Although it has long been recognized that deposition along meandering rivers is not restricted to convex banks (i.e., point bars), the consensus is that sediment deposition on concave banks of channel bends mostly occurs when meander bends translate downstream because erosion-resistant barriers inhibit their lateral migration. Using a kinematic model of channel meandering and time lapse satellite imagery from the Mamoré River in Bolivia, we show that downstream translation and associated concave bank deposition are essential, autogenic parts of the meandering process, and resulting counter point bars are expected to be present whenever perturbations such as bend cutoffs and channel reoccupations create short bends with high curvatures. The implication is that zones of concave bank deposition with lower topography, finer-grained sediment, slack water, and riparian vegetation that differs from point bars are more common than previously considered.

Meandering rivers are ubiquitous on the surface of the Earth and shaped the surface of Mars. Predicting channel migration and the related spatial distributions of erosion, sedimentation, and riparian vegetation is important for a series of problems like agricultural land management, bridge design, loss of real estate and infrastructure, stream restoration, and distribution of pore space in the resulting deposits and sedimentary rocks (e.g., Lorenz et al., 1985; Salo et al., 1986; Nakamura et al., 2014). While both modeling and observations suggest that the kinematics of meandering is more complicated than simple growth in bend amplitude (i.e., expansion), the classic model of largely coarse-grained deposition on convex point bars still dominates our basic understanding of these systems. In recent decades it has been increasingly recognized that deposition along concave banks and development of concave bank benches or counter point bars is common among many meandering rivers (Page and Nanson, 1982; Makaske and Weerts, 2005; Smith et al., 2011; Ielpi and Ghinassi, 2014; Durkin et al., 2015, 2017; Ghinassi et al., 2016). These features are often interpreted to result only under special circumstances such as downstream channel migration caused by decreased bank erodibility (Page and Nanson, 1982; Smith et al., 2009; Nicoll and Hickin, 2010; Ielpi and Ghinassi, 2014; Ghinassi et al., 2016) or by a reduction of stream power that inhibits the ability of the river to erode the banks (Makaske and Weerts, 2005). Although several modeling studies have emphasized that downstream migration is an intrinsic meandering process (e.g., Parker et al., 1982; Howard and Knutson, 1984; Sun et al., 1996; Chen and Duan, 2006), the implications of this idea for concave bank deposition and the development of counter point bars have not been explored in detail (Parker, 1996).

Counter point bars form adjacent to and downstream of typical point bars, where lateral accretion takes place along a concave bank of channel bends and results in sets of concave accretionary ridges (Smith et al., 2009; Fig. 1). This simple and broad geometric definition is identical to that of “concave bank deposits” (Willis and Tang, 2010); see Smith et al. (2009) for a discussion of related but more specific terms such as “eddy accretion deposits” and “concave bank bench deposits.” Counter point bars occur downstream of the inflection point that—strictly speaking—marks the downstream termination of the point bar. That being said, most counter point bars are direct, physical continuations of the upstream point bar (Fig. 1). The boundary between the two is often gradational, as curvature changes slowly in the vicinity of the inflection point. Many depositional concave banks are associated with a zone of flow separation along the river bank (Hickin, 1978; Makaske and Weerts, 2005; Blanckaert et al., 2013; Ghinassi et al., 2018). Few studies of modern and ancient meander belt deposits have identified counter point bar deposits, likely due to the qualitative and limited recognition criteria (Durkin et al., 2020). Although the number of well-documented field examples is limited, the existing data suggest that counter point bars are generally finer grained than adjacent point bars and can be dominated by silt- and mud-grade sediment (Hickin, 1979; Page and Nanson, 1982; Makaske and Weerts, 2005; Smith et al., 2009, 2011; Hooke and Yorke, 2011; Hubbard et al., 2011; Durkin et al., 2017, 2018, 2020; Fig. 2). In addition, counter point bars occupy lower surface elevations than adjacent point bars. As a result, the heterogeneity of meandering river deposits is probably significantly larger, and the distributions of permeability and porosity are more complex than simple point bar-based models imply (Smith et al., 2009; Durkin et al., 2017).

