Evolution of Meander Loops

JAMES C. BRICE Department of Earth Sciences, Washington University, St. Louis, Missouri 63130

ABSTRACT pound meander loops described in this report. However, attention here is confined to the form and evolution of these and other mean- The evolution of the meanders on reaches of 10 alluvial streams in der types that occur on natural streams, a matter that needs to be the United States is reconstructed, and a scheme for the evolution clarified before further speculations are made on the mechanics of and classification of meander loops, derived from a study of the meandering. meandering pattern of 125 alluvial streams, is proposed. In the main Meander-loop evolution, as described and illustrated here, has evolutionary trend, a low symmetrical arc of approximately con- been interpreted mainly from the sequential development of mean- stant curvature tends to increase in height but decrease in radius as it der scrolls as shown on aerial photographs and from the shift of grows. When its length exceeds its radius, the arc is termed a simple stream position as determined from sequential aerial photographs. symmetrical meander loop. A simple loop becomes asymmetrical by Most old maps have sufficiently great planimetric errors so that they the growth on its perimeter of a second arc of constant curvature, are not useful for this purpose. Aerial photographs and recent U.S. which is commonly tangent to the first and curved toward the same Geological Survey topographic maps have been acquired for about side of the stream. A simple loop becomes compound when a second 125 reaches of meandering streams in the United States, and loop arc on its perimeter has developed into a loop. Four main categories forms on all of these reaches have been studied. However, few river of loops (simple symmetrical, simple asymmetrical, compound flood plains show meander scrolls with sufficient clarity for interpre- symmetrical, and compound asymmetrical) and about 16 form types are proposed. The compound loops are regarded as aberrant forms of indefinite radius and length, but the meandering patterns can be analyzed into simple loops whose properties can be measured and treated statistically.

INTRODUCTION It is generally known that the evolution of meander loops on natural streams is likely to include downstream migration, increase in amplitude, and eventual cutoff at the neck; but specific informa- tion as to how the different loop forms evolve seems to be lacking in the literature. This paper presents a scheme for the evolution and classification of meander loops and demonstrates its application to some meandering streams in different geomorphic regions of the United States. The terms "meander" and "meander loop" are some- times used interchangeably, but according to the usage of Leopold and others (1964, p. 295), which is followed here, a meander con- sists of a pair of opposing loops. On natural streams, however, a meander loop is not usually paired with another loop of the same size and form. The hypothetical development of meanders is illustrated and described in a general way by Davis (1902) and by Lobeck (1939, p. 226). The relevant discussion in Leopold and others (1964) is di- rected mainly toward the mechanics of meandering rather than the evolutionary development of meander forms. Langbein and Leopold (1966) proposed that meanders tend toward the development of a stable form (a "sine-generated" curve) tha t minimizes the sum of the squares of the changes in direction for each successive unit length, but the stages by which such a form may develop are not discussed. An account of migration at certain bends of the lower Mississippi River is given by Carey (1969). Daniel (1971) analyzed the growth of several meanders on streams in Indiana, including the White River, from which he concluded that the process of channel movement in a meander system involves rotation and translation of meander loops and an increasing path ler.gth. Handy (1972) plotted the growth of a meander in the Des Moines River for a period of about 90 yr and showed that the rate of growth slowed gradually as the channel approached the edge of the meander belt. Lewin (1972) has described the late-stage growth of meanders on some gravel-bed rivers in Wales. As meander length increases, the meander form increases in complexity which is attributed to the effects of evenly spaced riffles on the stream bed. The late-stage Figure 1. Scheme for evolution and classification of meander loops. Flow direction is meanders illustrated by Lewin are apparently the same as the com- left to right for these and all subsequent illustrations.

