The bed topography data analyzed in this paper were collected in a 30 km long reach of the Elbe River (river kilometers Ekm 503.0–533.1) by the German Federal Waterways and Shipping Administration (WSV) on behalf of the German Federal Waterways Engineering and Research Institute (BAW). The reach, shown in Figure 2, is characterized by four successive bends (Ekm 503–507.5, Ekm 507.5–510, Ekm 510–517, and Ekm 517–522, respectively) which are followed by a nearly straight section (Ekm 523–533). The riverbed is composed of sand with a mean grain size of d_m ≈ 1.2 mm, a coefficient of uniformity of d₆₀ / d₁₀ = 2.16, and has an average slope of S = 0.00012. The width of the main channel between the banks is fairly constant along the reach, but the active width of the riverbed, defined here as the distance between opposite groin heads, changes. The river section between Ekm 508–521, which is also known as the “Elbe-Reststrecke,” was historically not trained to the same extent as the neighboring sections being the reason why the active width of this reach corresponds to about 220 m. In contrast, the active width of the reaches upstream and downstream of Ekm 508 and 521 corresponds to approximately 175 and 180 m, respectively (cf. Figure 2).

Alternate bars are known to form at about Ekm 512 and to fade again downstream of Ekm 521. This offers the unique opportunity to test the applicability of bar predictors and to investigate bar characteristics considering different geometrical boundary conditions. For this purpose, the reach has been subdivided into six sections that are depicted in Figure 2. Three 5 km long sections (AB1, AB2 and AB3; marked by filled green frames in Figure 2) that are characterized by alternate bars are situated in and directly downstream of the Reststrecke and were used to characterize bars and to investigate the impact of the river course on bar characteristics. These sections range from Ekm 512–517 (AB1), Ekm 517–522 (AB2) and Ekm 523–528 (AB3) (the reason for the gap between Ekm 522 and 523 is explained below). Sections AB1 and AB2 cover a bend to the left and to the right, respectively, while section AB3 covers a straight section. We note that the bars propagate further downstream but fade with increasing distance to the Reststrecke, being the reason that they can only be identified occasionally in section NB3 (Ekm 528–533). The other two sections (NB1 and NB2), in which alternate bars were absent, cover the ranges from Ekm 503–508 and Ekm 508–512, respectively. The sections NB1, NB2, and NB3 were used to test the applicability of bar predictors (cf. Section 2.3).

The bed topography data used in this study were surveyed annually during the period from 2007 to 2016 by the WSV with the sounding vessel Kugelbake. The vessel was equipped with an Atlas Fansweep 20/200 multibeam echosounder with a measurement frequency of 200 kHz and a Trimble GPS-system using the federal Real Time Kinetics (RTK) service “SAPOS-HEPS.” The sections where surveyed in one or two parts on different dates which are given in the supplementary data section. The data were combined and post-processed by the WSV and made available for the present study in gridded form with a spatial resolution of 1 x 1 m (spatial reference system EPSG 31468). It must be noted that the echo sounder data exhibited gaps in 2013 (AB1 and AB2) and 2012 (AB3). These gaps were the reason for defining the sections AB2 and AB3 with a distance of 1 km (c.f. Figure 2). Moreover, the survey of AB1 in 2011 was omitted from further analyses as the time lag between the partial surveys was larger than 14 days which might bias the results.

Discharge and water levels were available from the records of the nearby gauging station “Neu Darchau” situated downstream at Ekm 536 (Figure 2) on the left channel side (GK coordinates E:4425900 N:5900611). According to DGJ (2014), the river has a drainage area of 131,950 km² at the gauging station. The mean low discharge, mean discharge, and mean flood discharge correspond to MNQ = 276 m3/s, MQ =712 m3/s, and MHQ = 1,960 m3/s, respectively. The water stages W of MNW = 1.24 m, MW = 2.80 m and MHW = 5.78 m are given in relation to the gauge zero point (5.68 m above normal height null NHN) that corresponds to the mean bed elevation at the gauging site. Bankfull conditions in the investigated reach correspond to a water depth of ~5–6 m and prevail at discharges of ~2MQ–MHQ due to the locally varying floodplain elevations. The discharge and water levels at the gauging station are shown in Figure 3 for the period between 2006 and 2016 together with the dates of the echo sounder surveys for the sections AB1–AB3. We note that three major flood events, where MHQ was exceeded by approximately a factor of two, occurred in the investigated period in the years 2006, 2011 and 2013.

