These questions and more are addressed in this paper.
The authors wish to thank the geologist Prof. Akiva Flexer, the hydro-geologist Mr. Mordechai Movshovitz, as well as Dr. Alex Kuperman and Mrs. Ziva Schaham for their helpful comments.
The river meandering phenomenon
Rivers and rivulets run from high grounds to low grounds due to gravity. The speed of the current grows faster, as the river flows further downstream. Although it is natural to expect that the water will pave its way in a straight line, it is rare to find a long river running in a straight line for long. There are few and short sections in which the flow is almost a straight line. In a bird's eye photos, rivers usually look like meandering lines, and are similarly represented in geographical maps. The surprising fact is that river meandering follow rules. Moreover: in different parts and conditions, this phenomenon follows the same rules. In nature the rivers meander in similarly recurring patterns.
Figure 1: The Map of Turkey with the River Menderes
The origin of the term “meander” is the name Menderes of a river in Turkey (Figure 1). Tourist officials in Turkey point out to this river's valley as the cradle of many cultures.
Meanders in Colorado
Figure 2 illustrates the meanders of the Colorado River which runs in the south of the state of Utah. The picture was taken from an altitude of 1000 meters. These meanders were probably formed close to the beginning of the creation of the Colorado-Plateau, about a million years ago or more. The meanders grew bigger and deeper as the river paved its path through the layers of rocks and alluvium.
Figure 2: Meanders of the Colorado River
Meanders change their shape perpetually. The current state of the meanders is categorized by the numeric ratio between the length of the river bed (the watercourse length), and the length of the river in air-distance measure (the distance between its endpoints). This ratio is called: "The meander-ratio". The most common meander-ratio is approximately 3:2. The Jordan River is one of the rivers, which are most prominent in a high meander-ratio number, not only in comparison to other rivulets in Israel, but also in comparison to other rivers in the world. The meander-ratio of the Jordan River is approximately 2:1. The change of the Jordan meanders in the course of the last few decades has been influenced by factors, which are beyond the natural turn of events. Human interference in the proper and natural course of the river flow, by activating the Deganya dam in 1932 and the diversion of the Yarmuk River at the beginning of the sixties, reduced the maximum annual supply and the difference between it and the minimum flow. This interaction shortened many meanders, thus reducing its overall meander-ratio. Although this tendency will continue, it is reasonable to assume that the watercourse of the Jordan River will never turn completely straight, because the natural tendency in rivers around the world is not to straighten, rather to meander. (According to )
The mystery of meanders
The meander generating process in each river is dynamic. River meanders develop throughout the years, and perpetually change during long periods. The perpetual creation of meanders is an amazing natural phenomenon, which occurs all around the globe. Its various aspects link up to numerous earth research disciplines: geology, geophysics, geomorphology and more. Researchers investigated and thoroughly surveyed the geographical conditions of river flow that caused the meander creation and evolution. This is elaborated later on.
The surprising fact about meander creation is the specific and uniform geometric shape of all meanders, despite the different geophysical conditions. For instance the meanders, which develop in rich-alluvium environment , have the same shape as those that develop in alluvium-free environment, like icebergs' ravines and the Golf Stream. Moreover, even outside the water, one can observe a phenomenon, which is similar to the meander phenomenon. Figures 4 and 5 present such examples: In Figure 4 one observes a train track deformed as a result of an accident, in a shape which resembles a river meander. In Figure 5 one can see how it is possible to create a meander-shape by hand: one holds a flexible narrow steel strap in two points so that between those points the strap can get any shape without constraints. The strap will "choose" the shape that guaranties the minimum "bending effort". This shape resembles the meander shape in rivers. How can one explain the meanders' unique and unified shape? We shall discuss this question in the following section.
Figure 3: The meanders of the Jordan River (from , page 25)
Figure 4: train accident near Greenville, South Carolina, 1965 (from, page 67)
The creation of a meander shape with a flexible metal strap (From , page 66)
The Processes of Creation and Evolution of Meanders
River meandering is one of the most predominant, rhythmic geometric phenomenon on the surface of Earth. Meander study is one of the primary subjects of quantitative geomorphology (a science that investigates the shapes, which occur above the ground or beneath it). Numerous studies in this field prove that the processes of forming the shape of meanders are a lot more elaborated than what appears at first sight. Geomorphology gives a clear, detailed image of the causes and the development processes of meanders, from their "birth" till their "death". Within the scope of this article we limit ourselves to presenting the major framework of the subject. A further discussion can be found in ,  and .
