Geological Society of America Bulletin · rivers have large bends, and small rivers have small bends. An adequate theory of meanders should explain why all river bends seem to be

Geological Society of America Bulletin

doi: 10.1130/0016-7606(1960)71[769:RM]2.0.CO;2 1960;71, no. 6;769-793Geological Society of America Bulletin

LUNA B LEOPOLD and M. GORDON WOLMAN RIVER MEANDERS

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BULLETIN OF THE GEOLOGICAL SOCIETY OF AMERICA

VOL. 71. PP. 769-794, 7 FIGS.. 1 PL. JUNE 1960

RIVEE MEANDERS

BY LUNA B. LEOPOLD AND M. GORDON WOLMAN

ABSTRACT

Most river curves have nearly the same value of the ratio of curvature radius to channelwidth, in the range of 2 to 3. Meanders formed by meltwater on the surface of glaciers, andby the main current of the Gulf Stream, have a relation of meander length to channelwidth similar to rivers. Because such meanders carry no sediment, the shapes of curvesin rivers are evidently determined primarily by the dynamics of flow rather than byrelation to debris load.

Velocity distributions along river curves provide a generalized picture of flow charac-teristics. Evidence on flow resistance in curved channels suggests that a basic aspect ofmeander mechanics may be related to the distribution of energy loss provided by aparticular configuration or curvature. No general theory of meanders is as yet satisfac-tory, however; in fact, present evidence suggests that no single theory will explain the for-mation and characteristics of all meanders and that few of the physical principles in-volved have yet been clearly identified.

RESUME

Dans la plupart des courbes de rivieres la valeur du rapport entre le rayon de courbureet la largeur du lit, de Pordre de grandeur de 2 a 3, est presque toujours la meme. Lesmeandres formed par 1'eau de fonte & la surface des glaciers et par le principal Courantdu Gulf Stream, montrent une relation entre la longueur du mândre et la largeur duchenal semblable a celle des rivieres. Puisque de tels me'andres ne transportent pas desediments, on peut en conclure que la forme des courbes de rivieres est determinee princi-palement par la dynamique de Pecoulement d'eau plutot que liee a la charge sedimentaire.

La distribution des velocites le long des courbes de rivieres donne une image ge'neraledes caracteristiques d'ecoulement. Ce qu'on sait de la resistance a I'ecoulement dans leschenaux courbes suggere que la resistance minimum est un facteur de base dans lam&anique des me'andres. Toutefois, aucune the'orie ge'ne'rale des me'andres n'est encorecompletement satisfaisante. En fait, les donnees actuelles suggerent qu'aucune the'oriene suffira pas a expliquer tous les meandres, et que quelques-uns seulement des principesde physique en jeu ont deja etc reconnus.

ZUSAMMENTASSUNG

Die meisten Flufiwindungen haben etwa dasselbe Wertverhaltnis von Kriimmungs-radius zu Bettbreite, ungefahr zwei zu drei. Maanderwindungen, die durch Schmelzwasserauf der Oberflache der Gletscher entstehen, und solche, die durch die Hauptstromungdes Golfstromes bedingt sind, haben ein Verhaltnis von Maanderlange zu Bettbreite,welches dem der Flusse ahnlich ist. Da solche Maander kein Sediment mit sich fiihren,sind die Formen der FluBwindungen offensichtlich in erster Linie durch die Dynamik derStromung bedingt, weniger durch die Gegebenheiten des mitgefiihrten Materials.

Die Verteilung der Stromungsgeschwindigkeit entlang den FluBwindungen zeigt einverallgemeinertes Bild der Stromungsmerkmale. Was iiber den Stromungswiderstand ingewundenen Flufibetten beobachtet wurde, lafit die Vermutung aufkommen, dafi dergeringste Widerstand eine wichtige Ursache fur Maanderbildungen ist. Tatsachlich lassendie gegenwartig beobachteten Tatsachen vermuten, dafi vorlaufig noch keine einzigeTheorie alle Maanderbildungen erklaren kann, und dafi nur wenige der physikalischenGesetzmafiigkeiten, die dabei eine Rolle spielen, bisher gefunden worden sind.

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769



770 LEOPOLD AND WOLMAN—RIVER MEANDERS

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CONTENTS

TEXT

PageGeneral statement 770Geometry and patterns of channel

bends 771Curves in sediment-free channels 774Channel cross sections and longitudinal

profiles 776Dynamic and flow characteristics 778

Pattern of flow in a meander bend 778Generalized picture of flow in a meander— 779Channel shape and movement in relation to

the flow pattern 781Effect of flow pattern on deposition and

erosion 781Meander mechanics and physiographic

problems 784General discussion 784Initiation and development of meanders.... 784Problem of channel equilibrium 786

Direction for future work 788Bibliography 789Appendix. Shapes of meander waves in alluvial

plains 791

ILLUSTRATIONS

Figure Page1. Sketch to define terms used in describing

geometric characteristics of a meanderingchannel 772

2. Relations between meander length and chan-

nel width and meander length and meanradius of curvature 773

3. Planimetric map and bed topography of ameander of the Mississippi River at PointBreeze, Louisiana, the New Fork nearPinedale, Wyoming, and Duck Creeknear Cora, Wyoming 775

4. Lateral and downstream components ofvelocity at various cross sections in abend, Baldwin Creek, near Lander, Wyo-ming 778

5. Isometric view of generalized diagram offlow distribution in a meander, showingdownstream (open parabolas with arrows)and lateral (closely lined areas) compo-nents of velocity as vectors, and surfacestreamlines 780

6. Stratigraphy of point bar in relation to meas-ured cross sectional profiles during sixyears of observation, Watts Branch, 1mile northwest of Rockville, Maryland . 782

7. Map and cross section of a typical pointbar of the flood plain of Watts Branch,1 mile northwest of Rockville, Maryland 783

Plate Facing page1. Meanders on Dinwoody Glacier, Wyoming.. 776

TABLE

Table Page1. Empirical relations between size parameters

for meanders in alluvial valleys 773

GENERAL STATEMENT

The most characteristic features of all streamchannels, regardless of size, are the absence oflong straight reaches and the presence offrequent sinuous reversals of curvature. Thesebends, whether or not they display regularreversals and sufficient symmetry to warrantthe name meander, tend to be scaled versionsof a given set of proportions. The proportionsare determined by three dimensions: the

repeating distance or length of the curve, thewidth of the channel, and the radius of curva-ture. In brief, the size of the bend appears tobe proportional to the size of the river; largerivers have large bends, and small rivers havesmall bends.

An adequate theory of meanders shouldexplain why all river bends seem to be ofproportional sizes, and by what mechanical orphysical principle a bend becomes adjusted insize to the stream discharge. Such a theory



GENERAL STATEMENT 771

should also explain why some rivers meanderand others do not. It should explain whymeandering rivers occur on ice with high veloc-ity of flow and no sediment load, and whytortuous channels are so prevalent in tidalestuaries. A meander theory should also explainhow chance perturbations of the flow maypersist and lead to the development of a sinuouspattern.

The present paper does not provide anytheory. An attempt is made to analyze andintegrate what appear to be the most prevalentessential characteristics of meandering channelsin nature. At least some of these prevalentcharacteristics are probably basic to themechanics of meandering channels which areas yet poorly understood. These include theexternal features of the channel, as well as thedistribution of velocity and the pattern offlowing water. Such a summary may proveuseful in the eventual definition of the necessaryand sufficient conditions for meandering ofchannels in diverse natural environments.

There is a large published record on theoccurrence and physiographic setting ofmeandering channels, as well as on the formand theory of river curves. It is not our purposehere to cite this record exhaustively, but ratherto sift out those facts which appear to us mostpertinent, emphasizing at the same time thatopinions may differ as to what is pertinent.

Our studies of meandering streams and ofmeander mechanics lead us to write a somewhatdifferent kind of a paper than might usuallybe expected in a "review" series. Many newobservations are available, some previouslyunpublished, some recently published or inprocess of publication. We hope it will be ofsome service, especially to the reader who hasan interest but not a specialty in the subject,to have these new observations summarizedand related to earlier observations. We believethat the status of the problem will be clearer asa result, and these observations are, in ouropinion, of some importance in indicatingdirections toward which future work mightprofitably be directed.

We then attempt to show that the problemof river meanders can be broken down intotwo parts. The first concerns the mechanicaland hydraulic processes that govern the form,size, and probably the occurrence of meanders.The second includes the physiographic historyof a particular channel and the formation,maintenance, and possible dissolution of a

meandering pattern. Although these two partsare almost inseparable they do represent quitedifferent ways of viewing the meander problem.We hope it will be clear from this review thatmany important physiographic questionscannot be answered with satisfaction becausethere is insufficient knowledge at present aboutmeander mechanics.

The authors are much indebted to J. HooverMackin and W. W. Rubey for many construc-tive suggestions and for a critical reading of anearly draft of the manuscript.