Most of the existing methods to analyze the geometry of alternate bars have been developed to characterize gravel bars in straight waterways or laboratory flumes. They require the characterization of individual bars which can be achieved by visually identifying every single bar complex from photographs and by local measurements of the bed elevation (e.g., Tubino, 1991; Garcia and Niño, 1993) or by determining an averaged bar geometry in the investigated reach (e.g., Redolfi et al., 2020). These methods could not directly be applied in the present study, since the investigated reach is characterized by four successive bends (Figure 2) hampering the spatial analysis of bed elevations, in general, and in particular of alternate bars due the channel curvature. Moreover, the investigated reach of the Elbe River is characterized by a sand bed, and bars in sand bed rivers are often superimposed by smaller bedforms (e.g., Colombini and Stocchino, 2012; Rodrigues et al., 2015; Le Guern et al., 2019). The resulting rather complex bed morphology therefore complicates the unambiguous identification of bar characteristics compared to gravel bars. A further approach found in the literature to characterize and track bars is the use of satellite imagery (e.g., Adami et al., 2016). However, this approach requires the visibility of the bars which, in case of trained sand bed rivers such as the Elbe River, is not necessarily given as the bars are mostly submerged and the turbidity does not allow for a visual identification of the bars. Therefore, we developed an approach to estimate bar characteristics from statistical parameters derived from the available echo-sounder data which is described in the following.

Statistical methods for the analysis of bar characteristics and bar dynamics can be applied most easily using equidistant and rasterized bed elevation data. Using such data in straight sections, it is straightforward to apply more complex approaches to describe the bed texture using, e.g., 2D-structure functions (e.g., Aberle et al., 2010; Qin et al., 2019). However, the application of such methods to describe the bed structure in curved river sections is sophisticated, if not even impossible, if the data are not accordingly transformed to account for skew coordinates. Therefore, the bed elevation data was remapped to a curvilinear coordinate system whose principal axis follows the river centerline (Figure 4). The centerline coordinates and the river kilometers (Ekm) were provided by the WSV, and cross-sectional profiles perpendicular to the centerline were constructed using a spacing of Δx_c = 10 m. This spacing was arbitrarily chosen as the bar wavelength was expected to be more than 1,000 m so that the impact of small scale bedforms was assumed to be negligible. The cross-sectional profiles were constructed using a spacing of Δy = 1 m (cf. Figure 4) with the origin of the y-axis at the centerline. The defined cross-sectional profile points do not necessarily coincide with the gridded data points so that the elevation at the profile points were determined as follows. Beginning at the centerline (y(x_c) = 0), a circular area with a radius of 1.0 m was defined and the elevation of all gridded data points falling into this circle were averaged and assigned as corresponding bed elevation. This procedure was subsequently repeated for the further cross-sectional profile points using Δy = 1 m (c.f. Figure 4). The cross-sectional profile width for the further analysis was set to 100 m and covered hence a distance of 50 m to the left and right of the centerline. This width was chosen to limit the data to the riverbed unimpacted by groins and hence to exclude the associated scours at the groin heads as good as possible. The lateral interval of Δy = 1 m allowed for the construction of longitudinal profiles parallel to the centerline. For the 5 km long sections, this resulted in bed elevation data sets for each available scan consisting of 500 cross-sectional profiles (Δx_c = 10 m) with 101 data points. Defining an x_c-y coordinate system (note again that y is oriented differently for each profile but defines the distance to the centerline), it became possible to project the curved topography using a straight reference system enabling a further analysis of the data with respect to the centerline distance. Fitting a plane to the projected elevation data in each section, the longitudinal and transverse trend was removed and the reach averaged mean elevation was set to z_m = 0.

