A Discourse of Rivers
When Size Doesn’t Matter—
Self-Similarity Over Scale Changes
By Tamia Nelson
March 11, 2008
When I was a kid I loved playing in water. Any water. Puddles, drainage ditches, streams hardly big enough to float a good-sized tadpole…it made no difference to me. And when I played, I played hard. I was a little engineer, building dams across tiny rills and digging canals to drain water-filled potholes. I didn’t stop there, either. I used the spoil from my dredging to construct tiny islands. Then I built boats from twigs and scraps of wood and set them afloat in a watery world of my own making, watching intently as the little craft drifted wherever wind and current carried them.
Of course I was just playing at being an engineer, though I learned a lot about moving water in the process. But it wasn’t until I started paddling that I realized my childhood lessons could be “scaled up” to the real thing. And thereby hangs a tale. Lesson Number One…
Size Doesn’t Matter
Surprise! The tiny ribbons of snowmelt that twist and turn across the apron of sand in your driveway exhibit all the features of rivers everywhere, albeit on a smaller scale. With the exception of the pororoca, the tidal bore that sweeps upstream on the back of the highest of high tides, not even the mighty Amazon has a bigger bag of tricks. Yes, the Amazon packs a much greater punch. Obviously. After all, it drains most of a continent. But the dynamics are the same. For instance, an eddy is an eddy is an eddy. It doesn’t matter whether it forms behind a pebble in your drive, a boulder the size of a Greyhound bus midstream in a mountain torrent, or an island as big as Manhattan in one of the world’s largest rivers. An eddy is an eddy is an eddy. Period. Scientists have coined a phrase to describe this remarkable kinship: self-similarity over changes of scale. That’s a bit of a mouthful, to be sure, but the underlying idea is dead simple.
The implications for paddlers are profound, however. Consider this. If you want to know how the Amazon works, you don’t have to book a flight to Brazil. You just have to wait for the snow in your yard to start melting, or for a rainstorm to pass through. Then go out onto your driveway and look down. The Amazon lies at your feet—in miniature. It works the other way, too. Anything you can find in the Amazon you can find somewhere along the course of the rill in your drive (tidal bores excepted).
Maybe you need to be convinced. If so, check out this picture. (No, it wasn’t taken in my driveway. But it ought to illustrate my point, nonetheless.)
Look carefully. How big do you think the reversal in the photo is? It’s hard to judge, isn’t it? And why is that? Simply because the water alone doesn’t tell you enough. You need to put something else in the frame—a boat, say, or a tree, or a paddler—to give it scale. So let’s step back a little and look at the whole picture (the yellow rectangle identifies the location of the first shot):
Now you can see the trees on the opposite shore. That’s a lot better than nothing, but unless you can estimate their height you’ll need still more to gauge the magnitude of the drop and the size of the reversal. As it happens, however, I know the trees well, and I’m also acquainted with this river in all seasons. So I know that the drop in the picture is around four feet high, with the reversal extending about the same distance out. That’s not much, but it’s enough to get careless boaters into serious trouble. In fact, on the day I shot the picture, the river—it isn’t the Amazon!—was swollen with snowmelt. Only the most skillful (or foolhardy) paddler would attempt to run the drop under these conditions.
OK. Let’s look at another picture—the one on the left below. Was this shot from a chopper hovering over over the lip of a hundred-foot falls somewhere along the headwaters of the Amazon? Or from a sea-cliff on the coast of Hawai’i? Or did I find it closer to home? What do you think? Hard to tell, isn’t it?
Well, maybe it wasn’t that hard, after all. But you’ll have to admit I made it easy for you by also giving you the shot I cropped to make the mystery picture. (Again, the yellow rectangle shows the location of the first shot.) This time, it’s the cobbles on the far shore that help us out. In any case, my “hundred-foot falls” turns out to be nothing more than another ledge on The River that runs by my door. Without the second picture to clear thing up, however, you’d be hard pressed to tell this. It could be anywhere. It could even be a “hundred-foot falls.” That’s “self-similarity over scale” in action.
