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Meander

A meander is one of a series of regular sinuous curves in the channel of a river or other watercourse. It is produced as a watercourse erodes the sediments of an outer, concave bank (cut bank or river cliff) and deposits sediments on an inner, convex bank which is typically a point bar. The result of this coupled erosion and sedimentation is the formation of a sinuous course as the channel migrates back and forth across the axis of a floodplain.[1][2]

A stream bed following a tilted valley. The maximum gradient is along the down-valley axis represented by a hypothetical straight coast channel. Meanders develop, which lengthen the course of the stream, decreasing the gradient.
Meanders of the Rio Cauto at Guamo Embarcadero, Cuba
The Jordan River, near the Dead Sea, 1937

The zone within which a meandering stream periodically shifts its channel is known as a meander belt. It typically ranges from 15 to 18 times the width of the channel. Over time, meanders migrate downstream, sometimes in such a short time as to create civil engineering challenges for local municipalities attempting to maintain stable roads and bridges.[1][2]

The degree of meandering of the channel of a river, stream, or other watercourse is measured by its sinuosity. The sinuosity of a watercourse is the ratio of the length of the channel to the straight line down-valley distance. Streams or rivers with a single channel and sinuosities of 1.5 or more are defined as meandering streams or rivers.[1][3]

Origin of term edit

The term derives from the winding river Menderes located in Asia-Minor and known to the Ancient Greeks as Μαίανδρος Maiandros (Latin: Maeander),[4][5] characterised by a very convoluted path along the lower reach. As a result, even in Classical Greece (and in later Greek thought) the name of the river had become a common noun meaning anything convoluted and winding, such as decorative patterns or speech and ideas, as well as the geomorphological feature.[6] Strabo said: ‘...its course is so exceedingly winding that everything winding is called meandering.’[7]

The Meander River is south of Izmir, east of the ancient Greek town of Miletus, now Milet, Turkey. It flows through series of three graben in the Menderes Massif, but has a flood plain much wider than the meander zone in its lower reach. Its modern Turkish name is the Büyük Menderes River.[8]

Governing physics edit

 
Straight channel culminating in a single bend

Meanders are a result of the interaction of water flowing through a curved channel with the underlying river bed. This produces helicoidal flow, in which water moves from the outer to the inner bank along the river bed, then flows back to the outer bank near the surface of the river. This in turn increases carrying capacity for sediments on the outer bank and reduces it on the inner bank, so that sediments are eroded from the outer bank and redeposited on the inner bank of the next downstream meander.[9]

When a fluid is introduced to an initially straight channel which then bends, the sidewalls induce a pressure gradient that causes the fluid to alter course and follow the bend. From here, two opposing processes occur: (1) irrotational flow and (2) secondary flow. For a river to meander, secondary flow must dominate.

Irrotational flow: From Bernoulli's equations, high pressure results in low velocity. Therefore, in the absence of secondary flow we would expect low fluid velocity at the outside bend and high fluid velocity at the inside bend. This classic fluid mechanics result is irrotational vortex flow. In the context of meandering rivers, its effects are dominated by those of secondary flow.

Secondary flow: A force balance exists between pressure forces pointing to the inside bend of the river and centrifugal forces pointing to the outside bend of the river. In the context of meandering rivers, a boundary layer exists within the thin layer of fluid that interacts with the river bed. Inside that layer and following standard boundary-layer theory, the velocity of the fluid is effectively zero. Centrifugal force, which depends on velocity, is also therefore effectively zero. Pressure force, however, remains unaffected by the boundary layer. Therefore, within the boundary layer, pressure force dominates and fluid moves along the bottom of the river from the outside bend to the inside bend. This initiates helicoidal flow: Along the river bed, fluid roughly follows the curve of the channel but is also forced toward the inside bend; away from the river bed, fluid also roughly follows the curve of the channel but is forced, to some extent, from the inside to the outside bend.

The higher velocities at the outside bend lead to higher shear stresses and therefore result in erosion. Similarly, lower velocities at the inside bend cause lower shear stresses and deposition occurs. Thus meander bends erode at the outside bend, causing the river to becoming increasingly sinuous (until cutoff events occur). Deposition at the inside bend occurs such that for most natural meandering rivers, the river width remains nearly constant, even as the river evolves.[10]

In a speech before the Prussian Academy of Sciences in 1926, Albert Einstein suggested that because the Coriolis force of the earth can cause a small imbalance in velocity distribution, such that velocity on one bank is higher than on the other, it could trigger the erosion on one bank and deposition of sediment on the other that produces meanders[11] However, Coriolis forces are likely insignificant compared with other forces acting to produce river meanders.[12]

Meander geometry edit

 
Uvac canyon meander, Serbia
 
Meanders on the River Clyde, Scotland

The technical description of a meandering watercourse is termed meander geometry or meander planform geometry.[13] It is characterized as an irregular waveform. Ideal waveforms, such as a sine wave, are one line thick, but in the case of a stream the width must be taken into consideration. The bankfull width is the distance across the bed at an average cross-section at the full-stream level, typically estimated by the line of lowest vegetation.

As a waveform the meandering stream follows the down-valley axis, a straight line fitted to the curve such that the sum of all the amplitudes measured from it is zero. This axis represents the overall direction of the stream.

At any cross-section the flow is following the sinuous axis, the centerline of the bed. Two consecutive crossing points of sinuous and down-valley axes define a meander loop. The meander is two consecutive loops pointing in opposite transverse directions. The distance of one meander along the down-valley axis is the meander length or wavelength. The maximum distance from the down-valley axis to the sinuous axis of a loop is the meander width or amplitude. The course at that point is the apex.

In contrast to sine waves, the loops of a meandering stream are more nearly circular. The curvature varies from a maximum at the apex to zero at a crossing point (straight line), also called an inflection, because the curvature changes direction in that vicinity. The radius of the loop is the straight line perpendicular to the down-valley axis intersecting the sinuous axis at the apex. As the loop is not ideal, additional information is needed to characterize it. The orientation angle is the angle between sinuous axis and down-valley axis at any point on the sinuous axis.

 
Concave bank and convex bank, Great Ouse Relief Channel, England

A loop at the apex has an outer or concave bank and an inner or convex bank. The meander belt is defined by an average meander width measured from outer bank to outer bank instead of from centerline to centerline. If there is a flood plain, it extends beyond the meander belt. The meander is then said to be free—it can be found anywhere in the flood plain. If there is no flood plain, the meanders are fixed.

