RIVER MEANDERS s there sucli a thing as a straig..l they are iiot a necessary reason. For one posited on the same side of the stream river? Alinost :inyone can tliiuk of a thing, such irrrgularitics cannot account from wliicli it was eroded. The condi- river that is inore or less straight for for the rather consistent geometry of tions in which meanders will be formed :I certain distaiice, brit it is iinlikely that meanders. Moreover, laboratory studies in rivers can be stated rather simply, the straight portion is either very indicate that strcams meander even in albeit only in a gencral way: kleanders straight or very long. In fact, it is almost “itleal,” or highly regular, mediums [see will usually appcar wherever the river certain that the distance any river is i/hfrutioti on pgs 641. traverses :i gentle slope in a medium straight docs not cxccetl 10 times its That the irregularity of the medium consisting of fine-grained material that width at that point. has little to do with the formation of is easily erodctl and transported but has The sinuosity of river channels is mcanders is further demonstrated by sufficient cohesiveness to provide firm clearly apparent in maps and aerial pho- the fact thut meandering streams have banks. tographs, where the successive curves been observed in several naturally ho- A given series of meanders tends to 4 of a river of‘ten appear to havc a certain mogeneous mediums. Two examples are have a constant ratio between the wave- regularity. In niany instances the re- ocean currents (notably the Gulf length of the curve and the radius of peating pattern of curves is so pro- Stream) and water channels on the sur- curvature. The appear:ince of regularity nounced that it is the most distinctive face of a glacier. The meanders in both depends in part on how constant this characteristic of the river. Such curves cases are as regular and irregular as ratio is. In the two drawings on page 62 are called meanders, after a winding river meanders. the value of this ratio for the meander stream in Turkey knowii in ancient The fact that local irregularities can- that looks rather like ii sine wave (top) Greek times as the hiaiandros and today not account for the existence of river is five for the wavelength to one for the as the hlenderes. The nearly geometric meanders does not rule out other ran- radius; the more tightly looped meander regularity of river meanders lins at- dom processes as a possible explana- (bo~om)has a corresponding value of tracted the interest of geologists for tion. Chance may be involved in subtler three to one. A sample of 50 typical many years, and :it the US. Gcological and inore continuous ways, for example meanders on many different rivers and Survey wc havc devoted considerable in turbulent flow, in the manner in streams has yielded an average value study to the problem of understanding which the riverbed and banks are for this ratio of ahout 4.7 to one. An- the general mechanism that underlies formed, or in the interaction of the flow other pi-operty that is used to describe the phenomenon. In brief, we have and the bed. As it turns out, chance op- meanders is sinuosity, or tightness of found that meanders arc not mere ac- erating nt this level can explain the bend, which is expressed as the ratio cidents of nature but the form in which formation of regular meanders. It is a of the length of the channel in a given a river does thc lcast work in turning, p::rxlox of nature that such random ciirve to the wavelength of the curve. and hence are the most probablc form processes can produce regular forms, For the large majority of meandering a river can take. and that rcguliir processes often pro- rivcrs the value of this ratio ranges be- duce randoni forms. tween 1.3 to one and four to one. h4eanders commonly form in allu- Close inspcction of the photographs Regular Forms from Random‘ Processes ~ vium (watcr-deposited material, usually Nature of course provides many op- unconsolidated), but even when they portunities for a river to change direc- occur in other mediunis they are in- ENTRENCHED MEANDERS of the Colo- tion. Local irregularities in the bound- variably formed by a continuous process rado River in southern Utah were photo- ing medium as well as the chance of erosion, transport:ition and deposi- graphed from a height of about 3,000 feet. The me.inders were probably formed on the emplacement of boulders, fallen trees, tion of the inatcrial that composes the surface of a gently sloping floodplain at blocks of sod, plugs of clay and otlicr case medium. In every rnaterial is almut the time the entire Colorado Plateau obstaclcs can and do divert many rivers eroded from the coiicave portion of a began to rise :it least a million years ago. from a straight course. Although loc:il meander, transported downstream and The meanders later 1,ecame more developrd irregularities are a sufficient reason for deposited on the convex portion, or bar, as river cut deep into layers of sediment. :I river’s not being straight, however, of a meander. The material is often de- Mean do\rn5tre.im direction is toward right. 