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# River self-organisation inhibits discharge control on waterfall migration

## Abstract

The action of rivers within valleys is fundamentally important in controlling landscape morphology, and how it responds to tectonic or climate change. The response of landscapes to external forcing usually results in sequential changes to river long profiles and the upstream migration of waterfalls. Currently, models of this response assume a relationship between waterfall retreat rate and drainage area at the location of the waterfall. Using an experimental study, we show that this assumption has limited application. Due to a self-regulatory response of channel geometry to higher discharge through increasing channel width, the bed shear stress at the lip of the experimental waterfall remains almost constant, so there was no observed change in the upstream retreat rate despite an order of magnitude increase in discharge. Crucially, however, the strength of the bedrock material exhibits a clear control on the magnitude of the mean retreat rate, highlighting the importance of lithology in setting the rate at which landscapes respond to external forcing. As a result existing numerical models of landscape evolution that simulate the retreat of waterfalls as a function of drainage area with a fixed erodibility constant should be re-evaluated to consider spatial heterogeneity in erodibility and channel self-organisation.

## Introduction

Understanding the processes that control how rivers adjust their elevation, location and morphology is essential for the understanding of landscape evolution across multiple spatial and temporal scales1,2,3. Regional changes in base level, such as those resulting from a drop in relative sea level or the movement of a tectonic fault, locally place rivers out of equilibrium1,2,3. Often, these perturbations appear in the landscape as abrupt vertical or near-vertical changes in channel bed elevation (i.e., a waterfall, or localised reach of increased channel slope). Here, we use the collective term ‘knickpoint’3 to refer to these dynamic locations. Over time, knickpoints propagate upstream, dividing a downstream reach broadly in equilibrium with its base level and an upstream reach which is yet to adjust4,5, typically characterised by more diffusive hillslopes with longer residence times of soils6, more rounded hillslopes with reduced erosion rates7 and broader valley bottoms8 (Fig. 1). Wider landscape changes propagate from the river channel after knickpoint migration, as adjoining hillslopes tend to elongate and steepen as a result of the drop in the channel elevation downstream of the knickpoint. This potentially increases the rate of hillslope sediment transport and the susceptibility to landsliding9,10 (Fig. 1). More generally, the phenomena of an ‘upstream incision wave’ has also been identified in other geomorphological processes such as soil erosion on hillslopes11, rill/gully formation12,13, as well as in extra-terrestrial environments including valley formation on Mars14. As a result, knickpoint dynamics are a fundamental aspect of landscape evolution and understanding what drives them is key.

The physics of knickpoint erosion processes are thought to depend on the mechanism of retreat such as plunge-pool erosion15,16, undercutting17 or column toppling18,19. Additional important factors controlling the rate at which knickpoints retreat include bedrock strength20, bedrock structure21,22, rate of sediment transport1,22 and the frequency of extreme flood events19,23. However, the complexities of the mechanisms of knickpoint retreat are often ignored in the stream power incision model, the concept that underpins much of modern quantitative geomorphology24,25. According to the stream power incision model, the rate of upstream knickpoint propagation is assumed to be proportional to the catchment drainage area, a proxy for river discharge:

$$R=K{A}^{m},$$
(1)

where R is the rate of knickpoint retreat, K is a parameter directly dependent on the bedrock erodibility coefficient, A is the catchment drainage area at the knickpoint and m is the coefficient setting the power law scaling between retreat rate and drainage area4,5,25,26.

This simple model has successfully predicted knickpoint retreat in the Roan Plateau, Colorado, USA5 (Fig. 1B), and the North Island of New Zealand4. However, the stream power incision model does not accurately predict knickpoint retreat in several other landscapes. These exceptions include (i) Hawai’i, where the dominant knickpoint erosion process is block plucking/toppling21, (ii) Scotland, where a reduction in knickpoint retreat rate has been associated with a reduction in the availability of glacial sediment for transport22, and (iii) Iceland, where the action of high magnitude low frequency flood events dominates the long-term evolution of the landscape19. In this paper, we explore the utility of the stream power incision model to predict knickpoint retreat rate in a carefully controlled experimental setup, and draw new conclusions about the morphodynamics of transient incising channels.

## Approach

Experimental modelling is a powerful tool for investigating the dynamics of the processes that drive landscape evolution27,28,29,30,31. Experimental studies using an erodible substrate of a silica cement mix have led to breakthroughs in our understanding of the processes that drive the evolution of the Earth’s surface, from the impact of tectonic uplift and climate at the orogen scale32,33,34 to the processes that control the geometry of individual channels31. This experimental modelling has only recently been extended to include knickpoint erosion dynamics in cohesive material35,36. Some investigations have considered headcut erosion in non-cohesive sediments such as sand11,37,38 and others have explored toppling of bedrock columns at the knickpoint face in heavily jointed bedrock in order to calibrate numerical models18. These studies have not, however, been specifically designed to test the assumptions that underpin the stream power incision model of bedrock incision.