The formation of counter point bars has been linked to downstream translation of meander bends (Parker, 1996). Bend translation refers to a lateral shift in bend location that is not associated with a significant increase in its amplitude and arc length (e.g., Daniel, 1971; Hooke, 1984; Fig. 1). This stands in contrast to bend expansion that involves a growing bend amplitude and arc length while the two inflection points defining the upstream and downstream extent of the bend remain stationary. Although conceptual models of meander expansion often depict perfectly symmetric point bars that result from expansion and no translation or rotation (Ghinassi et al., 2016; Yan et al., 2017), numerical models (e.g., Howard and Knutson, 1984) and measurements in satellite imagery (Sylvester et al., 2019) suggest that bends with 100% expansion are rare. Downstream translation has been discussed in the context of unconfined meandering (e.g., Bridge, 1975; Hooke, 1975; Jackson, 1976; Fustic et al., 2012; Ghinassi et al., 2014; Ahmed et al., 2019), but it is rarely ascribed explicitly to autogenic processes that are inherent to meandering. In studies that focus on downstream translation, it is commonly attributed to reduced erodibility along the outer bank, which inhibits expansion (Smith et al., 2009; Willis and Tang, 2010; Ghinassi et al., 2018). Indeed, some of the most striking examples of downstream translation and concave bank deposition occur along rivers that are clearly confined by escarpments that are resistant to erosion (e.g., Page and Nanson, 1982; Hickin, 1986; Nicoll and Hickin, 2010; Figs. 1A1B). Although it has been recognized that counter point bars are also present in unconfined meander belts (Figs. 1B1C), they are still interpreted as the result of translation due to the presence of low-erodibility material along the outer bank (e.g., oxbow lake fill; Makaske and Weerts, 2005; Smith et al., 2009; Ghinassi et al., 2016). However, the existence of this kind of allogenic forcing is not always obvious; for example, there is no clear evidence in aerial imagery that the translation of the counter point bar of Smith et al. (2009) is caused by low bank erodibility (Fig. 1C). As suggested by Smith et al. (2009, their Fig. 1), the question arises: can significant downstream translation, and therefore concave bank deposition, take place without substrate variability? This is the first question we aim to address in this study.

A second question is whether it is possible to improve the existing approaches to identify counter point bars. Until now, the definition of and recognition criteria for such segments and their deposits have been qualitative. Here we present a simple parameter that is relatively easy to estimate and can be used to identify segments of a river channel where counter point bars will form in an objective and reproducible way.

A third question focuses on the possibility of going beyond simply subdividing meandering river deposits into point bars and counter point bars. That is, channel segments with deposition on the concave bank have variable geometries and probably variable deposits. Their two main characteristics, bank concavity (or planform curvature) and rate of accretion (or migration rate), are likely to vary across a range; yet this variability and its potential impact on counter point bar geometry and stratigraphy have not been explored. Although the relationship between curvature and migration rate has been the subject of numerous studies (e.g., Hickin and Nanson, 1975; Furbish, 1988; Hudson and Kesel, 2000; Güneralp and Rhoads, 2009), it is unclear how curvature-driven meander kinematics are related to bend translation, expansion, and the presence or absence of counter point bars.

In this study, we aim to address these questions by exploring the relationships between the kinematics of meandering, downstream bend translation, and counter point bars. We use a simple model of meandering to illustrate and quantify these relationships and put forward a new planform-based parameter for differentiating counter point bars from point bars. This parameter is a combined measure of the sign and magnitude of both curvature and migration rate, and it can also be used to estimate the likelihood of counter point bar occurrence. We apply the insights gained from modeling to Landsat time lapse imagery of the Mamoré River in Bolivia.

Mapping River Banks in Satellite Imagery

We selected a ∼375-km-long segment of the Mamoré River in Bolivia for investigating channel migration through time and tracking the development of counter point bars. The segment is covered by one Landsat scene with imagery of channel-defining quality available beginning in 1986 (Fig. 3). The Mamoré River drains an area of 600,000 km2 at an average discharge of 2980 m3/s (Thames et al., 1993; Aalto et al., 2003). Over the study interval, the average channel width is 376 m, and the bankfull depth is 12 m (Thames et al., 1993). The median grain diameter (d50) of suspended sediment is 9.0 µm, and bed material is 400 µm (Guyot et al., 1999). As the Mamoré River drains parts of the Andes, it carries a large sediment load, and its channel migration rates are among the highest measured on Earth, on the order of tens of meters per year (Constantine et al., 2014). One Landsat scene per year was downloaded from the U.S. Geological Survey Earth Explorer site (https://earthexplorer.usgs.gov/; accessed January 2021) for the time interval of 1986 through 2018. Most of the selected scenes were collected in July or August, as these scenes tend to have reduced cloud cover. Due to lack of data of reasonable quality, no scenes were used from 2002 and 2012. Although the channel belt of the Mamoré River is clearly affected by low erodibility along its boundaries, there are numerous meander bends that seem to be freely meandering (Fig. 3). Previous work on the Mamoré River and six other Amazonian rivers suggests that the migration of meander bends that are not in contact with the valley boundaries matches relatively well a simple model that assumes a homogeneous substrate with constant erodibility (Sylvester et al., 2019). The anthropogenic influence over the area of interest is limited, as larger settlements are located outside of the floodplain.

