Rapidly Migrating Secondary Bedforms Can Persist on the Lee of Slowly Migrating Primary River Dunes

Trains of secondary bedforms, superimposed on primary dunes, have been observed in rivers worldwide. To date, it has remained unclear how these secondary bedforms affect the sediment transport dynamics in a fluvial system. In this study, a field campaign was conducted to acquire a data set with a high spatial and temporal resolution, enabling to track secondary bedform migration. Results show that two distinct scales of bedforms are actively migrating. In parts of the study area, superimposed bedforms fully dissipate at the lee slope of the underlying primary dune, suggesting that the secondary bedforms contribute to the migration of the primary dune. In other parts, however, superimposed bedforms persist on the lee slope and in the trough of the primary dune. The steepness of the primary lee slope seems to control this. We calculated the sediment transport rates associated with both primary and secondary bedform migration, based on dune tracking. The sediment transport associated with secondary bedform migration is of the same order of magnitude, and even exceeds the transport associated with primary dune migration. This implies that secondary bedforms play a crucial role in the transport of bedload sediment, and cannot be ignored in the quantification of sediment transport based on dune tracking.

observed under steady flow conditions. To explain these observations, it has been argued that the primary dunes generate a boundary layer in which superimposed bedforms can develop and are stable (Ashley, 1990;Rubin & McCulloch, 1980;Smith & McLean, 1977). Venditti et al. (2005b) studied the dynamics of multiple scales of migrating bedforms under steady flow conditions through laboratory experiments and confirmed the superimposition of multiscale bedforms under steady flow conditions. They found that the small superimposed bedforms form downstream of the upstream, primary lee side and dissipate at the lee side of the next primary dune. Furthermore, they found that the superimposed bedforms migrate significantly faster than the larger dunes and that the associated sediment transport is comparable to the transport rates measured based on primary dunes. They concluded that sediment transport rates seem invariant to the observed scale of bedforms and that the migration of superimposed bedforms is the agency by which the primary dune migrates. In their study, the superimposed bedforms travel over the stoss side and collapse at the lee side.
It is unknown, however, if the same conclusion can be drawn for multiscale dune migration in field conditions. Dunes in flumes, formed in limited water depth and high Froude number flows, are typically asymmetrical and have steep lee side angles. In natural rivers, so-called low-angle river dunes are often formed Naqshband et al., 2014a;Van der Mark et al., 2008). Lee side angles observed in natural rivers can be even smaller than 10° (Cisneros et al., 2020;Hendershot et al., 2016;Ten Brinke et al., 1999). In contrast to high-angle dunes, flow separation for low-angle dunes is not or only intermittently present (Bradley et al., 2013;Holmes & Garcia, 2008;Kostaschuk & Villard, 1996;Kostaschuk et al., 2009;Kwoll et al., 2016). It is unclear if the bedform behavior observed by Venditti et al. (2005b) in flume experiments would occur under conditions in which low-angle dunes develop. Galeazzi et al. (2018) described a river section in which superimposed bedforms were in some cases limited to the stoss of the low-angle primary dunes, whereas in other cases they persisted over the lee slope. Galeazzi et al. (2018) did not study the dynamics of these bedforms. It is unknown if superimposed bedforms contribute to the migration of the primary dune or form an additional transport mechanism where they persist over the dune lee side.
The dynamics of superimposed bedforms in sand-bedded rivers have not often been studied in field conditions. One important reason for this is that usually the temporal resolution of repeated bed elevation measurements is not high enough to resolve the migration of small bedforms (Claude et al., 2012;Kleinhans et al., 2007). In addition, a high spatial resolution is required to enable proper estimates of the morphological properties of superimposed bedforms. For the purpose of investigating the dynamics of small-scale superimposed bedforms, we acquired a data set with an extraordinarily high spatial and temporal resolution. The aim of this study is to investigate multiscale bedform migration in a fluvial system and to determine the importance of secondary bedform migration in the quantification of sediment transport rates.