The river bed and the river valley
Each river flows in a river bed which is a natural channel, formed by the requirements of the stream. The modifications to the shape of the river bed are noticed mainly by its broadening and deepening as a result of the alluvium deposits, and by the amount of the water that it carries. Deepening the river bed creates the river valley. Its slopes grow taller as the river digs deeper. The slopes aren't usually adjacent to the river bed, but are detached from it by flat, low areas, that are over flown during a flood. Those areas, along with the river bed, form the valley floor. Sometimes the valley floor is very narrow and merges with the river bed. Such is the case with most of the rivulets in Israel. Sometimes the valley floor is very wide, so that the river bed is only a small part of it.
The impact of the valley floor structure on the meander progress
Many geomorphologic researchers studied the impact of river valley floor on the progress of meanders. Meandering progress depends on the conditions of the river flow, and on the rock surface its valley is made of. In areas that consist of material of low morphological valence (a non-consolidated material or that its consolidation quality is low, so that the erosion and alluvium processes act in an easier manner), meander development occur more frequently . Meanders that develop in a non-consolidated environment are called "free meanders" , because they are free to relocate their path from one river bank to the other. The Jordan River falls under this category. Solid rock areas, with a high morphological valence, support the development of "enclosed” meanders, their movement is limited and they are called "engraved meanders" . These are the meanders of the Sorek rivulet in Judea Mountains. In this paper we discuss mainly the free meanders.
Creation of a meander
A single meander is constructed by two arcs attached to each other that create together the shape of the letter S. Each arc is usually larger than 180 degrees. Figure 6 illustrates the general shape of a meander and its main characteristics. The bending of the path is fundamental to meander's development. The speed of the current close to the bank is usually slower than the one in mid-stream, because of the friction with the river bank. When a disturbance to the straight water flow occurs, as a result of an obstacle or a change in soil conditions in different parts, the water detour the obstacle and an arc is formed in the river bed. The water flow is accelerated and as a result the alluvium process intensifies in the external side of the arc. In other words: the water flowing towards the concaved bank (looking from mid-river) strikes at it, and bounces back away from it towards the convex bank. The process repeats itself later on to create another arc in the opposite direction. Passing through the inflection point from one arc to the next, all the forces activated in the water, reverse their direction, and the creation process repeats itself in the opposite direction to the previous arc. That concludes the creation of one meander. In this process a centrifugal force develops, which supports the increase of the arc's radius. While the concaved bank withdraws following the alluvium process, the deposits accumulate on the convexes bank. These build-ups create triangular-shape flat ridges, called vertex ridges.
Figure 6: A schematic shape of a meander (From , page 62).
W - width of the river bed;
λ - wave length;
L - length of the river bed;
rc - curvature radius
The meanders extend the watercourse of the river, and while doing so, cause a local reduction in its slope and a reduction of the flowing speed in this part of the river. As a result, there is a gradual tapering off the centrifugal force until it diminishes altogether. In this state the curvature radius stops growing. Therefore the curvature radius does not grow beyond a certain size, which depends on the slant of the mountain's slope, the width of the river and the amount of water in this part of the river. When the curvature radius reaches its maximum size, the meander is called a mature meander.
The more one descends down stream, the intensity of the stream grows, and therefore the meanders reach maturity when the curvature radius is larger. The process of meander creation is therefore a process which creates a balance in the stream's speed along the river.
In the process of the meander development the space between the arcs' arms grows. This space is surrounded by the river bed almost completely, and creates a tiny peninsula. The more the meander grows towards maturity, the more its arms start to reach each other and the peninsula between them becomes narrower and narrower, until a water strait connects their edges. This strait is called the meander's neck. During a flood the river water break out and cover the meander's neck. This way one of the arcs is cut off from the river. Although some water will remain in the truncated part for a while, it is disconnected from the river itself. In this condition, the river path straightens in the breaking point. The truncated part has a crescent shape and is called the meander's drain. The meander's drain collects alluvium and gradually dries up. The dried up meander's drain is called a dehydrated meander. As long as the conditions that form meanders exist, other meanders are generated instead of the ones that dried up. For every meander that is cut off as a result of a break out, a new meander is being formed in another place, so the length of the river bed remains almost constant.