For the subsequent analysis, we hypothesized that the alternate bars are most pronounced in the region defined by the 20 outer left and right longitudinal profiles. Since the investigated sections covered only a single river bend with rather mild curvature, or were straight, the longitudinal spacing between the points in the 20 outer right and left longitudinal profiles was assumed to be constant as these distances were much shorter than the bar lengths. We are aware of the fact that the profile points were, strictly speaking, not equidistantly spaced as the distance between successive points along each longitudinal profile depends on river curvature and centerline distance (cf. Figure 4). However, the main scope of our analysis is on the comparison of bar characteristics of the same (or geometrically similar) sections and not to develop predicting approaches for bar geometry. Thus, we assume that the resulting error in our analysis, which makes use of a reference system based on the channel centerline, is small. An advantage of the performed data transformation is that it enables the estimation of geometrical bar properties such as wavelength and height, and to investigate bar celerity on either channel side to reveal possible asymmetries due to the channel curvature. Moreover, it becomes straightforward to estimate the transverse bed slope (TS) of each cross-sectional profile by a linear regression. As alternate bars feature a regular pattern with a bank on one channel side and a scour on the other one (c.f. Section 1) the fluctuations of the TS mirror both the wavelength of the bar units (one bar on both channel sides) and the bar height.

The bar wavelength λ_B was estimated in a first step by calculating the power spectra of the 20 outer left and right longitudinal profiles, respectively, using a Hanning window. However, the spectra were difficult to analyze in more detail due to the large bar wavelengths and limited section length. Therefore, we decided to apply further methods to quantify the wavelength. In detail, we determined the averaged 1D-autocorrelation function of the outer right and left longitudinal profiles as well as the autocorrelation function of the TS. Only autocorrelation functions that showed a periodic pattern, which is characteristic for periodic input data like alternate bars (e.g., Bendat and Piersol, 2000), were further analyzed. The location of the first peak in the autocorrelation function for a spatial lag > 0 was used to estimate the bar wavelength.

The bar heights on the left and right side of the channel were characterized using the standard deviation (SD) of the longitudinal outer 20 right and left elevation profiles as a surrogate measure. We refrained from calculating the absolute bar height defined as the distance between the lowest and the highest elevation of one bar complex (e.g., Ikeda, 1984; Knaapen et al., 2001; Adami et al., 2016) as the identification of a single bar is rather subjective and can be easily biased by single extreme values (e.g., Redolfi et al., 2020). It is worth noting that restricting the analysis to the 100 m wide riverbed may underestimate bar height, as bars could be even higher in the omitted areas. However, since these areas are also affected by groin induced scours, the identification of the lowest elevation is associated with uncertainties so that an unambiguous quantification of bar heights would have been difficult. As mentioned before, the scope of our analysis was not the detailed quantification of bar characteristics but to determine the existence of bars and the dependency of bar geometry and dynamics on curvature and hydrological boundary conditions. Such insights may be gained using the outlined framework for the analysis of the data.

Bar celerity c was approximated by calculating the average 1D cross-correlation function of the 20 outer right and left longitudinal profiles, respectively, and the TS between successive years. The migration speed of the bars was evaluated by finding the spatial lag that resulted in the highest correlation coefficient and by dividing this value by the temporal lag between the subsequent datasets.

In order to link bar characteristics with hydraulic conditions, we used the water stage W at the gauging station as a surrogate measure for the discharge. This was due to the fact that the Elbe River exhibits large floodplains that convey an unknown part of the total discharge during overbank flow. This is partly accounted for by the water stage which increases less than the discharge once bank full conditions are exceeded in channels with a compound geometry (c.f. WMO, 2010). Keeping in mind the response time of large-scale morphological structures to changing hydrological and hydraulic boundary conditions, we used the average value of the water stage (termed W₆ in the following) that was recorded at the gauging station during 6 weeks before the particular echo sounder survey to characterize the hydraulic boundary conditions. The duration a period of 6 weeks was chosen arbitrarily as the time scale for the morphodynamical adaption of the bars to discharge and hence water depth as well as β are unclear. Using a period of 6 weeks, we assumed that the duration for the bars adapting to the discharge is larger than the time lag needed for the adaption of dunes, which has been reported to be several days (e.g., Allen, 1976; Brinke et al., 1999). Finally, bar celerity was correlated with the average water stage (hereafter referred to as W_m) measured in the period between the subsequent echo sounder surveys.