We detected channel banks using a quasi-automated workflow based on the Python package RivaMap (Isikdogan et al., 2015, 2017) to increase the speed of interpretation and improve reproducibility. The input to the RivaMap algorithm is the Modified Normalized Difference Water Index (MNDWI; Xu, 2006). In the MNDWI image, rivers are identified using the multiscale singularity index, which enhances curvilinear features. Once the curvilinear bodies of water are highlighted, river centerlines are generated through non-maxima suppression along the dominant orientation. For additional details on the RivaMap centerline generation, see Isikdogan et al. (2017).

Although the RivaMap package can be used to generate not only centerlines but river width data as well, it does not create a single, continuous centerline and left and right banks for the channel of interest. Therefore, we wrote a new workflow to (1) extract the centerline of the Mamoré River as an upstream-to-downstream ordered array of pixel locations and (2) extract the left and right banks as ordered arrays of the same length and orientation as the centerline. This workflow is not entirely automated, as both the centerline and the thresholded water index need to be manually edited using image editing software, but it is less time consuming and more reproducible than an entirely manual interpretation. To obtain relatively smooth centerlines from the pixel-based data points, we applied a Savitzky-Golay filter (Savitzky and Golay, 1964) with a footprint of 21 points (500 m); then, to reduce the noise in the curvature data, we resampled and smoothed the centerlines and the banks again using a B-spline representation. Despite smoothing, there are no obvious errors when the banks are compared with the water index map (Fig. 4B). The distance between consecutive points along the centerline was set to 25 m. Because we need to measure and visualize the spatial distribution of parameters like curvature and migration rate for both the channel and the deposits it leaves behind, we resampled the banks so that for every point on the centerline there is a corresponding point on each bank located along a direction perpendicular to the centerline (Figs. 4A4B). This can be done without any loss in the accuracy of bank locations if the new bank points are defined as the intersections of the bank lines with a segment that is perpendicular to the local channel centerline and longer than the local channel width (Figs. 4A4B). By using these points, the channel planform can be divided into small polygons of equal length along the centerline, and these polygons can then be used to visualize parameters characteristic of that location along the channel. While this approach works well most of the time, two issues arise that we addressed with additional processing. The first one is that in tight bends, lines perpendicular to the centerline often intersect each other before intersecting the bank (Fig. 4C; Shumaker et al., 2018). We wrote a script that identifies these locations and replaces the intersecting segments with segments that do not intersect (Fig. 4D). This is done by selecting equally spaced points on both banks along the channel segment of interest. The second issue is that in places where channel migration was especially fast, the channel might move more than one channel width during one timestep. If this happens, a gap is created between the two channels (Fig. 4E). We automatically identified these gaps and extrapolated the 25-m-wide polygons of the older channel so that they cover the gap as well (Fig. 4F). We performed these operations using the “shapely” Python package.

Once the channels are extracted from each Landsat scene, migration rates can be estimated using two consecutive channel centerlines. We do this by looking for the nearest point on the second centerline for each point on the first centerline. Ideally, the migration rate should be computed along a line perpendicular to the channel centerline, and this is not what a nearest point approach does. However, if the distances between the centerlines and between the points along the centerlines are not too large, the nearest point approach gives a good approximation of the migration distances in the perpendicular direction. To speed up the computation of migration rate, we used the dynamic time warping algorithm, as implemented in the ‘librosa’ Python package (McFee et al., 2020). This algorithm measures the similarity between two temporal sequences, and it relies on dynamic programming. It is often used to correlate two time series and can be applied to spatial sequences as well (e.g., Lisiecki and Lisiecki, 2002; Sylvester et al., 2019).

The 25-m-wide channel polygons can be clipped so that only the preserved deposits are shown, and these polygons can be colored according to any parameter such as curvature, migration rate, or bar-type index. Generating such maps for each time step allows for the creation of time lapse animations of channel evolution that are useful in understanding the channel kinematics (see animations in the Supplemental Material1).

To check the results of channel tracking, true color images were created from the Landsat bands and compared against the detected channel banks. However, detail is limited for the Landsat imagery as its pixel size is 30 m × 30 m. To compare maps of channel migration against the current distribution of water bodies, scroll bars, and other surface features, we created true color images with 3 m resolution from Planet Ortho Visual 4-band scenes courtesy of Planet Labs, Inc.

