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The Mathematics of River Meanders

Profile image of Nitsa Movshovitz-HadarNitsa Movshovitz-Hadar
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Abstract

All the rivers run into the sea; yet the sea is not full" (Ecclesiastics 1, 7). But why is it that they all go meandering to the sea? Why do rain drops usually fall in straight lines, and a ball, when let roll freely on a steep slope, rolls down the shortest way, while rivers don't flow in a straight line? Is there logic and order in river meandering? Is there a mathematical model, which can predict a meander's route? These questions and more are addressed in this paper.

Key takeaways
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  1. River meanders follow consistent geometric patterns, influenced by gravity and current dynamics.
  2. The typical meander-ratio is approximately 3:2, with the Jordan River having a ratio of 2:1.
  3. Meander development varies based on geological conditions, categorizing them as 'free' or 'engraved' meanders.
  4. A mathematical model employing calculus of variations helps predict meander shapes and behaviors.
  5. Meander evolution reflects a balance in river dynamics, maintaining consistent riverbed length despite changes.
Figures (12)
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. 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.
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. 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.
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 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.
————— ee a a ee 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.
————— ee a a ee 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.
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The creation of a meander shape with a flexible metal strap (From [4], page 66)
The creation of a meander shape with a flexible metal strap (From [4], page 66)
Figure 0: A schematic shape of a meander (From [4], page 62). W - width of the river bed; 4 - wave length; L - length of the river bed; r,- curvature radius
Figure 0: A schematic shape of a meander (From [4], page 62). W - width of the river bed; 4 - wave length; L - length of the river bed; r,- curvature radius
A A schematic description of the meander progress in six steps (From: [2], page 106) 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.
A A schematic description of the meander progress in six steps (From: [2], page 106) 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.
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. As is well known, between each two points there is an infinite number of simple, emnanth nathe and among tham thara ic af cnirea tha etrmiqght lina ae ran ha caan in
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. As is well known, between each two points there is an infinite number of simple, emnanth nathe and among tham thara ic af cnirea tha etrmiqght lina ae ran ha caan in
Image
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
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
Figure 12 (from [4], 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 é deterministic way, i.e. without any probabilistic considerations. We tum now to ¢ probability-based approach to building such model. Essentially, this approach corresponds to Von Schelling's approach, with some shortcuts [6]. 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 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: Comparing the theoretical curve to the real shape of meanders was based upon the shape of meanders in nature (the interested reader can investigate why these values imply a change).
Figure 12 (from [4], 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 é deterministic way, i.e. without any probabilistic considerations. We tum now to ¢ probability-based approach to building such model. Essentially, this approach corresponds to Von Schelling's approach, with some shortcuts [6]. 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 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: Comparing the theoretical curve to the real shape of meanders was based upon the shape of meanders in nature (the interested reader can investigate why these values imply a change).
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References (6)

  1. Levanoni Y. (1975) "Is the Jordan river about to straighten?" Nature and country. April, 1975.
  2. 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.
  3. Langbein, W.B., (1966). River Meanders -Theory of Minimum Variance. U.S. Geological Survey Professional Paper 422-H, pp.1-15.
  4. Leopold, L.B., Langbein, W.B.(1966). River Meanders. Scientific American 214, pp. 60-70.
  5. Callander, R.A., (1978). River Meandering. Annual Reviews Fluid Mechanics 10, pp.129-158.
  6. Von Schelling H. (1951). Most Frequent Particle Paths in a Plane. Transactions American Geophysical Union 32, pp. 222-226.

FAQs

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What defines the meander-ratio of a river and its implications?add

The meander-ratio, commonly 3:2, defines the ratio of riverbed length to the straight aerial distance. The Jordan River exemplifies a higher meander-ratio of approximately 2:1, influenced by human interventions since 1932.

How do meanders evolve differently in various geological environments?add

Meanders develop similarly across diverse environments, maintaining consistent geometric shapes, regardless of geological context. For instance, meanders formed in rich alluvium have shapes akin to those in icebergs' ravines, demonstrating uniformity in meander creation.

What are the key factors driving meander truncation processes?add

Meander truncation occurs when the river's flow breaches the meander neck during floods, disconnecting one arc. This results in crescent-shaped meander drains that eventually dry up while new meanders form downstream, keeping riverbed length stable.

What mathematical models aid in predicting changes in river meandering?add

Researchers utilized models based on Von Schelling's work, employing calculus of variations to analyze meander shapes. These models provide insights into determining optimal river paths and understanding the balance of flow dynamics.

How does meandering affect the flow speed and erosion along riverbanks?add

The meandering nature reduces local slope and flow speed, impacting sediment deposition and riverbank erosion. The curvature of meanders is constrained by these dynamics, influencing maximum curvature radius and overall river morphology.

About the author
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Nitsa Movshovitz-Hadar
Technion Israel Institute of Technology, Emeritus

http://edu.technion.ac.il/en/show_details.php?id=47

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