Now let’s move on. Waves leave their mark on the land beneath. You’ve probably seen ripple marks in the sand at the seashore. You can also see them in shallow sediments along the margins of ponds and lakes, and at the bottoms of streams. These are the calling cards left by passing waves—provided that the waves are relatively gentle, that is. (Dumping surf quickly scours away all evidence of its passing. Given enough time or a strong storm, it can scour away the beach, too.)
But that’s not the topic for today. Let’s get back to ripple marks. They also exhibit self-similarity over scale. Here’s an example:
One set of photos shows the bed of a little stream. The other two are aerial photos depicting a great swathe of a landscape. But which is which? Well, the color pictures on the left depict ripple marks left by a temporary stream that was only five feet wide at its widest point. On the other hand, the black-and-white photos were taken by NASA. They show ripples the size of hills. In fact, these ripples are hills. And they’re not just any hills. They’re the Channeled Scablands, relics of an ancient flood of glacial meltwater—actually, the most recent of many such ancient floods—that swept through eastern Washington state at the end of the last Ice Age. By contrast, the ripples in the bed of the temporary stream were only a few inches across. Yet unless you had other clues to go by, I’ll bet you’d have trouble telling the difference. ’Nuff said?
So much for reversals, waves, and ripple marks. Next, we’ll look at…
The Big Picture
And I mean the really big picture. If you enjoy reading maps as much as I do, then you’ve almost certainly noticed that rivers, like waves, leave their mark on the land, and that the patterns they carve in their journey to the sea are nearly as distinctive as fingerprints. The rivers draining the Adirondack massif look different on the map than those draining the Appalachians, for instance. Geologists call these different motifs “drainage networks,” and it turns out that, although each river is unique, every possible network can be assigned to one of nine basic types. The drawing above shows an example of each type, drawn from rivers around the world. Now look at any small-scale map—an atlas or road map will do fine. You’ll probably find that you can infer quite a lot about the waterways on the map without even wetting a blade. The shapes of the drainage networks give much of the game away, telling you a little bit about the underlying geology and the character of the rivers themselves. (Look closer and you’ll see other patterns superimposed on the nine basic networks. But this takes us into the realm of “channel morphology,” and that’s a subject for another day.)
Back to drainage networks. The next time you go for a walk in the rain down a dirt road, take a few minutes to investigate the ephemeral rivulets that crisscross the road beneath your feet. You’ll see an example on the left below. (You might find something like it in your driveway.) It’s paired with an aerial photo of a braided section of the Brahmaputra, one of the great rivers of Asia. I think you’ll agree that the family resemblance is striking, even though the braided channels on the left could be bridged by my hand.
And now, a final illustration
Here are two aerial photos, both depicting meandering streams. But the “stream” on the left is in fact the mighty Shenandoah upriver of Charles Town, West Virginia, while its companion on the right is a little creek in northern New York, no more than five yards wide. What do you think? Could you tell which was which if all other clues (vegetation, field boundaries, etc.) were airbrushed out of the photos? I know I couldn’t. That’s what’s meant by self-similarity across scale.
So—what’s the moral of my tale? Simply this. Even if the closest “real” river is miles from your home, you can learn a lot about moving water right on your doorstep. But the lesson doesn’t end there. If you want to avoid unpleasant surprises, you’ll need to bring all you’ve learned about scale along with you in your boat. That, too, is a subject for another time. Watch this space.
We’re paddlers. Water’s our medium. We get up close and personal with it. And the medium is the message. You don’t need to ride the Colorado or fly over the Amazon to study the interplay of water and land. You can go to school in your driveway. Just wait for the snow to start melting. Or for the next heavy rain to fall. Then watch what happens. That’s all. Where the dynamics of moving water are concerned, size really doesn’t matter.
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