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\begin{document}

\title{Influence of Spatial Rainfall Gradients on River Longitudinal Profiles
and the Topographic Expression of Spatio-temporally Variable Climate in
Mountain Landscapes}



\author[1,1]{Joel S Leonard}%
\author[1,1]{Kelin X Whipple}%
\affil[1]{Arizona State University}%


\vspace{-1em}



  
  \date{June 14, 2024}


\begingroup
\let\center\flushleft
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\endgroup





\selectlanguage{english}
\begin{abstract}
Mountain landscapes have dynamic climates that, together with tectonic
processes, influence their topographic evolution. While spatio-temporal
changes in rainfall are ubiquitous in these settings, their influence on
river incision is understudied. Here, we investigate how changes in
rainfall pattern should affect both the steady state form and transient
evolution of river profiles at the catchment scale using the stream
power model. We find that spatially varied rainfall can complicate
steady state relationships between mean rainfall, channel steepness and
fluvial relief, depending on where rainfall is concentrated in
catchments. As a result, transient profile adjustments to climate
changes may proceed contrary to typical expectations, which can
ultimately affect the apparent sensitivity of landscapes and erosion
rates to climate. Additionally, changes in rainfall pattern cause
inherently multi-stage transient responses that differ from responses to
uniform changes in rainfall. These results have important implications
for detecting transient responses to changes in rainfall pattern (and
more broadly climate), and for interpreting of landscape morphometrics
above and below knickpoints. Further, we find that disparate responses
by rivers that experience different rainfall conditions, particularly
trunk and tributary rivers, are an important factor in understanding
catchment-wide responses, and accounting for such disparities may be
important for detecting and quantifying landscape sensitivity to
variations in climate. Lastly, we show how explicitly accounting for
rainfall patterns in channel steepness indices, and thus variations in
erosional efficiency, has potential to help address challenges related
to spatially variable rainfall patterns and advance understanding of
landscape sensitivity to climate in mountain settings%
\end{abstract}%



\sloppy


\textbf{Influence of Spatial Rainfall Gradients on River Longitudinal
Profiles and the Topographic Expression of Spatio-temporally Variable
Climate in Mountain Landscapes}

Joel S. Leonard\textsuperscript{1} and Kelin X.
Whipple\textsuperscript{1 1}School of Earth and Space Exploration,
Arizona State University, Tempe, Arizona, USA

Corresponding Author: Joel S. Leonard (joel.leonard@asu.edu)

\textbf{Key Points}

\begin{itemize}
\tightlist
\item
  Spatially variable rainfall complicates steady state relationships
  between mean rainfall and conventional topographic and erosion
  metrics.
\item
  Transient responses to changes in rainfall pattern differ from uniform
  changes in rainfall, which affects how they may be detected.
\item
  Rainfall gradients can obscure the sensitivity of fluvial erosion to
  rainfall variations and impede quantification of climate sensitivity.
\end{itemize}

Keywords: \emph{River Profiles, Stream Power Model, Orographic Rainfall,
Erosional Efficiency, Erosion Rates, Channel Steepness}

\textbf{Abstract}

Mountain landscapes have dynamic climates that, together with tectonic
processes, influence their topographic evolution. While spatio-temporal
changes in rainfall are ubiquitous in these settings, their influence on
river incision is understudied. Here, we investigate how changes in
rainfall pattern should affect both the steady state form and transient
evolution of river profiles at the catchment scale using the stream
power model. We find that spatially varied rainfall can complicate
steady state relationships between mean rainfall, channel steepness and
fluvial relief, depending on where rainfall is concentrated in
catchments. As a result, transient profile adjustments to climate
changes may proceed contrary to typical expectations, which can
ultimately affect the apparent sensitivity of landscapes and erosion
rates to climate. Additionally, changes in rainfall pattern cause
inherently multi-stage transient responses that differ from responses to
uniform changes in rainfall. These results have important implications
for detecting transient responses to changes in rainfall pattern (and
more broadly climate), and for interpreting of landscape morphometrics
above and below knickpoints. Further, we find that disparate responses
by rivers that experience different rainfall conditions, particularly
trunk and tributary rivers, are an important factor in understanding
catchment-wide responses, and accounting for such disparities may be
important for detecting and quantifying landscape sensitivity to
variations in climate. Lastly, we show how explicitly accounting for
rainfall patterns in channel steepness indices, and thus variations in
erosional efficiency, has potential to help address challenges related
to spatially variable rainfall patterns and advance understanding of
landscape sensitivity to climate in mountain settings.

