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 (Qf >Qi ) but decrease in downstream locations
(Qf < 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).