October 5, 2010

How and Why rivers meander?

Filed under: Uncategorized — Rajnikant Khatsuria @ 12:17 pm




Formerly Additional Director

Central Water and Power Research Station, Pune, India

e mail:  rmkhatsuria@rediffmail.com

Meandering rivers have fascinated all sections of viewers; laymen, students, artists, poets, and scientists alike. It is believed that the river Sabarmati (meandering all along its course) in western India, has derived its name from a poetical expression- (saa bhramati) meaning- she (the river) is wandering. A meandering river originating form a valley formed by two mountains and a rising sun is the most likely sketch that a beginner in a drawing class would attempt. Crooked rivers with lazy loops and bends have been favorites of artists and photographers. Meandering river has been one of the most explored and investigated topics in hydraulic engineering. No less an intellectual luminary than Albert Einstein postulated in 1926 a theory explaining the process of meandering on the basis of simple physical laws. Since then understanding of the process has traversed from simple physics to stochastic, equilibrium and geomorphic theories on one hand and from empiricism to complex mathematical modeling on the other, and yet without a final word as to-why and how rivers meander!

According to Einstein, a slight change in the velocity of flow between the banks of a river (due to a bend in the river) would give rise to secondary circular currents in the plane perpendicular to the direction of the flow. Even where there is no bend, Coriolis force caused by the earth’s rotation can cause a small imbalance in velocity distribution such that velocity on one bank is higher than on the other. This can generate erosion on one bank and deposition of sediment on the opposite bank. The secondary currents cause the flow to proceed in the direction towards eroded portion until redistribution of velocity reverses the process. This result in formation of tortuous water course called meandering. Surprisingly, Einstein’s contribution to river engineering has not been acknowledged in the literature.

There are other theories as well. Stochastic theory ascribes random fluctuations in the flow velocity due to some obstacle or disturbance to affect velocity distribution across the section, for the formation of meanders. Equilibrium theory states that meandering is the process by which a river adjusts its gradient (length along the course divided by the drop in elevation) so that there is an equilibrium between the erodibility of the terrain and the erosive power of the stream. Geomorphic theory attributes tectonic features acting as obstacles to deflect the stream to cause meanders.

Attempts were also made in the past, to understand the process of initiation of meanders, on physical hydraulic models. Two such cases are known; Waterways Experiments Station (WES) at Vicksburg, USA and Central Water and Power Research Station, Pune, India. Experiments were conducted in wide and long sand laden trays, starting with the straight channel and varying discharge and silt load. However, there was no conclusive evidence of the cause of initiation of meanders, having observed that meanders may even develop spontaneously. In general, all the theories can explain how the meandering rivers continue to meander but fail to explain how meanders initiate.

It is believed that a necessary condition for the development of meander is erosion of the bed and its subsequent deposition further downstream. This is caused by the secondary currents, generated downstream of a bend, provided of course that the discharge and bed material are favorable for this process. The question, then is- how secondary currents are generated in a straight reach with almost symmetrical distribution of velocity? Explanations put forth were these: a sudden obstruction, external distribution, falling of a tree etc. These can no longer be acceptable, given the fact that meanders spontaneously formed even in straight experimental channels without any of these!

 Further discussing on this issue, it is argued that even if the cross section is symmetrical, the turbulence and the instantaneous maximum shear stress may be asymmetric. Experimental observations indicate this to be near the flanks, rather than in the centre of the channel. Thus, the most likely site for the initiation of bed erosion and subsequent deposition would be nearer to the flanks. This would force the otherwise parallel stream lines from the upstream, to deviate towards the opposite flank, which in turn cause another spot of bed erosion and subsequent deposition. If the above sequence of events is visualized graphically, it would look something like this:


Initiation of meandering


The course of events, when allowed to continue, would result in a meandering of river as shown below.


Equilibrium theory states that meandering is the process by which a river adjusts its gradient so that there is equilibrium between the erodibility of the terrain and the erosive power of the stream. This argument seems to be supported by the theory of entropy (having its basis in the second law of thermodynamics). Without going into details, it can be stated that a stream with a given discharge would tend to increase its entropy, which consequently would tend to reduce its average velocity. This can be achieved by flattening of its slope, which would require the stream to go in a tortuous way via meander.

At present, the above two postulations are generally accepted. However, the interesting aspect about meanders is that there is a consistency in their shapes and relationships among various geometric features, irrespective of the geomorphic conditions. The geometric features are shown in the drawing below.

The meander ratio or sinuosity index is the ratio of actual length along a meandering river (Lm) to the straight distance S between the end points (AB). It is an indication of quantification of meandering. Obviously, for a straight river course this ratio is equal to unity. A ratio varying from 1 to 1.5 defines the river course as sinuous and from 1.5 to 4 as meandering. Also, the following relationships hold good for a majority of rivers:

     Lm/W ~10;   Lm/Rc~5;   Rc/W~2

            It is observed that the meander length is a function of discharge also. Various investigators have studied this aspect and their findings could be generalized in the form  Lm  k Q0.5, where k is a constant and Q is the discharge. The value of k ranges from 1 to 31 depending on whether the discharge selected was dominant, mean, maximum or with a certain return period etc.

            The above relationship points to the dynamic nature of meandering. Any feature that brings about a change in discharge can alter the meander geometry; as for example, a dam constructed upstream or a diversion of discharge from the main stream or an unusual flood etc.

            Even when one concludes that meander results from the processes of erosion and deposition in alluvial rivers, there is no explanation as to why meanders also form on glaciers in the absence of sediment. Meanders are present even in oceans as formed by the well known Atlantic Gulf Stream, while passing along the coast of Florida towards north, alongside New England and then east ward to Europe. Meanders form even when there are no banks; as exemplified by the flow from a small nozzle on a slightly inclined glass plate. Interestingly, a discharge-meander length relationship similar to the one mentioned above holds good in this case also. How this could be explained?  There is no convincing answer for this. Perhaps, the theory of entropy can be extended to address these issues.






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