Channel processes under high-magnitude flow events are of central interest to river science and management as they may produce large volumes of sediment transport and geomorphic work. However, bedload transport processes under these conditions are poorly understood due to data collection limitations and the prevalence of physical models that restrict feedbacks surrounding morphologic adjustment. The extension of mechanistic bedload transport equations to gravel-bed rivers has emphasised the importance of variance in both entraining (shear stress) and resisting (grain size) forces, especially at low excess shear stresses. Using a fixed-bank laboratory model, we tested the hypothesis that bedload transport in rivers collapses to a more simple function (i.e. with mean shear stress and median grain size) under high excess shear stress conditions. Bedload transport was well explained by the mean shear stress (1D approach) calculated using the depth–slope product. Numerically modelling shear stress to account for the variance in shear stress (2D) did not substantially improve the correlation. Critical dimensionless shear stress values were back-calculated and were higher for the 2D approach compared to the 1D. This result suggests that 2D critical values account for the relatively greater influence of high shear stresses, whereas the 1D approach assumes that the mean shear stress is sufficient to mobilise the median grain size. While the 2D approach may have a stronger conceptual basis, the 1D approach performs unreasonably well under high excess shear stress conditions. Further work is required to substantiate these findings in laterally adjustable channels.

The adjustment of rivers to the imposed valley gradient, sediment supply, and discharge is of central interest to geomorphology and has implications for understanding and managing natural hazards and ecological habitats. In alluvial channels, the adjustment is facilitated by the movement of sediment arising via the interaction between the flow and deformable boundary

Researchers have dedicated considerable effort to deriving mechanistic bedload transport functions – typically empirically calibrated – that relate the rate of movement to a force balance between the flow and individual particles. Other approaches exist: for example, non-threshold approaches that do not utilise a critical shear stress

Considerably less is known about rivers under high relative shear stress conditions

We test the relative performance of 1D and 2D bedload transport functions under high relative shear stress conditions in a Froude-scaled physical model. The experiments have a widely graded sediment mixture and develop alternate bars under pseudo-recirculating conditions at a range of widths and discharges. We record total bedload volumes and bathymetry, and we perform 2D hydraulic modelling to apply several transport functions akin to

Experiments were performed in the Adjustable-Boundary Experimental System (A-BES) at the University of British Columbia (Fig.

Adjustable-Boundary Experimental System (A-BES) at the University of British Columbia, featuring camera rig (top right) and bank control system at a width of 30 cm.

The experiments utilised interlocking landscaping bricks to constrict the channel to various widths

Summary of unit discharges

Summary of experiments conducted in the A-BES. The DEM count excludes screeded bed. Experiment 1 is published in

At the beginning of each experiment the bulk mixture was mixed by hand to minimise lateral and downstream sorting, and then the in-channel area was screeded to the height of weirs at the upstream and downstream end using a tool that rolled along the brick surface. The flow was run at a low rate with little to no movement of sediment until the bed was fully saturated, and it was then rapidly increased to the target flow.

Three different types of data were collected throughout each experiment; surface photos, stream gauge measurements, and sediment output. A rolling camera rig positioned atop the A-BES consisted of five Canon EOS Rebel T6i DSLRs with EF-S 18–55 mm lenses (set at 30 mm) positioned at varying oblique angles in the cross-stream direction to maximise coverage of the bed, as well as five LED lights. Photos were taken in RAW format at 0.2 m downstream intervals, providing a stereographic overlap of over two-thirds. A total of 10 water stage gauges comprised of a measuring tape on flat boards were located along the inner edge of the bricks every 1 m. To minimise edge effects, gauges were not placed within 0.60 m of either the inlet or the outlet. The gauges were read at an almost horizontal angle, which, in conjunction with the dyed blue water, minimised systematic bias towards higher readings due to surface tension effects.

The data collection procedure was designed to maximise measurement accuracy as much as reasonably possible. Given that stream gauge data would later be paired with topographic data, the timing of gauge readings needed to closely coincide with surface photography. Every time photos were taken the bed was drained, as the surface water would distort the photos. These constraints necessitated a procedure in which manual stream gauge readings (to the nearest 1 mm) were taken 30–40 s before the bed was rapidly drained, which is around the minimum time it would take to obtain the readings. The bed was then photographed and gradually re-saturated before resuming the experiment, which took approximately 10 min.

