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BULLETIN OF THE GEOLOGICAL SOCIETY OF AMERICA
VOL. 71. PP. 769-784. 7 FIGS.. 1 PL. JUNE 1960
c I
IiIVElt MEAN11 E M
I3u [,DNA 11. l m m x ~ AND M. G o a u o ~ WOLMAN
ARSIXACT Most river CLL~VCS have nearly the same value o l thc iatio of cuivature radius to channel
width, in the range of 2 to 3. Memders formed 114 meltwater on thc surface of glaciers, and by the main current of the Gulf Stream, have a relation of meander length to channel width similar to rivers. Because such meanders carry no sediment, the 4iapes of curves in rivers are evidently dcteiminetl i)riinaiily I > p the tlynrunics of flo\v I athcr than hy relation to del,ris load.
teristics. Evidence on flow resistance in curved channels suggests that a Imic aspect of meander mechanics may be related to the distiibution of energy loss provided by a particular configuration or curvature. No general theory of meanders is as yet satisfac-
ry, however; in fact, present evidence suggests that no single theory will explain the for- ation and charactcristics of all meanders and that few of the physical principles in-
I Velocity distriliutions along river Luives provi(ic n generalized picture ol flow L h a i a c -
volved have yet been clearly identified.
RBsud de rivii'res la valeur clu rapport entre le rayon de courbin e de grandeur de 2 2 3, est presque toujours la m&me. Les fonte i la suiface des glaciers et par le principal courant e relation entre la longueur du mdancire et la largeur du 'viiires. Puisque de tels miandies ne transportent pas de que la forme des courbes de rivihres est determinie piinci- I'c'Loulerncnt d'eau plutbt que lite A la charge sddimentaiie. s le long des courbcs d e rivihres donne unc image g6nEralc ent. Ce qu'on sait de la rc'sistance A I'icoulenient dans les la resistance minimum est un facteur de base clans la
tefois, aucune thiorie gEnirale des indandres n'est encore n fait, les donnCes actuelles suggi'rent qu'aucune thEorie
ne suffira pas i expliquer tous les indandres, et que quclques-uns seulement des principes de physique en jeu ont ddji &t i reconlitis.
ZUSAMMENPASSTJNC Die meisten FluRwindungen habeu etwa dasselbe Wertveihaltnis von Krumiiiungs-
radius zu Bettbreite, ungefahr zwei zu drei. Mhanderwindungen, die durch Schmelzwssser auf der Oberflache der Gletscher entstchen, und solche, die durch die Hauptstromung des Golfstromes bedingt sind, haben eiu Verhaltnis von Mhanderlange zu Betthreite,
elches dem der Fldssc Bhnlich ist. Da solche Maander kein Sediment mit sich fuhren, nd die Formen der FluDwindungcn offensichtlicli in crster Linie durch die Dynaniik der romung bedingt, weniger durch die Gegebenheiten des mitgefuhrten Materials. Die Verteilung der Stromungsgeschwindigkeit entlang den FluDwindungen zeigt ein rallgemeinertes Bild der Stromungsmerkmale. Was uber den Stromungsln,itlcrstand in wundenen FluObetten heobachtet xvurde, 1LOt die Vermutung aufkommen, dain der ringste Widerstand eine wichtige Ursache fu r Maanderbildungcn ist. Tatsaclilich lassen e gegenwartig heohachteten Tdtsachen vermuten, dafi vorlaufig noch keine cinzigr eorie alle Maanderhildungen erklaren kann, und daO nur wenige der physikalischen
esetzmainigkeiten, die dabei eine Rolle spielen, bisher gefunden worden sind.
MEAHAPLI PEIC
PesioMe J I ~ H ~ E ~ ~ O E O J ~ R YI M. ropAoH Y O ~ ~ M ~ H
Fo.nbIininmno permin ~ ~ m y r r w i r 0 6 n a ~ a ~ i ~ I I O W ~ OJ[IIIINCOBOI% R ~ , J I I i ~ I I T I I O h HOIUeHIlR pn~ i iyca ~ c p i m i m m ~ IC mpIpiiHe mm,nnr2, rco,ne6aror~e~cr1 n I I O ~ R J ~ I C C 2 no 3 MearrApbr, oFpn30sna11bre mxoa B O H O ~ HCL IIOBC~SIIOCTII ne:jmtcoB EI
769
770 1,EOPOLI) IIND WOLRI;1N- R I V E R MEANDERS
CON 'IT NI'S
rage General statement . . . . . . . . . . . . . . . . . . . . . . 7 70 Gcornetry antl pattcrns of channel
bends.. . . . . . . . . . . . . . . . . . . . . 771 Curves in sediment-free channels . . . . . . . . . . . 774 Channel cross scctions and longitudinal
profiles.. . . . . . . . . . . . . . . . . . . . . . . . . 776 Dynamic ant1 flow characteristics. . . . . . . . . . . 778
Pattern of flow in a meander bend.. . . . . . . . 778 Generalized picture of flow in a meander. . . . 779 Channel shape and movement in relation to
the flow pattern. . . . . . . . . . . . . . . . . . . . . 781 EfTect ol flow pattern on deposition and
erosioii. . . . . . . . . . . . . . . . . . . 751 RIcander mechanics and physiographic
I"-ol,lellls . . . . . . . 784 . . . . . General discussion. . . . . . . . . . . . . 784 Initiation and dcvclopnient 01 nicantlcrs . . 784 Prohlem of channel equililxiuni . . . . . . . . 786
Direction for future work . . . . . . . . . . . . . . 788 Bibliography.. . . . . . . . . . . . . . . . . . . . . . . . 789 Apliendis. Shapes of meander waves in alluvial
plains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791
TI, I, USTKATIONS
Figurc R lg r
1. Sketch to define ternis used in describing geometric characteristics oi a meantiering cliannel . . . . . . . . . . . . . . . 772
2. Relations ljetween meancicr length antl chan-
The most characteristic features of all stream cli,tnncls, regardless (,I size, drc the abserice of long straight reaches and the presence of frequent siiiuous reversals of curvature. These Iiends, \vlictlier or not they display regular reversals and sufficient ipnimctry to warrant thc iiniiie nieander, teiid to be scaled versions ol <L given set ol proportion\. The proportions arc tlctermiiir.tl Iry three tlimensions: tlie
ne1 width and meander length and t11c:in radius of curvature.
3. Planimetric map antl I m l topography of a nieantlcr of the Mississippi River at Point Breeze, Louisiana, the New Fork Ilear Pinedale, Wyoming, and Duck ('rrck near Cora, Wyoming. . . . .
4. Lateral and downstream components of velocity at various cross sections in a bend, Baldwin Creek, near Landcr, Wyo- ming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ;
5. Isometric view of generalized diagram of flow distribution in a meander, sIio\ving downstream (open parabolas with arrows) and lateral (closely lined areas) compo- nents of velocity as vectors, and surface streainlincs. . . . . . . . . . . . . . . . . . . . . . .
