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Theories of stream meander causation: a review and analysis
Earth-Science Reviews, 10 (1974) 121--138 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
Theories of Stream Meander Causation: A Review and Analysis
Peter P. Sakalowsky Jr.
ABSTRACT
Sakalowsky Jr., P.P., 1974. Theories of stream meander causa,~ion: a review and analysis. Earth-Sci. Rev., 10: 121--138. Meandering is one of the most complex problems associated with the behavior of rivers. Several explanations (derived from both field and laboratory flume) have been advanced to account for stream meandering, among them being the earth's rotation, excess current energy, transverse oscillations, initial current deflection, local disturbances, changes in stages of discharge and bed load and helicoidal flow. Foremost among these is the effects of helicoidal flow within the stream flow. The purpose of this paper is to review and analyze selected meander theories that contribute either directly or inherently to current meander concepts which revolve around the effect of helicoidal flow. Prevalent ideas concern meandering as being dependent not only on stream flow but also on the rate of stream discharge, sediment load, size and type of sediment, channel roughness, depth, width, velocity of flow, and quality of water itself. The characteristics of helicoidal flow seem to be the best explanatian for the development of meanders. Theories for meandering do not clearly explain the cause of helicoidal flow. Most of its intricacies have been learned through laboratory flume experiments. Interestingly, earlier work on river meandering was the product of long days in the field; in more recent years such research has been undertaken in the laboratory. One feels that a return to the stream would be a healthy complement to flume measurements. Further study should include hypothesis testing in a variety of material both homogeneous and heterogeneous.
INTRODUCTION
M e a n d e r i n g is o n e o f t h e m o s t c o m p l e x p r o b l e m s a s s o c i a t e d w i t h t h e b e h a v i o r o f rivers. C u r r e n t l y n o t h e o r y a d e q u a t e l y e x p l a i n s t h e c a u s e s o f a l l u v i a l r i v e r m e a n d e r s . S o m e o f t h e e a r l i e r t h e o r i s t s i n c l u d e W.M. Davis, H.M. E a k i n , G . K . G i l b e r t , M.S.W. J e f f e r s o n , a n d F. M a t t h e s . S e v e r a l t h e o r i e s have been advanced to account for meandering. These include the earth's r o t a t i o n , excess energy in the c u r r e n t , transverse oscillations, local disturb a n c e s , i n i t i a l d e f l e c t i o n o f c u r r e n t , c h a n g e s i n stages o f d i s c h a r g e a n d b e d load, a s y m m e t r y of cross-section and b a n k material, a n d helicoidal flow.
122 The purpose of this paper is to review and analyze selected meander theories that contribute either directly or inherently to current meander concepts which revolve around the effect of helicoidal flow. It is not an histarical sketch of theory that relates to the stream meander. Consequently, a number of meander theories have been omitted even though they may have contributed something to the meander concept. Several present day theories that appear to be most feasible in explaining the development and causation of meanders are here discussed and critically analyzed. Authors include J.B. Tiffany, G.A. Nelson, G.H. Matthes, J.F. Friedkin, J. Thomson, M.S. Quraishy, C.A. Mockmore, S. Leliavsky, R.A. Bagnold, W.F. Tanner, H.W. Shen, S. K o m u m , L.B. Leopold, and W.B. Langbein. THEORIES AND EXPERIMENTS In an early experimental laboratory model Tiffany and Nelson (1939) attempted to determine how meanders developed in alluvial rivers. A preliminary test revealed that an initially straight channel lacking any disturbing factors which might cause a deflection of the directed force of water would remain straight. Subsequent experiments were made with a short entrance channel set at a 45 ° angle with the axis of the initial parabolic channel (Fig.l). Sinuous channels closely resembling actual meandering streams developed. Pools were scoured below concave banks, crossings were formed as shallow shoals between bends, point bars developed, and materials from the place of scour were deposited downstream on the same side (Fig.2). It was noted that channel development resulted from a combination of such ideal conditions as constant flow, uniform bed material and the absence of tributaries discharging variable bed load into the channel. When bed maPeter P. Sakalowsky Jr. was born in Worcester, Massachusetts in 1942. He received the B.S. Ed. from Worcester State College, the M.A. in Geography from Clark University, and the Ph.D. in Physical Geography from Indiana State University. Dr. Sakalowsky has been a faculty member in the Department of Geography at Southern Connecticut State College in New Haven, Connecticut since 1970. He teaches courses in Physical Geography, Climatology, and Geomorphology. His research interests are primarily in geomorphology and include research in the quantitative relationships between beach morphology and nearshore processes, historical geomorphological changes in coastal configuration, and the interrelationships between coastal development and man.
123
1 l 1 l DISCHARGE
)
J
(~"~ .
.
.
.
B
.
Fig. 1. Laboratory flumes. Discharge into a straight flume is represented by A; discharge into a short channel which enters main flume at 45 ° angle is represented by B. CONCAVE BANK
POINT
PROFILE
WAT E R
TOP
SURFA CE
VIEW
SHOAL
POOL
SHOAL
PO0 L
Fig. 2. Characteristics of a meandering channel. (Lower two diagrams after Leopold and Langbein, 1966.)
terial, primarily sand, was introduced into the stream, it produced an accelerated effect on the development of the bends. These tests furnished little direct mathematical applicability to largerscale problems. However, t h e y did provide data on natural meander tendencies of an unconsolidated fluvial channel. It still is essentially a description of the test model and a presentation of the manner by which meanders originate and migrate. The authors did not adopt quantitative methods or empirical formulae with which to express their experimental methodology and results.
124
Helicoidal flow The first r ec or ded explanation of helicoidal flow of water around the bend of an alluvial stream is f o u n d in the work of James T h o m s o n , 1876 (Chorley et al., 1964, p.607). T h o m s o n f o u n d that flow along a meander bend was slower along the outside of the bend and faster on the inside, which contributes to the meandering of the stream. In curves the surface water moves to w a r d the out er bank w he r eby a piling up occurs. In this flow m ov emen t, the water pressures increase on the outside bend because of centrifugal force associated with the greater depth. Since friction at the b o t t o m layers reduces the velocity, t h e r e b y causing a lower centrifugal force, the b o t t o m layers of water move towards the inner bank of the bend. Thus mud is deposited on the inside of the bend while erosion occurs on the outside of the bend as a result of the swift-moving surface water, and meandering ensues (p.608).
Bank building and caving Matthes (1941) sets f o r t h certain fundamentals relating to the dynamics of meandering streams. These fundamentals are a result of extensive observations on different-size natural streams as well as studies on experimental models. The term meander is applied to any S-shaped channel pattern, formed in alluvial materials, whose location and shape are constantly shifting and adjusting as the channel migrates along the longitudinal axis of the valley. Matthes suggests five variables which appear to be essential factors in river meandering: valley slope, bed-load, stream discharge, bed resistance, and transverse oscillations (changes in the slope of the water surface at right angles to the axis o f flow; p.633). Matthes asserts t hat a meandering channel normally builds and refashions its bed and banks out of the material which it transports. Most of the load collected at a caving bank is deposited on the first bar downstream, causing it to grow outward as a convex bank (Fig.3). Since bank caving is principally related to the falling flood stage, only during this period do convex bars receive their greatest deposition and build out, causing swifter water to hug the concave shore, resulting in bank undermining.
Fig. 3. Sediment distribution along a curved channel. Sediment carried from A to B and X to Y on the right and left sides of the channel respectively without moving across the stream.
125 In experimentation Matthes noted that helicoidal flow takes place in distorted models and in channels that are deep relative to width. However, his experiments indicate that streams engaged in meandering in alluvial deposits are shallow in cross-section and show an absence of helicoidal motion. He finds that helicoidal flow is most improbable, observing no evidence of helicoidal motion in the meandering portion of the Mississippi River or in other meandering streams. The author mentions the effect of hydraulic gradient on stream meandering as being "a succession of modifications of valley slope into gradients adapted to the transportation of both water and sediment under peculiar bed resistance and variable flow conditions" which maintains a state of equilibrium (p.635). This is one of the poorest relationships in meandering. Major departures from a normal meander pattern are usually attributed to the interference with the methodological transfer of bank material causing the river to migrate along a path of least resistance. Matthes' theory revolves around the concept of bank building and caving due to changes in direction of stream energy. The five variables he advocates as being influential on stream meandering are not thoroughly analyzed. Transverse oscillation appears to be the major cause of bank caving which in turn provides material for bank building. However, Matthes fails to show the relationship between this variable and the others. Furthermore, he discounts helicoidal flow as having any effect. Perhaps, helicoidal flow for Matthes is hydraulic gradient and transverse oscillation. He does not conduct conclusive tests on this idea but merely states that helicoidal flow was not observed. No mention is made of the type of observation or experiment to prove or disprove the existence of helicoidal flow. In any event, Matthes did contribute a significant idea in meander causation, since bank caving and bank building can be readily observed. Bank erosion and stream overloading In a laboratory study of the meandering of alluvial rivers, Friedkin (1945) noted that meandering results primarily from local bank erosion and consequent local overloading with ensuing deposition of the heavier sediments moving along the river bed. In other words, meandering is essentially the trading of sediments from eroding banks to depositional bars. Friedkin writes "every phase of meandering represents a changing relationship between three closely related variables: the flow and hydraulic properties of the channel, the a m o u n t of sand moving along the bed, and the rate of bank erosion" (p.4). In experimentation meanders began when a sand bar resulting from bank erosion was created which in turn caused a local disturbance of flow. The caving bank overloaded the stream causing deposition and affecting stream capacity. As a result of this overloading, together with the limited capacity of moving water to carry sediment along its bed, a flow
126 disturbance was created which ultimately lead to the occurrence of meanders. Friedkin thus states that, " t h e only requirement for meandering is bank erosion" (p.4). The laboratory experiments showed that each bend developed as a result of deposition of sand on the inside of the bend. When flow was directed against a bank, the water piled up along concave banks. This deposition of sand on the inside of bends promotes a confinement of water which is necessary for erosive forces to develop. "The natural process might be likened to the oscillatory course taken by a ball which has started down a grooved incline so that it oscillates from side to side" (p.4), which is similar to Matthes' concept of transverse oscillations. " T h e diversion of sand from thalweg to bar as it enters the bend is one of the most important phenomena of meandering streams. It results in the building of convex bars in the bends" (p.6). This process of sand diversion is related to the diversion of slowmoving sand-carrying b o t t o m currents. Contrary to Matthes' study, Friedkin noted that helicoidal flow exists in bends. This was evidenced by a tendency of eroded particles to cross the channel. However, the force of the downstream currents carried the particles to bars below before they crossed the channel. In concurrence with Matthes' study, it was noted that erosion of banks, paths of sand movement, and location of deposition changed markedly with changes in the stage of flow -filling of crossings during high flows and scouring of crossings during low flows. "Probably the most important concept of a meandering river is that over each square foot of bed, shoaling or deepening takes place depending upon the relationship between the sand entering that area and the ability of the flow over that area to carry sand. Predominance of either, or a change in either, w i t h o u t a corresponding change in the other will cause deposition or scour" ( p . l l ) . The d o m i n a n t erosional agent is water turbulence along the banks. In this n o t e w o r t h y study, Friedkin attempts to isolate each variable among those regarded as basic factors of river meandering. His work comprises detailed observations together with detailed quantitative information concerning the problem of meandering. Furthermore, the use of green sand along concave banks on the left of the channel and red sand along the right has demonstrated the validity of the trading concept. This trading concept states that sediment derived from a concave-shaped cut bank is deposited mainly on the next bar downstream along the convex bank. A major shortcoming of the study is that all the experiments involved the discharge entering the flume at an angle; the incoming flow is thereby deflected into the main stream. This deflection presents conditions for meanders to start even before the experiment begins its natural course. It has been observed in laboratory models and in natural streams that an obstruction causes initial deflection. An explanation has not been given in these experiments as to why the discharge enters the flume at an angle and what effect this angular discharge has on meander development. Nevertheless, the
127 well-detailed experiments and observations together with detailed quantitative t r e a t m e n t represent a major advance in the understanding of stream meanders.