The available data were used to test the applicability of the approaches of Crosato and Mosselman (2009), Ahmari and Da Silva (2011) and Redolfi (2021) as bar predictors for the investigated sand bed reach. To allow a comparison between the results, the latter two approaches were rearranged to calculate β_c. The approach of Ahmari and Da Silva (2011) can be formulated as follows:

where H is the section averaged water depth and d is the characteristic grain size that was assumed to correspond to d_m. The approach of Crosato and Mosselman (2009) was rearranged to:

where b characterizes the degree of non-linearity of the sediment transport formula (-) and was assumed to b = 4 following the suggestion for sand bed rivers by Crosato and Mosselman (2009). C is the Chézy coefficient (m^0.5/s), u_m is the section averaged flow velocity (m/s), Δ is the relative submerged density (-) assumed to be Δ = 1.65 and d is the characteristic grain size (m). We note that the bar mode for critical conditions was assumed to be m = 0.5 and the equation was multiplied by 0.5 compared to the original formula as Crosato and Mosselman (2009) defined β = B/H instead of β = 0.5B/H.

The approaches were applied for discharges that approximately represent bankfull conditions (2MQ ≤ Q ≤ MHQ c.f. Section 2.1) as these are often assumed to represent the formative discharge (c.f. Section 1). We refrained from applying the recent approach of Carlin et al. (2021) that considers the whole discharge spectrum to circumvent the assumption of a formative discharge due to the following reasons: i) it is not a simple predictor, ii) limited hydrological data impeded the calculation of a representative probability density function of the discharge, iii) the discharge fractions routed via the floodplains for overbank flow are unknown (c.f. Section 2.2), and iv) the river transports an increasing amount of bed material in suspension with higher discharges, especially at discharges exceeding bankfull conditions, i.e., a discharge that is considered in the approach but a transport mechanism which is not yet covered by it.

The application of the approaches revealed a critical issue regarding the river width B. This parameter is clearly defined in channels with a rectangular cross section like hydraulic flumes, where it is also independent from the discharge. However, this is not necessarily the case in natural rivers that often exhibit gently sloped banks complicating the definition of the correct width and hence obscuring the definition of the width to depth parameter β. Moreover, groins restrict the active width of the riverbed, i.e., the fraction of the bed contributing to bed load transport, so that it differs from the river width. This is why we defined B as the average width of the active riverbed in the investigated section (c.f. Section 2.1), as this definition is more straightforward and independent of the bank geometry and discharge (c.f. Table 1).

For the application of the approaches, we finally assumed a rectangular cross section of the main channel with the width of the active bed B and a constant mean grain size of 1.2 mm. The active bed width B, the discharges and corresponding water depths H, as well as the water surface slope S_w were available in a longitudinal resolution of 100 m and were taken from the Elbe River data base “HyMoInfo” provided by BAW. The values were respectively averaged over the six investigated subsections. An overview over the discharge dependent value-ranges as well as the active bed width B is provided in Table 1.