Various mathematical formulae relate the variables of the meander geometry. As it turns out some numerical parameters can be established, which appear in the formulae. The waveform depends ultimately on the characteristics of the flow but the parameters are independent of it and apparently are caused by geologic factors. In general the meander length is 10–14 times, with an average 11 times, the fullbank channel width and 3 to 5 times, with an average of 4.7 times, the radius of curvature at the apex. This radius is 2–3 times the channel width.[14]

 
Meander of the River Cuckmere in East Sussex, Southern England

A meander has a depth pattern as well. The cross-overs are marked by riffles, or shallow beds, while at the apices are pools. In a pool direction of flow is downward, scouring the bed material. The major volume, however, flows more slowly on the inside of the bend where, due to decreased velocity, it deposits sediment.[15]

The line of maximum depth, or channel, is the thalweg or thalweg line. It is typically designated the borderline when rivers are used as political borders. The thalweg hugs the outer banks and returns to center over the riffles. The meander arc length is the distance along the thalweg over one meander. The river length is the length along the centerline.[15]

Formation edit

 
Life history of a meander

Once a channel begins to follow a sinusoidal path, the amplitude and concavity of the loops increase dramatically. This is due to the effect of helical flow which sweeps dense eroded material towards the inside of the bend, and leaves the outside of the bend unprotected and vulnerable to accelerated erosion. This establishes a positive feedback loop. In the words of Elizabeth A. Wood: "...this process of making meanders seems to be a self-intensifying process...in which greater curvature results in more erosion of the bank, which results in greater curvature..."[16]

The cross-current along the floor of the channel is part of the secondary flow and sweeps dense eroded material towards the inside of the bend.[17] The cross-current then rises to the surface near the inside and flows towards the outside, forming the helical flow. The greater the curvature of the bend, and the faster the flow, the stronger is the cross-current and the sweeping.[18]

Due to the conservation of angular momentum the speed on the inside of the bend is faster than on the outside.[19]

Since the flow velocity is diminished, so is the centrifugal pressure. The pressure of the super-elevated column prevails, developing an unbalanced gradient that moves water back across the bottom from the outside to the inside. The flow is supplied by a counter-flow across the surface from the inside to the outside.[20] This entire situation is very similar to the Tea leaf paradox.[21] This secondary flow carries sediment from the outside of the bend to the inside making the river more meandering.[22]

As to why streams of any size become sinuous in the first place, there are a number of theories, not necessarily mutually exclusive.

Stochastic theory edit

 
Meander scars, oxbow lakes and abandoned meanders in the broad flood plain of the Rio Negro, Argentina. 2010 photo from ISS.

The stochastic theory can take many forms but one of the most general statements is that of Scheidegger: "The meander train is assumed to be the result of the stochastic fluctuations of the direction of flow due to the random presence of direction-changing obstacles in the river path."[23] Given a flat, smooth, tilted artificial surface, rainfall runs off it in sheets, but even in that case adhesion of water to the surface and cohesion of drops produce rivulets at random. Natural surfaces are rough and erodible to different degrees. The result of all the physical factors acting at random is channels that are not straight, which then progressively become sinuous. Even channels that appear straight have a sinuous thalweg that leads eventually to a sinuous channel.

Equilibrium theory edit

In the equilibrium theory, meanders decrease the stream gradient until an equilibrium between the erodibility of the terrain and the transport capacity of the stream is reached.[24] A mass of water descending must give up potential energy, which, given the same velocity at the end of the drop as at the beginning, is removed by interaction with the material of the stream bed. The shortest distance; that is, a straight channel, results in the highest energy per unit of length, disrupting the banks more, creating more sediment and aggrading the stream. The presence of meanders allows the stream to adjust the length to an equilibrium energy per unit length in which the stream carries away all the sediment that it produces.

Geomorphic and morphotectonic theory edit

Geomorphic refers to the surface structure of the terrain. Morphotectonic means having to do with the deeper, or tectonic (plate) structure of the rock. The features included under these categories are not random and guide streams into non-random paths. They are predictable obstacles that instigate meander formation by deflecting the stream. For example, the stream might be guided into a fault line (morphotectonic).[25]

Associated landforms edit

Cut bank edit

A cut bank is an often vertical bank or cliff that forms where the outside, concave bank of a meander cuts into the floodplain or valley wall of a river or stream. A cutbank is also known either as a river-cut cliff, river cliff, or a bluff and spelled as cutbank.[1] Erosion that forms a cut bank occurs at the outside bank of a meander because helicoidal flow of water keeps the bank washed clean of loose sand, silt, and sediment and subjects it to constant erosion. As a result, the meander erodes and migrates in the direction of the outside bend, forming the cut bank.[26][27]

As the cut bank is undermined by erosion, it commonly collapses as slumps into the river channel. The slumped sediment, having been broken up by slumping, is readily eroded and carried toward the middle of the channel. The sediment eroded from a cut bank tends to be deposited on the point bar of the next downstream meander, and not on the point bar opposite it.[28][26] This can be seen in areas where trees grow on the banks of rivers; on the inside of meanders, trees, such as willows, are often far from the bank, whilst on the outside of the bend, the tree roots are often exposed and undercut, eventually leading the trees to fall into the river.[28][29]

Meander cutoff edit

 
The Rincon on Lake Powell in southern Utah. It is an incised cutoff (abandoned) meander.

A meander cutoff, also known as either a cutoff meander or abandoned meander, is a meander that has been abandoned by its stream after the formation of a neck cutoff. A lake that occupies a cutoff meander is known as an oxbow lake. Cutoff meanders that have cut downward into the underlying bedrock are known in general as incised cutoff meanders.[1] As in the case of the Anderson Bottom Rincon, incised meanders that have either steep-sided, often vertical walls, are often, but not always, known as rincons in the southwest United States.[30] Rincon in English is a nontechnical word in the southwest United States for either a small secluded valley, an alcove or angular recess in a cliff, or a bend in a river.[31]

Incised meanders edit

 
Glen Canyon, US

The meanders of a stream or river that has cut its bed down into the bedrock are known as either incised, intrenched, entrenched, inclosed or ingrown meanders. Some Earth scientists recognize and use a finer subdivision of incised meanders. Thornbury[32] argues that incised or inclosed meanders are synonyms that are appropriate to describe any meander incised downward into bedrock and defines enclosed or entrenched meanders as a subtype of incised meanders (inclosed meanders) characterized by a symmetrical valley sides. He argues that the symmetrical valley sides are the direct result of rapid down-cutting of a watercourse into bedrock.[1][33] In addition, as proposed by Rich,[34] Thornbury argues that incised valleys with a pronounced asymmetry of cross section, which he called ingrown meanders, are the result of the lateral migration and incision of a meander during a period of slower channel downcutting. Regardless, the formation of both entrenched meanders and ingrown meanders is thought to require that base level falls as a result of either relative change in mean sea level, isostatic or tectonic uplift, the breach of an ice or landslide dam, or regional tilting. Classic examples of incised meanders are associated with rivers in the Colorado Plateau, the Kentucky River Palisades in central Kentucky, and streams in the Ozark Plateau.[33][35]

 
Goosenecks of the San Juan River, SE Utah. There is a cut-off meander at right center.