Geological Society of America Bulletin, v. 85, p. 581-586, 6 figs., April 1974

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tation of meander evolution, and meanders on Few rivers grow at a fast enough rate for evolutionary trends to be discerned in the maximum time span (about 30 yr) between photographs. The White River and the East Fork White River in Indiana have both distinct meander scrolls and a rapid rate of meander growth, and several examples are therefore drawn from these rivers. Neither meander loops that are incised in bedrock nor alluvial loops th at are \ * J Y //!/ in contact with valley sides have been used as examples; nor have I used meanders that might have evolved with the "underfit" condi- / Xl/ 0 5 KM tions postulated by Dury (1964). With the exception of the Maamee River meanders, all meanders used as examples have probably evolved within the past 500 yr, as indicated either by historical records or by extrapolation of measured migration rates. " Jillw Meander loops and meandering reaches are represented on the illustrations herein by a single line that represents the stream center- m '••Nfcfji ^^ line. Where a stream centerline is traced from maps or aerial photographs, it applies to the stream at a stage near the median point on Mx.7^ V/ i^^tAo M the flow duration curve, which is an approximation to the modal stage and to the "normal" stage used by the Topographic Division of the U.S. Geological Survey for representation of streams on maps. The centerline at beads is drawn halfway between the shoreline of the point bar and the cut bank on the outside of bends. Reconstruction of former stream centerlines from meander scrolls has been based on the assumptions that a stream centerline is parallel I . • • • f to a contemporaneous meander scroll and that contemporaneous 0 500 M scrolls of adjoining loops can be correlated on the basis of trend and position. All map references, given in the captions of illustra:ions and elsewhere, apply to U.S. Geological Survey topographic maps. Figure 3. Evolution of simple asymmetrical meander loops. A, Mississippi River On all illustrations, the flow direction is from left to right and the betveen New Madrid, Missjuri, and Dyersburg, Tennessee (Dyersburg 1:250,000); metric scale is used. cent :rlines prior to 1944 intei preted from chronological sequence of alluvial deposits as Circular arcs are inscribed on the loops of Figure 1 for compara- illus rated by Fisk (1944, PI. j 2, Sheet 3). B, White River near Petersburg, Indiana (Iona 7%'I; centerline of 1937 (heiviest line) from aerial photograph and prior centerlines tive purposes to indicate the general geometry and symmetry ol the intei preted from meander sciolls. C, Elkhorn Fiver, Scribner, Nebraska (Scribner 7Vi' idealized forms. For the evolutionary scheme, an approximate ft of and Uehling 7 V;':. Redrawn from Bentall and others (1971, p. 20). circular arcs to segments of the meandering pattern is suffic ent. Such approximation can be quickly ascertained by fitting one of a successive small segments of the stream centerline against distance as series of concentric circles, inscribed on transparent film, to the measured along the stream, according to the useful method devised pattern. Curvature can also be studied by plotting the angular devia- by Langbein and Leopold (1966). Segments of the resulting plot, tion from mean downstream direction (or simply the azimuth) of which have a uniform slope, represent arcs of uniform curvature. Langbein and Leopold |p. H4) reported that the radius of curvature is nearly constant for £ full third of the length of a meander loop. Plots made in the cou-se of this study indicate that segments of constant curvature are much more prominent in the geometry of most meanders, which apparently have not attained the ultimate equilibrium form propesed by Langbein and Leopold. A quantitative analysis of the whole meandering pattern on natural streams must be preceded by a careful consideration of what forms are to be measure d. This paper is intended to provide a basis for such analysis by def ning and classifying meander loops and by demonstrating the genstic relations between loops of different conf guration. The fit of circular arcs to a meandering reach and the technique proposed for quantitative analysis of the pattern are demonstrated by an example at the end of this paper.

