5.1.2 Multi-stage Adjustment
It has long been recognized that spatially uniform changes in rainfall should promote transient changes in erosion rate everywhere across a landscape, which cause morphological adjustments to sweep upstream to restore erosional equilibrium (e.g., Tucker & Slingerland, 1997; Whipple & Tucker, 1999). Our model is fully consistent with this expectation under such conditions (e.g., Cases 1 & 2). In addition, we have shown that responses to changes in the rainfall pattern are variable in both space and time (e.g., Cases 3 & 4). As a consequence, following any non-uniform change in rainfall pattern, distinct initial morphological and erosional changes always precede the upstream sweeping adjustments that ultimately restore equilibrium. Contrary to expectations for a uniform change in rainfall, we find that catchments characteristically exhibit a relatively protracted, multi-stage, and spatio-temporally variable response to a single temporal change in rainfall pattern (Figure 8; Movie S5). We emphasize that this behavior is a general characteristic of any spatially variable change in rainfall pattern and is not exclusive to those that induce the complex transient responses. This leads to novel expectations for how transient responses to changes in climate should be expressed across a landscape and has potentially important implications for detecting transient landscape responses to climate changes.
The initial stage of morphological adjustment begins synchronously across the entire river basin following a change in rainfall pattern. At the onset, local erosion rate is everywhere a function of the local relative change in discharge. As this initial stage proceeds, spatial variations in erosion rate along the trunk river produce morphological changes along its length that progress at different rates (variable erosional efficiency). Importantly, this means that initial, or “relict”, conditions are often not preserved upstream of slope-break knickpoints on the trunk profile; the profile is progressively modified as adjustment proceeds even upstream of the main knickpoint. Indeed, the resemblance of “unadjusted” profile segments upstream from the main knickpoint to their initial state diminishes with time during the transient response, and thus with relative position upstream. This contrasts with spatially uniform changes in rainfall (and erosional efficiency; e.g., Cases 1&2), or uplift rate that does not affect erosional efficiency, that allow preservation of relict morphological characteristics (e.g., ksn ) upstream of migrating transient knickpoints, as is often assumed in analysis and inversions of river profiles (e.g., Clark et al., 2006; Fox et al., 2014; Gallen et al., 2013; Goren et al., 2014; Kirby & Whipple, 2012; Miller et al., 2013; Schoenbohm et al., 2004; Whittaker et al., 2007). More broadly, this contrasts with the notion that adjustments to climate change should simply propagate upstream from base level as is expected for other external changes (e.g., uplift rate). While there is a signal of transient adjustment that indeed migrates upstream, significant amounts of surface uplift, as observed in Case 3 (Movie S3), or incision, as observed in Case 4 (Movie S4), along with changes in channel steepness can occur prior to arrival of this signal. Nevertheless, these changes are in response to the change in climate. Additionally, spatio-temporally variable adjustments along the trunk profile dictate that individual tributaries experience temporally variable rates of base-level fall until the trunk profile reaches a new equilibrium at their confluence (Movie S3 & S4).
The continuous, yet variable nature of base-level fall imposed by the trunk river on tributaries during the initial stage of adjustment generally results in a broad adjustment zone characterized by smooth variations in channel steepness in tributary catchments. Indeed, tributaries in our model located in upstream positions, where this initial adjustment stage is relatively long-lived (compared to tributaries located near the trunk outlet), experience significant changes in slope and relief without formation of any knickpoints (e.g., Figure 8b; Movies S3 & S4). Importantly, this shows that tributaries are not insulated from effects of spatially variable changes in rainfall (variable erosional efficiency) along their trunk river, even if they experience essentially uniform rainfall throughout their history. Furthermore, they may appear relatively well-adjusted (graded) during periods of transient adjustment despite significant deviation from both initial and final steady state conditions (Figure 8b; also see Tributary 2, Movies S3 & S4).
