Novel hypothesis on the occurrence of sandbars Open Access

The morphologies of river channels can be classified into three categories: braiding, meandering, and straight.¹ In all three channel categories, geometric shapes called sandbars can spontaneously form on a riverbed. Sandbars are three-dimensional (3D) shapes with alternating deposition and scouring in longitudinal and transverse directions. Considering this spontaneous periodic shape, the sandbar formation can be attributed to self-assembly.² Knowing how and why sandbars form is a scientifically important subject that contributes to the elucidation of self-assembly in general.

Understanding the mechanism of the sandbar formation has also engineering value as follows. A bar wavelength is several times larger than the channel width, and the bar height is scaled to the flow depth, causing the flow to meander. During floods, flow deflection becomes more pronounced, resulting in large-scale channel fluctuations. In a worst-case scenario, flowing water can breach embankments, thus flooding cities and causing extensive damage. For this reason, there is an urgent need to understand how sandbars spontaneously form.

Field observations of rivers have long been conducted to understand the mechanism of the sandbar formation.^1,3 Recent studies have clarified various sandbar properties by measuring bedforms with multibeam sonar and flow velocities with acoustic Doppler current profilers (ADCP),⁴ as well as elucidating sandbars responses from several years of observations.^5–7 Earlier this year, Branß et al.⁸ proposed a method for estimating sandbar characteristics based on statistically derived geometric parameters. However, river measurements are limited by spatial- and temporal-scale constraints, and it is difficult to understand sandbar behavior under a wide variety of conditions such as hydraulic parameters, sand supply, and channel geometry. Previous studies have thus used flume experiments, theoretical analysis, and numerical analysis as alternative methods.

Studies using laboratory flumes have proposed equations for estimating basic physical quantities, such as bar wavelength, height, and celerity.^9–11 Various sandbar responses were also investigated, including the flow rate as a variable,¹² sand supply as a variable,^13,14 and long-time experiments.¹⁵ Moreover, the origin of sandbars has been investigated. A current finding showed that one of the origins of sandbars is multimodal 3D bedforms, which obliquely intersect a channel.¹⁶ They are also called rhomboid bars,¹⁷ rhomboid beach patterns,¹⁸ and oblique dunes,¹⁹ but in this paper, we refer to these 3D bedforms as “rhomboid bars” following Ikeda.¹⁷ Rhomboid bars may help initiate sandbars because they occur early in the process of the sandbar formation and have a 3D shape. However, rhomboid bars have relatively small wavelengths and heights, which fact makes them difficult to observe. A unified understanding of the mechanism by which rhomboid bars form from flatbeds has not yet been established,¹⁸ and our understanding of the sandbar initiation is poor.

Theoretical methods using the linear stability analysis have shown that conditions for the occurrence of sandbars depend on the width-to-depth ratio of a river channel.^20–25 This analysis also provides that the width-to-depth ratio is an indicator of whether the river channel is braiding or meandering.²⁶ In the analysis, the following sequence of processes is hypothesized for the occurrence of sandbars.²² First, a small perturbation is introduced to the initial bed and hydraulic parameters. If the width-to-depth ratio β exceeds a critical value $β c$ here, the bottom perturbation changes the flow pattern and sediment transport. Then, the bottom perturbation develops through interaction with the flow. At this time, a mechanism that selects bottom wavenumbers engages, and sandbars are formed. However, there are currently no measurement data to substantiate this process, and how sandbars actually occur has not been explained.

Numerical studies have shown that placing obstacles such as humps or introducing small random perturbations to the initial bed are effective at reproducing the occurrence and development of sandbars.^27–29 It is also known that the initial bedforms in numerical analyses are similar to the rhomboid bars seen in flume experiments.³⁰ However, it remains to be proven how well the numerical results correspond to the actual physical mechanisms and formation processes, because there is no measurement method that can be compared with the processes described by numerical analyses.

Theoretical and numerical analyses assume that bottom instability causes sandbars. Conversely, a recent study³¹ measured both the water surface and the bottom surface during the sandbar formation process and then demonstrated that the water surface contributes more to the determination of the flow depth distribution than the bottom surface in the initial phase of the sandbar formation. If the water surface fails to maintain a flat condition given initially before bedforms occur, the bottom instability may be caused by the water surface.