Curve development in straight channels Quraishy (1944), studying the origins of curves in rivers, agreed with T h o m s o n ' s t h e o r y of the d e v e l o p m e n t of curves as a result of secondary current flow. The presence of such factors as an obstacle, initial curvation, or initial a s y m m e t r y of cross-sections is not necessary t o the occurrence of curves. Based on observations and experiments with irrigation channels and canals in India, it was n o t e d that curves originated even in channels whose sides were quite straight and whose bed was very even (p.36). Curves developed as a result of certain interactions between the moving water and the sediment particles. In a series of flume experiments vigorous local scour of the beds close to the channel sides and alternating f r o m left to right was observed. This was at t r i but e d to the effect of eddies within the stream at points where a deficiency of energy and m o m e n t u m occurred primarily close to the vertical b a n k s . The sediment appeared t o be scooped out and deposited towards the center of the stream in a systematic manner resembling a fish-scale pattern. This deposition pattern was t erm ed skew scales. As these skew scales increased in size and depth, t h e y were t e r m e d skew shoals and represented those points of low or shallow water. As the skew shoals enlarged, the channel sides began to be scoured alternately opposite the widest part of the shoals (Fig.4). This marked the initiation of the curves (p.38). Thus, a t y p e of secondary flow was set up -- the faster fluid moving outwards eroding material f r o m the bank. Once this action was initiated, the effect became progressively intensified as the outer bend increased its curvature, ultimately causing a sinuous channel to develop.
A
Fig. 4. Fish-scalelike shoals A and scours B along a straight channel. Scours occur opposite widest part of shoals. Quraishy recognized the existence of helicoidal flow in the flume experiments. Yet it was not proved that this t y p e of flow was of prime importance in causing meanders to develop. None of the experiments were p e r f o r m e d to study specifically the effect of helicoidal flow. The d e v e l o p m e n t of bars (skew scales and shoals) and scours (pools or deeps) was recognized, but Quraishy does not objectively indicate their cause. Although helicoidal flow
128 was recognized, m o s t of the e x p e r i m e n t s were c o n c e r n e d with the interrelationship o f s t r e a m flow and s e d i m e n t , w h i c h never was fully explained. T h e results o f e x p e r i m e n t s were n o t q u a n t i t a t i v e l y analyzed, n o r were variable factors i n t r o d u c e d o t h e r t h a n a u n i f o r m w a t e r s u p p l y and sand t o r e p r e s e n t bedload. These t w o main f a c t o r s were n o t varied t o show c o n d i t i o n s over a variety of e x p e r i m e n t s such as t h e e f f e c t of high- and low-water stages. Perhaps, t h e significance o f this w o r k is the r e c o g n i t i o n of the d e v e l o p m e n t of shoals and scours and their ensuing e f f e c t o n river curvature. This c o u l d be r e f e r r e d t o as a t h e o r y o f a l t e r n a t e bars for the d e v e l o p m e n t o f river meanders. Flow in stable channels In a series o f l a b o r a t o r y e x p e r i m e n t s , M o c k m o r e ( 1 9 4 4 ) studied the flow o f fluids a r o u n d bends in Stable channels. T h e e x p e r i m e n t a l results s h o w e d t h a t spiral m o t i o n in fluids w o u l d be absent if t h e r e were streamline flow o f equal filamental v e l o c i t y at the e n t r a n c e t o t h e bend, and if friction c o u l d be eliminated, causing w a t e r to act as a p e r f e c t fluid (p.593). T h e r e f o r e , o n e of the principal causes o f spiral f l o w in either closed or o p e n channels is the existence of f r i c t i o n on channel walls and beds. This causes higher filamental velocities near the c e n t e r of the c h a n n e l and slower velocities along the walls and bed. T h e second principal cause o f spiral flow is centrifugal f o r c e which causes a d e f l e c t i o n of w a t e r particles f r o m a straight line m o t i o n . F u r t h e r e x p e r i m e n t a t i o n revealed t h a t in the latter half of bends along c o n v e x banks t h e r e were regions of u n s t e a d y flow w h e r e b y the w a t e r m o v e d u p s t r e a m at times. This o c c u r r e d at a p o i n t slightly b e y o n d the m i d - p o i n t of the b e n d and above m i d - d e p t h w h e r e t h e streamlines a p p e a r e d to leave the inside wall o f the channel. Considering the velocity c o m p o n e n t of a thin layer of fluid along t h e w a t e r surface, it is a p p a r e n t t h a t as the particles are moving d o w n s t r e a m t h e y are also slowly edging t o w a r d t h e outside o f the bend. T h e particles e x p e r i e n c e a t y p e o f diverging channel effect, with the streamlines diverging f r o m t h e inside wall causing a b a c k f l o w (Fig.5). Thus,
Fig. 5. Distribution of filamental velocities and secondary currents in a closed channel. The filamental velocities are greatest on the outside of the bend. Secondary currents develop along the inside of the bend at A. The cross-sectional diagram represents the currents at point A. (After Mockmore, 1944).
129
in the lower p o r t i on of the area of backflow the velocity c o m p o n e n t of the water particles was toward the inside of the bend. This resulted in a less t u r b u l e n t region than the water nearer the surface (p.599). These theoretical and quantitative experiments on the behavior of fluids in curves have lead to several conclusions: spiral flow exists along the bends of b o th open and closed channels with usually a double spiral in a closed channel; the filamental velocities are greater near the outside bend than at the inside bend, which results in greater erosive power along the outside bank; the physical characteristics of spiral m o t i o n cont ri but e to the movem en t of bed load bot h d o w n s t r e a m and toward the inside of bends; and, as a result of spiral motion, at about three-fourths of the way around the inside bend, there is a t e n d e n c y for the d e v e l o p m e n t of an eddy or slack water which is conducive to sediment deposition and bar form at i on (p.617). Mo ck mo r e presents a significant study of flow around bends. Although the experiments are c o n d u c t e d in a laboratory, it is recognized that a study of flow conditions in the bend of a natural stream involves many com pl ex factors, such as the difficulty of obtaining accurate measurements of filamental velocities at specified points and m a n y irregularities in the stream channel as well as the instability of channel beds themselves. Mathematical analysis is offered to illustrate and prove the existence of spiral flow as well as its effect on the inside and outside of meander bends. Particle m o v e m e n t is observed through the use of dyes, rice grains, and chemical globules. The filament of m a x i m u m velocity is initially and for most of the j o u r n e y around the bend, near the inside. This is in contrast to most studies in h y d r o l o g y which reveal that the filament of m a x i m u m velocity is always on the outside of the bend. The m o v e m e n t of water around a bend is affected by m a n y factors in addition to the rate of flow. These are the relative curvature of the bend, the shape of the channel cross section, the length of the bend, the a m o u n t of discharge, the a m o u n t and t y p e of sediment and channels roughness. The interrelationship between these factors and the flow of water with the resultant meander devel opm e nt was not investigated. F u r t h e r m o r e , the experiments and analyses were c o n d u c t e d with a rectangular pipe which render the results of d o u b t f u l worth. Channels in sediment flumes or in nature are not rectangular. Maximum velocity filaments could possibly be closer to the inside bend if the meander g e o m e t r y is not proport i onal to the discharge conditions, which might be the case with a rectangular channel. The fact that he used a rectangular channel rather than a r o u n d e d channel is the major objection to M ockm or e's experiments, and tenders the validity of his results questionable.
