40 1. x .a 2. Boo Bello Fourche River at Wyo.-S. Dak. State line 3...... 2 150 59 G6 2. 3 1.1 ...... Belle Fourche River near Fruitdale S. Dak-...... 3 250 66 112 2. 2 1. 7 Belle Fourche River near Sturgis S: Dak ...... 2 500 134 228 2. 2 1.7 ...... Belle Fourche River near Elm &rings, S. Dak ...... 16 447 112 IfiX 2. I 1. 5 7.210 Kansas River Basin Arikaree River at Haigler Nebr ...... 15 31.7 45 27. 4 1.4 .5 ’ 1,41i Republican River at Culdertson Nebr ...... lfi 226 120 101 2. 2 .x ‘ 10,600 Republican River near Bloominiton, Nebr...... in 714 230 390 l.R 1. 7 ‘20,Roo Republican River at Clay Center Kans ...... 30 1,093 272 544 2. 0 2. 0 4 24,570 Smoky Hill River at Elkader Kins...... 5 36 41 24. 6 I. 5 .fi 3.555 Smoky Hill River near Ellis, Kans ...... 5 Gfi. 3 HI 48 1. 4 .8 5,630 Smoky Hill River near Russell Kans...... 4 227 104 114 2.0 I. I fi, 965 Smoky Hill River at Ellsworth: Kans...... 34 215 90 12fi 1. 7 1. 4 7, 580 Smoky Hill River near Langley. Kans ...... 4 362 92 193 1.9 2. 1 7,857 Smoky Hill River at Lindsborg, Kans ...... 17 292 80 192 1.5 2. 4 n. 110 Smoky Hill River at Enterprise, Kans...... 13 1,419 102 73fi 1.9 7.2 19, 200 Kansas River at Oeden. Kans...... 30 2,514 342 1,300 1.9 3. 8 45,24n l

Missouri River at Bismarck, N. Dak...... 21 20,320 1,202 7,327 2. 9 6. 1 1&6, 4Gfl Missouri River at Pierre. S. Dak ...... m 22,080 962 n. 718 2. 5 9. 1 243. Mx) Missouri Hiwr lit St. Joseph, 310 ...... 20 35,440 863 9,927 3. 6 11.5 424, Nil Missouri Rivrr at Kansas City. 410...... 20 43,710 1,093 12,750 3. 4 11. 7 489,200 Missouri River at Ilrrrnarin. Mil. .. m 69,170 1,578 22,9.50 3.0 14. 5 528.200 Mississippi River Basin Mississippi River at Alton, I11 ...... 12 96,670 1,750 32, 500 3.0 18. A 171,500 Mississippi River at St. Louis, Mo ...... 14 1%,700 1, ,586 44,440 3.8 28. 0 701.0~ Mississippi River at Memphis, Tenn ...... 13 454.800 1,950 99,400 4. 6 51.0 932,800 Mississippi River near Vicksburg, Miss...... 16 554,600 2,610 105,650 5.3 40. 1 1,144,500 ..- .

1 Located at a dam. 1 Concrete walls on side. 8 Poor records. 4 Includes some noncontributing area. 8 Long-term mean should be about 5,200 cfs. 56 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

APPENDIXB.-Average channel characteristics and average suspended-sediment load at jixed discharges [Data on width, average depth, velocity, and suspended load represent approximations to mean values of their respective factors at the discharge specified. Exponents b, 1, and m are dehed by the muations on p. 81

Rate of increase with discharge (0) Velocity iuspended Station load (tons VPS) per day) Im Remarks - ____ Cheyenne River near Hot Springs, S. Dak ...... ___._._.-... I. 1 2. 2 1.3 2.6 For 4 below 800 CIS. 1. 7 3. 1 215 2. 5 3.9 __- Middle Loup River at St. Paul, Nebr ...... - . . . ___ - 0. R I. 6 1.0 2.0 1.3 2. 5 ______1. 5: 3. 3 .31 I '3i ...... 2.6 Rio Qrande at San Acacia, N. Mex.. - 84 1.0 .5s .13 For 0 below 700 cfs. "," 140 1.1 3.3 } .a 1 1 182 1.4 3.8 2.17wo OOO 218 2.0 4.6 ___ Rio Qrande at Sau Felipe, N. Mex ...... -...... 0.8 1.7 4lm I1 1. 1 2. 1 1.6 2.6 2.3 3.2 ____ I- ___

