## 1. Introduction

[2] Rapid mountain river incision through bedrock is an inherently stochastic process resulting from the long-term summation of flow and sediment discharge events at highly variable rates and frequency [*Hartshorn et al.*, 2002; *Howard*, 1998; *Turowski et al.*, 2008b]. While the actual incision processes remain difficult to apprehend in situ and are the subject of ongoing research [*Hancock et al.*, 1998; *Hartshorn et al.*, 2002; *Lamb et al.*, 2008; *Sklar and Dietrich*, 2004; *Turowski et al.*, 2007], there is no ambiguity on the inhibiting effect of a thick alluvial cover (several meters) on bed incision. An extreme case is the damming by large landslides or debris flows that reduces the downstream supply of coarse sediment and locally inhibits bedrock incision for several days to years [*Benda and Dunne*, 1997a; *Korup et al.*, 2006; *Lancaster and Grant*, 2006; *Ouimet et al.*, 2007]. More commonly, in rapidly eroding areas, the thickness of sediment stored in bedrock channels is known to vary from daily to yearly time scales [*Schuerch et al.*, 2006; *Turowski et al.*, 2008b]. For instance, in Taiwan, typhoon Longwang (October 2006, return time ∼ 30–50 years) deposited up to 8 m of sediment during a 3 day flood in the Liwu river, whose short-term averaged incision rate is of the order of 5 mm/yr (Figure 1) [*Hartshorn et al.*, 2002; *Turowski et al.*, 2008b]. Bed incision was likely completely inhibited for several months, while subsequent lower flow events were progressively removing the in-channel stored sediment at high rate: only 1–2 m of sediment were protecting the lowest part of the channel 5 months after the typhoon (Figure 1). This example illustrates one of the models postulated by *Howard* [1998] for the long-term dynamics of mixed bedrock-alluvial channels (channels with a moderate exposure of bedrock and significant alluvial cover deposits elsewhere). It underlines two important aspects of short-term alluvial cover dynamics in bedrock channels [*Benda*, 1990; *Hartshorn et al.*, 2002; *Hovius et al.*, 2000; *Turowski et al.*, 2008b]: (1) alluvial cover thickness can vary extremely rapidly in steep mountain rivers, and (2) channel bed incision can be negligible during large flood events. How these short-term dynamics propagate through time into long-term inhibition of bed incision is not clearly understood. This is the central question addressed in this paper. In the next paragraphs I detail the elementary mechanisms and couplings governing bedrock channel evolution that are likely relevant to this problem.

[3] The fluctuations of the volume of sediment deposited in channels over short to intermediate time scales are tied to the relationship between (1) the transport capacity of the channel set by the combination of discharge characteristics (mean, variability) and channel geometry (slope, width, cross section, roughness, grain size distribution, …) and (2) the frequency-magnitude distribution of sediment supply events to the channel [*Benda and Dunne*, 1997a; *Hovius et al.*, 2000]. Over geological time scales, bedrock channels are forced to incise to follow relative base-level fall. To do so, their long-term transport capacity _{t} must be on average larger than or equal to the long-term flux of sediment supply _{s} over all grain size classes. Erosion and transport processes driving river geometrical change (slope and width) are expected to operate in order to bring the channel into a steady state configuration allowing long-term bedrock incision and sediment transport to operate at the rates imposed by base-level fall and upstream sediment supply.