GEOMETRY AND PATTERNS orCHANNEL BENDS

Because most channels are not straight butsinuous, the forms which they display in planview are, in detail, infinitely varied. In ourexperience it is unusual for a reach of naturalstream to be straight for a distance exceeding10 channel widths (Leopold and Wolman,1957, p. 53). Yet certain common features arediscernible. These features, common to manyindividual bends, probably have some signifi-cance in the mechanics or hydraulics of mean-ders, but this significance is only poorly under-stood. The geometric characteristics of achannel seen in plan view are defined byFigure 1.

We use the term "arc distance" to mean thelength measured along the channel center linefrom one point of inflection to the next. "Sinu-osity" we use to mean the ratio of arc distancetojhalf the meander length.

The geometry of meander curves has beenthe object of extensive statistical study,examples of which are papers by Jefferson(1902), Inglis (1937; 1949, Pt. 1, p. 143),Bates (1939), and Leopold and Wolman (1957,p. 58). An additional independent sample inthe appendix includes measurements of riversin which the presence of a well-defined floodplain indicates that the channel form andpattern have developed reasonably free ofbedrock control. These data for rivers ofmoderate to large size were derived frompublished topographic maps and for smallstreams from plane-table maps made by theauthors. The appendix also includes some datafrom published laboratory studies of meander-ing streams in erodible materials.

Studies of the geometry of the patterns ofmeandering rivers give generally similar results.




The most consistent correlations are betweenmeander length, channel width, and radius ofcurvature, as shown by Figure 2, plotted from

are all so close to unity that the relationsbetween meander length (wave length),amplitude, radius of curvature, and channel

Axis ofbend

Location ofpoint bar

Concave bank

L = meander length (wave length)A = amplitude

rm— mean radius of curvature

FIGURE 1.—SKETCH TO DEFINE TERMS USED IN DESCRIBING GEOMETRIC CHARACTERISTICSor A MEANDERING CHANNEL

the data in the appendix. Because of thevariability in the dimensions of channels innature, considerable scatter of plotted data isexpected. Despite the scatter the relationsamong the factors appear to hold through avery large range of stream size, from laboratorystreams a foot wide to the Mississippi Rivernearly a mile wide. Various authors haveexpressed the relations for rivers in alluvialflood plains by regression equations, a few ofwhich are presented in Table 1. Table 1 alsoincludes relations derived from the data in theappendix.

The exponents in the regression equations

width may be considered linear. Amongdifferent groups of data fairly consistent valuesare obtained for the coefficient relating meanderlength to channel width; meander lengthranges from 7 to 10 times the channel width.This ratio holds in general not only for meanderson a broad flood plain but also characterizessinusoidal or meandering valleys incised intoconsolidated materials (Dury, 1958, p. 109).

A marked inconsistency is exhibited, how-ever, in the ratio of amplitude to channelwidth. Amplitude correlates only poorly withmeander length. We tentatively interpretthese findings to mean the amplitude of



GEOMETRY AND PATTERNS OF CHANNEL BENDS 773

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CHANNEL WIDTH, IN FEET MEAN RADIUS OF CURVATURE, IN FEET

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EXPLANATIONo

Meanders of rivers andin flumes

xMeanders of Gulf Stream

•Meanders on glacier ice

FIGURE 2.—RELATIONS BETWEEN MEANDER LENGTH AND CHANNEL WIDTH (LEFT DIAGRAM)AND MEAN RADIUS OF CURVATURE (RIGHT DIAGRAM)

TABLE 1.—EMPIRICAL RELATIONS BETWEEN SIZE PARAMETERS FOR MEANDERS IN ALLUVIALVALLEYS

Inglis (1949, pt. 1, p. 144, Jeffersondata)

Inglis (1949, pt. 1, p. 149, Bates data)Leopold and Wolman (this paper)

Meander length tochannel width

L = 6.610° •"

L = 10.9W1-01

Amplitude tochannel width

A = 18.6a)°-9»A = 10. 9wlM

A = 2.7IB1-1

Meander length toradius of curvature

L = 4.7rm°-98

meander loops is determined more by erosioncharacteristics of stream banks and by otherlocal factors than by simple hydraulic princi-ples.

The U. S. Waterways Experiment Stationstudies (Friedkin, 1945, p. 15) showed that inuniform material amplitude did not progres-sively increase nor did meander loops cut offas the meanders migrated downstream. Friedkininterpreted this to mean that "natural cut-offsresult from local differences in the erodibilityof bank materials (p. 16)."

These considerations appear to support theconcept that a limit to the amplitude ofmeanders is provided by the formation ofcut-offs. The rate of downstream migrationrelative to the rate of growth of the amplitudeof bends probably depends on the distributionof shear against the stream banks, which inturn is a function of channel shape. The latteris determined at least partly by the character-istics of the flood-plain alluvium and the movingdebris.

We believe relations between channel width,




meander length, and radius of curvature, incontrast to amplitude, are to a great extentindependent of bed and bank materials and arerelated in some unknown manner to a moregeneral mechanical principle.

Examining the correlations presented aboveone can write

L = 10.9w1M = 4.7fm°-98.

Assuming the exponents to be unity andsolving for the ratio of radius of curvature towidth,

10.92.3.

Computing the value of rm/w river by riverfrom the data in the appendix rather thanequating the least-square regression lines, itis found that in the sample of 50 rivers themedian value is 2.7, mean 3.1, and two-thirdsof the cases lie between 1.5 and 4.3. About onequarter of the values lie between 2.0 and 3.0.The tendency for similarity in the ratio ofcurvature to width makes all rivers look quitesimilar on planimetric maps, as suggested byFigure 3. In fact when one inspects a planimet-ric map of a river without first glancing atthe map scale it is not immediately obviouswhether the river is large or small owing tothis tendency for similar ratio of radius towidth regardless of river size.

As will be mentioned later, the value of theratio of radius of curvature to channel widthhas an important bearing on resistance toflow, and this appears to offer some clue as tothe hydraulic basis for the observed geometricsimilarity among channels of different sizes.

CURVES IN SEDIMENT-FREE CHANNELS

Meandering channels are carved in glacierice by streams of meltwater flowing on theglacier surface. To obtain measurements ofsuch channels we went to one of the Dinwoodyglaciers on the east side of the crest of theWind River Range in Wyoming (Lat. 43°12',Long. 109°38"). The meandering channels onthis small valley glacier can be seen in Plate 1.Although concentrated primarily upon thesteeper slope downstream, meandering channelsbegan on the relatively flat portion of theprofile, that is, near the point of inflection fromconcave to convex portions of the surfaceprofile. The channels extended downstream tothe convex-upward part of the glacier snout.

Some of the meandering channels wereincised 6 to 12 feet into the ice, and those

which appear on Figure 1 of Plate 1 are ofthis kind. Satisfactory measurements of theseincised meanders could not be obtained, butthe channel shown in Figure 2 of Plate 1incised 1 to 2 feet, appeared to be comparableto the more deeply incised streams.

Comparison of the meandering stream withincipient channels being formed by waterflowing over the ice surface led us to concludethat an irregular sheet of water graduallyconcentrated in a shallow flat channel at someplace where local factors favored melting. Themeandering form developed as the flowingwater gradually melted a channel downward.Channels in the ice only a foot wide and 2inches deep had no well-developed meanderform, but another channel of similar widthbut 1 foot deep showed a well-developedsinusoidal pattern.

Although occasional rocks and pebblesoccurred in the channel none were in motionat time of observation. In some bends a fewpebbles had collected on the inside of thecurve in the position of the point bar in ordinarystreams. The surface of the glacier in thelocality was strewn with cobbles and rocks in arandom manner owing to the fact that thechannel measured was near the edge of apoorly developed medial moraine.

In several places terrace remnants were cutinto the ice and abandoned as downcuttingprogressed. No pebbles were seen on theseterrace remnants. The meanders appeared to becut by melting alone with no effective helpfrom abrasion by transported pebbles.

Over-all dimensions (in feet) and flowparameters of the channel pictured in Figure 2of Plate 1 were as follows:

Channel width 2.3Channel incision below ice

surface 1.0Width of surface of flowing

water 2.0Mean depth of flowing water,

dm 0.25Average velocity from meas-

urements of flow, » 5.3 ft/sec.Average meander length 20Average channel slope, s 0.023Discharge 2.4 cfsFroude number v/ Vgdm 1.9

(where g is the accelerationof gravity)

The Froude number, 1.9, was well withinthe range of supercritical or shooting flow. Inshooting flow it is usual to observe surfacewave trains and the diagonal crisscross of



GEOMETRY AND PATTERNS OF CHANNEL BENDS 775

Mississippi River at Point Breeze, Louisiana^from Corps of Engineers data

FLOOD PLAINApproximate elev 3 unit

CONTOUR INTERVAL H MEANHEIGHT OF BANKS

ZERO AT MEAN BED

Average width60 feet

FLOOD PLAINApproximate elev 92 feet

New Fork River near Pinedale,Wyoming

Q § , 2 ° 0 FEET

CONTOUR INTERVAL 0.5 FEET

DATUM ARBITRARY

Duck Creek near Cora, Wyoming

FLOOD PLAINApproximate elev 94.0 feet

FIGURE 3.—PLANUOETRIC MAP AND BED TOPOGRAPHY op A MEANDER or THE MISSISSIPPI RIVERAT POINT BREEZE, LOUISIANA, THE NEW FORK NEAR PINEDAIE, WYOMING, AND

DUCK CREEK NEAR CORA, WYOMINGScales are chosen so that meander length is equal on the printed page.




surface waves reflected off the channel sides.If these were present they were masked by theviolent turbulence, although they were probablythe cause of a marked hump in water-surfaceelevation extending along the channel centerline in some places. The water was conspicu-ously superelevated on the outside of thecurves.

