Flow directionality of pristine meandering rivers isembedded in the skewing of high-amplitudebends and neck cutoffsXingyan Guoa,b, Dong Chena,b,1, and Gary Parkerc,d,1

aKey Laboratory of Water Cycle and Related Land Surface Processes, Institute of Geographic Sciences and Natural Resources Research, Chinese Academy ofSciences, 100101 Beijing, China; bCollege of Resources and Environment, University of Chinese Academy of Sciences, 100049 Beijing, China; cDepartment ofCivil & Environmental Engineering, University of Illinois, Urbana, IL 61801; and dDepartment of Geology, University of Illinois, Urbana, IL 61801

Contributed by Gary Parker, September 9, 2019 (sent for review June 25, 2019; reviewed by Alessandra Crosato and Bart Vermeulen)

Information concerning the dynamics of river meandering isembedded in their planforms. Here, we focus on how bendskewing varies with increasing sinuosity, and how flow di-rection is embedded in bend skewing. It has often been thoughtthat upstream-skewed bends are dominant within a suffi-ciently long reach. These bends may allow a reasonable infer-ence as to the direction of flow. Here we consider this issueusing 20 reaches of freely meandering alluvial rivers that are inremote locations, generally far from human influence. We findthat low-amplitude bends tend to be downstream-, rather thanupstream-skewed. Bends with sinuosity greater than 2.6, how-ever, are predominantly upstream-skewed. Of particular inter-est are the neck cutoffs, all chosen to be relatively recentaccording to their position related to the main channel: 84%of these are upstream-skewed. Neck cutoffs, which have likelyevolved directly from bends of the highest sinuosity, representthe planform feature most likely to have flow direction embed-ded in them. The field data suggest that meander bends withoutexternal forcing such as engineering works tend to evolve fromdownstream-skewed low-sinuosity bends to upstream-skewedhigh-sinuosity bends before cutoff. This process can be repro-duced, to some extent, using models coupling sedimentarydynamics with flow dynamics.

alluvial rivers | meandering | flow direction | planform skewing | bends

Meandering rivers are common, esthetically pleasing, andof engineering importance. The planform of this ubiqui-

tous river pattern possesses a high degree of regularity (1–5).Meandering channels are seen in a wide variety of settings, in-cluding alluvial rivers (4–8), mixed bedrock–alluvial rivers (9,10), tidal channels (11, 12), and flows over ice (13, 14). Here wefocus on alluvial meandering. Such streams have planforms witha composite of organized and irregular variation representingboth autogenic and allogenic effects (1, 15–17). We ascertain towhat degree flow directionality is embedded in the planforms ofmeandering rivers and their neck cutoffs.The streamwise-symmetrical sine-generated curve has been

proposed as an idealized description of river meander formbased on the assumption of minimum variance (18). This re-lation is

θ = θosinðksÞ, [1]

where θ = bend angle between the longitudinal (streamwise)and downvalley direction, θo = angle amplitude, s = centerlinestreamwise axis, and k = 2π/λ, where λ = arc length wavelengthand k = arc length wavenumber. This planform shape is shownin Fig. 1A.Although widely used in physical and numerical models,

streamwise-symmetrical curves are a simplification of bend shape,and therefore may not be suitable for highly sinuous channels oroxbow lakes, which often show planform asymmetry (6, 19–21).

Adding third-order terms allows for planforms with skewing andfattening (22):

θ = θ0 sinðksÞ + θ30�Jscosð3ksÞ − Jf sinð3ksÞ

�. [2]

Here Jf = coefficient of fattening and Js = coefficient of skewing.Fig. 1B shows a bend with a positive coefficient of skewing. Thebend doubles back upon itself and points upstream (upstreamskewing). Fig. 1C shows a bend that points downstream (down-stream skewing). The recognition of skewing is not specificallydependent on [2]; it can be inferred from river planform. Thetendency for upstream skewing was recognized as early as 1955(23). Fig. 1D shows a Fisk (24) map of the Mississippi River; itshows both upstream and downstream skewing. We assume thatwhile skewing may be generated by floodplain inhomogeneities (16,17, 25), it may also be an autogenic tendency inherent to the mor-phodynamics of meandering alluvial rivers. Indeed, it was suggestedbased on empirical information that upstream skewing might pre-dominate for meandering bends in the Ishikari River, Japan (26).Our focus is on the tendency of bends to be skewed. It has

been suggested that upstream skewing is the most natural stateof alluvial meandering rivers (22, 27). The patterns predicted inrefs. 22 and 27 may, however, be overly biased toward upstreamskewing (28). One explanation for this limitation is that the