The visual inspection of the bed elevation data showed, in agreement with previous observation of the WSV, the formation of bars just downstream of Ekm 512 in section AB1. The existence of bars can be validated by investigating the cross-sectional distribution of the longitudinally averaged bed elevations which, in case of existence of alternate bars, should be characterized by a bell-shaped distribution (Fujita and Muramoto, 1985). Such a distribution can be identified for the sections AB1, AB2, and AB3 (using the mean value of all available scans) while it is absent or not as pronounced for sections NB1, NB2, and NB3 (Figure 6A). Moreover, the distributions indicate that alternate bars were most distinct in section AB2. The longitudinal distribution of the transverse slope (TS) shows a periodic behavior in AB1, AB2, and AB3, which is also a strong indicator for the presence of bars (Figure 6B). The small peaks that can be observed in this figure in the AB-sections gradually shift downstream from year to year indicating downstream migration of the bars. Moreover, the distinctness of bars is shown by the temporal variance of the TS (right axis in Figure 6B; moving average over 50 values along the river axis). The increasing variance in AB1 indicates the formation and growth of bars in this section, whereas the highest and nearly constant values in AB2 suggest that bars are quasi fully developed. The clear peak in variance at Ekm 521 correlates with the end of the river bend in this section and similar peaks can be seen at Ekm 509 and Ekm 515 i.e., locations also located at the end of a river bend. In AB3 the temporal variance of TS decreases and hence indicates the beginning decay of the bars. In contrast, only stationary point bars could be identified in the sections NB1 and NB2, i.e. in the sections upstream of AB1 (Ekm 503–512). In these sections, the longitudinally averaged distribution of the bed elevations does not show a bell-shape and the longitudinal distribution of the TS does not show a periodic behavior and is characterized by a low temporal variance. The absence of bars in section NB2 is an interesting fact, as the river bed already widens at the beginning of NB2 (Ekm 508) and has hence the same width as AB1 where bars are present. We therefore hypothesize that the absence of bars can be attributed to the river course as this is the major difference between the river sections. In fact, the river widening at Ekm 508 is located in a 2 km long sharp left bend (radius ≈ 800 m) directly followed by a right bend (c.f. Section 2) that likely impede the initial formation of bars. In contrast river curvature in AB1 and AB2 is smaller so free bars can coexist with local point bars (c.f. Section 1). In section NB3, located downstream of AB3 (Ekm 528), bars could only be identified in certain years using the aforementioned approaches. In the other years, the bed in this section was dominated by smaller bedforms and stationary bars with a wavelength larger than 2,000 m, which can be inferred from Figure 6B. This observation can be linked to a successive dampening of the alternate bars due to narrowing width of the active bed by the larger groins.

A further interesting aspect is that a trend between the origin of bar formation and the W₆-parameter was identified (Figure 7). The origin of bar formation was determined by identifying the first apparent peak in the transverse slope distribution, and this peak was shifted up to 3.5 km downstream within AB1 with increasing W₆ parameter, i.e. increasing discharge. We infer that this behavior can be attributed to the flow field upstream of AB1 that is governed by the bends and associated upstream point bars situated at Ekm 509 and 511 (c.f. Figure 2). Despite being fixed, the geometry of the point bars changed throughout the years (not shown here).

Figures 8A–C present the bar wavelength λ_B determined from the autocorrelation analysis of the bed profiles and TS, respectively, as a function of time for sections AB1–AB3. These figures reveal both different patterns and orders of magnitude. In AB1, λ_B was fairly constant over the years (λ_B ≈1,000 m) while in AB2 it decreased from λ_B ≈1,300 m in 2007 to λ_B ≈1,000 m in 2011 and increased to λ_B ≈1,500 m in 2016. Bars in AB3 showed the opposite trend compared to AB2; λ_B corresponded to about 1,000 m in 2007, increased to λ_B ≈1,250 m in 2011 before it decreased to λ_B ≈1,000 m in 2016. Note that the sections were situated next to each other and were surveyed within a short time frame in nearly every year. Thus, these figures indicate at the effect of different boundary conditions on bar characteristics and dynamics and fits to the variance pattern of the TS discussed before (c.f. Section 3.1).