The measurements of meander length andwidth of this stream cut in ice are plotted onFigure 2. The meander length of channelsdeveloped in ice bears the same relation tochannel width as in ordinary meanderingstreams. Measurements made by D. G. Ander-son of meanders of small streams on the ice ofChamberlain Glacier, Alaska, confirm thisconclusion (Oral communication to Leopold,1958).

Data on meanderlike phenomena fromanother source deserve mention as possiblyhaving some relation to the present problem.In the Gulf stream of the North Atlanticoceanographers have found bands of relativelyhigh speed which have a meandering pattern inplan view. Stommel (1954, p. 887) describedthese as "horizontal eddy and meander forma-tions," and Von Arx (1952, p. 213) used the term"meandering current." These currents appearboth in cross sections of the velocity field andin planimetric maps of the temperature field.Maps of the temperature field at a depth of200 meters have been published by Stommel(1954, Fig. 2) and by Fuglister (1955, chart 3a).From some of these published maps roughmeasurements of meander length may beobtained and, less satisfactorily, estimates ofcurrent width. Better estimates of currentwidth are obtainable from velocity crosssections published by Von Arx (1952, Fig. 2)and Worthington (1954, Fig. 9).

Four measurements of the width and meanderlength were possible from the published data,and these are plotted on Figure 2. Althoughthey fall slightly below the line drawn throughthe points representing river data, the graphsuggests that meandering currents in the GulfStream bear certain analogies to river meanders.Although frictional flow in rivers should not

be compared to geostrophic flow of an oceancurrent, the observed similarity in meanderdimensions appears worthy of attention.

The importance of meanders in ice and inocean currents lies in the fact that both flowsystems exhibit meanderlike phenomena inthe absence of sediment debris. The establish-ment of meander length-width relationssimilar to those of sediment-laden riverssuggests that sediment alters or affects butdoes not cause the meander pattern. (For acontrasting view see Matthes, 1941.)

CHANNEL CROSS SECTIONS ANDLONGITUDINAL PROFILES

From a study of bends of the GaronneRiver, Fargue (1908, p. 25) stated as a generalrule of river behavior that the shallowestsections occur downstream from the crossover,and the deepest sections downstream fromthe axis of the bend or point of maximumcurvature. The topographic maps of samplemeanders in Figure 3 show in a general waythe features described by Fargue and manyworkers after him. These same figures, however,also indicate that the location of deeps andshallows may be highly variable. For example,data from 25 bends on Buffalo Creek nearBuffalo, New York, indicate that the deeppoints are usually found downstream from thepoint of maximum curvature but do occur aswell upstream from this point (Parsons, 1959).On a reach of the Popo Agie River near Hudson,Wyoming, containing four successive bends ofamazing symmetry deeps occurred upstreamfrom the point of maximum curvature and notdownstream.

In the reach of the Mississippi River shownin Figure 3 the deepest portion occurs at theaxis of curvature in one bend but somewhatdownstream from this axis in the other bend.In the New Fork the deepest portion in onebend occurs practically at the point of inflection.The reach on Duck Creek more nearly conformsto Fargue's generalization.

There seems little doubt that the depth ofwater at a bend of a river is closely correlated

PLATE 1.—MEANDERS ON DINWOODY GLACIER, WYOMING

FIGURE 1.—In middle foreground are meandering channels of meltwater streams carved in the glacierice. View southwesterly from Sentinel Peak; photograph by Mark F. Meier

FIGURE 2.—Channel carries meltwater from glacier. The water flowed at high velocity and was violentlyagitated.



BULL. OEOL. SOO. AM., VOL. 71 LEOPOLD AND WOLMAN, PL. 1

--------- - --

MEANDERS ON DINWOODY GLACIER, WYOMING



CHANNEL CROSS SECTIONS AND LONGITUDINAL PROFILES 777

with the bend curvature. Using 103 measure-ments of mean depth and corresponding radiiof curvature on the River Elbe, Leliavsky(1955, p. 118) shows that the two factors beara simple linear relation and that depth increasesinversely as a function of radius of bend.

The point of inflection in a river curve isclosely associated with a shallow portion ofthe reach, or a depositional bar on the bed.The occurrence of a bar or riffle in the bed of astraight river reach should correspond morpho-logically therefore to the point of inflection of ameandering channel. It is on this basis thattwice the distance between successive riffles ina straight reach was compared with the wavelength of meanders (Leopold and Wolman,1957, p. 55).

Not only do the riffles in straight channelstend to be in a position analogous to comparableshallows in a meandering reach, but there alsois some tendency for the shape of the bars tobe the same. In some straight reaches successivegravel or sand bars which constitute the rifflesassume the form of lobate wedges slopingalternately toward one bank and then theother. In some meanders the bar near thepoint of inflection slopes sharply across thechannel toward the deep near the concavebank downstream. This is apparent on thetopographic map of the New Fork (Fig. 3) andis accentuated at low flow when water flowsdiagonally across the channel down the steepestlocal bed slope. At high flow there is rapiddivergence as the cross-sectional area expands.

The longitudinal profile of bars, whetherassociated with the point of inflection of ameander, or a riffle in a straight channel, isusually moundlike rather than dunelike and isasymmetric in cross section. Perhaps ideallythe bar has a dunelike profile with a short, steepdownstream face. The topography of thebend in the Mississippi River (Fig. 3) suggeststhis, but many are less regular.

The bar is asymmetric in cross section in adifferent sense on upstream and downstreamfaces. In a meander, as one moves downstreamtoward the point of inflection, the surface of thebar slopes laterally toward the concave bank,but immediately having passed the crest ofthe bar the asymmetry changes, and thesurface of the bar slopes toward the oppositebank.

For this reason the channel cross section in ameandering reach is symmetrical only for avery short distance. We estimate that on theaverage the distance through which the crosssection is essentially symmetrical is only about

one-tenth wave length. This short distance canbe seen on topographic maps of Figure 3.

The importance of the configuration of thebars and their relation to the deeps lies in therelation of bed topography to bed shear. Riverpilots and others know that in large rivers thecrossings (points of inflection) tend to scour atlow flow and fill at high flow; the bends (pools)tend to fill at low flow and scour at high (Straub,1942, p. 617; Lane and Borland, 1954, p. 1075).

Local bed shear must be greater at points ofscour than where filling occurs, but directmeasurements of bed shear are few, and littleis known in detail about the relation of bedtopography to stress on the bed.

Lane and Borland (1954, p. 1079) explainthe observed sequence of scour and fill as aconsequence of the different shape of crosssection in pool and riffle. They say that thecrossing or riffle has a larger cross-sectionalarea than the pool during high stages and asmaller area during low stages. Laursen andToch (1954, p. 1085) state that width oftentends to be greater at a crossing than at abend. The expansion in width they believewould cause deposition on the wide crossingwhile scour occurs in the narrower bend.

We have made measurements of slope and ofdepth over pool and riffle separately at variousstages of flow in an attempt to compute approxi-mately the variation of mean stress. At lowflow the bend is a still pool over which thewater has a relatively flat slope. In contrast,in the riffle the water tends to plunge over thesteep downstream face of the bar. As stage in-creases, the slope over the pool or bend in-creases, while that over the bar decreases untilnear bankfull stage when the water surfaceattains a continuous uniform slope, and allobvious surface effects of the shallow bar havebeen erased.

With increasing stage, depth increases inboth bend and crossing. The product of depthand slope, then, increases with stage morerapidly in pool than at the riffle or crossing.

In the one stream measured by us, SenecaCreek near Dawsonville, Maryland, the com-puted mean shear is greater in the riffle at lowflow and greater in the pool at high flow(Leopold, in press). This observation is inqualitative agreement, then, with the conceptof scouring in the crossing at low flow and inthe bend at high flow. The analysis is incom-plete, however, inasmuch as variations in thedistribution of velocity in successive crosssections have not been included in the computa-tion of shear, and additional observations




Location of sections FEETrO

Nott: Velocity isoplethsIn feet per second

10 FEETi i i—i—i—I

Section 1

VLEFT BANK ^ RIGHT BANK

Section 3

LEFT BANK 2 RIGHT BANK

DOWNSTREAM COMPONENT DOWNSTREAMCOMPONENT

LATERAL COMPONENT LATERAL COMPONENT

Section 2 Section 4

LEFT BANK RIGHT BANK LEFT BANK* RIGHT BANK

DOWNSTREAM COMPONENT DOWNSTRECOMPONENT

RIGHT y RIGHT RIGHT

LATERAL COMPONENT LATERALCOMPONENT

FIGURE 4.—LATERAL AND DOWNSTREAM COMPONENTS OF VELOCITY AT VARIOUS CROSS SECTIONSIN A BEND, BALDWIN CREEK, NEAR LANDER, WYOMING

elsewhere are necessary before any generaliza-tion can be made. Mean depth and slope,even over a short reach, probably do notmeasure the true shear exerted on any localarea of bed. Direct measurement of shearstress in various parts of a bend at variousstages is needed.