Development of shoals and deeps Through field and l a bor at or y observations and analyses, Leliavsky (1955) has a t t e m p t e d to explain river meandering as the t e n d e n c y of rivers to create a succession of shoals and deeps primarily as a result of perm anent , local,
130 secondary circulations within t he water. In model experiments a straight channel became sinuous by eroding concave banks and building convex bars. Deeps f o r med in the concave curves while shoals developed along the inflection points between the curves (p.94). In further investigations of natural alluvial channels, it was observed t ha t contiguous stream lines are never parallel to one anot her or to the river banks. Additionally, the greater the curvature of stream trajectories, the deeper the scoured channel beneath. On the bases of his experiments and observations as well as the observations and experiments of others, Leliavsky developed a theoretical explanation of meandering. A straight water channel marks the starting point. Initial channel d e f o r m a t i o n occurs by the presence of a scour or sediment deposit along the edge of the straight channel. This channel obstruction forces stream lines to deviate f r om their original straight courses to curved ones wh er eb y a centrifugal force is developed. This centrifugal effect represents the main cause of a helicoidal current which erodes soil from the concave bank, transports the sediment across the channel, and deposits it on the convex bank, thus initiating and progressively intensifying the process of meandering. Therefore, the slightest irregularity in channel form which causes a shift of stream lines from straight to curved courses creates a focal point for the erosion process which ultimately leads to the devel opm ent of a meander (p.122). F u r t h e r m o r e , the piling up of surface water on the outside bend contributes to the centrifugal effect which is an i m p o r t a n t factor in the mechanics of helicoidal flow. The downward c o m p o n e n t of the helicoidal current at the concave bank of a meandering stream is highly erosive. The same current carries the eroded sediment across the channel, depositing it as a wide shallow bar on the convex bank. Thus one bank erodes, the o t her extends, causing the river to shift towards the eroded, concave bank. This is the fundamental process of meandering (Fig.6). Through the stud5 of theoretical and experimental models, field observations and tests, and the use of quantitative methods and empirical formulae, Leliavsky has established t ha t helicoidal flow is the i m p o r t a n t cause of meandering. His methods and procedures are a significant a t t e m p t f ..~'~..:..':'V.. N
, ~:.:.. : • .':L:.,:..
LEFT
SIDE
Fig. 6. Cross-section of a m e a n d e r b e n d . A r e p r e s e n t s the inside of the b e n d ; B r e p r e s e n t s the o u t s i d e o f the b e n d w h e r e surface w a t e r piles up. T h e diagram o n the right s h o w s s e d i m e n t being carried f r o m o n e side o f the c h a n n e l across to the o p p o s i t e side.
131 to prove the existence and effect of helicoidal flow b o t h in model experiments and in natural rivers. However, the results of these studies indicate that sediment is moved f r om t he concave bank across the channel to be deposited on the convex bank. Friedkin's investigations illustrated quite conclusively th at most of the sediment eroded from the concave bank is carried d o wn s tr eam and deposited on the same side of the stream. This appears to be the major discrepancy in Leliavsky's t h e o r y on the distribution of sediment. His work, however, has established the existence of helicoidal currents in b o t h the lab o r at or y and the field.
Transverse flow Th r o u g h a series of l abor a t or y experiments, Bagnold (1960) sought to understand and explain the de ve l opm ent of river meanders. It was observed that a transverse flow occurs when water flows t hrough a curved circular pipe. This t y p e of flow occurs when frictional drag is exerted on the tangential flow. The radial acceleration on the flow creates an increase of pressure on the outside of the bend and a corresponding decrease on the inside. Within the main flow the centrifugal pressure gradient balances the radial pressure gradient. However, an imbalance along the periphery remains as a result of b o u n d a r y friction, causing a reduct i on in the flow velocity and a corresponding decrease in the radial pressure gradient along the periphery. " H e n c e water must flow radially inwards over the periphery, dow n the radial pressure gradient and its place is taken by an out w ard flow across the centerline of the cross section. The effect of the transverse flow is to reduce the internal shear due to the direction of the flow around the b e n d " (p.135). In order to occur and to reach o p t i m u m velocity, transverse flow requires angular acceleration within the confines of the bend. Resistance to flow through the channel arc is the sum of the energy e x p e n d e d to create the transverse flow and the energy dissipated by wall friction. Therefore, the total resistance to flow owing to the channel bend is the sum of two forces necessary to overcome wall friction and inertia (p.136). As a result a stage is ultimately reached w he r eby flow along the inner b o u n d a r y becomes unstable and breaks away leaving a zone of unstable and confused water. Generally, diverging flow has been responsible for the breakaway and the ensuing unstable region. With an increase in divergence, flow becomes rapidly unstable in the region of the diverging boundaries resulting in sudden increases in energy dissipation. As the flow moves t hrough the bend it tends to decrease along the inside region of greatest curvature. As a result, the shear rate, shear stress in the fluid, and pressure gradient in the direction of flow are reduced along the inside of the bend. Ultimately the pressure gradient along the inside curve becomes r e duc e d to such a level t hat a backward m o v e m e n t occurs. Thus, flow in the inside curve becomes unstable resulting in local eddying accompanied by increased energy dissipation (Fig.7).
132 A
Fig. 7. Stream velocity in a curved c h a n n e l . With increased curvature m a x i m u m flow velocity o c c u r s at A. R e d u c e d velocity along the inside o f the b e n d results in a q u i e t z o n e of u n s t a b l e flow at B, w h e r e energy is dissipated in e d d y i n g .
This co mb ina t i on of transverse and helicoidal flow accompanied by secondary circulations accounts for the development of meanders. The increased curvature causes an increase in divergence and a sudden increase of energy dissipation on the outside of the bend, causing erosion to occur. The eddying currents and areas of slack water along the inside bend provide conditions for deposition to occur. Thus the meander is born and continues to grow in the manner outlined above. Bagnold's study is an extension of earlier investigations of helicoidal flow. Most of his conclusions result f r om model studies with a seeming lack of application to natural streams. However, the experiments are c o n d u c t e d carefully and t h e y are quantitatively analyzed. Nevertheless, nothing new has been presented. Several of t he experiments did provide further conclusive p r o o f for the existence and effect of helicoidal flow. Sorely needed is more direct application of laboratory findings to natural streams. One major outgrowth of Bagnold's work was the development and application of the c o n c e p t of least ef f or t to stream flow which was later used by Leopold. Inherent helicoidal currents Tanner (1960) has investigated the presence of helicoidal flow in a stream and its effect as a possible cause of meandering. A meander pattern develops b e c a u s e helicoidal currents ar~ inherent in straight-flowing streams, rather than the c o n c e p t that helicoidal currents develop after the meander pat t ern has begun to f o r m (p.994). This concept was simply tested in the laboratory using a piece o f glass that had been covered with dust over several years. When the glass plate was covered with dust, t urbul ent flow and meandering occurred, while on clean glass laminar flow was observed. When the helicoidal current becomes established, t h e main stream flow tends to flow parallel to the direction of the bed current rather than parallel to the direction of the surface current. This characteristic of the flow is primarily a result of the effects of channel curvature. Once helicoidal flow
133 began the descending side of the helix showed strengthening while the ascending side showed retardation, resulting in a deflection of the stream toward the slower, rising side. Thus the spiral motion within the helicoidal current causes the main stream to be diverted away from the m a x i m u m slope until the effect of gravity is largely reduced. Then, the altered gravity component causes an alteration of the channel and current characteristics resulting in the formation of a n e w helicoidal current spiraling in the opposite direction. These characteristics of helicoidal flow, then, cause the channel course to become sinuous. Thus, the sequence of events for the establishment of meanders is turbulence, helicoidal flow, and meandering. A major shortcoming of this t h e o r y is the factor or factors necessary to establish helicoidal current (p.995). In later experimentation, Tanner (1963) observed that roughness was an additional factor that contributed to meander development. Thus the new sequence of meander development is roughness, turbulence, helicoidal flow, and meandering. Verification of spiral flow was made with tracers. Further observations concluded that in a large river system where the load was mobile, coherent, and abundant, meandering resulted. The turbulence factor associated with meanders results from an adverse pressure gradient associated with sudden changes in channel depth or width or changes in velocity, or changes effected by obstruction to flow (p.41). In a subsequent study, Tanner (1968) further establishes the fact that meandering is a product of secondary overturn--helicoidal flow. The variability of b o t t o m roughness contributes to spiral development by causing a variation in stream flow velocity between the b o t t o m region and the surface region. Thus, "a helix in the earth's gravity field must have a curved axis, inasmuch as the descending waters will be accelerated relative to ascending fluid" (Tanner, 1968, p.960). Additional factors that contribute to meander development include stream gradient, grain size, cohesiveness of sediment, stream discharge and the channel pattern. The initial development of meandering appears as a pattern of pools (deep zones) and riffles (shallow zones) occurring in the stream channel (see Fig.2). The formation of deeps and shoals is related to flow velocities within the bend of a stream. Deeps form in zones of maximum flow velocity which occurs near channel walls while shoals form in zones of lower flow velocity. On meandering streams the deeps occur on the outside banks of meander loops where undercutting is occurring as a result of the maximum flow velocity along this part of the bend. The shoals develop between the deeps of each bend along the inside bank as a result of lower flow velocity allowing sediment deposition. The development of the sinuous maximum-velocity flow lines (helicoidal flow) and the ensuing pool and riffle b o t t o m profile may be related to the characteristics of channel-wall sediment. A single patch would be sufficient to obstruct the stream flow, causing a deviation of flow lines with an ensuing overcompensated counter deviation and a continuing and progressive series
134 of deviations. This t y p e of mechanism can operate only where a bend causes the initial flow deviation. Helicoidal flow and a resultant distribution of stream velocities occurs, causing erosion and deposition in such a manner that meanders develop. Tanner attributes the cause of meandering to helicoidal flow which is initiated by an obstruction to flow, causing a deviation of flow lines with a thread of m a x i m u m flow velocity developing. The effect of other variables such as stream gradient, grain size, sediment cohesiveness, stream discharge and channel pattern on meander devel opm ent is also considered. Many of the conclusions are based on laboratory experiments, although Tanner did succeed in observing helical flow in natural streams with the aid of floats and tracers. F u r t h e r m o r e , the conclusions reached in these studies recognize in addition to helical flow the i m p o r t a n c e of ot her factors such as roughness and turbulence in causing meandering to occur. The use of empirical m eth o d s additionally enhance the experiments and results.