As noted above, it was initially either argued or presumed that an incised meander is characteristic of an antecedent stream or river that had incised its channel into underlying strata. An antecedent stream or river is one that maintains its original course and pattern during incision despite the changes in underlying rock topography and rock types.[32][33] However, later geologists[36] argue that the shape of an incised meander is not always, if ever, "inherited", e.g., strictly from an antecedent meandering stream where its meander pattern could freely develop on a level floodplain. Instead, they argue that as fluvial incision of bedrock proceeds, the stream course is significantly modified by variations in rock type and fractures, faults, and other geological structures into either lithologically conditioned meanders or structurally controlled meanders.[33][35]

Oxbow lakes edit

The oxbow lake, which is the most common type of fluvial lake, is a crescent-shaped lake that derives its name from its distinctive curved shape.[37] Oxbow lakes are also known as cutoff lakes.[1] Such lakes form regularly in undisturbed floodplains as a result of the normal process of fluvial meandering. Either a river or stream forms a sinuous channel as the outer side of its bends are eroded away and sediments accumulate on the inner side, which forms a meandering horseshoe-shaped bend. Eventually as the result of its meandering, the fluvial channel cuts through the narrow neck of the meander and forms a cutoff meander. The final break-through of the neck, which is called a neck cutoff, often occurs during a major flood because that is when the watercourse is out of its banks and can flow directly across the neck and erode it with the full force of the flood.[28][38]

After a cutoff meander is formed, river water flows into its end from the river builds small delta-like feature into either end of it during floods. These delta-like features block either end of the cutoff meander to form a stagnant oxbow lake that is separated from the flow of the fluvial channel and independent of the river. During floods, the flood waters deposit fine-grained sediment into the oxbow lake. As a result, oxbow lakes tend to become filled in with fine-grained, organic-rich sediments over time.[28][38]

Point bar edit

A point bar, which is also known as a meander bar, is a fluvial bar that is formed by the slow, often episodic, addition of individual accretions of noncohesive sediment on the inside bank of a meander by the accompanying migration of the channel toward its outer bank.[1][26] This process is called lateral accretion. Lateral accretion occurs mostly during high water or floods when the point bar is submerged. Typically, the sediment consists of either sand, gravel, or a combination of both. The sediment comprising some point bars might grade downstream into silty sediments. Because of the decreasing velocity and strength of current from the thalweg of the channel to the upper surface of point bar when the sediment is deposited the vertical sequence of sediments comprising a point bar becomes finer upward within an individual point bar. For example, it is typical for point bars to fine upward from gravel at the base to fine sands at the top. The source of the sediment is typically upstream cut banks from which sand, rocks and debris has been eroded, swept, and rolled across the bed of the river and downstream to the inside bank of a river bend. On the inside bend, this sediment and debris is eventually deposited on the slip-off slope of a point bar.[1][26][27]

Scroll-bars edit

Scroll-bars are a result of continuous lateral migration of a meander loop that creates an asymmetrical ridge and swale topography[39] on the inside of the bends. The topography is generally parallel to the meander, and is related to migrating bar forms and back bar chutes,[40] which carve sediment from the outside of the curve and deposit sediment in the slower flowing water on the inside of the loop, in a process called lateral accretion. Scroll-bar sediments are characterized by cross-bedding and a pattern of fining upward.[41] These characteristics are a result of the dynamic river system, where larger grains are transported during high energy flood events and then gradually die down, depositing smaller material with time (Batty 2006). Deposits for meandering rivers are generally homogeneous and laterally extensive unlike the more heterogeneous braided river deposits.[42] There are two distinct patterns of scroll-bar depositions; the eddy accretion scroll bar pattern and the point-bar scroll pattern. When looking down the river valley they can be distinguished because the point-bar scroll patterns are convex and the eddy accretion scroll bar patterns are concave.[43]

Scroll bars often look lighter at the tops of the ridges and darker in the swales. This is because the tops can be shaped by wind, either adding fine grains or by keeping the area unvegetated, while the darkness in the swales can be attributed to silts and clays washing in during high water periods. This added sediment in addition to water that catches in the swales is in turn is a favorable environment for vegetation that will also accumulate in the swales.

Slip-off slope edit

Depending upon whether a meander is part of an entrenched river or part of a freely meandering river within a floodplain, the term slip-off slope can refer to two different fluvial landforms that comprise the inner, convex, bank of a meander loop. In case of a freely meandering river on a floodplain, a slip-off slope is the inside, gently sloping bank of a meander on which sediments episodically accumulate to form a point bar as a river meanders. This type of slip-off slope is located opposite the cutbank.[44] This term can also be applied to the inside, sloping bank of a meandering tidal channel.[45]

In case of an entrenched river, a slip-off slope is a gently sloping bedrock surface that rises from the inside, concave bank of an asymmetrically entrenched river. This type of slip-off slope is often covered by a thin, discontinuous layer of alluvium. It is produced by the gradual outward migration of the meander as a river cuts downward into bedrock.[46][47] A terrace on the slip-off slope of a meander spur, known as slip-off slope terrace, can formed by a brief halt during the irregular incision by an actively meandering river.[48]

Derived quantities edit

 
Meanders, scroll-bars and oxbow lakes in the Songhua River

The meander ratio[49] or sinuosity index[50] is a means of quantifying how much a river or stream meanders (how much its course deviates from the shortest possible path). It is calculated as the length of the stream divided by the length of the valley. A perfectly straight river would have a meander ratio of 1 (it would be the same length as its valley), while the higher this ratio is above 1, the more the river meanders.

Sinuosity indices are calculated from the map or from an aerial photograph measured over a distance called the reach, which should be at least 20 times the average fullbank channel width. The length of the stream is measured by channel, or thalweg, length over the reach, while the bottom value of the ratio is the downvalley length or air distance of the stream between two points on it defining the reach.

The sinuosity index plays a part in mathematical descriptions of streams. The index may require elaboration, because the valley may meander as well—i.e., the downvalley length is not identical to the reach. In that case the valley index is the meander ratio of the valley while the channel index is the meander ratio of the channel. The channel sinuosity index is the channel length divided by the valley length and the standard sinuosity index is the channel index divided by the valley index. Distinctions may become even more subtle.[51]

Sinuosity Index has a non-mathematical utility as well. Streams can be placed in categories arranged by it; for example, when the index is between 1 and 1.5 the river is sinuous, but if between 1.5 and 4, then meandering. The index is a measure also of stream velocity and sediment load, those quantities being maximized at an index of 1 (straight).