DYNAMIC AND FLOW CHARACTERISTICS

Pattern of Flow in a Meander Bend

Relatively few detailed measurements of thedistribution of flow in a natural meander havebeen published (Blue et al., 1934; Eakin, 1935;Leliavsky, 195S, p. 96-100; Van Til and Tops,



DYNAMIC AND FLOW CHARACTERISTICS 779

1953, p. 19-23). The velocity distributionswhich follow are intended both as additions tothe slim store of data and as illustrations ofthe general characteristics of flow in bends innatural channels. Figure 4 shows downstreamand transverse components of velocity at theaxis of the bend on Baldwin Creek near Lander,Wyoming. Point velocities were measured byPrice current meter and horizontal angles by avane attached to the wading rod. Measuredangles were used in conjunction with themeasured velocity to compute the lateralcomponents of velocity. Measurements weremade at a high but less than bankfull stage.

At least in narrow channels a cross-channelvelocity component is directed toward theconvex bank or point bar near the bed, andtoward the concave bank near the surface.Continuity requires, then, that surface waterplunge downward near the concave bank andthat some bed water emerge at the surfacenear the convex bank.

This circulatory motion in the cross-sectionalplane of a channel, as clearly explained byThomsen (1879), results from the largercentrifugal force on fast-moving surface parcelsthan on slower-moving ones near the bed. Themotion gives to an individual water parcel apath resembling a helix.

Despite the tendency for circulatory motionin the stream cross section, surface streamlines (Fig. 5) show that no single parcel ofwater crosses completely from the convex tothe concave bank. The maximum cross-channelmotion does not exceed perhaps two-thirds ofthe channel width in any given meander bend.This accounts for the observation by Friedkin(1945, p. 5) in laboratory models that materialeroded from one bank tends to deposit on apoint bar downstream on the same side of thestream.

Matthes (1941, p. 634) stated that in wide,shallow rivers helical circulation does notexist. It is our opinion that in rivers of largewidth/depth ratio the helical motion exists,but not as a single rotating cell involving thewhole channel (Nemenyi, 1946). The existenceof several cells of circulation in the TammBend data from the Mississippi can be seen inthe distribution of the lateral components ofvelocity published by Eakin (1935). As in theatmosphere, circulation cells of large magnitudetend to break down into smaller cells. The"polar-front" cell and the "trade-wind" cell ofRossby (1941, p. 610) are directly driven bydynamic forces which we consider analogous to

the direct helical motion in river bends due tothe combined effect of frictional and centrifugalforces. Rossby's "frictionally driven" circula-tion cell in the middle latitudes furnishes, inour opinion, an analogy to those circulation cellsseen in the Eakin data of the Mississippi whichrotate in a direction opposite to the helicalmotion expected in a single cell occupying theentire cross section.

It is well documented that the water surfaceis superelevated near the concave bank of achannel bend (Blue et al., 1934; Eakin, 1935;Leliavsky, 1955, p. 123; and others). Theamount of the superelevation is proportional to

where v is the mean velocity, w is channelwidth, rm the mean radius of curvature, and gthe acceleration of gravity (Leliavsky, 1955,p. 122).

This superelevation is a consequence of thecurved path of water flowing around a bendand is a correlative of the helical motion.

Generalized Picture of Flow in a Meander

Measurements in meandering streams and incurved flumes (for example Mockmore, 1944,p. 569) allow the construction of a generalizedpicture of the flow pattern in a meander (Fig. 5).The isometric view of the two principal com-ponents of velocity at various positions in thebend show the main features. The scale is suchthat superelevation of the water surface in thebend does not show on the diagrams but isimplied by the velocity distribution.

At the crossover or point of inflection of thebend (Fig. 5, section 1), the cross-sectionalshape is not completely symmetrical but isslightly deeper near the bank which wasconcave in the bend immediately upstream.Downstream from the crossover the section (2)becomes approximately symmetrical, but,because this section is in the bend, some cross-current component exists in accord with thecurving stream lines.

The velocity in a meander crossover is notsymmetrically distributed. As would beexpected, proceeding downstream from theaxis of the bend the thread of maximumvelocity is much closer to the concave bankthan to the center of the channel. The highvelocity, moreover, continues to hug this sidethrough the point of inflection of the curve.



Generalized surfacestreamlines

Generalized velocitydistribution

FIGURE 5.—ISOMETRIC VIEW OF GENERALIZED DIAGRAM or FLOW DISTRIBUTION IN A MEANDERShowing downstream (open parabolas with arrows) and lateral (closely lined areas) components of

velocity as vectors, and surface stream lines. All sections viewed from a changing position to the left of andabove the individual section.

780




The asymmetry in the velocity field persistsdownstream from the point of reversal ofcurvature, at least as far as the most nearlyrectangular and symmetrical channel crosssection.

Downstream from the symmetrical crosssection the maximum downstream velocityapproaches the concave bank. The highestvelocity at any point occurs near the concavebank just downstream from the axis of bend.Individual filaments of water accelerate anddecelerate along a stream line. The maximumvelocity is not always associated with thesame parcels of water.

Measured distributions of velocity show alsothat the maximum-point velocity in the bendis depressed below the water surface. Theseobservations agree with the measurements madeby Mockmore (1944, p. 596) in a curved flume.

An effect of increasing discharge in a meanderreach is to move downstream the position onthe concave bank where the thread of maximumvelocity impinges. The stream lines of highestvelocity do not touch the concave bank butapproach it and then swerve into a coursemore or less parallel to that bank.

Channel Shape and Movement in Relationto the Flow Pattern

The slight lack of congruence of stream-linecurvature with bank curvature leads to thetendency for the locus of point-bar depositionto occur downstream from the axis of thebend. A building point bar tends to concentratecaving of the bank downstream from the axisof bend and thus accounts for the progressivedown-valley movement of meander bends.This same tendency exists in all river curves,even in those lacking sufficient symmetry to becalled meanders, and this accounts in part fornormal movement of river channels and theconsequent construction of flood plains.

Material slumping into the bed by bankcaving is caught in the transverse componentand carried toward the middle of the stream.If the location from which it is derived is farenough downstream in the bend, insufficientlength of reach remains for the material to getall the way across the channel. Instead it movesthrough the crossover reach without havingcrossed the channel. Once into the reversedcurve the particle moving near the bed isdrawn toward the same bank from which itstarted rather than to the opposite bank. Thisdoes not rule out the likelihood that muchfine material which is not carried near the bed

of the stream will be deposited at a downstreamlocation on the side opposite to the one fromwhich it was derived.

The plunging or diving motion of surfacewater near the concave bank is one factortending to depress the point of maximumvelocity below the surface near the axis of thebend. Bank friction also has a similar effect.Because the highest velocity gradient on bothbed and bank occurs where the thread ofhighest velocity is nearest the boundary, thismaximum velocity gradient is equivalent tomaximum shear stress. Its location accounts,then, for deepening of the bend reach near theconcave bank. This scour, together with build-ing of the point bar, explains the triangularshape of the cross section.

Increasing the depth of water in the channelreduces the vertical velocity gradient. In achannel that is very deep relative to its width,the effect of bed friction becomes relativelysmall, and the velocity is nearly constantwith depth except close to the bed. Thus thehelical circulation becomes negligible. Flow in acurved channel of great depth and smallwidth approaches potential flow—that is, thedownstream velocity varies across the channelinversely as the radius of curvature of thestream lines. This was demonstrated byWattendorf (1935, p. 574) in a deep, narrow,curved flume. In the absence of the circulationin cross section the thread of high velocityhugs the convex bank and does not cross thechannel. Thus the mechanism for building thepoint bar is probably absent in channels ofextremely small width-depth ratio.

Effect of Flow Pattern on Depositionand Erosion

The vigorous cross currents near the bedcan transport considerable quantities of bedmaterial from the concave to the convex sideof the channel. On a meander bend of BaldwinCreek (Fig. 4) depth-integrated samples ofsuspended load were taken at each of fiveverticals spaced across the stream and repeatedin each of five cross sections along a meanderwave. A nearly uniform concentration ofsediment existed across the section in a cross-over. In the sections located in the curvedreach the concentration increased markedlynear the convex bank.1 A similar distributionof sediment was observed by Eakin (1935,

1 Dr. John P. Miller assisted the senior authorin obtaining these measurements.




p. 471) in his samples taken at Tamm Bend onthe Mississippi River. This increased concen-tration near the convex bank is attributed tothe fact that water in this location has, to

The path of particles which are deposited on abuilding point bar has, of course, only a smallcomponent across the stream compared with alarge down-stream component. The lobate

102 10 12 14 24 2616 18 20 22DISTANCE, IN FEET

SucMtthM surface profile* of meender channel and point tar, 1953-59

30 32 34

24DISTANCE, IN FEET

26 28 30

Mean elevation turtle*of strumbw), 1953-59

Urge scale crow-section of stratigraphy In relation to profile*

FIGURE 6.—STRATIGRAPHY OP POINT BAR IN RELATION TO MEASURED CROSS-SECTIONALPROFILES DURING Six YEARS OF OBSERVATION, WATTS BRANCH, 1 MILE NORTHWEST

OF ROCKVILLE, MARYLANDKey to stratigraphic units is as follows:A Gravel, mostly 3-8 mm with considerable 8-20 mm and a few 64 mmB Olive-gray clayey silt, with organic matter, small mica flakesC Orange-brown, mottled sandy silt with some clay, lenses of leaflike organic mterial, micaD Coarse sand, brown-stained, with pebbles up to 8 mm, some fine roots in situ, lenses of siltE Brown sandy silt gradually changing upward in places into fine sandF Fine sand with some silt

some extent at least, come laterally in a pathclose to the bed and is comprised, therefore,of water elements with relatively high con-centrations of sediment.