Channel roughness and flow resistance Shen (1961) c o n d u c t e d laboratory experiments in which he observed the characteristics of channel bed patterns as a factor in flow resistance which ultimately may cause a river to meander. The development of different bed patterns is closely related to the rate of sediment transport, which is a function of the ratio between the resistance of the grain to m o t i o n and the force ex er ted on a grain by flow (p.17). F u r t h e r m o r e , the differences of the shear stress between the rough and s m o o t h e r bed surfaces induce secondary circulations that are perpendicular to the direction of main flow. As a result the direction of the b o t t o m current is such that it is always flowing t ow ard the rougher surface. The upstream flow system creates a circulation that results in a scour or deep forming along one side of the channel. A second circulation occurs in the opposite direction as a result of flow resistance being caused by channel roughness. As the flow moves downstream, the rotation of these secondary circulations increases to such a magnitude that a scour hole is created. The sediment is moved from the left-hand t o the right-hand side of the channel (see Fig.6). At the same time, a second circulation in the opposite direction occurs as a result of bank roughness at the left-hand side. This process is repeated gradually downstream. Shen bases his postulations on Hans A. Einstein and Huon Li's work with the vorticity equation in t ur bul ent viscous flow. The initial condition in straight u n if o r m flow is that the secondary current be zero everywhere. A secondary circulation will occur whenever the equation value differs from zero. This situation could develop near the frictional b o u n d a r y where lines of co n s tan t velocity are not parallel (p.20). Shen and Komura (1968) in a later experimental study observed that all eddies that developed between the free water surface and the channel wall
135 stretched asymmetrically in the direction of flow, whereby a circulation developed which tended to move water up along and out from the bank toward the center of the channel. Thus a clockwise circulation was created along the left bank and counterclockwise along the right bank. In a clockwise circulation, surface water, having a higher velocity, tends to move toward the right bank while the lower-velocity b o t t o m water tends to move toward the left bank. As a result of the higher velocity water along the right bank, a large shear stress is created there. Eventually the initial clockwise circulation would be reduced to zero whereby a counterclockwise circulation would result. With the newly created counterclockwise circulation, sediment would be transported from the left to the right bank. This process would continually repeat, resulting in the development of alternate scour holes and bars and, consequently, the development of meanders (p.1002). The resulting meander pattern is a function of channel roughness and the variation of flow conditions as discussed above. This work is, perhaps, the best explanation of the meander as caused by helicoidal flow. The experiments are enhanced by quantitative methods and some field observations. There appears to be one discrepancy, however. The results indicate sediment movement across the channel, which is in contradiction to Friedkin's study showing most sediment being deposited on the same side of the stream. There is agreement with Leliavsky on the transportation and distribution of the sediment. Furthermore, the authors do not indicate the a m o u n t of sediment carried across the channel, nor the mode of transportation, whether it be suspension or bed load. The main focus of the work is with the mechanics of helicoidal flow and the quite conclusive proof of its existence and effect. In addition, the close relationship between channel roughness and helicoidal flow is explored. These few experiments concern alluvial streams containing relatively homogeneous material. There is need for study of streams having different sediment types as well as heterogeneous sediments. M i n i m u m variance
Leopold and Langbein (1966) postulated from the theory of minimum variance that meandering is the most probable form of channel geometry. In fact, variances in bed shear and friction are uniformly lower in a curved stretch than a straight stretch. Meanders will usually appear wherever the river flows through fine-grained alluvium that is easily eroded and transported while still being cohesive enough to maintain firm banks. The development of this meander t h e o r y is based upon the random walk method. A random walk between two points traces a curve termed "sinegenerated" curve which closely approximates the shape of real river meanders (p.62). A sine-generated curve has the smallest variation of changes of direction -- sums of the squares of this change are less than for any other regular curve of the same length. Furthermore, it is also the curve of
136 minimum total w or k in bending (p.63). A river changes its course by its ability to erode, transport, and deposit the material t hat composes the river channel. Where the flow velocity gradient is higher against the channel bank, local eddies are created whereby a c o n c e n t r a t i o n of energy occurs which localizes erosion. This circulation system occurs as the surface water plunges toward the bed near the concave bank while the b o t t o m water rises t ow ard the surface near the convex bank. This circulation along with downstream m o v e m e n t gives each particle a roughly helicoidal flow which reverses its direction of r o t a t i on with each successive meander. The ultimate result of this circulation pat t er n is erosion on the outside bend and deposition on the inside bend which leads to the devel opm e n t of a meander. The meandering pattern is the most probable result of a combination of processes that tend to eliminate concentrations of energy loss while attempting to reduce total energy loss to a minimum. Thus, a sine-generated curvature assumed by meanders is the most suitable curvature for this energy relationship. Leopold and Langbein did not emerge with a new t h e o r y of meandering. The existence and effect of helicoidal flow together with many ot her factors was recognized. The efforts of this study were concerned mainly with establishing that a sine-generated curve is the path of least resistance. Energy expenditure is at a m i ni m um in this t y p e of curve. Ultimately the characteristic curvature of a meander will be a sine-curve. The work is primarily theoretical. Additional field testing is required to establish p r o o f of sinecurved meanders. However, the c o n c e p t of sine-curvature and the m e t h o d of the r a n d o m walk has merit for further applied research. CONCLUSIONS
On the basis of these varied theories on the devel opm ent of meanders, the following co mb i ne d conclusions are presented. (1) Most theories of meandering are the result of experimental laboratory studies. (2) Several studies showed that channel devel opm ent resulted from a c o mb in atio n of ideal conditions, which were: constant flow, uniform bed material, and no variable bed-load discharge by tributary channels. (3) Valley slope, bed-load, discharge, bed resistance and transverse oscillations work in com bi na t i on to cause a sinuous channel to develop. (4) Meandering develops by an initial deflection causing material to be eroded on concave banks and deposited on convex banks due to changes in direction of stream energy. (5) Material eroded on the concave bank is deposited on the same side further downstream. (6) Most present-day theories attribute meandering to helicoidal flow; at times it is accompanied by secondary circulations. (7) Meandering stream channels are characterized by series of deeps and shoals or pools and riffles.
137 (8) Bed shear, friction of bed and walls, width, depth, velocity, slope, and roughness all are important to meandering. (9) Several authors feel helicoidal flow is inherent in straight stream channels; therefore, helicoidal flow is already present before meandering develops. The theories for meandering do not clearly explain the cause of helicoidal flow. Most of the theorization is based upon laboratory experimentation, but lacks thorough exploration in the field. Authors like Matthes and Friedkin explain meandering as a change in the direction of stream energy but do not fully account for it. They mention helicoidal flow but discount its importance either through poor observations, lack of adequate testing, or lack of comprehension. One ponders why more consideration was not given to the concept of helicoidal flow which had been presented several years earlier by such men as Thomson, Lacy, Matthes, Mockmore, and others.
Development of meanders The characteristics of helicoidal flow seem to be the best explanation for the development of meanders. However, most of its intricacies have been learned through flume experiments in the laboratory. It is clearly evident that more work needs to be done with natural streams. New procedures and methods need to be developed to test this theory in natural streams with all the varied and uncontrolled factors present. Prevalent ideas concern meandering as being dependent not only on the stream flow but also on the rate of stream discharge, sediment load, size and type of sediment, channel roughness, depth, width, velocity of flow, and quality of the water itself. Since there is a close interrelationship of all these factors, they must be jointly considered for the development of a theory. A theory dealing solely with the characteristics of water flow is inadequate i; the other factors are not considered. Furthermore, studies should include hypothesis testing in a variety of material both homogeneous and heterogeneous. The laboratory has its place for isolating factors of importance and developing theories on the type and manner of occurrence. However, meaningful concepts should be able to stand the test when applied to many varied natural meandering streams. Perhaps, no one theory would adequately explain the complexities of meandering but rather a combination of theories that vary with individual streams and channel mediums. REFERENCES Bagnold, R.A., 1960. Some aspects of the shape of river meanders. U.S. Geol. Surv. Prof. Pap., 282-E:135--144. Chorley, R.J., Dunn, A.J. and Beckinsale, R.P., 1964. The History of the Study of Landforms; or the Development of Geomorphology. Methuen, London, 1:607--609. Friedkin, J.F., 1945. A Laboratory Study of the Meandering of Alluvial Rivers. U.S. Waterways Experiment Station, Vicksburg, Miss., 39 pp.
138
Leliavsky, S., 1955. An Introduction to Fluvial Hydraulics. Constable, London, 257 pp. Leopold, L.B., 1960. River meanders. Geol. Soc. Am. Bull., 71 (6):769--794. Leopold, L.B. and Langbein, W.B., 1966. River meanders. Sci. American, 214:60--70. Leopold, L.B. and Wolman, M.G., 1957. River channel patterns: braided, meandering, and straight. U.S. Geol. Surv. Prof. Pap., 282-B: 39--85. Matthes, G.H., 1941. Basic aspects of stream meanders. Trans. Am. Geophys. Un., 22:632--636. Mockmore, C.A., 1944. Flow around bends in stable channels. Am. Soc. Civil Eng. Trans., 109:593--628. Quraishy, M.S., 1944. The origin of curves in rivers. Current Science, 13 (2):36--39. Schumm, S.A., 1960. The shape of alluvial channels in relation to sediment type. U.S. Geol. Surv. Prof. Pap. 352-B:17--30. Schumm, S.A., 1963. Sinuosity of alluvial rivers on the Great Plains. Geol. Soc. Am. Bull. 74(9):1089--1099. Shen, H.W., 1961. A Study on Meandering and other Bed Patterns in Straight Alluvial Channels. Water Resources Center, Contrib. 33, Hydraulic Laboratory, Univ. California, Berkeley, Calif., 68 pp. Shen, H.W., and Komura, S., 1968. Meandering tendencies in straight alluvial channels. J. Hydraulics Div., Am. Soc. Civ. Eng., 94, (H 44), Proc. Pap. 6042:997--1016. Tanner, W.F., 1960. Helicoidal flow, a possible cause of meandering. J. Geophys. Res., 65(3):993--995. Tanner, W.F. 1963. Spiral flow in rivers, shallow seas, dust devils, and models. Science 139:41--42. Tanner, W.F., 1968. Rivers -- meandering and braiding. In: R.W. Fairbridge (Editor), The Encyclopedia of Geomorphology. Reinhold, New York, N.Y., pp.957--963. Tiffany, J.B. and Nelson, G.A., 1939. Studies of meandering of model streams. Trans. Am. Geophys. Un., 20:644--649. (Accepted for publication October 10, 1973)
Theories of Stream Meander Causation: A Review and Analysis
Peter P. Sakalowsky Jr.