60 CONCAVE BANK and maps that accompany this article will show that typical river meanders CONVEX BANK do not exactly follow any of thc familiar curves of elementary geometry. The portion of the meiinder near the axis of bend (the center of the curve) does POINT OF INrLECTION resemble the arc of a circle, but only approximately. Neithcr is the curve of a meandcr quite a sine wave. Geiiernlly the circular segment hi the bend is too long to be well described by a sine * wave. Thc straight segment at the point of inHcctioii-the point where the curva- AXIS OF GCND ture of the cha~iiielchanges direction- prevents a meander from being simply a series of connected semicircles. Sine-generat cd Curves WIDTH OF CHANNEL (W) = 1 We first recognized the principal WAVELENGTH (A) - 11 5 characteristics of the actual curve traccd LENGTH OF CHANNEL (L) - 165 out by a typical river meander in the RADIUS OF CURVATURE (rc) :23 course of a niatl~e~r~aticalanalysis aimed at generating meander-like curves by means of “random walk” techniques. A random walk is a path described by successive moves oil a surface (for ex- aiiiple a shect of graph paper); each move is generally a fixed unit of dis- tance, but the direction of any move is determined by some random process (for example the turn of a card, the throw of a die or the sequence of a table of random numbers). Depending e on the purpose of the experiment, there is usually at least one constraint placed on the direction of the move. In our random-walk study one of the con- straints we adopted W:IS that the path was to begin at some point A and end at some other point B in a given num- -W ber of steps. In other words, the end points and the length of the path were fixed but the path itself was “free.” The mathematics involved in finding the average, or most probable, path taken by a random walk of fixed length had been worked out in 1951 by Her- nraiin von Schelling of the General Electric Company. The exact solution is expressed by an elliptic integral, but I WIDTH OF CHANNEL (W) = 1 in our case a sufficiently accurate ap- WAVCLENGTH (A) ~ 6‘3 proximation states that the rnost prob- LFNGlII OF CHANNEL (L) 248 able geometry for a river is one in wliicli RADIUS OF CURVAlURE (r,) = 23 the angular direction of the channel at any point with respect to the menn down-valley direction is a sine func- PROPERTIES used to describe river meanders are indicated for two typical meander tion of the distance menstired along curves. A series of nieanders has a regular appearance on a map whenever there tends to be the channel [sce illustration on opposite a constant ratio between the wavelength (A) of the curve and its radius of curvature (rc). The value of this ratio for the meander that looks rather like a sine wave (top) is five to pagel. one; the more tightly looped meander (bottom) has a corresponding value of three to one. The curve that is traced out by this An average value for this ratio is about 4.7 to one. Sinuosity, or tightness of bend, is ex- most probable random walk between pressed as the ratio of the length of the channel (L) in a given rurve to the wavelength of two points in ;I river valley wc named curve. The value of this ratio for the top curve is 1.4 to one and for the bottom curve 3.6 a “sine-gcnernted” curve. As it happens, to one. On the average the value of this ratio ranges between 1.3 to one and four to one, this curve closely approximates the 62 .- shape of real river ineandcrs [ ,sw illrrs- a bundle to the car bcds. The train, possible. This example is particularly tration on ncxt pa'gc].At the axis of bend pulled by five locomotives, collided with appropriate to our discussion of river the channel is directed in the mean a bulldozer and was derailed. The vio- meanders because, like river meanders, down-valley direction and the angle of lent compressive strain folded the train- the bent rails deviate in a random way deflection is zero, whereas at the point load of rails into a drastically foreshort- from the perfect symmetry of a sine- of inflection the angle of deflection ened snakelike configuration.