modeling high sinuosity meanders in a small flume · modeling high sinuosity meanders in a small...

Ž .Geomorphology 25 1998 19–30

Modeling high sinuosity meanders in a small flume

Charles E. Smith )

730 Applause Pl., San Jose, CA, 95134, USA

Received 10 September 1997; revised 9 November 1997; accepted 6 December 1997

Abstract

Meandering channels with exposed point bars and sinuosities near 2.0 can spontaneously develop in a mix ofdiatomaceous earth and kaolinite clay. The experimental streams that were studied were as small as 4 cm wide. They showedmany of the characteristics found in large meandering streams such as migration of channels, formation of point bars, clayplugs, chutes and bar scrolls. Most of the meander series began as a first sharp bend that induced subsequent bends downstream. Sediment transport, specifically bed load, together with the slope of the floodplain, were the dominant influences, ifnot the cause of the channel instabilities that led to channel migration and bend formation. Bank cohesion allowed migratingchannels to assume sinuous shapes and to maintain fairly uniform widths. The experiments were conducted in a small flumeand used simple equipment. The use of light, fine grained materials in flume experiments may prove to be valuable inlearning more about the conditions of soil, slope and flow which produce various meandering planforms. q 1998 ElsevierScience B.V. All rights reserved.

Keywords: meanders; flume studies; modeling; braided streams

1. Introduction

Examples of meandering patterns are commonamong natural streams, but the reproduction of welldefined, highly sinuous channels in laboratory flumeshas proved elusive. Though much knowledge hasbeen gained from earlier flume studies, such as

Ž . Ž .Tiffany and Nelson 1939 , Friedkin 1945 andŽ .Schumm and Kahn 1972 , the resulting channel

patterns have been acknowledged as being generallystraight with a sinuous thalweg.

Perhaps the most successful recent attempt tomodel high sinuosity river meanders has come from

Ž .Jin and Schumm 1986 . They used a floodplain

) E-mail: [email protected]

made of a base of sand and topped with a layer ofkaolinite clay and fine sand. Because of theirprogress, experiments were developed to exploreadditional techniques that could be useful in model-ing meanders in a laboratory environment. This pa-per presents the results of experiments that simulatedthe formation of well defined, highly sinuous mean-ders in a small flume using light, fine grained materi-als.

2. Experiments

The experiments were conducted in a flume 3 mlong and 1.2 m wide that had wooden sides, aplywood base and was lined with a removable plastictarpaulin. The frame of the flume had a hinge near

0169-555Xr98r$19.00 – see frontmatter q 1998 Elsevier Science B.V. All rights reserved.Ž .PII S0169-555X 98 00029-4

( )C.E. SmithrGeomorphology 25 1998 19–3020

the midpoint for adjustment of the slope. The waterwas circulated by small pump, with discharge levelsbeing determined by volume per time measurements.Sediment combinations used in the experiments haveincluded rock flour with kaolinite, cornstarch, corn-

Ž .starch with calcined white China clay CWC , di-Ž .atomaceous earth DE , and perhaps the most suc-

cessful to date, diatomaceous earth with calcinedŽ .white China clay DEqCWC .

The experimental materials all had a powder-likeconsistency. Particle sizes for kaolinite, rock flourand CWC were distributed in the 4 to 10 mm range.The cornstarch grains averaged 12 mm in diameterwith a 4 mm standard deviation. The range of sizesfor DE was considerably wider than the other materi-als. The mean diameter for separate particles wasestimated to be 35 mm with a standard deviation of14 mm. Because of surface roughness, the individualdiatoms could interlock to form loose clusters thatwere 100 mm or more in size.

The sediments were chosen to scale down thealluvial properties needed for meanders to form inthe reduced flows of a flume. The intent of the lowdensity and fine grain size of the materials was tofacilitate transport, especially to the shallows of in-cipient point bars. To ensure enough cross-sectionalstrength in the channel to support curves and lateralreaches, bank cohesion is needed. A binding agentsuch as clay is necessary, as was indicated by the

Ž .studies of Schumm 1960 . But the cohesion mustnot be so strong as to stop erosion entirely. For smallflows, CWC clay seemed a good choice, being lesssticky than other types of kaolinite.