\textbf{Plain Language Summary}

Rainfall in mountain landscapes often varies with elevation; a pattern
known as orographic rainfall. Rivers that sculpt these landscapes rely
on rainfall for their erosive power, where more rainfall typically means
greater erosive power. Rainfall also affects how steep these rivers are,
which in turn affects the steepness of the topography around them. Here,
we investigate how concentrating rainfall at higher and lower elevations
-- representing two common orographic rainfall patterns that may be
enhanced or relaxed by climate change -- influences the steepness
mountain rivers, erosion patterns, and thus the evolution of mountain
topography. We show that these orographic rainfall patterns complicate
simple expected relationships among metrics commonly used to quantify
the role of rainfall (and more broadly climate) on the topography of
mountain landscapes. Further, we show that rivers respond in unexpected
ways to changes in orographic rainfall patterns, as would occur
following a change in climate, suggesting that common wisdom about how
rivers and mountain landscapes respond to changing climates is
incomplete.

\textbf{1. Introduction 1.1 Motivation}

Advances in tectonic geomorphology require quantitative understanding
about relationships among climate, tectonics, and erosion. In temperate
mountain landscapes, studies of bedrock rivers provide important
insights into interactions between these processes (e.g., D'Arcy \&
Whittaker, 2014; Harel et al., 2016; Kirby \& Whipple, 2012; Lague,
2014; Olen et al., 2016; Scherler et al., 2017; Whipple \& Tucker, 1999;
Whittaker, 2012). However, despite longstanding theoretical support for
the notion that climate, like tectonics, has a fundamental role in
influencing erosion (e.g., Bonnet \& Crave, 2003; Howard \& Kerby, 1983;
Lague, 2014; Molnar, 2001; Perron, 2017; Rinaldo et al., 1995; Tucker \&
Slingerland, 1997), a general relationship between climate and erosion
has proven elusive (Perron, 2017; Whittaker, 2012). Here, we explore the
extent to which this conundrum may reflect limitations in the current
framework describing how climate-related signals should be expressed in
landscapes that, in turn, may impede recognition of diagnostic
characteristics of landscape response to climate change.

Orographic precipitation patterns are ubiquitous in mountain landscapes.
In general, they develop from the interaction of humid air masses with
topographic relief and can create dramatic spatial and elevation
dependent gradients in precipitation (see Roe, 2005 for an overview).
For instance, the Olympic, Sierra Nevada, and Wasatch ranges in western
North America all experience orographically enhanced precipitation with
increasing elevation (e.g., Barros \& Lettenmaier, 1994; Barstad \&
Smith, 2005; Roe, 2005). Alternatively, large tracts along eastern and
southern flanks of the Andes and Himalaya, respectively, become more
arid as elevation increases (Anders et al., 2006; Bookhagen \& Burbank,
2010; Bookhagen \& Strecker, 2008; Burbank et al., 2003). While numerous
factors affect orographic precipitation patterns in detail, broadly
speaking, atmospheric moisture content, topographic characteristics
(e.g., relief), and general circulation patterns are primary physical
controls on their development (e.g., Held \& Soden, 2006; Roe, 2005; Roe
et al., 2008; Trenberth et al., 2003). Because atmospheric moisture
content depends strongly on temperature (i.e., Clausius-Clapeyron
relationship; Held \& Soden, 2006; Roe, 2005; Trenberth et al., 2003),
shifts in temperature that accompany changes in climate must influence
these precipitation patterns (e.g., Mutz et al., 2018; Roe \& Baker,
2006; Siler \& Roe, 2014). Therefore, if erosional processes in these
landscapes are generally sensitive to spatial and/or temporal variations
in precipitation, then changing characteristics of orographic
precipitation patterns with changes in climate should importantly
influence mountain landscape evolution.