Significance

The bends of alluvial meandering rivers often double back onthemselves, showing skewing. This skewing may be directedupstream or downstream. How skewing evolves as bends de-velop remains incompletely understood. Our analysis showsthat, on 20 reaches of nearly pristine alluvial meandering riv-ers, downstream skewing dominates when the bends are rel-atively straight, but upstream skewing increasingly dominatesas bend sinuosity increases. This provides a guide for inter-preting bend evolution and offers a useful comparison withnumerical models. The results suggest that rivers often carry animprint of the direction of the flow that created them, throughthe shape of their high-amplitude bends and neck cutoffs. Thisprovides a reference tool for estimating paleoflow direction onEarth and other planets.

Author contributions: X.G., D.C., and G.P. designed research; X.G. performed research;X.G. and G.P. analyzed data; and X.G. and G.P. wrote the paper.

Reviewers: A.C., UNESCO-IHE Institute for Water Education; and B.V., Universityof Twente.

The authors declare no competing interest.

Published under the PNAS license.1To whom correspondence may be addressed. Email: dchen@igsnrr.ac.cn or parkerg@illinois.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1910874116/-/DCSupplemental.

First published November 4, 2019.

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model in question leaves out some subtle points pertaining tomorphodynamics in meander loops. The authors of ref. 29 con-sidered vegetation–river interaction in their simulations, andfound that some transverse vegetation distributions along a rivermay reduce upstream skewing and induce downstream skewing.A more advanced model (see ref. 30 for comprehensive refer-ences) suggests that downstream skewing can occur if the channelis sufficiently wide. The difference in skewing regime is charac-terized in terms of subresonant (resulting in upstream skewing) orsuperresonant (resulting in downstream skewing) behavior. Thisresonance refers to the interaction of processes that form alternatebars and processes that form bend point bars.Most models of meander migration relate the bank erosion

rate to the near-bank streamwise flow velocity (e.g., refs. 22, 27,28, and 30). A bank erosion model has also been deduced byanalyzing the conservation of near-bank sediment mass (31). Inthis model, the bank erosion rate is assessed in terms of the gra-dient of the longitudinal sediment transport rate and the strengthof the secondary current. The authors used this model to simulatethe evolution of meandering from mildly sinuous to highly sinuous(32). In this simulation, single loops first rotate downstream andthen gradually rotate back to an upstream-skewed configuration.A recent numerical model of meander evolution (33) includes

a more detailed description of erosional bank processes (natu-ral armoring), depositional bank processes (vegetal encroachment),and self-formed width. Under unsteady flow with two cyclicallyalternating discharges, this model captures rich and complexmeandering planforms in a train of several bends, including per-sistent upstream skewing, persistent downstream skewing, transi-tion from downstream skewing to upstream skewing, and vice versa.In addition to modeling results, field observations also show a

range of behavior. An inspection of 277 bends of 15 rivers con-firmed the statistical preference for upstream skewing (19).

These ideas and methods have been used to interpret paleoflowdirection of channels on Mars (34). In ref. 20, 57 natural streamsegments were observed: these meanders tended to be skewedslightly upstream. The authors of ref. 35 examined continuousloops in 4 natural rivers (1 in a temperate region and 3 in tropicalregions) and pointed out that most bends in their 3 tropicalreaches did not show obvious skewing. In summary, the occur-rence of, and the mechanisms behind the skewing phenomenonare not yet thoroughly documented and understood.Here we study bend skewing by analyzing the planforms of 20

long reaches of freely migrating alluvial rivers with wide flood-plains in remote locations that are in as pristine a condition aspossible. This allows us to observe the planforms of self-generatedby rivers with negligible anthropogenic effects. Besides naturalreaches, several representative planforms from the numericalsimulations of refs. 28 and 36, using a version of the theory sum-marized in ref. 30, are also analyzed.

