3.2 Transient River Profile Response to Changes in Rainfall Patterns
According to the SPM, transient responses to climate change are primarily driven by changes in discharge that, in turn, affect erosional efficiency. In response to climate change, in addition to changes in mean rainfall, increases or decreases in rainfall may occur in different positions within a catchment, for example by strengthening or relaxing existing orographic rainfall distributions (Roe et al., 2003; Roe & Baker, 2006). Any such change in the pattern of rainfall fundamentally changes how discharge accumulates and can be expected to drive adjustments in the form of river profiles.
Changes in discharge at a given location following a temporal change in rainfall pattern reflect changes upstream average rainfall conditions. (Hereafter we use subscripts i and f , respectively, to denote initial and final steady states, before and after a temporal change in rainfall pattern.) While integrating upstream conditions somewhat buffers discharge from localized variations in rainfall upstream, because it accumulates non-linearly downstream relatively modest systematic variations in rainfall can exert a strong influence. Indeed, contrary to spatially uniform changes in rainfall that cause monotonic changes in discharge everywhere, we find that for a wide range of temporal changes in rainfall patterns discharge may increase in upstream locations (f >Qi ) but decrease in downstream locations (f < Qi ), or vice versa.
We refer to the position of such a reversal (e.g. from increasing to decreasing discharge or vice versa) as xsc . At this position discharge remains constant, and thus equilibrium river slope does not change following a temporal change in rainfall pattern (at x = xsc , Qf =Qi and Sf =Si ). As we will show, transient responses to temporal changes in rainfall pattern that cause such reversals have distinctive qualities. For now, we note an interesting feature where upstream of xsc initial and final steady state profiles begin to converge (see Figure 2). Thus,xsc marks a local maximum elevation difference between initial and final steady state profiles. This convergent behavior contrasts with expectations for spatially uniform changes in rainfall where the difference in channel bed elevation increases monotonically upstream from the outlet (Figure 1a).
Assuming spatially uniform rock uplift rate andKp , the maximum difference in elevation along the profile between initial and final steady states, ΔzSc , can be expressed:
\({z}_{\text{Sc}}=\ \left(\frac{U}{K_{p}}\right)^{1/n}\int_{x_{b}}^{x_{\text{sc}}}{\left(Q_{f}-Q_{i}\right)^{-m/n}\text{\ dx}}\). (8)
In some circumstances, initial and final steady state profiles can intersect at a position x­zc (Figure 2), determined by:
\(0=\left(\frac{U}{K_{p}}\right)^{1/n}\int_{x_{b}}^{x_{\text{zc}}}{\left(Q_{f}\ {-\ Q}_{i}\right)^{-m/n}\text{\ dx}}.\)(9)
Notably, xzc marks a location where the net adjustment to reach steady state elevation changes along the profile from enhanced incision to surface uplift, or vice versa. Temporal changes in rainfall patterns that produce xzc are those that lead to positive relationships between spatially averaged mean rainfall and fluvial relief (Figure 1b).