The extracted channel banks and centerlines and the code used for analyzing them are available at https://github.com/zsylvester/channelmapper.

Kinematic Model

Counter point bars are defined as locations of concave bank accretion (Smith et al., 2009). The concave/convex nature of a channel at any given location can be quantified using a curvature estimate, and the speed of bank accretion for a channel with invariant width is identical to migration rate. Thus, curvature and migration rate are the two parameters that need to be measured if we want to quantify the location and evolution of counter point bars.

To develop a better understanding of how curvature and migration rate are related, we rely on the simple kinematic model of Howard and Knutson (1984). If migration rate were only a function of local curvature, the two vectors would have the same orientation and convex banks would be depositional while concave channel banks would be erosional everywhere (Fig. 5A). The location of maximum migration would coincide with the location of maximum curvature, and the inflection points would coincide with the locations of no migration (Fig. 5A). However, simple physical arguments, modeling, and observations all suggest that migration rate is a function of the local and upstream curvatures as well (Howard and Knutson, 1984; Sylvester et al., 2019). In this model, migration rate is a function of the weighted sum of the upstream curvatures. Simple models like this have been successfully used in recent years to understand cutoff-related knickpoints (Finnegan and Dietrich, 2011; Sylvester and Covault, 2016) and the impact of erodibility on bedrock valley evolution (Limaye and Lamb, 2014). Although the Howard and Knutson (1984) model is indeed kinematic, it can be derived from the physics-based model of Ikeda et al. (1981) (Sun et al., 1996). In the approach adopted by Howard and Knutson (1984), curvature is used in the form of a “nominal migration rate” that is estimated using an empirical relationship between curvature and migration rate. The idea is that this empirical relationship would be based on field measurements, like those of Hickin and Nanson (1975). We have argued elsewhere (Sylvester et al., 2019) that the relationship between curvature and migration rate is not as complicated as the data of Hickin and Nanson (1975) suggest. Therefore, we simplify the Howard and Knutson (1984) approach and define the nominal migration rate M0 as the product of the dimensionless curvature (W/R, where W is channel width and R is radius of curvature) and the migration rate constant kl:
formula

The predicted migration rate M1 is the weighted sum of upstream curvatures:

formula

where Ω and Γ are weighting parameters with values of −1 and 2.5, and G(ξ) is an exponential weighting function:
formula
where α is a function of friction factor Cf and spatially averaged water depth D:
formula

In each time step, points along the centerline are moved in a direction perpendicular to the centerline using migration rates computed with Equation (2). The sign of local curvature is arbitrary. However, the nominal migration rate is in phase with the curvature and has the same sign (Equation 1), whereas the predicted migration rate (Equation 2) is the weighted sum of the nominal rate and has a phase shift relative to curvature. As a result, it will have segments where its sign is the opposite of the sign of curvature (Fig. 5B). The migration rate constant kl is fixed along the centerline and throughout the simulation; this also means that no spatial variability in erodibility is considered. To account for neck cutoffs, a function was written that connects two points along the channel centerline if they get closer to each other than a critical distance. The model is sensitive to the spacing between points along the centerline; this spacing is kept less than or equal to half of the channel width. A number of points at the beginning and end of the centerline are fixed; this helps to keep the channel belt in roughly the same position and avoids strong edge effects, especially on the downstream end of the model.

Using this model, it is possible to investigate how curvature and migration rate change through time along a channel and the potential processes that result in downstream translation and counter point bar formation.

The Bar-type Index

It is useful to combine curvature and migration rate into a single parameter that captures (1) their magnitudes and (2) whether they are of the same or opposite sign. To capture the second parameter, we relied on signed versions of both curvature and migration rate (Figs. 5B and 6). Migration rate is considered positive when a channel segment moves to the right of its current location looking downstream. Similarly, curvature is considered positive when it would cause migration to the right if migration was only a function of local curvature. The simplest parameter that fits the description above is the product of the dimensionless curvature W/R and normalized migration rate M/kl, where kl is the migration rate constant:
formula