This study revisited the conventional hypothesis on the occurrence of sandbars. In Sec. II, we measured the process of the sandbar formation at high density and high frequency in a laboratory flume and quantified the wavenumbers of the water surface and bottom surface. In Sec. III, we conducted flume experiments on a fixed bed and showed that the standing waves existed on the water surface before the occurrence of small bedforms. In Sec. IV, we discussed what generates the standing waves. In this section, we also proposed a novel hypothesis on the occurrence of sandbars. Finally, Sec. V provides a summary of this paper.

The channel used for the experiment is shown in Fig. 1. The flume was a straight rectangular section made of fiber reinforced plastic with a channel width B = 45 cm and a total length of 12 m. The length of the moving bed section was 10 m. The channel was constructed with 5 cm high weirs at the upstream and downstream ends of the moving bed section. In this experiment, the channel was given a flatbed of uniformly spread silica sand with an average grain size d = 0.76 mm as the initial condition. To minimize artificial influences on the occurrence of bedforms, an erosion zone of 2 m in length was placed at the upstream end of the channel. These are the so-called free bars conditions.^32,33 Water supply was provided from the upstream end, the flow rate Q was assumed to be steady, and an electromagnetic flow meter was used to confirm that the steady state was maintained.

Figures 2 and 3 show the plane view of the bed level and water level measured by ST in cases 1 and 2, respectively. The channel width is shown only at 42 cm due to the measurement limitations of ST near the sidewalls. As described above, measurements were conducted at 1-min intervals, but the intervals shown were adjusted for space constraints. Note that the range of the bed level varies with time due to coloring.

The focus is on phenomena before and immediately after the water is supplied. Figure 4 displays a limited section because the initial phenomenon is relatively small. Figure 4 (left) shows that the initial bottom shape is flat. However, Fig. 4 (middle) shows that bottom undulations already occur 1 min from the beginning of the experiments in both cases. The wavelength of the bottom undulations is approximately 5 cm. At the same time, wave trains are generated on the water surface. The shape of the wave trains [Fig. 4 (right)] is similar to that of the bottom undulations, and the wavelength of the water surface is also 5 cm. As seen in the legend, the wave height of the water surface is larger than that of the bedforms. This is also evident from the results of the 2D wavenumber analysis described below.

After 2 min from the beginning of the experiments in Figs. 2 and 3, wave heights gradually increased both at the bottom and water surfaces. According to previous studies, the bedforms are similar to dunes and antidunes.³⁷ Stability analyses also assume the so-called dune-covered beds^{21–24,38,39} as the initial shape of the sandbar formation. The shapes of both the bottom surface and the water surface are uniform in the transverse direction, and there appears to be a 2D shape. Bedforms that were a 2D shape transitioned over time to a 3D shape that obliquely intersected the channel. According to previous studies, oblique bedforms are similar to rhomboid bars.¹⁷ Rhomboid bars grew in wavelength and height, eventually forming alternate bars. At the final stage of the experiments, in contrast to the early stage of the experiments, the wave height of the water surface is smaller than that of the bottom surface.

The method using correlation coefficients by Moteki et al.³¹ makes it clear whether the water surface or the bottom surface contributes to the flow depth distribution at each stage of sandbar initiation and development. This method also classifies the process of the sandbar formation into three phases. To obtain a more quantitative view of the forming process, the correlation coefficients were quantified for our measurement results.

Figure 5 shows the results, where $r wl$ and $r bl$ are the correlation coefficients between the water surface and flow depth, and the bottom surface and flow depth, respectively, taking absolute values. The horizontal axis is the dimensionless time. The black dotted lines approximately divide the process of the sandbar formation into three phases, based on a study by Moteki et al.³¹ These results are averaged over five measurements. The figures show that $r wl$ is large in phase I, while $r bl$ is large in phase III. This means that in phase I, the wave height of the water surface waves is larger than that of the bedforms, and conversely, in phase III, the wave height of the sandbars is larger than that of the water surface waves. These results are consistent with the qualitative discussion in Sec. II C 1.

To quantify bedforms generation immediately from the beginning of the experiment, a 2D planar wavenumber analysis was performed for the dry bed, bottom surface, and water surface 1 min from the beginning of the experiments. The results of the analysis are shown in Fig. 6. The amplitudes are averaged over five measurements. The target sections are all shown dry and at 1 min in Figs. 2 and 3. The figures show that there are almost no wavenumbers in the dry bed in both cases. In contrast, at 1 min from the beginning of the experiments, the amplitude of the bottom surface is large in the area marked by the pink circle. The longitudinal wavenumber in the area is approximately 20 m⁻¹, which is approximately 5 cm in wavelength. The analysis of the water surface shows that amplitudes with the same wavenumber as the bedforms are dominant and are larger than those of the bedforms. These results are consistent with the results seen in the plane view of Fig. 4.