Results and DiscussionWe set strict criteria for data collection, as stated inMaterials andMethods, to ensure that our study reaches are nearly pristine,with wide, vegetated floodplains and abundant evidence of chan-nel migration. Of our 20 sample reaches, 8 are in South America, 7are in North America (mostly northern Canada), 2 are in Africa, 2are in Siberia, and 1 is in Papua New Guinea. The number ofbends per reach ranges from 56 to 311, with a mean of 149. Theaverage width ranges from 40 m to 9 times that. The reaches aresummarized in Fig. 2 A and B. The width, length, and countryinformation are listed in SI Appendix, Table S1. A subreach of anexample reach, the Chulym River, a tributary of the River Ob inRussia, is shown in Fig. 2C. It flows west by south on a pristinefloodplain with numerous scrolls and cutoffs.We detected and connected successive inflection points to di-

vide the reaches into bends. This segmentation is described inMaterials and Methods. Due to the complexity of the planforms,segmentation into individual bends was not always straightforward.As explained below, 311 bends from our initial set of 2,976 bendswere discarded, leaving 2,645 bends and 411 recent neck cutoffsfor analysis. (We do not consider chute cutoffs here.) Detailsconcerning the reaches can be found in SI Appendix, Table S2.The next step is evaluating bend-skewing angle, as described in

Materials and Methods. A positive skewing angle denotes up-stream skewing and a negative skewing angle denotes down-stream skewing. Viewed in total, 60% of our channel bends aredownstream skewed, a higher percentage than would be expec-ted from the literature quoted above. We further organize thedata into bins of increasing bend sinuosity, as measured by theinflection-to-inflection arc length to the length of the straightline connecting the inflection points.Fig. 3 characterizes skewing in terms of percentage upstream

skewed, mean, median, and 75% and 25% quartiles. The corre-sponding probability distributions for each sinuosity bin are givenin SI Appendix, Fig. S1.Fig. 3A shows 1) the number of upstream-skewed bends (dark

blue), downstream-skewed bends (light blue), and total bends(dark + light) for each sinuosity bin, and 2) the correspondingpercentage of bends that are upstream skewed (right axis). Thecorresponding numbers for neck cutoffs are shown on the farright of the plot.According to Fig. 3A, a large proportion of our bends tend to

have low sinuosity, and most of these bends are downstreamskewed. This explains why downstream skewing could dominatefrom an overall perspective even when a superficial look suggestsupstream skewing. On the other hand, the total number of bendswith higher sinuosity is relatively small, but the percentage ofupstream-skewed bends increases with increasing sinuosity forthese bends. Downstream skewing dominates up to a sinuosity of2.2, and upstream skewing increasingly dominates for higher

Fig. 1. Features of bend skewing in idealized models, as well as a historicalmap showing several skewed bends (A) Sine-generated centerline of meanderwith no skewing. (B) Upstream-skewed centerline. (C) Downstream-skewedcenterline. (D) Meandering Mississippi River (channel 07 near Vicksburg, MS)from a Fisk (19) map showing varying modes of skewing.

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sinuosity. By a sinuosity of 3.8 to 4.2, the bin of highest sinuosity,74% of bends are upstream skewed.It is likely that high-sinuosity bends continue to evolve to cut

themselves off. In the case of the neck cutoffs, 84% of 411 cut-offs are upstream skewed. This indicates that in pristine alluvialmeandering rivers, the direction of the flow can be reasonablyestimated from the pattern of skewing of recent neck cutoffs, aswell as the bends of highest sinuosity.Fig. 3B shows mean and median skewing angle as a function of

