Meandering Rivers


            The most notable feature of the Shenandoah Valley as seen from the surrounding mountains is the circuitous path of the north and south branches of the Shenandoah River as they pass on either side of Massanutten Mountain.  They meander, a word which derives from the serpentine flow of rivers; the Menderes River in western Asia Minor is the etymological root, proverbial for its winding course (it was called Maiandros in Ancient Greece, the source language).  The pronounced and universal sinuosity of meandering rivers has been the subject of scientific curiosity  among naturalists, geologists and potamologists (river specialists) for many years. This has led to serious study with concomitant hypothesizing as to the scientific explanation for the phenomenon. While there are theories as to causality, there is no accepted doctrine.


            The fundamental physics of river flow is pedestrian; water runs downhill. The extension of this principle to a wandering path also seems fairly obvious. Local irregularities in the streambed, like large rocks, downed trees and variegated sedimentary riparian substrate would act to divert the otherwise inexorable downward trajectory of the water. Once the flow is diverted by an impediment, it gets complicated, the vagaries and complexities of fluid flow coming into play. But before that can be adequately addressed, some lexical and dimensional terms must be introduced. 




            What is the nature of a meander? It consists of three parts: the outward side of the river bend, which is called the concave portion, the inner side of the river bend or convex portion and the straight area between successive bends, which is called the point of inflection because it marks the point at which the curvature changes or inflects from one direction to the other. A river section has a length which is considered to be the distance the river travels over the ground and not the crow-fly distance between two points. It also has a width and a depth. Because a meandering river has alternating bends, it is like a wave and therefore has an amplitude, a wavelength, and a radius of curvature. The amplitude is the distance along the ground that the river bend extends away from the overall downward path of the  river. The wavelength is the distance along the ground between successive bends that are in the same direction. The radius of curvature is like the radius of a circle; for a bend in a river, it is the radius of the arc that the bend makes.


            Returning now to the formation of river bends; when an impediment to the downward flow of water is encountered, the river will flow around it; not symmetrically, but favoring one side. The water that takes the longer route will need to go faster. This is the same principle that creates the lift in an airfoil, the upper surface is curved so the air must go faster, creating a pressure drop. The faster flow will cause more erosion on that side of the obstruction and eventually, a concave bank will be formed. The silt that is removed by the erosion will be transported downstream where it will settle out in an area where the flow has abated, eventually forming a bar. As this process continues, the concave banks become more eroded and the sediment is deposited along the sides of the river, generally at the convex curve of the next bend. The flow of the river is therefore fastest and hugs the bank at on the concave side and crosses over to the other bank at the infection point. Because the river's flow has streamlines that are not congruent with the shape of the banks, it is given a special name, thalweg, the line of fastest flow in a river. It is from the Old German thal meaning "valley" and weg meaning "way;" it also means the line of greatest slope in a valley, that the river generally follows.


            No less an intellectual luminary than Albert Einstein cogitated on the origin and mechanism of the meander. Applying his noted "thought experiments" to the problem, he compared the river to a cup of tea, the leaves at the bottom of the cup like the sediments of a river. When the tea is stirred, the leaves collect in the center of the bottom of the cup. The explanation is couched in the fundamentals of physics. As the liquid is rotated, a centrifugal force is created (the force is the one you feel in going around a corner). If all of the liquid moved at the same speed, there would be no effect. However, the friction caused by the sides and bottom of the cup act to reduce the rotational motion and hence the centrifugal force, which will now be lower at the bottom. This sets up an outward flow at the top that goes outward until it strikes the wall, the flow extending down to return at the bottom of the cup, where the force is less. In a river, the downstream flow of the river results in a helical flow, the centrifugal force at the top pushing the water outward and down to drive the sediments downstream. Einstein further postulated that the maximum erosion would occur downstream of the inflection point so that the meander would over the course of time migrate down-river, a noted and observable phenomenon.