For comparison, the wetted densities of selectedalluvial combinations are shown in Table 1. Wetted

Table 1Wetted densities of experimental sediments

Material Wetted MixtureŽ .density %mass of tot.

specific.gravity

water 1.0 yrock flourqkaolinite 1.82 kaol. 15%cornstarchqCWC clay 1.28 CWC 20%DEqCWC clay 1.35 CWC 30%Ždiatomaceous

.earthqclay

Table 2Experimental parameters of the stream flow

DEq Cornstarchq Rock flourqCWC CWC kaolinite

Ž .Q mlrs 9 35 45slope 0.015 0.020 0.025Ž .u cmrs 15–20 18–22 18–25Ž .d cm ;0.5 ;0.5 ;0.7

Ž .'Froude a ur gd 0.68–0.91 0.82–1.00 0.69–0.95Ž .Ž .Reynolds a udrvisc 750–1000 900–1100 1250–1750

density is defined as the mass per volume with wateradded to form the sediment consistency of an experi-mental run. Table 2 lists typical values for experi-mental parameters. Included are the flow, slope, and

Ž .velocity levels, Q, slope and u , which produced themost sinuous patterns with well defined channels.The parameter d is the mean measured depth of thechannel and g is the constant for gravity. Thecorresponding Froude and Reynolds numbers havebeen calculated.

A typical run was started by mixing the sedimentcomponents in the desired ratios. The mixture wasthen premoistened and smoothed in the flume. Astraight initial channel was created, the slope ad-justed and the water flow started.

The water was introduced without an initial bendat levels between 5 and 50 mlrs. For most runs,flow was kept at a constant, or bankfull level. Forsome tests it was decreased or increased for periodsof days to study the effects on planform. An hour ortwo after the start of an experiment, after the streamhad established its initial channel, moist sedimentwas added at the stream head by hand twice per dayat roughly 12-h intervals. It was arranged in smallheaps to either side of the water inlet so as to erodegradually into the stream over several hours.

The intermittent method of the sediment supplyhad no observable effect on whether the streammeandered or not for the featured alluvial combina-tions. In several instances, a meandering patternrequired only a few hours to develop. Under suchconditions of continuous observation, the sedimentflow could be kept reasonably constant. It was muchmore important that the sediment levels within thestream were sufficient to promote channel instabilitywithout causing the stream to overflow its banks.

( )C.E. SmithrGeomorphology 25 1998 19–30 21

The transported sediment that reached the foot of theflume was allowed to move down the drain with thewater to a collection tank. By measuring the amountof sediment in the collection tank, the typical flow ofsediment was shown to average 10 mlrh for theexperiment shown in Figs. 1 and 2.

While some of the runs needed only a few hoursto begin meandering, others, such as those usingrock flour, required two to three weeks. Even for theDEqCWC mixture, a week or more might be neededfor the stream to create a flood plain of sufficient

Ž .breadth 4–8 channel widths and consistency tosupport meander formation. Given a sufficient slopeand sediment supply, any of the experimental mixeshaving enough cohesion to maintain a well defined,single thread channel, were likely to produce highsinuosity bends.

3. Results

The mixtures, rock flourqkaolinite, cornstarchqCWC, and DEqCWC were all effective in pro-ducing well defined channels and meanders in asmall flume. The most effective at producing a seriesof meanders has been the DEqCWC combinationmixed in a ratio of 5:2 by weight. The slope wasnear 0.015. Experience indicates considerable leewayin the allowable CWC percentage. Ratios of 5:1 to2:1 worked well for discharge levels of about 8–25mlrs. Photographs of the various stages of one ofthe more successful runs are shown in Figs. 1 and 2.

A distinguishing feature of these experiments wasthe development of deeps and shallows related to thechannel curves and crossings. The pools of the curvesare the deepest parts of the system as shown by theprofile in Fig. 3a. The related map, Fig. 3b, showsthe locations of the pools.

Along the channel, the average distance betweenpools was 15 cm. in the example shown. Given achannel width of 4 cm, the pool spacing is about 4widths, which somewhat smaller than the 5–7 widthsoften seen in rivers, as described in Leopold et al.Ž .1964 .

The measured cross sections of the channel areshown in Fig. 4a, with the respective locations inFig. 4b. The thalweg is offset from the centerline inthe direction toward the concave bank. The asymme-

try is not as pronounced as is usual for naturalstreams. In Fig. 4a a few morphologic features oftenassociated with meandering streams are indicated.Labeled are portions of abandoned channels, clayplugs, scrolls on points bars and an example of aconcave bank bench.