We call this quantity the “bar-type index” (BTI), which is a dimensionless parameter that can be used to identify the location and likelihood of counter point bar development. Counter point bars are likely to develop where the bar-type index is negative and has a relatively large absolute value. As both dimensionless curvature and normalized migration rate tend to have values between −1 and 1, the magnitude of BTI is often, but not always, bounded between these two values. Its value depends somewhat on how curvature, migration rate, and the migration rate constant are estimated. A modeled channel segment can be used to illustrate the typical spatial distribution of curvature, migration rate, and bar-type index (Fig. 6). Curvature and migration rate have opposite signs in several locations as a result of the spatial lag between the two curves: the migration rate curve is shifted downstream relative to curvature. Occurrence of a lag is an important aspect of meandering that is predicted by theory (e.g., Sun et al., 1996; Seminara, 2006) and confirmed by measurements in time lapse satellite imagery (Sylvester et al., 2019). In the kinematic model used here, the lag is due to the fact that migration rate is not only a function of the local curvature (in this case there would be no lag), but it is the convolution integral of upstream curvatures (Equation 2; Howard and Knutson, 1984; Furbish, 1991). The same relationship between migration rate and curvature exists in physics-based models as well, such as the bend equation of Ikeda et al. (1981). Because integration of a typical fluvial curvature function results in a similarly-shaped curve with a constant phase shift, it is expected that the phase lag is approximately the same along a river segment with limited changes in discharge, a model prediction that is confirmed in the rivers of the Amazon Basin (Sylvester et al., 2019). Downstream displacement of the impingement of the maximum velocity zone and maximum shear stress on the outer bank relative to the bend apex has also been observed in natural velocity fields (Leopold and Wolman, 1960; Jackson, 1975; Dietrich et al., 1979; Julien and Anthony, 2002) and in numerical models (e.g., Nelson and Smith, 1989; Legleiter et al., 2011).

Because this phase lag tends to be significantly smaller than the half wavelength of the channel bends, curvature and migration have opposite signs (and the BTI is negative) along segments that are shorter than segments with the same sign. In other words, the kinematic model results in a larger area covered by point bar deposits than the area occupied by counter point bar deposits. The length of the segments with negative BTI is the same as the phase lag; as the phase lag tends to be about the same along the channel, the length of the counter point bars tends to stay the same in the model. However, the magnitude of the BTI is highly variable, ranging from only slightly negative values to well-defined peaks of about −0.2 (Fig. 6). Large absolute values of the BTI are associated with bends of high overall curvature; as migration rate is a quasi-linear function of curvature, these bends also have high migration rates.

Mamoré River Examples

We computed migration rates and bar-type indices for 31 images spanning the time interval 1986–2018. Although we used Landsat scenes that capture different stages of the river and this results in variability of the measured width and migration values (Fig. S1, Supplemental Material; see footnote 1), we decided to maximize the number of cloud-free scenes so that changes along channel segments with high migration rates are captured in detail. In general, the migration rate curve follows channel curvature relatively well with a downstream phase lag (e.g., Fig. S2, Supplemental Material; see footnote 1; see also Sylvester et al., 2019) unless there is a significant increase in channel width in the image pair under consideration (Fig. S1, Supplemental Material). Overall, the bar-type indices recorded in the maps of channel migration come from robust changes in the location of the channel bank regardless of river stage. To compute the bar-type indices, we used a constant channel width of 350.0 m, which is the median of all channel widths, and a migration rate constant of 60.0 m/yr, which is the 90th percentile of the migration rates measured between scenes with limited stage variability.

Within the larger reach studied, we selected three areas for a more detailed analysis of channel evolution (Fig. 3). In the first example, a large meander bend underwent a neck cutoff sometime between July 2009 and July 2010 (Fig. 7). As a result, two short and relatively sharp bends formed (bends 1 and 2 in Fig. 7E). After 2010, bend 2 rapidly migrated downstream and left behind a ∼3-km-long zone with concave bank deposits. The BTI is negative as well in this zone, with the largest absolute values (close to −4.2) recorded soon after the cutoff event in the oldest part of the counter point bar (Fig. 7F). The satellite images and the MNDWI map suggest that this zone tends to have low elevations as indicated by the presence of a number of lakes. The largest lake formed close to the cutoff location and corresponds to the lowest BTI values (Fig. 7G). Although the apex of bend 1 (defined as a channel segment between two inflection points) also moved downstream, its concave bank was not depositional; after the cutoff event, bend 1 underwent an increase in wavelength and decrease in curvature. Further downstream, at bends 3 and 4, two additional zones of concave bank deposition also developed during the same time period (Figs. 7E7F). At least one of these bends (bend 4) is clearly constrained by the valley boundary. Additional zones of concave bank deposition and downstream translation, albeit with lower absolute BTI values, are also present between bends 3 and 4, and they occur along short bends that are the result of a significant cutoff/reoccupation event (Fig. 7C). This new channel segment occupied a preexisting tie channel that previously linked a number of oxbow lakes (Fig. 7B).