Although there is some evidence that riflingin a spirally welded pipe tends to increase theability of a flow to transport sediment overthat of a cast-iron pipe (M. L. Albertson, oralcommunication to authors), evidence frompreliminary experiments (L. M. Brush, oralcommunication to Wolman) indicates thattransport in a meandering channel is muchreduced over that in a straight channel ofsimilar dimensions and boundary roughness.This effect of curvature on transport is inaccord with the added energy losses accompany-ing increased curvature.

The cross-channel component of bed flowtoward the convex bank is probably the princi-pal mechanism for the building of a point bar.

form of a point bar may be compared to asector of a very flat cone. A bed particle, takinga path along a chord of this cone, may climbthe inclined plane or ramp presented by thecone. Thus, even some of the coarser debris ofthe channel bed will be deposited on the pointbar at an elevation considerably higher thanthe average level of the stream bed.

Variation in discharge would lead one toexpect that a thin layer of coarse material maybe deposited on the point bar at one time andat another time a layer of fine material may bedeposited upon the coarse. On the average,fine material will be carried higher up the barthan will coarse material, so the point barwould show on the average a rough gradationof size from coarse near the base to fine nearthe flood-plain level. In many instances, then,a growing point bar would not merely lapover the coarse material on the stream bed.




60 FM

Steep or wrticil bank Contour Intml 0.5 hotArtitrtry dttum

FIGURE 7.- -MAP AND CROSS SECTION OF A TYPICAL POINT BAR OF THE FLOOD PLAIN OF WATTSBRANCH, 1 MILE NORTHWEST OF ROCKVILLE, MARYLAND

After Wolman and Leopold (1957)

An unusual opportunity to verify thisreasoning was provided by a series of crosssections made in a meandering reach of WattsBranch near Rockville, Maryland. During 6years of observation the stream has movedlaterally a distance equal to one channel width.At the end of this period of observation atrench was dug in the point bar normal to thechannel. The stratigraphy revealed in this

trench superimposed on known surface positionsis sketched in Figure 6. This trench was locatedon the right side of the stream on section x-x'of Figure 7.

Approximate contact surfaces betweenmaterials of different textures are more or lessparallel to past surface profiles. Coarse materialcharacteristic of the bed is observed in aposition well above the channel bed. In addi-




tion, there is no doubt that deposition occurredup to the level of the water surface at bankfullstage. In the same period deposition over theflat flood plain by overbank flow has been toosmall to measure. These observations appearto confirm the authors' (Wolman and Leopoldt1957) hypothesis that point-bar building i,the primary process of flood-plain developmensin flood plains of this type.

Bank erosion is greatly influenced by wettingof the bank materials. Arroyos cut in fine-grainedalluvium experience most bank cutting after,not during, flow. The wetting causes laterslumping (Leopold and Miller, 1956, p. 5).Bank erosion is also enhanced by returnseepage of water which infiltrated the banksduring high flow. Upon lowering the stage thebalancing pressure of the water in the channelis released, and the banks slump or collapse(Inglis, 1949, pt. 1, p. 152). A study of bankcutting in Watts Branch near Rockville,Maryland, showed that a combination of bankwetting and ice-crystal formation promotedthe greatest bank erosion (Wolman, 1959,p. 214). Although the largest dischargesoccurred in summer, the winter provided morethorough soil wetting which, in combinationwith freeze and thaw, led to maximum bankerosion.

As bank erosion occurs in the bend of ameander, over a period of time it is usual foran approximately equal amount of depositionto occur on the opposite bank. This generalequality of deposition and erosion is the reasonwidth and cross-sectional area remain aboutthe same as the channel moves laterally acrossthe flood plain.

MEANDER MECHANICS AND PHYSIOGRAPHICPROBLEMS

General Discussion

The vagaries of nature provide endlessopportunities for perturbations in the flow—local bank erosion, chance emplacement of aboulder, fallen trees, or blocks of othervegetation—any one of which would alter thepath of a straight channel. Thus one needhardly inquire why a stream channel is notstraight. On the other hand, a random succes-sion of chance perturbations might be expectedmerely to result in random bends of differentpatterns. Although this situation describesmany channels, the existence of beautifullysymmetrical meander bends and the remarkablesimilarity of bends in rivers of different sizes

and physiographic settings must be explained.There is not yet available any theory or dynam-ical principle which explains qualitatively thecharacteristic patterns common to meanderingchannels. In the absence of such a generalprinciple, however, attempts have been madeto explain at least qualitatively how symmet-rical successive bends begin and grow, and howsize of bends is related to the size of the river.

Initiation and Development of Meanders

A large body of experience and literature onriver regulation has been built up by Europeanengineers on rivers in Europe, India, andAfrica. This experience has covered a fargreater span of time and of field conditions thanhas American practice. Regarding meanders,Leliavsky recently summarized the salientconcepts developed (1955, esp. p. 111-141).Apparently the consensus of these workers isthat the effect of helical flow is the dominantfactor. Leliavsky expresses it (p. 128) as follows:

"For some reason or other, a small abrasion ...in the ... bank of a straight channel is supposed tohave taken place. The water particles movingalongside the eroded portion of the bank follow acurved trajectory and develop, consequently, acentrifugal force. This force, in turn, gives rise to alocal heUcoidal current, which intensifies the originalabrasion and works its way deeper and farther intothe shore,... until the whole channel becomesfinally involved in the process and the erodedmaterial accumulates on the opposite bank. This,then, is the birth of a meander."

Prus-Chacinski (1954) also argues that heli-cal flow is the basic mechanism leading tomeandering. He showed, further, that, by intro-ducing an artificial secondary flow at the entryto the first bend, it is possible to produce variouskinds of secondary circulations in the nextsuccessive bend which, in turn, affect the circu-lation in the next bend, and so on. He demon-strates that the downstream effects' of a givencirculation pattern are quite persistent, oftenthrough several successive bends. The "cause"of meandering Prus-Chacinski ascribes to anydisturbance which produces an initial secondarycirculation.

It seems clear that helical flow may play animportant role in the process of deposition on apoint bar. A building point bar helps concen-trate shear against the concave bank and thuspromotes bank caving and channel movement.Even denying that helical flow exists in widerivers, Matthes' concept of meander develop-ment is closely allied with that just mentioned.



MEANDER MECHANICS AND PHYSIOGRAPHIC PROBLEMS 785

Matthes (1941, p. 633) indicates that bankcutting and orderly transfer of sediment to itsplace of deposition on point bars is a principalrequirement for meandering. He observed thatmaterial tends to be deposited on point bars onthe same side of the channel as that of theconcave bank from which it was eroded. Thus,by bank erosion of a concave bank and con-current deposition on a point bar across thechannel, the channel will move laterally anddownstream. Thus, the meander wave maymigrate downstream generally maintaining itsconfiguration if the materials are uniform.Friedkin's concept of the process (1945, p. 4)is essentially similar to that of Matthes (1941).

Only a few observers have studied this proc-ess of meander formation and development inlaboratory channels. Quraishy (1944) describedthe development of a sinuous channel from aninitially straight one in a sand-bed flume. Ofparticular interest was the formation of a seriesof small dunelike ripples of sand which he called"skew shoals." These developed on the initiallyflat bed after the sand grains had been in motionfor a short time. These skew shoals were regu-larly spaced and alternated in position alongthe channel sides. After their full developmentthey obstructed the path of water and forced itto assume a sinuous course.

This kind of mechanism is highly suggestivebecause it produced alternating bends, spaceduniformly along an initially straight channel atdistances depending on the spacing of deposi-tional barlike features on the bed. Although theQuraishy experiments do not indicate how theshear force of flowing water interacts quantita-tively with bed debris to govern size and spacingof the dunelike features, it implies that thegeometry of meander waves might be a functionof bed forms.

The coincidence of the spacing of pools andriffles in straight reaches and meander lengthsin rivers of comparable size might well be theresult of a principle of bar or dune formation,as yet unknown, allied in some manner to skewshoal formation. Wolman and Brush (in press)found, however, that in noncohesive sandssimilar skew shoals formed only at supercriticalflow (Froude number, based on mean depth andvelocity, greater than 1.0). Although their wideshallow channels did not meander, Friedkin(1945, p. 4) produced a meandering channelfrom an initially straight one and concludedthat bank erosion alone was sufficient to initiatemeandering as long as channel widening andshallowing did not proceed too rapidly. In thestudy by Wolman and Brush (in press) helical

flow was observed in the channel bends pro-duced from the skew shoals.