3.1. Formation of point bars

The build up of point bars resulted almost entirelyfrom bed load. Often, the channel was broader andshallower on its approach to a bend with the bulk ofthe sediment flow moving along the inside of theturn. The presence of the sediment train took upspace within the channel and shifted the water flowslightly toward the opposite bank, enhancing erosionthere. The process was similar to one that has been

Ž .described by Dietrich and Smith 1984 . As the outerbank eroded, even by a few millimeters, grains of thesediment could be seen coming to rest on the bar.Bank migration continued. The water flowing overthe edge of the point bar gradually became shalloweruntil particle movement on the margin of the bar wasno longer evident. Over time, the local water levelcontinued to subside leaving part of the new depositabove water. The mechanism by which the subsi-dence took place was not readily apparent. Possiblecauses could include channel deepening near the baror slight changes in slope because of local degrada-tion or aggradation. Bar deposition could be slowand steady but was often episodic in nature, manifestby the growth of dune-like structures along the barmargin.

For the pictured experiment, exposure of a cen-timeter wide lens of bar could take from less than anhour to as long as a week. Whether by dune orsteady accretion, bar stabilization was dependent onchannel migration away from the bar. The resultingdecreased flow of water over the bar was necessaryfor sediment adhesion and consolidation.

An example of a point bar slow in emerging isseen in Fig. 2c, indicated by the arrow. Most of thebar remained under water for approximately 120 h,even though it was covered by a steady flow ofsediment. Fig. 1b shows that the radius of the adja-cent bend of the channel eventually enlarged enoughto lower the water level slightly, halting sedimentmovement and exposing the bar.


Ž .Fig. 1. a,b,c Three photographs showing a series of meanders that formed in a mixture of diatomaceous earth and calcined white Chinaclay. After approximately 500 h, the flow was stopped to obtain profile data. Channel migration had slowed to virtually nothing after the

Ž .first 250 h. The planform appeared to be near a state of static equilibrium. In a , a broad view of the stream channel. Flow is toward theŽ . Ž . Ž .viewer. The measuring scale is 30.48 cm 1 ft in length. b shows a close-up view of the channel section under study. c provides a more

detailed view of the meandering channel. The flow goes from left to right.


Ž .Fig. 1 continued .

Suspended load, seen largely as water turbidity,did not contribute much to the overall height of thepoint bars. It did, however, help secure point barsediments already in place. Usually, as the flow ofwater over a new deposit ebbed, the deposit becamegrayer as clay gradually settled upon it—or perco-lated up through it. The clay appeared to give de-posits a hysteresis-like quality. The shear stressneeded to erode a stabilized bar was observed to behigher than that needed to create it initially. Erosionresistant deposits can more readily deflect the waterflow laterally across the valley slope, enhancingstream sinuosity. It is possible that suspended clay,moving into the sediment matrix, makes the pointbars more durable.

Evidence in the literature that supports the idea ofhysteresis, can be found in the studies of cohesive

Ž .deposits found in estuaries. Partheniades 1965 ;Ž . Ž .Parchure and Mehta 1985 ; Mehta et al. 1989 and

others discuss in detail the processes of depositionand consolidation for cohesive sediments. The fig-

Ž .ures in Nicholson and O’Connor 1986 clearly illus-trate how consolidation increases the resistance ofdeposits to subsequent erosion. Similar reports, pub-

lished specifically for river systems, are harder tofind. When mentioned, consolidation effects are dis-

Ž .cussed only briefly, as in Section 9 of Parker 1978 .The present experiments indicate that, in addition toerosion, transport, and deposition, consolidation pro-cesses play an important role in the evolution ofriver channel patterns.

Suspended load was important in the creation ofthe clay plugs that formed across sections of slackwater adjoining the main channel. Examples of clayplugs are labeled in Fig. 4b. Accretion was slow,taking a week or more to approach the water surface.Deposits were medium gray in appearance indicatinghigh clay and low DE content in contrast to the moretan shade of mixed sediments. Once in place, theplugs were tougher to erode than bars.