6 . Stratigraphy of point bar in relation to meas- ured cross sectional profiles during six years of observation, Watts Branch, 1 mile northwest of Rockville, Maryland
7. Map and cross section of a typical point har of the flood plain of Watts L(ranch, 1 mile northwest of Rockville, R.laryl;mtl ;
Plate Facing g
1. Meanders on 1)inwoocly Glacier, Wyoming , ;
. . . . . . . . . . i
I
;
TAIX E
Table I'aac 1. Ini11irical relations I)etrveen size i'ararnetcrs
for meanders in alluvial valleys . . Ti3
repeating tlist.aiice or lcngth of the curvr., the width ol thc channel, and the radius of c u r w ture. In brief, the size of the bend apprars to bc proportional to tlie size of the rivcr; large rivers have large bends, and sinal1 rivers have sinall bends.
i i i i adequate theory oi meandcrs slioul[I explain why all river bends seem to h e oi proportioiial sizes, arid by what ~iiccliaiiical 01
physical principle a I~cnd Iiccoincs ailjustetl in size to the strcain discharge. Such a thcory
GIi~NIi;I<AI,
also explain why sonic iivcrs mednder tliers do not. It should explain why ering rivers occur on ice with high vcloc- flow and no sctliment load, and why
us channels are so prevalent in tidal ies. A meander thcory should also explain
icc perturbations of the flow may d lead to the devclopmcnt of a sinuous
present papcr does not provide any An attcmpt is inadc to analyze and
te wliat appear to be the most prevalent ial characteristics 01 meandcring channels
At least some of these prevaleiit tics arc probahly basic to the of mcandcring channels which are rly understood. These include the
aturcs of the channcl, as well as the ion of velocity and the pattein of watcr. Such a summary may prove
in thc eventual definition ol the necessary ufficient conditions for mcandcring of
in diverse nalural cnvironmciits. is a large publishcd record on the
ce and physiographic setting of ing channcls, as well as on the form ry of rivcr curves. I t is not our purpose ite this record exhaustively, but rathcr
out thosc facts which appear to us most ent, emphasizing a t the samc time that ns may differ as to wliat is pertincnt.
studies of meandering streams and of er mechanics lead us to write a somewhat it kind of a paper than might usually
tcd in a “review” series. Many ncw ions are availablc, sonic previously hed, sonic recently published or in
blication. We hope it will be of , especially to the reader who lias u t not a spccialty in the subject, sc new observations summarized
latcd to earlier observations. We belicvc c status of tlie problem will bc clcarcr as
It, and tlicsc obscrvatioris arc, in our n, of some importance in indicating
s toward which iuture work might c directed. attcnipt to show that tlic problem
eanders can be brolicn down into The first concerns the mechanical
raulic processes that govern the form, probably tlic occurreiice of meanders. nd includcs the physiographic history rticular channel and the formation, iicc, and possiblc dissolution of a
S‘I~A’lF 771
meandering pattern. Although tlicsc two parts are almost inseparable they do represent quite different ways of viewing thc meander problem. We hope i t will be clear from this revicw that many important physiographic questions cannot be answered with satisfaction bccausc thcre is insufficient knowledge a t preseiit about meander mechanics.
The authors are much indebted tu J. Hoover MacBin and W. W. Rubey for many construc- tive suggestions and for a critical reading of an early draft of tlie manuscript.
(;EOMETRY ANI) I’ATrERNS OP CHANNEL BENDS
Hccausc most channels are not straight bill sinuous, the forms which thcy display in plan view arc, in detail, irifinitcly varied. In our experience i t is unusual for a reach of natural stream to be straight for a distance cxcccding 10 channel widths (Leopold and Wolman, 1957, p. 53). Yet certain common features are discernible. These features, coninion to many individual bends, probably have some signifi- cance in the inechanics or hydraulics of mcan- ders, but this significance is oiily poorly under- stood. Thc geometric characteristics of a channel seen in plan view are defined by Figure 1.
We use the term “arc distance” to iiican the length measured along the channel center line lrom one point of inflection to the ncxt. “Sinu- osity” we use to mean tlic ratio ol arc tlistancc toihalf the meander length.
The geometry of meander curves lias been the object of extensive statistical study, cxaniples of which are papers by Jcfferson (1902), lnglis (1937; 1949, Pt. 1, p. 143), Bates (1939), and Lcopold and Wolman (1957, p. 58). An additional independent sample in the appendix includes n~casurcments of rivers in which the presence of a well-defined flood plain indicates that the channel forin and pattern have developed reasonably frec of bedrock control. These data for rivers of moderate to large size were derived from published topographic maps and for small streams lroiii plane-table maps made by thc authors. The appendix also includes soinc data from publishcd laboratory studics of meander- ing streams in erodiblc materials.
Studies of the geometry of the patterns of meandering rivers give generally similar results.
~
772
Point of inflection or crossover
L = meander length (wave length) A =amplitude
rm= mean radius of curvature
the data in thc appendix. Hccausc of the width may bc considcrctl 1 variability in thc dimensions of channels in different groups of data fairly co
773
ABLE l . - ~ M P l R l C A L h3LATIONS BETWEEN SIZE PARAMETERS FOR MEANDERS IN ALLUVIAL VALLEYS
_ _ ~ __- - ___
Meander length to channel width channel width radius of curvature
1949, pt. 1, p. 144, Jefferson A = 1 8 . 6 ~ ~ ~ ~ il = 1 0 . 9 ~ ' "' A = 2 . 7 ~ " L = 4.7r,,,0
ps is determined more by erosion These considerations appear to support the cs of stream banks and by other concept that a limit to the amplitude of
than by simple hydraulic princi- meanders is provided by the formation of cut-offs. The rate of downstream migration relative to the ratc of growth of the amplitude of bcnds probably depcnds on the distribution of shear against the stream hanks, which in turn is a function of channcl shape. The latter is determined at least partly by the character- istics of the flood-plain alluvium and the moving debris.
We believe rclations between channel width,
U. S. Waterways Experiment Station (Friedkin, 1945, p. 15) showed that in
terial amplitude did not progres- nor did meander loops cut off migrated downstream. Fricdkin
s to mean that "natural cut-offs a1 differences in the erodibility
materials (p. M)."
7 74 LEOPOLD AND WOLMAN-RIVER MEANDERS
meander length, and radius of curvature, in contrast to amplitude, are to a great extent indepcndcnt of bed and hank materials and are related in some unknown manner to a more general mechanical principle.
Examining the correlations presented above one can write
L = 1 0 . 9 ~ ~ "l = 4 . 7 ~ ~ 0 98.
Assuming the exponents to be unity and solving for the ratio of radius of curvature to width,
= 2.3. 10.9
r,/w = - 4.7
Computing the value of mVJw river by river from the data in the appeiidix rather than equating the least-square regression lines, i t is found that in the sample ol 50 rivers the median value is 2.7, mean 3.1, and two-thirds of the cases lie between 1.5 and 4.3. About one quarter of thc values lie between 2.0 and 3.0. The tendency for similarity in the ratio of curvature to width makes all rivers look quite similar on planimetric maps, as suggested by Figure 3. In fact when one inspects a planimet- ric niap ol a river without first glancing a t the map scale i t is not immediately obvious whether the river is large or sniall owing to this tendency for similar ratio of radius to width regardless of river size.