ABSTRACT
Sakalowsky Jr., P.P., 1974. Theories of stream meander causa,~ion: a review and analysis. Earth-Sci. Rev., 10: 121--138. Meandering is one of the most complex problems associated with the behavior of rivers. Several explanations (derived from both field and laboratory flume) have been advanced to account for stream meandering, among them being the earth's rotation, excess current energy, transverse oscillations, initial current deflection, local disturbances, changes in stages of discharge and bed load and helicoidal flow. Foremost among these is the effects of helicoidal flow within the stream flow. The purpose of this paper is to review and analyze selected meander theories that contribute either directly or inherently to current meander concepts which revolve around the effect of helicoidal flow. Prevalent ideas concern meandering as being dependent not only on stream flow but also on the rate of stream discharge, sediment load, size and type of sediment, channel roughness, depth, width, velocity of flow, and quality of water itself. The characteristics of helicoidal flow seem to be the best explanatian for the development of meanders. Theories for meandering do not clearly explain the cause of helicoidal flow. Most of its intricacies have been learned through laboratory flume experiments. Interestingly, earlier work on river meandering was the product of long days in the field; in more recent years such research has been undertaken in the laboratory. One feels that a return to the stream would be a healthy complement to flume measurements. Further study should include hypothesis testing in a variety of material both homogeneous and heterogeneous.
INTRODUCTION
M e a n d e r i n g is o n e o f t h e m o s t c o m p l e x p r o b l e m s a s s o c i a t e d w i t h t h e b e h a v i o r o f rivers. C u r r e n t l y n o t h e o r y a d e q u a t e l y e x p l a i n s t h e c a u s e s o f a l l u v i a l r i v e r m e a n d e r s . S o m e o f t h e e a r l i e r t h e o r i s t s i n c l u d e W.M. Davis, H.M. E a k i n , G . K . G i l b e r t , M.S.W. J e f f e r s o n , a n d F. M a t t h e s . S e v e r a l t h e o r i e s have been advanced to account for meandering. These include the earth's r o t a t i o n , excess energy in the c u r r e n t , transverse oscillations, local disturb a n c e s , i n i t i a l d e f l e c t i o n o f c u r r e n t , c h a n g e s i n stages o f d i s c h a r g e a n d b e d load, a s y m m e t r y of cross-section and b a n k material, a n d helicoidal flow.
122 The purpose of this paper is to review and analyze selected meander theories that contribute either directly or inherently to current meander concepts which revolve around the effect of helicoidal flow. It is not an histarical sketch of theory that relates to the stream meander. Consequently, a number of meander theories have been omitted even though they may have contributed something to the meander concept. Several present day theories that appear to be most feasible in explaining the development and causation of meanders are here discussed and critically analyzed. Authors include J.B. Tiffany, G.A. Nelson, G.H. Matthes, J.F. Friedkin, J. Thomson, M.S. Quraishy, C.A. Mockmore, S. Leliavsky, R.A. Bagnold, W.F. Tanner, H.W. Shen, S. K o m u m , L.B. Leopold, and W.B. Langbein. THEORIES AND EXPERIMENTS In an early experimental laboratory model Tiffany and Nelson (1939) attempted to determine how meanders developed in alluvial rivers. A preliminary test revealed that an initially straight channel lacking any disturbing factors which might cause a deflection of the directed force of water would remain straight. Subsequent experiments were made with a short entrance channel set at a 45 ° angle with the axis of the initial parabolic channel (Fig.l). Sinuous channels closely resembling actual meandering streams developed. Pools were scoured below concave banks, crossings were formed as shallow shoals between bends, point bars developed, and materials from the place of scour were deposited downstream on the same side (Fig.2). It was noted that channel development resulted from a combination of such ideal conditions as constant flow, uniform bed material and the absence of tributaries discharging variable bed load into the channel. When bed maPeter P. Sakalowsky Jr. was born in Worcester, Massachusetts in 1942. He received the B.S. Ed. from Worcester State College, the M.A. in Geography from Clark University, and the Ph.D. in Physical Geography from Indiana State University. Dr. Sakalowsky has been a faculty member in the Department of Geography at Southern Connecticut State College in New Haven, Connecticut since 1970. He teaches courses in Physical Geography, Climatology, and Geomorphology. His research interests are primarily in geomorphology and include research in the quantitative relationships between beach morphology and nearshore processes, historical geomorphological changes in coastal configuration, and the interrelationships between coastal development and man.
123
1 l 1 l DISCHARGE
)
J
(~"~ .
.
.
.
B
.
Fig. 1. Laboratory flumes. Discharge into a straight flume is represented by A; discharge into a short channel which enters main flume at 45 ° angle is represented by B. CONCAVE BANK
POINT
PROFILE
WAT E R
TOP
SURFA CE
VIEW
SHOAL
POOL
SHOAL
PO0 L
Fig. 2. Characteristics of a meandering channel. (Lower two diagrams after Leopold and Langbein, 1966.)
terial, primarily sand, was introduced into the stream, it produced an accelerated effect on the development of the bends. These tests furnished little direct mathematical applicability to largerscale problems. However, t h e y did provide data on natural meander tendencies of an unconsolidated fluvial channel. It still is essentially a description of the test model and a presentation of the manner by which meanders originate and migrate. The authors did not adopt quantitative methods or empirical formulae with which to express their experimental methodology and results.
124
Helicoidal flow The first r ec or ded explanation of helicoidal flow of water around the bend of an alluvial stream is f o u n d in the work of James T h o m s o n , 1876 (Chorley et al., 1964, p.607). T h o m s o n f o u n d that flow along a meander bend was slower along the outside of the bend and faster on the inside, which contributes to the meandering of the stream. In curves the surface water moves to w a r d the out er bank w he r eby a piling up occurs. In this flow m ov emen t, the water pressures increase on the outside bend because of centrifugal force associated with the greater depth. Since friction at the b o t t o m layers reduces the velocity, t h e r e b y causing a lower centrifugal force, the b o t t o m layers of water move towards the inner bank of the bend. Thus mud is deposited on the inside of the bend while erosion occurs on the outside of the bend as a result of the swift-moving surface water, and meandering ensues (p.608).
Bank building and caving Matthes (1941) sets f o r t h certain fundamentals relating to the dynamics of meandering streams. These fundamentals are a result of extensive observations on different-size natural streams as well as studies on experimental models. The term meander is applied to any S-shaped channel pattern, formed in alluvial materials, whose location and shape are constantly shifting and adjusting as the channel migrates along the longitudinal axis of the valley. Matthes suggests five variables which appear to be essential factors in river meandering: valley slope, bed-load, stream discharge, bed resistance, and transverse oscillations (changes in the slope of the water surface at right angles to the axis o f flow; p.633). Matthes asserts t hat a meandering channel normally builds and refashions its bed and banks out of the material which it transports. Most of the load collected at a caving bank is deposited on the first bar downstream, causing it to grow outward as a convex bank (Fig.3). Since bank caving is principally related to the falling flood stage, only during this period do convex bars receive their greatest deposition and build out, causing swifter water to hug the concave shore, resulting in bank undermining.
Fig. 3. Sediment distribution along a curved channel. Sediment carried from A to B and X to Y on the right and left sides of the channel respectively without moving across the stream.
125 In experimentation Matthes noted that helicoidal flow takes place in distorted models and in channels that are deep relative to width. However, his experiments indicate that streams engaged in meandering in alluvial deposits are shallow in cross-section and show an absence of helicoidal motion. He finds that helicoidal flow is most improbable, observing no evidence of helicoidal motion in the meandering portion of the Mississippi River or in other meandering streams. The author mentions the effect of hydraulic gradient on stream meandering as being "a succession of modifications of valley slope into gradients adapted to the transportation of both water and sediment under peculiar bed resistance and variable flow conditions" which maintains a state of equilibrium (p.635). This is one of the poorest relationships in meandering. Major departures from a normal meander pattern are usually attributed to the interference with the methodological transfer of bank material causing the river to migrate along a path of least resistance. Matthes' theory revolves around the concept of bank building and caving due to changes in direction of stream energy. The five variables he advocates as being influential on stream meandering are not thoroughly analyzed. Transverse oscillation appears to be the major cause of bank caving which in turn provides material for bank building. However, Matthes fails to show the relationship between this variable and the others. Furthermore, he discounts helicoidal flow as having any effect. Perhaps, helicoidal flow for Matthes is hydraulic gradient and transverse oscillation. He does not conduct conclusive tests on this idea but merely states that helicoidal flow was not observed. No mention is made of the type of observation or experiment to prove or disprove the existence of helicoidal flow. In any event, Matthes did contribute a significant idea in meander causation, since bank caving and bank building can be readily observed. Bank erosion and stream overloading In a laboratory study of the meandering of alluvial rivers, Friedkin (1945) noted that meandering results primarily from local bank erosion and consequent local overloading with ensuing deposition of the heavier sediments moving along the river bed. In other words, meandering is essentially the trading of sediments from eroding banks to depositional bars. Friedkin writes "every phase of meandering represents a changing relationship between three closely related variables: the flow and hydraulic properties of the channel, the a m o u n t of sand moving along the bed, and the rate of bank erosion" (p.4). In experimentation meanders began when a sand bar resulting from bank erosion was created which in turn caused a local disturbance of flow. The caving bank overloaded the stream causing deposition and affecting stream capacity. As a result of this overloading, together with the limited capacity of moving water to carry sediment along its bed, a flow
126 disturbance was created which ultimately lead to the occurrence of meanders. Friedkin thus states that, " t h e only requirement for meandering is bank erosion" (p.4). The laboratory experiments showed that each bend developed as a result of deposition of sand on the inside of the bend. When flow was directed against a bank, the water piled up along concave banks. This deposition of sand on the inside of bends promotes a confinement of water which is necessary for erosive forces to develop. "The natural process might be likened to the oscillatory course taken by a ball which has started down a grooved incline so that it oscillates from side to side" (p.4), which is similar to Matthes' concept of transverse oscillations. " T h e diversion of sand from thalweg to bar as it enters the bend is one of the most important phenomena of meandering streams. It results in the building of convex bars in the bends" (p.6). This process of sand diversion is related to the diversion of slowmoving sand-carrying b o t t o m currents. Contrary to Matthes' study, Friedkin noted that helicoidal flow exists in bends. This was evidenced by a tendency of eroded particles to cross the channel. However, the force of the downstream currents carried the particles to bars below before they crossed the channel. In concurrence with Matthes' study, it was noted that erosion of banks, paths of sand movement, and location of deposition changed markedly with changes in the stage of flow -filling of crossings during high flows and scouring of crossings during low flows. "Probably the most important concept of a meandering river is that over each square foot of bed, shoaling or deepening takes place depending upon the relationship between the sand entering that area and the ability of the flow over that area to carry sand. Predominance of either, or a change in either, w i t h o u t a corresponding change in the other will cause deposition or scour" ( p . l l ) . The d o m i n a n t erosional agent is water turbulence along the banks. In this n o t e w o r t h y study, Friedkin attempts to isolate each variable among those regarded as basic factors of river meandering. His work comprises detailed observations together with detailed quantitative information concerning the problem of meandering. Furthermore, the use of green sand along concave banks on the left of the channel and red sand along the right has demonstrated the validity of the trading concept. This trading concept states that sediment derived from a concave-shaped cut bank is deposited mainly on the next bar downstream along the convex bank. A major shortcoming of the study is that all the experiments involved the discharge entering the flume at an angle; the incoming flow is thereby deflected into the main stream. This deflection presents conditions for meanders to start even before the experiment begins its natural course. It has been observed in laboratory models and in natural streams that an obstruction causes initial deflection. An explanation has not been given in these experiments as to why the discharge enters the flume at an angle and what effect this angular discharge has on meander development. Nevertheless, the
127 well-detailed experiments and observations together with detailed quantitative t r e a t m e n t represent a major advance in the understanding of stream meanders.