Field Campaign
A field campaign was conducted on October 11, 2018 in the river Waal, the main distributary branch of the Dutch river Rhine (Figure 1). A bathymetric survey was conducted over a reach of approximately 500 m. Within this area, depths range between −1 and −4 m + NAP. The water level during the campaign was 2.2 m + NAP at the city of Tiel, approximately 5 km upstream and 1.1 m + NAP at Sint Andries, approximately 7 km downstream. Herein, NAP refers to Normaal Amsterdams Peil, which is the Amsterdam Ordnance Datum. During the campaign, large primary dunes were present with lengths predominantly between 50 and 180 m and heights up to 1.6 m, and superimposed secondary bedforms with lengths and heights mostly below 5.5 and 0.35 m, respectively.
Bed elevations were measured using a boat-mounted multibeam echosounder (MBES). In addition to the bathymetric survey, the field campaign included eight repeated measurements of a cross-sectional area. The time interval between each of the eight measurements was approximately 15 min. Each repeated measurement consisted of five parallel boat tracks, such that the bed areas covered by the multibeam were partly overlapping. Together, these tracks covered an area of about 220 × 50 m. During a measurement cycle, the boat started in one corner of the survey area. It navigated the five tracks and then went back to the initial position, to start the next cycle. In addition to an MBES, an acoustic Doppler current profiler (ADCP) was mounted on the boat to measure flow velocities.
The data of the dedicated field campaign were complemented by MBES data made available by the Dutch Ministry of Infrastructure and Environment, Rijkswaterstaat (RWS). These data are part of a larger data set, which was acquired to monitor the river bed to ensure that the water depth is sufficient for ships to pass through. The RWS data were acquired with an interval of six days, on October 9 and October 15. At the location of the field site, information on the bed grain size distribution is also available, based on samples that have been taken on November 24, 2017 using a Hamon grab (de Ruijsscher et al., 2020b;Eleftheriou & Moore, 2013). The 50th (D 50 ) and 90th percentile sediment diameter (D 90 ) were determined following the approach of de Ruijsscher et al. (2020a) and are shown in Figure 1   the bathymetry as surveyed on October 11, 2018. The arrow indicates the flow direction, which is from the northeast to the southwest. The earth coordinates (Easting, Northing) are in the EPSG:28992 system; (c) the discharge at Tiel, approximately 5 km upstream of the study area. On October 11, Q = 710 m 3 s −1 .

Data Processing
The MBES data were processed as point clouds with a spatial resolution of 1 × 1 × 1 cm. The coordinates were transformed to a Cartesian coordinate system that aligns with the central axis of the river, and has its origin in the lower left corner of the bathymetric map, as indicated in Figure 1. The point clouds were subsequently interpolated on an equidistant grid with a cell size of 0.1 m through inverse distance weighted interpolation. The interpolation was done with the Geospatial Data Abstraction Library (GDAL) (GDAL/ OGR contributors, 2020). For the bathymetric map, the outline of the grid is indicated in Figure 1.
The coordinates of the repeated transects were transformed to the same Cartesian coordinate system as the bathymetric map and interpolated on the same 0.1 m equidistant grid. In addition to the spatial interpolation, the repeated transects were linearly interpolated in time. After the spatial interpolation, at each grid cell, eight elevation (z) values were available, with their corresponding time stamps. Through linear interpolation, the elevation values at a grid node were interpolated to eight predefined timestamps with a fixed time interval of 14.4 min. The temporal interpolation was done, because the time intervals between point measurements within a measured transect were relatively large compared to the time intervals between repeated transects.
The MBES data made available by RWS were processed as 1 × 1 m grids. The coordinates of the grid nodes were transformed and the values were interpolated on a 1 × 1 m grid with the same origin and orientation as that of the bathymetric map.
Further processing of all gridded data is performed for each individual transect, where each transect is aligned with the central axis of the river, and corresponds to one y−coordinate of the grid.
The ADCP data were processed using the Matlab toolbox ADCPTools (Vermeulen et al., 2014). The ADCP data were acquired with a bin size of 0.25 m and a blanking distance at the transducer of 0.25 m. Velocity values in the lowest 6% of the measured water column were excluded from further analysis. The remaining flow velocities were fitted on a mesh with a horizontal resolution of 5 m and a vertical resolution of approximately 0.25 m. The obtained velocity values were interpolated toward the surface through nearest neighbor interpolation and toward the bed through linear interpolation, assuming zero-velocity at the bed. The extrapolated values were included in computations of the depth-mean velocity.
A value for the shear velocity was calculated for each velocity profile following the approach of Hoitink et al. (2009), assuming the velocity profiles can be described by the law of the wall up to the surface: where u * is the shear velocity, κ is the Von Karman constant, σ is the relative depth, and U is depth-mean velocity. Fitted parameters with an adjusted R 2 < 0.9 were excluded from further analysis. The shear velocity was used to compute the bed shear stress:    2 * b u .

