Dynamic bedrock channel width during knickpoint retreat enhances undercutting of coupled hillslopes

Mountain landscapes respond to transient tectonic and climate forcing through a bottom‐up response of enhanced bedrock river incision that undermines adjoining hillslopes, thus propagating the signal from the valley bottom to the valley ridges. As a result, understanding the mechanisms that set the pace and pattern of bedrock river incision is a critical first step for predicting the wider mechanisms of landscape evolution. Typically, the focus has been on the impact of channel bed lowering by the upstream migration of knickpoints on the angle, length and relief of adjoining hillslopes, with limited attention on the role of dynamic channel width. Here, we present a suite of physical model experiments that show the direct impact of knickpoint retreat on the reach‐scale channel width, across a range of flow discharges (8.3 to 50 cm3 s−1) and two sediment discharges (0 and 0.00666 g cm−3). During knickpoint retreat, the channel width narrows to as little as 10% of the equilibrium channel width, while the bed shear stress is >100% higher immediately upstream of a knickpoint compared to equilibrium conditions. We show that only a fraction of the channel narrowing can be explained by existing hydraulic theory. Following the passage of a knickpoint, the channel width returns to equilibrium through lateral erosion and widening. For the tested knickpoint height, we demonstrate that the lateral adjustment process can be more significant for hillslope stability than the bed elevation change, highlighting the importance of considering both vertical and lateral incision in landscape evolution models. It is therefore important to understand the key processes that drive the migration of knickpoints, as the localized channel geometry response has ongoing implications for the stability of adjoining hillslopes and the supply of sediment to the channel network and export from landscapes onto neighbouring depositional plains.

river incision is a critical first step for predicting the wider mechanisms of landscape evolution. Typically, the focus has been on the impact of channel bed lowering by the upstream migration of knickpoints on the angle, length and relief of adjoining hillslopes, with limited attention on the role of dynamic channel width. Here, we present a suite of physical model experiments that show the direct impact of knickpoint retreat on the reach-scale channel width, across a range of flow discharges (8.3 to 50 cm 3 s À1 ) and two sediment discharges (0 and 0.00666 g cm À3 ). During knickpoint retreat, the channel width narrows to as little as 10% of the equilibrium channel width, while the bed shear stress is >100% higher immediately upstream of a knickpoint compared to equilibrium conditions. We show that only a fraction of the channel narrowing can be explained by existing hydraulic theory. Following the passage of a knickpoint, the channel width returns to equilibrium through lateral erosion and widening. For the tested knickpoint height, we demonstrate that the lateral adjustment process can be more significant for hillslope stability than the bed elevation change, highlighting the importance of considering both vertical and lateral incision in landscape evolution models. It is therefore important to understand the key processes that drive the migration of knickpoints, as the localized channel geometry response has ongoing implications for the stability of adjoining hillslopes and the supply of sediment to the channel network and export from landscapes onto neighbouring depositional plains.

K E Y W O R D S
analogue experiments, bedrock river, channel width, cohesive substrate, hillslope, knickpoint, lateral erosion