3.2. General scenario for meander formation

The general sequence of meander developmentfor DE, rock flour and cornstarch is shown in Fig. 5.The progression of the pattern is similar to that of anexperiment, involving surface tension meanders, de-

Ž .scribed in Davies and Tinker 1984 and illustrated


Ž .Fig. 2. a,b,c Shown are the earlier stages of the same experiment depicted in Fig. 1. They show in part the development of the meanderingŽ .pattern over time. a shows the experiment nine hours after start and two hours after the initial kink at the upper right had caused the next

Ž . Ž .bend downstream to form. b shows the channel four hours after the onset of meandering. c shows the planform about 120 h aftermeandering first began. The bar, indicated by the arrow, is still largely inundated. An additional 50 h were required for the bend to shiftlaterally enough for the bar to emerge above water.

by Fig. 2a–e in that paper. For the present experi-ments, the usual chain of events began with a reachof the initially straight channel drifting laterally as aresult of shoaling along one bank. The shoal itselfmight be from 7 to over 25 channel widths in length.A kink in the channel gradually formed that becamelarger and more acute as the flow eroded into thebank. The bank material, because of clay content,was resistant enough to deflect the flow in a crossslope direction, creating a growing bend opposite theinitial kink. The new bend in turn enlarged enough toinduce another bend down slope and so on. As thesequence grew, the bends increased in amplitude andwave length. Repeated trials indicated that only onekink was necessary to produce a meander sequence.

The observed shoaling mechanism may be con-nected with ‘highly elongated, partially beached

Ž .oblique dunes’ referred to in Parker 1996 . Parkersuggested that such dunes give rise to the scrolls

often seen on point bars. In the present experiments,dune-like structures were sometimes seen; a numberof point bars did have a scroll-like appearance.

3.3. More obserÕations for Õarious alluÕial combina-tions

In the cases of rock flour and DE the individualmeander loops tend to become asymmetrical in shape,but in the opposite sense to those of Davies and

Ž . Ž .Tinker 1984 and Parker et al. 1982 . They areconvex in the upslope direction but still tend tomigrate downslope.

In the pictured experiment with DEqCWC,stream migration was most rapid right after themeander sequence began to form, but slowed gradu-ally over time. It took about 180 h from the forma-tion of the initial kink to what looked to be full


Ž . Ž .Fig. 3. a The long thalweg profile for the featured meander sequence. b The planimetric map showing the locations of the deeps markedA to H.

sequence amplitude. At this point, even significantincreases to sediment load only resulted in slight,localized changes. Sediment continued to flow but

without much effect. Given the constant flow ofwater, the stream appeared to be in a state near staticequilibrium.


Bends in rock flourqkaolinite formed much moreslowly, even though higher flow levels of 45 mlrswere used. Steeper slopes were required to create awell defined channel, typically around 0.025. Manybends had an abrupt, sawtooth appearance. Neverthe-

less, after a month, a loop with a goose neck formedbefore the run was discontinued. Though no alternatebar series were seen, the bed in places containedshallow pools spaced from 1 to 5 channel widthsapart. They were not as deep as pools at bends and

Ž .Fig. 4. a Channel cross sections measured at various points along stream. The direction of water flow is into the page. The average widthŽ .to depth ratio is around 5, well below the minimum value of 10 considered necessary for alternate bar formation. b The locations of the

cross sections. They where chosen to show channel shape along typical straight and curved reaches. Also labeled are structures of interestoften associated with meandering streams.


Ž .Fig. 4 continued .

looked similar to those described by Keller andŽ .Melhorn 1973 . The presence of the pools did not

have a perceptible effect on planform changes.

Fig. 5. The sequence of meander development most often seen inthe experiments.

CornstarchqCWC produced interesting results,although it is recommended that chlorine bleach orsimilar agent be added to the water to suppress thegrowth of mold. Flow rates were usually around 35mlrs. Bends drifted downslope rapidly, at up to 10cmrh, often becoming more compressed as theymoved. They tended to be rounded in shape and welldefined in cross section. Because of the rapid migra-tion, only two or three point bars formed before theloops cut-off or disappeared at the foot of the flume.A virtually straight channel often would result. Aftersome hours, a new bend would begin as a reachshoaled again to one side.

3.4. The effect of the slope of the floodplain onchannel pattern

The experiments made clear that sufficient longi-tudinal slope was necessary for well defined mean-dering channels to form. The slope of the floodplainneeded to be steep enough to promote channelizationand channel instabilities, but not so steep as toinduce braiding. This requirement agrees with the

Ž .experimental findings of Schumm and Kahn 1972Ž .as well as the field work of Schumm et al. 1972

Ž .and Martinson 1983 . After a considerable numberof trials, slopes of around 0.015 were found to workwell for lighter the sediments, such as cornstarch anddiatomaceous earth. For denser materials, such asrock flour, steeper slopes of around 0.020 to 0.025were usually needed. If the slope were too shallow,the resulting pattern could vary from a water flowthat spread out over the floodplain, to wide non-uni-form channels, to channels that were fairly welldefined and transported sediment, but were of lowsinuosity and prone to bifurcation.