In mountainous settings, transverse rivers tend to cross orographic
precipitation gradients, which are generally oriented orthogonally to
the topographic trend of the range. Their tributaries, on the other
hand, typically experience a relatively muted range in precipitation due
to their orientation and/or smaller areal extent. Consequently, rivers
of different size, orientation, and position often experience
dramatically different precipitation conditions. Within large river
basins, these differences may be substantial. Exposure to orographic
precipitation patterns is expected to systematically affect river
profile concavity; increases in precipitation with distance upstream
lowers profile concavity, while the opposite trend increases profile
concavity (Han et al., 2014, 2015; Roe et al., 2002, 2003; Ward \&
Galewsky, 2014). Changes in concavity driven by temporal changes in
orographic precipitation patterns require longitudinally variable
amounts of incision. Furthermore, because transverse rivers set
erosional base level for their tributaries, any along-stream variation
in incision exhibited by transverse rivers during this adjustment will
necessarily drive spatially and temporally variable base level histories
for tributaries. Developing a framework that accommodates such
variability and its influence on river profile evolution is a
fundamental need.

\textbf{1.2 Approach and Scope}

Here, we investigate how spatio-temporal changes in precipitation may
influence erosion and topography of mountain landscapes using the stream
power model (SPM). First, we show how simple spatial gradients in
rainfall, resembling typical orographic precipitation patterns (i.e.,
increasing or decreasing downstream; herein referred to as bottom-heavy,
and top-heavy, respectively), influence river profile form at steady
state in one dimension. Next, we use a quasi-two-dimensional numerical
model to simulate the response of a transverse river network to a change
in rainfall pattern, which we compare to better-understood spatially
uniform changes in rainfall. Finally, we discuss some implications for
studies set in mountain landscapes. A comprehensive analysis of the
co-evolution of orographic rainfall patterns and topography is beyond
the scope of this paper. Instead, we focus on characterizing the
controls on landscape response to imposed changes in rainfall patterns,
highlighting where expectations differ from uniform changes in rainfall,
and implications of those differences.

\textbf{2. Methods 2.1 Model Description}

We explore the influence of longitudinal rainfall gradients on large
transverse rivers first using a simple 1-dimensional river incision
model. We model erosion as detachment-limited (Howard, 1994; Roe et al.,
2002; Whipple \& Tucker, 1999) following a general form of the SPM:

\(E=KA^{m}S^{n},\ \) (1a)\(K=K_{p}{\overset{\overline{}}{P}}^{m},\) (1b)\(E=K_{p}Q^{m}S^{n},\) (1c)

where \emph{E} is the erosion rate; \emph{K}
and\emph{K\textsubscript{p}} are erosional efficiency
coefficients;\emph{A} is upstream drainage area; \emph{S} is the channel
slope;\(\overline{P}\) is the upstream average rainfall rate; \emph{Q}
is water discharge and is calculated as \(\overline{P}\)\emph{A} ,
which assumes that all rainfall is converted to runoff; and \emph{m}
and\emph{n} are positive constant exponents (Table 1). We use \emph{n} =
2 and \emph{m} = 1 for all model runs as values of \emph{n}
\textgreater{} 1 appear more appropriate in many settings (e.g., Adams
et al., 2020; Harel et al., 2016; Lague, 2014). First-order results do
not rely on choices of \emph{m} or \emph{n} providing the ratio between
the two is approximately maintained, but the nonlinear dependence of
erosion rate on slope (i.e., \emph{n} = 2) affects details of the
transient behavior. Also, because \emph{m} = 1, \emph{K} is directly
proportional to both\(\overline{P}\) and \emph{Q} . We explicitly
treat the influence of climate on erosional efficiency (e.g., Adams et
al., 2020; Roe et al., 2002) such that \emph{K\textsubscript{p}} is
independent of rainfall, but still encapsulates a number of factors
including rock properties and details of erosional processes (Royden \&
Perron, 2013; Whipple \& Tucker, 1999). Rock uplift rate (\emph{U} ) and
\emph{K\textsubscript{p}}are spatially and temporally uniform and
invariant across model runs.