            One could from the foregoing explanation conclude the meanders form when a river flows down a gentle slope where the sediment of the substrate is of a consistency to allow for erosion by the water source. There is something more fundamental going on, however. Meanders form on glaciers where there is no sediment. Meanders form when the water flows down a car's window glass. Meanders form in the open ocean; the Gulf Stream passes through the more static waters of the Atlantic in a pattern that has the same geometry as a meander.  It is the consensus of scientific thought that energy plays a role in the formation of meanders. To understand how this might be, it is necessary to look at the shape of a meandering bend and the parameters of the riparian system to which it belongs.


            Leopold Luna was a Chief Hydrologist of the U. S. Geological Survey and is considered to have been one of the foremost authorities on potamology in North America. He was a keen observer of rivers, performing field measurements and deducing mathematical relationships that quantified the behavior of rivers. He noted that meandering rivers possess markedly similar dimensions, that the wavelength of the bend was about 10 times the width w and about 5 times the radius of curvature rc ; the ratio of the radius of curvature to the width was thus about 2 to 1. Luna computed the w/rc for 50 rivers and found that they were quite similar, explaining why meandering rivers looked about the same on planimetric maps. This means that a wider river bends a lot less, a phenomenon that was also noted by Einstein, who attributed it to the slower helical flow that would erode the banks less. This is also important to the notion of the amount of energy in the flow stream.


            The shape of a meander is distinctive; it is not a segment of a circle nor is it a regular sinusoidal wave. Luna called the shape a sine-generated curve. To understand this, picture yourself in a boat on a river (without tangerine trees and marmalade skies) with a compass in one hand. As you proceed down the meandering river taking periodic compass readings, you will find that your direction (heading angle ψ) will vary according to the sine of the distance you have traveled, or, in mathematical terms:


                                                                            heading angle ψ  =   csin(s)


Where s is the distance down the river (in radians) and c is a constant that depends on the maximum angle that the meander bend makes with the down-river axis. It depends on a number of factors that include the topography and the river flow; a value of less than one yields a gently undulating stream and a value of about two yields the more characteristic horseshoe shaped meander.


             The sine-generated curve has three properties that provide some indication of the reason why rivers meander: it is the path of minimum bending stress, the path of minimal directional variance, and the path that represents the most likely random walk. That the sine-generated curve is a fundamental property of physics may be demonstrated with a strip of steel that, if fixed at both ends and bent into an arc, will assume the characteristic horseshoe shape.  From the mathematical standpoint, the minimization of bending stress relates to the fact that this stress is proportional to the square of the curvature at each point; this  has a logical appeal. The curvature is defined as the inverse of the radius of curvature, so that a small radius bend has a large curvature and therefore, more stress. Since the sum the squares of the curvature at each point along a sine-generated curve is a minimum, so is the bending stress. The directional variance has similar provenance.


            The random walk is a bit more difficult to explain, but is important in that it is necessary, though not sufficient, for understanding the apparent randomness of river meanders. The basic idea is simple. A person (or water molecule) starts at a point a, takes off at a random angle with a pace length L, repeating the process with a new angle but the same pace length until the final point b is reached. If one then seeks to determine which of these random walks are most frequent, it turns out that those drawn from a normal probability distribution (also known as Gaussian for the German mathematician Frederick Gauss) provide a nontrivial solution. Though the actual solution is not determinate, the sine-generated curve is a close approximation. At the macroscopic level of the meandering of a river, the difference is, quite literally, lost in the weeds.


            It is therefore all about energy. As with any physical process, the lowest energy state, that which is most nearly at rest, is the equilibrium solution. And this is what the water is doing, albeit in a turbulent and seemingly random manner. But there is order in the chaos. The incongruous symmetry of the meandering river the result. Given the complexities and unknowns of this relatively mundane and tellurian phenomenon, it is not surprising that those who try to predict anthropomorphic effects on the climate are reduced to probabilities, for that is a vastly more difficult problem for which there will likely not be a simple sine-generated solution, which is, after all, only an approximation.