As will be nicntioned later, thc value of the rat io of radius of curvature to channel width has an important bcariiig on resistance to flow, and this appcars to offer some clue as to the hydraulic basis for the observed geometric similarity among channels of different sizes.
CURVES IN SLDINENI-FREE CJTANNELS
Meandering channels arc carved in glacicr ice by streams of mcltwater flowing on the glacier surface. To obtain measurements .,I such channels we went to o11c of the Dinwoody glacicrs on the cast side of the crest of the Wind River Rangc in Wyoming (Lat. 43'12', Long. 109O38"). The mcandering channels 073
this small valley glacier can be seen in Plate 1. Although concentrated primarily upon the steeper slope downstream, meandering channels began on tlic rclativcly flat portion of the piofdc, that is, ncar thc point of inflection from concavc to convex portions ol the surface profile. The channels extended downstream to the convex-upward part of the glacier snout.
Some of the meandering cha~inels were incised 6 to 12 feet into the ice, and those
which appear on Figure 1 of Plate this kind. Satisfactory measurements incised meanders could not be obtai the channel shown in Figure 2 of incised 1 to 2 feet, appeared to be comp to the more deeply incised streams.
Comparison of the meandering strea incipient channels being formed by flowing over the ice surface led us to that an irregular sheet of water concentrated in a shallow flat channel place where local factors favored melt meandering form developed as the water gradually melted a channel do Channels in the ice only a foot wid inches deep had no well-developed form, but another channel of similar but 1 foot deep showed a well-dev sinusoidal pattern.
Although occasional rocks and occurred in the channel none were in at time of observation. In some b pebbles had collected on the ins curve in the position ol the point liar streams. The surface of the glac locality was strewn with cobbles and random manner owing to the fac channel measured was near the edge poorly developed medial moraine.
In several places terrace renina into the ice and abandoned as progressed. No pebbles were se terrace remnants. The meanders app cut by melting alone with no eff from abrasion by transported pebbles.
Over-all dimensions (in feet) and parameters of the channel pictured in Fi of Plate 1 were as follows:
Channel width Channel incisiou below ice
surface Width oi surface of flowing
water Mean depth ol flowing watcr,
d,, Average velocity from ineas-
urenients of flow, v Average meander length Average channel slope, s Discharge Froude number v/ z/gd,
(where g is the acceleration of gravity)
The Froude number, 1.9, was we the range of supercritical or shooting shooting flow it is usual to observe wave trains and the diagonal criss
GEOMETRY AND PATTERNS OF CHANNEL BENDS 775
Bississippi River at Point Breeze, Louisiana from Corps of Engineer's data ,h
FLOOD PLAIN Approximate elev 3 units
4 CCNTOUR INTERVAL ?? MEAN
HEiGHT OF BANKS ZERO AT MEAN BED
,verage width 60 feet
Wyoming New Fork River near Pinedale,
0 2OC FEET
CONTOUR INTERVAL 0.5 FEET DATUM ARBITRARY
FLOOD PLAIN Approximate elev 92 feet
A h Duck Creek near Cora, Wyoming
\
!E 3.- AT
-PLANIMETRIC MAP AND BED TOPOGRAPHY OF A MEANDER OF TILE MISSISSIPY POINT BREEZE, LOUISIANA, THE NEW FORK NEAR PINEDALE, WYOMING, AND
DUCK CREEK NEAR CORA, WYOMING Scales are chosen so that meander length is equal on the printed page.
81 RIVER
776
surface waves reflected off the channel sides. If these wcre present they were masked by the violent turbulence, although they were probably the cause of a marked hump in water-surface elevation extending along the channel center line in some places. The water was conspicu- ously superelevated 011 the outsidc of the curves.
The measurements of meander length aiid width of this stream cut in ice are plotted 011
Figure 2. ‘The meander length of channels tlevcloped in ice bears the same relation to channel width as in ordinary incandering streams. Measurements made by D. G. Ander- son of meanders of small streams on the ice of Chamberlain Glacier, Alaska, confirm this conclusion (Oral communication to Leopold, 10.58).
Data on meanderlike phenomena from another source deserve mention as possibly having some relation to the present problem. In the Gulf stream of the North Atlantic oceanographers have found bands of relatively high speed which have a meandering pattern in plan view. Stommel (1954, p. 887) described these as “horizontal eddy and meander forma- tions,” and Von Arx (1952, p. 213) used the term “meandering current.” These currents appear both in cross sections of the velocity field and in planimetric imps of the temperature field. Maps of thc temperature field a t a depth of 200 meters have been published by Stommel (1954, Fig. 2) and by Fuglistcr (1955, chart 3a). From some of these published maps rough measurements of meander length may be obtained and, less satisfactorily, estimates of current width. Better estimates of current width are obtainable from velocity cross sections published by Von Arx (1952, Fig. 2) and Worthington (1954, Fig. 9).
Four measurements of the width aiid meander length were possible from the published data, and these are plotted on Figure 2. Although they fall slightly below the line drawn through the points representing river data, the graph suggests that meandering currents in the Gulf Stream bear certain analogies to river meanders. Although frictioiial flow in rivers should not
~ _ _ _
FIGURE ice. View
FIGURE agitated.
LEOPOLD AND WOLMAN-RIVER MEANDERS
be compared to geostrophic flow of an current, the observed similarity in dimensions appears worthy of a
The importance of meanders in . ocean currents lies in the fact that systems exhibit meanderlike phen the absence of sediment debris. The es ment of meander length-width r similar to those of sediment-laden suggests that sediment alters or affects does not cause the meander pattern. ( contrasting view J P ~ Matthes, 1941.)
CHANNEL CROSS SECTIONS AND LONGITUDINAL PROFILES
From a study of bends of the Rivcr, Fargue (1908, p. 25) stated as a rule of river behavior that the sections occur downstream from the and the deepest sections downstre the axis of the bend or point of curvature. The topographic inaps meanders in Figure 3 show in a ge the features described by Fargue a workers after him. These same figures, also indicate that the location of shallows may be highly variable. F data from 25 bends on Buffalo Buffalo, New York, indicate tha points are usually found downstream point of maximum curvature but do o well upstream from this point (Parsons On a reach of the Pop0 Agie River near Wyoming, containing four successive b amazing symmetry deeps occurred from the point of maximum curvatur downstream.
I n the reach of the Mississippi Ri in Figure 3 the deepest portion occ axis of curvature in one bend but downstream from this axis in the ot I n the New Fork the deepest portion bend occurs practically a t the point of in The reach on Duck Creek more nearly co to Fargue’s generalization.