3.5. The role of sediment flow in meander formation

In addition to the slope of the floodplain, sedi-ment flow provided the other key impetus for theinstability that led to meander formation in the flumeexperiments. The shoaling that produced the initialkink required ongoing deposition from a supply ofsediment, as did the subsequent growing point bars.When sediment no longer was introduced at theflume head, enlargement of meanders and evolution


of the planform soon slowed to imperceptibility.When sediment input was high enough, the channelwidth to depth ratio increased markedly and streambraiding resulted, similar to what was reported by

Ž .Stebbings 1963 . The most effective levels of sedi-ment flow covered 60% to 80% of the channel bedon straight reaches but tended to keep to the insidehalf of the channel around the turns. Migration couldtake place while channel definition was maintained.

Ž .Analytical treatments such as Parker 1976 , haveemphasized the primary role of sediment transport instream meandering and braiding.

The presence of periodic alternate bars with 10channel widths separation or less, often viewed as aprecursor to meander development, was notably ab-sent in the featured experiments. Deeps or poolscould almost always be associated with the curve ofa reach. The observed channel kinks may prove to beelongated forms of alternate bars. They occurredsingularly however, with no discernible periodicityalong the stream course. Moreover, the lengths ofresulting meanders were much shorter, without ap-parent correlation to the reach length of the initial

Ž .shoaling. In his field work, Ikeda 1989 has notedan absence of alternate bars in certain previouslymeandering streams that have been artificiallystraightened.

One explanation for the lack of recognizable alter-nate bar patterns during the experiments is the lowwidth to depth ratios of most of the stream channels.A value of 10 or less has been common for alluvialmixes with clay content of greater than around 15%.In trials with low levels of cohesion, however, ratiosof greater than 20 were observed. The sediment flowwas high in both volume and velocity relative toother test runs. Though a tendency to braid was oftenevident, regular, near-periodic alternate bar patternswere not obvious in any of these channels.

3.6. Effects of the leÕels of water flow

For DEqCWC, meander geometries were in ac-cord with the observations of Leopold and MaddockŽ .1953 and others. At flow levels of 9 mlrs, mean-der wave lengths averaged a little over 8 channelwidths. A value around 7 was common for theamplitude to width ratio. The values for Reynoldsnumber were calculated from stream velocity and

depth measurements and found to be in the 750–1000range. Alluvial meandering does not require fullturbulence to occur. The Froude numbers for allexperiments were subcritical.

A limited range of water flow occurred for whichthe 5:2 DEqCWC mix would allow defined chan-nels and meandering. If discharge were increasedenough, braiding would result. Discharges below 25mlrs produced fairly uniform channels. Point barsbuilt up and emerged readily. At levels around 35mlrs, meandering was still seen, but channel widthbecame much less uniform, varying by a factor of 4or more around bends. At still higher discharges, thestream course became even less well defined andsinuosity decreased. Multiple channels developed.The sediment cohesion was no longer sufficient toforce the flow into a single thread channel.

4. Conclusion

The experiments described in this paper, thoughlargely qualitative, demonstrate that high sinuositymeanders in alluvial media can be produced by smallstreams in a laboratory setting. Only simple equip-ment and combinations of low density, fine grainedmaterial are needed. The resulting planforms showmany similarities and several differences in compari-son to larger rivers. Similar attributes include scrolledpoint bars, concave bank benches, clay plugs, etc.Differences are found in meander shape, much lowerReynolds numbers and slopes that are steeper than inmost natural streams.

The experiments show how meanders can de-velop. They illustrate the importance of slope, sedi-ment flow and sediment cohesiveness in the forma-tion of meanders as well as in enabling a stream tomaintain a well defined channel. Future work shouldmore clearly quantify the role of cohesion in mean-der formation. Cohesion seems responsible for a‘hysteresis effect’ with regard to observed sedimenterosion, deposition and consolidation characteristicsas channels migrate. The lack of a strong tendencyfor periodic alternate bar formation in the experi-ments could use some additional analysis. Explana-tions involving channel width to depth ratios aloneare not sufficient.


Acknowledgements

The author wishes to thank Luna B. Leopoldwhose kind help and encouragement made this paperpossible. And thanks also to Gary Parker and ChrisPaola for their encouragement and discussions con-cerning the experiments.