Figure 7: A schematic description of the meander progress in six steps (From: , page 106)
An invitation to an investigation journey in rivers
The surprising and amazing fact is that all around the world, in different local conditions, rivers and rivulets "prefer" to deviate from the "righteous path" and to meander in order to reduce the banks' alluvium. While meandering, rivers tend to balance the current speed along the river bed. We have nothing left to do, but to admire the hidden wisdom of the creation accomplishment. In our small and lovely country there are many rivulets, large and small, and one river, which is well-known all around the world: the Jordan River. All of them meander, of course. Which one twists the most and which one is the straightest?
The main characteristic of a meandering river is the ratio of deviation from a straight line. This characteristic is called the meander ratio. To determine this characteristic one has to know the starting point of the rivulet, its ending point and mainly the aerial distance D between the two. One has to know the length L of the river bed as well, which is the distance one covers walking along the bank from its beginning to its end. The ratio L/D is the meander-ratio of the river and it characterizes the deviation of the streamline from a straight line. Other important factors are related to locating the meanders in the rivulets, and to describing them according to the attributes given in Figure 2.
The Ministry of Environment, the Jewish National Fund and the "rivers preservation administration" publish a magazine by the name: "Our country's rivulets", presenting the state of rivulets, reconstruction projects and developing plans around them. In the fall and in the spring, and even in clear winter days, it is worthwhile to go for mathematical journeys in rivulets, watch the meanders and familiarize yourself with their surroundings. If by those journeys more information that characterize the rivulets streams will be collected from around the country, we'll be able to find the most twisted one. Through statistical evaluations, it is possible to calculate the average meander-ratio of the rivulets in the country and the average deviation from it, and much more.
A Mathematical Model of the Meandering Phenomenon
The geomorphologic analysis of the meandering phenomenon as summarized in the previous section, explains the exterior form, which is a result of the creation mechanism. Still in this analysis format, one doesn't get answers to specific questions which are related to the shape of the meanders, like: why do river meanders which run in totally different geophysical conditions have the same geometric shape? Why do meander shapes appear in non-liquid environments (Figures 4 and 5). Sometimes answers to questions like these can be retrieved from a mathematical investigation of universal principles of water flowing, in other words with the help of a mathematical model built to investigate the meander phenomenon.
Various and different mathematical models can be developed for a certain phenomenon. They differ from one another by the extent of their affinity to the properties of the phenomenon being investigated. A mathematical model of the river meander phenomenon, if exists, can help predicting the changes in the flowing path, and by doing so can allow human being's interfering in order to prevent disasters and increase the benefit that the river carries as a source of life to mankind, animals and vegetation around it.
Callander investigated several mathematical models of rivers' meanders. The mathematical model, presented here is basically the fruit work of Von Schelling , followed by Leopold and Langbein . The approach to the mathematical model presented in this article is different from Von Schelling's in a sense that it is not probabilistic. It is rather based solely on the classical calculus of variations . After presenting a model based on this approach, which is deterministic, we present a probabilistic approach to building the same model.
Mathematical Model Development
Let A and B be two given points in a plane. Let us call the line, straight or curved, that connects these points: a path from A to B. Let us call the path a simple path, if in the movement from A to B along the path, one passes every point once and only once, meaning that there are no loops and no "holes" on it. A simple path will be called a smooth path, if in every point on it there is a straight tangent line, with a slope that changes smoothly (without "jumps"), when a point of tangency is moving along the path. From here on we assume that every path we discuss is a simple and smooth path, as is natural to assume for a path of a river flow.
Figure 8: Simple, smooth paths between two points
As is well known, between each two points there is an infinite number of simple, smooth paths and among them there is, of course, the straight line, as can be seen in Figure 8.
Parametric equation of a path
Let B be a point in the plane x O y. Consider a simple, smooth path from O to B. Let us denote by L the length of the path. We now try to reach a parametric formula of that path.
