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Up a Lazy River

Meandering through a classic theory of why rivers meander

Brian Hayes

River of Randomness

The random-walk model of river meandering is one example of how a local rule—in this case aimless wandering—might give rise to large-scale regularity. The premise is that over decades or centuries, a river channel can drift over its floodplain, twisting and shifting at random (although always subject to certain constraints, such as not flowing uphill). Any configuration is possible, but all of the most likely ones, according to Leopold, look something like a sine-generated curve.

Trajectories of random walks...Click to Enlarge Image

The mention of random walks in this context both intrigued and confused me. Random walks have become a common notion in recent years, and yet the kind of walk that yields the sine-generated curve was not one I had encountered before. From Leopold's description I was not able to grasp all the details. He referred to earlier work by Hermann von Schelling, but the crucial document was a 1964 technical report from the General Electric Company, which I had a hard time tracking down. Eventually I found a copy at the Smith College library.

The process studied by von Schelling is one in which a walker takes a step of unit length, turns through a randomly selected angle, takes another step in the new direction, and so on. Not just any such walk qualifies, however. To be admitted, a walk must begin by leaving point a at a specified angle; it must end by reaching point b; and in between it must cover a specified distance L. Among random paths that satisfy these constraints, von Schelling asked what the most frequent or likely paths might look like. If the walker chooses each step's direction from a uniform probability distribution (so that any angle is equally likely), von Schelling got no nontrivial answer. But he did find a solution for a walk where the turning angle at each step is drawn at random from a normal, or Gaussian, distribution with a mean of zero.

Von Schelling's mathematical solution takes the form of an integral that he found difficult to evaluate. The sine-generated curve is an approximation to the value of this integral—inexact, but quite close within the range of parameter values of interest for river meanders. Strictly speaking, the properties of minimal squared curvature and minimal directional variance have been proved only for the exact curve defined by the integral, not for the approximation. At the level of detail needed for describing river channels, however, the discrepancy is of no consequence.

From a computational point of view, the trouble with these most-frequent random walks is that they're not nearly frequent enough. The naive algorithm for generating examples of such paths calls for launching many walkers from point a, all in the appropriate initial direction, and then discarding all walks except those that happen to reach point bafter exactly Lunit-length steps. There are infinitely many walks that satisfy these criteria, and yet the probability of ever seeing one is zero. Life is too short to wait for such events.

In order to get some rough idea of what individual von Schelling walks might look like, I have tried a sloppier algorithm. Instead of insisting that a walk end precisely at point b, I accept any walk that takes the requisite number of steps and lands within one further step of b. Even with this relaxed criterion, the algorithm is practical only for fairly short walks.

Superimposing a few hundred of these walks produces quite a frizzy hairball, but taking the average of all the paths yields a smooth arc that resembles a sine-generated curve. One peculiarity of the average walk is its asymmetry: It leans one way or the other, depending on the departure angle at point a. The reason is that we have specified the direction of the initial segment but put no constraint on the final step at b. This may have been an oversight in the way the problem was formulated by von Schelling. (On the other hand, for what it's worth, many river meanders exhibit systematic asymmetry, typically crossing the valley at a sharper angle on the upstream leg.)

A deeper perplexity awaits when we go in search of von Schelling's "most likely" or "most frequent" random walk. Should we look for it among the individual walk trajectories, or in the average of all such walks? Which of these is the right model for a river meander? Often, the terms "typical" and "average" are nearly synonymous, and Leopold clearly thought that the average would be representative of the population; "the most probable path is the average path of a random walk," he wrote. In other words, if you choose a random walk at random, it will probably be much like the average of all random walks. Von Schelling offered an analogy with thermodynamics, where uncommon events (such as perfume returning to its bottle) are so utterly improbable that we invent laws of physics to forbid them. It's a fundamental assumption, he wrote, "that in our environment random walks are approaching most frequent paths in an overwhelming majority of cases." But then he added: "This is far from being self-evident."

It's certainly not evident in the little sample of walks I generated. Not one of the individual walks looks anything like the average of all the walks. If we imagine a river channel wandering over a floodplain according to this algorithm, wouldn't a snapshot made at some arbitrary moment be likely to resemble a single random walk, rather than the average? But it's the average of the walks that corresponds more closely to the sine-generated curve and to the shapes seen in real landscapes.

Admittedly, the algorithm that generated these specimens is inexact, at best. Von Schelling's calculations call for taking a limit as the step size goes to zero, and my simulations are nowhere near that limit. Also, it should be noted that individual walks can be made more like the average walk by reducing the standard deviation of the angular distribution—by squeezing the randomness out of the random walk. Still, as von Schelling noted, it's far from self-evident that the typical path will ever come to resemble the average path.

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