References

Davies, T.R.H., Tinker, C.C., 1984. Fundamental characteristicsof stream meanders. Bull. Geol. Soc. Am. 95, 505–512.

Dietrich, W.E., Smith, J.D., 1984. Bedload transport in a riverŽ .meander. Water Resour. Res. 20 10 , 1355–1380.

Friedkin, J.F., 1945. A laboratory study of the meandering ofalluvial rivers. U.S. Waterways Experiment Station, Vicks-burg, MS, 40 pp.

Ikeda, H., 1989. Sedimentary controls on channel migration andorigin of point bars in sand-bedded meandering rivers. In:

Ž .Ikeda, S., Parker, G. Eds. , River Meandering. AmericanGeophysical Union, Washington, DC, pp. 51–68.

Jin, D., Schumm, S.A., 1986. A new technique for modelling riverŽ .morphology. In: Richards, K.S. Ed. , Proc. First Internat.

Geomorphology Conf. Wiley, Chichester, pp. 680–691.Keller, E.A., Melhorn, W., 1973. Bedforms and fluvial processes

in alluvial stream channels: Selected observations. In: Mori-Ž .sawa, M. Ed. , Fluvial Geomorphology. Publications in Geo-

morphology, State Univ. of N.Y., Binghamton, pp. 253–283.Leopold, L.B., Maddock, Jr., T., 1953. The hydraulic geometry of

stream channels and some physiographic implications. U.S.Geol. Survey Prof. Paper 252, 56 pp.

Leopold, L.B., Wolman, M.G., Miller, J.P., 1964. In: Freeman,Ž .W.H. Ed. , Fluvial Processes in Geomorphology. San Fran-

cisco.

Martinson, H.A., 1983. Channel changes of Powder River be-tween Moorhead and Broadus, Montana, 1939 to 1978, U.S.Geol. Survey, Water Res. Inv. Rept. 82-4128, 62 pp.

Mehta, A.J., Hayter, E.J., Parker, W.R., Krone, R.B., Teeter,A.M., 1989. Cohesive sediment transport 1: process descrip-

Ž .tion. J. Hydr. Eng. ASCE 115 8 , 1076–1093.Nicholson, J., O’Connor, B.A., 1986. Cohesive sediment transport

Ž .model. J. Hydr. Eng. ASCE 112 7 , 621–639.Parchure, T.M., Mehta, A.J., 1985. Erosion of soft cohesive

Ž .sediment deposits. J. Hydr. Eng. ASCE 111 10 , 1308–1326.Parker, G., 1976. On the cause and characteristic scales of mean-

Ž .dering and braiding in rivers. J. Fluid Mech. 76 3 , 457–480.Parker, G., 1978. Self-formed straight rivers with equilibrium

banks and mobile bed: Part 2. The gravel river. J. Fluid Mech.Ž .89 1 , 127–146.

Parker, G., 1996. Some speculations on the relation betweenchannel morphology and channel-scale flow structures. In:

Ž .Ashworth, P.J., Bennet, J.L., Eds. , Coherent Flow Structuresin Open Channels. Wiley, New York, pp. 429–432.

Parker, G., Sawai, K., Ikeda, S., 1982. Bend theory of rivermeanders: Part 2. nonlinear development of finite amplitudebends. J. Fluid Mech. 76, 457–480.

Partheniades, E., 1965. Erosion and deposition of cohesive soils.Ž .J. Hydr. Div. ASCE 91 1 , 105–139.

Schumm, S.A., 1960. The shape of alluvial channels in relation tosediment type. U.S. Geol. Survey Prof. Paper 352-B, pp.17–30.

Schumm, S.A., Kahn, H.R., 1972. Experimental study of channelpatterns. Bull. Geol. Soc. Am. 83, 1755–1770.

Schumm, S.A., Kahn, H.R., Winkley, B.R., Robins, L.G., 1972.Variability of river patterns. Nature 237, 75–76.

Stebbings, J., 1963. The shapes of self-formed model alluvialchannels. Proc. Inst. Civ. Eng. 25, 485–510.

Tiffany, J.B., Nelson, G.A., 1939. Studies of meandering of modelŽ .streams. Trans., Am. Geophysical Union Part IV , pp. 644–

649.

modeling high sinuosity meanders in a small flume · modeling high sinuosity meanders in a small...

Documents