3.3 Sensitivity Analysis of 1-D River Profiles to Changes in Rainfall Patterns
Next, we explore a simple example scenario to evaluate the sensitivity of discharge and fluvial relief to changes in rainfall pattern. We introduce this analysis here using a steady state profile adjusted to spatially uniform rainfall (Figure 3). While idealized, this simple case is well suited to developing intuition about more complex scenarios, like strengthening or relaxing existing orographic rainfall patterns, which as we show in section 5.1 produce analogous responses to those we discuss here.
We define different fields bounding rainfall gradients that result in different classes of behavior (Figure 3a). Boundaries demarcating these fields are independent of U , Kp , m , and n provided m /n is unchanged. Channel length has a negligible influence for channels longer than a few kilometers whereA >> Ac , and only minor influence for different m /n ratios (~0.4-0.6). The h exponent in Hack’s Law (Equation 2) can influence field boundaries, as indicated in Figure 3a; however, the effect is minor for typical h values (1.67 ≤h ≤ 2; e.g., Rigon et al., 1996). Contours ofQf /Qi illustrate the extent to which a given rainfall gradient represents a net wetter (Qf /Qi > 1) or drier (Qf /Qi< 1) condition (Figure 3b). Contours ofRf /Ri describe the extent to which fluvial relief increases (Rf/Ri > 1) or decreases (Rf /Ri < 1). These contours show that steady state fluvial relief is sensibly correlated with discharge, but the relationship is complex when rainfall is not spatially uniform.
White fields (Figure 3, both panels) encompass rainfall gradients – and spatially uniform changes in rainfall – where the profile would experience wetter or drier conditions everywhere. Transient adjustments to such gradients generally mimic adjustments to spatially uniform increases or decreases in rainfall, although spatially variable changes are expected to affect adjustments differently than uniform changes in detail (e.g., see section 5.1.2).
Light grey fields (Figure 3a) encompass rainfall gradients where relative changes in discharge and equilibrium slope would invert along the profile, but initial and final steady state profiles would not intersect (i.e., produce xsc , but notxzc ­). The mode of transient adjustment is variable in space and time upstream ofxsc (variably E > U orE < U ; Figure 2). Despite this, the net change in fluvial relief is inversely related to the change in mean rainfall at steady state, consistent with expectations for spatially uniform changes in rainfall. xsc marks the position of the absolute maximum elevation difference between initial and final steady states in these cases, not the channel head. Therefore, while each point along the profile experiences net incision or surface uplift to reach steady state, the largest differences in elevation between initial and final steady states are along the central part of the profile.
Dark grey fields (Figure 3a) encompass rainfall gradients that produce both xsc and zc and are characterized by the most complex transient responses (e.g., Figure 2). Implied spatial patterns of relative changes in discharge and slope follow as for light grey fields, and modes of transient adjustment are similarly spatio-temporally variable upstream fromxsc . The distinguishing feature of these gradients is that the resulting steady state fluvial relief is positively related to the change in spatially averaged mean rainfall (e.g., Figures 1b, 2), contrary to expectations for spatially uniform changes in rainfall. This results from the non-linear influence of discharge on channel slope and the cumulative influence of downstream slopes on channel elevation. The absolute maximum difference in elevation between initial and final steady states may either be atxsc or at the channel head in these cases depending on specific characteristics of the change in rainfall gradient.
This analysis reveals several interesting ways that changes in rainfall pattern influence river profiles differently than expected for uniform changes. Note, we define our usage of ‘complex’ transient responses hereafter to include all responses that result in an along-stream inversion in the change in discharge, unless we specify otherwise (i.e., cases where xsc exists; both grey fields in Figure 3a). First, changes in longitudinal rainfall gradients that result in complex transient responses appear relatively common and do not require large changes in rainfall patterns or total rainfall. That these complex responses arise readily from a range of changes in rainfall patterns suggests that they may be a typical aspect of landscape evolution in mountain settings. For instance, this scenario implies topographic growth of incipient mountain ranges may be supported or suppressed by the orographic rainfall patterns they generate, depending on where rainfall is concentrated, even if they experience more total rainfall as a result (e.g., Roe et al., 2003).
Among changes in rainfall pattern where complex responses result in a positive relationship between the change in mean rainfall and fluvial relief (dark grey fields in Figure 3a), changes to top-heavy and bottom-heavy conditions have an asymmetric influence on fluvial relief. Top-heavy gradients in this category always inhibit growth of fluvial relief (Rf /Ri always < 1) and bottom-heavy gradients promote topographic growth, but incremental changes in bottom-heavy gradients result in greater increases (Figure 3). Incremental increases in rainfall upstream (top-heavy) suppress the rate at which slope increases upstream, limiting potential elevation change. In contrast, incremental decreases in rainfall upstream (bottom-heavy) enhance increases in slope upstream and support greater elevations.
To get a sense for potential magnitudes of topographic changes that changes in climate might produce, we tested a wide range of parameters that may be applicable to major mountain ranges (e.g.,Kp , U, h , L ). We find that a temporal change in rainfall pattern alone may support as much as ~102–103 m of change in fluvial relief in the opposite direction expected from the spatially averaged change in mean rainfall (e.g., an increase in relief associated with an increase in mean rainfall). The same climate change may also drive up to ~101–102 m of enhanced incision or surface uplift along downstream and central portions of river profiles in the manner consistent with conventional expectations for the change in mean rainfall. This spatially segregated behavior may be particularly important for understanding how adjustments to climate changes are expressed, sediment transport out of mountain catchments, and fluvial terraces to name a few examples. While we do not treat the latter two points further here, they nevertheless warrant more research.
Taken together, this analysis supports the notion that spatially variable changes in rainfall pattern can readily and importantly influence landscape form and processes in ways that fundamentally differ from expectations for spatially uniform changes in rainfall.