Separation of Bedform Scales
The superimposed bedforms were separated from the underlying morphology, including primary dunes, to study the superimposed bedforms and primary dunes separately. The raw data were filtered based on a LOESS (locally estimated scatterplot smoothing) algorithm (Cleveland, 1979;Greenslade et al., 1997;Schlax & Chelton, 1992). Because the high-frequency fluctuations in the signal associated with secondary bedforms have a relatively large amplitude, sufficient data points have to be included in the scatter plot prior to smoothing. In sections where the underlying morphology does not change much over short spatial scales, LOESS performed well. However, at locations where the change in elevation is significant, especially at primary lee sides, the filtering method does not perform well, as the LOESS curve does not match the primary morphology.
The unintended smearing of the primary dune morphology was addressed by introducing breaks at the locations where the unmodified LOESS curve significantly departs from the dune trough and from the slope of the dune lee. An illustration of the applied method is provided in Figure 2. In the first panel ( Figure 2: Step 1), an example of a bed elevation signal (black) is shown, along with the unmodified LOESS curve (red). In the second step, the locations are identified where a break should be applied, based on this LOESS curve. These breaks were identified based on two criteria. The first criterion is that the break should be in the center of a primary dune trough. The centers of the primary dune troughs were initially approximated as local minima of the unmodified LOESS curve. The second criterion is that the maximum slope of the LOESS curve between the trough and the nearest, upstream crest, the next local maximum of the LOESS curve, should be higher than a specified value, here set at 1.72°. An example is shown in Figure 2: Step 2.
Subsequently, in the third to the fifth step the exact location where the breaks in the trough centers are set is updated (Figure 2: Steps 3-5). The breaks should be located at the transition from the trough to the lee. In the third step, the minimum z − value of the original signal near the break is located (within 15 m up or downstream of the break). In the fourth step, all local minima are identified within a window between the breakpoint found in the previous step and 15 m upstream. If there are local minima that have a maximum vertical distance of 0.05 m with the absolute minimum, the most upstream of these local minima is selected ZOMER ET AL.  Step 1: The unmodified LOESS curve does not match the primary morphology at the primary lee side and trough; Step 2: Detection of the break location based on the LOESS curve, which is to be adjusted. Steps 3-5: The break location is updated toward the slope discontinuity in the bed elevation signal, the transition from trough to the primary lee. This location is treated as a break in the filtering.
Step 6: The algorithm is used for incremental LOESS filtering between the break locations. Steps 1-3 and 6 show the same bed elevation profile. For Steps 4 and 5, different elevation profiles are shown. LOESS, locally estimated scatterplot smoothing. Step Step 3 Step 6: LOESS curve is fitted stepwise as the new break location. This step is especially relevant if secondary bedforms are present in the trough. In the fifth step, the z − values from the breakpoint to 1 m upstream are considered. If the next value is less than 5 mm higher (corresponding to a slope of 2.9°) the breakpoint is updated to that location.
In the sixth and final step, the LOESS curve is fitted stepwise, starting at the most downstream section, up to the first break. This implies that the LOESS curve is fitted through the stoss downstream of the break. For the next section, the LOESS curve is forced to connect to the previous LOESS curve that ends at the breakpoint. This is achieved by adding artificial data points at the break. In this way a LOESS curve for each section of a transect is fitted.

Bedform Properties
The properties of secondary bedforms are based on the filtered data and were determined following the method of Van der Mark and Blom (2007). In this method, troughs and crests are identified as minima and maxima between zero crossings. The bedform length is defined as the distance between two troughs. The bedform height was calculated both from trough to the next bedform crest and from crest to the next bedform trough. The height of a bedform was subsequently defined as the average of these two. To compute the lee side slope, 1/6th part of the slope was removed both at the crest and at the trough edge of of the slope. The slope angle was then approximated by a linear fit though the remaining central part ( Van der Mark & Blom, 2007). Random irregularities also cross the nulline and are as a result wrongly identified as bedforms. All identified bedforms of which the height was smaller than four times the standard deviation of local elevation values (a total of 15 values) were excluded from the results.
Primary dune length and height were also determined following Van der Mark and Blom (2007). The primary dune height was calculated from the trough to the upstream crest. The lee side slope was determined as the maximum bed elevation slope between the trough and the upstream crest. This approach was followed because, first, the primary dune morphology was filtered and did not include irregularities that can affect the observed maximum lee side slope. A second reason is that flattened crests of primary dunes in the study area would lead to large discrepancies between locally observed lee side slopes and the lee side slopes determined following the approach of Van der Mark and Blom (2007). Toward the left and right outer edges of the primary dunes, where the dune transitions to a flat bed, dune detection results in small heights and lee side angles. To exclude these values, datapoints with heights smaller than 10% of the maximum dune height were filtered out.