The results of Sec. II showed that undulations on both the water surface and bottom surface already existed 1 min from the beginning of the experiment. Since the amplitude of the water surface undulations was larger than that of the bottom undulations, the water surface undulations may have influenced the occurrence of the bottom undulations commonly called bedforms. At this point, however, it is unclear whether undulations occur on the water surface first. Clarifying this process has significant implications for revisiting the hypothesis on the occurrence of sandbars. In this section, we investigated the behavior of the water surface in fixed-bed channel.

The channel used for the fixed-bed experiment is shown in Fig. 7. The same flume used in Sec. II was used, but as mentioned above, the bottom was not covered with sand. The bottom of the flume is painted and has almost no unevenness, and it can be regarded as a smooth surface. The geometry of the water surface was measured by ST as in the previous experiments, and surface flow velocities were measured by a simple particle tracking velocimetry (PTV) technique because it is necessary to investigate the mechanism of occurrence from multiple angles, accounting for not only the shape but also the flow velocities. This PTV method and its accuracy are outlined in the  Appendix.

Figures 8(a) and 8(b) show photographs of the water surface under the conditions described in Sec. III A. The photographs show that intersecting patterns on the water surface fully covered whole of the channel in both cases, even though the bed surface is a smooth. The heights of the patterns appeared to be larger in case 1 than in case 2. Although the water surface appears to be irregular, the water surface is hardly moving and seems to be standing waves. We also visually confirmed that the patterns were present in the flume outside the photographed area.

Figures 9(a) and 9(b) show the measurement results of the water surface measured by ST. The figures showed that waves with a wavelength of approximately 5 cm and a height of several millimeters occur in both cases. These wavelengths and heights are generally consistent with the characteristics of the patterns observed on the water surface of the moving bed in Figs. 2 and 3.

Figures 11(a) and 11(b) show the longitudinal cross sections of Figs. 9(a) and 9(b). The water level is nondimensionalized by $h ¯$⁠. The number of cross sections is 43 in the channel width direction. The range of the figure is limited to the area not affected by the slight tilt of the measured plane view in the longitudinal direction. It is clear that the maximum amplitude in case 1 is approximately 20% of $h ¯$ and that the maximum amplitude in case 2 is approximately 10% of $h ¯$⁠. Figure 12 shows the result for the 2D wavenumber analysis of the water surface, as shown in Fig. 6. The figure shows that the dominant wavenumbers on the fixed bed are similar to that on the moving bed.

In the u value of case 1, the intersecting patterns, also seen in Fig. 8(a), are existed. The same intersecting patterns were observed in v/u. In the u value of case 2, the waves with a wavelength of approximately 5 cm are observed, similar to those observed by ST. In contrast, they were not observed in v/u.

As in Sec. III B 1, Figs. 14(a) and 14(b) show longitudinal plots of the dimensionless values of u with $u ¯$⁠. The display area is the white dotted line on the contour of u for each condition in Figs. 13(a) and 13(b) to remove missing measurements and asymmetry constraints. These figures show that the maximum amplitudes ranged from 2 to 3% in both cases.

Figures 15(a) and 15(b) show the results of the wavenumber analysis of u. The analysis area is the same as in Fig. 14 and is within the white dotted line on the u of Fig. 13. In case 1, the white circles in the figure indicate that these wavenumbers are correspond to the intersecting patterns. In case 2, the white circles in the figure show that the dominant wavenumbers similar to Fig. 12. The results are consistent with those seen in the plane view.

In this section, the experimental results show that the water surface had intersecting patterns similar to standing waves before bedforms occur where $β > β c$⁠. This result could renew the conventional hypothesis on the occurrence of sandbars.

Here, we clarify why the intersecting patterns on the water surface occur despite a flat bottom. Some obstacles in the sidewalls or inside the channel in supercritical flow generate shock waves.⁴¹ If the wave angles are the same with and without obstacle, the intersecting patterns are most likely to be shock waves and caused by the sidewalls. We intentionally generated waves by placing obstacles in the channel and measured the angle it made with the sidewall.

A cube with 5-cm-long sides was placed along the left sidewall, as shown in Fig. 16. Surface flow velocities were captured by PTV as shown in Fig. 17. The hydraulic conditions for this experiment are the same as those for case 1, but we refer to it here as case 1′. Figures 16 and 17 show that waves are generated from the block.