See also edit

References and notes edit

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  4. ^ "Meander". Merriam-Webster. Retrieved July 12, 2012.
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  7. ^ Strabo, Geography, Book 12 Chapter 8 Section 15.
  8. ^ Gürbüz, Alper; Kazancı, Nizamettin (2019). "The Büyük Menderes River: Origin of Meandering Phenomenon". Landscapes and Landforms of Turkey. World Geomorphological Landscapes. pp. 509–519. doi:10.1007/978-3-030-03515-0_29. ISBN 978-3-030-03513-6. S2CID 134826361.
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  13. ^ The technical definitions of this section rely heavily on Julien, Pierre Y. (2002). River Mechanics. Cambridge University press. pp. 179–184. ISBN 0-521-52970-0. In addition concepts are utilized from Graf, Walter (1984). Hydraulics of Sediment Transport. Water Resources Publications. pp. 261–265. ISBN 0-918334-56-X.
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  18. ^ Hickin 2003, p. 434.
  19. ^ Hickin 2003, p. 432. "In the absence of secondary flow, bend flow seeks to conserve angular momentum so that it tends to conform to that of a free vortex with high velocity at the smaller radius of the inner bank and lower velocity at the outer bank where radial acceleration is lower."
  20. ^ Hickin 2003, p. 432. "Near the bed, where velocity and thus the centrifugal effects are lowest, the balance of forces is dominated by the inward hydraulic gradient of the super-elevated water surface and secondary flow moves toward the inner bank."
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  42. ^ G. Wasser (2005). "A Comparison Of Meandering River Deposits From The Middle Belly River And Horsefly With Recent Milk River Valley Deposits; Central And Southern Alberta". Calgary, Alberta: Canadian Natural Resource Limited.
  43. ^ Norman D. Smith and John Rogers (1999). Fluvial Sedimentology (6 ed.). blackwell publishing. ISBN 0-632-05354-2.
  44. ^ Scheffers, A.M., May, S.M. and Kelletat, D.H., 2015. Forms by Flowing Water (Fluvial Features). In Landforms of the World with Google Earth. (pp. 183–244). Springer, Amsterdam, Netherlands. 391 pp. ISBN 978-94-017-9712-2
  45. ^ Keck, R., Maurer, D. and Watling, L., 1973. Tidal stream development and its effect on the distribution of the American oyster. Hydrobiologia, 42(4), pp. 369–379.
  46. ^ Davis, W.M., 1913. Meandering valleys and underfit rivers. Annals of the Association of American Geographers, 3(1), pp. 3–28.
  47. ^ Crickmay, C.H., 1960. Lateral activity in a river of northwestern Canada. The Journal of Geology, 68(4), pp. 377–391.
  48. ^ Herrmann, H. and Bucksch, H., 2014. Dictionary Geotechnical Engineering/Wörterbuch GeoTechnik: English-German/Englisch-Deutsch. Springer, Berlin, Germany. 1549 pp. ISBN 978-3-642-41713-9
  49. ^ Shaw, Lewis C. (1984). Pennsylvania Gazetteer of Streams Part II. Bulletin No. 16. Commonwealth of Pennsylvania, Department of Environmental Resources. p. 8. OCLC 17150333.
  50. ^ Gordon, Nancy D.; Thomas A. McMahon; Christopher J. Gippel; Rory J. Nathan (2005). Stream Hydrology: an Introduction for Ecologists: Second Edition. John Wiley and Sons. pp. 183–184. ISBN 0-470-84357-8.
  51. ^ Singh, R.Y. (2005). "Interface drainage analysis of a water divide". In Jansky, Libor; Haigh, Martin J.; Prasad, Hushila (eds.). Sustainable Management of Headwater Resources: Research from Africa and India. Tokyo, New York: United Nations University Press. pp. 87–106. ISBN 92-808-1108-8.

Bibliography edit

  • Hickin, Edward J. (2003). "Meandering Channels". In Middleton, Gerard V. (ed.). Encyclopedia of Sediments and Sedimentary Rocks. Kluwer Academic Encyclopedia of Earth Sciences. Dordrecht; Boston: Kluwer Academic Publishers. pp. 430–434. ISBN 1-4020-0872-4.
  • Leopold, Luna B.; Langbein, W.B. (June 1966). "River Meanders". Scientific American. 214 (6): 60. Bibcode:1966SciAm.214f..60L. doi:10.1038/scientificamerican0666-60.
  • Thonemann, P., The Maeander Valley: A historical geography from Antiquity to Byzantium (Cambridge, 2011) (Greek Culture in the Roman World Series).

External links edit

  • Movshovitz-Hadar, Nitsa; Alla Shmuklar (2006-01-01). "River Meandering and a Mathematical Model of this Phenomenon". Physicalplus. Israel Physical Society (IPS) (7).