Werner (1951) among others2 has attemptedto develop a general equation to describemeander formation and geometry. He (1951,p. 899) has suggested that the initiation ofmeanders is caused by "some local impulse ordisturbance" which creates a transverse oscilla-tion in the straight stream channel. Expressingvelocity in terms of roughness, slope, anddepth, as given by the Manning equation (p.900), he derives an equation in which meanderlength is a function of initial width, channelroughness, slope, depth, and a coefficient pro-portional to sediment load.

An assumption is made that sediment pro-longs the period of oscillation, inasmuch as anymechanism relying principally on inertialoscillations of the water within the channelwidth (seiche effects) gives wave periods muchtoo short to account for meanders. The equa-tion agrees qualitatively with respect to theinterrelationships of some parameters. Itindicates, for example, that meandering will beinhibited at high slopes, a conclusion whichappears to be borne out in natural rivres(Leopold and Wolman, 1957, Fig. 46). On theother hand, the equation does not appearto describe quantitatively the observed relationof meander length to channel width. Further,if sediment load is not a necessary condition ofmeandering, the equation is invalid, for, whenthe load is zero, meander length is zero, andhence meandering should not occur.

Hjulstrom (1942, p. 252; 1949, p. 84) pre-sented an expression for meander length basedon seiche theory. He computed wave lengthas a product of seiche period and wave celerity,period being that of a seiche having a fetchequal to the width of the meander belt andwater depth equal to that of the river. Wavecelerity he considered to be the mean down-valley velocity of the river. The wave lengthso determined was also made a function of acoefficient of turbulent friction.

The resulting expression provided Hjulstroma basis for reasoning about the effects ofchanging discharge on wave length throughthe interacting effects of water depth, velocity,

8 The Coriolis force or the lateral deflection dueto the rotation of the earth has frequently beencited as the cause of meandering, but calculationsby Ludin (1926) and others indicate that thevirtual radius of curvature attributable to theCoriolis force in a stream with a velocity of 3 feetper second would be on the order of 8 miles at alatitude of 60°.




and turbulence, and his argument showed keeninsight and understanding of field conditions(1949, p. 86-88). But the formula has thedisadvantage of making meander lengthdependent on wave amplitude; in our opinion,measurement data do not demonstrate this.

Although a simple comprehensive expressionis still wanting, it appears that the forcesdetermined by the velocity distribution,including the helical circulation, are all that isnecessary to account for (1) the shape of thecross section in a meander, (2) the depositions!and erosional pattern, and (3) the progressivedown-valley migration of the meander. Theseobservations are not new, but it is importantto emphasize the following idea. Althoughpoint-bar formation and associated erosion ofthe opposite bank are necessary if a straightchannel is to develop curves, the concept ofhelical flow, as Leliavsky (1955, p. 128) recog-nized, does not seem to explain how the second-ary circulation determines the characteristicdimensions or proportions of meanderingchannels. The existence of meanders on glacierice also implies that erosion and deposition maybe a collateral, not the governing principle ofmeander development and movement. Becausethe hydraulic or mechanical significance of thepattern of curvature is closely tied to thefundamental physiographic questions, weconsider this aspect of meander mechanics inthe following section.

Problem of Channel Equilibrium

The preceding discussion was concernedprincipally with the initiation and developmentof a meander, elements that can be discussedin terms of a short reach of river or a singlemeander wave. There are a host of broaderproblems of channel adjustment to externalcontrols which might be thought of as physio-graphic problems for want of a more specificterm.

To begin, one might ask how meandering ofa channel relates to the fundamental processof stream adjustment and stream equilibrium.It is generally believed that channel equilibriumis constantly approached, although rarelyattained, by a process of continual adjustment.To use the words of Rubey (1952, p. 129),

" .... with changing conditions, the stream isconstantly cutting or filling and modifying itsslope, velocity and cross section so as eventuallyto accomplish the imposed work with the leastexpenditure of energy."

If, indeed, a principle of least work is in-volved—for it is not yet proven—how doesthe development of a meandering patternhelp accomplish this objective? Is the reductionof gradient achieved by increasing lengthrelative to a straight channel necessary becauseof an excess of energy? Or, as was once believed,is meandering the aimless wandering of achannel too sluggish to accomplish any workof erosion? Does a river reach a stage at whichvertical erosion is negligible and thence expendits excess energy on lateral erosion by meander-ing?

Under what changes of conditions would ariver change its pattern from meandering tononmeandering or vice versa? What would bethe effect of an increase or decrease in dis-charge or in sediment load from the drainagebasin?

Opinions on some of these questions havebeen published, but data or measurements aremeager. Existing data may answer a particularquestion but do not explain why the observedresult was obtained. In the following paragraphssome of these questions are considered alongwith related observations from the literature.Results of recent work, where applicable, arecited, and we suggest what appears to ussome of the directions in which future work isneeded.

Many of these queries relate to a single,fundamental question—what has the meander-ing pattern to do with energy expenditure?

Inglis (1949, p. 158) states that,

"Meandering is Nature's way of damping outexcess energy during a wide range of varying flowconditions, the pattern depending on the grade ofmaterial, the relation between discharge and charge(load) and the rate of change of discharge andcharge."

Schoklitsch (1937, p. 149) earlier statedwhat appears to be the same idea, that meanderformation

"might be due to the fact that the slope in suchstretches is too great and is not in equilibrium withthe size of the bed-sediment grains."

Water-surface slope of a river is a measureof energy expenditure. Whereas the equilibriumslope is closely related to the size of particleson the bed, there are other factors involved.Hack (1957, p. 61) confirms quantitatively thegenerally held belief that channel slope iscontrolled to a great extent by bed grain sizeand discharge, but to what extent channelshape enters is still unclear. As Rubey explains



MEANDER MECHANICS AND PHYSIOGRAPHIC PROBLEMS 787

(1952, p. 131), channel shape may adjustjointly with slope. At constant discharge ifthe principal effect of bed particle size is onchannel roughness or frictional drag, thenother forms of drag need to be considered.Total channel resistance is materially influencedby form drag of various kinds, pools and riffles,bars and dunes, and channel curvature.

There is reason from the hydraulic standpointto believe that meandering may in part be afunction of frictional drag and thence energyloss. More energy loss occurs per unit of lengthin a curved than in a straight channel of thesame depth and cross section, owing to eddying,secondary circulation, or increased rate ofshear. These eddy losses result from deflectionof the water to a new direction as it movesaround a curve or bulge. It is known fromhydraulic experiments in pipes (King, 1954,Fig. 87) that energy loss first decreases andthen increases with a decrease in the ratioradius of curvature/pipe diameter. Althoughany curved channel offers greater resistanceto flow than a straight one, the minimumincrease in resistance is about 40 per cent, andthis applies within a narrow range of the ratioof radius of curvature/diameter when thatratio has a value of 2 to 3.

It was pointed out earlier that in meanderingchannels the comparable ratio, radius ofcurvature/channel width, is relatively conser-vative. Values of this ratio also tend to fall inthe range of 2 to 3.

Bagnold (in press) suggests an explanationof this minimum increase in resistance at thisvalue. He postulates that, as radius of curvatureis reduced and becomes about 2 to 3 times thechannel width, an eddy or zone of reverseflow forms just downstream from the bulge orconvex bank. With the appearance of thereverse eddy the local width is constricted,and there is a local increase in effective radiusof curvature and a net decrease in energy loss.

To the extent that further work confirmsthe indication that the modal value of thisratio is in the range 2-3, meanders tend to becharacterized by a geometric pattern whichhappens to offer the smallest energy loss ofany configuration of curved channel. Thesignificance of this observation is unknown,but it suggests that some principle related toenergy conservation does operate in themeander mechanism.

Assuming that a bend actually does tend todevelop a configuration such that the energyloss due to the bend is a minimum, it does not

follow that the total energy expenditure hasbeen minimized. When a given discharge fallsthrough some specified vertical distance, acertain amount of energy is transferred frompotential form to some other form. If the waterdoes not accelerate (velocity remains aboutconstant downstream), then this potentialenergy is expended as work or heat. The sameamount of energy is spent whether the watermoves in a straight channel on a steep slopeor in a longer curved channel at a smallergradient. The energy expenditure per foot ofchannel length is smaller in the longer curvedchannel than in the shorter straight one. Thequestion, then, is how much energy is utilizedper foot of channel length and in what form itis used.

This energy may be spent in moving particlesof debris, or it is otherwise dissipated into heat.The energy may be spent in removing particlesfrom the bank and transporting them (bankerosion) or in transporting bed or suspendedparticles. If this energy is concentrated in sucha manner that more of it goes into movingparticles from one place than from another,then local scour will occur there and depositionelsewhere. For a channel to be in equilibriumscour must balance fill within the reach inquestion, and, further, the energy must be soexpended that the net amount of debris cominginto the reach must equal the net amountcarried out of the reach.