Why do the intersecting patterns occur on the water surface even though the bottom surface is flat? We showed that the intersecting patterns covered the entire channel and seemed to generate from the sidewalls. We also showed that the pattern angles were approximately matched with and without the block. Moreover, in general, the planar distribution of the flow velocity is parabolic, with a smaller velocity near the sidewalls and a larger velocity near the center of the channel. Considering the above, we suppose that the intersecting patterns are generated by sidewalls, but we cannot go into further detail.

We showed that the intersecting patterns on the water surface occurred on the smooth bed under sandbar forming conditions. The patterns initially occur on the water surface, and then, the patterns led to occur bedforms, even under conditions in which sandbars do not form where $β < β c$⁠, under conditions in which sediment transport occurs where $τ * > τ * c$⁠. However, it is unclear whether intersecting patterns occur under conditions whereby sediment transport cannot occur where $τ * < τ * c$⁠.

We also investigated the behavior of water surfaces where $τ * < τ * c$⁠, to provide a unified explanation for the process of the sandbar formation. Case 3, in which sediment transport does not occur, was prepared and is shown in Table I. Under this condition, water surfaces were measured and analyzed as in Sec. III. The results are shown in (c) of each figure in Figs. 8–15. From Eq. (9), the Froude number was calculated to be 0.83, which indicates subcritical flow. From Eq. (10), the Reynolds number was calculated to be 2136.

Figure 8(c) qualitatively shows that the flow is more stable in case 3 than in cases 1 and 2. Figure 9(c) shows that flow appears to have less turbulence in the cross-sectional direction, but the waves occurred with a certain wave height in the longitudinal direction. These waves were stationary. Figure 11(c) shows that the wave height in case 3 is lower than in case 1 but slightly higher than in case 2. In Fig. 12(c), the result of wavenumber analysis also shows that the dominant wavenumbers in case 3 are similar to those in cases 1 and 2. We confirmed that the waves are gravity-dominated as shown in Fig. 10. The depth-to-wavelength ratio $h ¯ / λ$ was 0.9/5, indicating that the waves were classified as shallow-water waves, as in cases 1 and 2. On the other hand, Fig. 13(c) shows that there is no distribution of u and v/u corresponding to the waves. Figure 15(c) also shows that there are no dominant wavenumbers. However, Fig. 14(c) shows that the wave height is approximately the same as in case 2 when viewed as a percentage.

The above results were compared across conditions in Fig. 18. The figures quantify the scale of the wave height and flow velocity distribution of the water surface waves under the above conditions. The Shields number on the horizontal axis is calculated based on the measured values of the smooth surface condition. The Shields number is thus slightly smaller than those of the rough condition shown in Table I. Even under smooth surface conditions, however, $τ *$ exceeds $τ * c$ in cases 1 and 2, and $τ *$ does not exceed $τ * c$ in case 3. Whether the surface is smooth or rough is therefore not an essential issue when comparing the hydraulic conditions set up in this study for the Shields number.

The figure shows that there is no significant difference in the water surface whether $τ *$ exceeds $τ * c$ or not. In other words, the water surface waves can occur even when $τ * < τ * c$ with a similar magnitude as those observed under the condition of $τ * > τ * c$⁠.

We synthesize the results and then propose a novel hypothesis on the inception of sandbars. Here, we collectively define the water surface waves observed in Fig. 9 and the wave trains on the water surface observed in Fig. 4 (right) as “standing waves.” We also define the bottom undulations with a wavelength of approximately 5 cm shown in Fig. 4 as “small bedforms.” The results of this study are systematically presented as a schematic diagram on the right of Fig. 19. On the left of the figure, a schematic diagram is drawn based on the explanation for the occurrence of sandbars by Tubino et al.²² The fundamental difference between the two hypotheses is the primary surface that first deviates from a flat state: whether it is the water surface or the bed surface. Theoretical analysis assumes that when $β > β c$⁠, the bottom becomes unstable, causing the flow to have a distribution. However, our measurement results showed that the standing waves already existed before the occurrence of small bedforms. Our investigations also showed that when $τ * > τ * c$⁠, the wavenumbers of small bedforms coincided with those of standing waves. This result indicates that the shape of the standing waves is projected onto that of small bedforms.

As can be seen from the photographs and PTV results, these standing waves are probably caused by the sidewalls. Sidewalls also physically confine water and sediment, and sandbars would not be formed without them.^16,17,41 In experimental flumes and straight river channels, sidewalls may contribute to the occurrence and development of sandbars.