SCHEME FOR EVOLUTION AND CLASSIFICATION The scheme of Figure 1 applies to about 125 reaches of meandering rivers in the United States, from which examples corresponding to the different loop forns will be cited. The sequence of forms A through C in Figure 1 tepresents an evolutionary trend in which meander ioops originate as stream segments of approximately constant curvature, or low circular arcs, that increase in height with time. Length of an arc i:; measured along a chord drawn between point; of tangency with adjoining arcs. Height (or amplitude) is measured along a perpendicular from the apex of the arc to the Figure 2. Evolution of simple symmetrical meander loops. A, Chena River, Alaska midpoint of the chord. A simple meander loop is arbitrarily defined (Fairbanks D-l); centerline of 1969 (solid line) traced from aerial photograph; pr or as an arc whose length exceeds its radius and is not greater than centerlines (dashed and dotted lines) interpreted from meander scrolls. B, East Fork about seven times its hei,*ht (A of Fig. 1). Also, the curvature of a White River, Seymour, Indiana (Seymour 7W); centerlines from aerial photographs. C, Leaf River, Hattiesburg, Mississippi (Hattiesburg 7'h' and Carterville 7Vi')> traced simple loop has a central angle that exceeds 60° or about 1 radian C from aerial photographs. (57.3 ). As the evolution proceeds (B and C of Fig. 1), simple loops

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tend to increase in height and decrease in radius. The limbs of Carolina (Leesburg 7Vi'); Kissimmee River, Florida (Okeechobee opposing loops tend to depart from tangency before either loop NW 7V2'); and Rogue River, Michigan (Sparta 7V2'). attains a central angle of 180° (a half circle of an arc) and become Form G of Figure 1 looks rather contrived, mainly because natural connected by nearly straight segments. If the radius of a simple loop loops having this arrangement of arcs are usually asymmetrical. Its does not decrease with increasing height, and adjacent loops are form is attributed to the development of an arc on each limb of an closely spaced, a U-shaped loop such as form D of Figure 1 develops. extended simple loop. Examples that correspond reasonably well For loops connected to adjacent loops by nearly straight segments, are on the Goodpaster River in Alaska (Big Delta A-3) and the Red as in C through G of Figure 1, the chord connects the midpoints of River in Minnesota (Halstad 7V2'). these segments. As before, length is measured along the chord, and Asymmetry of simple meander loops, which is very common, is height is measured along a perpendicular from the apex of the loop usually caused by the development of a second arc that is tangent to to the midpoint of the chord. For each of the simple symmetrical the first and curved toward the same side of the stream, as illustrated loops in Figure 1, a single loop includes only the upper half, approx- in K of Figure 1. Such an asymmetrical loop is not classified as imately, of the illustration; its height is about one-half the height of compound unless both arcs meet the arbitrary criterion for a loop; the illustration. that is, the chord is longer than the radius. On elongated loops, two Examples of simple symmetrical loops corresponding to A secondary arcs may develop symmetrically (G of Fig. 1), but again through D of Figure 1 are on the following river reaches, as the loop is not classified as compound unless it includes two simple represented on the indicated U.S. Geological Survey quadrangle loops. The radius of an asymmetrical loop cannot be expressed maps: Big River, Iowa (Akron 7V2'); Iowa River, Iowa (Hills 7xh')\ accurately by a single value, but it is approximately an average of the Neuse River, North Carolina (Deep Run 15'); Savannah River, radii of its components. Asymmetry can also arise by the encroach- Georgia (Mechanic Hill IVi')-, and Sabine River, Texas (Merryville ment of an adjoining loop (L of Fig. 1), which occurs more fre- 15'). quently on the upstream side. The asymmetry of simple meander Elongation of simple loops, as measured by decrease in loops is not consistent relative to flow direction, although the length/height ratio, does not continue indefinitely. The maximum smaller arc occurs more frequently on the upstream side. The flow value of length/height ratio for a loop is about 7, and the average direction of a stream cannot be reliably inferred from the value is about 2.2. Loops that have ratios in the range of 1 to 0.7 look geometrical form of meander loops. Among the many