Experimental river delta size set by multiple floods and backwater hydrodynamics

The experiments were conducted with fresh water in the river-ocean facility at the Caltech Earth Surface Dynamics Laboratory. The experimental arrangement consisted of a 7-cm-wide, 7-m-long alluvial river, which drained into an “ocean” basin (5 m × 3 m) with 10-cm-deep standing water (Fig. 2). Sediment and water were fed at the upstream end of the alluvial river, and the sea level was held constant during the experiments using a programmable standpipe at the downstream end of the ocean basin. Two experiments were conducted, with each experiment starting with no sediment in the ocean basin. In the first experiment (experiment A), sediment and water were fed at a constant rate, and the alluvial river was allowed to build its own delta through time. In contrast, in the second experiment (experiment B), we oscillated between a 40-min low flow and 15-min high flow for 160 cycles, such that the delta grew under persistent backwater and drawdown hydrodynamics. With a quasi-2D, coupled hydrodynamic and morphodynamic model as a guide (18, 20, 30), the experimental water and sediment discharge in experiment B were set such that both discharge events created an equivalent bed slope for normal-flow conditions in which sediment transport was at capacity—an important aspect of our experiments considering that imbalances in sediment supply and water discharge in itself can cause adjustment of the equilibrium bed slope (50). The discharge events were chosen to produce backwater hydrodynamics and deposition during low flow (that is, M1 regime) and drawdown and erosion during high flow (that is, M2 regime) (20, 30, 51). In addition to achieving backwater and drawdown dynamics, the water and sediment fluxes were chosen such that the experiments produced subcritical flows, relatively low bed slopes (typical of large natural deltas), relatively small backwater lengths, and measurable changes between the flow depths for low flow and high flow, respectively (Table 1). For the constant discharge experiment, the flow discharge was chosen to be equal to the low-flow conditions of the variable discharge experiment; however, we increased the sediment infeed rate relative to experiment B to speed up the deltaic evolution, maintaining the subcritical flow conditions (Table 1 and fig. S2). To achieve the correct scaling of flow velocity (Fr number scaling), bed slope, and backwater length, we needed a relatively long alluvial river section (7 m; Fig. 2), such that water and sediment were fed at the upstream end, which is well within the normal-flow reach of the alluvial river.

Figures S2 and S3 show the instantaneous water surface and bed surface profiles along with the computed Froude numbers and the measured flow depths during both experiments, which highlight the subcritical flow conditions in both experiments. We achieved the Froude number scaling in our experiments in part by using crushed walnut shells of uniform grain diameter (D = 0.7 mm) as the sediment. Because crushed walnut shells (ρ_s = 1300 kg/m³) are significantly lighter than sand (ρ_s = 2700 kg/m³), flows with relatively low velocities mobilized sediment, resulting in sediment transport and deltaic evolution. During our experiments, sediment transport primarily occurred as bed load with some suspension during the high-flow conditions. The computed Rouse numbers $$(\mathit{P}=\raisebox{1ex}{${\mathit{w}}_{\mathit{s}}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{\kappa }{\mathit{u}}_{*}$}\right.)$$, where w_s = 32 mm/s is the settling velocity of the sediment [calculated by Chatanantavet and Lamb (30)], u_* is the shear velocity [computed as (gh_nS)^1/2, where g is the gravitational acceleration, h_n is the normal-flow depth, and S is the bed slope], and κ = 0.41 is the von Karman constant, were 4.5 and 3.8 for low flow and high flow, respectively, and 3.6 for the constant discharge experiment (Table 1). The bed was mostly planar, and no form drag correction was performed. All our experiments fell within the range of intermittent suspension (P < 6.1 for large-particle Reynolds numbers) (52).