Bedforms such as dunes and antidunes are expected to occur even under conditions where sandbars do not occur; therefore, it is appropriate to set the bifurcation point in the latter stage of phase I to determine whether β exceeds $β c$⁠, as shown on the right of Fig. 19. Previous studies^22–24,38 have performed stability analyses considering not only flatbeds but also dune-covered beds as the initial bottom shape. Future works should conduct experiments under the $β < β c$ condition to clarify the transition process to phase II.

This study revisited a conventional hypothesis that assumes inherent instability on bottom is the inception of sandbars. We performed two different flume experiments. First, we conducted flume experiments on a moving bed that provided conditions for the development of sandbars. In the experiments, we measured both the water surface and bottom surface at high density and high frequency during the experiment. The results showed that sandbars do not form suddenly, but rather that small bedforms transition to rhomboid bars, and then, the rhomboid bars transition to sandbars. The results of 2D wavenumber analysis showed that the dominant wavenumbers of the water surface and bottom surface agreed and that the amplitude of the water surface was larger. Second, we conducted fixed-bed experiments under the same conditions as a moving bed to ascertain the behavior of the water surface. The results of the fixed-bed experiments showed that standing waves were observed on the water surface even when the experimental conditions were steady and the flatbed channel was straight. A two-dimensional wavenumber analysis showed that the dominant wavenumbers of the standing waves and initial small bedforms were in good agreement. We also suggested that the standing waves may be originated from the sidewalls. The whole set of results indicated that the standing waves were already present on the water surface before bedforms occurred. The standing waves on the water surface are the most prominent factors that lead to the occurrence of sandbars.

The work was supported by JSPS KAKENHI (Grant Nos. JP21H04596, JP20K20543, JP22H00512, JP21K05826, JP21K18788 and JP21K05826). We would like to thank AJE (https://secure.aje.com) for the English language editing. We also thank T. Shioya and A. Suzuki for technical assistance with measurements by PTV and with an LDV.

The authors have no conflicts to disclose.

Shohei Seki: Conceptualization (equal); Data curation (lead); Investigation (lead); Writing – original draft (lead). Daichi Moteki: Conceptualization (equal); Software (equal); Writing – review & editing (equal). Hiroyasu Yasuda: Conceptualization (equal); Funding acquisition (lead); Project administration (lead); Resources (lead); Supervision (lead); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

This section presents an overview of PTV measurements. The PTV measurements were made according to the following procedure. First, light tracers were sprayed on the water surface of the channel. Video images of the moving object were taken to obtain the tracer's position at each time point. Then, the surface flow velocity was calculated based on the tracer's distance traveled and the frame-per-second (fps) rate of the video. Styrofoam spheres with an average grain size of 2 mm were used as tracers. The specific gravity of the tracers was so low that they did not sink into the water during measurement. To record the tracer's moving bodies, the camera was placed 2 m directly above the flume, facing downward, and calibrated, with the resolution set to full HD (1920 × 1080) and the fps set to 40. In the video, a tracer is represented by 5 × 5 pixels, which is sufficient resolution to identify the feature of the particles during the particle tracking described below. During the particle tracking, the 2D flow velocity field is obtained from the video using Optical Flow in OpenCV software.⁴² 

Validation of the PTV-measured velocities was performed by comparing them with the velocities measured with laser Doppler velocimeter (LDV). Case 4 in Table I outlines the hydraulic conditions for the experiment. A model was created with fiber-reinforced plastic from the measured data of one wavelength of sandbars formed in the experiment on the moving bed. As shown in Fig. 20, this is the fixed-bed condition with the model placed on the bottom of the channel. The twelve measurement positions were set up as shown in Fig. 20, and measurements with an LDV were conducted at intervals of 1 mm from the bottom surface to near the water surface at each position. More than 10 000 measurements were conducted at arbitrary plane positions and depths within the flow with an LDV to obtain a reliable flow velocity distribution for comparison. In the PTV measurements, tracers were sprayed until the number of flow velocity measurements at each measurement location reached several hundred, and the average of these measurements was obtained.

Figure 21 shows the measurement results by PTV and with an LDV at each position. The figure shows that the surface flow velocity measured by PTV and the near-surface flow velocity measured with an LDV are different at the leading edge of the sandbar model, such as Point 3 and Point 6. The reason for the difference is likely the stepped flow at the leading edge of the sandbar model. In contrast, at other locations where flow fluctuations are relatively small, the two velocities are in good agreement. These results mean that the validity of PTV is sufficient, at least for near-uniform flow, as utilized in this study.