Here, we present results from experiments designed to quantify the role of discharge (Am) and relative bedrock erodibility (K) in setting knickpoint retreat rate and also to explore the origin of discrepancies between observed empirical data and the knickpoint retreat dynamics forecast by the stream power incision model19,21,22. Three mixtures of homogenous, cohesive, fine-grained silica cement were used to simulate an erodible bedrock substrate in a 0.65 × 0.3 × 0.15 m box flume fitted with an adjustable outlet that was used to generate knickpoints by instantaneous base level fall (Fig. 2A; experimental set-up described in Methodology section and SI section 1). Erosion of the silica substrate occurs through hydraulic shear by clear water rather than abrasion by transported sediment27. We used two types of silica grains in the mix: angular grains and spherical beads. The proportion of angular grains to spherical beads sets the relative cohesion of the material (Fig. S1), with increasing proportion of beads reducing the strength of the bonds between particles. In the experiments presented here, those where the silica mix contained 16.4% of spherical beads have the highest relative cohesion (Fig. S1).

The initial configuration of all experiments featured a channel with a vertical face 2 cm in height at the downstream edge of the flume. Experiments used discharges between 1.66–50 cm3/s, regulated via a pump on the inflow pipe. The width of the box flume was sufficient for a channel to form with a self-regulated channel geometry, unlike the one-dimensional experiments where width is fixed36. During the course of the experiments (typical duration: 60–120 minutes), the knickpoints retreated up the channel to the water inlet (Fig. 2A). High-resolution topography of the channel was surveyed using a terrestrial laser scanner (green laser that penetrates water) at 120 second intervals. This produced mm-scale quantifications of topographic characteristics, such as channel geometry and channel slope during the course of the experiment (Fig. 2B). The location of the knickpoints in the laser scans was extracted manually from the point clouds in order to measure the topographic change and calculate the retreat rate (Fig. 2C; Methods section). Hydraulic information such as water depth and shear stress was calculated using the Floodos 2D hydrodynamic model operating directly on a digital elevation model created from the laser scans (SI Section 2; ref.39). The channels were free to migrate across the width of the box, and occasionally they flowed along the side of the box for a short distance at the end of experiments; this did not affect the measurements of knickpoint retreat rate.

## Results

In these experiments, the main control on the knickpoint retreat rate is the relative cohesion of the substrate material (Fig. 3A), with the fastest retreat rates occurring in the silica mix with the highest proportion of spherical silica beads. The experimental datasets show that knickpoint retreat rate is insensitive to river discharge, suggesting m = 0 (Fig. 3A). These results do not agree with the predictions of the stream power incision model. The knickpoint retreat occurred through a combination of plunge pool undercutting, cantilever failure of the overhanging knickpoint face and hydraulic detachment on steep segments. The constant knickpoint retreat rate with increasing discharge cannot be explained by a systematic change in the knickpoint geometry. Steepened reaches, vertical faces and knickpoints with undercutting plunge pools occurred across the range of discharges, and with similar knickpoint retreat rates (Fig. 3). Increasing discharge has an impact on the geometry of the knickpoints in the experiments, with wider incised channels occurring at the knickpoint lip under higher discharges (Fig. 4A), but there is no clear associated increase in shear stress within the channel upstream of the knickpoint (Fig. 4B).

## Discussion

In this study, there is a clear link between knickpoint retreat rate and the relative strength of the bedrock material (Fig. 3A), consistent with previous field, experimental and numerical studies of knickpoint erosion11,20,36, bedrock channel incision27 and landscape evolution40. The strength of the bedrock substrate contributes to the value of the bedrock erodibility coefficient K in the stream power incision model4. In modelling studies, K is often assumed to be constant over large spatial areas despite spatial heterogeneities in the resistance of bedrock to erosion, potentially caused by lithology type27, or spatial patterns in chemical-weathering rates induced by climate41. We suggest that further consideration of variability in the bedrock erodibility coefficient should be an important component of future modelling studies, due to the potential dominant role in setting the knickpoint retreat rate.