The second example also involves a bend cutoff; similar to the occupation of the tie channel described above, the river took advantage of a pre-existing narrow channel to cut off a bend that was far from a stage where a neck cutoff would normally be expected to occur (Fig. 8). This narrow channel linked an old oxbow lake with the main channel in 1994 and took over most of the discharge by August 1995 (Figs. 8B8C). As a result, a sharp bend (bend 1) formed on the downstream side of the cutoff bend. This change caused the downstream bend (bend 2) to switch from expansion and point bar deposition to downstream translation and significant concave bank deposition on the opposite side of the river (Figs. 8D8G). Like in case 1, the BTI is negative in this zone of translation and has its highest absolute value for deposits formed immediately after the cutoff event. Another similarity with case 1 is the evolution of bend 1: its wavelength increases, and its curvature decreases through time; the associated BTI value is mostly positive, and the satellite images suggest typical sand-rich point bar deposition along a majority of the bend (Figs. 8D8H). The translational zone of bend 2 is covered by a number of lakes, and the largest lake is located at the oldest part of the counter point bar (Fig. 8H). In the 2018 Planet Labs true color image, the transition from positive to negative bar-type indices seems to correspond to the transition from sand-rich to sand-poor deposition (Figs. 8G8H).

The third example has the most complicated history. Two large bends were abandoned sometime between July 1987 and August 1988 (Fig. 9). The abandonment of the first bend took place as the main river channel eroded into an old oxbow lake and reoccupied one side of the oxbow and the tie channel that linked the lake to the river (Figs. 9B9C). The second meander was abandoned when a preexisting narrow and sinuous channel took over most of the discharge from the main channel. This probably happened because the orientation of the narrow channel coincided with the new orientation of the reoccupied branch of the oxbow lake. As a result of these abandonments and reoccupations, a new channel segment formed that had numerous short bends, some of them with high curvatures (Fig. 9C). In the years following the channel reorganization, most of these bends have translated downstream and formed deposits predominantly along the concave banks (Figs. 9D9F). Strings of lakes seem to track the areas with strongly negative values of the BTI (Fig. 9G), and the channel width is anomalously large at these locations, especially in images collected during a lower stage (Fig. 9E). During the 30 years since the cutoffs occurred, the channel planform has simplified significantly as several of the short bends disappeared; the original 10 segments with distinct changes in the sign of curvature have been replaced by four much larger wavelength and overall lower-curvature bends (Fig. 9F).

Counter Point Bars in the Kinematic Model

The simple kinematic model outlined above can be used to investigate why some channel bends are dominated by expansion whereas, others undergo significant downstream translation and generate counter point bars. A first important observation is that downstream translation is present in simple models of meandering due to the phase shift between curvature and migration rate. If migration rate were only a function of local curvature, the curvature and migration rate series would be in phase and all meander bends would be purely expansional (e.g., Fig. 5A). However, if migration rate is the weighted sum of upstream curvatures, the point of maximum migration is shifted downstream from the point of maximum curvature (Fig. 5B). The length of this phase lag shows limited variability for a river with similar parameters; this is true in the model (Figs. 56) and at least in some rivers (Sylvester et al., 2019). In contrast, the size of meander bends is highly variable: mature meanders are much larger than those that have formed only recently due to a cutoff or some other perturbation. Large meanders are dominated by expansion because the maximum migration takes place within a few channel widths downstream from the bend apex; short bends undergo translation as the point of maximum migration falls on the downstream side of the bend, close to the inflection point (Fig. 10A). This applies to short bends of any kind and not only to bends related to conventional neck cutoffs (Fig. 10B); a common pattern that we observed in the Mamoré River is that during partial avulsions small channels in the floodplain take over most of the main river discharge, and the inherited short-wavelength bends and/or those created at the junctions with the main channel start migrating downstream after the “occupation” event (Figs. 79).

Meander bends that form through neck cutoff often have a short arc length and high curvatures (Figs. 1011). The point of maximum migration falls on the downstream side, and this results in downstream translation. Due to the increased sediment supply associated with the cutoff (Zinger et al., 2011; Schwenk and Foufoula-Georgiou, 2016) and the newly generated high curvatures, the migration rates—and therefore the translation rates—are relatively high. It is not only the bend that has formed through the cutoff process that undergoes translation; the high curvatures are transmitted downstream to the next bend as well and, at least in some cases, even to the third bend. As a result, several zones of concave bank deposition and potential counter point bar formation develop, all of which are located downstream from the cutoff location (Fig. 11). Through time, the cutoff-related meanders become larger; the rate of translation decreases, and expansion starts to dominate. These changes are reflected in a gradual decline of the absolute value of the BTI for the counter point bars (Figs. 7F and 8G).