Sediment Transport Associated With Bedform Migration
The sediment transport rates associated with both primary and secondary bedform migration were calculated based on dune tracking. Various methods based on dune tracking have been developed, commonly based on the one dimensional Exner equation for mass conservation of sediment, valid for steady flow: where p is the porosity of the bed (−), η is the local bedform elevation with respect to the base level of the bedform, η = z − z 0 , (m), and q is sediment transport per unit width (m 2 s −1 ).
Based on the one-dimensional Exner equation (Equation 2), the following equation for the sediment transport associated with bedform migration can be derived (McElroy & Mohrig, 2009): where V c is the dune migration rate (ms −1 ) and q 0 a constant of integration. The derivation of Equation 3 has been described by McElroy and Mohrig (2009) and is based on the assumption that migrating bedforms are unchanging under steady flow conditions. Further assuming that the bedload transport is zero in the trough (q = 0 at η = 0), q 0 = 0, leads to: Often, q is integrated over the bedform, introducing the dune height and a shape factor β, replacing η. Here, we did not compute the transport rate per bedform, but computed an average q s per transect.
V c , the bedform migration rate, is computed for each transect through the cross-correlation of two bed elevation scans. The migration rate is then the distance traveled divided by Δt. The local bedform elevation, η, is the bed elevation with respect the bedform base level z 0 . The bedform base level was computed through linear interpolation between bedform troughs. The porosity, p, is assumed to be 0.36 (Allen, 1985).
The eight repeated measurements of the cross-sectional area ( Figure 1)  The local bedform elevation, η, was computed per transect between the first and last trough (so only for bedforms fully included in the measured area). The bed elevations were computed for each of the eight repeated measurements. The averaged values were used to compute the sediment transport.
The sediment transport rate associated with primary dune migration was computed based on the MBES data acquired on 9 and 15 October. Here, a moving-average dune migration rate was computed.
The sediment transport rates found based on dune tracking were compared to transport rates calculated with a transport formula. Here, we used the equation of Engelund and Hansen (1967). This equation was selected because it is applicable to a sandy, dune-covered bed. Following this approach, the nondimensional sediment discharge Φ is computed as: where θ is the Shields stress computed from the bed shear stress values based on the ADCP measurements and f is the friction factor. The bed sediment discharge per unit width [m 2 s −1 ] then reads: with g is the acceleration of gravity [ms −2 ], s is the relative density of sediment grains (ρ s /ρ w ) and D 50 is the median grain size [m]. The values of D 50 across the channel were computed based on the bed samples presented in Figure 1. The y-coordinates of the samples were transformed into the Cartesian coordinate system of the grids. The samples taken in the western and eastern river sections were averaged to get one value in the western section and one value in the eastern section. Together with the central sample, bs3, values for D 50 were obtained over the full river width, i.e. as a function of the y-coordinate, by linearly interpolating in between sample locations and extrapolating toward the river banks using nearest neighbor extrapolation.
The transport equation of Engelund and Hansen (1967) computes the total transport of bed sediments, including the suspended sediment transport. To enable good comparison with the estimates based on dune tracking, which only relate to the bedload mode, we assessed whether bedload transport was the dominant sediment transport mode. A common parameter to assess the relative importance of bedload transport to suspended load transport is the ratio of the bed shear velocity and the particle fall velocity, u * /w s , also referred to as the suspension number (Bagnold, 1966). Bedload transport is the dominant transport mode for u * /w s < 1. According to Van Rijn (1984), initiation of suspension starts at u * /w s = 0.4. The particle fall velocity was calculated following Cheng (2009): with the drag coefficient C D defined as: where D * is the dimensionless grain diameter.

Morphology of Superimposed Bedforms
The morphology of the superimposed bedforms, which has been separated from the underlying bathymetry, is shown in Figure 3. The secondary bedforms are superimposed on lunate primary dunes. The purple areas indicate where primary lee slopes are positioned. The lee sides of the primary dunes are defined as slopes with an aspect angle between 112.0° (southeast) and 247.5° (southwest). The western section is deeper than the eastern section, as visualized in Figure 1. Figure 3 indicates a high lateral variability of the superimposed bedforms. In the western part of the river section, roughly between y = 0-70 m, superimposed bedforms are very small and some regions are poorly resolved by the MBES measurements. The middle section of the riverbed (y = 70-130 m) is covered with larger, two-dimensional bedforms. Here, the superimposed bedforms are clearly visible also on the lee slopes of primary dunes. In the eastern river section (y = 130-175 m), superimposed bedforms are also clearly visible, but here they are much more irregular in shape and size. At x = 365 m, a triangular section is visible, directly downstream of a primary lee slope, where superimposed bedforms have dissipated.
In Figure 4, four bed elevation profiles are shown. The first plot shows a profile at y = 45 m. Very small secondary bedforms are superimposed on an asymmetrical dune with a steep lee side slope. Directly downstream of the lee side slope, superimposed bedforms are absent. The second and third plot show profiles from the central river section, where secondary bedforms are larger. These profiles show small primary lee side angles over which the secondary bedforms persist. In the third profile, at y = 120 m, the superimposed bedforms are significantly smaller downstream of the primary lee side. They grow again in the downstream direction. The fourth plot shows a profile at y = 150 m. Downstream of primary lee side slopes, superimposed bedforms entirely disappear and start to develop roughly 25 m further downstream.  Figures 5g-5j show the primary dune properties, indicating that dune lengths do not significantly vary per section, but primary dunes are highest and steepest in the western section, whereas lee side slopes in the central and eastern sections are comparatively low. Figure 6 shows the digital elevation models (DEMs) of difference of two repeated elevation scans, where Δz = z new − z old . Positive values indicate a net deposition of sediment and negative values indicate a net erosion between two elevation scans. If the time span between two elevation scans is appropriate, the DEM of difference reflects bedform migration, visualizing sediment deposition on the bedform lee and erosion of the stoss.