There xcms little doubt that the water at a bend of a river is closely c
- ~ _ _ _
PLATE 1 -MEANDERS ON DINWOODY GLACIER, WYOMING
1 .-In middle foreground are meandering channels of meltwater streams carved in the southwesterly from Sentinel Peak; photograph by Mark F. Meier 2.--Channcl carries meltwater from glacier. The water flowed at high velocity and was v
CHANNEL CROSS SECTIONS 4
e bend curvature. Using 103 measure of mean depth and corresponding radii
ure on the River Elbe, Leliavsky 118) sbows that the two factors bear near relation and that depth increases as a function of radius of bend. 'nt of inflection in a river curve is sociated with a shallow portion of , or a depositional bar on the bed. retice of a bar or riffle in the bed 01 a
t river reach should correspond morpho- herefore to the point of inflection ol a
channel. It is on this basis that 'stance between successivc rimes in each was compared with the wave neanders (Leopold and Wolman,
only do the riffles in straight channels be in a position analogous to comparable s in a meandering reach, but there also
endency for the shape of the bars to In some straight reaches successive
nd bars which constitute the riffles form of lobate wedges sloping
toward one bank and then the some meanders the bar near the
inflection slopes sharply across the 1 toward the deep near the concave
m. This is apparent on the of the New Fork (Fig. 3) and t low flow when water flows the channel down the steepest A t high flow there is rapid cross-sectional area expands.
longitudinal profile of bars, whether ted with the point of inflection of a r, or a riille in a straight channel, is moundlike rather than dunelike and is
etric in cross section. Perhaps ideally has a dunelike profile with a short, steep ream face. The topography of the
the Mississippi River (Fig. 3) suggests t many are less regular. bar is asymmetric in cross section in a t sense on upstream and downstream n a meander, as one moves downstream
t of inflection, the surface of the ally toward the concave bank, y having passed the crest of
asymmetry changes, and the bar slopes toward thc opposite
s reason the channel cross section in a reach is symmetrical only for a istance. We estimate that on the distance through which the cross
sentially symmetrical is only about
LND SITUDIN PROFU 777
one-tenth wave length. This short distance can be seen on topographic maps of Figure 3.
The importance of the configuration of the bars and their relation to the deeps lies in the relation of bed topography to bed shear. River pilots and others know that in large rivers the crossings (points of inflection) tend to scour a t low flow and fill at high flow; the bends (pools) tend to fill at low flow and scour a t high (Straub, 1942, p. 617; Lanc and Borland, 19.54, p. 1075).
Local bed shear must be greater a t points of scour than where filling occurs, but direct measuiemcnts of bed shear are few, and little is known i n detail about the relation of bed topography to stress on the bed.
Lane and Borland (1954, p. 1079) explain the observed sequence of scour and fill as a consequence of the different shape of cross section in pool and riffle. They say that the crossing or riffle has a larger cross-sectional area than the pool duting high stages and a smaller area during low stages. Laurseii and Toch (19.54, p. 1085) state that width often tends to be greater at a crossing than at a bend. The expansion in width they believe would c a u x deposition on the wide crossing while scour occurs in the narrower bend.
We have made measurements of slope and of depth over pool and riffle separately a t various stages of flow in an attctnpt to compute approxi- mately the variation of mean stress. A t low flow the 'riend is a still pool over which the water has a relatively ilat slope. I n contrast, in the riffle the water tends to plunge over the steep downstream face of the bar. As stage in- creases, the slope over the pool or bend in- creases, while that over the liar decreases until near bankfull stage when the water surface attains a continuous uniform slope, and all obvious surface effects of the shallow bar have been erased.
With increasing stage, depth increases in both bend and crossing. The product of depth and slope, then, increases with stage more rapidly in pool than at the riffle or crossing. In the one stream measured by us, Seneca
Creek near Dawsonville, Maiyland, the com- puted mean shear is greater in the riffle a t low flow and greater in the pool at high flow (Leopold, in press). This observation is in qualitative agreement, then, with the concept of scouring in the crossing at low flow and in the bend a t high flow. The analysis is incorn- plete, however, inasmuch as variations in the distribution of velocity in successive cross sections have not been included in the computa- tion of shear, and additional observations
Section 1
" DOWNSTREAM COMPONENT
Section 2
QOWNSTREAM COMPONENT
COMPONENT T"
w LATERAL COMPONENT
Section 4
FIGURE 4.-hTERAL AND DOWNSTRXAM cOMPON%NTS OP VELoClTY AT VARIOUS CROSS SECTIOl IN A BEND, BALDWIN CREEK NEAR LANDER, WYOMING
C,>~;wIITIc . aLL "LLLdd"', ""I"-... Y.
tion can be made. Mean dept even over a short rea-' ---' measure the trvp shpnr area of bed. I stress in variou, ywsbu
stages is needed.
h and slope, cn, pruaably do not .- -___-_ exerted on any local
lirect measurement of shear distribution of flow in a nat c nor+< of a bend a t various
Pattern of Flow in a Meaqzder Relatively few detailed mea
been published (Blue et al., Leliavsky, 1955, p. 96-100; Van
DYNAMIC AND FLO
. 19-23). The velocity distributions are intended both as additions to e of data and as illustrations of haracteristics of flow in bends in nels. Figure 4 shows downstream se componcnts of velocity a t the nd on Baldwin Creek near Lander,
Point velocities were measured by n t meter and horizontal angles by a
to the wading rod. Measured sed in conjunction with the city to compute the lateral
velocity. Measurements were
component is directed toward the ik or point bar near the bed, and
concave bank near the surfacc. requires, then, that surface water nward near the concave bank and bed water emerge a t the surface
irculatory motion in the cross-sectional of a channel, as clearly explained by
oving surface parcels es near the bed. The dual water parcel a
eam cross section, surface stream how that no single parcel of ompletely from the convex to
The maximum cross-channel xceed perhaps two-thirds of in any given meander bend. the observation by Friedkin
oratory models that material one bank tends to deposit on a nstream on the same side of the
s (1941, p. 634) stated that in wide, rivers hclical circulation does not is our opinion that in rivers of large pth ratio the helical motion exists,
a single rotating cell involving the ne1 (Nemenyi, 1946). The existence cells of circulation in the Tamm from the Mississippi can be seen in
ibution of the lateral components of blishecl by Eakiii (1935). As in the
irculation cells of large magnitude k down into smaller cells. The cell and the “trade-wind” cell of , p. 610) are directly driven by s which we consider analogous to
IW CHARACTERISTICS 779
the direct helical motion in river bends due to the combined effect of frictional and centrifugal forces. Rossby’s “frictionally driven” circula- tion cell in the middle latitudes furnishes, in our opinion, an analogy to those circulation cells seen in the Eakin data of the Mississippi which rotate in a direction opposite to the helical motion expected in a single cell occupying the entire cross section.
It is well documented that the water surface is superelevated near the concave bank of a channel bend (Blue et al., 1934; Eakin, 1935; Leliavsky, 1955, p. 123; and others). The amount of the superelevation is proportional to
Ub grm
where ZJ is the mean velocity, w is channel width, r, the mean radius of curvature, and g the acceleration of gravity (Leliavsky, 1955,
This superelevation is a consequence of the curved path of water flowing around a bend and is a correlative of the helical motion.
-
p. 122).