The mean standard deviation of the analyzed outer right and left bed elevation profiles, which serves as surrogate for relative bar height, is presented in Figures 8D–F for AB1, AB2 and AB3, respectively. In contrast to the wavelength, the standard deviation is nearly constant over the years but differs in between the investigated sections and channel sides. On average, the standard deviations of the right and left outer profiles in AB1 corresponded to 0.63 m and 0.39 m, respectively, whereas in AB2 these values were 0.60 and 0.82 m, respectively. Interestingly, both λ_B and SD were larger on the right channel side in AB1 and on the left channel side in AB2. We note also that the underlying autocorrelation functions used to estimate λ_B were more periodic on the right and left in AB1 and AB2, respectively (not shown here). As AB1 and AB2 cover a left and a right turn of the river, this is a strong indicator that bars on the outer side of the bend tend to be longer and relatively higher than their counterparts on the inner bank. Hence, this shows that the bar characteristics are affected by channel curvature, and this can be associated with the interaction of the free bars with the point bar in the inner bend, which has been reported to slow down free bars depending on channel curvature (c.f. Section 1). The findings in the nearly straight section AB3 substantiate these observations, as the differences in λ_B between both sides were minor and the SD of both sides were equal (0.61 m). Moreover, the analysis of the bed elevation data in AB1 and AB2 revealed that bars in the inner bend consistently changed their shape while moving through the bend. Bars approaching the bend exhibited a migration front (lee side) inclined toward the river centerline, whereas this inclination changed throughout the bend. Bars downstream of the bend apex had fronts inclined toward the inner bank. This effect can be seen in Figure 5A showing that in 2007 the migration front of the first bar on the inner bank was inclined toward the river bank, and this inclination was reversed after its migration through the bend in 2010. This change in geometry can again be associated with the interaction of the migrating bar with the point bar that locally led to a higher transverse slope (c.f. Struiksma et al., 1985) successively diverting bed load transport toward the outer bend (c.f. Section 1).

The bar celerity is presented in Figure 9 for the investigated sections and the observation period. In analogy to the bar wavelength, bar celerity changed over the years and exhibited different temporal trends in the investigated sections. In AB1, bar celerity decreased over the years from ~430 m/year in 2007–2008 to 300 m/year in the period 2015–2016. In AB2, bar celerity increased almost linearly from 330 m/year in 2007–2008 to 600 m/year in 2010–2011 and decreased to a minimum of 241 m/year in 2015–2016, hence showing the opposite trend to λ_B in this section. Section AB3 shows an overall trend of decreasing bar celerity over the years with a celerity of ~400 m/year in 2007–2008 and 340 m/year in 2015–2016. However, in 2010–2011 and 2013–2014 bar velocities where much higher (~500 m/year). It is worth mentioning that we observed the general trend that bars with smaller celerity exhibited longer wavelengths and vice versa (not shown here), an observation that fits to investigations of Crosato et al. (2011) and Jang and Shimizu (2005).

As shown before, bar characteristics changed over years, and this can be linked to the hydraulic boundary conditions in the period before the surveys of the riverbed. This can be investigated by Figure 10 showing λ_B and SD plotted as a function of the average water stage W₆.

Figures 10A, B show that λ_B in AB1 is almost independent of the hydraulic conditions, whereas, in AB2, λ_B decreases with increasing water depth. It is interesting to note that this decrease is more distinct for the bars situated on the outer bend i.e., the left side in AB2. A similar trend can be observed for the standard deviation which decreases with increasing water levels on the left side in AB2 (Figure 10E) and also on the right side in AB1 (Figure 10D), whereas the standard deviation on respective other side remains nearly constant. Hence, this indicates that the geometry of bars in the outer bend is more affected by changes in the discharge than the ones in the inner bend. Moreover, the general trend in AB2 that increasing water stages (discharges) lead to decreasing bar dimensions substantiates findings of Redolfi et al. (2020) who found similar relations in their flume experiments for fully developed bars. Interestingly, neither the bar wavelength nor the bar height seemed to correlate with the water level in section AB3 and did not show differences between the channel sides. This indicates once more that bars in AB2 were quasi fully developed whereas this was not the case in AB1 and AB3, where bars grew and were successively damped, respectively.

Figure 11 shows the annual bar migration plotted against the average water stage W_m and reveals that bar celerity exhibited also a dependency on the hydrological conditions. In all investigated sections, bar migration rates increased with increasing W_m, which can be expected due to increased bed shear stress and hence sediment transport. Bar migration rates were fairly similar for the same W_m values indicating that bar celerity was mainly driven by the hydrological boundary conditions and was hence less affected by the geometrical characteristics. This is partly contradictory to the findings of Jang and Shimizu (2005) who reported a decrease in bar celerity with increasing channel width, as such a trend cannot be observed in the wider section AB3 compared to AB2. It is important to note that Jang and Shimizu (2005) focused on channel widening due to lateral erosion whereas the bed width of Elbe River is restricted by groins. Hence this may indicate on the important of such instream structures on bar dynamics. Moreover, Figure 11 shows that bar celerity correlated well with averaged hydrological conditions indicating that discharge variations only had a minor impact on average bar dynamics. This fits to the findings that short flood events do not affect bar characteristics (Carlin et al., 2021).