sinuosity. Also shown are the 25 and 75 percentiles. The lengths ofthe sinuosity bins on the horizontal axis vary because we calculatedthese statistics using the same amount of bends within each bin.Counting from the left, solid circles 1 through 9 at the center ofeach bin represent 250 bends, and the last solid circle representsthe remaining 215 (we have 2,465 channel bends in total). Thefigure documents a fairly smooth transition from predominantlydownstream skewed at low sinuosity to predominantly upstreamskewed at high sinuosity.The idea that upstream skewing predominates has a long

history (23, 26). But, since downstream-skewed bends are notrare, we cannot say definitively that bends innately skew upstream.Indeed, different field researchers give different results (19, 20,35). Our result that bend skewing evolves from downstream toupstream with increasing sinuosity does not necessarily contradictearlier theories. If a sufficiently long reach is evaluated, upstreamand downstream-skewed bends might occur with roughly equalfrequency. But, if we focus only on high-amplitude bends, or shortreaches with high-amplitude bends, upstream skewing may mani-fest itself as dominant.

An early theory of meandering predicts that upstream skewingis an inherent tendency (22, 27, 37). This theory, however, con-siders only point-bar morphodynamics, and does not considerfree bars. The analysis cannot be correct in general, becausemost of the bends analyzed here, and in particular those at lowsinuosity, are downstream skewed. High-amplitude bends, how-ever, may conform more to the conditions of the theory. A moreadvanced theory (e.g., refs. 30, 38, and 39) includes the effect offree bars as well as point bars. These two bar classes can go intoresonance when the channel width–depth ratio β reaches aspecific value βc (dependent on other parameters such as grainsize). In subresonant conditions (β < βc), the theory predictsupstream-skewed bends, and in superresonant conditions (β > βc)it predicts downstream-skewed bends.In order to investigate this issue, we obtained the results of

meander simulations conducted with a version of the above nu-merical model, under 3 different conditions (29, 36). All simula-tions assume that the form drag associated with hydraulic resistanceis set by dunes. The planforms of the simulations are presented inFig. 4A. The run conditions can be found in the original references.Fig. 4C shows a comparison of the results for skewing versus

sinuosity for the 3 model simulations and each of our 20 naturalreaches. It shows that the behavior of natural reaches falls inbetween the subresonant and superresonant simulations. Thesubresonant (dune1 and chaos) cases somewhat overpredictupstream skewing compared to the field data. The superresonantcase (dune6), on the other hand, substantially underpredictsupstream skewing. Overall, the subresonant simulations betterapproximate the observed field relation between skewing and

Fig. 2. Overview of the study reaches, and an example of the characteristic floodplain type our sample reaches flow through. (A) Planforms of all 20 reachesused in this study. (B) Location of all 20 reaches (C) Detailed view of the planform of a subreach of the Chulym River, Russia used in this study.

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sinuosity, with upstream skewing dominating at high sinuosity.We have plotted the natural cases so that wider lines representwider reaches; it is seen that river width does not play a signifi-cant role in skewing. The width differences are relatively easy toquantify using remote sensing, but other differences that may beimportant, including bed grain size, flow depth, flow field, andbank strength must be sampled on the ground.As seen in Figs. 3 and 4C, our field cases do not suggest that

strongly superresonant behavior is common at high sinuosity. Thismay be because most of our reaches were chosen to be in humidareas with heavily vegetated floodplains. This condition may favorthe development of narrow channels, pushing them into the sub-resonant regime. It is possible that downstream skewing at highamplitude may be favored in wider streams with more sparsefloodplain vegetation.Among the field data, however, there is an outsider in Fig. 4C.

For this reach, the percent upstream skewed in the bin withsinuosity ranging from 3.4 to 3.8 drops below the curve of thesuperresonant case. This reach, in distinction to all of the othernatural cases, shows predominantly downstream-skewed bendsin the high-sinuosity range. It is a reach of the Tana River inKenya, Africa. The reason for this abnormity is not clear. Theclimate is relatively dry (average annual precipitation is 350to 470 mm per year), the vegetation is relatively sparse (40),and the floodplain (about 4 km wide, measured from GoogleEarth) is relatively narrow compared to the other reaches.The reach might be locally affected by bedrock: see Fig. 4B.Recently, a reach of the Wabash River, USA has been