Each discharge phase was divided into a series of segments between which the data were collected. The procedure occurred in 5, 10, 15, 30, 60, and 120 min segments with four repeats of each (i.e.

Throughout the experiments, sediment falling over the downstream weir was collected in a mesh bucket, drained of excess water, weighed damp to the nearest 0.2 kg, placed on the conveyor belt at the upstream end, and gradually recirculated at the same rate it was output, as opposed to a “slug” (i.e. all at once) injection. Based on a range of samples collected across the experiments, we determined the weight proportion of water to be approximately 5.8 % and applied this correction factor to obtain approximate dry weights. There was no initial feed of sediment, although this no-feed period was only 5 min. The experiments are best described as pseudo-recirculating as sediment was measured and recirculated at the end of the 5 and 10 min segments and, for longer segments (i.e. 30, 60, 120 min), every 15 min.

Using the images, point clouds were produced using structure-from-motion photogrammetry in Agisoft MetaShape Professional 1.6.2 at the highest resolution, yielding an average point spacing of around 0.25 mm. A total of 12 spatially referenced control points and additional unreferenced ones were distributed throughout the A-BES, which placed photogrammetric reconstructions within a local coordinate system and aided in the photo-alignment process. Using inverse distance weighting, the point clouds were converted to digital elevation models (DEMs) at 1 mm horizontal resolution.

Despite the use of control points, the DEMs contained a slight arch effect in the downstream direction whereby the middle of the model was bowed upwards, which was an artefact of the photogrammetric reconstruction (see doming:

For each DEM, 10 wetted cross-sections were reconstructed using the water surface elevation data, which were then used to estimate reach-averaged hydraulics. For more detailed spatial analysis, the flow conditions of water depth and shear stress were reconstructed using a 2D numerical flow model (Nays2DH) to the final DEM of each discharge phase. The selection of the final DEM was arbitrary as any DEM over the steady-state portion of the experiment could have been selected. Nays2DH is a two-dimensional, depth-averaged, unsteady flow model that solves the Saint-Venant equations of free surface flow with finite differencing based on a general curvilinear coordinate system (further details can be found in

To minimise rounding errors associated with the relatively shallow depths in our experiments, the DEM size and discharge were adjusted to the prototype scale (i.e. using a length scale ratio of 25). The estimated water depths, shear stresses, and velocities from Nays2DH were then back-transformed to the model scale (Table

Summary of reach-averaged hydraulics (from the 2D flow model) and sediment transport (from measurements). Parameters are as follows.

The results of the flow model were quantitatively validated by comparing measured reach-averaged hydraulic depths (

Measured versus modelled mean hydraulic depth

The channels were formed under constant-discharge conditions for 4–16 h, beginning from either a screeded bed or a morphology developed at a lower discharge. Each experimental phase comprised an initial adjustment period during which morphology, hydraulics, and sediment transport were nonstationary. This adjustment period, which varied from minutes to an hour, was followed by a steady-state period during which these characteristics fluctuated around a mean value (see

Width-averaged bedload transport over time in two experiments with different widths but similar reach-averaged shear stress:

We determined a representative sediment transport rate for each experimental phase by averaging output over the final 3 h period (Table

We examined the correlation between the observed representative sediment transport rate and two formulations of excess shear stress based on the

We aimed to investigate the concepts underlying 1D and 2D bedload transport equations rather than to refine them. Subsequently, we ignored the parameter

This equation was modified to integrate across the distribution of local shear stresses,

Optimised values of

The correlation between

Under the imposed channel widths (0.30–0.60 m) and unit discharges (2.22–7.50 L m

Channel area at the conclusion of experiment 3b (

The depth–slope method of calculating mean shear stress estimated higher values compared to the numerical model (7 %–23 %) and also higher values of critical dimensionless shear stress in the corresponding transport functions (

Estimated values of

Local shear stresses at or below the mean were estimated to exceed

Frequency distributions of mean-normalised flow depth and shear stress (over each

Despite following similar frequency distributions, modelled local flow depth and shear stress were not strongly coupled spatially (Fig.