| INTRODUCTION
The dynamic adjustment of bedrock river channels to transient forcing controls the mechanism and pace of wider landscape evolution (e.g. Burbank et al., 1996;DiBiase et al., 2015;Duvall et al., 2004;Lague, 2014;Whittaker et al., 2007;Yanites, 2018), with increasing recognition that landscape-scale studies require an understanding of the coupling of in-channel and hillslope processes (e.g. Hurst et al., 2019). A common manifestation of this phenomenon is the bottom-up landscape adjustment to changes in external forcing (e.g. tectonic uplift rate or base-level fall) through the upstream migration of knickpoints often in the form of waterfalls or channel reaches with a heightened bed slope. At a knickpoint, the channel bed elevation drops relatively suddenly so as it migrates upstream, the signal of base-level fall (e.g. increased uplift rate) is transferred to the adjoining hillslopes, leading to their steepening, lengthening, increased relief (Gallen et al., 2011;Grieve et al., 2016;Hurst et al., 2013) and a shift in the pattern of basin hypsometry (Gallen et al., 2011). As a result, hillslopes can become destabilized and contribute more sediment to the fluvial system  due to increased hillslope potential energy leading to enhanced mass-wasting events such as landsliding, rockfalls or debris flows (Gallen et al., 2011;Golly et al., 2017;Korup & Schlunegger, 2007;Mackey et al., 2014;Reinhardt et al., 2007). Therefore, the physical processes that control the dynamic adjustment of bedrock channel morphology during knickpoint retreat have important implications for adjoining hillslopes and wider landscape evolution over short and long timescales.
Channel morphology is expected to change at, and upstream of, waterfalls/knickpoints due to variability associated with the flow hydraulics (e.g. Haviv et al., 2006). Rouse (1936) performed a series of experiments to demonstrate the presence of a flow-acceleration zone upstream of a freefall lip (i.e. the brink of a waterfall), as a consequence of a pressure gradient induced by the shift from hydrostatic to atmospheric pressure dominating (Flores-Cervantes et al., 2006;Haviv et al., 2006;Lapotre & Lamb, 2015). The magnitude of flow acceleration is a function of the Froude number upstream of the waterfall (Rouse, 1936) and impacts the flow velocity, water depth and shear stress acting on the bed over a distance of two to four times the normal flow depth (Stein & Julien, 1993), under the assumption of a fixed channel width (Haviv et al., 2006): where τ lip is the shear stress at the waterfall lip, τ upstream is the shear stress upstream of the flow acceleration zone (i.e. at 'normal' conditions), Fr upstream is the Froude number upstream of the flow acceleration zone and ε is an empirical constant determined experimentally by Rouse (1936), with a value of 0.4. Haviv et al. (2006) also demonstrated the impact of the flow acceleration zone on the upstream channel slope, with the formation of an oversteepened reach possible when the rate of erosion at the waterfall face is lower than the rate of erosion at the waterfall lip. While the hydraulic theory above predicts that a zone of higher shear stress and flow velocity acceleration is primarily a phenomenon of vertical waterfalls, it was also documented experimentally by Gardner (1983) and  for break-in-slope knickpoints. It is logical, if the flow acceleration upstream of a waterfall is sufficient to enhance erosion to the point of producing a long oversteepened convex reach upstream (e.g. Haviv et al., 2006), that the channel morphology and typical scaling of the channel width and depth would also be affected. For example, due to the principle of conservation of mass and the relationship between hydraulic radius, slope and flow velocity in Manning's equation, it is possible that channels can narrow with increasing channel slope (Finnegan et al., 2005). Such observations have been made for natural channels in convex knickzones, with channel widths reported to be narrower than would be expected under typical bedrock river hydraulic scaling (Lague, 2014) in the French Alps (Valla et al., 2010), Turkey (Whittaker & Boulton, 2012), the Italian Apennines (Whittaker et al., 2007) and Taiwan, where the Da'an river channel narrowed substantially during a period of rapid knickpoint retreat after the Chi-Chi earthquake (Cook et al., 2013). Despite these observations, and the hydraulic theory that predicts flow acceleration zones upstream of knickpoints, there remains an incomplete understanding of channel width variation in transiently adjusting reaches of bedrock channels (i.e. knickpoints and knickzones) and the possible implications for the stability of adjoining hillslopes.
In this paper, we investigate the role of lateral adjustment of channel geometry during transient knickpoint migration for the first time and present a set of systematic analogue flume experiments to document the extent to which channel width adjustment is a significant process within the transient response of channels to changes in external forcing.