meander, other, uses, disambiguation, meander, series, regular, sinuous, curves, channel, river, other, watercourse, produced, watercourse, erodes, sediments, outer, concave, bank, bank, river, cliff, deposits, sediments, inner, convex, bank, which, typically,. For other uses see Meander disambiguation A meander is one of a series of regular sinuous curves in the channel of a river or other watercourse It is produced as a watercourse erodes the sediments of an outer concave bank cut bank or river cliff and deposits sediments on an inner convex bank which is typically a point bar The result of this coupled erosion and sedimentation is the formation of a sinuous course as the channel migrates back and forth across the axis of a floodplain 1 2 A stream bed following a tilted valley The maximum gradient is along the down valley axis represented by a hypothetical straight coast channel Meanders develop which lengthen the course of the stream decreasing the gradient Meanders of the Rio Cauto at Guamo Embarcadero CubaThe Jordan River near the Dead Sea 1937The zone within which a meandering stream periodically shifts its channel is known as a meander belt It typically ranges from 15 to 18 times the width of the channel Over time meanders migrate downstream sometimes in such a short time as to create civil engineering challenges for local municipalities attempting to maintain stable roads and bridges 1 2 The degree of meandering of the channel of a river stream or other watercourse is measured by its sinuosity The sinuosity of a watercourse is the ratio of the length of the channel to the straight line down valley distance Streams or rivers with a single channel and sinuosities of 1 5 or more are defined as meandering streams or rivers 1 3 Contents 1 Origin of term 2 Governing physics 3 Meander geometry 4 Formation 4 1 Stochastic theory 4 2 Equilibrium theory 4 3 Geomorphic and morphotectonic theory 5 Associated landforms 5 1 Cut bank 5 2 Meander cutoff 5 3 Incised meanders 5 4 Oxbow lakes 5 5 Point bar 5 6 Scroll bars 5 7 Slip off slope 6 Derived quantities 7 See also 8 References and notes 9 Bibliography 10 External linksOrigin of term editThe term derives from the winding river Menderes located in Asia Minor and known to the Ancient Greeks as Maiandros Maiandros Latin Maeander 4 5 characterised by a very convoluted path along the lower reach As a result even in Classical Greece and in later Greek thought the name of the river had become a common noun meaning anything convoluted and winding such as decorative patterns or speech and ideas as well as the geomorphological feature 6 Strabo said its course is so exceedingly winding that everything winding is called meandering 7 The Meander River is south of Izmir east of the ancient Greek town of Miletus now Milet Turkey It flows through series of three graben in the Menderes Massif but has a flood plain much wider than the meander zone in its lower reach Its modern Turkish name is the Buyuk Menderes River 8 Governing physics edit nbsp Straight channel culminating in a single bendMeanders are a result of the interaction of water flowing through a curved channel with the underlying river bed This produces helicoidal flow in which water moves from the outer to the inner bank along the river bed then flows back to the outer bank near the surface of the river This in turn increases carrying capacity for sediments on the outer bank and reduces it on the inner bank so that sediments are eroded from the outer bank and redeposited on the inner bank of the next downstream meander 9 When a fluid is introduced to an initially straight channel which then bends the sidewalls induce a pressure gradient that causes the fluid to alter course and follow the bend From here two opposing processes occur 1 irrotational flow and 2 secondary flow For a river to meander secondary flow must dominate Irrotational flow From Bernoulli s equations high pressure results in low velocity Therefore in the absence of secondary flow we would expect low fluid velocity at the outside bend and high fluid velocity at the inside bend This classic fluid mechanics result is irrotational vortex flow In the context of meandering rivers its effects are dominated by those of secondary flow Secondary flow A force balance exists between pressure forces pointing to the inside bend of the river and centrifugal forces pointing to the outside bend of the river In the context of meandering rivers a boundary layer exists within the thin layer of fluid that interacts with the river bed Inside that layer and following standard boundary layer theory the velocity of the fluid is effectively zero Centrifugal force which depends on velocity is also therefore effectively zero Pressure force however remains unaffected by the boundary layer Therefore within the boundary layer pressure force dominates and fluid moves along the bottom of the river from the outside bend to the inside bend This initiates helicoidal flow Along the river bed fluid roughly follows the curve of the channel but is also forced toward the inside bend away from the river bed fluid also roughly follows the curve of the channel but is forced to some extent from the inside to the outside bend The higher velocities at the outside bend lead to higher shear stresses and therefore result in erosion Similarly lower velocities at the inside bend cause lower shear stresses and deposition occurs Thus meander bends erode at the outside bend causing the river to becoming increasingly sinuous until cutoff events occur Deposition at the inside bend occurs such that for most natural meandering rivers the river width remains nearly constant even as the river evolves 10 In a speech before the Prussian Academy of Sciences in 1926 Albert Einstein suggested that because the Coriolis force of the earth can cause a small imbalance in velocity distribution such that velocity on one bank is higher than on the other it could trigger the erosion on one bank and deposition of sediment on the other that produces meanders 11 However Coriolis forces are likely insignificant compared with other forces acting to produce river meanders 12 Meander geometry edit nbsp Uvac canyon meander Serbia nbsp Meanders on the River Clyde ScotlandThe technical description of a meandering watercourse is termed meander geometry or meander planform geometry 13 It is characterized as an irregular waveform Ideal waveforms such as a sine wave are one line thick but in the case of a stream the width must be taken into consideration The bankfull width is the distance across the bed at an average cross section at the full stream level typically estimated by the line of lowest vegetation As a waveform the meandering stream follows the down valley axis a straight line fitted to the curve such that the sum of all the amplitudes measured from it is zero This axis represents the overall direction of the stream At any cross section the flow is following the sinuous axis the centerline of the bed Two consecutive crossing points of sinuous and down valley axes define a meander loop The meander is two consecutive loops pointing in opposite transverse directions The distance of one meander along the down valley axis is the meander length or wavelength The maximum distance from the down valley axis to the sinuous axis of a loop is the meander width or amplitude The course at that point is the apex In contrast to sine waves the loops of a meandering stream are more nearly circular The curvature varies from a maximum at the apex to zero at a crossing point straight line also called an inflection because the curvature changes direction in that vicinity The radius of the loop is the straight line perpendicular to the down valley axis intersecting the sinuous axis at the apex As the loop is not ideal additional information is needed to characterize it The orientation angle is the angle between sinuous axis and down valley axis at any point on the sinuous axis nbsp Concave bank and convex bank Great Ouse Relief Channel EnglandA loop at the apex has an outer or concave bank and an inner or convex bank The meander belt is defined by an average meander width measured from outer bank to outer bank instead of from centerline to centerline If there is a flood plain it extends beyond the meander belt The meander is then said to be free it can be found anywhere in the flood plain If there is no flood plain the meanders are fixed Various mathematical formulae relate the variables of the meander geometry As it turns out some numerical parameters can be established which appear in the formulae The waveform depends ultimately on the characteristics of the flow but the parameters are independent of it and