Generalized Picture of Flow iit a Meander
Measurements in meandering streams and in curved flumes (for example Mockmore, 1944, p. 569) allow the construction of a generalized picture of the flow pattern in a meander (Fig. 5). The isometric view of the two principal com- ponents of velocity a t various positions in the bend show the main features. The scale is such that superelevation of the water surface in the bend does not show on the diagrams but is implied by the velocity distribution.
At the crossover or point of inflection of the bend (Fig. 5, section l), the cross-sectional shape is not completely symmetrical but is slightly deeper ncar the bank which was concave 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 the curving stream lines.
The velocity in a meander crossover is not symmetrically distributed. As would be expected, proceeding downstream from the axis of the bend the thread of maximum velocity is much closer to the concave bank than to the center of the channel. The high velocity, moreover, continues to hug this side through the point of inflection of the curve.
2
3
-+ 4
3
Generalized surface streamlines
5 Generalized velocity
d s t r i bution FIGURE 5.-kOKETRIC VIEW OF GENERALUED DIAGRAM OF FLOW 1)ISTRIBUTION I N A MEA Showing downstream (open parabolas with arroycs) and lateral (closely lined areas) cornpoll
velocity as vectors, and surface stream lines. All sections viewed from a changing position to the left above the individual section.
780
DYNAMIC AND FLO
ar and symmetrical channel cross
iistreain from the symmetrical cross
t any point occurs near the concave t downstream from thc axis of bend.
ents of water accelerate and a stream line. The maximum always associated with the
ess parallel to that bank.
?we1 Shape and Movetneizt iit Relation to the Flow Pattern
slight lack of congruence of stream-line ure with bank curvature leads to the
e bank downstream froin the axis thus accounts for the progressive movement of meander bends.
movement of river channels and the
transverse component middle of the stream. ich i t is derived is far
moving near the bed is same bank from which i t to the opposite bank. This the likelihood that much is not carried m a r the bed
w CH AIZACTF :RIST 'ICs 781
of the stream will be deposited a t a downstream location on the side opposite to the one from which i t was derived.
The plunging or diving motion of surface watcr near the concave bank is one factor tending to depress the point of maximum velocity below the surface near the axis of the bend. Bank friction also has a siinilar effect. Because the highest velocity gradient on both bed and bank occurs where the thread of highest velocity is nearest the boundary, this maximum velocity gradient is equivalent to maximum shear stress. I ts location accounts, then, for deepening of the bend reach near the concave bank. This scour, together with build- ing of the point bar, explains the triangular shape of the cross section.
Increasing the depth of water in the channel reduces the vertical velocity gradient. In a channel that is very deep relative to its width, the effect of bed friction becomes relatively small, and the velocity is nearly constant with depth except close to the bed. Thus the helical circulation becomes negligible. Flow in a curved cliannel of great depth and small width approaches potential flow-that is, the downstream velocity varies across the channel inversely as the radius of curvature of thc stream lines. This was demonstrated by Wattendorf (1935, p. 574) in a deep, narrow, curved flume. I n the absence of the circulation in cross section the thread of high velocity hugs the convex bank and does not cross the channel. Thus the mechanism for building the point bar is probably absent in channels of extremely small width-depth ratio.
&fed of Flow Pattern o n DepositiotA and Erosion
The vigorous cross currents near the bed can transport considerable quantities of bed material from the concave to the convcx side of the channel. On a meander bend of Baldwin Crcek (Fig. 4) depth-integrated samples of suspended load were taken a t each of five verticals spaced across the stream and repeated in each of five cross sections along a meander wave. A nearly uniform concentration of sediment existed across the section in a cross- over. I n the sections located in the curved reach the concentration increased markedly near the convex bank.l A similar distribution of sediment was observed by Eakiii (1935,
Dr. John P. Miller assisted the senior author in obtaining these measurements.
782 1,EOPOLD AND WOLMAN-RIVER MEANDERS
p. 4’11) in his samples taken at Tamm Bend on the Mississippi River. This increased concen- tration near the convex bank is attributed to
The path of particles which ar building point bar has, of cour component across the stream c
Mean elevation sutf of streambed, 1953
Large scala cross.sectlon of StratiEraphy in relation to pmfi1.S
FIGURE STRATIGRAPHY OF POINT BAR IN RELATION TO MEASURED CROSS-SECTION PROFILES DURING SIX YEARS OF OBSERVATION, WATTS BRANCH, 1 MILE NORTEWEST
OF ROCKVILLE, MARYLAND Key to stratigraphic units is as follows: A Gravel, mostly 3-8 mm with considerable 8-20 mm and a few 64 mm B Olive-gray clayey silt, with organic matter, small mica flakes C Orange-brown, mottled sandy silt with some clay, lenses of leaflike organic mterial, mica D Coarse sand, brown-stained, with pebbles up to 8 mm, some fine roots in sit%, lenses of silt E Brown sandy silt gradually changing upward in places into fine sand F Fine sand with some silt
in a spirally wcldrd pipe tcnds to increase the ability of a flow to transport sediment over
the channel bed will be deposi bar a t an elevation considera
similar dimensions and boundary roughness. fine material will be carried This effect of curvature on transport is in than will coarse material, accord with the added energy losses accompany- would show on the average ing increased curvature.
pal mechanism for the building of a point bar. over the coarse material on the s
DYNAMIC AND FLOW CHARACTERISTICS 783 X
784 LEOPOLD AND WOLMAN--RIVER MEANDERS
tion, there is no doubt that deposition occurred up to the level of the water surface at bankfull stage. I n the same period deposition over the flat flood plain by overbank flow has been too small to measure. These observations appear to confirm the authors’ (Wolman and Leopoldt 1957) hypothesis that point-bar building i, the primary process of flood-plain developmens in flood plains of this type.
Bank erosion is greatly influenced by wetting of the bank materials. Arroyos cut in fine-grained alluvium experience most bank cutting after, not during, flow. The wetting causes later slumping (Leopold and Miller, 19.56, p. 5). Bank erosion is also enhanced by return seepage of water which inftltrated the banks during high flow. Upon lowering the stage the balancing pressure of the water in the channel is released, and the banks slump or collapse (Inglis, 1949, pt. 1, p. 152). A study of bank cutting in Watts Branch near Rockville, Maryland, showed that a combination of bank wetting and ice-crystal formation promoted the greatest bank erosion (Wolman, 1959, p. 214). Although the largest discharges occurrcd in summer, the winter provided more thorough soil wetting which, in combination with freeze and thaw, led to maximum bank erosion.
As bank erosion occurs in the bend of a mcander, over a period of timc it is usual €or a n approximately equal amount of deposition to occur on the opposite bank. This general equality of deposition and erosion is the reason width and cross-sectional area remain about the same as the channel moves laterally across the flood plain.