characterized as a bedrock–alluvial river (41). This river ismostly alluvial, but is also affected by occasional bedrockoutcrops. We do not assert that bedrock must lead to down-stream skewing. But, these outcrops distort the shape of theadjacent bends to varying degrees. In fact, some of our othersample reaches might also be affected by bedrock to a minordegree, but still they show subresonant behavior at high am-plitude. Removing the Tana River from our database does notcause any significant change in the results of Fig. 3 (SI Ap-pendix, Fig. S2). Meandering bedrock–alluvial streams in whichbedrock is ubiquitous are briefly discussed in SI Appendix,Fig. S3.Meanders in confined valleys have been documented in ref.

42. In contrast to our free meander reaches, these confined bendtrains are remarkably upstream skewed and tend to migratedownstream as a coherent waveform. Another significant dif-ference between these planforms and our alluvial reaches isthat the former seldom generate cutoffs. Cutoffs, the dynamicprocesses of which are still incompletely understood, shouldplay a vital and controlling role in the long-term planformdynamics of meandering rivers. By sporadically eliminatingportions of itself, meander planform sinuosity fluctuatesaround a critical state (7, 43–45). Meanwhile, cutoffs generateintermittent noise that is able to influence the river both up-stream and downstream (7, 46, 47). They also contribute tofloodplain heterogeneity through cutoff infilling (7, 16, 17, 48).Further, ref. 21 suggests that similar cutoff geometries sharesimilar dynamic histories, which means that early formative

Fig. 3. Relationships between bend skewing and bend sinuosity. (A) Number of upstream skewed bends and downstream-skewed bends in each sinuosityrange (left axis) and percentage of upstream-skewed bends (right axis) vs. sinuosity (horizontal axis). Results for cutoffs are given at the far right. (B) Meanand median bend skewing angle as well as the 75% and 25% quartiles for each sinuosity bin as a function of sinuosity. Note fromMaterials and Methods thatangle θ is negative for downstream-skewed bends and positive for upstream-skewed bends.

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dynamics may be inferred to some extent from snapshots ofcutoff planforms. Indeed, our result shows that at least onepiece of information about flow dynamics, i.e., flow direction, isembedded in cutoff planforms.While there have been major advances in the theory of me-

ander migration over the years, no single theory predicts thebehavior shown in Fig. 3. Here we consider 3 theories of me-ander migration that can be used to numerically model plan-form evolution, introduced above but here organized as followsfor clarity: the bend theory (22, 27, 37); the bar-bend theory (17,28, 30, 36, 49); and the bend-gradient theory (31, 32). The bendtheory agrees with the data presented here in that the high-amplitude bends of our dataset tend to be upstream skewed,but it also incorrectly predicts that in the absence of floodplaininhomogeneities, all bends should be upstream skewed. Thebar-bend theory captures the general trend of the data only inthe subresonant case, where it somewhat overpredicts upstreamskewing. And, since width is assumed to be constant in space inthis theory, a reach can be either in the subresonant (narrow)regime or superresonant regime (wide), but not both simulta-neously. The bend-gradient theory predicts that a bend can evolvefrom low-amplitude downstream skewing to high-amplitude up-stream skewing, but only a single calculational example is pre-sented. Evidently there is more progress to be made in explainingthe evolution of skewing in meandering rivers, including auto-genic and allogenic tendencies (16, 29, 48, 50).

Many theories use the assumption of constant channel width(50, 51). It may be that a consideration of an imposed variationof width (52) or self-formed width variation (53–55) can betterexplain the evolution from downstream skewed at low amplitudeto upstream skewed at high amplitude. As in ref. 33, super-resonant effects and subresonant effects may coexist within areach when channel width is self-formed.We conclude the following based on our study of 20 reaches of

alluvial meandering rivers: 1) Low-sinuosity bends tend to bedownstream skewed. 2) High-sinuosity bends tend to be upstreamskewed, and both the frequency and mean angle of skewing in-crease with increasing sinuosity. 3) Recent cutoffs tend to be themost likely to be upstream skewed. These results can serve asgoalposts for future modeling efforts.

