A meander is one of a series of regular sinuous curves in the channel of a stream. 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.
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.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. Charlton, R., 2007. Fundamentals of fluvial geomorphology. Routledge, New York, New York. 234 pp.
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.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.
Origin of term The term derives from the winding river Menderes located in Asia Minor and known to the Ancient Greeks as Μαίανδρος Maiandros (Latin: Maeander), 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. Strabo said: ‘...its course is so exceedingly winding that everything winding is called meandering.’Strabo, Geography, Book 12 Chapter 8 Section 15. The river is south of Izmir, east of the ancient Greek city of Miletus. 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.
Governing physics
Meanders are a result of the interaction of water flowing through a curved channel with the underlying stream bed. This produces helicoidal flow, in which water moves from the outer to the inner bank along the bed, then flows back to the outer bank near the surface of the stream. 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 deposited on the inner bank of the next downstream meander.
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 stream 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 there is 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 streams, 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 stream. In the context of meandering streams, a boundary layer exists within the thin layer of fluid that interacts with the stream 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 stream from the outside bend to the inside bend. This initiates helicoidal flow: along the stream 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 stresss 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 stream to become 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.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&isAllowed=y
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 However, Coriolis forces are likely insignificant compared with other forces acting to produce meanders.
Geometry
The technical description of a meandering watercourse is termed meander geometry or meander planform geometry.The technical definitions of this section rely heavily on In addition concepts are utilized from 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.
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 floodplain, it extends beyond the meander belt. The meander is then said to be free—it can be found anywhere in the floodplain. If there is no floodplain, 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.
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.
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 stream length is the length along the centerline.
Formation
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..."
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.. “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.” 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.. Due to the conservation of angular momentum the speed on the inside of the bend is faster than on the outside.. "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."
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.. "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." This phenomenon is similar to the tea leaf paradox. This secondary flow carries sediment from the outside of the bend to the inside making the river more meandering. 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