Curve development in straight channels Quraishy (1944), studying the origins of curves in rivers, agreed with T h o m s o n ' s t h e o r y of the d e v e l o p m e n t of curves as a result of secondary current flow. The presence of such factors as an obstacle, initial curvation, or initial a s y m m e t r y of cross-sections is not necessary t o the occurrence of curves. Based on observations and experiments with irrigation channels and canals in India, it was n o t e d that curves originated even in channels whose sides were quite straight and whose bed was very even (p.36). Curves developed as a result of certain interactions between the moving water and the sediment particles. In a series of flume experiments vigorous local scour of the beds close to the channel sides and alternating f r o m left to right was observed. This was at t r i but e d to the effect of eddies within the stream at points where a deficiency of energy and m o m e n t u m occurred primarily close to the vertical b a n k s . The sediment appeared t o be scooped out and deposited towards the center of the stream in a systematic manner resembling a fish-scale pattern. This deposition pattern was t erm ed skew scales. As these skew scales increased in size and depth, t h e y were t e r m e d skew shoals and represented those points of low or shallow water. As the skew shoals enlarged, the channel sides began to be scoured alternately opposite the widest part of the shoals (Fig.4). This marked the initiation of the curves (p.38). Thus, a t y p e of secondary flow was set up -- the faster fluid moving outwards eroding material f r o m the bank. Once this action was initiated, the effect became progressively intensified as the outer bend increased its curvature, ultimately causing a sinuous channel to develop.
A
Fig. 4. Fish-scalelike shoals A and scours B along a straight channel. Scours occur opposite widest part of shoals. Quraishy recognized the existence of helicoidal flow in the flume experiments. Yet it was not proved that this t y p e of flow was of prime importance in causing meanders to develop. None of the experiments were p e r f o r m e d to study specifically the effect of helicoidal flow. The d e v e l o p m e n t of bars (skew scales and shoals) and scours (pools or deeps) was recognized, but Quraishy does not objectively indicate their cause. Although helicoidal flow
128 was recognized, m o s t of the e x p e r i m e n t s were c o n c e r n e d with the interrelationship o f s t r e a m flow and s e d i m e n t , w h i c h never was fully explained. T h e results o f e x p e r i m e n t s were n o t q u a n t i t a t i v e l y analyzed, n o r were variable factors i n t r o d u c e d o t h e r t h a n a u n i f o r m w a t e r s u p p l y and sand t o r e p r e s e n t bedload. These t w o main f a c t o r s were n o t varied t o show c o n d i t i o n s over a variety of e x p e r i m e n t s such as t h e e f f e c t of high- and low-water stages. Perhaps, t h e significance o f this w o r k is the r e c o g n i t i o n of the d e v e l o p m e n t of shoals and scours and their ensuing e f f e c t o n river curvature. This c o u l d be r e f e r r e d t o as a t h e o r y o f a l t e r n a t e bars for the d e v e l o p m e n t o f river meanders. Flow in stable channels In a series o f l a b o r a t o r y e x p e r i m e n t s , M o c k m o r e ( 1 9 4 4 ) studied the flow o f fluids a r o u n d bends in Stable channels. T h e e x p e r i m e n t a l results s h o w e d t h a t spiral m o t i o n in fluids w o u l d be absent if t h e r e were streamline flow o f equal filamental v e l o c i t y at the e n t r a n c e t o t h e bend, and if friction c o u l d be eliminated, causing w a t e r to act as a p e r f e c t fluid (p.593). T h e r e f o r e , o n e of the principal causes o f spiral f l o w in either closed or o p e n channels is the existence of f r i c t i o n on channel walls and beds. This causes higher filamental velocities near the c e n t e r of the c h a n n e l and slower velocities along the walls and bed. T h e second principal cause o f spiral flow is centrifugal f o r c e which causes a d e f l e c t i o n of w a t e r particles f r o m a straight line m o t i o n . F u r t h e r e x p e r i m e n t a t i o n revealed t h a t in the latter half of bends along c o n v e x banks t h e r e were regions of u n s t e a d y flow w h e r e b y the w a t e r m o v e d u p s t r e a m at times. This o c c u r r e d at a p o i n t slightly b e y o n d the m i d - p o i n t of the b e n d and above m i d - d e p t h w h e r e t h e streamlines a p p e a r e d to leave the inside wall o f the channel. Considering the velocity c o m p o n e n t of a thin layer of fluid along t h e w a t e r surface, it is a p p a r e n t t h a t as the particles are moving d o w n s t r e a m t h e y are also slowly edging t o w a r d t h e outside o f the bend. T h e particles e x p e r i e n c e a t y p e o f diverging channel effect, with the streamlines diverging f r o m t h e inside wall causing a b a c k f l o w (Fig.5). Thus,
Fig. 5. Distribution of filamental velocities and secondary currents in a closed channel. The filamental velocities are greatest on the outside of the bend. Secondary currents develop along the inside of the bend at A. The cross-sectional diagram represents the currents at point A. (After Mockmore, 1944).
129
in the lower p o r t i on of the area of backflow the velocity c o m p o n e n t of the water particles was toward the inside of the bend. This resulted in a less t u r b u l e n t region than the water nearer the surface (p.599). These theoretical and quantitative experiments on the behavior of fluids in curves have lead to several conclusions: spiral flow exists along the bends of b o th open and closed channels with usually a double spiral in a closed channel; the filamental velocities are greater near the outside bend than at the inside bend, which results in greater erosive power along the outside bank; the physical characteristics of spiral m o t i o n cont ri but e to the movem en t of bed load bot h d o w n s t r e a m and toward the inside of bends; and, as a result of spiral motion, at about three-fourths of the way around the inside bend, there is a t e n d e n c y for the d e v e l o p m e n t of an eddy or slack water which is conducive to sediment deposition and bar form at i on (p.617). Mo ck mo r e presents a significant study of flow around bends. Although the experiments are c o n d u c t e d in a laboratory, it is recognized that a study of flow conditions in the bend of a natural stream involves many com pl ex factors, such as the difficulty of obtaining accurate measurements of filamental velocities at specified points and m a n y irregularities in the stream channel as well as the instability of channel beds themselves. Mathematical analysis is offered to illustrate and prove the existence of spiral flow as well as its effect on the inside and outside of meander bends. Particle m o v e m e n t is observed through the use of dyes, rice grains, and chemical globules. The filament of m a x i m u m velocity is initially and for most of the j o u r n e y around the bend, near the inside. This is in contrast to most studies in h y d r o l o g y which reveal that the filament of m a x i m u m velocity is always on the outside of the bend. The m o v e m e n t of water around a bend is affected by m a n y factors in addition to the rate of flow. These are the relative curvature of the bend, the shape of the channel cross section, the length of the bend, the a m o u n t of discharge, the a m o u n t and t y p e of sediment and channels roughness. The interrelationship between these factors and the flow of water with the resultant meander devel opm e nt was not investigated. F u r t h e r m o r e , the experiments and analyses were c o n d u c t e d with a rectangular pipe which render the results of d o u b t f u l worth. Channels in sediment flumes or in nature are not rectangular. Maximum velocity filaments could possibly be closer to the inside bend if the meander g e o m e t r y is not proport i onal to the discharge conditions, which might be the case with a rectangular channel. The fact that he used a rectangular channel rather than a r o u n d e d channel is the major objection to M ockm or e's experiments, and tenders the validity of his results questionable.