Materials and MethodsData Collection.We used Landsat images and Google Earth Pro to collect dataon pristine meandering rivers with well-vegetated, wide alluvial floodplainsand a wealth of cutoffs and scrolls documenting channel migration. TheLandsat Product Identifier of the images that have been used are listed in SIAppendix, Table S1. The centerline coordinates of the 20 reaches used in thisstudy are available in Dataset S1. We chose remote locations in order toexclude the effect of human interference. (This was not always entirelypossible: one reach is affected by a mine far upstream and another is af-fected by forestry practices). And, as discussed in Results and Discussion, theinfluence of bedrock cannot be totally discounted. This issue is addressed inmore detail in SI Appendix.

Fig. 4. Comparison of bend skewing patterns for different model simulations and natural reaches. (A) Representative planforms from the simulations of refs.29 and 36. Case “chaos” (36) is an extensive subresonant simulation. Case “dune1” (29) is a subresonant simulation. Case “dune6” (29) is a fully superresonantsimulation. (B) A segment of the Tana River possibly affected by bedrock. (C) Comparison of percent upstream skewed bends and cutoffs for each of the fieldreaches, along with the results for three numerical simulations.

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Bend Segmentation. To segment successive bends, we first calculate curvatureas a function of streamwise channel length as

C =�dxds

d2yds2

−dyds

d2xds2

�.

"�dxds

�2

+�dyds

�2#−3=2

. [3]

The zero-crossing points define the inflections. Since the calculation ofcurvature is notably affected by noise, it is necessary to use a numerical filterto smooth out spurious bends (56, 57).

Different smoothing filters have different effects on the shape and size ofdeveloping meanders (57). We carried out a comparison of cubic spline in-terpolation and the curvature smoothing method in ref. 56 for a reach ofthe Bai River, China (Fig. 5D). The plot shows only minor differences be-tween these two smoothing methods, which means that either method issufficient for the purpose of determining inflection points.

Following ref. 56, we smoothed the curvature series and identified all ofthe inflection points. Segmentation was done by connecting these inflectionpoints successively. Due to the complexity of the planforms, segmentation

into individual bends was not always straightforward. Some of the segmentswere too short or too straight for quantification. We disqualified bendswhose amplitudes were less than the average channel width, as well asbends whose lengths were less than 4 times the width. Although this cri-terion might be somewhat arbitrary, it is appropriate here since our focus ison higher-amplitude bends.

Skewing Angle Evaluation. The method we use to evaluate bend skewingangle is illustrated in Fig. 5C. We drew 3 straight lines. Line P1P2 extends fromupstream inflection point to downstream inflection point of each bend. LineAM connects the bend apex, which is defined as the point of the highestcurvature, to the midpoint of line P1P2. Line MO extends normal to line P1P2from its midpoint. The angle θ from line MO to line AM defines the skewingangle. A positive value of θ corresponds to upstream skewing, and vice versa.

ACKNOWLEDGMENTS. This work was supported by the National Key R&DProgram of China (2017YFC0405203) and the National Natural Science Foun-dation of China (51779242). We are grateful to Alessandro Frascati for data

Fig. 5. Illustration of the transformation of a reach in order to extract information related to bend skewing. (A) The meandering Bai River, Sichuan Province,China. (B) Straight lines from inflection point to inflection point connect the bends. (C) The assemblage of straight lines is straightened into a single base line.The red lines from the center of the base line for each bend to the bend apex allow determining the angle of skewing. The Bai River is not included in themain compendium of bends in this analysis. (D) Comparison of cubic spline interpolation and the curvature smoothing method in ref. 56 for error reduction inthe computed local curvature.

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pertaining to the numerical simulation of meandering channel evolutionusing the bar-bend theory. X.G. thanks Houpu Yang for financial supportand Chenge An for inspiring discussions. G.P. expresses his thanks to Xudong

Fu of Tsinghua University, and the H. W. Johnson Professorship for supportto visit China in order to work with X.G. and D.C. We thank the reviewers fortheir insightful reviews.

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