| Flume setup and experimental conditions
Due to the slow rates of knickpoint migration and bedrock channel adjustment in the natural environment, field observations often provide important insights into the response of the bedrock channel geometry to external forcing (e.g. Whittaker et al., 2007) and knickpoint formation processes (e.g. Groh & Scheingross, 2021) but lack insights into the temporal dynamics of these processes. We therefore performed a suite of analogue physical modelling experiments where the spatial and temporal scales of bedrock erosion processes are reduced (Paola et al., 2009). The laboratory modelling approach allows a systematic investigation of channel width evolution during knickpoint retreat, and whether input parameters (water and sediment discharge; Q and Q s , respectively) have an impact on the magnitude and rate of any channel width adjustment.
Experiments were performed using the 80 Â 30 cm Bedrock River Experimental Incision Tank at the Université de Rennes. The flume setup has been described in detail previously , Baynes, Lague, & Kermarrec, 2018, Baynes et al., 2020 and implements a 'similarity of process' analogue modelling approach whereby no formal scaling of the experiments with a particular natural location is sought. The experiments presented here do not therefore represent scaled experimental versions of any particular natural river, but the appropriate process representation within the flume allows the relative impact of the initial boundary conditions on the channel morphodynamics to be explored (Baynes, van de Lageweg, et al., 2018;Hooke, 1968;Paola et al., 2009). The scaling of the width and slope of the experimental channels with discharge follows the patterns observed in natural channels , ensuring that the findings from the experimental channels are transferrable to the natural environment despite erosion being driven primarily by hydraulic shear rather than sediment impacts.
We used a cohesive mix of granular silica cement, spherical beads (ratio 3:1 granular silica to spherical beads, both 45 μm grain size) and water (18% of total mix volume) to represent a cohesive bedrock substrate in the experimental channels ( Figure 1). At the beginning of the experiments, we flowed water over the silica surface with a 2 cm initial base-level fall to allow the channel to self-form an equilibrium geometry (width and slope) constrained within silica 'bedrock' banks ( Figures 1a and c). We performed 11 experiments (Table 1) at water discharges ranging from 8.33 to 50 cm 3 s À1 (0.5 to 3 L min À1 ), with two scenarios inputting coarse sand sediment of 250 μm grain size: (i) Q s = 0 g of sediment per cm 3 of water (g cm À3 ); (ii) Q s = 0.00666 g of sediment per cm 3 of water (g cm À3 ). For each experiment, we trig- We selected this approach for knickpoint channel width extraction to facilitate appropriate data extraction for knickpoints where the reach of active incision was relatively long (i.e. a steepened reach) or for a single vertical step undergoing parallel retreat.

| Impact of transient incision on channel geometry
At equilibrium conditions before the base level was dropped by 3 cm, the width of the experimental channel increases with Q ( Figure 2a) following a power law (all experiments: W Eq = 11.9Q 0.52 ; R 2 = 0.53) consistent with typical natural bedrock channels (Lague, 2014). The experiments that had an additional input Q s were wider (W Eq = 13.5Q 0.54 ) than those without an input Q s (W Eq = 11.9Q 0.48 ), matching previous experimental results (Baynes et al., 2020).

Reach-scale adjustment
Due to the reduced spatial scale and analogue processes at work in the flume experiments, the temporal evolution of the channel geometry is accelerated compared to natural bedrock systems. As a result, we are able to observe the coincident channel geometry adjustment as the knickpoints retreat over the course of the experiments (up to 150 min) rather than interpreting such change in the natural environment. Here, we exploit this experimental capability to explore the processes that drive the difference in channel geometry at and around knickpoints compared to equilibrium conditions.  (Figure 4). For a given Q, WR at the knickpoint is smaller for the experiments with Q s = 0.00666 g cm À3 than with Q s = 0 g cm À3 (Figure 4), implying a greater narrowing effect for channels that contain additional input sediment (also seen in Figure 2).  Figures 4a and h).

| DISCUSSION
The experimental results presented here demonstrate a clear link between channel width variability and the retreat of knickpoints within bedrock channels. Here, we discuss the drivers of channel change at knickpoints (Section 3.1), the implications for adjoining hillslopes (Section 3.2), insights into the response time of landscapes (Section 3.3) and finally the role of sediment (Section 3.4).