river reaches distinctly elongated. The lowest ratios measured were in the range of on which simple asymmetrical loops occur are the following: Au 0.6 to 0.5, and these occurred on rivers incised into alluvium, of Sable River, Michigan (Red Oak 7V2'); Lumber River, North which the Milk River in Montana (Glasgow IVz'), the Red River in Carolina (Fair Bluff 7V2'); Red River, Louisiana (Campti 15'); and Minnesota (Bygland IVi'), and the Smoky Hill River in Kansas Suwannee River, Georgia (Fargo SW 7Vi'). (Marquette 7Vi') are examples. Somewhat less elongated simple Compound loops evolve by the gradual development of a second loops are usually on densely vegetated flood plains. Both incision arc on the perimeter of a simple loop. This evolution can be demon- and vegetation are factors that inhibit cutoffs and thus permit the strated with sequential aerial photography and is also indicated by elongation of loops. Examples of elongated simple loops forms that are gradational between simple and compound. A com- corresponding with E and F of Figure 1 are: on the following river pound loop includes two or more simple loops, which may either reaches: Pearl River, Louisiana (Sun 15'); Wateree River, South be tangent or connected by nearly straight lines. Successive simple loops are commonly curved toward the same side of the stream, an important aspect of meander geometry that seems to have been previously noticed only by Winkley (1972, p. 68). The terms "simple" and "compound" have been applied to meanders by Hack (1965, p. B31), but, according to his usage, compound meanders have a harmonic quality, consisting of smaller meanders superimposed on meanders of a distinctly longer wavelength. The compound meanders described here have no harmonic quality and are not necessarily larger than the simple meanders with which they occur. Designation as first-order compound meanders should serve to distinguish them from the compound meanders defined by Hack, whose meanders can then be described as second order. Schumm (1963, p. 1091) has also noted the occurrence of a meander pattern of low amplitude and wavelength, superimposed on a larger pattern. Compound meanders have been investigated by Foweraker (1963), who made special reference to their occurrence as incised forms on rivers in the Ozark region. Most compound loops are asymmetrical and include no more than two simple loops, both of which curve toward the same side of the stream, as in M, N, and O in Figure 1. Examples of such loops occur on the following river reaches: Red River, Arkansas (Barkman 7V2'); Rum River, Minnesota (Wyanette 7Vi')\ Rough River, Ken- tucky (Dundee 7xh')\ Blacks Fork, Wyoming (Bryan 7xh' and Blue Point 7V2'); and Animas River, Colorado (Durango East 7Vi'). Compound loops that include several simple loops (P of Fig. 1) are commonly (but not exclusively) on river reaches that are either incised or in a densely vegetated flood plain. In Alaska, examples occur both in tundra (Kanuti River, Bettles 1° x 3°) and in forest cover (Chena and Little Chena Rivers, Fairbanks D-l). Other exam- Figure 4. Evolution of compound meander loops. A, Hess Creek, Alaska (Livengood ples, apparently associated with incision, are the Solomon River, C-5); solid line, traced from aerial photograph, is cent:rline prior to cutoff of loop, Kansas (Niles 7V2'); Saline River, Kansas (Trenton 7V2'); Milk which took place about 1967; prior centerlines interpreted from meander scrolls. B, River, Montana (Glasgow 7xh')\ and Souris River, North Dakota Maumee River upstream from Antwerp, Ohio (Woodburn North 7W); centerline of 1969 (solid line) from aerial photograph, prior centerliiies interpreted from meander (Karlsruhe 7V2'). There may be some question as to whether a form scrolls. C, East Fork White River (Seymour 7%'), from aerial photographs. such as P on Figure 1 is in fact a loop or not. Genetically, all

