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 andθ ≠ θref 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/n ≈θref , 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.