High curvatures drive river meandering

One of the longand widely held ideas about the dynamics of meandering rivers is that migration slows down in bends with higher curvatures. High-resolution measurements of migration rates of more than 1600 bends in time-lapse Landsat satellite images, covering more than 4000 km of seven rapidly migrating meandering rivers in the Amazon Basin, suggest that the variation of migration rate closely follows that of the local channel curvature. However, locations of maximum migration rate are shifted downstream relative to peak curvature, with a phase lag that shows limited variability for the same river segment. A quasi-linear relationship exists between the two if this lag is taken into account. Overall, bends with the highest curvatures show the highest migration rates; exceptions with limited migration seem to be related to the low erodibility of the outer bank, not the hydrodynamics of the flow. The implication is that one of the most important ways river migration is rejuvenated and meander ing patterns are reshuffled is the generation of high-curvature bends through cutoffs. INTRODUCTION Meandering rivers are among the most dynamic sedimentary systems on Earth. Mean­ der bends of large rivers, especially ones with high sediment discharge, can migrate several meters per year (Constantine et al., 2014). Understanding and predicting how meanders change through time have major implications for a myriad of engineering and geological problems, including management of agricultural land, loss of infrastructure, bridge design, and distribution of heterogeneities in porous sedi­ ments and sedimentary rocks. The meander­ ing process consists of erosion on the outer bank and deposition on the inner bank, which are the result of an asymmetric distribution of flow velocity and shear stress in curved chan­ nel segments (e.g., Dietrich et al., 1979). Iner­ tial forces drive the high­velocity core of the river toward the outer bank; the strength of this effect is dependent on the bend curvature (e.g., Leopold and Wolman, 1960). In theory, the larger the curvature (1/R, where R is the radius of curvature), the larger is the centrifugal force and the shear stress exerted on the outer bank. Therefore, bends with high curvature should also have the highest migration rates. However, early studies of the relationship between curva­ ture and migration rate suggested that migration rate reaches its highest value when the radius of curvature is about two to three times the width of the river (W; Hickin and Nanson, 1975; Nanson and Hickin, 1983, 1986). High­curvature bends with an R/W value < 2 appeared to migrate more slowly, potentially due to a higher energy expen­ diture and the development of a separation zone on the upstream side of the bend (Hickin, 1978). In parallel with these field measurements, theo­ retical work has clearly shown that the velocity excess at the outer bank depends not only on the local curvature, but it is also a function of the weighted sum of upstream curvatures (Ikeda et al., 1981; Howard and Knutson, 1984; Fur­ bish, 1988). The integration of upstream curva­ tures results in a phase lag between curvature and migration rate, and this phase lag has not been considered in most previous studies. Although Furbish (1988) argued that migration rate is a monotonic function of curvature if bend length is taken into account, the idea that high­curva­ ture bends show limited migration persists and is frequently explored in field studies, despite the increasing scatter in the plot of bend­averaged migration rate versus R/W as more data points are added (e.g., Knighton, 1998; Hudson and Kesel, 2000; Lagasse et al., 2004; Hooke, 2007; Sylvester and Covault, 2016; Finotello et al., 2018; Strick et al., 2018). High curvatures also seem to be associated with low migration rates in numerical modeling results (Crosato, 2009). Although significant progress has recently been made in understanding the impact of sedi­ ment discharge on bend­averaged migration rates between different rivers (Constantine et al., 2014) and as a result of cutoffs (Schwenk and Foufoula­Georgiou, 2016), the bend­scale vari­ ability of migration rates has received limited attention. Time­lapse satellite imagery provides an opportunity to reevaluate the relationship between curvature and migration rate at a tem­ poral and spatial resolution that was not possible before. Therefore, we digitized channel center­ lines from Landsat satellite images of seven rivers in the Amazon Basin and estimated local migration rates through correlating centerlines. DATA AND METHODS Rivers of the Amazon Basin that drain the Andes have high water and sediment discharges and are among the fastest­migrating meandering rivers on Earth (Constantine et al., 2014). Tribu­ taries with low­relief drainage basins are migrat­ ing at slower rates, but they still show a measur­ able change over the last ~30 yr, the time period for which Landsat imagery is available. Here, we focused on nine segments