Development of shoals and deeps Through field and l a bor at or y observations and analyses, Leliavsky (1955) has a t t e m p t e d to explain river meandering as the t e n d e n c y of rivers to create a succession of shoals and deeps primarily as a result of perm anent , local,
130 secondary circulations within t he water. In model experiments a straight channel became sinuous by eroding concave banks and building convex bars. Deeps f o r med in the concave curves while shoals developed along the inflection points between the curves (p.94). In further investigations of natural alluvial channels, it was observed t ha t contiguous stream lines are never parallel to one anot her or to the river banks. Additionally, the greater the curvature of stream trajectories, the deeper the scoured channel beneath. On the bases of his experiments and observations as well as the observations and experiments of others, Leliavsky developed a theoretical explanation of meandering. A straight water channel marks the starting point. Initial channel d e f o r m a t i o n occurs by the presence of a scour or sediment deposit along the edge of the straight channel. This channel obstruction forces stream lines to deviate f r om their original straight courses to curved ones wh er eb y a centrifugal force is developed. This centrifugal effect represents the main cause of a helicoidal current which erodes soil from the concave bank, transports the sediment across the channel, and deposits it on the convex bank, thus initiating and progressively intensifying the process of meandering. Therefore, the slightest irregularity in channel form which causes a shift of stream lines from straight to curved courses creates a focal point for the erosion process which ultimately leads to the devel opm ent of a meander (p.122). F u r t h e r m o r e , the piling up of surface water on the outside bend contributes to the centrifugal effect which is an i m p o r t a n t factor in the mechanics of helicoidal flow. The downward c o m p o n e n t of the helicoidal current at the concave bank of a meandering stream is highly erosive. The same current carries the eroded sediment across the channel, depositing it as a wide shallow bar on the convex bank. Thus one bank erodes, the o t her extends, causing the river to shift towards the eroded, concave bank. This is the fundamental process of meandering (Fig.6). Through the stud5 of theoretical and experimental models, field observations and tests, and the use of quantitative methods and empirical formulae, Leliavsky has established t ha t helicoidal flow is the i m p o r t a n t cause of meandering. His methods and procedures are a significant a t t e m p t f ..~'~..:..':'V.. N
, ~:.:.. : • .':L:.,:..
LEFT
SIDE
Fig. 6. Cross-section of a m e a n d e r b e n d . A r e p r e s e n t s the inside of the b e n d ; B r e p r e s e n t s the o u t s i d e o f the b e n d w h e r e surface w a t e r piles up. T h e diagram o n the right s h o w s s e d i m e n t being carried f r o m o n e side o f the c h a n n e l across to the o p p o s i t e side.
131 to prove the existence and effect of helicoidal flow b o t h in model experiments and in natural rivers. However, the results of these studies indicate that sediment is moved f r om t he concave bank across the channel to be deposited on the convex bank. Friedkin's investigations illustrated quite conclusively th at most of the sediment eroded from the concave bank is carried d o wn s tr eam and deposited on the same side of the stream. This appears to be the major discrepancy in Leliavsky's t h e o r y on the distribution of sediment. His work, however, has established the existence of helicoidal currents in b o t h the lab o r at or y and the field.
Transverse flow Th r o u g h a series of l abor a t or y experiments, Bagnold (1960) sought to understand and explain the de ve l opm ent of river meanders. It was observed that a transverse flow occurs when water flows t hrough a curved circular pipe. This t y p e of flow occurs when frictional drag is exerted on the tangential flow. The radial acceleration on the flow creates an increase of pressure on the outside of the bend and a corresponding decrease on the inside. Within the main flow the centrifugal pressure gradient balances the radial pressure gradient. However, an imbalance along the periphery remains as a result of b o u n d a r y friction, causing a reduct i on in the flow velocity and a corresponding decrease in the radial pressure gradient along the periphery. " H e n c e water must flow radially inwards over the periphery, dow n the radial pressure gradient and its place is taken by an out w ard flow across the centerline of the cross section. The effect of the transverse flow is to reduce the internal shear due to the direction of the flow around the b e n d " (p.135). In order to occur and to reach o p t i m u m velocity, transverse flow requires angular acceleration within the confines of the bend. Resistance to flow through the channel arc is the sum of the energy e x p e n d e d to create the transverse flow and the energy dissipated by wall friction. Therefore, the total resistance to flow owing to the channel bend is the sum of two forces necessary to overcome wall friction and inertia (p.136). As a result a stage is ultimately reached w he r eby flow along the inner b o u n d a r y becomes unstable and breaks away leaving a zone of unstable and confused water. Generally, diverging flow has been responsible for the breakaway and the ensuing unstable region. With an increase in divergence, flow becomes rapidly unstable in the region of the diverging boundaries resulting in sudden increases in energy dissipation. As the flow moves t hrough the bend it tends to decrease along the inside region of greatest curvature. As a result, the shear rate, shear stress in the fluid, and pressure gradient in the direction of flow are reduced along the inside of the bend. Ultimately the pressure gradient along the inside curve becomes r e duc e d to such a level t hat a backward m o v e m e n t occurs. Thus, flow in the inside curve becomes unstable resulting in local eddying accompanied by increased energy dissipation (Fig.7).
132 A
Fig. 7. Stream velocity in a curved c h a n n e l . With increased curvature m a x i m u m flow velocity o c c u r s at A. R e d u c e d velocity along the inside o f the b e n d results in a q u i e t z o n e of u n s t a b l e flow at B, w h e r e energy is dissipated in e d d y i n g .
This co mb ina t i on of transverse and helicoidal flow accompanied by secondary circulations accounts for the development of meanders. The increased curvature causes an increase in divergence and a sudden increase of energy dissipation on the outside of the bend, causing erosion to occur. The eddying currents and areas of slack water along the inside bend provide conditions for deposition to occur. Thus the meander is born and continues to grow in the manner outlined above. Bagnold's study is an extension of earlier investigations of helicoidal flow. Most of his conclusions result f r om model studies with a seeming lack of application to natural streams. However, the experiments are c o n d u c t e d carefully and t h e y are quantitatively analyzed. Nevertheless, nothing new has been presented. Several of t he experiments did provide further conclusive p r o o f for the existence and effect of helicoidal flow. Sorely needed is more direct application of laboratory findings to natural streams. One major outgrowth of Bagnold's work was the development and application of the c o n c e p t of least ef f or t to stream flow which was later used by Leopold. Inherent helicoidal currents Tanner (1960) has investigated the presence of helicoidal flow in a stream and its effect as a possible cause of meandering. A meander pattern develops b e c a u s e helicoidal currents ar~ inherent in straight-flowing streams, rather than the c o n c e p t that helicoidal currents develop after the meander pat t ern has begun to f o r m (p.994). This concept was simply tested in the laboratory using a piece o f glass that had been covered with dust over several years. When the glass plate was covered with dust, t urbul ent flow and meandering occurred, while on clean glass laminar flow was observed. When the helicoidal current becomes established, t h e main stream flow tends to flow parallel to the direction of the bed current rather than parallel to the direction of the surface current. This characteristic of the flow is primarily a result of the effects of channel curvature. Once helicoidal flow
133 began the descending side of the helix showed strengthening while the ascending side showed retardation, resulting in a deflection of the stream toward the slower, rising side. Thus the spiral motion within the helicoidal current causes the main stream to be diverted away from the m a x i m u m slope until the effect of gravity is largely reduced. Then, the altered gravity component causes an alteration of the channel and current characteristics resulting in the formation of a n e w helicoidal current spiraling in the opposite direction. These characteristics of helicoidal flow, then, cause the channel course to become sinuous. Thus, the sequence of events for the establishment of meanders is turbulence, helicoidal flow, and meandering. A major shortcoming of this t h e o r y is the factor or factors necessary to establish helicoidal current (p.995). In later experimentation, Tanner (1963) observed that roughness was an additional factor that contributed to meander development. Thus the new sequence of meander development is roughness, turbulence, helicoidal flow, and meandering. Verification of spiral flow was made with tracers. Further observations concluded that in a large river system where the load was mobile, coherent, and abundant, meandering resulted. The turbulence factor associated with meanders results from an adverse pressure gradient associated with sudden changes in channel depth or width or changes in velocity, or changes effected by obstruction to flow (p.41). In a subsequent study, Tanner (1968) further establishes the fact that meandering is a product of secondary overturn--helicoidal flow. The variability of b o t t o m roughness contributes to spiral development by causing a variation in stream flow velocity between the b o t t o m region and the surface region. Thus, "a helix in the earth's gravity field must have a curved axis, inasmuch as the descending waters will be accelerated relative to ascending fluid" (Tanner, 1968, p.960). Additional factors that contribute to meander development include stream gradient, grain size, cohesiveness of sediment, stream discharge and the channel pattern. The initial development of meandering appears as a pattern of pools (deep zones) and riffles (shallow zones) occurring in the stream channel (see Fig.2). The formation of deeps and shoals is related to flow velocities within the bend of a stream. Deeps form in zones of maximum flow velocity which occurs near channel walls while shoals form in zones of lower flow velocity. On meandering streams the deeps occur on the outside banks of meander loops where undercutting is occurring as a result of the maximum flow velocity along this part of the bend. The shoals develop between the deeps of each bend along the inside bank as a result of lower flow velocity allowing sediment deposition. The development of the sinuous maximum-velocity flow lines (helicoidal flow) and the ensuing pool and riffle b o t t o m profile may be related to the characteristics of channel-wall sediment. A single patch would be sufficient to obstruct the stream flow, causing a deviation of flow lines with an ensuing overcompensated counter deviation and a continuing and progressive series
134 of deviations. This t y p e of mechanism can operate only where a bend causes the initial flow deviation. Helicoidal flow and a resultant distribution of stream velocities occurs, causing erosion and deposition in such a manner that meanders develop. Tanner attributes the cause of meandering to helicoidal flow which is initiated by an obstruction to flow, causing a deviation of flow lines with a thread of m a x i m u m flow velocity developing. The effect of other variables such as stream gradient, grain size, sediment cohesiveness, stream discharge and channel pattern on meander devel opm ent is also considered. Many of the conclusions are based on laboratory experiments, although Tanner did succeed in observing helical flow in natural streams with the aid of floats and tracers. F u r t h e r m o r e , the conclusions reached in these studies recognize in addition to helical flow the i m p o r t a n c e of ot her factors such as roughness and turbulence in causing meandering to occur. The use of empirical m eth o d s additionally enhance the experiments and results.