The bar predictors defined by equations 1 and 2 as well as the approach of Redolfi (2021) were applied to the six sections (c.f. Table 1), and the results are shown in Table 2 in terms of the width to depth parameter β and the threshold values β_c calculated by the particular approaches. Similar β_c-values were obtained for both investigated discharges and the different river sections when applying the approach of Redolfi (2021). However, β varies with the discharge and complicates the interpretation of the results as for 2MQ the approach forecasts β > β_c, and hence the occurrence of bars in all sections, whereas for MHQ the prediction nearly mirrors the situation apparent from the field observations (c.f. Section 3.1). Assuming MHQ for bankfull conditions, the approach predicts the formation of bars in the sections NB2, AB1, and AB2 (Ekm 508-522), where β = 16.4–18.1 and β_c = 15–15.3. This fits with the field observations except for NB2 (Ekm 508–512), where bars are absent, although the active bed width of B = 222 m is comparable to the active bed width in AB1. This can be attributed to the effect of the river curvature (c.f. Section 3.1). In section NB1 (upstream of the Reststrecke), β = 12.9 is below the calculated threshold β_c = 14.8 and, hence the approach suggests the absence of bars which can be confirmed by the field observations. Finally, the sections AB3 and NB3 (downstream of the Reststrecke) are characterized by β = 13.7–13.9, i.e. values lower than β_c that indicate the absence of bars. However, when considering that bars perturbate into theses sections from upstream the approach mirrors the field observation as bars slowly decay in these sections (c.f. Section 3.1).

The criterium of Ahmari and Da Silva (2011) results in a constant threshold value of β_c = 10 for all sections and discharges as H/d was always larger than 130 (see Equation 1). The minimum width to depth parameter in all investigated sections was β_min = 12.9, thus the approach suggests the formation of bars in all sections and for both discharges. This does not fit to the observations and is most likely caused by the empirical nature of the approach and the limited data basis for conditions with high relative submergences.

The critical width to depth ratios β_c of the rearranged Crosato and Mosselman (2009) approach (calculated by assuming m = 0.5 as the minimum value for bar-formation) differed between the investigated sections (Table 2). Similar to the Redolfi-approach, this approach forecasts bars in all sections assuming 2MQ as bankfull discharge, whereas the results are more nuanced for MHQ. For MHQ, the approach yields β_c < β for the sections NB1-2 and AB1-2 (Ekm 503–522) and hence forecasts the existence of alternate bars. While alternate bars could be identified in sections AB1 and AB2 (Ekm 512–522), the predictions for NB1 and NB2 (Ekm 503–5,012) could not be confirmed by the field observations. For the latter, this might be related to the bends situated in this area, as for the Redolfi-approach (cf. Section 3.1). However, bar formation is also suggested for NB1 which shows the smallest β values of all investigated sections. This can be linked to the different water-surface slopes of the sections which are highest in NB1 and indicates that the approach of Crosato and Mosselman is more sensitive to this parameter than the Redolfi-approach. Downstream of the Reststrecke in sections AB3 and NB3, β_c values corresponded to β_c = 14.9–15.2. They were hence similar to the ones of the Redolfi approach and fit the field data considering the decay of alternate bars in these sections.

In general, the bar predictors of Redolfi (2021) and Crosato and Mosselman (2009) can be applied to forecast bars in sand bed rivers like Elbe River. However, such complex field conditions bare the difficulty to define adequate values for river width and bankfull discharge. Moreover, when applying these approaches to river sections, one has to keep in mind that boundary conditions like river curvature or the migration of bars from upstream are not considered in the approaches, so the prediction can differ from the situation in the field.