Let P(x, y) be a point on the path. Let l denote the length of the path between O and P. Let φ be the angle between the positive direction of the x- axis and the path's tangent in point P (Figure 9).
Figure 9: The path's direction in an arbitrary point P
Let P'(x+Δx, y+Δy) be a point on the path that is close to P. Let us mark by Δl the length of the path between P and P' (Figure 10). These definitions imply that:
These equations hold for each value of l in the segment 0<l<L. When the value of l changes, the values x, y, φ become functions of the variable l. The functions x(l), y(l) are the location functions of the point which moves along the path, and the function φ(l) is a direction function of the path.
The right hand sides of the above two equations are trigonometric functions of the direction function φ(l), while the functions x(l), y(l) are their primitive functions:
Figure 10: The characteristics of the path between P and P'
Let t designate a variable parameter in the segment (0, L).
Because x(0) =y(0)=0, we may write the last equations like this:
The two equations (1) are the parametric equations of the path which passes between the points O and B, where φ(l) is its direction function.
The curvature of a path
According to the initial assumptions, the function φ(l) is continuous in the segment 0 ≤ l ≤ L. This assumption is justified by the fact that in nature, the waters in rivers and rivulets usually change their flow direction gradually, without sudden jumps. Moreover: the speed of change of the flow direction changes in a continuous and gradual fashion, also. Thus, we can assume, in addition to the continuity of the function φ(l) in the segment 0 ≤ l ≤ L, the continuity of its derivative in this segment, as well.
Let P be a point on the path, which corresponds to the value l, and let P' be a point on the path which corresponds to the value l'=l+Δl. The average curvature of the arc between the point P and P' is:
A minimum curvature of a path
We are interested in finding a path (or paths), which leads from the origin O to a given point B and possesses minimal average curvature, as it is natural to assume that the river "looks for" this kind of a path. Clearly, if we consider an absolute minimum, i.e. a minimum within all possible paths from O to B, the solution is immediate and it is the straight segment OB, because
for a straight segment. Therefore we'll search a local minimum and not an absolute one.
Hence, the question we are going to investigate is:
Among the curved (simple and smooth) paths from O to B, with a given length (L>|OB|), is there a path for which ĉ is minimal (has a local minimum)?
Since the square root function increases within its domain, and the value of L is constant, the question above is the same as the following question:
Among the curved paths from O to B, with a given length L>|OB|,
is there a path providing a (local) minimum for the following integral:
According to the assumption, the function φ(l) is continuous and with a continuous derivative in the segment 0 ≤ l ≤ L.
Let us denote by X1,Y1 the coordinates of point B, which is the end point of all the paths. According to the parametric equations (1) which describe the coordinates of the point on the path from O to B, we have:
According to the above, one can re-phrase the minimum problem of the paths' curves connecting two given points on a plane in the following manner:
Among all the functions φ(l) that are continuous, have a continuous derivative and satisfy (3), (4) in the segment [0, L], find a non-constant function providing a (local) minimum to the integral (2).
In the next paragraphs we bring a theoretical solution to the problem, based upon the calculus of variations, and compare the theoretical results with real processes. Later on we discuss a probabilistic approach to constructing a mathematical model of the meander phenomenon.
Some additional details are given in the Appendix.
Applying Lagrange multiplier method for solving the problem
In order to relate to the integral (2) and to the constraints (3), (4) in a unified manner, we perform an operation suggested by Lagrange.
Let us construct a new expression derived from (2), (3), and (4) in the following way:
where the coefficients μ, λ, called Lagrange multipliers, are arbitrary numbers.
According to one of the theorems of the calculus of variations, if a function φ(l), which is continuous and has a continuous derivative in the segment [0 L], provides a minimum to the integral (2) under constraints (3), (4), this function provides a minimum as well to the integral (5), with the appropriate values of multipliers μ, λ which do not vanish simultaneously.
Using Euler equation for further development
Let F designate the integrand in integral (5):
Thus a second-degree differential equation is obtained, in which the unknown is the function φ(l). Note that the equation contains trigonometric functions of the functional variable φ(l).
Solutions of the differential equation
Von Schelling  found the formula, which describes in an implicit form the infinite number of solutions of the equation (8).
where l and φ are variables and α, ω are positive real constants (parameters) .