MEANDER MECIIANICS AND PIIYSIOGRAPHIC PROBLEMS
General Discztssioit
The vagaries of nature provide cndless opportunities for perturbations in thc flow- local hank erosion, chance emplacement of a bouldcr, fallen trees, or blocks of other vegetation-any one of which would alter the path of a straight channel. Thus one need hardly inquire why a stream channel is not straight. On the other hand, a raiidom succes-
merely to result in random bends of different patterns. Although this situation describes many channels, the existence of beautifully symmetrical meander bends and the remarkable similarity of bcnds in rivers of different sizes
sion of chancc pcrturbations iiiight bc expccted
and physiographic settings must be e There is not yet available any ical principle which explains characteristic patterns comm channels. In the absence o principle, however, attempts
size of bends is related to the size of the
Initiation and Development of Mea
A large body of experience and litera river regulation has been built up by Eu engineers on rivers in Europe In Africa. This experience has c greater span of time and of field c has American practice. Regard Leliavsky recently summarized concepts developed (1955, esp. p. Apparently the consensus of these that the effect of helical flow is the factor. Leliavsky expresses i t (p. 128) a
“For some reason or other, a small abra
have taken place. The water alongside the eroded portion of curved trajectory and develop centrifugal force. This force, in t local helicoidal current, which int abrasion and works its way deep the shore, . . . until the whole
material accumulates on the opposite ba then, is the birth of a meander.”
Prus-Chacinski (1954) also argues cal flow is the basic mechanism 1 meandering. He s ducing an artifici to the first bend, ’ kinds of second successive bend w
circulation pattern are quite persi through several successive bends. The ‘ of meandering Prus-Chacinski ascribes disturbance which produces an initial sec circulation.
It seems clear that helical important role in the process
trate shear against the concave promotes bank caving and channel m Even denying that helical flow exist rivers, Matthcs’ concept of meander inent is closely allied with that just m
point bar. A building point
786 LEOPOLD AND WOLMAN-RIVER MEANDERS
and turbulence, and his argument showed keen insight and understanding of field conditions (1949, p. 86-88). Bu t the formula has the disadvantage of making meander length dependent on wave amplitude; in our opinion, measurement data do not demonstrate this.
Although a simple comprehensive expression is still wanting, i t appears that the forces determined by the velocity distribution, including the helical circulation, are all that is necessary to account for (1) the shape of the cross section in a meander, (2) the depositional and erosional pattern, and (3) the progressive down-valley migration of the meander. These observations are not new, but i t is important to emphasize the following idea. Although point-bar formation and associated erosion of the opposite bank are necessary if a straight channel is to develop curves, the concept of helical flow, as Leliavsky (1955, p. 128) recog- nized, does not seem to explain how the second- a ry circulation determines the characteristic dimensions or proportions of meandering channels. The existence of meanders on glacier ice also implies that erosion and deposition may be a collateral, not the governing principle of meander development and movement. Because the hydraulic or mechanical significance of the pattern of curvature is closely tied to the fundamental physiographic questions, we consider this aspect of meander mechanics in thc following section.
Problem of Chanrccl Equilibrium
The preceding discussion was concerned principally with the initiation and development of a meander, elements that can be discussed in terms of a short reach of river or a single meander wave. There are a host of broader problems of channel adjustment to external controls which might be thought of as physio- graphic problems for want of a more specific term.
To begin, one might ask how meandering of a channel relates to the fundamental process of stream adjustment and stream equilibrium. It is generally believed that channel equilibrium is constantly approached, although rarely attained, by a process of continual adjustment. To use the words of Rubey (1952, p. 129),
“ . . . . with changing conditions, the stream is constantly cutting or filling and modifying its slope, velocity and cross section so as evrntually to accomplish the imposed work with the least expenditure of energy.”
If, indeed, a principle of least work volved-for i t is not y the development of a help accomplish this objec of gradient achieved b relative to a straight chann of a n excess of energy? Or, as was once is meandering the aimless wandering channel too sluggish to accomplish an of erosion? Does a river reach a stage a vertical erosion is negligible and thence its excess energy on lateral erosion by me ing?
Under what changes of conditions w river change its pattern from meande nonmeanderiiig or vice versa? What wo the effect of an increase or decreas charge or in sediment load from the basin?
Opinions on some of these quest been published, but data or measure meager. Existing data may answer a question but do not explain why the result was obtained. I n the followingp some of these questions are considered with related observations from Results of recent work, where cited, and we suggest what some of the directions in which needed.
Many of these queries relate to fundamental question-what has the ing pattern to do with energy expe
Inglis (1949, p. 158) states that ,
“Meandering is Nature’s way of d excess energy during a wide range of conditions, the pattern depending on material, the relation between discha (load) ;?nd the rate of change of charge.
Schoklitsch (1937, p. 149) earli what appears to be the same idea, tha formation
“might be due to the fact that the stretches is too great and is not in eq ;he size of the bed-sediment grains.”
of energy expenditure. Whe slope is closely related to t on the bed, there are other Hack (1957, p. 61) confirms generally held belief that controlled to a great extent and discharge, but to wh shape enters is still unclear.
Water-surface slope of a river is a
MEANDER MECHANICS AND
131), channel shape may adjust ith slope. At constant discharge if
cipal effect of bed particle size is on roughness or frictional drag, then
rms of drag need to be considered. inel resistance is materially influenced rag of various kinds, pools and riffles,
dunes, and channel curvature. is reabon from the hydraulic standpoint
ha t meandering may in part be a frictional drag and thence energy ncrgy loss occurs per unit of length than in a straight channel of the
and cross section, owing to eddyir.~, circulation, or increased rate of
se eddy losses result irom deflection er to a new direction as i t moves
a curve or bulge. It is known from c experiments in pipes (King, 1954,
that energy loss first decreases and eases with a decrease in the ratio
rvature/pipe diameter. Although urved channel offers greater resistance
than a straight one, the minimum in resistance is about 40 per cent, and lies within a narrow range of the ratio s of curvature/diameter when that a value of 2 to 3. pointed out earlier that in meandering
the comparable ratio, radius of e/channel width, is relatively conser- alues of this ratio also tend to fall in
) suggests an explanation crease in resistance a t this that, as radius of curvature
mes about 2 to 3 times the eddy or zone of reverse stream from the bulge or
the appearance of the eddy the local width is constricted,
is a local increase in effective radius re and a net decrease in energy loss. extent tha t further work confirms tion that the modal value of this the range 2-3, meanders tend to be
rized by a geometric pattern which ns to offer the smallest energy loss of
figuration of curved channel. The nce of this observation is unknown,
suggests that some principle related to conservation does operate in the
ing that a bend actually does tend to a configuration such that the energy to the bend is a minimum, i t does not
PHYSIOGRAPHIC PROBLEMS 787
follow that the total energy expenditure has been minimized. When a given discharge falls through some specified vertical distance, a certain amount of energy is transferred from potential form to some other form. If the water does not accelerate (velocity remains about constant downstream), then this potential energy is expended as work or heat. The same amount of energy is spent whether the water moves in a straight channel on a steep slope or in a longer curved channel at a smaller gradient. The energy expenditure per foot of channel length is smaller in the longer curved channel than in thc shorter straight one. The question, then, is bow much energy is utilized per foot of cliannel length and in what form i t is used.