Crucially, however, our experiments do not follow the pattern of faster knickpoint retreat associated with greater discharge, as predicted by the stream power incision model (Fig. 3A). In the experiments, erosion of the silica paste is driven by the hydraulic shear of the water detaching the particles from the bed surface. Therefore, the bed shear stress is critical in controlling the knickpoint retreat rate, but does not vary significantly between the experiments despite an order of magnitude increase in discharge. The shear stress within the channel is a function of the channel geometry (width, slope) and the discharge (SI Section 4). The experiments indicate that any increase in the shear stress driven by an increase in the discharge is offset by changes in geometry (Fig. 4A). This channel self-regulation acts to maintain the shear stress at a constant value (Fig. 4B), a similar response to the self-regulation of alluvial channel geometry that allows floods to slightly exceed the critical shear stress for bedload transport42,43. Due to the different relative cohesion of the silica mixes, the similar shear stresses across the experiments can explain why the mean retreat rate can be significantly faster for a given discharge during experiments with weaker material. Channel geometry (i.e., channel width) in natural rivers is expected to emerge as a balance of the local discharge, sediment supply and uplift rate25,33,44,45, with a timescale of adjustment as short as 103–104 years following changes in mean discharge or sediment composition46. Channel geometry can adjust to variations in discharge or sediment supply through lateral erosion of banks or vertical incision of the channel bed47, and we suggest that these processes lead to the self-regulation of the shear stress exerted on the bed42,48.

The power-law relationship between width and discharge in the experiments (Fig. 4A) is similar to the relationship between channel width and catchment drainage area in the Roan Plateau, Colorado (SI section 5), where the stream power model has been shown to accurately predict the rate of knickpoint retreat (R = 1.37 × 10−8A0.54 m/yr; ref.5). Recent studies have suggested that the upstream bedload sediment flux is an important driver of knickpoint dynamics16,22,49,50, but this is an element not included in our simple experiments. In some natural settings, the unit bedload sediment flux will increase with catchment size due to the greater sediment source area associated with larger catchments (e.g. more hillslopes connected to the channel51, providing more ‘tools’ for abrasion27). We suggest that this relationship can, in locations where erosion is dominated by abrasion, result in an apparent drainage area dependency on knickpoint retreat rate as predicted by the stream power incision model (e.g., ref.5). However, this relationship does not always hold, due to sediment comminution with transport distance52, barriers affecting sediment transport (e.g., tectonic uplift in the Da’an River, Taiwan1) or the exhaustion of available sediment in postglacial landscapes (e.g. Scotland22). From our experimental data, and the demonstrated importance of the physical processes in driving knickpoint erosion in natural settings, we suggest that patterns of knickpoint retreat are a function of the erodibility of the bedrock substrate (Fig. 2) and the complex drivers controlling abrasion or plucking/toppling. Due to the ability of channels to self-regulate the bed shear stress through their geometry, we suggest that modelling knickpoint retreat using the discharge/drainage area as a key component fails to capture the complexities of the processes driving this landscape response mechanism.

## Conclusions

Migrating knickpoints are key drivers of change within landscapes, because they are located at the dynamic boundary between a downstream reach that is in equilibrium with the new forcing conditions and an upstream reach that is yet to adjust. As such, accurate modelling of their behaviour is fundamentally important for understanding past, present and future landscape evolution. Here we demonstrate that the lithological strength is a strong control on the rate that knickpoints retreat upstream. Due to a self-regulatory mechanism of channel width to increased discharge, shear stress within the channel can remain constant, resulting in constant rates of knickpoint retreat over an order of magnitude of discharges. Thus, we suggest caution is required when using a simple scaling with drainage area or a uniform value of bedrock erodibility to interpret past knickpoint retreat as an indicator of past changes in external forcing on landscapes.

## Methods

### Experimental set-up

The experiments were carried out in the experimental modelling laboratory at Géosciences-Rennes, Université de Rennes 1, where previous microscale landscape evolution studies using silica cement have been undertaken32,33,34. The experiments presented here were carried out in a 0.65 × 0.3 × 0.15 m box flume, containing a homogenous substrate mix of 18% water, and different proportions of angular silica (D50 = 40 μm) and spherical silica beads (D50 = 40 μm). After mixing, the material was transferred into the box flume and homogenised using a high frequency concrete vibrator to re-liquefy the paste and remove any air trapped during the mixing process. The material was then left to settle for one hour before experiments began. At the end of the mixing process, the silica mixture was cohesive and water infiltration was negligible. SI section 1 contains more detail regarding the relative cohesion of the different mixes (Fig. S1), and demonstrates the relevance of the experiments to natural channels, including the demonstration of a power law relationship, with exponent 0.5, between the experimental channel width and discharge (Fig. S2).