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Figure 5. Sequential aerial photographs of White River at Edwardsport, Indiana (Plainville 7lh'). Top photogra )h, dated October 7,1966, has scale 6 percent smaller than bottom photograph, dated September 20, 1937. Lettered circles indicate corresponding reference points. Both photographs from U.S. Department of Agriculture. first-order compound loops are considered to have evolved from limb of an extended simple loop. Reaches having at least one loop simple loops, and their identification depends either on demonstra- that corresponds reasonably well to J of Figure 1 can be cited: White tion of this evolution from available evidence or on the probability River, Arkansas (De Vails Bluff 15'); Little Wind River, Wyoming that such evolution could have occurred. Thus, the distance across (Arapahoe lxh'Y, and Licking River, Kentucky (Farmers IVi'). the neck of a first-order compound loop cannot be greater than the diameter of the largest simple loop in the population. For example, EXAMPLES OF E\OLUTION the reaches shown in B and C of Figure 2 are each curved in a broad Simple Symmetrical Loops arc, but the distance across the "necks" of these arcs is too great for The evolution of simple symmetrical loops is illustrated in Figure them to have developed from simple loops. The large arc in C of 2 by examples frorr three rivers of similar size but located in dis- Figure 2 may represent a half wavelength of the type of compound tinctly different geomorphic regions. In the example from the Chena meander described by Hack (1965), but full or repeated wavelengths River (A of Fig. 2), tie distance of downstream translation is greater of such meanders were rarely observed on the streams used for this between the earlier stages of evolution than between the last two study. stages, and the radius of loops has decreased with increasing Of the compound symmetrical loops, H and I of Figure 1 represent amplitude. The center loop in the sequence has not migrated forms that are fairly common. Examples are on the Laramie River, perpendicular to flew direction and therefore owes its increase in 1 Wyoming (Bosler 7V2'); Rock River, Iowa (Rock Valley 7 /?.'); Rio amplitude to the latsral migration of adjoining loops. The loops in Grande River, Colorado (Mount Pleasant School 7Vi'); Congaree the 1969 centerline (solid line) have the configuration of circular River, South Carolina (Saylors Lake 7V2'); and Red River, arcs, which are tangent for the two loops at left (compare with B of Minnesota (Perley IV2'). Form J of Figure 1 is somewhat Fig. 1) but connect« d by short straight reaches for the other loops hypothetical, but it illustrates a fairly common tendency and is (compare with C of Fig. 1). One loop (second from left) is slightly attributed to the development of a pair of symmetrical arcs on each asymmetrical.

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0 0.4 0.8 L2 L6 2 i:.4 ZB 12 16 4 4.4 43 5L2 5J6 6 6.4 68 7.2 7.6 DISTANCE ALONG RIVER, IN KM Figure 6. A, Fit of circular arcs to 1968 centerline (dotted) of White River, as traced from the aerial photograph in Figure 5, at top; dashed lines are chords, which represent the length of simple loops. B, Plot of river distance, as measured along centerline, versus azimuth of centerline, as measured clockwise from true north.

In the example from the East Fork "White River (B of Fig. 2), the Also shown is a tendency for radius to decrease with increasing loop at extreme left migrated laterally about 140 m between 1937 amplitude and for asymmetry to increase. The two loops at left are and 1968, with decrease in radius but with little or no downstream simple asymmetrical, analogous in form to K of Figure 1. The other translation. Most of the loops downstream have undergone rapid three loops are classified as simple symmetrical, although a secon- downstream translation, accompanied by i n increase in amplitude dary arc of large radius has developed on the downstream limb of the and a decrease in radius. Two of the loops at right had no distinct loop at center. By 1956, chutes had developed across the point bars precursors in 1937 and evidently grew rapidly in response to of the loop at center and its opposing loop downstream. meander development immediately upstream. The White River meander loops in B of Figure 3 evolved from a The Leaf River (C of Fig. 2) provides an example, at left, of the nearly straight reach that resulted from the cutoff of a section of the evolution of symmetrical loops after a cutoff. These seem to have river course. During earlier stages, a tendency for radius to decrease evolved rapidly and probably did not pass through the earliest stages with increasing amplitude can be seen, but the two largest loops have in the scheme of Figure 1. The train of symmetrical meanders at right expanded during their later stages of development. The loops of B of of center has undergone fairly uniform downstream translation, Figure 3 are similar in form to K, L, and M of Figure 1 and are with a tendency toward increase in amplitude and decrease in radius. transitional between simple asymmetrical and compound asymmet- The loop at extreme right, although slightly asymmetrical, is an