It is generally agreed that meanderingchannels are often stable or in quasi-equilib-rium. They may be so even though, over aperiod of time, a meander wave moves graduallydownstream. The slope, discharge, and channelshape tend to become adjusted so that theabove requirements are fulfilled. Adjustmentsin channel shape occur through erosion ordeposition which in turn affect velocity, depth,and width. Specific hydraulic requirementsrelating depth, slope, velocity, and totalresistance, including resistance offered by bedconfiguration (form resistance), bed and bankgrains (skin resistance), and channel curvature(a particular kind of form resistance) mustalso be maintained.

Meandering is one way in which erosion anddeposition may change the distribution,location, and amount of energy expenditureper unit of channel boundary. By lengtheningthe channel between two points at differentelevations, the energy expenditure per foot oflength is reduced. By bank erosion, point-barbuilding, and by scour and fill, the channel




cross section is adjusted, and the energyexpenditure is redistributed. Presumably abruptdiscontinuities in the rate of energy expenditurein a reach of channel are less compatible withconditions of equilibrium than is a more orless continuous or uniform rate of energy loss.It may well be that a meandering channel ismost stable when the energy loss due to curva-ture is at a minimum. Such a conclusion isperhaps implied in the modal distribution ofvalues of the ratio of radius of curvature towidth in natural streams.

If we view curvature as simply one methodof altering the distribution of energy expendi-ture in a given length of channel, it is dearthat the pattern of meandering will respondto changes in discharge and load. It is wellknown that an increase in discharge in ameandering channel will increase the channelwidth and will increase the size of the meanderbends. Decrease in discharge will graduallyreverse the process.

In the natural rivers, geologic and strati-graphic evidence clearly demonstrates thatduring late Pleistocene time an increasingdischarge markedly decreased the gradient ofOster-Dal River and enlarged the width ofchannel as well as the size of meander beds(Wenner and Lannerbro, 1952, p. 108). Duringthis gradual degradation of the valley floorthe meandering pattern persisted. Thus themeander pattern where it exists in natureappears to be a persistent attribute of theriver.

Change in load will cause aggradation ordegradation and thus change in channel slopeand size of bend (Friedkin, 1945, p. 7-9); anincrease in slope will produce an increase inmeander length and amplitude. Schoklitsch(Shulits, 1935, p. 644-646) and Bagnold (1960)have postulated that at high dischargessediment transport is a function of the rate ofwork done per foot of channel length, or powerintensity. In a meandering channel in equilib-rium increasing tightness of bend (curvature)through its effect on the rate of energy expendi-ture decreases the rate of transport. Forequilibrium, then, a balance must be maintainedbetween curvature and transport quite apartfrom any change in intensity of energy lossbrought about by a change in length.

DIRECTION FOR FUTURE WORK

No wholly adequate explanation of meander-ing is yet available. Probably no single simplemechanism will suffice to explain all aspects ofmeandering. Although there is general agree-

ment on the manner in which bank erosionand point-bar formation are related to theorderly transfer of sediment which is basic tomeandering, no physical or mechanical principlehas been identified which explains qualitativelythe size and geometry of meander curves.

There is need to investigate oscillatoryforces which might explain more adequatelythe manner in which an initial bulge or depres-sion in a stream bank leads to a symmetricalreversal of curvature.

Although the available velocity distributionsin channel bends do permit general descriptionsof the flow, many characteristics, includingorientation and position of the helical patterns,are as yet poorly defined. More detailed meas-urements in natural channels are required todefine the loci of energy losses and their relationto flow resistance and localized erosion ofbed and banks.

The way in which a natural channel distrib-utes the energy loss as between boundaryfriction, form resistance, curvature, andtransport is little understood. Without suchunderstanding it is virtually impossible toexplain or predict the behavior of a meanderingchannel. Although some studies of energylosses in curved channels have been made(Allen, 1939; Leopold et al., in press), it appearsto us that laboratory as well as field studies onthe distribution of energy expenditure instraight and curved alluvial channels areneeded. It would be particularly desirable tomap the distribution and magnitude of bound-ary shear in bends of different patterns butwith similar cross section and depth. Theseobservations must in turn be related to themechanics of sediment transport in meanderingchannels of different patterns.

The principal unsolved problem with regardto the pattern of flow and its relation to erosionand deposition is in the area of the mechanicsof sediment transport. Present theories areinadequate to explain the transport of hetero-geneous sizes under the variety of conditionsfound in nature. The stress needed to induceand maintain motion is probably different forscattered rocks on a sand bed, uniform gradingfrom sand to cobbles, graded cobbles andboulders without sand, or a few isolatedboulders on a cobble bed. At the same time,objective criteria are needed to describe theability of varying bank materials to withstanderosive stresses. A quantitative explanation ofthe meandering process will require balancingthe erosive stress produced by tangential shearof the flowing water and the comparableresisting stress provided by the bank material.



DIRECTION FOR FUTURE WORK 789

It is particularly important that criticalfield observations be tied to theoretical andlaboratory studies. A few fundamental conceptscan unify a vast amount of empirical observa-tion if the observations include those criticalmeasurements which make possible an adequatetest of theory.

BIBLIOGRAPHY

Allen, Jack, 1939, The flow in a tortuous stretch ofa river and in a scale model of the same: Jour.Inst. Civil Engineers, v. 11, p. 115-132

Bagnold, R. A., 1960, Sediment discharge andstream power—a preliminary announcement:U. S. Geol. Survey Circ. 421, 15 p.

• in press, Some aspects of the shape of rivermeanders: U. S. Geol. Survey Prof. Paper 282

Bates, R. E., 1939, Geomprphic history of theKickapoo region, Wisconsin: Geol. Soc.America Bull., v. 50, p. 819-879

Blue, F. L., Jr., Herbert, j. K., and Lancefield, R. L.,1934, Flow around a river bend investigated:Civil Eng., v. 4, p. 258-260

Dury, G. H., 1958, Tests of a general theory ofmisfit streams: Inst, of British GeographersTrans., pub. 25, p. 105-118

Eakin, H. M., 1935, Diversity of current-directionand load-distribution on stream-bends: Am.Geophys. Union Trans., pt. II, p. 467-472

Fargue, L., 1908, La forme du lit des rivie'res 4fond mobile: Gauthier-Villars, Paris, 187 p.

Friedkin, J. F., 1945, A laboratory study of themeandering of alluvial rivers: U. S. Water-ways Expt. Sta., 40 p.

Fuglister, F. C., 1955, Alternative analyses ofcurrent surveys: Deep-Sea Research, v. 2, p.213-229

Hack, J. T., 1957, Studies of longitudinal streamprofiles in Virginia and Maryland: U. S. Geol.Survey Prof. Paper 294-B, p. 45-97

Hjulstrom, F., 1942, Studien Uber das maander-problem: Geog. Annaler, B. XXIV, p. 233-2691949, Climatic changes and river patterns, in

Glaciers and climate: Geog.Annaler, h. 1-2, p.83-89

Inglis, C. C., 1937, The relationships betweenmeander belts, distance between meanders onaxis of stream, width, and discharge of riversin flood plains and incised rivers: Ann. Rept.(Tech.), Central Board of Irrigation (India)1938-39, p. 491949, The behavior and control of rivers and

canals: Central Waterpower Irrigation andNavigation Res. Sta., Poona, Res. Pub. 13,v. 1 and 2, 486 p.

Jefferson, M., 1902, Limiting width of meanderbelts: Nat. Geog. Mag., v. 13, p. 373-384

King, H. W., 1954, Handbook of hydraulics, 4thed.: New York, McGraw-Hill Book Co., Inc.,400 p.

Lane, E. W., and Borland, W. M., 1954, River-bedscour during floods: Am. Soc. Civil EngineersTrans., v. 119, p. 1069-1079

Laursen, E. M., and Toch, A., 1954, Discussionof Lane and Borland, River-bed scour duringfloods: Am. Soc. Civil Engineers Trans., v. 119,p. 1084-1087

Leliavsky, Serge, 1955, An introduction to fluvialhydraulics: London, Constable and Co., 257 p.

Leopold, L. B., hi press, The gravel bar: someobservations on its role in channel morphology:Internal. Geol. Cong., Copenhagen

Leopold, L. B., and Miller, J. P., 1956, Ephemeralstreams, hydraulic factors and their relationto the drainage net: U. S. Geol. Survey Prof.Paper 282A, p. 1-37

Leopold, L. B., and Wolman, M. G., 1957, Riverchannel patterns—braided, meandering, andstraight: U. S. Geol. Survey Prof. Paper 282B,p. 39-85 H

Leopold, L. B., Bagnold, R. A., Wolman, M. G.,and Brush, L. M. Jr., in press, Flow resistancein sinuous and irregular channels: U. S. Geol.Survey Prof. Paper 282

Ludin, A., 1926, Influence of the rotation of theearth on rivers: Die Wasserkraft, v. 21, p.216, partial translation by Sam Shulits, 1958

Matthes, G. H., 1941, Basic aspects of streammeanders: Am. Geophys. Union Trans., v.22, p. 632-636

Mockmore, C. A., 1944, Flow around bends instable channels: Am. Soc. Civil EngineersTrans., v. 109, p. 593H528

Nemenyi, P. F., 1946, Discussion of Vanoni, V.,Transportation of suspended sediment bywater: Am. Soc. Civil Engineers Trans., v. Ill,p. 116-125

Parsons, D. A., 1959, Observations of flood floweffects on channel boundaries: U. S. Dept.Agriculture, Agric. Research Service, mimeo-graphed

Prus-Chacinski, T. M., 1954, Patterns of motionin open-channel bends: Assoc. Internat.d'Hydrologie, pub. 38, v. 3, p. 311-318

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Rubey, W. W., 1952, Geology and mineral re-sources of the Hardin and Brussels quadrangles(in Illinois): U. S. Geol. Survey Prof. Paper218,175 p.