Channel roughness and flow resistance Shen (1961) c o n d u c t e d laboratory experiments in which he observed the characteristics of channel bed patterns as a factor in flow resistance which ultimately may cause a river to meander. The development of different bed patterns is closely related to the rate of sediment transport, which is a function of the ratio between the resistance of the grain to m o t i o n and the force ex er ted on a grain by flow (p.17). F u r t h e r m o r e , the differences of the shear stress between the rough and s m o o t h e r bed surfaces induce secondary circulations that are perpendicular to the direction of main flow. As a result the direction of the b o t t o m current is such that it is always flowing t ow ard the rougher surface. The upstream flow system creates a circulation that results in a scour or deep forming along one side of the channel. A second circulation occurs in the opposite direction as a result of flow resistance being caused by channel roughness. As the flow moves downstream, the rotation of these secondary circulations increases to such a magnitude that a scour hole is created. The sediment is moved from the left-hand t o the right-hand side of the channel (see Fig.6). At the same time, a second circulation in the opposite direction occurs as a result of bank roughness at the left-hand side. This process is repeated gradually downstream. Shen bases his postulations on Hans A. Einstein and Huon Li's work with the vorticity equation in t ur bul ent viscous flow. The initial condition in straight u n if o r m flow is that the secondary current be zero everywhere. A secondary circulation will occur whenever the equation value differs from zero. This situation could develop near the frictional b o u n d a r y where lines of co n s tan t velocity are not parallel (p.20). Shen and Komura (1968) in a later experimental study observed that all eddies that developed between the free water surface and the channel wall
135 stretched asymmetrically in the direction of flow, whereby a circulation developed which tended to move water up along and out from the bank toward the center of the channel. Thus a clockwise circulation was created along the left bank and counterclockwise along the right bank. In a clockwise circulation, surface water, having a higher velocity, tends to move toward the right bank while the lower-velocity b o t t o m water tends to move toward the left bank. As a result of the higher velocity water along the right bank, a large shear stress is created there. Eventually the initial clockwise circulation would be reduced to zero whereby a counterclockwise circulation would result. With the newly created counterclockwise circulation, sediment would be transported from the left to the right bank. This process would continually repeat, resulting in the development of alternate scour holes and bars and, consequently, the development of meanders (p.1002). The resulting meander pattern is a function of channel roughness and the variation of flow conditions as discussed above. This work is, perhaps, the best explanation of the meander as caused by helicoidal flow. The experiments are enhanced by quantitative methods and some field observations. There appears to be one discrepancy, however. The results indicate sediment movement across the channel, which is in contradiction to Friedkin's study showing most sediment being deposited on the same side of the stream. There is agreement with Leliavsky on the transportation and distribution of the sediment. Furthermore, the authors do not indicate the a m o u n t of sediment carried across the channel, nor the mode of transportation, whether it be suspension or bed load. The main focus of the work is with the mechanics of helicoidal flow and the quite conclusive proof of its existence and effect. In addition, the close relationship between channel roughness and helicoidal flow is explored. These few experiments concern alluvial streams containing relatively homogeneous material. There is need for study of streams having different sediment types as well as heterogeneous sediments. M i n i m u m variance
Leopold and Langbein (1966) postulated from the theory of minimum variance that meandering is the most probable form of channel geometry. In fact, variances in bed shear and friction are uniformly lower in a curved stretch than a straight stretch. Meanders will usually appear wherever the river flows through fine-grained alluvium that is easily eroded and transported while still being cohesive enough to maintain firm banks. The development of this meander t h e o r y is based upon the random walk method. A random walk between two points traces a curve termed "sinegenerated" curve which closely approximates the shape of real river meanders (p.62). A sine-generated curve has the smallest variation of changes of direction -- sums of the squares of this change are less than for any other regular curve of the same length. Furthermore, it is also the curve of
136 minimum total w or k in bending (p.63). A river changes its course by its ability to erode, transport, and deposit the material t hat composes the river channel. Where the flow velocity gradient is higher against the channel bank, local eddies are created whereby a c o n c e n t r a t i o n of energy occurs which localizes erosion. This circulation system occurs as the surface water plunges toward the bed near the concave bank while the b o t t o m water rises t ow ard the surface near the convex bank. This circulation along with downstream m o v e m e n t gives each particle a roughly helicoidal flow which reverses its direction of r o t a t i on with each successive meander. The ultimate result of this circulation pat t er n is erosion on the outside bend and deposition on the inside bend which leads to the devel opm e n t of a meander. The meandering pattern is the most probable result of a combination of processes that tend to eliminate concentrations of energy loss while attempting to reduce total energy loss to a minimum. Thus, a sine-generated curvature assumed by meanders is the most suitable curvature for this energy relationship. Leopold and Langbein did not emerge with a new t h e o r y of meandering. The existence and effect of helicoidal flow together with many ot her factors was recognized. The efforts of this study were concerned mainly with establishing that a sine-generated curve is the path of least resistance. Energy expenditure is at a m i ni m um in this t y p e of curve. Ultimately the characteristic curvature of a meander will be a sine-curve. The work is primarily theoretical. Additional field testing is required to establish p r o o f of sinecurved meanders. However, the c o n c e p t of sine-curvature and the m e t h o d of the r a n d o m walk has merit for further applied research. CONCLUSIONS
On the basis of these varied theories on the devel opm ent of meanders, the following co mb i ne d conclusions are presented. (1) Most theories of meandering are the result of experimental laboratory studies. (2) Several studies showed that channel devel opm ent resulted from a c o mb in atio n of ideal conditions, which were: constant flow, uniform bed material, and no variable bed-load discharge by tributary channels. (3) Valley slope, bed-load, discharge, bed resistance and transverse oscillations work in com bi na t i on to cause a sinuous channel to develop. (4) Meandering develops by an initial deflection causing material to be eroded on concave banks and deposited on convex banks due to changes in direction of stream energy. (5) Material eroded on the concave bank is deposited on the same side further downstream. (6) Most present-day theories attribute meandering to helicoidal flow; at times it is accompanied by secondary circulations. (7) Meandering stream channels are characterized by series of deeps and shoals or pools and riffles.
137 (8) Bed shear, friction of bed and walls, width, depth, velocity, slope, and roughness all are important to meandering. (9) Several authors feel helicoidal flow is inherent in straight stream channels; therefore, helicoidal flow is already present before meandering develops. The theories for meandering do not clearly explain the cause of helicoidal flow. Most of the theorization is based upon laboratory experimentation, but lacks thorough exploration in the field. Authors like Matthes and Friedkin explain meandering as a change in the direction of stream energy but do not fully account for it. They mention helicoidal flow but discount its importance either through poor observations, lack of adequate testing, or lack of comprehension. One ponders why more consideration was not given to the concept of helicoidal flow which had been presented several years earlier by such men as Thomson, Lacy, Matthes, Mockmore, and others.
Development of meanders The characteristics of helicoidal flow seem to be the best explanation for the development of meanders. However, most of its intricacies have been learned through flume experiments in the laboratory. It is clearly evident that more work needs to be done with natural streams. New procedures and methods need to be developed to test this theory in natural streams with all the varied and uncontrolled factors present. Prevalent ideas concern meandering as being dependent not only on the stream flow but also on the rate of stream discharge, sediment load, size and type of sediment, channel roughness, depth, width, velocity of flow, and quality of the water itself. Since there is a close interrelationship of all these factors, they must be jointly considered for the development of a theory. A theory dealing solely with the characteristics of water flow is inadequate i; the other factors are not considered. Furthermore, studies should include hypothesis testing in a variety of material both homogeneous and heterogeneous. The laboratory has its place for isolating factors of importance and developing theories on the type and manner of occurrence. However, meaningful concepts should be able to stand the test when applied to many varied natural meandering streams. Perhaps, no one theory would adequately explain the complexities of meandering but rather a combination of theories that vary with individual streams and channel mediums. REFERENCES Bagnold, R.A., 1960. Some aspects of the shape of river meanders. U.S. Geol. Surv. Prof. Pap., 282-E:135--144. Chorley, R.J., Dunn, A.J. and Beckinsale, R.P., 1964. The History of the Study of Landforms; or the Development of Geomorphology. Methuen, London, 1:607--609. Friedkin, J.F., 1945. A Laboratory Study of the Meandering of Alluvial Rivers. U.S. Waterways Experiment Station, Vicksburg, Miss., 39 pp.
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Leliavsky, S., 1955. An Introduction to Fluvial Hydraulics. Constable, London, 257 pp. Leopold, L.B., 1960. River meanders. Geol. Soc. Am. Bull., 71 (6):769--794. Leopold, L.B. and Langbein, W.B., 1966. River meanders. Sci. American, 214:60--70. Leopold, L.B. and Wolman, M.G., 1957. River channel patterns: braided, meandering, and straight. U.S. Geol. Surv. Prof. Pap., 282-B: 39--85. Matthes, G.H., 1941. Basic aspects of stream meanders. Trans. Am. Geophys. Un., 22:632--636. Mockmore, C.A., 1944. Flow around bends in stable channels. Am. Soc. Civil Eng. Trans., 109:593--628. Quraishy, M.S., 1944. The origin of curves in rivers. Current Science, 13 (2):36--39. Schumm, S.A., 1960. The shape of alluvial channels in relation to sediment type. U.S. Geol. Surv. Prof. Pap. 352-B:17--30. Schumm, S.A., 1963. Sinuosity of alluvial rivers on the Great Plains. Geol. Soc. Am. Bull. 74(9):1089--1099. Shen, H.W., 1961. A Study on Meandering and other Bed Patterns in Straight Alluvial Channels. Water Resources Center, Contrib. 33, Hydraulic Laboratory, Univ. California, Berkeley, Calif., 68 pp. Shen, H.W., and Komura, S., 1968. Meandering tendencies in straight alluvial channels. J. Hydraulics Div., Am. Soc. Civ. Eng., 94, (H 44), Proc. Pap. 6042:997--1016. Tanner, W.F., 1960. Helicoidal flow, a possible cause of meandering. J. Geophys. Res., 65(3):993--995. Tanner, W.F. 1963. Spiral flow in rivers, shallow seas, dust devils, and models. Science 139:41--42. Tanner, W.F., 1968. Rivers -- meandering and braiding. In: R.W. Fairbridge (Editor), The Encyclopedia of Geomorphology. Reinhold, New York, N.Y., pp.957--963. Tiffany, J.B. and Nelson, G.A., 1939. Studies of meandering of model streams. Trans. Am. Geophys. Un., 20:644--649. (Accepted for publication October 10, 1973)