apparently are caused by geologic factors In general the meander length is 10 14 times with an average 11 times the fullbank channel width and 3 to 5 times with an average of 4 7 times the radius of curvature at the apex This radius is 2 3 times the channel width 14 nbsp Meander of the River Cuckmere in East Sussex Southern EnglandA meander has a depth pattern as well The cross overs are marked by riffles or shallow beds while at the apices are pools In a pool direction of flow is downward scouring the bed material The major volume however flows more slowly on the inside of the bend where due to decreased velocity it deposits sediment 15 The line of maximum depth or channel is the thalweg or thalweg line It is typically designated the borderline when rivers are used as political borders The thalweg hugs the outer banks and returns to center over the riffles The meander arc length is the distance along the thalweg over one meander The river length is the length along the centerline 15 Formation edit nbsp Life history of a meanderOnce a channel begins to follow a sinusoidal path the amplitude and concavity of the loops increase dramatically This is due to the effect of helical flow which sweeps dense eroded material towards the inside of the bend and leaves the outside of the bend unprotected and vulnerable to accelerated erosion This establishes a positive feedback loop In the words of Elizabeth A Wood this process of making meanders seems to be a self intensifying process in which greater curvature results in more erosion of the bank which results in greater curvature 16 The cross current along the floor of the channel is part of the secondary flow and sweeps dense eroded material towards the inside of the bend 17 The cross current then rises to the surface near the inside and flows towards the outside forming the helical flow The greater the curvature of the bend and the faster the flow the stronger is the cross current and the sweeping 18 Due to the conservation of angular momentum the speed on the inside of the bend is faster than on the outside 19 Since the flow velocity is diminished so is the centrifugal pressure The pressure of the super elevated column prevails developing an unbalanced gradient that moves water back across the bottom from the outside to the inside The flow is supplied by a counter flow across the surface from the inside to the outside 20 This entire situation is very similar to the Tea leaf paradox 21 This secondary flow carries sediment from the outside of the bend to the inside making the river more meandering 22 As to why streams of any size become sinuous in the first place there are a number of theories not necessarily mutually exclusive Stochastic theory edit nbsp Meander scars oxbow lakes and abandoned meanders in the broad flood plain of the Rio Negro Argentina 2010 photo from ISS The stochastic theory can take many forms but one of the most general statements is that of Scheidegger The meander train is assumed to be the result of the stochastic fluctuations of the direction of flow due to the random presence of direction changing obstacles in the river path 23 Given a flat smooth tilted artificial surface rainfall runs off it in sheets but even in that case adhesion of water to the surface and cohesion of drops produce rivulets at random Natural surfaces are rough and erodible to different degrees The result of all the physical factors acting at random is channels that are not straight which then progressively become sinuous Even channels that appear straight have a sinuous thalweg that leads eventually to a sinuous channel Equilibrium theory edit In the equilibrium theory meanders decrease the stream gradient until an equilibrium between the erodibility of the terrain and the transport capacity of the stream is reached 24 A mass of water descending must give up potential energy which given the same velocity at the end of the drop as at the beginning is removed by interaction with the material of the stream bed The shortest distance that is a straight channel results in the highest energy per unit of length disrupting the banks more creating more sediment and aggrading the stream The presence of meanders allows the stream to adjust the length to an equilibrium energy per unit length in which the stream carries away all the sediment that it produces Geomorphic and morphotectonic theory edit Geomorphic refers to the surface structure of the terrain Morphotectonic means having to do with the deeper or tectonic plate structure of the rock The features included under these categories are not random and guide streams into non random paths They are predictable obstacles that instigate meander formation by deflecting the stream For example the stream might be guided into a fault line morphotectonic 25 Associated landforms editCut bank edit Main article Cut bank A cut bank is an often vertical bank or cliff that forms where the outside concave bank of a meander cuts into the floodplain or valley wall of a river or stream A cutbank is also known either as a river cut cliff river cliff or a bluff and spelled as cutbank 1 Erosion that forms a cut bank occurs at the outside bank of a meander because helicoidal flow of water keeps the bank washed clean of loose sand silt and sediment and subjects it to constant erosion As a result the meander erodes and migrates in the direction of the outside bend forming the cut bank 26 27 As the cut bank is undermined by erosion it commonly collapses as slumps into the river channel The slumped sediment having been broken up by slumping is readily eroded and carried toward the middle of the channel The sediment eroded from a cut bank tends to be deposited on the point bar of the next downstream meander and not on the point bar opposite it 28 26 This can be seen in areas where trees grow on the banks of rivers on the inside of meanders trees such as willows are often far from the bank whilst on the outside of the bend the tree roots are often exposed and undercut eventually leading the trees to fall into the river 28 29 Meander cutoff edit Main article Meander cutoff nbsp The Rincon on Lake Powell in southern Utah It is an incised cutoff abandoned meander A meander cutoff also known as either a cutoff meander or abandoned meander is a meander that has been abandoned by its stream after the formation of a neck cutoff A lake that occupies a cutoff meander is known as an oxbow lake Cutoff meanders that have cut downward into the underlying bedrock are known in general as incised cutoff meanders 1 As in the case of the Anderson Bottom Rincon incised meanders that have either steep sided often vertical walls are often but not always known as rincons in the southwest United States 30 Rincon in English is a nontechnical word in the southwest United States for either a small secluded valley an alcove or angular recess in a cliff or a bend in a river 31 Incised meanders edit Main article Entrenched river nbsp Glen Canyon USThe meanders of a stream or river that has cut its bed down into the bedrock are known as either incised intrenched entrenched inclosed or ingrown meanders Some Earth scientists recognize and use a finer subdivision of incised meanders Thornbury 32 argues that incised or inclosed meanders are synonyms that are appropriate to describe any meander incised downward into bedrock and defines enclosed or entrenched meanders as a subtype of incised meanders inclosed meanders characterized by a symmetrical valley sides He argues that the symmetrical valley sides are the direct result of rapid down cutting of a watercourse into bedrock 1 33 In addition as proposed by Rich 34 Thornbury argues that incised valleys with a pronounced asymmetry of cross section which he called ingrown meanders are the result of the lateral migration and incision of a meander during a period of slower channel downcutting Regardless the formation of both entrenched meanders and ingrown meanders is thought to require that base level falls as a result of either relative change in mean sea level isostatic or tectonic uplift the breach of an ice or landslide dam or regional tilting Classic examples of incised meanders are associated with rivers in the Colorado Plateau the Kentucky River Palisades in central Kentucky and streams in the Ozark Plateau 33 35 nbsp Goosenecks of the San Juan River SE Utah There is a cut off meander at right center As noted above it was initially either argued or presumed that an incised meander is characteristic of an antecedent stream or river that had incised its channel into underlying strata An antecedent stream or river is one that maintains its original course and pattern during incision despite the changes in