Laboratory rivers adjust their shape to sediment transport

An alluvial river builds its own bed with the sediment it transports; its shape thus depends not only on its water discharge but also on the sediment supply. Here we investigate the influence of the latter in laboratory experiments. We find that, as their natural counterpart, laboratory rivers widen to accommodate an increase of sediment supply. By tracking individual particles as they travel downstream, we show that, at equilibrium, the river shapes its channel so that the intensity of sediment transport follows a Boltzmann distribution. This mechanism selects a well-defined width over which the river transports sediment, while the sediment remains virtually idle on its banks. For lack of a comprehensive theory, we represent this behavior with a single-parameter empirical model which accords with our observations.

Figure 1

Experimental setup. (a) Laboratory river, with a top-view camera and inclined laser sheet. $Q_w$ and $Q_s$ denote the flow and the sediment discharges, respectively. (b) Close view on the river bed. White dashed lines materialize banks. A few grain trajectories are plotted in pink.

Figure 2

Evolution of the sediment discharge in a laboratory river. Blue dots: Sediment discharge measured with particle tracking (10-min average). Dashed line: Exponential relaxation with 45-min time constant (fitted to data). Sediment supply is $0.6\pm 0.1\mathrm{g}$/min.

Figure 3

Bed elevation and sediment-flux profile for runs 1, 4, and 5 (sediment supply increases from left to right). (a) River bed in the absence of sediment supply. Dashed line: Cosine shape predicted by the threshold theory for $L=0.06d_s$ and $S=0.0046$ [Eq. (4)]. [(c) and (e)] River beds with sediment supply. Dashed line: Theory of Sec. 6. [(b), (d), and (f)] Experimental sediment-flux profiles. Dashed line: Empirical model of Sec. 6. Brown arrows in (c) and purple arrows in (d) illustrate gravity and diffusion sediment fluxes, respectively. Data are available in the supplemental material [31].

Figure 4

(a) Histogram of pixel hue in a single frame for run 3. (b) Sediment-flux profile for different grain colors (red, blue, and orange). Dashed black line: Average.

Figure 5

Boltzmann distribution in our rivers. Colors correspond to runs (Table 1). Black line shows Eq. (3) without fitting parameter (${\lambda }_B=0.12d_s$ after Ref. [30]).

Figure 6

Model of an active river (Sec. 6). (a) Cross section. Blue: River flow. Brown: Sediment bed. (b) Sediment-flux profile.

Figure 7

Average intensity and width of the sediment-flux profiles. (a) Active width of the channel $W_T$ as a function of the sediment discharge. Blue points: Experimental rivers. Dashed line: Equation (14) fitted to data. (b) Average sediment flux ${\overline{q}}_s$ as a function of sediment discharge. Blue points: experimental rivers. Dashed line: Equation (14) fitted to data. (c) Average intensity of the profile as a function of the active width of the channel. Blue points: Experimental rivers. Dashed line: Equation (13) fitted to data.

Figure 8

Width and depth of laboratory rivers. (a) Slope as a function of sediment discharge [Eq. (20)]. (b) Depth as a function of sediment discharge. Gray line: Banks depth $D_B$ [Eq. (19)]. Pink line: Transport depth $\overline{D}$ [Eq. (21)]. Points: Experiments. Inset: Discontinuity $\delta$ as a function of the sediment discharge [Eq. (22)]. (c) Width as a function of sediment discharge. Gray line: Total width predicted by the model of Sec. 6. Gray points: Experimental width. Pink line: active width [Eq. (14)]. Purple points: Experimental active widths.

Figure 9

Aspect ratio of laboratory rivers as a function of their sediment discharge. Each point corresponds to an experimental run, for which we divide the total width of each river [Eq. (23)], by its depth. Colors are those of Fig. 5. Numbers correspond to experimental runs. Dashed line: Prediction of the model developped in Sec. 6.

Figure 10

Calibration of the bed-elevation measurement. (a) Laser sheet on calibrated stairs. (b) Laser line on the stairs (green) and its numerically found location (green line). (c) Linear relation between the height of a step and the corresponding laser deviation [Eq. (25)].

Figure 11

Measurement of bed elevation and water depth. (a) Side view of the laser sheet projected onto the river bed. Flow from left to right. When the channel is bare, the laser line is located on $x_{\mathrm{laser}}$. When it is filled with fluid, the laser is deviated to $x_{\mathrm{laser}}^{\prime }$ by the air-liquid interface. (b) Location of the laser line while the bed is filled with fluid (green line). (c) Location of the laser line on a bare bed (green line). (d) Cross section. Brown: Sediment bed. Blue: Fluid.