## 1. Motivation

[2] Changes in global climate affect the mean water discharge in rivers, and its temporal distribution between rare extreme events (floods and droughts) and frequent normal events [*Ely*, 1997]. This is likely to have an impact on long-term river incision [*Molnar*, 2001; *Tucker and Bras*, 2000; *Tucker and Slingerland*, 1997] and landscape lowering. Climate-driven changes in erosion modify the flux of sediment to depositional basins [*Clift et al.*, 2002; *Harris and Mix*, 2002; *Zhang et al.*, 2001], the weathering drawdown of atmospheric CO_{2} [*Millot et al.*, 2002] and the sequestration of organic carbon [*France-Lanord and Derry*, 1997], with feedbacks to global climate [*Raymo and Ruddiman*, 1992; *Ruddiman and Preil*, 1997]. These changes also govern landscape morphology at local [*Rinaldo et al.*, 1995; *Tucker and Bras*, 2000; *Tucker and Slingerland*, 1997; *Whipple et al.*, 1999] and continental scales, and can potentially change exhumation and tectonic deformation of mountain belts [*Hilley and Strecker*, 2004; *Molnar and England*, 1990; *Whipple and Meade*, 2004]. Progress in our understanding of the feedbacks between atmospheric and lithospheric processes, as well as the stratigraphic signature of past climate changes critically depends on our quantitative understanding of the role of discharge and discharge variability in continental erosion.

[3] Bedrock rivers are central to this problem. They incise the substrate, drive mass wasting, and remove the erosion products. Many studies have modeled the dynamics and geometry of bedrock river channels with incision laws assuming a simple relation between the rate of vertical incision, channel slope and discharge [*Braun and Sambridge*, 1997; *Crave and Davy*, 2001; *Howard*, 1994; *Kooi and Beaumont*, 1996; *Lague et al.*, 2000; *Tucker and Slingerland*, 1994; *Willgoose et al.*, 1991]. In these studies, fluvial incision is commonly scaled with the mean annual discharge or an effective discharge with a given recurrence time, and these parameters are thought to reflect the long-term integration of discharges of different magnitude and frequency. This approach would be valid if, for example, the effective discharge were identical for all incision rates and/or discharge probability distributions. However, this is unlikely when the chosen incision law is nonlinear with respect to discharge, and climate change is characterized by shifts in the relative frequencies of large and small events. This problem has been addressed by using simplified frequency-magnitude distributions of discharge [*Molnar*, 2001; *Snyder et al.*, 2003b; *Tucker*, 2004; *Tucker and Bras*, 2000], and it has emerged that the constant effective discharge model is indeed inappropriate when the threshold of erosion is nonnegligible. Then, its application would result, for example, in an erroneous prediction of the scaling of channel slope with rock uplift rate. Various authors have independently argued that thresholds of detachment or entrainment cannot be neglected [*Lague et al.*, 2003; *Lague and Davy*, 2003; *Snyder et al.*, 2003b; *Tucker*, 2004] and that these thresholds have a profound impact on landscape geometry and dynamics [*Baldwin et al.*, 2003; *Lague et al.*, 2003; *Molnar*, 2001; *Tucker*, 2004]. If correct, these findings compel a fuller investigation of the impact of discharge variability on longitudinal channel geometry.

[4] Building on the work by *Snyder et al.* [2003b], *Tucker and Bras* [2000], and *Tucker* [2004], we use a realistic, stochastic distribution of daily discharge coupled with a deterministic incision law to derive approximate analytical solutions that predict the theoretical scaling between channel slope, drainage area, long-term incision rate and discharge characteristics (mean and variability) at steady state. We focus on detachment-limited models in which channel dynamics are set by the rate of bedrock incision. This rate is a power function of the basal shear stress, above a critical value required to entrain bed load or detach bedrock [*Howard*, 1994; *Howard and Kerby*, 1983]. This choice is justified by the finding that incision proportional to basal shear stress can reasonably predict incision rates in rapidly cutting rivers [*Lavé and Avouac*, 2001]. Moreover, it remains to date the most widely used model of mountain channel evolution [*Finlayson et al.*, 2002; *Howard*, 1994; *Roe et al.*, 2003; *Snyder et al.*, 2003b; *Sobel et al.*, 2003; *Whipple and Meade*, 2004], and as such deserves a thorough study of the importance of runoff variability. We do not restrict our study to a particular set of incision law parameters, but rather explore predictions for various parameter combinations.

[5] We start by introducing the stochastic model of daily discharge, the incision law and the derivation of the long-term incision rate. Then, we calculate the steady state channel slope-drainage area relationship and its dependency with (1) the nonlinearity of the incision law (possibly reflecting the type of bedrock incision process [*Whipple et al.*, 2000]), (2) the value of the critical shear stress, (3) the cross-sectional geometry of the channel (the principle control on the variation of flow width with discharge), and (4) the incision rate, mean runoff, and runoff variability. Finally, we explore the relative importance of mean runoff and runoff variability in setting the steady state geometry of bedrock channels.