It is impossible to bring this formula to an explicit form, since the integral in (9) can't be expressed with the help of analytic functions in a closed (finite) form. On the basis of Von Schelling's solution, Leopold and Langbein  found explicit approximate solutions of the differential equation (8) of the form:
The graph of function (10) is obtained by stretching the graph of the function sin l, along the coordinate axes. Formula (10) describes a family of functions, which are the sine functions with a period L, and an amplitude ω.
The development of formulas (9), (10) is shown in the appendix.
A theoretical curve of meanders
According to Leopold and Langbein , the function (10) describes approximately the change of meander direction. According to this result, when (10) is substituted in (2), the shape of the meanders is approximately described by the parametric equations:
where t is a parameter varying in the segment [0, L].
Leopold and Langbein named curve (11): Sine generated curve. The integrals in equations (11) cannot be calculated directly, only by numeric methods. Figure 11-A below, illustrates a curve constructed by Mathcad software for some given values of L and ω. It is important to state that the method of finding curve (11) is based on the necessary condition for a local minimum of integral (2) under constrains (3), (4). Therefore, it is to be additionally checked, whether among all the curves in the neighborhood of the sine-generated curve, this curve actually provides the minimum value for integral (2). A computer-assisted investigation in this direction is presented in the next paragraph.
Computerized verification of the theoretical results
Let us define two direction functions on the segment 0 ≤ l ≤ L: a trigonometric function
and a polynomial function of the third degree φ(l) the graph of which within
0 ≤ l ≤ L is of a shape similar to the graph of fo(l). Then, we compare the average curvatures of the two paths, constructed by the parametric equations (1) for the two functions φ0(l), φ(l). If in spite of changing the parameter values ω,L we'll see by examination, that the average curvature of the "sine generated curve" is smaller than the curvature of the curve generated by φ(l), this will support the theory, of course.
The polynomial function φ(l) can be constructed as follows: choose a third degree polynomial β(l) which vanishes, like the function φ0(l), at the endpoints and center of the segment 0 ≤ l ≤ L :
We chose a polynomial function φ(l) which resembles the trigonometric function φ0(l) in a sense that both have the same zero points l=0, l=L/2, l=L, and the same amplitude ω. Figures 11-A, 11-B present the curves (which represent paths) created by the parametric equations
orrespondingly, for the values L = 3, ω = 1.5. The value of the square of the average curvature ĉ of the curve, calculated by the integral of the square of the derivative of the path generator in the segment: 0 ≤ l ≤ L , appears later on.
A curve generated by the polynomial
if this curve represents the shape of meanders. A computerized experiment indicates that for ω values greater than 2, there are loops in the sine generated curves. This configuration produces non-simple paths, in contrast to our initial hypothesis, which was based upon the shape of meanders in nature (the interested reader can investigate why these values imply a change).
Comparing the theoretical curve to the real shape of meanders
Many attempts have been done by meander researches in rivers to compare the shape of meanders in reality to the theoretical "sine generated" curve. The outcome of these comparisons regarding two rivers is presented in Figure 12 on the left hand side. The wide line represents the meander shape as found in surveys of river paths in nature. The broken line almost adjacent to it is the "sine generated" theoretical line, i.e. the curve that has as its direction function:
appears in a thin continuous line in the coordinate system. In addition, few discrete values of the direction function in reality are shown by points in the coordinate system. These were obtained empirically by measuring the direction of the river flow.
Figure 12 (from , page 64): On the left: A comparison between the theoretical curves and the shapes of meanders in reality. On the right: a comparison between the direction function of a theoretical path and the values of direction function of the path in reality
A probabilistic approach to constructing a mathematical model for the meander phenomenon
The mathematical model of the meander phenomenon, presented above, was built in a deterministic way, i.e. without any probabilistic considerations. We turn now to a probability-based approach to building such model. Essentially, this approach corresponds to Von Schelling's approach, with some shortcuts .
Let us assume that a particle moves with a constant speed (=1) on a plane, but its direction changes at random in time. The particle direction at a certain point in fixed time is measured by the angle φ, between the particle speed vector and the positive direction of the x axis. Let us mark by Δφ the direction change. Assume, to start with, that changes of the particle direction happen at equal time intervals.
