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.