Sinuous rivers.

The windings of rivers have long fascinated their human observers. For example, Aboriginal legend explains the sinuous pattern of the modern Finke River (Fig. 1) as the creation of the immense and powerful Rainbow Serpent as he emerged during the Dreamtime from deep waterholes. Recently in PNAS (1), a new theory for the general origin of such sinuous flow patterns was published, which follows from a long tradition in seeking a scientific explanation for the winding patterns of rivers. Science itself was arguably born when the first explanatory/genetic hypotheses were formulated in the sixth century BCE by the pre-Socratic philosopher Thales (c. 624 BCE– 546 BCE) in the Ionian Greek city of Miletus (2). The site of ancient Miletus, now in southwestern Turkey, overlooks a river that in ancient times was known as the Maeander. From this name comes the word “meander,” which since classical Greek time has applied to the description of anything winding, including especially, of course, the curving patterns of rivers. We now know that channelized flows of water, lava, and other fluids have incised such patterns into the surfaces of our solar system’s rocky planets and moons (3). Responding to the need for a general theory to explain all these sinuous, threadlike flow patterns, Lazarus and Constantine’s (1) theory employs a numerical model of flow routing to demonstrate that flow resistance R (involving a kind of landscape roughness) relative to surface slope S provides the fundamental control on sinuosity, which for rivers is defined as the long, winding path length of the river channel divided by the relatively short and direct path length down the valley or other surface into which that sinuous river has incised. Although Lazarus and Constantine’s (1) theory is directed at a general explanation for sinuous flow patterns in all types of channelized flows, the long history of scientific interest in those patterns has focused mainly on the flow type exhibited by alluvial rivers, which have beds and banks composed of the same types of sediment that the rivers commonly transport along their channels. Attributes of such rivers are well illustrated in the notebooks of Leonardo da Vinci (1452–1519), who even depicted a meandering river over the right shoulder of the Mona Lisa in his famous painting. Leonardo da Vinci’s notebooks contain numerous sketches showing the swirling movements of secondary currents in water, and there is one diagram that shows how meanders can migrate in a downstream direction. Moreover, da Vinci’s interest in winding rivers was not confined to the aesthetic and the scientific; he also studied rivers for purposes of practical engineering. Indeed, much of the subsequent progress on understanding river meandering was made by engineers. As described in the classic 1955 book, An Introduction to Fluvial Hydraulics (4) by Serge Leliavsky (1891–1963), two general schools of practical engineering research developed in regard to alluvial rivers. One school was empirical, using quantitative measures of river properties. For example, an observation that da Vinci had made qualitatively was quantified in the 19th century: there is a regular downstream decrease in the size of sedimentary particles on a streambed that closely follows the downstream decrease in the slope of that stream. This relationship, known as “Sternberg’s Law” (5), was used by Armin Schoklitsch (1888–1969) to infer an explanation for river sinuosity. Presuming from this “law” that a river’s slope must be adjusted to the diameter of sediment transported, Schoklitsch (6) reasoned that, if this slope of transport is less than the average surface slope of the plain into which the river channel is incised [i.e., the slope S used in Lazarus and Constantine’s (1) theory], then it will be necessary for the river to assume a winding path to make its channel slope equal to the slope that is appropriate for the transported sediment size. Because the channel slope is the ratio of vertical fall to the distance measured along the winding path of the channel, dividing this number into the valley or surface slope, S, which is ratio of the same vertical fall to the direct path down that valley, will yield a ratio of the distance measured along the winding path of the channel to the distance along the more direct path down the valley, which is by definition the sinuosity of the river. Thus, like Lazarus and Constantine’s (1) theory, the Schoklitsch theory (6) places emphasis on sinuosity in relation to surface slope, but unlike Lazarus and Constantine’s (1) theory, it does so in relation to the sediment size that the forces of the river are transporting instead of the land-surface roughness R that is opposing those forces. The Schoklitsch explanation for meandering accords with the observation that meandering generally takes place in the lower courses of rivers, where sediments are relatively fine-grained and the corresponding slopes are relatively flat. However, as noted by Leliavsky (4), the theory is not very useful to hydraulic engineers who need a rational, mechanical formulation of the problem, which is the motivation for the second school of alluvial river engineering. An early example from this school was the causative explanation for river meandering proposed by the Scottish civil engineer James Thomson (1822–1892), elder brother to William Thomson (later to become Lord Kelvin). Thomson’s paper, “On the origin of windings of rivers in alluvial plains, with remarks on the flow of water round bends in pipes” (7), was communicated to the Royal Society by his brother (who was then a member) on March 14, 1876. The paper describes the phenomenon that, as water moves around a river bend, a helical Fig. 1. Oblique aerial view of the gorge of the Finke River winding through the Krichauff Ranges of central Australia. A new generic theory of channel sinuosity explains such patterns in terms of flow resistance relative to surface slope.