rical loops. example of the form illustrated in G of Figure 1. The frequent cutoffs and somewhat irregular pattern of the Elk- horn (C of Fig. 3) are attributed to the fact that its banks are sandy Simple Asymmetrical Loops and not cohesive. Most Elkhorn loops originate at cutoffs, and their The Mississippi River reach in A of Figure 3 illustrates the rela- evolution is not very systematic. The upward extending asymmetri- tively rapid downstream translation of low-amplitude loops (two cal loop of the 1965 centerline, at left, has decreased in radius and loops at left) and the less rapid translation of higher amplitude loops. has become more asymmetrical between 1941 and 1965, whereas its

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opposing loop downstream (somewhat flattened at the apex by a measured. The radii havs a symmetrical log-normal size distribution railroad embankment) has greatly expanded during this period. The with a range from 100 tD 700 m and a median of 220 m. Numerals loop forms on all the centerlines correspond satisfactorily with :he along the centerline of A of Figure 6 correspond to distances along scheme of Figure 1, if some allowance is made for irregularities. the river as shown in B of Figure 6. The straight dashed lines, fitted by eye, represent segments of constant curvature. Compound Loops Compound loops are characteristic of many meandering Alas .-can CONCLUSIONS rivers. Particularly good examples occur along Hess Creek, a Most of the meander loops on a group of 125 alluvial stream tributary of the Yukon. No chronology is available for the Hess reaches correspond to one or another of about 16 form types, and Creek example illustrated in A of Figure 4, but lateral migration some general trends in evolution have been distinguished. In the rates measured for the period 1951 to 1969 (obtained by main evolutionary trend, a simple symmetrical loop increases in comparison of aerial photographs) indicate that the time required height (amplitude) by lateral migration of an arc of approximately for evolution of the loop was 200 to 300 yr. As interpreted from constant curvature at its apex. It may then become asymmetrical by meander scrolls, evolution began with a flat loop of large radius, the development on its perimeter of a second arc that is tangent to which first decreased in radius with increasing amplitude and then the first and curved toward the same side of the stream. On some expanded as secondary arcs developed on its perimeter. The first streams, loops tend nol to evolve past this stage; on others, simple secondary arcs were on the same side of the river (compare with loops evolve into compound loops, v/hich expand in an apparently form O in Fig. 1), but a pair of alternating arcs subsequently random way by the growth of the simple loops on their perimeter. developed on the upstream perimeter of the loop. Cutoff of loops, whether simple or compound, results mainly from The Maumee River (B of Fig. 4) is incised into a glacial drift plain diverse loop orientations and directions of growth. (the former bed of glacial Lake Maumee) to a depth of about 40 ft, The simple loops have a single point bar and migrate as functional and its present rate of lateral migration is such that no changes could units, but the compound loops evolve only by migration of the be observed by comparison of aerial photographs taken 30 yr apart. individual simple loop; on their perimeter. Moreover, the radius, Nevertheless, the evolution of the meander loops is rather clearly length, and height of most compound loops are indefinite. The indicated by the pattern of meander scrolls on a narrow flood p.ain compound loops are regarded as aberrant forms whose contribution (or low terrace) that stands about 15 ft above the river. Of the five to sinuosity is assumed, after cutoff, by new simple loops. The full loops illustrated in B of Figure 4, the two at right and the one at meandering patterns a n be analyzed into simple loops, whose size- left are regarded as simple asymmetrical; the remaining two (the pair frequency distribution can be determined. The relations between just to the left of center) are regarded as compound symmetrical, meander size and properties such as river width and discharge can although they are not perfectly symmetrical. The development of then be evaluated more accurately, and some further insight into the two secondary arcs on the periphery of each of the compound loops mechanics of meandering may be gained. is apparent from the evolutionary sequence of meander scrolls. In