| Drivers of channel change at knickpoints
The systematic narrowing of bedrock channels associated with high tectonic uplift rate, active faulting or landslide dams has been observed in the natural environment (e.g. Burbank et al., 1996;Cook et al., 2013;Duvall et al., 2004;Ouimet et al., 2008;Whittaker et al., 2007), or within numerical model outputs (e.g. Yanites, 2018) and is thought to be associated with an intrinsic mechanism induced by high local slope that leads to focused flow, high shear stresses and stream power, bedrock scour and a narrower channel with a smaller F I G U R E 4 Analysis for the evolution of the width ratio and channel incision for the channel mid-point (50% distance from inlet to outlet) through time. Black points show the value of the width ratio, with the black line a five-point moving mean. Red line shows the channel incision at the mid-point through time, calculated as the difference between the minimum bed elevation for the timestep and the minimum bed elevation at the preceding timestep. Shaded grey area shows the time period when the knickpoint is present at the channel midpoint (i.e. time between steepened reach upstream of the knickpoint first reaching the mid-point and downstream limit of the pool migrating past the mid-point). width-to-depth ratio (Finnegan et al., 2005;Whittaker et al., 2007).
The rapid headward retreat of a knickpoint in the Da'an River, Taiwan (Cook et al., 2013(Cook et al., , 2014(Cook et al., , 2020 is the most relevant natural analogue for the experimental results presented here due to the discrete generation of a single knickpoint following the uplift of an anticline feature during the 1999 Chi-Chi earthquake. Cook et al. (2013) documented the controls on the rapid retreat knickpoint rate and subsequent channel geometry adjustment at the reach scale, although the magnitude and pattern of channel narrowing and widening was not the main focus of their work. Before the Chi-Chi earthquake, the Da'an River flowed across an $450 m-wide braidplain (W Eq ) over the anticline and the initial knickpoint retreat cut an incised gorge 14 m deep (KP H ) and 31 m wide (W KP ), 7% of the pre-disturbance width (Cook et al., 2013). In these experiments, we demonstrate the same phenomenon of channel narrowing and widening as observed in the Da'an River by Cook et al. (2013), demonstrating the relevance of our results for nat- is similar for equilibrium channels as well as knickpoints (b $ 0.5), matching the findings of Duvall et al. (2004). However, the value of K is approximately three times smaller for the knickpoints (Figure 2 (2018), where a channel will tend towards a condition that leads to the greatest vertical incision.
We can speculate that the excess higher shear stress observed beyond what is predicted by the hydraulic theory of the acceleration zone is due to the requirement for the channel to increase its transport capacity and bed shear stress in order to transport the optimum material eroded from the knickpoint in addition to the sediment supplied from upstream in the most efficient way possible. Narrowing of channels in order to increase the transport capacity has been numerically modelled, as suggested for the mobilization of landslide material following a sudden input of sediment , and we suggest a similar mechanism is occurring during knickpoint retreat. The hydraulic theory of Rouse (1936) does not take into account knickpoint generation and longer-term shift of a channel from equilibrium conditions. Rather, it predicts the hydraulics at knickpoints themselves, so it could be expected that any hydraulic-induced width variability is superimposed on the more significant transport capacity-induced width variability.
The set of experiments that included an additional input of coarse sediment had a higher bed shear stress both at equilibrium and knickpoint conditions (Figure 2c), driven by the requirement to transport both the eroded material from the knickpoint as well as the sediment load from upstream. There is a larger difference between τ Eq and τ KP than between the Q s = 0 g cm À3 and Q s = 0.00666 g cm À3 experiments, highlighting that in the cases tested here, the transient nature of channels is more significant for their geometry than variability in upstream sediment supply.