Middle Loup River at Arcndia, Nebr- ~ ...... ~..~..-~.... 0. 9 1. 5 189 1.3 2.0 4,i .I3 .52 .3f 1. 2.6 11 2. O0O 217 2. , 3.3 26. OOO ~~ ~ ____ Smoky Hill River near Ellis, Kans ...... ~.... . -... 250 115 1. 2 1.8 For 0 below 400 cfs. 500 148 1.6 2. 1 - ___~___ --~ Powder River at Arvada, Wyo ... 105 1.0 2.3 500260 116 1.5 2. n 1.OOO 125 2.3 3.5 For 0 above cfs. _____ -~~~____--2, OOO 135 3.4 4.3 Moroau River near Faith, 5. Dak ...... __ -.. . . -. -... . -. -. -. . . -. 26o iin 1.3 1.6 1, mo 500 129 1.9 2.0 1. OOO 135 2.8 2. 6 For 0 above 400 cfs. , 2.m 140 4. 1 3. 5 24. m -. White River near Oglala, E. Dak ..... - . ______..-1.1 1. Y 1.9 2. 1

Belle Fourche River helow Moorcroft. Wyo ...... ~...-.-.....-.. . 69 2. 1 2.0 1.700 2.9 2.3 .31 .44 .25 4.0 2.7 ai1) 5.5 3.2 110. OOO -~ Virgin River at Virgin, Utah _.._._._.__.______... 2.7 15, OOO 3.7 .45 Curves poorly defined. 5. 1 .29 ___ 6. 9 1,600, OOO ____ Rio Pueroo at Rio Puerco, N. Mex .._....._._...___._.-.-.-.-.--.- 1.2 3.1 !E/-< 1.4 4.1 .I6 .41 1.5 5.4 2.2 6. n ______-___- 1.5 1.7 1.6 2.0 1.7 2.3 .51 For above 1,OOO cfs. 2.3 3.3 .46 Q I-- Saline River near Russell. Ran8 ...... 66 1.8 2.0 1.350 85 2.4 2.4 .38 .25 3.1 2.9 4.2 3.4 28.200 I-- I-- Smoky Hill River at Ellsworth, Kans ..__._..._._.___..__-...--.. 1.9 1,620 2.2 .I4 .M) .% 2.6 li:zI} 3.1 32,400 - __ ~ Republican River near Bloomington, Nebr...... 92 1.8 1.5 For Q below 1,000 CIS. 1.9 2.0 2.0 2.7 2,000 210 3.2 2.9 __ Bighorn River at Kane, Wyo...... __..__...... ~...... -..... 152 1.3 1. 7 . 51 . 43 1. OOO 168 2.3 2. m 175 3.2 Bighorn River at Thermopolis, Wyo ...__._.______.._._.--....-..1.7 1.6 2.1 2.4 31 1 .55 2.7 3.5 .- Bighorn River at Manderson, Wyo .....______._____._._..._.__.. 130 I. 9 7n 11 2.3 2. n 3. 4 2,000 210 ___. Grand River at Shadehill. S. Dak ..._._...... _.__.-...... 2.0 1.6 2, 430 .25 2.2 2.2 l;;q} .38 36 2.8 2.8 3.8 3.4 26,500 INDEX

