3. Longitudinal Profiles (1-D)
3.1 Influence of Longitudinal Rainfall Gradients on River Profiles at
Steady State
Where rainfall is spatially uniform, topographic metrics (e.g., fluvial
relief, channel steepness) at steady state are expected to vary
inversely and monotonically with mean rainfall (Figure 1a). However,
spatially variable rainfall patterns complicate these expectations, as
shown in Figure 1b, where comparisons between rivers that experience
different rainfall patterns instead result in positive relationships
between these topographic metrics and mean rainfall. This reversal
reflects limitations of using spatially averaged metrics where climate
is spatially variable (e.g., in most mountain landscapes).
Systematic longitudinal variations in rainfall require that upstream
average rainfall values change systematically downstream, which
similarly affects erosional efficiency (K ), and thus equilibrium
channel slope (Equation 1). Where such spatial variations exist, mean
values of rainfall and ksn therefore depend on
where they are measured. In contrast, where equation 1 holds andm /n = θref ,ksn-q is independent of changes in mean rainfall
(Figure 1). Comparison of SPM equations for ksnand ksn-q at steady state (E = U )
further clarifies this difference:
\(k_{\text{sn}}=\ \left(\frac{U}{K_{p}{\overset{\overline{}}{P}}^{m}}\right)^{1/n}\),
(6a)\(k_{sn-q}=\ \left(\frac{U}{K_{p}}\right)^{1/n}\). (6b)
For spatially uniform rock uplift rate (U ) andKp , steady state fluvial relief (R ) is
proportional to the upstream integrated discharge (Han et al., 2015; Roe
et al., 2003; Royden & Perron, 2013). Integrating equation 1c from base
level (xb ) upstream to the channel head
(xh ), it can be shown that:
\(R=\ \left(\frac{U}{K_{p}}\right)^{1/n}\int_{x_{b}}^{x_{h}}{Q^{-m/n}\text{\ dx}}\).
(7)
This demonstrates clearly how fluvial relief depends on the cumulative
effect of discharge and implies that fluvial relief does not necessarily
scale monotonically with discharge or rainfall measured at any single
position, or averaged along any segment of a profile, except under the
special condition where rainfall is spatially uniform (Gasparini &
Whipple, 2014; Han et al., 2015). This is an important result,
particularly for understanding the topographic evolution of mountain
landscapes because it suggests that considering how rainfall patterns,
specifically, have changed with time is critical to predicting responses
to changes in climate. For instance, shifts toward ‘wetter’ climates may
support topographic growth, contrary to expectations and even in the
absence of any change in tectonics, depending on where rainfall is
concentrated, or vice versa.