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 xzc 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.