Bedform Migration
In Figure 6, the left plot shows the DEM of difference based on MBES data acquired on 9 and 15 October. This plot shows the migration of the larger, primary dunes over a period of 6 days. The two plots on the right show two DEMs of difference based on the measurements repeated within one day. These plots indicate the migration of secondary bedforms over a time period of approximately 28.8 min and 100.8 min. The DEMs of difference indicate that in the same time period, both primary and superimposed, secondary bedforms are actively migrating. Figure 7a shows a cross-sectional overview of the streamwise velocity. The flow velocity is significantly higher across the western river section, which is also reflected in the depth-averaged flow velocity shown in Figure 7b. Figure 7b further shows the bed shear stress calculated based on the velocity measurements and the ratio of the bed shear velocity u * and the particle fall velocity w s , which ranges between 0.12 and 0.54, indicating that bedload transport is the dominant transport mode. Sediment transport rates associated with primary and secondary bedform migration have been calculated based on the bedform migration celerity V c and the bedform elevation with respect to the bedform base level η = z − z 0 . The celerity V c of secondary bedforms was computed per transect and presented in Figure 7c. The bedform elevation η was averaged longitudinally within each transect. Both bedform celerity and the average bedform elevation vary laterally, with a sharp transition around y = 70 m. The smaller secondary bedforms in the western section of the river migrate much faster than the larger secondary bedforms in the central and eastern sections.

Sediment Transport Rates Associated With Bedform Migration
(d) Figure 7d shows the bedload sediment transport rates per unit width (q s ) associated with primary and secondary bedform migration based on V c and η. The propagated uncertainty of the calculated transport rate associated with secondary bedform migration (  q s ) is calculated according to: assuming the errors of η and V c are uncorrelated. The quantification of uncertainty includes the variability related to errors in the temporal interpolation of the repeated bed elevation measurements.
There are gaps apparent in the secondary transport calculations; this occurs at locations where bedform celerity is not properly calculated, because either superimposed bedforms are not present, or are not properly resolved. The sediment transport rates associated with primary dune migration were calculated based on the MBES data of October 9 and October 15, 2018.
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10 of 20 The sediment transport rates, q s , for primary and secondary bedform migration are of the same order of magnitude. Primary transport is largest around y = 40 m and y = 150 m and decreases in the central river section. Transport associated with secondary bedform migration significantly exceeds that by primary dunes in the central section of the river. The transport associated with secondary bedform migration peaks around y = 25 m and y = 100 m, correlating to peaks in dune celerity (at y = 100 m) and η (at y = 25 m).
Figure 7d further shows the calculated bed sediment transport based on Engelund and Hansen (1967). The calculated transport is of the same order of magnitude as the sediment transport associated with bedform migration. Around y = 20 m the transport peaks, which is not reflected in the calculations based on bedform migration. The peak corresponds to a peak in the estimated bed shear stress (Figure 7b).

Discussion
Results of bed elevation measurements at different time-scales have been presented. The results indicate that two distinct scales of bedforms are actively migrating within the same river section. The smaller scale bedforms are superimposed on larger, primary dunes and can persist across the lee sides of the primary dunes. The sediment transport associated with the more rapid migration of the smaller superimposed bedforms is of the same order of magnitude as the transport associated with primary dune migration. Averaged over the width of the river, it even shows to be larger.
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11 of 20  (e) sediment transport associated with primary and secondary bedform migration and the bed sediment transport rate calculated following Engelund and Hansen (1967). The vertical light blue lines indicate the propagated uncertainty related to the sediment transport based on tracking of secondary bedforms.