Here, again, φ indicates the direction function of the particle path, and L indicates the length of the path. The question asked previously about the direction change set becomes a question about a direction function: For which direction function φ(l) the integral (12) takes a minimum value?
If in addition we require that the path ends at a given point x1,y1, we have to add boundary conditions:
The integral (12) and the conditions (13), (14) are correspondingly identical to the integral (2) and to the conditions (3), (4). Thus, we have reached the same minimum problem as obtained by the deterministic approach. All the solutions to this problem are presented in formula (9). It contains two parameters and defines an infinite number of curves in a plane possessing minimum average curvature.
The development above yields yet another special characteristic of these curves as most probable paths in random movement in a plane. These paths are illustrated in Figure 13. One may see some of them in the free movement of smoke rising up from a cigarette.
Let us think now about the river bed changes as a random process, influenced by a variety of random factors. Since curves of meanders are good approximation to the curves from the family above, one may say that the meandering path of a river does not only possess the minimum average curvature, but is also the most practical among all the paths that a river can "choose" to flow in its random movement.
Figure 13: The most common paths of random movement in a plane (From , page 225)
Finding solutions of the differential equation (8)
(one may find the value of ω experimentally from the realistic meander shape). Let us choose as h the average value of φ(l), i.e.: let us assume that h = 0. The values C, U we'll choose according to two conditions:
(a) The expression in the right hand side of equation (ii) is non-negative.
(b) The solution of equation (ii) is a periodic function in the variable l.
Condition (a) is based on mathematical considerations and condition (b) on considerations linked to the physical shape of the meanders.
One can ensure the fulfillment of these conditions by choosing:
where φ , l are variables and ω, α are real positive constant numbers.
Formula (v) describes in an implicit form an infinite set of solutions of equation (iv). It is not possible to bring this formula to an explicit form, because the integral on the left hand side of (v) cannot be expressed using analytical functions in a closed (final) form.
Now, let us calculate the integral (v) approximately and as a result find explicit approximated solutions of the differential equation (iv).
For cosine difference we have the identity
Let us assume that ω is a real, positive, small enough number. Then each sine argument in the formula (vi) is also small. Applying the well-known approximate equality sin X~X to formula (vi), and recalling that the smaller the x value, the smaller the approximation error is, we get the following approximate equality:
The last equation represents infinite number of explicit approximate solutions of the differential equation (iv) according to the infinite possible values of the parameters α, β. The deviation of these approximate solutions from the accurate solutions of equation (iv) is small when ω is sufficiently small. To choose particular solutions among all these solutions, one needs to lay more assumptions. Let us, for instance, assume that the path is tangent to the x axis at each of its ends, i.e., φ(l) = 0 where l = 0 and l = L.
Then, based upon (viii), we get:
. Levanoni Y. (1975) "Is the Jordan river about to straighten?" Nature and country. April, 1975.
. Alluvium is a term that expresses all the substance collected by the river through sweeping away or dissolved in its water.
Calculus of variations is a field of mathematics which deals with functions of functions - or functionals (as opposed to ordinary calculus which deals with functions of numbers). The interest is in extremal functions: those making the functional attain a maximum or minimum value.
. Shetner Y. (1970) "Geomorphology - the external pattern of the outride". Kiriat Sefer publishers Ltd. Jerusalem. Chapter: "The river valley and its shape". Page: 97-113.
.. Langbein, W.B., (1966). River Meanders – Theory of Minimum Variance. U.S. Geological Survey Professional Paper 422-H, pp.1-15.
.. Leopold, L.B., Langbein, W.B.(1966). River Meanders. Scientific American 214, pp. 60-70.
.. Callander, R.A., (1978). River Meandering. Annual Reviews Fluid Mechanics 10, pp.129-158.
.. Von Schelling H. (1951). Most Frequent Particle Paths in a Plane. Transactions American Geophysical Union 32, pp. 222-226.
Footnote: The feature article above is a translation of a paper in Hebrew published in 2000 in Aleh - The Israel Journal for High School Mathematics Teachers, vol. 25, pp. 62-76. We are grateful to the Aleh Editor, Dr. Hamutal David, for granting permission to reproduce this article in English translation.