I'm continually amazed at the number of every day things we have no clues about. Gradually my faith that my betters know how things work, and that I could in principle know too, is being beaten out of me.

I first encountered the mysteries of meanders in an article by Luna B. Leopold and Walter Langbein, published 40 years ago in Scientific American. They gave a lucid account of how meanders form and why they assume their characteristic sinuous shapes. I was a student at the time, and the article made a lasting impression. Not that I was inspired to go off and pursue a career in potamology, but the Leopold-Langbein theory of meanders was an eye-opener all the same. It brought home to me the curious fact that the world is a comprehensible place: You can look at a landform, say, and expect to understand what you see. The patterns of nature make sense, if you know how to read them.

The column examines the Leopold-Langbein theory of meanders in some depth and finds it unsatisfying. The discussion is worth a read, but concludes:

Let me return to the question with which I began this column: Why doesn't a river just take the shortest path to the sea? From the point of view of a drop of water moving with the current, there is no paradox in the existence of meanders. The water follows the local gravitational gradient, which always points downriver. But how does that gradient get twisted into such tortuous shapes? The issue is not how the channel guides the river but how the river carves the channel.

Simple curves, random walks and optimization principles may not be enough to answer such questions. We may need to get into the nitty-gritty of erosion, deposition and sediment transport. Leopold dealt with these matters in his accounts of meanders, as others had before him, going back a century or more. The basic idea is that once a bend has formed, differential erosion and deposition tend to exaggerate it. Water flows more rapidly near the outer bank, which therefore tends to wash away. Meanwhile the slower current near the inner bank drops its load of sediment, forming a "point bar." The net effect is to shift the channel in a way that widens the bend.

Computer simulations of this process have produced some very realistic-looking meanders. The models are detailed and elaborate, incorporating dozens of subtle effects—cross-channel currents, graded sediment, variations in bank erodibility. The output reproduces not only the static form of natural meanders but also their evolution.

Is that the answer, then: What we need to understand meandering is not abstract mathematics but a bucket of sand and silt? I would be willing to leave it at that but for one extraordinary fact: Rivers meander even when they carry no sediment, and even when they have no banks! Meltwater streams atop glaciers, with no sand to deposit in point bars, meander much like other rivers. And the Gulf Stream, flowing unconfined in the open ocean, also meanders in a way remarkably like that of a river carving its way through continental alluvium. It appears there may be some principle at work that transcends the particular dynamics of the erosion-deposition cycle.

Reviewing the state of meander studies in 1998, David Knighton of the University of Sheffield concluded, "There is no general agreement as to how or why streams meander." That's a bit of a step backward from where I began—with admiration for Luna Leopold's simple and elegant theory. But I haven't lost my admiration, or given up on simple and elegant explanations. Although meanders have so far wriggled out of my grasp, I still think the universe will turn out to be a comprehensible place.

He has a touching faith, one I used to have as well. But I'm not as sure as I once was. I can't think of a compelling reason why the universe would be a comprehensible place for humans. Maybe it is, maybe it isn't. We have no good evidence one way or the other.

Update:

At his blog, bit-player, Brian struggles with another mystery.

Euclid famously said, “There is no royal road to geometry.” Among the non-royal roads, the computational pathway is notably muddy, rutty and potholed. . .

The animus behind this entire rant is a feeling that I must be missing something obvious, that a problem like finding the intersection of two line segments shouldn’t be this hard. The difficulty I’m talking about is not computational but conceptual. There are lots of hard computational problems—graph coloring, say, or factoring integers—for which one can write a very tidy and perspicuous program. True, the program may have a running time that exceeds your lifespan, but it’s easy to describe what needs to be done. Programs for geometric problems, in contrast, seem often to be efficient but hideous, with tangled logic, an abundance of special cases, and hidden numerical perils. Why is that? Nature seems to have no trouble at all detecting intersections or collisions. If two wires cross on a circuit board, you can count on blowing a fuse no matter what the slopes of the conductors. Why can’t we compute the same result so effortlessly?

The comments to that post offer some answers. For those who despair that the blogosphere is a lunatic asylum with all the crazies shouting at once this might be illuminating to read. Some of the lunatics have gathered in quiet corners to have interesting discussions.

Update: Turbulent Feedback

The story is told of many giants of modern physics, but most plausibly of Heisenberg, that, on his death-bed, he remarked that the two great unsolved problems were reconciling quantum mechanics and general relativity, and turbulence. "Now, I'm optimistic about gravity..."

So what, you may ask, is the fabled "problem of turbulence"? In essence, this: what on Earth do our statistics and our equation have to do with each other? A solution to the problem of turbulence would be, more or less, a valid derivation from the Navier-Stokes equation (and statements about the appropriate conditions) of our measured statistics. Physicists are very far from this at present. Our current closest approach stems from the work of Kolmogorov, who, by means of some statistical hypotheses about small-scale motion, was able to account for the empirical laws I mentioned. Unfortunately, no one has managed to coax the hypotheses from the Navier-Stokes equation (sound familiar?) and the hypotheses hold exactly only in the limit of infinite Reynolds number, i.e. they are not true of any actual fluid.

So what's to do? Well, all sorts of things, including more or less direct simulations of flows by cousins of cellular automata called "lattice gasses" (which is how I connect to the subject, though very vaguely). One approach uses the vorticity (the curl of the velocity field, which tells us about how the fluid swirls), since it turns out to be possible to identify some (more or less) simple objects in the flow, called vortex lines or vortex tubes, work out how they interact (there's a Hamiltonian), and then use statistical mechanics to calculate various emergent properties --- which, if you use just the right approximations, and tolerate negative temperatures (which are not impossible, and actually hotter than infinity) gives you the Kolmogorov laws. This could've been custom-tailored for my philosophical and methodological biases, which makes me suspicious, as do all the leaps in the approximation scheme used. (For the pro-vorticity case, see Chorin; reasons for caution are discussed by Frisch, pp. 189f.)

If people must find analogies for society, ecosystems, etc., from physics and engineering, turbulence is probably a better one than feedback.