A channel was cut (depth ~2 mm) into the top of the silica paste to focus the flow, and the slope of the flume was set to 4°. Initially, water was input into the channel at the required discharge (1.66–50 cm3/s, set within 2% by a regulated pump on the inflow pipe) and, in all except one set of experiments (with 27.3% beads), the water was run for enough time (>1 hour) for the channel width and slope to adjust to equilibrium conditions. After stable conditions had been reached in the equilibrium experiments, a 2 cm high vertical knickpoint was generated at the downstream end of the channel by dropping the height adjustable outlet (Fig. 2), to represent an instantaneous base level fall. After initial trials, 2 cm was chosen as an adequate balance between generating waterfall-like geometry and optimizing the number of base level drops achievable with the total volume of silica paste.

After the baselevel drop, the upstream knickpoint retreat was monitored using a terrestrial laser scanner (Leica Scanstation 2) programmed to collect a high resolution (sub millimetre) point cloud of the topography every 120 seconds. Additionally, a time lapse camera was mounted directly above the flume and collected imagery every 60 seconds. In some experiments, due to the cohesivity of the silica, the channel began to undercut the adjoining banks. This overhanging material was removed manually so the laser scanner could maintain line-of-sight with all parts of the channel bed (visible in Fig. 2A downstream of the knickpoint). In practice, this meant that no major bank collapse occurred in the channel, and the majority of the transported material comes from channel erosion at the knickpoint location. Bank collapse would only occur downstream of the knickpoints due to elevation change, so the manual removal of bank material will not have impacted the results for knickpoint retreat presented here. Data processing, extraction and analysis of the point clouds were carried out using the open source software CloudCompare 2.753.

The analysis of knickpoint location and retreat rate was carried out on a subset of the point cloud data in order to accelerate the timescale of data processing. A 5 mm wide swath was extracted from the length of the thalweg in the channel for each point cloud. The location of the knickpoint lip and base was identified from each of these swaths visually, based on the inflection point of the channel slope at the upstream and downstream edges of the steepened reach. The locations of the knickpoints were compared between subsequent point clouds and the retreat rate calculated by averaging the distance between the lip and base of subsequent knickpoints and dividing it by the time between scans (extracted from the time-stamp of the point cloud collection by the laser scanner). Estimated measurement error is of the order of 1 mm, but the large number of measurements mitigate this. The consistency in retreat rates over short and long distances during each experiment suggests that the measurement error does not have a significant impact on the overall retreat rate calculations. The data analysis was performed directly on the point clouds rather than more traditional ‘DEM-of-Difference’ algorithms because some of the knickpoints developed undercutting plunge pools, a feature that would have been lost by converting the point clouds into raster layers. The knickpoint width measurements were performed at the lip of the knickpoint and the channel slope measurements were performed by calculating the change in elevation over a measured distance of ~2 channel widths upstream of the knickpoint lip. Hydraulic information (shear stress, water depth) was calculated using the Floodos hydrodynamic model39 over digital elevation models of the topography (SI Section 2).

### Data availability

The experimental data is available from E.R.C.B on request. The Floodos numerical model is available for download: https://osur.univ-rennes1.fr/eros/

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## Acknowledgements

We would like to thank Jean-Jacques Kermarrec for his technical assistance during the experiments, and also for his work in developing the experimental apparatus. Jens Turowski and Alzeena Bata are also thanked for their past work in developing the experimental techniques employed here. The research was funded by Natural Environment Research Council Ph.D. Studentship No. NE/H525270/1 (to E.R.C.B.) and Marie Sklodowska-Curie Individual Fellowship No. 703230 (to E.R.C.B.).

## Author information

### Affiliations

1. #### School of GeoSciences, University of Edinburgh, Edinburgh, UK

• Edwin R. C. Baynes
• , Mikaël Attal
• , Linda A. Kirstein
•  & Andrew J. Dugmore
2. #### Université de Rennes, CNRS, Géosciences Rennes - UMR 6118, F-35000, Rennes, France

• Edwin R. C. Baynes
• , Dimitri Lague
•  & Aurélien Gangloff

### Contributions

E.R.C.B., D.L., M.A., L.A.K. and A.J.D. contributed to the research design. E.R.C.B. and D.L. planned and conducted the knickpoint experiments. E.R.C.B. processed the laser scans and extracted the knickpoint retreat rates. E.R.C.B. and D.L. led the data analysis and wrote the paper, with input from all other authors.

### Competing Interests

The authors declare that they have no competing interests.

### Corresponding author

Correspondence to Edwin R. C. Baynes.