Relationship between local mean-normalised flow depth and shear stress across all experiments, produced by randomly sampling 10 % of cells from each flow model. Contour lines represent 2D kernel density estimation, and vertical dashed lines indicate the range of flow depths used to threshold the flow model.

We present the correlation between bedload transport and the four different representations of excess shear stress in Fig.

Correlation between excess shear stress and observed bedload transport (averaged over the final 3 h of each experiment) using the four approaches outlined in Table

These experiments had several advantages over traditional field and flume datasets in modelling and recording channel processes. Although the experiments did not model lateral adjustment, the smaller scale ratio (

We evaluated four different bedload transport functions based on the correlation between excess shear stress and observed volumes of bedload transport, averaged over the final 3 h of each experimental phase. We first focus our discussion on three of these approaches in increasing order of sophistication (A1, B1, then B2) and then explain their relative effectiveness. Finally, we discuss the conceptual differences between 1D and 2D bedload transport functions.

Most bedload transport functions index the applied excess shear stress using the mean depth–slope product as these data are relatively easy to collect in field contexts

In recent decades, technological advancements in remote sensing and hydraulic modelling have allowed researchers to directly model bed shear stress, thus providing a potentially more accurate estimate. This advancement is utilised in the B1 approach (1D modelled shear stress), which accounted for the effect of sinuosity, flow resistance, and energy losses to the channel banks. Accounting for these additional factors may explain the 13 % reduction in RMSE (0.44) compared to approach A1. Further advancements have led to the proliferation of 2D hydraulic models and some 2D bedload transport equations, which aim to account for the proportion of the bed participating in transport and the spatial variation in shear stress

Numerical modelling of shear stress and accounting for its frequency distribution led to similarly strong correlations between bedload transport and excess shear stress compared to the mean depth–slope product method. The ability of the mean shear stress to effectively capture variation in bedload transport is consistent with empirical evidence. In a reanalysis of data from Oak Creek, OR,

The four approaches demonstrated key differences based on how shear stress was calculated (depth–slope product vs. numerical modelling) and more importantly the formulation (1D vs. 2D). Both estimates of mean shear stress were linearly related to unit discharge, but those based on the depth–slope product were 7 %–23 % higher (Fig.

Despite having similar prediction errors, the 1D and 2D functions provided considerably different estimates of critical dimensionless shear stress. Using the 2D approach, estimates of

The differences between estimates of

By conceptualising transport as a function of mean shear stress, 1D equations may inflate the importance of relatively moderate shear stresses and deflate values of

The results may have implications for non-threshold approaches to predicting bedload transport in natural gravel-bed rivers

Further work is required to investigate differences in 1D and 2D estimates of

We investigated the performance of 1D and 2D bedload transport functions under high relative shear stress conditions in a Froude-scaled physical model. The analysis highlights the effectiveness of highly reductionist bedload transport functions based only on median grain size and mean shear stress calculated using the depth–slope product. Numerically modelling shear stress to account for flow resistance and energy losses from the channel planform and banks did not substantially reduce prediction error, nor did accounting for the relative importance of shear stresses across the frequency distribution. The results suggest that bedload transport may collapse to a more simple function (i.e. with average shear stress and grain size) under high excess shear stress conditions. Given that the channels herein have limited lateral mobility, our conclusions are most applicable to channels where lateral adjustment is suppressed. Further work is required to examine the effect of planform adjustments (widening, meandering), for which small-scale laboratory experiments serve as an effective research tool. The 1D and 2D approaches provided substantially different estimates of critical dimensionless shear stress, reflecting differences in how these approaches conceptualise excess shear stress. Estimates of

Raw hydraulic and sediment transport data from Tables 3 and 4 and Figs. 2, 3, and 6–8 are available at Zenodo (

DLA was responsible for conceptualisation, data collection, formal analysis, visualisation, and writing. BCE was responsible for supervision, review, and editing.

The contact author has declared that neither of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We would like to thank Rob Ferguson, Lucy MacKenzie, Will Booker, and two reviewers whose suggestions have greatly improved the paper.

This work was supported by graduate scholarships provided by the Canadian and Australian governments, as well as a postgraduate writing-up award (Albert Shimmins Fund) from the University of Melbourne.

This paper was edited by Rebecca Hodge and reviewed by Chenge An and one anonymous referee.