underlying rock topography and rock types 32 33 However later geologists 36 argue that the shape of an incised meander is not always if ever inherited e g strictly from an antecedent meandering stream where its meander pattern could freely develop on a level floodplain Instead they argue that as fluvial incision of bedrock proceeds the stream course is significantly modified by variations in rock type and fractures faults and other geological structures into either lithologically conditioned meanders or structurally controlled meanders 33 35 Oxbow lakes edit Main article Oxbow lake The oxbow lake which is the most common type of fluvial lake is a crescent shaped lake that derives its name from its distinctive curved shape 37 Oxbow lakes are also known as cutoff lakes 1 Such lakes form regularly in undisturbed floodplains as a result of the normal process of fluvial meandering Either a river or stream forms a sinuous channel as the outer side of its bends are eroded away and sediments accumulate on the inner side which forms a meandering horseshoe shaped bend Eventually as the result of its meandering the fluvial channel cuts through the narrow neck of the meander and forms a cutoff meander The final break through of the neck which is called a neck cutoff often occurs during a major flood because that is when the watercourse is out of its banks and can flow directly across the neck and erode it with the full force of the flood 28 38 After a cutoff meander is formed river water flows into its end from the river builds small delta like feature into either end of it during floods These delta like features block either end of the cutoff meander to form a stagnant oxbow lake that is separated from the flow of the fluvial channel and independent of the river During floods the flood waters deposit fine grained sediment into the oxbow lake As a result oxbow lakes tend to become filled in with fine grained organic rich sediments over time 28 38 Point bar edit Main article Point bar A point bar which is also known as a meander bar is a fluvial bar that is formed by the slow often episodic addition of individual accretions of noncohesive sediment on the inside bank of a meander by the accompanying migration of the channel toward its outer bank 1 26 This process is called lateral accretion Lateral accretion occurs mostly during high water or floods when the point bar is submerged Typically the sediment consists of either sand gravel or a combination of both The sediment comprising some point bars might grade downstream into silty sediments Because of the decreasing velocity and strength of current from the thalweg of the channel to the upper surface of point bar when the sediment is deposited the vertical sequence of sediments comprising a point bar becomes finer upward within an individual point bar For example it is typical for point bars to fine upward from gravel at the base to fine sands at the top The source of the sediment is typically upstream cut banks from which sand rocks and debris has been eroded swept and rolled across the bed of the river and downstream to the inside bank of a river bend On the inside bend this sediment and debris is eventually deposited on the slip off slope of a point bar 1 26 27 Scroll bars edit Scroll bars are a result of continuous lateral migration of a meander loop that creates an asymmetrical ridge and swale topography 39 on the inside of the bends The topography is generally parallel to the meander and is related to migrating bar forms and back bar chutes 40 which carve sediment from the outside of the curve and deposit sediment in the slower flowing water on the inside of the loop in a process called lateral accretion Scroll bar sediments are characterized by cross bedding and a pattern of fining upward 41 These characteristics are a result of the dynamic river system where larger grains are transported during high energy flood events and then gradually die down depositing smaller material with time Batty 2006 Deposits for meandering rivers are generally homogeneous and laterally extensive unlike the more heterogeneous braided river deposits 42 There are two distinct patterns of scroll bar depositions the eddy accretion scroll bar pattern and the point bar scroll pattern When looking down the river valley they can be distinguished because the point bar scroll patterns are convex and the eddy accretion scroll bar patterns are concave 43 Scroll bars often look lighter at the tops of the ridges and darker in the swales This is because the tops can be shaped by wind either adding fine grains or by keeping the area unvegetated while the darkness in the swales can be attributed to silts and clays washing in during high water periods This added sediment in addition to water that catches in the swales is in turn is a favorable environment for vegetation that will also accumulate in the swales Slip off slope edit Main article Slip off slope Depending upon whether a meander is part of an entrenched river or part of a freely meandering river within a floodplain the term slip off slope can refer to two different fluvial landforms that comprise the inner convex bank of a meander loop In case of a freely meandering river on a floodplain a slip off slope is the inside gently sloping bank of a meander on which sediments episodically accumulate to form a point bar as a river meanders This type of slip off slope is located opposite the cutbank 44 This term can also be applied to the inside sloping bank of a meandering tidal channel 45 In case of an entrenched river a slip off slope is a gently sloping bedrock surface that rises from the inside concave bank of an asymmetrically entrenched river This type of slip off slope is often covered by a thin discontinuous layer of alluvium It is produced by the gradual outward migration of the meander as a river cuts downward into bedrock 46 47 A terrace on the slip off slope of a meander spur known as slip off slope terrace can formed by a brief halt during the irregular incision by an actively meandering river 48 Derived quantities edit nbsp Meanders scroll bars and oxbow lakes in the Songhua RiverThe meander ratio 49 or sinuosity index 50 is a means of quantifying how much a river or stream meanders how much its course deviates from the shortest possible path It is calculated as the length of the stream divided by the length of the valley A perfectly straight river would have a meander ratio of 1 it would be the same length as its valley while the higher this ratio is above 1 the more the river meanders Sinuosity indices are calculated from the map or from an aerial photograph measured over a distance called the reach which should be at least 20 times the average fullbank channel width The length of the stream is measured by channel or thalweg length over the reach while the bottom value of the ratio is the downvalley length or air distance of the stream between two points on it defining the reach The sinuosity index plays a part in mathematical descriptions of streams The index may require elaboration because the valley may meander as well i e the downvalley length is not identical to the reach In that case the valley index is the meander ratio of the valley while the channel index is the meander ratio of the channel The channel sinuosity index is the channel length divided by the valley length and the standard sinuosity index is the channel index divided by the valley index Distinctions may become even more subtle 51 Sinuosity Index has a non mathematical utility as well Streams can be placed in categories arranged by it for example when the index is between 1 and 1 5 the river is sinuous but if between 1 5 and 4 then meandering The index is a measure also of stream velocity and sediment load those quantities being maximized at an index of 1 straight See also editBaer s law Billabong Crevasse splay Helicoidal flow Jet stream Meander cutoffs in Avulsion river Meander scar Riffle pool sequenceReferences and notes edit a b c d e f g h i Neuendorf K K E J P Mehl Jr and J A Jackson J A eds 2005 Glossary of Geology 5th ed Alexandria Virginia American Geological Institute 779 pp ISBN 0 922152 76 4 a b Charlton R 2007 Fundamentals of fluvial geomorphology Routledge New York New York 234 pp ISBN 0 415 33453 5 Leopold L B Wolman M G Wolman M G and Wolman M G 1957 River Channel Patterns Braided Meandering and Straight United States Geological Survey Professional Paper no 282B US Government Printing Office Washington DC 47 pp Meander Merriam Webster Retrieved July 12 2012 Leong Goh Cheng 1995 10 27 Certificate Physics And Human Geography Indian Edition Oxford University Press pp 41 42 ISBN 978 0 19 562816 6 Meander Online Etymology