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.