of seven rivers (Fig. 1) that show only limited contact with the edges of their incised valleys, which are usually loca­ tions with a significant reduction in erodibility compared to the alluvium deposited inside the valley (e.g., Nicoll and Hickin, 2010). For each river, we selected the time intervals between the two scenes as a function of overall migration rate. Channel banks and centerlines were digi­ tized using a quasi­automated workflow based on the Rivamap river analysis and mapping Python package (Isikdogan et al., 2017). Migration rates were calculated using a dynamic time warping algorithm (e.g., Lisiecki and Lisiecki, 2002) that correlates each point along the first channel cen­ terline to the closest point on the second center­ line. These point­by­point measurements were taken at 25 m intervals. To evaluate these measurements in a kine­ matic context, it is useful to summarize some key characteristics of the meandering process. A fundamental aspect of the meandering phenom­ enon is that the location of maximum migra­ tion does not coincide with the bend apex (e.g., Seminara, 2006); instead, it is often located CITATION: Sylvester, Z., Durkin, P., and Covault, J.A., 2019, High curvatures drive river meandering: Geology, v. 47, p. 263–266, https:// doi .org /10 .1130 /G45608.1 Manuscript received 17 September 2018 Revised manuscript received 11 December 2018 Manuscript accepted 10 January 2019 https://doi.org/10.1130/G45608.1 © 2019 The Authors. Gold Open Access: This paper is published under the terms of the CC-BY license. Published online 6 February 2019 Downloaded from https://pubs.geoscienceworld.org/gsa/geology/article-pdf/4650321/263.pdf by guest on 17 March 2019 264 www.gsapubs.org | Volume 47 | Number 3 | GEOLOGY | Geological Society of America downstream of the point of maximum curvature. This idea can be best illustrated using the con­ cept of “nominal migration rate” (Fig. 2; How­ ard and Knutson, 1984), which is the migration rate that would be expected if bank erosion were only a function of local curvature. For the pur­ pose of this study, we simplified the approach adopted by Howard and Knutson (1984) and defined the nominal migration rate as the prod­ uct of the dimensionless curvature (W/R) and the migration rate constant kl:


INTRODUCTION
Meandering rivers are among the most dynamic sedimentary systems on Earth.Mean der bends of large rivers, especially ones with high sediment discharge, can migrate several meters per year (Constantine et al., 2014).Understanding and predicting how meanders change through time have major implications for a myriad of engineering and geological problems, including management of agricultural land, loss of infrastructure, bridge design, and distribution of heterogeneities in porous sedi ments and sedimentary rocks.The meander ing process consists of erosion on the outer bank and deposition on the inner bank, which are the result of an asymmetric distribution of flow velocity and shear stress in curved chan nel segments (e.g., Dietrich et al., 1979).Iner tial forces drive the highvelocity core of the river toward the outer bank; the strength of this effect is dependent on the bend curvature (e.g., Leopold and Wolman, 1960).In theory, the larger the curvature (1/R, where R is the radius of curvature), the larger is the centrifugal force and the shear stress exerted on the outer bank.Therefore, bends with high curvature should also have the highest migration rates.However, early studies of the relationship between curva ture and migration rate suggested that migration rate reaches its highest value when the radius of curvature is about two to three times the width of the river (W; Hickin and Nanson, 1975;Nanson andHickin, 1983, 1986).Highcurvature bends with an R/W value < 2 appeared to migrate more slowly, potentially due to a higher energy expen diture and the development of a separation zone on the upstream side of the bend (Hickin, 1978).In parallel with these field measurements, theo retical work has clearly shown that the velocity excess at the outer bank depends not only on the local curvature, but it is also a function of the weighted sum of upstream curvatures (Ikeda et al., 1981;Howard and Knutson, 1984;Fur bish, 1988).The integration of upstream curva tures results in a phase lag between curvature and migration rate, and this phase lag has not been considered in most previous studies.Although Furbish (1988) argued that migration rate is a monotonic function of curvature if bend length is taken into account, the idea that highcurva ture bends show limited migration persists and is frequently explored in field studies, despite the increasing scatter in the plot of bendaveraged migration rate versus R/W as more data points are added (e.g., Knighton, 1998;Hudson and Kesel, 2000;Lagasse et al., 2004;Hooke, 2007;Sylvester and Covault, 2016;Finotello et al., 2018;Strick et al., 2018).High curvatures also seem to be associated with low migration rates in numerical modeling results (Crosato, 2009).