The Superimposition of Secondary Bedforms on the Stoss and Lee of Primary River Dunes
Previous research offers two distinct explanations for the development of trains of superimposed bedforms on primary river dunes. A first explanation for the superimposition of small dunes or ripples on primary sand-bedded dune lee sides is that the superimposed bedforms are the only active bedform scale, whereas the primary dunes have become inactive and are a relict of prior, high discharge conditions (Allen & Collinson, 1974;Martin & Jerolmack, 2013;Myrow et al., 2018). In this case, the superimposed bedforms erode the primary crests and fill up the dune troughs, flattening the dune and reducing the lee side slope (Martin & Jerolmack, 2013).
At the study site, the discharge has been stable for over 7 months, rendering it unlikely that the larger dunes are a relict from the last discharge peak that faded away at end of January. However, it has been shown that especially after floods with short falling limbs, primary dunes can be in disequilibrium with the prevailing flow conditions for long periods of time (Leary & Ganti, 2020;Myrow et al., 2018). Following a flood, during which large dunes have formed, a large mass redistribution is necessary for large dunes to adapt or disappear, whereas the sediment transport capacity is low. Myrow et al. (2018) proposed a framework to assess whether bedforms in a system are in equilibrium or disequilibrium with the prevailing flow conditions, namely the dimensionless bedform disequilibrium number, T*: where T f is the duration of the prevailing flow and T t is the bedform turnover timescale: with β ≈ 0.55. For T* < 1, bedforms are in disequilibrium with the flow, whereas for T* > 1, bedforms have sufficient time to adapt to the changing flow conditions.
We apply this framework to investigate whether the primary dunes observed in the study area can be considered relicts of prior peak discharge, or, alternatively, are in (near-)equilibrium with the prevailing flow.
We computed T* based on the hydrograph presented in Figure 1, the primary dune properties ( Figure 5) and the sediment transport rates (Figure 7). We computed T t for all datapoints and present the distribution of the obtained values in Figure 8, as well as the hydrograph and the time it takes for T P50 * and T P95 * to reach a value of one (based on the 50th and 95th percentile of T t , respectively). Figure 8 demonstrates that the primary dunes have had ample time to adapt from the peak discharge at the end of January 2018.
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13 of 20 Further considering that the primary dunes have high lee side angles at their central axes and the fact that the primary dunes are actively migrating over the full river width (Figures 6 and 7), the superimposition of secondary bedforms here cannot be explained as cannibalism of relict dunes.
When the superimposition of secondary bedforms is observed under steady flow, they are expected to have developed in an internal boundary layer that re-establishes on the primary dune stoss. The development of the superimposed bedforms is then similar to the development of bedforms on a flat bed (Ashley, 1990;Rubin & McCulloch, 1980;Smith & McLean, 1977). Findings by Venditti et al. (2005b) based on flume experiments support this theory. In their flume experiments, the sand sheets that are superimposed on the stoss of primary dunes are similar to those developing over a flat bed (Venditti et al., 2005a(Venditti et al., , 2005b. The near-bed flow dynamics in this case are similar between primary dunes and a flat bed. Bed elevation profiles shown in Figure 4 reveal that where superimposed bedforms are not present in the primary dune trough, they start to develop some distance downstream of the trough. This might be at a distance where the boundary layer has sufficiently re-established itself, downstream of the flow reattachment point. The data presented herein shows that at some locations the superimposed bedforms dissipate in the lee of primary dunes, which confirms previous observations by Venditti et al. (2005b). At other locations, however, superimposed bedforms travel over the dune lee and trough. Figure 4 indicates an influence of the primary lee slope. To further study this, the primary lee slope and height (trough to crest) were determined (Figure 9, left panel). The properties are calculated for every 10th transect of the bathymetric map. It was visually determined in which areas secondary bedforms were either or not superimposed on the dune lee and trough. Figure 9 indicates that at primary lee slopes larger than approximately 11°, no superimposed bedforms occur. This is a lower value than that found by Galeazzi et al. (2018). They studied compound dunes with and without superimposed bedforms on the lee side and found that for dunes with superimposed bedforms on the lee side the primary lee side angle is generally below 15-18°.
We suggest two mechanisms through which the primary lee side slope controls whether or not superimposed bedforms dissipate when entering the lee side slope. First, for primary lee slopes near or at the angle of repose, superimposed bedforms might simply disintegrate because of sediment avalanching. Second, the flow structure over the primary dunes might play an important role. For dunes with steeper lee side angles, the boundary layer developed over the dune stoss side does not persist over the lee side. Short-scale flow expansion and deceleration over the dune lee leads to flow separation as well as high turbulence production (Kwoll et al., 2016;Naqshband et al., 2014b), causing the disintegration of the secondary bedforms reaching the lee. For low-angle dunes on the other hand, it has been shown that flow separation is not or intermittently present and turbulence production is significantly lower (Kwoll et al., 2016). This can allow superimposed bedforms to pass through the primary troughs.
Prior research indicates that (permanent) flow separation disappears at a lee side slope ranging from 11° to 20° (Bradley et al., 2013;Kwoll et al., 2017;. The slope for which permanent flow separation is present, has been indicated to depend on dune height relative to the depth, where flow separates at lower slopes for dunes with a larger relative dune height . The angle for which flow separates might further depend on the bed grain size distribution and the prevailing flow velocity.
Figure 9a also shows datapoints for which superimposed do not persist over low lee side angles. These datapoints are primarily located in the western sections, where superimposed bedforms are absent in the proximity of steep lee side angles or not well resolved by the MBES measurements over larger areas.