C of Figure 4, the centerline of 1937 (dotted) traces at lower ACKNOWLEDGMENTS center a good example of a compound loop that includes several This work has beer supported by a grant from the U.S. Army simple loops. One compound loop in the 1969 centerline (second Research Office — Durham. from right) has developed since 1937 from a simple symmetrical loop. REFERENCES CITE ) The transition from simple to compound and the growth of simple Bentall, R., and others, 1971, The Elkhorn River basin of Nebraska: Re- loops as components of compound loops are illustrated by sequen- source Atlas no. 1, Nebraska Univ. Conservation and Survey Div., 50 p. tial aerial photographs in Figure 5. The downward extending loop Carey, W. C., 1969, Forriation of flood plain lands: Am. Soc. Civil Engineers above and just to the right of c was simple asymmetrical in 193 7 but Proc., Jour. Hydrav lies Div., v. 95, paper 6574, no. HY 3, p. 981-994. had become compound by 1966. The upward extending loop above Daniel, J. F., 1971, Char nel movement of meandering Indiana streams: U.S. c was simple prior to 1937 (compare with E of Fig. 1), as indicated by Geol. Survey Prof. Paper 732-A, A1-A18. its meander scrolls, but had become distinctly compound by 1966. Davis, W. M., 1902, Ri/er terraces in New England: Harvard Univ. Mus. The compound loop at d can be compared with N and O of Figure 1, Comp. Zoology Bull., v. 38, Geol. Ser. V, p. 281-346. and the downward extending compound loop that adjoins it is Dury. G. H., 1964, Prir ciples of underfit streams: U.S. Geol. Survey Prof. Paper 452-A, p. A1-A67. similar to P of Figure 1. Note that each simple loop in the 1966 Fisk, H. N., 1944, Geological investigation of the alluvial valley of the lower photograph has a distinct point bar, exposed because of the low Mississippi River: Vicksburg, Miss., U.S. Army Corps of Engineers, flow (470 cfs as compared with an average discharge of 4,521 cfs). Mississippi River Comm., 78 p. The relation of circular arcs to the White River reach of Figure 5 Foweraker, J. C., 1963. Quantitative studies in river sinuosity with special (at top) is demonstrated by two different techniques in Figure 6, and reference to incised meanders of Ozark rivers [Ph.D. thesis]: St. Louis, the method used to delineate simple meander loops for measurement Mo., Washington Jniv., 124 p. is illustrated. The geometry of most meandering reaches is less Hack, J. T., 1965, Postglacial drainage evolution and stream geometry in the complicated than this, but the fit of the circular arcs here is some- Ontonagon area, Michigan: U.S. Geol. Survey Prof. Paper 484, 84 p. Handy, R. L., 1972, Alii vial cutoff dating from subsequent growth of a mean- what better than average. The circular arcs in A of Figure 6 were der: Geol. Soc. An erica Bull., v. 83, p. 475^80. fitted by superimposing a series of closely spaced concentric circles Langbein, W. B., and Leopold, L. B., 1966, River meanders — Theory of drawn on frosted acetate film. Radius of meander loops is also minimum variana : U.S. Geol. Survey Prof. Paper 422-H, p. H1-H15. measured in this way. Simple loops are marked off by chords Leopold, L. B., Wolman, M. G., and Miller, J. P., 1964, Fluvial processes in (dashed in A of Fig. 6), which are positioned in continuity from loop geomorphology: San Fransisco, W. H. Freeman Co., 522 p. to loop except where this would cause a loop to have limbs of Lew in, J., 1972, Late-stage meander g;rowth: Nature Phys. Sci., v. 240, p. unequal length about its axis. Loop length, which is measured along 116. the chord, is equivalent to half the wavelength of a meander as Lobeck, A. K., 1939, Geomorphology: New York, McGraw-Hill Book Co., defined by Leopold and others (1964, p. 295). Radius is regarded as 731 p. a more primary form property than length, because loops of the Schurr.m, S. A., 1963, Sinuosity of alluvial rivers on the Great Plains: Geol. Soc. America Bui ., v. 74, p. 1089-1100. same radius can have different lengths, depending on their stage of Winkley, B. R., 1972,b Shen, H. W., ed., River mechanics, v. I, chapt. 19: Ft. development. As a measure of loop size, radius is also preferred Collins, Colo., Water Resources Pub., 79 p. because the selection of chords for measurement is more subjective than the selection of arcs. The reach in Figure 6 is part of a longer MANUSCRIPT RECEIVE:} BY THE SOCIETY JUNE 4, 1973 reach for which the radii of all simple loops (a total of 115) were REVISED MANUSCRIPT RECEIVED OCTOBER 15, 1973 Printed in U.&A.

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