Although significant progress has recently been made in understanding the impact of sedi ment discharge on bendaveraged migration rates between different rivers (Constantine et al., 2014) and as a result of cutoffs (Schwenk and FoufoulaGeorgiou, 2016), the bendscale vari ability of migration rates has received limited attention.Timelapse satellite imagery provides an opportunity to reevaluate the relationship between curvature and migration rate at a tem poral and spatial resolution that was not possible before.Therefore, we digitized channel center lines from Landsat satellite images of seven rivers in the Amazon Basin and estimated local migration rates through correlating centerlines.

DATA AND METHODS
Rivers of the Amazon Basin that drain the Andes have high water and sediment discharges and are among the fastestmigrating meandering rivers on Earth (Constantine et al., 2014).Tribu taries with lowrelief drainage basins are migrat ing at slower rates, but they still show a measur able change over the last ~30 yr, the time period for which Landsat imagery is available.Here, we focused on nine segments of seven rivers (Fig. 1) that show only limited contact with the edges of their incised valleys, which are usually loca tions with a significant reduction in erodibility compared to the alluvium deposited inside the valley (e.g., Nicoll and Hickin, 2010).For each river, we selected the time intervals between the two scenes as a function of overall migration rate.Channel banks and centerlines were digi tized using a quasiautomated workflow based on the Rivamap river analysis and mapping Python package (Isikdogan et al., 2017).Migration rates were calculated using a dynamic time warping algorithm (e.g., Lisiecki and Lisiecki, 2002) that correlates each point along the first channel cen terline to the closest point on the second center line.These pointbypoint measurements were taken at 25 m intervals.
To evaluate these measurements in a kine matic context, it is useful to summarize some key characteristics of the meandering process.A fundamental aspect of the meandering phenom enon is that the location of maximum migra tion does not coincide with the bend apex (e.g., Seminara, 2006); instead, it is often located downstream of the point of maximum curvature.This idea can be best illustrated using the con cept of "nominal migration rate" (Fig. 2; How ard and Knutson, 1984), which is the migration rate that would be expected if bank erosion were only a function of local curvature.For the pur pose of this study, we simplified the approach adopted by Howard and Knutson (1984) and defined the nominal migration rate as the prod uct of the dimensionless curvature (W/R) and the migration rate constant k l : (1) Predicted migration rates can be estimated as the weighted sum of upstream curvatures: where Ω and Γ are weighting parameters with values of -1 and 2.5, s is the current location along the centerline, ξ is the alongchannel dis tance upstream from that location, and G(ξ) is an exponential weighting function: where α is a function of friction factor C f and water depth D, and k is a constant that equals 1: Using this approach, and assuming a con stant value for D for each river segment, we can optimize the value of C f so that the phase shift between the actual and predicted migration curves is at a minimum.Then, the migration rate constant k l can be estimated by minimizing the difference between the absolute values of the actual and predicted migration rates.Migra tion rates that would be expected with the sim plest meandering model can be estimated using Equation 2. Because large tributaries can sig nificantly change both the water and sediment discharge, and these have an impact on migra tion rates (Constantine et al., 2014), we split all of the rivers into a few segments that did not include large tributaries.
When measuring migration rates in bends that have a strong translational component, it is important to consider the difference between bank migration and bend migration (Figs. 2B and 2C)."Bank migration" refers to the rate of bank erosion and accretion measured along a direc tion perpendicular to the banks or the center line.Bend migration can be measured by linking points of similar curvature on the two centerlines, and it is significantly different from bank migra tion in bends showing downstream translation.In this study, we focused on bank migration, the primary physical process behind meandering, and a key component of the model used here.
Data and code (Jupyter notebooks) are available at https:// github .com/zsylvester /curvaturepy.See the GSA Data Repository1 for full results and more detail on the methods used.

RESULTS
The curvature and migration rate series are remarkably similar to each other for all river segments, although there are some obvious departures.We use a short segment of the Juruá River to illustrate the pointbypoint relation ship between the two series (Fig. 3).Although the nature of the change in channel locations is not immediately obvious in map view, a plot of the curvature and migration rate series side by side highlights their similarity and the need to consider the phase lag (Fig. 2) when trying to understand the relationship between the two.In the case of the Juruá River, the mean phase lag is 1073 m, with a standard deviation of 269 m.The phase lag shows a relatively low variability for the same river segment but varies between 2.1× and 4.7× the average channel width across different rivers.Virtually all bends show a down stream shift of the maximum migration rate rela tive to the bend apex.