Although our experiment B was not scaled to any natural delta, it is useful to cast our flow variability in light of natural flow variability on deltas. For example, if we were to treat our low flow as a bankfull discharge (flood event with recurrence interval of 2 years) on the Mississippi delta, then the high flow would correspond to a flood event with a recurrence interval of ~30 years based on the normal-flow depths of the two events in our experiments [for example, (20)]. In this framework, experiment A would correspond to a delta that evolved under constant bankfull discharge conditions. The duration of each flow in experiment B was long enough such that significant deposition or erosion occurred during each flow event but short enough to prevent the bed from reaching morphodynamic equilibrium with that flood event. This not only allows for the persistence of transient backwater hydrodynamics but also forces the experimental alluvial river to be in a state of perpetual disequilibrium, which has been argued to be the case for most natural coastal rivers (30). Thus, the key difference between our experiments A and B was that experiment A did not have persistent backwater and drawdown hydrodynamics; in the absence of multiple flood events, the bed in experiment A aggraded such that near-normal-flow conditions with uniform water velocities existed everywhere (fig. S6). Thus, subcritical flow is a necessary, but not sufficient, condition to produce long-lived gradually varied flow; in addition, the flow depth must deviate from the normal-flow depth, which in experiment B occurred through the action of multiple floods of different discharge. The similarity between our experiment A and alluvial fans and fan deltas is drawn because, in the absence of persistent backwater hydrodynamics, the avulsions in experiment A were driven by the change in confinement alone, which has been argued to be the primary mechanism for initiating avulsions on fans and fan deltas [for example, (15)].

An ultrasound distance meter (Massa) and a laser triangulation sensor (Keyence) were mounted on a motorized 3D positioning instrument cart with submillimeter positioning accuracy. The ultrasound probe was used to measure the water surface elevation during experiments, and the laser sensor was used to measure bed topography when the flow was switched off. We collected water surface elevation twice during each flow event in experiment B—once each at the beginning and end of the flow event. Bed surface elevation of the long profile of the channel was collected along with the water surface elevation during the flow, which was later empirically corrected for the refraction index of light in water. Errors associated with both the index-of-refraction correction and instrument error are ±0.1 mm in the vertical dimension. In experiment B, water flow was switched off at the end of each flow event, and bed surface elevation data were collected for the whole delta. This procedure allowed us to evaluate the amount of bed elevation change that occurred during each flow event. In experiment A, similar measurements were made at irregular intervals throughout the experiment.

Six cameras were mounted on the frame bordering the experimental facility (Fig. 2), whose viewing area covered the whole deltaic plain (Figs. 3 and 4 and movies S1 and S2). These cameras simultaneously captured overhead images of the deltaic evolution at a temporal resolution of 1 min for the whole duration of the experiments. Multiple orthogonal tape measures were laid down on the ocean basin floor, and when the basin was dry, cameras were used to capture overhead images, which were used to calibrate the concatenation of the six images from these multiple cameras. During each flow event, a fluorescent dye pulse was introduced just upstream of the tank boundary, and the dye transport was captured using overhead videos at a rate of 59 frames per second (fps) (movie S3). These dye videos were then used for estimating the water surface velocity through the backwater reach for both experiments.

Table 1 summarizes the measured and given parameters for experiments A and B. The water discharge was directly measured during the experiments using an inline flow meter, whereas the sediment discharge was measured using a sediment hopper that was calibrated using a balance and stopwatch. The normal-flow depth (h_n) was computed by differencing the bed elevation and water elevation data from the narrow section within the normal-flow reach of the experimental river (averaged over the first 3 m of the narrow section over the duration of the experiment). The normal-flow Froude number was computed as follows: $$\text{Fr}=\mathit{U}/\sqrt{\mathit{g}{\mathit{h}}_{\mathit{n}}}$$ , where g is the gravitational acceleration and U is the depth-averaged flow velocity. The depth-averaged flow velocity within the normal-flow reach of the alluvial river was computed by the ratio of the input water discharge and the product of the channel width (7 cm) and depth of flow (h_n). Note that the depth-averaged flow velocity is always less than the water surface velocity reported using dye videos in Fig. 6B and fig. S6.

The bed elevation and water surface elevation data were filtered, where spurious spikes were removed through a spatial slope threshold (~2% of the data). Because our refraction index correction of the laser scans for submerged bed works only for standing-water case, we did not use the bed elevation measurements collected during the flow events. Instead, bed elevation measurements at the end of each flow event were used to characterize the patterns of deposition and erosion that occurred during each event. Further, anomalous data points whose elevation was lower than the ocean basin floor were removed (<1% of data). To denoise the data, we applied a moving-median filter with a kernel size of 1.5 cm to both the bed and water surface profiles. These methods were used for processing the 3D scans of the delta surface at the end of each flow event. Finally, the data removed during the filtering procedure were replaced using linear interpolation.