Adjustment to quasi-steady-state conditions along the trunk river, which may or may not be associated with the upstream migration of a significant or obvious slope-break knickpoint, defines the beginning of the second – and final – stage of adjustment in our model landscapes (Figure 8). The rate of base-level fall experienced by a given tributary stabilizes upon local adjustment of the trunk profile, representing a distinct change from the initial stage where the rate of base-level fall is temporally variable. Depending on circumstances, this change may be abrupt and produce a discrete knickpoint that sweeps upstream through the tributary catchment. The change in base-level fall rate is the dominant signal exhibited during this second adjustment stage, although it acts upon the profile state reached during the initial adjustment stage, and it is largely a function of the shape and migration rate of the trunk knickpoint. Both factors are controlled by the integrated response of the trunk profile to this point, and therefore do not relate to the change in rainfall pattern in a direct manner. Therefore, the dominant signal passed to tributaries during this second stage, and any knickpoints that form as a result, generally do not reflect the change in rainfall locally within the tributary catchment, and their relationship to the regional rainfall pattern experienced by the trunk stream is complex. This is directly contrary to expectations for spatially uniform changes in rainfall where changes in slope above and below knickpoints should scale with the magnitude of the change in rainfall (e.g., Whipple, 2001).
Extrapolating these observations to natural, inherently more complex river networks, suggests that broad adjustment zones comprising multiple knickpoints might be associated with a given change in rainfall pattern – in contrast to the single knickpoint or knickzone expected to accompany a spatially uniform change in rainfall magnitude (e.g., Case 1 & 2). For instance, if second-order rivers (sensu Hack, 1957) experience spatially variable rainfall patterns in addition to the trunk, then we expect third-order rivers should experience an additional pair of adjustment stages. This implies that the full transient response to changes in rainfall pattern may be expressed in a complex fashion, and potentially across a large areal extent, in large river basins (e.g., Figure 8; Movie S5). If true, this multi-stage adjustment behavior may ultimately pose a significant, still unresolved, challenge to recognizing and quantifying transient responses to changes in climate in many settings.
5.2 Recognizing the Influence of Climate on Topography and Erosion Rates 5.2.1 Steady State Relationships among Channel Steepness, Erosion Rate, and Erosional Efficiency
The SPM makes specific predictions about the relationships among channel steepness, erosion rate, and erosional efficiency (K ) at steady state (Equation 6), and as shown by curves in Figures 4-6. Because the role of climate is encapsulated in K , it is important to remember that a uniform K value implies that the influence of climate is uniform over the spatial and temporal scales of interest. Further, the expectation that basin-average topographic metrics likeksn should relate to rainfall in a simple way generally relies on an assumption of a spatially uniform K value. Rainfall gradients systematically affect this expectation, where bottom-heavy gradients result in higher ksn(lower apparent erosional efficiency), while top-heavy gradients result in lower ksn (higher apparent erosional efficiency) (Figure 9). The magnitude of this effect (as a percentage of actual erosional efficiency) varies with strength of the rainfall gradient. However, ksn values also vary with erosion rate (uplift rate at steady state; Equation 6a). Therefore, while subtle rainfall gradients affect apparent erosional efficiency to a proportionally lesser degree, they can still substantially influence observed ksn values, even at steady state, where uplift rates are higher. This can be important in natural settings where uplift rates, erosional efficiency, and the form of their relationship to topography are generally unknown.
This analysis has two related and important implications. First, interpretation of the controls on topography (e.g., meanksn , mean gradient, relief, etc.) in terms of climate sensitivity, uplift, and/or rock properties using measurements from catchments that experience the same mean rainfall, but different rainfall patterns, is not necessarily valid even at steady state. Second, it predicts weaker correlations (more dispersed) between topographic metrics and erosion rate than would be expected if rainfall were always uniformly distributed, as is implied by use of basin-average rainfall (Figure 9), simply from neglecting the rainfall pattern. This prediction applies even before considering any geologic uncertainties (e.g., at quasi-steady-state or potentially transient?), analytical uncertainty, and even if no other variations in K exist. Because rainfall gradients create systematic, rather than random, dispersion around relationships expected for uniformly distributed rainfall, there is not necessarily any expectation that larger datasets will more accurately resolve variations in erosional efficiency unless catchments where rainfall is uniform are isolated, or a correction is made for the influence of spatially variable rainfall (e.g., usingksn-q ). Rather, compilation of topographic measurements from basins that do and do not experience rainfall gradients can, in and of itself, obscure or, depending on circumstances, distort the actual influence of rainfall on erosional efficiency (Figure 9). This result is of particular importance for designing future efforts to empirically test the SPM in natural settings.