[4] However, we still lack a proper understanding of the role played in channel dynamics by short-term fluctuations in alluvial cover and incision. Recent theoretical work has demonstrated that the simple inclusion of a transport threshold to initiate incision combined with stochastic variations of daily discharge results in strongly nonlinear relationships between steady state slope and incision rate compared to the predictions arising from models without daily fluctuations [*Lague et al.*, 2005; *Snyder et al.*, 2003; *Tucker and Bras*, 2000]. Yet, these studies did not factor in two potentially important factors: (1) the inhibiting effect of daily variation of alluvial cover thickness and (2) the potential variation of channel width with incision rate and sediment supply. This latter effect has been document in the field [*Duvall et al.*, 2004; *Lavé and Avouac*, 2001; *Whittaker et al.*, 2007] and experimentally [*Finnegan et al.*, 2007; *Johnson and Whipple*, 2007; *Turowski et al.*, 2006]. It has motivated the development of analytical [*Turowski et al.*, 2007] and numerical models [*Stark*, 2006; *Turowski et al.*, 2009; *Wobus et al.*, 2006] of channel width evolution and steady state geometry. These models show that channel width dynamics result from the competition between vertical bed incision (narrowing tendency) and lateral bank incision (widening tendency) [*Stark*, 2006; *Turowski et al.*, 2009; *Wobus et al.*, 2006]. Present-day measurement of erosion distribution in the Liwu river [*Hartshorn et al.*, 2002; *Turowski et al.*, 2008b] shows that bank erosion rates were higher than bed erosion rates during major flood events (return time of 10 and more years). *Turowski et al.* [2008b] showed that the variation of shear stress distribution with discharge cannot explain this distribution. They concluded that bed incision reduction by an alluvial cover developing at high discharges is dominantly governing the ratio between bed and bank incision rates. Consequently, it is expected that the temporal fluctuations of static alluvial cover on the bed will have a significant impact on channel width evolution. Whether and how this short-term complexity can be averaged out over long time scales is a fundamental question that has not yet been addressed by theoretical work.

[5] Channel width is also an important factor controlling the space available to store sediment, and consequently the long-term incision efficiency reduction. In agreement with this, a theoretical analysis using a constant discharge model and assuming that steady state channel geometry minimizes potential energy, predicts an increase of steady state bedrock channel width with sediment supply rate [*Turowski et al.*, 2007, 2009]. A similar result has been predicted for the relationship between valley width and sediment supply rate in debris flow dominated environments [*Lancaster*, 2008]. Although it pertains to a slightly different environment than narrow rivers for which the channel/valley width ratio is about one for the mean annual discharge (Figure 1), and does not factor in the variability of discharge, it underlines the importance of channel width variations in accommodating various rates of sediment supply.

[6] In modeling studies, the inhibiting effect related to sediment transport is called the cover effect *C*_{v}. It varies between 0 (no incision) and 1 (no cover), and is always expressed as a function of the ratio between flux of sediment supply to the channel and sediment transport capacity *Q*_{s}/*Q*_{t}. Note that as discharge and sediment variability have never been explicitly accounted for in any previous work, long-term _{s}/_{t} and daily equivalent *Q*_{s}/*Q*_{t} have been treated as equal. Many ad hoc models dedicated to long-term channel dynamics lump the details of temporal- and reach-scale spatial variations of alluvial cover assuming that the long-term cover effect _{v} decreases linearly with _{s}/_{t} [*Beaumont et al.*, 1992; *Gasparini et al.*, 2006; *Sklar and Dietrich*, 1998; *Tucker and Slingerland*, 1994]:

Two other theoretical models developed at the flood time scale invoke more specific effects: (1) in the linear decline model [*Sklar and Dietrich*, 2004] the development of patches of sediment progressively covering the bed leads to equation (1) (except that it is expressed with daily variable *Q*_{s}/*Q*_{t}), and (2) in the exponential decline model [*Turowski et al.*, 2007] the development of alluvial patches and the increase of near bed sediment concentration increases grain-grain collisions to the detriment of grain-bed impacts. A probabilistic argument shows that in that case