We identified both the avulsion locations and the shorelines on the concatenated images of deltaic evolution whose pixel coordinates were transformed into the coordinates of the instrument cart following calibration using tape measures. For experiment B, we defined an avulsion as the development of a new channel where the daughter (new) channel captured flow even during low-flow events along with the parent channel being partially or completely abandoned. This definition precludes the characterization of crevasse splays as avulsion events in our experiment. Crevasse splays were common during high-flow events that resulted in overbank flooding and channelization on the floodplain; however, splays were not active during low-flow events. In experiment A, crevasse splays were uncommon and avulsions were defined by the creation of the new channel. Channel abandonment was abrupt in experiment A.

We computed the avulsion lengths through time using the overhead images of the deltaic evolution during both experiments. The concatenated photograph, which corresponds to each avulsion, was first identified, and the pixel coordinates of the avulsion site and the river mouth were extracted (Fig. 3). The avulsion site was defined as the location of the levee breach that resulted in formation of a new channel (Fig. 3 and fig. S4). Because we defined the avulsion site as a point, the error associated with the calculation of the avulsion length can be a maximum of the channel widths, which were, on an average, 10 and 25 cm for experiments A and B at the location of the avulsion sites, respectively. This corresponds to an overall error of approximately 0.06L_b and 0.08L_b for experiments A and B, respectively. We note that this error is a maximum and the actual error should be less than this value for all the computed avulsion lengths (for example, fig. S4). We then measured the streamwise distance of the avulsion site from the river mouth, that is, the shoreline, to quantify the avulsion length for each avulsion cycle in both experiments. We converted the time stamp on each of the avulsion pictures into an experimental run time. We normalized the experimental run time with the time it would take to build a radially symmetric, semicircular delta of size 0.5L_b (T_delta)where h_ocean = 10 cm is the depth of water in the ocean basin, ρ_s is the density of crushed walnut shells, and Q_s is the sediment mass feed rate for each experiment (Table 1). This procedure allowed us to account for differences in the delta growth rate that were simply due to differences in the sediment feed rates between the experiments and, consequently, directly compare the location of avulsions across the experiments (Fig. 5).

We used the bed topography data collected at the end of each flow event to estimate the amount of in-channel sedimentation during each flow event and through an avulsion cycle. Using the overhead images and dye videos of the delta as a guide, we first traced the river channel thalweg on the topography data and extracted the bed elevation perpendicular to the thalweg to get the channel cross-section topography at a spatial resolution of 1 cm. We then computed the cross-section averaged riverbed elevation at each location along the channel by estimating the cross-sectional area and evaluating the depth of the rectangular channel with the same width and area as the computed cross-sectional area. This procedure allowed us to account for the width changes in the channel during each flow. We then measured the vertical change in the cross-section averaged bed elevation through each flow event (that is, during low- and high-flow events) and also during an avulsion cycle, which was defined as the time from which a new channel was formed until it avulsed to a new location. The red and blue lines in Fig. 6C show the amount of in-channel sedimentation that occurred during each high- and low-flow event during the avulsion cycle, and the green curve shows the in-channel sedimentation that occurred through the whole avulsion cycle. The in-channel sedimentation during an avulsion cycle results from the combination of in-channel sedimentation from multiple low- and high-flow events during the avulsion cycle. The avulsion cycle shown in Fig. 6 was composed of six low-flow and five high-flow events, and the avulsion occurred during the sixth high-flow event. The shaded red and blue colors in Fig. 6C show the standard error (SE) in normalized in-channel sedimentation across high- and low-flow events, respectively. Thus, the green line in Fig. 6C is a summation of in-channel sedimentation that occurred during six low-flow and five high-flow events, whose average values along with SE are shown by blue and red curves, respectively.

We used the overhead dye videos (frame rate of 59 fps) to compute the water surface velocity on the deltaic plain in both experiments (movie S3). For each dye video, we used the color contrast between the dye (fluorescent) and the deltaic surface (brown) to automatically identify the location of the dye in each frame of the video. Each frame was then converted into a binary matrix of “dye” and “no dye,” and we manually kept track of the location of the dye front in each frame of the video. Once the dye front was located, we estimated the distance traveled by the dye along the channel thalweg in successive frames and converted this computed distance into instantaneous surface velocity using central differencing. Finally, the instantaneous surface velocity data were smoothed using a moving-window average filter over nonoverlapping windows of size 25 cm to produce Fig. 6B and fig. S6. The shaded region in these figures corresponds to the SE resulting from the spatial averaging. The water surface velocity computed using the dye videos is higher than the depth-averaged flow velocity, as expected.