2.2 Analysis of River Profiles and Erosion Rates
We quantify river profile form and responses to changes in rainfall patterns using channel steepness indices and erosion rates. These metrics are commonly used, and often in tandem, to study influences of climate and/or tectonics in mountain settings (Adams et al., 2020; Bookhagen & Strecker, 2012; Cyr et al., 2014; DiBiase et al., 2010; Duvall, 2004; Godard et al., 2014; Insel et al., 2010; Kober et al., 2015; Morell et al., 2015; Olen et al., 2016; Ouimet et al., 2009; Portenga et al., 2015; Safran et al., 2005; Scherler et al., 2014; Vanacker et al., 2015; Willenbring et al., 2013).
A widely used metric to analyze river profiles, interpret erosion rates, and make comparisons to the SPM is the normalized channel steepness index, ksn :
\(k_{\text{sn}}=SA^{\theta_{\text{ref}}},\) (3)
where θref is the reference concavity index (Wobus et al., 2006). We use a value of θref = 0.5, which is common and consistent with our choice ofm /n , and also with SPM predictions thatθref = m/n ≈ 0.5 where rock uplift rate (U ) and erosional efficiency (K ) are uniform (Tucker & Whipple, 2002). As previously noted, the SPM predicts that orographic rainfall gradients should produce longitudinal variations in Kthat, in turn, affect the concavity index, θ (Han et al., 2014, 2015; Roe et al., 2002, 2003). Any such variations are systematically reflected in the spatial pattern of ksn andθθref is expected. Importantly, however, many studies relate upstream-average values ofksn to measured spatially-averaged erosion rates, which relies on quasi-uniform (or linear) upstreamksn to be valid (Wobus et al., 2006). In cases where systematic longitudinal variations in K affect the downstream pattern of ksn (i.e., upstreamksn varies non-linearly), the meaning of such an average is not obvious.
To address this, we use discharge, rather than drainage area alone, to calculate a modified channel steepness indexksn­-q (Adams et al., 2020):
\(k_{sn-q}=SQ^{\theta_{\text{ref}}}.\) (4)
Like ksn­ , ksn­-q is an empirically supported metric independent from the SPM. In principle, however, ksn­-q is analogous to Erosion Index (EI) used by Finlayson et al., (2002) provided that m/nθref , such that EI = (ksn-q )n . Also, asksn is the slope of χ-transformed river profiles in χ-elevation space, if χ is redefined to include precipitation to estimate discharge, slopes of χ-transformed profiles would instead represent ksn-q (Royden & Perron, 2013; Yang et al., 2015). To the extent that the SPM captures the influence of discharge on erosional efficiency, it predicts that along-stream variations in ksn-q should scale with local erosion rate, precisely as it does for ksn whereK is spatially uniform. Hereafter, we useksn and ksn-q to refer to upstream averaged values, consistent with their common usage in catchment-mean erosion rates analyses, unless we specifically state that they represent local values.
Millennial-scale catchment-averaged erosion rates measured, for example, using cosmogenically-derived 10Be found in quartz in alluvial sediment (e.g. Bierman & Steig, 1996; Brown et al., 1995; Granger et al., 1996), seek to quantify erosion rates at the river basin scale. At steady state, spatially averaged erosion rate, local incision rate, and rock uplift rate are equivalent; however, during periods of transient adjustment these values differ, complicating interpretations (Willenbring et al., 2013; Wobus et al., 2006). To make our results more portable to studies of natural landscapes, we calculate the spatially averaged erosion rate (Eavg ) in addition to the instantaneous vertical incision rate (E ):
\({E_{\text{avg}}}_{j}=\frac{\sum_{x_{h}}^{j}\left(E_{j}\ \bullet\left(A_{j}-A_{j-1}\right)\right)}{A_{j}},\)(5)
where j corresponds to a downstream node of the profile, andxh is the channel head.