This energy may be spent in moving particles of debris, or it is otherwise dissipated into heat. The energy may be spent in removing particles from the bank and transporting them (bank erosion) or in transporting bed or suspended particles. If this energy is concentrated in such a manner that more of i t goes into mo\7ing particles from one place than from another, then local scour will occur there and deposition elsewhere. For a channel to be in equilibrium scour must balance fill within the reach in question, and, further, the energy must be so expended that the net amount of debris coming into the reach must equal the net amount carried out of the reach.
I t is generally agreed that meandering channels are often stable or in quasi-eyuilib- rium. They may be so even though, over a period of time, a meander wave moves gradually downstream. The slope, discharge, and channel shape tend to become adjusted so that the above requirements are fulfilled. Adjustments in channel shape occur through erosion or deposition which in turn affect velocity, depth, and width. Specific hydraulic requirements relating depth, slope, velocity, and total resistance, including rcsistance offered by bed configuration (form resistance), bed and bank grains (skin resistance), and channel curvature (a particular kind of form resistance) must also be maintained.
Meandering is one way in which erosion and deposition may change the distribution, location, and amount of energy expenditure per unit of channel boundary. By lengthening the channel between two points a t different elevations, the energy expenditure pcr foot of length is reduced. By bank erosion, point-bar building, and by scour and al, the channel
788 LEOPOT,D AND
cross section is adjusted, and the energy expenditure is redistributed. Presumably abrupt discontinuities in the rate ol energy expenditure in a reach of channel are less compatible with conditions of equilibrium than is a more or less contiiiuous or uniform rate of energy loss. I t may well be that a meandering channel is most stable when the energy loss due to curva- ture is a t a minimum. Such a conclusion is perhaps implied in the modal distribution of values of the ratio of radius of curvature to width in natural streams.
If we view curvature as simply one method of a1 tering the distribution of energy cxpendi- ture in a given length of channel, i t is clear that the pattern of meandering will respond to changes in discharge and load. I t is well known that an increase in discharge in a meandering channel will incrcasc the channel width and will increase the size of the meander bends. Decrease in discharge will gradually reverse the process.
In the natural rivers, geologic and strati- graphic evidence clearly demonstrates that during late Pleistocene time an incrcasing discharge markedly decreased the gradient of Oster-Dal River and enlarged the width of channel as well as the size of meander beds (Wenner and Lannerbro, 1952, p. 108). During this gradual degradation of the valley floor the meandering pattern persisted. Thus the meander pattern where i t exists in nature appears to he a persistent attribute of thc river.
Change in load will cause aggradation or degradation and thus changc in channel slope and sise of bend (Friedkin, 1945, p. 7-9); an increase in slope will produce an increase in meander length aid amplitude. Schoklitsch (Shulits, 1935, p. 644-616) arid Bagnold (1960) have postulated that a t high discharges sediment transport is a function of the rate c,; work done per foot of chaniiel length, or power intensity. In rt meandering channel in equilib- rium increasing tightness of bend (curvature) through its effecl on the rate of energy cxpendi- ture decreases the rate of transport. For equilibrium, then, a balance inubt be rnaintained between curvature and transport quite apart from any changc in intensity of energy l o s ~ brought about by :I change in length.
DIRECTION FOR FUTIJRE WORK
No wholly adequate explanation of meandcr- ing is yet available. Probably 110 single simple nicchanisin will suffice to explain all aspects of meandering. A1 hinueh there is general agree-
N-RIVER MEANDERS
nient on the manner in which bank and point-bar formation are related orderly transfer of sediment which is meandering, no physical or mechanical p has been identified which explains quali the size and geometry of meander cu
There is need to investigate osc forces which might explain more a the manner in which an initial bulge sion in a stream bank leads to a sy reversal of curvature.
Although the available velocity distribu in channel bends do permit general descrip of the flow, many characteristics, in orientation and position of the helical are as yet poorly defined. More detai ureinents in natural channels are re define the loci of energy losses and the to flow resistance and localized e bed and banks.
The way in which a natural chann utes the energy loss as between bo friction, form resistance, curvature transport is little understood. Withou understanding it is vir explain or predict the behavior of a mea channel. Although some studies of losses in curved channcls have been (Allen, 1939; Leopold et al., in press), i t to us that laboratory as well as field stu the distribution of energy expendit straight and curved alluvial chaii needed. It would be par map the distribution and ary shear in bends of d' with similar cross sectio observations must in turn mechanics of sediment transport in channels of different patterns,
to the pattern of flow and its relation and deposition is in the area of the of sediment transport. Present th inadequate to explain the transport geneous sizes under the variety of found in nature. The stress needed and maintain motion is probably di scattered rocks on a sand bed, uniform from sand to cobbles, graded cobb boulders without sand, or boulders on a cobble bed. At objective criteria are needed ability of varying bank materi erosive stresses. A quantitative e the meandering process will requir the erosive stress produced by tangeu of the flowing water and the co resisting stress provided by the bank
The principal unsolved problem with
DIRECTION FOR
ory studies. A few fundamental concepts fy a vast amount of empirical observa- the observations include those critical
BIBLIOGRAPHY
, 1939, The flow in a tortuous stretch of r and in a scale model of the same: Jour.
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7, Studies of longitudinal stream
., 1937, The relationships between belts, distance between meanders on
tream, width, and discharge of rivers plains and incised rivers: Ann. Rept. Central Board of Irrigation (India)
Central Waterpower Irrigation and ion Res. Sta., Poona, Res. Pub. 13,
, 1954, River-bed
, 1954, Discussion r-bed scour during eers Trans., v. 119,
FUTURE WORK 789
Leliavsky, Serge, 1955, An introduction to fluvial hydraulics: London, Constable and Co., 257 p.
Leopold, I,. B., in press, The gravel bar: some observations on its role in channel morphology: Internat. Geol. Cong., Copenhagen
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Leopold, L. B., and Wolman, M. G., 1957, River channel Datterns-braided, nieanderiw.. and straight: U. S. Geol. Surveir Prof. Pape;Z82B,
Leopold, L. B., Bagnold, R. A., Wolman, M. G., and Brush, L. M. Jr., in press, Flow resistance in sinuous and irregular channels: U. S. Geol. Survey Prof. Paper 282
Ludin, A., 1926, Influence of the rotation of the earth on rivers: Die Wasserkraft, v. 21, p. 216, partial translation by Sam Shulits, 1958
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Mockmore, C. A., 1944, Flow around bends in stable channels: Am. SOC. Civil Engineers Trans., v. 109, p. 593-628
Nemenyi, P. F., 1946, Discussion of Vanoni, V., Transportation of suspended sediment by water: Am. Soc. Civil Engineers Trans., v. 111, p. 116-125
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I’rus-Chacinski, T . M., 1954, Patterns of motion in open-channel bends: Assoc. Internat. d’Hydrologie, pub. 38, v. 3, p. 311-318
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Stommel, H., 1954, Circulation in the North Atlantic Ocean: Nature, v. 173, p. 886893
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565-598
LEOPO1,U AND WOLMA
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N-RIVER MEANDERS
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U. S. GEOLOGICAL SURVEY, WASHINGTON, D THE JOHNS HOPKINS UNIVERSITY, BALTI MD.