A long-term simulation of river meandering that leaves behind a broad belt of deposits allows us to visualize the preservation of both convex and concave bank deposits (Fig. 12). As counter point bars tend to be associated with finer-grained, heterogeneous deposits (Smith et al., 2009; Hubbard et al., 2011; Durkin et al., 2018), there is significantly more heterogeneity in the channel belt stratigraphy than would be expected if only oxbow lake fills were considered (e.g., Colombera et al., 2017). Because downstream translation is characteristic of relatively young meander bends, counter point bars are more common along the most recent channel location. However, deposits with negative bar-type indices are preserved further away from the river as well. They are always located along erosional surfaces that result from the translation; these surfaces often directly juxtapose deposits of unrelated bars and are important when considering pore space connectivity between different bars (Fig. 12). This connectivity is reduced if mud-rich, counter point bars are associated with the erosional surface.

The kinematic model reproduces remarkably well many of the observations we made using the time lapse satellite imagery: the pronounced downstream translation of small meander bends and the related concave bank deposition; the impact of large curvatures not just at the cutoff location but further downstream as well, resulting in more than one translation zone; and the role of cutoffs and local channel reorganizations in creating the small, predominantly translational bends. Some of the examples discussed here (Figs. 79) highlight the importance of floodplain channels not just in terms of surface-water connectivity between a river and its floodplain (Czuba et al., 2019) but as the starting points for rapid and significant channel planform reorganizations or annexational avulsions (Edmonds et al., 2016) that reset the channel evolution over larger distances than those characteristic of simple neck cutoffs. The satellite imagery also suggests that not all translation inside the channel belt is due to “kinematic” perturbations like cutoffs/channel avulsions. Zones of significant translation and concave bank deposition are also present in places that are not obviously related to recent cutoffs or channel reoccupations (Fig. 7). These zones are either due to older perturbations that cannot be observed anymore, or they are the result of intra-channel belt variations in bank composition.

While the observed post-cutoff high migration and translation rate is in part explained by the high curvatures (Sylvester et al., 2019), increased sediment supply associated with cutoff events is also shown to play a role in the increased migration rate (Schwenk and Foufoula-Georgiou, 2016). Measured migration rates along several rivers in the Amazon Basin support this conclusion: cutoff-related bends often show significantly higher migration rates than what is expected from a curvature-based prediction alone (Sylvester et al., 2019).

Although our analysis of the satellite imagery and the kinematic model provides insights into the relationships between channel kinematics and counter point bar deposition, it does not address the cause of finer-grained deposition along concave banks. The satellite images suggest that concave bank deposition is often associated with anomalously large river widths (e.g., Figs. 8C8G), probably as a result of low deposition rates at these locations; the concave bank cannot keep up with the rapidly eroding inner bank of the river. The presence of numerous lakes along these translational zones (Figs. 79) is also evidence for lower rates of deposition and lower elevations. A flow separation zone often develops in locations with concave bank deposition (e.g., Page and Nanson, 1982; Vietz et al., 2012), and it has been suggested that this separation zone results in energy loss and reduced migration rates (Hickin, 1978; Blanckaert, 2011). However, along the Mamoré River, bends with well-defined counter point bars have some of the highest migration rates. Therefore, the development of the flow separation zone and the associated fine-grained deposition is likely to be the effect, not the cause, of unusual local channel kinematics (see also Sylvester et al., 2019) and results from increased channel migration rates. The presence of the flow separation zone helps explain the finer-grained nature of some of the counter point bar deposits, as only suspended sediment is carried across the eddy line and into the separation zone (e.g., Hickin, 1979; Rubin et al., 1990).

Even in the absence of flow separation zones, downstream fining in point bars has been observed in nature (Bridge and Jarvis, 1982; Julien and Anthony, 2002) and predicted by theory and numerical models (Parker and Andrews, 1985; Sun et al., 2001). Increasing curvature and topographic steering of the flow by a large point bar results in cross-channel flow and high shear stress on the upstream side of the bar that significantly decreases the flow and shear-stress values along the downstream side of the bar because the high-velocity core is pushed to the outer bank (Dietrich and Smith, 1983; Nelson and Smith, 1989; Fig. 12A). Although studies of grain size variability on point bars have focused so far on relatively coarse-grained systems (Bridge and Jarvis, 1982; Parker and Andrews, 1985; Sun et al., 2001; Julien and Anthony, 2002), it is likely that the abundance of silt and mud in the downstream parts of some point bars—and in counter point bars—is due to the same process. If the decrease in shear stress is large enough, it is possible that the amount of bedload that makes it to the downstream end of the bar is significantly reduced. In addition to the reduction in grain size, this could also result in reduced sedimentation rates, which in turn might be responsible for the lower elevations of counter point bars. Higher shear stresses and occasional erosion on the upstream side of the bar also result in a downstream shift in the location of the point bar relative to the bend apex; this has been observed in nature (e.g., Jackson, 1975; Fig. 12B) and in numerical models (Nelson and Smith, 1989; Sun et al., 2001). The phase lag between curvature and migration rate could be a result of this phenomenon. However, the length of the lag seems to be unrelated to bend or bar size, which suggests that the convolutional nature of the bend instability is more important. This is essentially the same question as the bar push/bank pull problem (van de Lageweg et al., 2014; Mason and Mohrig, 2018).