Lateral Variation in Secondary Bedform Properties
The secondary bedforms vary laterally in size and shape. The bedforms are superimposed on large three-dimensional lunate dunes which are steepest over their central axes and have much more gradual lee side slopes toward their sides, which extend from both the west and east sections toward the center (Figures 3-5).
The underlying river bed also varies in elevation with higher elevation at the eastern river bank compared to the western bank ( Figure 1). Figures 1 and 7 further indicate strong lateral variation in the flow velocity and the bed grain size distribution.
We quantified the linear relationships between the secondary bedform height on the one hand and the local depth, primary dune properties and flow properties (  , b u ) on the other. The Pearson correlation coefficients (R) are presented in Figure 9b. Figure 9b indicates a moderate correlation with depth and primary height. There is no significant correlation (p > 0.05) with the primary dune length and the bed shear stress, τ b . Correlation is strongest with the primary lee side slope and depth averaged flow velocity. The bed grain size is not included in the analysis, because within the study area the number of samples is limited.
We first discuss the correlation with primary dune height and lee side slopes, which are inevitably linked. We hypothesize that the correlation with secondary bedform height depends on the flow structure over and downstream of primary dunes, which depends on the relative dune height and lee side angle. Downstream of steep lee side slopes, a stronger turbulent wake develops starting from the flow reattachment point (Bennett & Best, 1995;Kwoll et al., 2016;Venditti & Bennett, 2000). This turbulent wake confines the internal boundary layer on the downstream dune stoss and by that might limit the vertical space in which secondary bedforms can develop. Downstream of low lee side angles, the wake is significantly weaker (Best & Kostaschuk, 2002;Kwoll et al., 2016;. The flow over the crests of low-angle dunes shows more resemblance to flow over a flat bed (Kwoll et al., 2016). Liu et al. (2020) showed the co-evolution of developing sand waves and the near bed flow, indicating the importance of the boundary layer for the development and growth of bedforms.
Second, the secondary bedform height correlates with lateral variations in the depth-averaged flow velocity and local river bed elevation. Though the upstream 4-5 km of the river channel is relatively straight, in general, the curvature of the Waal river affects the river bed morphology, with stable point bars located in inner bends (de Ruijsscher et al., 2020). Within the study area, the eastern section is shallower, whereas the western section is deeper, but decreases in depth in the downstream direction. At the western bank, a point bar develops approximately 0.5 km downstream (de Ruijsscher et al., 2020). The river bed morphology and weak curvature upstream might explain the flow distribution, with higher flow velocities along the western bank.
Previous studies have indicated that an increase in water depth seems more favorable to the superimposition of secondary bedforms (Galeazzi et al., 2018;Reesink et al., 2018), which is different from what we observe here. The flow velocity might affect the overall flow structure over the primary dunes and through that affect the development of superimposed bedforms.
Bed samples further indicate that the bed sediments are coarser in the western section of the river compared to the eastern section ( Figure 1). Wilbers (2004) has attributed this typical pattern in grain size variation to the effect of navigation. Draw down currents generated by passing ships lead to a delivery of fine sediments from eroding beaches to the main channel. This effect is more pronounced for the (south-)eastern part of the river, because the ships sailing upstream, from the port of Rotterdam to Germany, typically stay close to the southeastern river bank and carry a heavier load. The variation in grain size has previously been connected to a lateral variation in dune properties in the river Waal (Wilbers, 2004), where primary river dunes in the southeastern section are lower and shorter (Wilbers & Ten Brinke, 2003). This matches with our observations and might indirectly affect the secondary bedform size, which have been shown to negatively correlate with the primary dune height. Whether the superimposed bedform properties are also directly affected by the bed grain size is not clear.
In a final note regarding the lateral variation of bedform size, the bedforms in the central section have a large relative height compared to the primary dunes and have steep lee side angles ( Figure 5). Here, the superimposed bedforms might be dominant in controlling the local flow structure. It has also been shown before that superimposed bedforms with a relatively large height compared to their host (25-30%) reduce the hosts bedform lee slope (Reesink & Bridge, 2007), creating even more favorable conditions for secondary bedform growth.