where *v* is a cover factor dependent on bed topography and equal to one for a flat bed [*Turowski et al.*, 2007]. In the exponential model, bed incision is never strictly speaking completely inhibited, even if *Q*_{s} > *Q*_{t}. This arises from the assumption that, starting from a bare bedrock configuration, immobile patches of sediment protecting the bed only develop theoretically once *Q*_{s} > *Q*_{t}. It leads to the theoretical distinction between a static cover effect (immobile patches of sediment) and a dynamic cover effect (related to the increase of near bed sediment concentration reducing grain-bed impacts and/or mobile patches of sediment) [*Turowski et al.*, 2007]. Experimental results show that for a constant supply of sediment, the fraction of bed covered by immobile patches of sediment increases with *Q*_{s}/*Q*_{t} [*Chatanantavet and Parker*, 2008], and a linear or exponential decay could equally fit the data [*Turowski*, 2009]. Yet, in natural systems, the alluvial cover thickness in a bedrock channel is not only a function of the instantaneous value of *Q*_{s}/*Q*_{t}, but also strongly dependent on past history of sediment deposition (Figure 1). This led various authors to postulate a cover effect as a function of the thickness of sediment deposited on the bed [*Hancock and Anderson*, 2002; *Howard*, 1998; *Stark et al.*, 2009]. The prerequisite (or consequence) to use *Q*_{s}/*Q*_{t} -dependent cover models for long-term dynamics is to assume either (1) that *Q*_{s}/*Q*_{t} < 1 for all discharge events or (2) that the long-term integrated effect of alluvial cover variability _{v} = *f*(_{s}/_{t}) is captured by the same functional relationship as the short-term relationship *C*_{v} = *f*(*Q*_{s}/*Q*_{t}). The latter assumption has never been tested, and is the central problem tackled in this study. The former assumption might be valid for bedrock channels with very low rates of sediment supply and negligible alluvial deposits (the “bedrock channels” as defined by *Howard* [1998]). However, the ubiquitous existence of alluvial deposits in bedrock channels, especially in mountain belts (where arguably understanding bedrock channel dynamics matters most), demonstrates that there is at least a range of discharge events for which *Q*_{s}/*Q*_{t} > 1. As a consequence it cannot be assumed that *Q*_{s}/*Q*_{t} -dependent cover models can be safely upscaled to longer time scales using an effective discharge approach [*Cowie et al.*, 2008; *Crosby et al.*, 2007; *Gasparini et al.*, 2006; *Sklar and Dietrich*, 2006; *Turowski et al.*, 2007; *Whipple and Tucker*, 2002]. A proper upscaling should at least be tried once to verify this assumption and define the minimum time scales at which an effective model can be defined.

[7] In this study, I address the long-term resulting cover effect _{v} of short-term stochastic supply of water and sediment to channels by using a new numerical model of bedrock channel width and profile evolution calculated at daily time scale (code SSTRIM, Stochastic Sediment Transport and River Incision Model). The modeling strategy that I followed is based on three elementary mechanisms that are likely fundamental for the long-term dynamics of bedrock channels: (1) combination of transport threshold and daily stochastic variations in water discharge and sediment supply, (2) free evolution of width and slope as a function of bed and bank incision rate and (3) explicit treatment of alluvial thickness evolution through time and its consequence on the bed incision reduction.

[8] I start by describing the numerical model and how the cover effect at daily time scale can be cast in terms of alluvial cover thickness. Then, I use this model to explore the steady state geometrical configuration of a model bedrock channel reach submitted to a uniform uplift rate, and its relationship to changes in _{s}, changes in the variability of water discharge, and the degree of nonlinearity between sediment supply rate and water discharge. Model results are divided in two parts: first, I show how the long-term cover effect operates at short time scales, and the specific role of extreme events. Then, I focus on the resulting long-term cover effect law at steady state _{v} = *f*(_{s}/_{t}) and how it compares with the linear and exponential decrease cover models. As these two models are deficient, I finally suggest improved modeling strategies to simulate bedrock channel dynamics over the long term.

[9] I have limited the scope of this study mainly to steady state channels, because it allows me to study the impact of boundary conditions and specific features of the model definition (static versus dynamic cover for instance) on cover response, in a simpler framework. Nevertheless, at the end of this study I discuss the applicability of the steady state derived cover effect law _{v} = *f*(_{s}/_{t}) during transient channel adjustment.