A simple kinematic model of meandering allows us to explore the post-cutoff evolution of channel bends over longer time scales; the results suggest that autogenic counter point bars are more common along the current river location as more mature meanders are preferentially preserved in the older channel belt stratigraphy (Fig. 13). However, autogenic counter point bars are still likely to form a significant part of the stratigraphic record of meandering channels and to have a major impact on the distribution of large-scale heterogeneity in these depositional systems. Although in this study we have focused on examples from the Mamoré River, the “unconstrained” concave bank deposits of the Peace River (Figs. 1E1F), the Koyukuk River (Figs. 1C1D), and other North American examples (Durkin et al., 2018) suggest that they are not restricted to rivers in tropical or subtropical climates.

The autogenic development of counter point bars also has ecological implications. Compared to typical point bars, river segments with counter point bars are characterized by lower elevations, larger river widths, the common presence of stagnant water bodies, and relatively high bank migration rates. These characteristics have a strong impact on the rate of establishment of vegetation, the distribution of biomass and species (e.g., Nanson and Beach, 1977; Perucca et al., 2006), and the preservation of organic carbon in fluvial deposits (Torres et al., 2017).

Time lapse satellite imagery covering more than 30 years of change along a 375-km-long stretch of the Mamoré River in Bolivia and a simple kinematic model of meandering suggest that translation and the associated concave bank deposition are intrinsic features of river meandering. Previous work has demonstrated that erosion-resistant barriers that force downstream translation of meander bends are common causes of concave bank deposition and counter point bar formation (Smith et al., 2009); we have shown here that this phenomenon is also common in the absence of significant changes in erodibility. In other words, the answer to the first question we set out to address is affirmative. Significant downstream translation, and therefore concave bank deposition, can take place without substrate variability.

The main cause of this type of downstream translation is the phase lag between curvature and migration rate. In small enough bends the point of maximum migration falls close to the downstream inflection point of the bend, and this results in downstream translation (Fig. 10). Over a segment of such bends, there is deposition on the concave bank and erosion on the convex bank. The length of this segment is set by the lag between curvature and migration rate; therefore, concave bank deposits and counter point bars that originate this way tend to have a characteristic length. On the other hand, large, mature bends are dominated by expansion as the phase lag is small relative to the arc length of the meander. Such bends often also have relatively short segments with concave bank deposition; however, the migration rates are typically among the lowest in the system.

To address the second and third questions, whether it is possible to improve the existing approaches to identify counter point bars and to quantitatively characterize their variability, we introduced a dimensionless parameter called the “bar-type index” (BTI) that can be used to characterize and predict the location of concave bank deposition and counter point bars. The BTI is the product of dimensionless curvature and normalized migration rate; it is negative where concave bank deposition occurs. The higher the migration rate and curvature, the higher the absolute value of the BTI. Curvature and migration rate are related, and short bends with high curvatures and high migration rates have strongly negative BTI values. These bends are perturbations of the channel that form at locations of channel reorganization such as cutoffs and channel reoccupations. Further work is needed to verify whether the BTI correlates with sand content or grain size of the corresponding deposits, but our observations suggest that there is a good correlation between negative values of the BTI and areas with low elevations and lakes in satellite images.

We are grateful for discussions with Jake Covault and Cole Speed, constructive comments by two anonymous reviewers, satellite imagery provided by USGS/NASA and Planet Labs Inc., and to sponsors of the Quantitative Clastics Laboratory research consortium at the Bureau of Economic Geology, The University of Texas at Austin (http://www.beg.utexas.edu/qcl).

1Supplemental Material. Animations illustrating the evolution of counter point bars in the study area and in forward models. Please visit https://doi.org/10.1130/GSAB.S.13667684 to access the supplemental material, and contact editing@geosociety.org with any questions.
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