Schoklitsch, Armin, 1937, Hydraulic structures,v. 1: Am. Soc. Mech. Engineers, 504 p.

Shulits, Sam, 1935, The Schoklitsch bed-loadformula: The Engineer, v. 139, p. 644-646, 687

Stommel, H., 1954, Circulation in the NorthAtlantic Ocean: Nature, v. 173, p. 886-893

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Thomsen, J., 1879, On the origin of windings ofrivers in alluvial plains: Royal Soc. LondonProc., v. 25, p. 5-6

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Von Arx, W. S., 1952, Notes on the surface velocityprofile and horizontal shear across the widthof the Gulf Stream: Tellus, v. 4, p. 211-214

Wattendorf, F. L., 1935, A study of the effect ofcurvature on fully developed turbulent flow:Royal Soc. London Proc., ser. A, v. 148, p.565-598

Wenner, C. G., and Lannerbro, R., 1952, Themeander field at Mora against the backgroundof the geology of the Siljan basin and the




history of the Mora district: Ymer., h. 2, p.81-109

Werner, P. W., 1951, On the origin of river mean-ders: Am. Geophys. Union Trans., v. 32, p.898-902

Wolman, M. G., 1959, Factors influencing erosionof a cohesive river bank: Am. Jour. Sci., v.257, p. 204-216

Wolman, W. G., and Brush, L. M., Jr., in press,Laboratory study of equilibrium channelsin non-cohesive sands: U. S. Geol. SurveyProf. Paper 282

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U. S. GEOLOGICAL SURVEY, WASHINGTON, D. C.;THE JOHNS HOPKINS UNIVERSITY, BALTIMORE,MD.

MANUSCRIPT RECEIVED BY THE SECRETARY OF THESOCIETY, OCTOBER 29, 1959

PUBLICATION AUTHORIZED BY THE DIRECTOR, U. S.GEOLOGICAL SURVEY



APPENDIX 791

APPENDIX. SHAPES OF MEANDER WAVES IN ALLUVIAL PLAINS



APPENDIX.—SHAPES OF MEANDER WAVES IN ALLUVIAL PLAINS(Note: Reaches of river chosen from maps to represent (a) a single meander length in which the S3 curve was reasonably symmetrical; (b) the samples

would include a large range in sizes of rivers. Each sample is considered representative of the local reach of river.)

River and location

Sacramento R nr Chico, Calif. . .Ninnescah R nr Belle Plain, KansNew R nr Brawley CalifMissouri R nr Buckner MoMissouri R nr Buckner Mo

Kansas R nr Eudora

Kissimmee R nr Okeechobee, FlaArkansas R nr Mulvane, KansColorado R nr Blythe Calif

San Joaquin R nr Patterson, Calif. . . .James R nr Forestburg, S. D. . . ...James R nr Clayton, S. D. ... ...Souris RMissouri R nr Lexington, Mo

Sacramento R nr Glenn Calif.

Red R nr Campti, La.

Henrys Fork nr Menan, Idaho ....Henrys Fork nr Menan, Idaho . . . .Cedar R nr Belgrade, Neb ...Cedar R nr Belgrade, NebMississippi R nr Wynnburg, Tenn

Eleva-tion(ft)

1101200150700690

790

181190270

35122011901480690

88+

100

4810481217001650265±

Range inwidth

(ft)

250-55080-16070-90

350-1550550-1950

220-800

85-155220-360450-1650

120-28060-9080-15030-60

450-1150

250-450

400-1200

180-300110-20060-13090-150

2132-5460

Meanwidth0*0(ft)

550*260*250*

1850*1600*

1000*

180*500*

1050*

400*160*230*150*

1350*

950*

1400*

340*280*200*210*

4940*

Meanderlength

(X)(ft)

670013501475

14 60037 100

12,750

10505650

25,300

2400222015501100

24,400

7600

11,400

2700195030502300

38,580

Arc dis-tance

(ft)

840017001750

24 30041 800

14 400

29007900

33 300

4550360026002500

29,000

10 800

18,700

6100330045502900

60,050

Meanradius

(n»)(ft)

2050360210

315015 550

4300

20513507300

610450265160

6000

2300

2130

850410750575

9620

Amplitude(A)(ft)

2150400400

88507700

2850

125021508900

16001275900720

7100

3200

7000

230012001450820

27,400

Sinu-osity

1.251.261.181.661.13

1.12

2.761.401.32

1.901.621.682.271.19

1.42

1.64

2.261.691.491.261.55

Map quadrangle(scale = 1:24,000)

Ord Ferry, Calif.Belle Plain, Kans.Brawley Calif.Buckner Mo.Buckner, Mo.Camden, Mo.Lawrence E. Kans.Eudora, Kans.Okeechobee N.W., Fla.Mulvane, Kans.Blythe N E. Calif.

Ariz.Brush Lake, Calif.Forestburg, S. D.Tschetter, S. D.Voltaire, N. D.Camden, Mo.Lexington, Mo.Glenn Calif.Llanos Seco, Calif,(scale 1:62,500)Campti, La.Menan Buttes, IdahoMenan Buttes, IdahoBelgrade, Neb.Belgrade, Neb.*** Map 6, miles 92-102



Wind R nr Dubois, Wyo., Dumb CowboyReach

Squaw Cr nr Lander, WyoLittle Pipe Cr nr Westminster, MdBaldwin Cr nr Lander, WyoLittle Sandy nr Elkhorn, WyoBuffalo Fork at Black Ranger StaBaldwin Cr nr Lander, Wyo

Mississippi R nr Smithland, LaMississippi R nr Lake Providence, LaMississippi R nr Rosedale, MissNew Fork nr Pinedale, Wyo. (Hailstone

Reach)Model, U. S. Waterways Expt. Sta

Model, U. S. Waterways Expt. Sta

Model, U. S. Waterways Expt. Sta

Moyen R nr Castets, France. . .Coosa RiverKansas RiverRed RiverHenrys Fork, nr Menan, Idaho.Sacramento River

Kissimmee River. . .Sacramento River...James RiverSouris River, N. D..Souris RiverSouris RiverSouris RiverRed River, La

1585

145

440780100

481085

18120

12201610159015601430

40

22-48

10-1810-1810-1670-11512-24

4900-21802200-58004300-2440

48-76

1.2-3.2

2.3-3.0

627-1120650-400850-170500-1000350-150600-220

130-110700-350

3081214138015

333035003210

623.3

1.6

2.6

870520510750250410

120525110140140120100600

20582145185248990165

65,00053,50040,300

74534

28.2

29.5

90408976

16,40010,00027006100

1100834022201800184084011407400

3101151922452731230205

75,40075,80047,000

96043.5

36.0

34.2

12,20018,50026,10017,5005500

11,000

210011,500260032004000160024008800

411735457630843

15,7009900

11,200

1638.9

7.2

7.0

20001700315020006501700

22514006004404802203001640

944058725531048

19,85011,40011,760

22513.0

9.5

8.5

376067008550500020003700

6003600600

10401240500800

2100

1.511.401.32

321024

1.24

1.161.421.17

1.291.28

1.28

1.16

1.362.061.591.752.041.80

1.911.381.171.782.171.902.101.19

Plane-table map, authorsPlane-table map, authorsPlane-table map, authorsPlane-table map, authorsPlane-table map, authorsPlane-table map, authorsPlane-table mapLeopold and Miller*** Maps 38, 38, miles 770-785*** Maps, 26, 27, miles 550-565*** Maps, 19, 20, miles 387-396

Plane-table map, authorsMeandering of alluvial

rivers, Friedkin, 1945Test 4Plate 41TestlPlate 29Test3Plate 37

Fargue, 1908Riverside, Ala.Eudora, Kans.Campti, La.Menan Buttes, IdahoGlenn and LlanoSeco, Calif.Okeechobee N.W., Fla.Ord Ferry, Calif.Forestburg, S. D.** Sheet A, mile 12, 1926** Sheet A, mile 40, 1926** Sheet Cm mile 123, 1926** Sheet K, mile 290, 1930*** Mile 33.0

* Estimated bankfull width**Plan and profile of Souris (Mouse) River, International Boundary to Verendrye, N. D. (Advance sheets): Department of the Interior, Geological Survey, printed

1926*** Mississippi River Commission, 1935, Maps of the Mississippi River, Cairo, 111., to the Gulf of Mexico, La.; scale, 1:62,500: Vicksburg



Documents

Geological Society of America Bulletin · rivers have large bends, and small rivers have small bends. An adequate theory of meanders should explain why all river bends seem to be