| Hillslope-channel coupling
The impact of headward knickpoint migration on adjoining hillslopes through the vertical reduction of the hillslope 'base-level' elevation can be significant in triggering the onset of active hillslope processes downstream of the knickpoints (e.g. Gallen et al., 2011;Reinhardt et al., 2007). The phases of lateral channel adjustment associated with knickpoint retreat highlighted in the experiments presented here (narrowing then widening) indicates an additional mechanism for the destabilization of the adjoining hillslopes. As the channel geometry relaxes back to equilibrium conditions following the knickpoint migration, the width can increase by >50% (and up to 80% where channels have a high sediment load; Figure 2). Channel widening through bank erosion or gorge wall retreat has been shown to destabilize hillslopes through undermining of the hillslope toe (Golly et al., 2017;Harvey, 2001;Kondolf et al., 2002), and steepening of the hillslope beyond the threshold angle for failure (Larsen & Montgomery, 2012 Þtan θ where θ is the hillslope angle. In Figure 7, we used a hillslope angle of 30 , a value appropriate for representing hillslopes at a typical angle of repose (e.g. Whittaker et al., 2007) and identified the experiments where KPH WEqÀWKP ð Þtan θ < 1 to indicate when the lateral widening component is more important for hillslope destabilization than the magnitude of the channel incision (i.e. the knickpoint height). The lateral adjustment process is relatively more important than vertical incision for hillslope destabilization in the experiments with higher Q (Figure 7a). We suggest that this is due to the lower values of KPH WEq at higher Q, when KP H is fixed across all experiments (Figure 7b). This equation could also be applied in natural field settings where knickpoint width, knickpoint height, hillslope angle and equilibrium channel width can be estimated, and we can suggest that hillslope destabilization driven by lateral channel undercutting can be particularly prevalent in circumstances where the channel width is relatively large compared to the knickpoint height (when KPH WEq < $ 0:7).
The hillslope response to knickpoint migration can be mapped to the same three phases described above and shown conceptually in (2) As the knickpoint propagates past the point location, rapid vertical bed incision leads to rapid reduction in the hillslope base level that triggers hillslope destabilization (Gallen et al., 2011;Mackey et al., 2014). (3)   3.4 | Role of in-channel sediment supply in setting the degree of hillslope destabilization Previous field and experimental observations (e.g. Baynes et al., 2020) and modelling studies ( (1), during (2) and after (3) the passage of a knickpoint past the hillslope toe. Adapted from Hurst et al. (2013) and Baynes, Lague, & Kermarrec (2018). [Color figure can be viewed at wileyonlinelibrary.com] shown that channels with higher sediment supplies are typically wider for a given discharge, thought to be due to the 'cover effect' that protects the bed from vertical incision and encourages lateral erosion of the banks by particle impacts (the 'tools effect'; Sklar & Dietrich, 2004). Here, we show that sediment-rich channels under transport-limited conditions are not only wider under equilibrium conditions (Baynes et al., 2020), they also undergo a more dynamic response to transient forcing (i.e. knickpoint retreat), as they narrow proportionally more from their equilibrium state compared to low Q s channels (Figures 2 and 4). Absolute KP W values are similar for both Q s = 0 and 0.00666 g cm À3 experiments at a given discharge (Figure 2a), which is expected as the higher slope at the knickpoint increases the sediment transport capacity significantly , such that the higher sediment supply has a negligible impact on the channel width under knickpoint conditions compared to equilibrium (Baynes et al., 2020). As the channels return to W Eq following the passage of the knickpoint and the slope of the channel is reduced (during phase 3 in Figure 8), channels with a high Q s and therefore the bed are protected by the 'cover effect' widened to a larger extent, and therefore potentially have a stronger role in undermining the adjoining hillslopes through bank erosion (Kondolf et al., 2002). There exists a potential positive feedback, whereby channels with a higher Q s experience a more pronounced wave of lateral adjustment during knickpoint retreat, leading to an increased likelihood of hillslope destabilization following the passage of a knickpoint. In turn, the heightened hillslope instability supplies more sediment to the channel through mass wasting Gallen et al., 2011), completing the positive feedback loop with a wider equilibrium channel state before the migration of subsequent knickpoints in response to further base-level falls. Such an observation from the experiments presented here highlights an additional complexity resulting from the presence of sediment in bedrock channels and highlights its importance for both erosion processes and wider landscape response.

| CONCLUSION
Headward-migrating knickpoints are key features of transiently adjusting landscapes through the translation of bottom-up signals of base-level fall or changes in tectonic uplift through the river network and the adjoining hillslopes. In addition to the rapid vertical incision associated with knickpoint passage, we show here-using a suite of analogue flume experiments-that a wave of lateral adjustment of channel geometry (narrowing by up to 80% compared to equilibrium conditions, followed by a widening after the passage of a knickpoint) can have a long-lasting impact on the stabilization of hillslopes. The relative degree of channel narrowing/widening is independent of the water discharge, but channels with a high sediment load experience a greater extent of narrowing/widening compared to channels with a lower sediment load, and the lateral adjustment can be more important for the destabilization of hillslopes than the magnitude of vertical incision. Rates of lateral adjustment are typically slower than rates of upstream knickpoint migration, therefore having a longer-term impact on the stability of hillslopes than rapid bed-elevation change. These observations enhance the importance of understanding the key processes that drive the migration of knickpoints, as the localized channel geometry response has ongoing implications for the stability of adjoining hillslopes and, therefore, the supply of sediment to the channel network and export from landscapes.

ACKNOWLEDGEMENTS
We thank Jean-Jacques Kermarrec for his assistance with the laboratory experiments. We also thank two anonymous reviewers for their comments, which improved a previous version of this manuscript. This research was funded by the European Union's Horizon

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