If this phase lag is accounted for, both the nominal and predicted migration rates correlate well with the measured (actual) migration rates, for all river segments, with mean Pearson's r correlation coefficients of 0.76 and 0.75, respec tively (Fig. 4; Table DR1).The scatter slightly increases at larger values, but there is no indica tion that migration rates would decrease beyond a critical high curvature.
Instead, there are significant numbers of data points that have anomalously large migration rates.A visual inspection of rap idly migrating highcurvature bends suggests that the majority are located downstream of recent cutoffs (green dots in Fig. 4).Most of the tight bends with potential separation zones along the upstream concave banks also fall into this category; that is, they do not show reduced migration rates.Low migration rates would be expected if the idea of increased flow resistance applied (Hickin and Nan son, 1975).The number of data points with lower than expected migration rates is also significant (red dots in Fig. 4).Part of this variability is likely due to the overall hetero geneous nature of the bank material (Güneralp and Rhoads, 2011).However, in many cases, the cause of the reduced migration rates can be more ex plicitly identified.Meanders that are impinging on erosional scarps related to river incision (red dots in rightside panels of Fig. 4) suggest that these low migration rates are caused by a reduction in erodibility associ ated with the edges of incised valleys.These edges and the related lowmigration bends can be easily recognized in topographic maps derived from Shuttle Radar Topography Mis sion (SRTM) elevation data.

IMPLICATIONS FOR RIVER MEANDERING
Previous work has demonstrated the impor tance of upstream curvatures (Ikeda et al., 1981;Howard and Knutson, 1984;Furbish, 1988;Güneralp and Rhoads, 2009), variations in bank erodibility (Sun et al., 1996;Güneralp and Rhoads, 2011;Bogoni et al., 2017), and changes in sediment discharge (Constantine et al., 2014;Schwenk and FoufoulaGeorgiou, 2016) when estimating migration rates.The simple kinematic model of meandering used in our analysis focuses only on the impact of upstream curvatures and cannot reproduce processes like compound meander formation without cutoffs (Frascati and Lanzoni, 2009) or upstream migration under "superresonant" conditions (Seminara et al., 2001).However, it has a small number of parameters and pro vides a simple yet powerful framework for the analysis of the curvature-migration rate relationship.Our results suggest that the link between meander curvature and migration rate is relatively simple.One of the important impli cations of this link is that meander cutoffs play a key role in rejuvenating river migration.Most meanders start out after a cutoff event as small, sharp bends that rapidly migrate downstream and later undergo a slower expansion associ ated with lower curvatures (see also Schwenk et al., 2015).
To summarize our findings, highresolution estimates of channel migration along seven large, relatively rapidly migrating rivers in the Amazon Basin suggest that (1) there is a quasilinear relationship between local curva ture and migration rate; (2) as expected from theory, the locations of the maximum curvature and migration are separated by a lag that is roughly constant for each river segment; and (3) the migration is shifted downstream rela tive to bend curvature by a distance that is 2.1 to 4.7 times larger than the channel width.On average, 57% of the variance in migration rates is predicted by curvature alone if the phase lag is considered.For a 250kmlong segment of the Jutaí River, this proportion is 74%.Con trary to the idea that channel migration slows down at high curvatures, we find that some of the sharpest bends are the ones that migrate with the highest speed.Some of the bends that are related to recent cutoffs show anomalously large migration rates, and these anomalies sup port the observations of Schwenk and Foufoula Georgiou (2016).Lowerthanexpected migra tion rates seem to be related to the reduced erodibility of the boundaries of the incising meander belts.The simple curvature-migra tion rate relationship observed in these rivers is likely to break down in systems with chute cutoffs, midchannel bars, and ones that are affected by significant erodibility variations.Highresolution mapping of additional rivers from different settings is needed to understand the general applicability of our approach and our findings.However, our results do suggest that some of the classic examples of meander ing rivers on Earth display surprisingly simple and predictable migration patterns, in contrast with widely held ideas about the complexity of meander kinematics.

Figure 4 .
Figure 3. A: Channel locations of a short segment of Juruá River (Amazon Basin) in A.D. 1987 (blue) and in 2017 (red).Migration that is related to curvatures of bends defined on 1987 channel (blue arcs) is manifested downstream, in segments defined by points of no migration (red arcs).B: Curvature for 1987 channel (blue) and migration rate between 1987 and 2017 (red) along same segment of Juruá River.Bend numbers are same as in A. Note similarity between two curves and downstream shift of migration rate relative to curvature.See the Data Repository (see footnote 1) material for similar plots for all river segments.