4 L Page Page Acknowledgments 111 Lacey. Gerald. equations quoted ...... 43 At a station. defined 4 Lane, E . W., quoted ...... 29 Load, bed, defhed...... 19 B in relation to roughness ...... 48 measurements...... 29 Bed configuration. in relation to roughness...... 40-42 significance...... 30 Belle Fourche river^ ...... 15 suspondel, defined ...... 19 ...... 11 Bighorn Riverand tributariqdata effect on channel roughness ...... 36 Blench,Thomas, quoted ...... 45 In relation to roughness ...... 3642,4R method of sampling ...... 21 C Loup River ...... 15 Canals. stable. similarity to aradrd strrams ...... 43-48 slopeof ...... 45 M Change of discharge downstream, de6ned ...... 4 Channelcontrol, defined ...... 8 Mackin, J . H., quoted ...... 29.33-34,45,46,47,48, 51 Channel shape, relation to widthdepth ratio ...... 9 Madm irrigation canals ...... 15 Channel shapefactors, interrclationsat given discharge ...... 28-30 Manning equatlpn (equation 4) ...... 14,35, 48 Channel-shape adjustment, flood in Colorado River.- ...... 33,34, 37 Manning roughness factor ...... 40, 41 flood in Rio Qrande ...... 33,34, 37 Maumee River basin ...... 16-18 flood in San Juan River ...... 30-33,34,37 Mean depth . See Depth . Colorado River, adjustment of channel during Rood ...... 33,34, 37 Mean velocity . See Velocity. suspnded-load measurements...... 30 Middle Loup River, bed-load mt.snrements~ ...... 30 Current-meter data ...... 5 Mhsissippi River, lower, data ...... 13 Curve, cumulative frequency ...... 2 main stem ...... 15 flowduration . See Curve, cumulative frequency . MIssourl River, main trunk, data ...... 13

D S Das,Ishar, quoted ...... 45 Siobrara Rivr.r, hed.load measurements ...... 29 Davis,W . M.,quoted ...... 31 Depth, defined ...... 5 P Discharge, cumulative frequency ...... 3 daily mean ...... 2 Plan of report ...... 2 frequency...... 3 Powder River, data on tributaries-...... 10 given, in relation to channel-shape factors ...... 73-24 Purpow ofreport ...... 1 in relation to depth (equation 2) ...... 8 in relation to suspended load at cross section ...... IS21 R in relation to suspended load in a downstream direction.^- ...... 21-23 in relation to velocity (equation 3) ...... 8. 14 Regime channels. defined ...... A7 in relation to wetted perimeter ...... 43 Republican and Kausa? Rivers ...... 15 in relation to width (equation 1) ...... 8 Rlo Grandr, adjustment of channel during flood ...... 33,34,37 mean annual ...... 3 Roughness, defined for this report ...... 40 median ...... 3 fart.ors affecting ...... 40 records, method of selection^..^...... 8 in relation to bed configuration-...... 4M2 variable, in relation to channrl-shape factors ...... 24-28 in relation to particle size...... 48 in relation to sand sizes ...... 40, 42 E in relation to suspended lond ...... 36-40.42 Einstein equation ...... 28 Roughness factor. See Manning equation . Equation 1 ...... 16 Rubey, W . W-.,cited ...... 46 2 ...... 16.43 3 ...... 16. 43 S 4...... 35 36 San Juan River, adjustment of channel during flood ...... 30-33,34.37 Equations determining longitudinal profile of streams ...... 50-51 Scloto Rivrr basin ...... 16-1R Equilibrium, in channels. requisites for ...... 51 Sediment rating curve, defined ...... 20 summarized ...... 51 Sedimentation, of Lake Mead ...... 30 Emdibilityofbankand bed ...... 51 Shear, characteristics ...... 51 Exponents, b .J, m,average values ...... 9. 16 Slope, effects snmmarized ...... 52 relation to velocity and depth in a downst.ream dirrctlon ...... 49-51 F Stage, denned ...... 4 Flumes, experimental ...... 28 Streams, graded defined ...... 45. 51 French Broad River.-...... 15 method of identifying ...... 47 slope of ...... 45 Q proflles, eight equations for determining longitndinal ...... 50-51 Qage height . See Stage . in equilibrium and all others ...... 50 Gaging station, location of...... 8 Summary and interpretation ...... 51-52 Gilbert, 0 . K .. dataonexperimental flumes ...... 28 Symbols erplalnod ...... 11' Glossary of terms used ...... IV Orlllith, W . M., quoted ...... 29 T-Y

H-K Tombigbee River ...... 15 Hoover Dam, effect on channel shape ...... 37-39 Velocity...... 5 In a downstream direction, deflned ...... 4 Yellowstone and Blghorn Rivers ...... 15 Kansas River system, data ...... 12 Yuma, Ark, gaging station . 5ec Hoover Dam . 57 * U.S. GOVERNMENT PRINTING OFFICE: 1959 0-497780