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Up a Lazy River

Meandering through a classic theory of why rivers meander

Brian Hayes

Around the Bend

Leopold brought a distinctively quantitative and mathematical style to the study of rivers. For example, he formulated scaling laws that describe how the cross section of a natural channel changes as a function of the volume of water flowing through it. He even did some computer simulations—without a computer! Using shuffled decks of cards or tables of random numbers, he carried out probabilistic studies of landform features such as the branching of a drainage network.

The sine-generated curve...Click to Enlarge Image

The serpentine shapes of meanders certainly invite mathematical analysis. Although in nature the curves are highly irregular—no two alike, perhaps—Leopold argued that they all derive from a specific underlying form, which he called a sine-generated curve.

Imagine you are canoeing down a meandering river with a compass in hand, making note of your heading at regular intervals. According to Leopold, your direction should vary sinusoidally as a function of the distance you have traveled along the river centerline. This is not to say that the shape of the river itself is a sine curve; rather, the sine function specifies the heading. The governing equation is:


Here q is the heading angle, measured with respect to the mean down-valley direction (the path the river would follow if it did not meander at all); s is distance along the stream centerline; and w is the maximum angle that the path makes with the down-valley axis. For small values of w, less than 90 degrees, the sine-generated curve has gentle undulations, so that the river weaves back and forth but at all times maintains a down-valley component of motion. At w=90 degrees, the path of the stream crosses perpendicular to the valley axis. At still larger values of w, the lobes of the curve become horseshoe-shaped, and for part of each meander cycle the river's course takes it back up the valley. A little beyond w=120 degrees, adjacent lobes of the curve begin to overlap. On graph paper the lines merely cross, but in a river this event signals the development of a "cutoff," diverting the flow and leaving behind a stranded oxbow lake.

The sine-generated curve looks like a plausible candidate for describing meanders, at least within a limited parameter range. But what made Leopold so sure it was the one right candidate? His argument goes as follows. Take two points a and b connected by a stretch of river of length L, where L is greater than the straight-line distance from a to b. Now think of all the ways of bending and folding this segment of river into a smooth curve without changing its length or detaching it from its end points. Among all such paths, the sine-generated curve has three interesting properties: It is the path of minimal bending stress, it is the path of minimal variance in direction, and it is the path representing the most likely random walk. I shall first discuss the two minimization principles and return later to the random walks.

A thin strip of spring steel...Click to Enlarge Image

The bending stress of a river is the work or energy that has to be expended to make its path deviate from a straight line. At each point along the route, the bending stress is proportional to the square of the curvature at that point. For a straight segment, bending stress and curvature are both zero; they increase as a turn gets sharper. Among all smooth, length-L curves from a to b, the sine-generated curve has the smallest squared curvature summed over the entire path.

Directional variance is a similar concept. As you follow the river from ato b, measure at each point along the way how much your heading deviates from the mean down-valley direction, then compute the sum of the squares of these angles. Again, the sine-generated curve yields the smallest possible total.

These properties of the sine-generated curve are mildly surprising. I would have guessed that an arc of a circle—the most symmetrical curve—would have the lowest squared curvature and directional variance, but that is not the case. (Of course a straight line is superior, but that solution is forbidden by the length constraint.)

Leopold offers a simple demonstration of how the sine-generated curve emerges as a natural solution to a problem of minimizing work or energy. If you hold the ends of a strip of spring steel so that it forms a horseshoe-shaped loop, the metal spontaneously adopts the form of a sine-generated curve. I couldn't resist trying this myself. I found that it works reliably only for single loops. If you try to fold the spring into multiple meanders, the configuration is unstable.

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