Dictionary Retrieved July 12 2012 Strabo Geography Book 12 Chapter 8 Section 15 Gurbuz Alper Kazanci Nizamettin 2019 The Buyuk Menderes River Origin of Meandering Phenomenon Landscapes and Landforms of Turkey World Geomorphological Landscapes pp 509 519 doi 10 1007 978 3 030 03515 0 29 ISBN 978 3 030 03513 6 S2CID 134826361 Callander R A January 1978 River Meandering Annual Review of Fluid Mechanics 10 1 129 158 Bibcode 1978AnRFM 10 129C doi 10 1146 annurev fl 10 010178 001021 Weiss Samantha Freeman April 2016 Meandering River Dynamics Doctoral dissertation Retrieved from Ideals https www ideals illinois edu bitstream handle 2142 92706 WEISS DISSERTATION 2016 pdf sequence 1 amp isAllowed y Albert Einstein river meandering Hans Einstein sediment transport Victor Miguel Ponce Archived from the original on 2017 11 19 Martinez Alberto A March 2014 The questionable inventions of the clever Dr Einstein Jozsef Illy The practical Einstein Experiments patents inventions Baltimore Johns Hopkins University Press 2012 xiv 202pp 60 00 HB Metascience 23 1 49 55 doi 10 1007 s11016 013 9819 x S2CID 169290222 The technical definitions of this section rely heavily on Julien Pierre Y 2002 River Mechanics Cambridge University press pp 179 184 ISBN 0 521 52970 0 In addition concepts are utilized from Graf Walter 1984 Hydraulics of Sediment Transport Water Resources Publications pp 261 265 ISBN 0 918334 56 X Leopold L B Langbein W B 1966 River meanders Scientific American 214 6 60 73 Bibcode 1966SciAm 214f 60L doi 10 1038 scientificamerican0666 60 JSTOR 24930965 a b Leopold Luna Wolman M Gordon 1957 River channel patterns Braided meandering and straight Professional Paper 282 B United States Geological Survey p 50 doi 10 3133 pp282B Wood Elizabeth A 1975 Science from Your Airplane Window 2nd Revised Edition New York Courier Dover Publications p 45 ISBN 0 486 23205 0 Hickin 2003 p 432 One of the important consequences of helical flow in meanders is that sediment eroded from the outside of a meander bend tends to move to the inner bank or point bar of the next downstream bend Hickin 2003 p 434 Hickin 2003 p 432 In the absence of secondary flow bend flow seeks to conserve angular momentum so that it tends to conform to that of a free vortex with high velocity at the smaller radius of the inner bank and lower velocity at the outer bank where radial acceleration is lower Hickin 2003 p 432 Near the bed where velocity and thus the centrifugal effects are lowest the balance of forces is dominated by the inward hydraulic gradient of the super elevated water surface and secondary flow moves toward the inner bank Bowker Kent A 1988 Albert Einstein and Meandering Rivers Earth Science History 1 1 45 Bibcode 1988ESHis 7 45B doi 10 17704 eshi 7 1 yk72n55q84qxu5n6 Retrieved 2016 07 01 Callander R A 1978 River Meandering Annual Review of Fluid Mechanics 10 129 58 Bibcode 1978AnRFM 10 129C doi 10 1146 annurev fl 10 010178 001021 Scheidegger Adrien E 2004 Morphotectonics Berlin New York Springer p 113 ISBN 3 540 20017 7 Riley Ann L 1998 Restoring Streams in Cities A Guide for Planners Policymakers and Citizens Washington DC Island Press p 137 ISBN 1 55963 042 6 D Alessandro Leandro Miccadei Enrico Piacentini Tommaso November 2008 Morphotectonic study of the lower Sangro River valley Abruzzi Central Italy Geomorphology 102 1 145 158 Bibcode 2008Geomo 102 145D doi 10 1016 j geomorph 2007 06 019 a b c d Reineck H E and Singh I B 2012 Depositional sedimentary environments with reference to terrigenous clastics Springer Science amp Business Media New York New York 551 pp ISBN 9783642962912 a b Chant Robert J 2002 Secondary circulation in a region of flow curvature Relationship with tidal forcing and river discharge Journal of Geophysical Research 107 C9 3131 Bibcode 2002JGRC 107 3131C doi 10 1029 2001jc001082 a b c d Fisk H N 1944 Geological investigation of the alluvial valley of the lower Mississippi River War Department Corps of Engineers Mississippi River Commission Vicksburg Mississippi 78 pp Fisk H N 1948 Fine grained Alluvial Deposits and Their Effects on Mississippi River Activity War Department Corps of Engineers Mississippi River Commission Vicksburg Mississippi 2 Vols 82 pp Shoemaker E M and Stephens H G 1975 First photographs of the Canyon Lands in Fassett J E ed pp 111 122 Canyonlands Country A Guidebook of the Four Corners Geological Society Eighth Field Conference September 22 25 1975 Four Corners Geological Society Durango Colorado pp 278 Merriam Webster Incorporated 2017 Dictionary by Merriam Webster America s most trusted online dictionary last accessed November 22 2017 a b Thornbury W D 1954 Principles of Geomorphology John Wiley amp Sons New York New York 618 pp a b c d Fairbridge R W 1968 Incised meander In Fairbridge R W ed pp 548 550 The Encyclopedia of Geomorphology Encyclopedia of Earth Sciences Series Vol 3 McGraw Hill Company Inc New York New York 1295 pp Rich J L 1914 Certain types of stream valleys and their meaning The Journal of Geology 22 5 pp 469 497 a b Barbour J R 2008 The origin and significance of sinuosity along incising bedrock rivers Doctoral dissertation Columbia University New York New York 172 pp Hack J T and Young R S 1959 Intrenched meanders of the North Fork of the Shenandoah River Virginia United States Geological Survey Professional Paper 354 A 10 pp Hutchinson G E 1957 A treatise on limnology v 1 Geography Physics and Chemistry Wiley 1015p a b Toonen W H Kleinhans M G and Cohen K M 2012 Sedimentary architecture of abandoned channel fills Earth Surface Processes and Landforms 37 4 pp 459 472 Woolfe and Purdon Purdon Richard 1996 Deposits of a rapidly eroding meandering river terrace cut and fill in the Taupo Volcanic Zone New Zealand Journal of Geology and Geophysics 39 2 243 249 Bibcode 1996NZJGG 39 243W doi 10 1080 00288306 1996 9514708 K Whipple September 2004 Alluvial channels and their landforms Surface Processes and Landscape Evolution Sam Boggs Jr 2003 Principles of Sedimentology and Stratigraphy 4 ed NJ Pearson Prentice Hall ISBN 0 13 099696 3 G Wasser 2005 A Comparison Of Meandering River Deposits From The Middle Belly River And Horsefly With Recent Milk River Valley Deposits Central And Southern Alberta Calgary Alberta Canadian Natural Resource Limited Norman D Smith and John Rogers 1999 Fluvial Sedimentology 6 ed blackwell publishing ISBN 0 632 05354 2 Scheffers A M May S M and Kelletat D H 2015 Forms by Flowing Water Fluvial Features In Landforms of the World with Google Earth pp 183 244 Springer Amsterdam Netherlands 391 pp ISBN 978 94 017 9712 2 Keck R Maurer D and Watling L 1973 Tidal stream development and its effect on the distribution of the American oyster Hydrobiologia 42 4 pp 369 379 Davis W M 1913 Meandering valleys and underfit rivers Annals of the Association of American Geographers 3 1 pp 3 28 Crickmay C H 1960 Lateral activity in a river of northwestern Canada The Journal of Geology 68 4 pp 377 391 Herrmann H and Bucksch H 2014 Dictionary Geotechnical Engineering Worterbuch GeoTechnik English German Englisch Deutsch Springer Berlin Germany 1549 pp ISBN 978 3 642 41713 9 Shaw Lewis C 1984 Pennsylvania Gazetteer of Streams Part II Bulletin No 16 Commonwealth of Pennsylvania Department of Environmental Resources p 8 OCLC 17150333 Gordon Nancy D Thomas A McMahon Christopher J Gippel Rory J Nathan 2005 Stream Hydrology an Introduction for Ecologists Second Edition John Wiley and Sons pp 183 184 ISBN 0 470 84357 8 Singh R Y 2005 Interface drainage analysis of a water divide In Jansky Libor Haigh Martin J Prasad Hushila eds Sustainable Management of Headwater Resources Research from Africa and India Tokyo New York United Nations University Press pp 87 106 ISBN 92 808 1108 8 Bibliography editHickin Edward J 2003 Meandering Channels In Middleton Gerard V ed Encyclopedia of Sediments and Sedimentary Rocks Kluwer Academic Encyclopedia of Earth Sciences Dordrecht Boston Kluwer Academic Publishers pp 430 434 ISBN 1 4020 0872 4 Leopold Luna B Langbein W B June 1966 River Meanders Scientific American 214 6 60 Bibcode 1966SciAm 214f 60L doi 10 1038 scientificamerican0666 60 Thonemann P The Maeander Valley A historical geography from Antiquity to Byzantium Cambridge 2011 Greek Culture in the Roman World Series External links edit nbsp Look up rincon in Wiktionary the free dictionary nbsp Wikimedia Commons has media related to Meanders Movshovitz Hadar Nitsa Alla Shmuklar 2006 01 01 River Meandering and a Mathematical Model of this Phenomenon Physicalplus Israel Physical Society IPS 7 Retrieved from https en wikipedia org w index php title Meander amp oldid 1188209956, wikipedia, wiki, book, books, library,

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