Sediment Transport Rates Associated With Secondary and Primary Bedform Migration
The transport rate associated with secondary bedform migration is of the same order of magnitude as that inferred from primary dune migration, and partly even exceeds the latter estimate (Figure 7). Though smaller, the secondary bedform migrate faster than the primary dunes. It has been shown before that small-scale bedforms tend to move faster and large-scale bedforms slower (Coleman & Melville, 1994;Guala et al., 2014Guala et al., , 2020. The observations presented here add field evidence to a suggestion made by Venditti et al. (2005b), who argued that, under conditions where secondary bedforms are superimposed on high-angle dunes, sediment transport rates are likely invariant to bedform scale. In addition, when comparing migration rates of smaller and larger secondary bedforms, the average bedform elevation, η , inversely correlates to migration celerity, V c (R = − 0.68, p < 0.05).
The bed sediment transport was also estimated following Engelund and Hansen (1967) (Figure 7d). Since the ratio of the bed shear velocity to the particle fall velocity is low (Figure 7b), the bed sediments are predominantly transported as bedload, enabling comparison with the estimates based on dune tracking. The calculated transport is of the same order of magnitude as the sediment transport rates calculated based on dune tracking. The values are comparable over the full river width, except for a peak around y = 20 m. The peak in the calculated transport results from a peak in the bed shear stress at that location. This peak in bed shear stress does not correspond to any peak in sediment transport associated with bedform migration. Though the transport based on Engelund and Hansen (1967) is similar to that based on dune tracking and thus supports the validity of the used dune tracking method, uncertainty in the calculated transport limits a precise, quantitative comparison.
To assess what the overall impact is of multiscale bedform migration on the total sediment transport inferred from tracking of primary dunes only, the question arises to what extent the transport rate associated with secondary bedform migration contributes to primary dune migration. When superimposed bedforms travel over the stoss side and dissipate across the primary dune lee side, they are fully accounted for in bedload transport rates inferred from primary dune migration tracking. In the other extreme, when superimposed bedforms uninterruptedly travel through the primary dune troughs, transport rates inferred from primary dune tracking are an underestimation. Here, at least a subtle interaction is likely to occur. Secondary bedforms may cause the migration of a primary dune by causing net erosion of the primary stoss, which is when the secondary bedforms grow, and causing net deposit of sediment at the lee side, which is when the secondary bedforms shrink (Figure 10). The sediment transport associated with secondary bedform migration then varies depending on the position on the primary dune. To further investigate this, high-resolution MBES measurements are needed at different locations on the primary dune, such that the spatial pattern in sediment transport associated with secondary bedform migration can be studied.
Even though the interaction between bedform scales is not clear and it is unknown what the additional contribution of secondary bedforms is to the total sediment transport, the presented results demonstrate that sediment transport calculations based on tracking of primary dunes only can lead to significant underestimations. The temporal resolution of repeated bed elevation scans chosen for dune tracking confines the bedform scale for which the associated transport rate can be calculated. Before dune tracking can be applied in river sections with multiscale bedform migration, it needs to be understood how separate scales interact, and by that, to what extent transport rates associated with distinct scales overlap or add up.

Conclusions
The aim of this study is to investigate the morphodynamics of multiscale fluvial bedforms and the associated implications for sediment transport rates. For this purpose, bed elevation surveys with a high spatiotemporal resolution were conducted in the Dutch river Waal, and compared to MBES data acquired on a lower temporal resolution. Results indicate that both primary dunes and superimposed, secondary bedforms actively migrate over the full river width. The bedload sediment transport associated with the migration of the smaller, superimposed bedforms equals or even exceeds the bedload transport rates associated with primary bedform migration. This is because the trains of smaller bedforms migrate with a much higher celerity than primary dunes.
Previous research suggested that secondary bedforms fully dissipate on the primary lee slopes and thus contribute to migration of the primary dunes. Our results demonstrate that secondary bedforms can travel over the lee side of primary dunes. One of the controlling factors determining whether superimposed bedforms dissipate at the primary lee side or not, is the primary dune morphology, specifically the primary lee side angle, where on lee side angles below 11° secondary bedforms can persist. ZOMER ET AL. Low-angle river dune with secondary bedforms superimposed.
Migrating secondary bedforms grow on the primary stoss and shrink on the lee.
As a result, the primary river dune migrates downstream.

Time
Interaction of dune dynamics at multiple scales determines the quality of bedload sediment transport estimations inferred from tracking of primary dunes. It remains to be discovered what conditions enable the development of superimposed bedforms within a fluvial system, and how well-developed superimposed bedforms in turn affect flow dynamics and the overall sediment transport in a system.