MANUSCRIPT RECEIVED BY THE SECRETARY OF SOCIETY, OCTOBER 29, 1959
PUBLICATION AUTHORIZED BY THE DIRECTOR, GEOLOGICAL SURVEY
APPENDIX
APPENDIX. SHAPES OB MEANDER WAVES IN ALLUVIAL P L A ~ S
791
APPENDUC.-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, Kans.. . . . . . . New R nr Brawley, Calif.. . . . . . . . . , . . . . Missouri R nr Buckner, Mo.. . . . . . . . . , . . Missouri R nr Buckner, Mo.. . . . . . . . . . .
Kansas R nr Eudora
3 Kissimmee R nr Okeechobee, Fla.. . . . . . . Arkansas R nr Mulvane, &CIS.. . . . . . . . . , Colorado R nr Blythe, Calif.. . . . . . . . . . . .
San Joaquin R nr Patterson, Calif.. . . . . . . James R nr Forestburg, S. D.. . . . . . . . . . . James R nr Clayton, S. D.. . . . . . . . . . , . . . Souris R. . . . . , . . . . . . . . . . . . . . . . . . . . . Missouri 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 K nr Belgrade, Neb. . .
Eleva- tion (ft)
110 1200 150 700 690
790
18 1190 270
35 1220 1190 1480 690
88+
100
4810 4812 1700 1650
Range in width
(ft)
250-550 80-160 70-90
350-1550 550-1950
220-800
85-155 220-360 450-1650
120-280 60-90 80-150 30-60
450-1150
2 50-450
400-12oc
180-300 110-200 60-130
Mean width (W) (ft)
S O * 260* 250*
1850* 1600 *
Meander length
(A) (ft)
6700 1350 1475
14,600 37,100
1000* 12,750
180* 1050 500* 5650
1050* 25,300
400* , 2400 160* 2220 230* 1550 l50* 1100
1350* 24,400
340* 280* 200*
2700 1950 3050
Arc dis- tance (ft)
8400 1700 1750
24,300 41,800
14,400
2900 7900
33,300
4550 3600 2600 2500
29,000
10,800
18,700
6100 3300 4550
Mean radius (4 (it)
2050 360 210
3150 15,550
4300
205 1350 7300
610 450 265 160
6000
2300
2130
850 410 7 50
implitude ( A 1 (it)
2150 400 400
8850 7700
2850
1250 2150 8900
1600 1275 900 720
7 100
3200
7000
2300 1200 1450
Sinu- osity
1.25 1.26 1.18 1.66 1.13
1.12
2.76 1.40 1.32
1.90 1.62 1.68 2.27 1.19
1.42
1.64
2.26 1.69 1.49
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, Idaho Menan Buttes, Idaho Belgrade, Neb. Belgrade, Neb. *** Map 6, miles 92-102
p \.mw, II, ~ ~
, ^ * A
Little Pipe Cr n r Westminster, J l d Baldwin Cr nr Lander, Wyo.. . . . . . . . . . . . Little Sandy nr Elkhorn, Wyo.. . . . . . . . . .
Baldwin Cr nr Lander, Wyo. . . . . . . . . . . Buffalo Fork a t Black Ranger Sta.. . . . . . .
Mississippi R nr Smithland, La.. . . . . . . . . Mississippi R nr Lake Providence, La., . . . Mississippi R nr Rosedale, Miss.. . . . . . . . .
Model, U. S. Waterways Expt. Sta.. . . . . .
Model, U. S. Waterways Expt. Sta.. . . . . .
New Fork nr Pinedale, Wyo. (Hailstont Reach) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Model, U. S. Waterways Expt. Sta. . . . . . .
Moyen R nr Castets, France. . . . . . . . . . .
.I
2
Coosa River.. . . . . . . . . . . . . . . . . . . . . . . . . . Kansas River. . . . . . . . . . . . . . . . . . . . . . . Red River. . . . . . . . . . . . . . . . . . . . . . . . . . . . Henrys Fork, nr Menan, Idaho. . . . . . . . . Sacramento River. . . . . . . . . . . . . . . . . . . . . .
Kissimmee River, . . . . . . . . . . . . . . . . . . . . . Sacramento River.. . . . . . . . . . . . . . . . . . . . . James River.. . . . . . . . . . . . . . . . . . . . . . . . . . Souris River, N. D. . . . . . . . . . . . . . . . . . . . . Souris River.. . . . . . . . . . . . . . . . . . . . . . . . Souris River., . . . . . . . . . . . . . . . . . . . . . . . Souris River.. . . . . . . . . . . . . . . . . . . . . . . . Red River, La.. . . . . . . . . . . . . . . . . . . . . . .
15 85
1 45
440 780 100
4810 85
18 120
1220 1610 1590 1560 1430
40
10-1s 10-18 10-16 70-115 12-24
4900-2180 2200-5800 4300-2440
48-76 2-4.8
1.2-3.2
2.3-3.0
627-1120 650-400 850-170 500-1000 350-150 600-220
130-110 700-350
12 14 13 80 15
3330 3500 3210
62 3.3
1 . 6
2.6
870 520 510 750 250 410
120 525 110 140 140 120 100 600
145 185 248 990 165
65,000 53,500 40,300
7 45 34
28.2
29.5
9040 8916
16,400
2700 6100
1100 8340 2220 1800 1840 840
1140 7400
10,000
192 245 273
1230 205
75,400 75,800 47,000
960 43.5
36.0
34.2
12,200 18,500 26,100 17,500
5500 11,000
2100 11,500
2600 3200 4000 1600 2400 8800
35 45 76
308 43
15,700 9900
11,200
163 8.9
7.2
7 .O
2000 1700 3150 2000 650
1700
225 1400 600 440 480 220 300
1640
58 72 55
310 48
19,850 11,400 11,760
225 13.0
9.5
8.5
3760 6700 8550 5000 2000 3700
600 3600 600
1040 1240 500 800
2100
1.32 1.32 1.10 1.24 1.24
1.16 1.42 1.17
1.29 1.28
1.28
1.16
1.36 2.06 1.59 1.75 2.04 1.80
1.91
1.17 1.78 2.17 1.90 2.10 1.19
1.38
Plane-table map, authors Plane-table map, authors Plane-table map, authors Plane-table map, authors Plane-table map Leopold and Miller *** Maps 38,38, miles 770-785 *** Maps, 26, 27, miles 550-565 *** Maps, 19, 20, miles 387-396
Plane-table map, authors Meandering of alluvial ) Test 4
rivers, Friedkin, 1945 1 Plate 41
\:yzie129
1 gEe337 Fargue, 1908 Riverside, Ala. Eudora, Kans. Campti, La. Menan Buttes, Idaho Glenn and Llano Seco, Calif. Okeechobee N.W., Fla. Ord Ferry, Calif. Forestburg, S. D. ** Sheet A, mile 12, 1926 ** Sheet A, mile 40, 1926 ** Sheet Cm miIe 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
*** Mississippi River Commission, 1935, Maps of the Mississippi River, Cairo, Ill., to the Gulf of Mexico, La.; scale, 1:62,500: Vicksburg 1 1926