We define drainage area (\emph{A} ) following Hack, (1957):

\(A=k_{a}x^{h}+A_{c},\) (2)

where \emph{x} is distance along the channel downstream from the
drainage divide, \emph{k\textsubscript{a}} and \emph{h} are constants,
and \emph{A\textsubscript{c}} is the upstream drainage area at the
channel head -- equal to 1 km\textsuperscript{2}. Channel length
(\emph{L} ) and drainage area are fixed and do not evolve over the
course of a model run.

For simplicity, we model orographic precipitation as constant gradients
in rainfall (i.e., linear changes with distance). Although a constant
gradient is a simplification, it is a reasonable approximation to
commonly observed orographic rainfall patterns, which can be both top-
and bottom-heavy (e.g., Anders et al., 2006; Bookhagen \& Burbank, 2010;
Bookhagen \& Strecker, 2008; Roe, 2005). Further, the framework we
develop from these simple rainfall patterns is generally applicable to
addressing more complex versions of the fundamental problem we address
here -- the effects of spatially concentrated rainfall.

\textbf{2.2 Analysis of River Profiles and Erosion Rates}

We quantify river profile form and responses to changes in rainfall
patterns using channel steepness indices and erosion rates. These
metrics are commonly used, and often in tandem, to study influences of
climate and/or tectonics in mountain settings (Adams et al., 2020;
Bookhagen \& Strecker, 2012; Cyr et al., 2014; DiBiase et al., 2010;
Duvall, 2004; Godard et al., 2014; Insel et al., 2010; Kober et al.,
2015; Morell et al., 2015; Olen et al., 2016; Ouimet et al., 2009;
Portenga et al., 2015; Safran et al., 2005; Scherler et al., 2014;
Vanacker et al., 2015; Willenbring et al., 2013).

A widely used metric to analyze river profiles, interpret erosion rates,
and make comparisons to the SPM is the normalized channel steepness
index, \emph{k\textsubscript{sn}} :

\(k_{\text{sn}}=SA^{\theta_{\text{ref}}},\) (3)

where \selectlanguage{greek}\emph{θ\textsubscript{ref}}\selectlanguage{english} is the reference concavity index
(Wobus et al., 2006). We use a value of \selectlanguage{greek}\emph{θ\textsubscript{ref}}\selectlanguage{english} =
0.5, which is common and consistent with our choice of\emph{m} /\emph{n}
, and also with SPM predictions that\selectlanguage{greek}\emph{θ\textsubscript{ref}}\selectlanguage{english} =
\emph{m/n} [?] 0.5 where rock uplift rate (\emph{U} ) and erosional
efficiency (\emph{K} ) are uniform (Tucker \& Whipple, 2002). As
previously noted, the SPM predicts that orographic rainfall gradients
should produce longitudinal variations in \emph{K}that, in turn, affect
the concavity index, \selectlanguage{greek}\emph{θ} \selectlanguage{english}(Han et al., 2014, 2015; Roe et al., 2002,
2003). Any such variations are systematically reflected in the spatial
pattern of \emph{k\textsubscript{sn}} and\selectlanguage{greek}\emph{θ} \selectlanguage{english}[?]
\selectlanguage{english}\selectlanguage{greek}\emph{θ\textsubscript{ref}}\selectlanguage{english} is expected. Importantly, however, many
studies relate upstream-average values of\emph{k\textsubscript{sn}} to
measured spatially-averaged erosion rates, which relies on quasi-uniform
(or linear) upstream\emph{k\textsubscript{sn}} to be valid (Wobus et
al., 2006). In cases where systematic longitudinal variations in
\emph{K} affect the downstream pattern of \emph{k\textsubscript{sn}}
(i.e., upstream\emph{k\textsubscript{sn}} varies non-linearly), the
meaning of such an average is not obvious.

To address this, we use discharge, rather than drainage area alone, to
calculate a modified channel steepness
index\emph{k\textsubscript{sn-q}} (Adams et al., 2020):

\(k_{sn-q}=SQ^{\theta_{\text{ref}}}.\) (4)

Like \emph{k\textsubscript{sn}} , \emph{k\textsubscript{sn-q}} is an
empirically supported metric independent from the SPM. In principle,
however, \emph{k\textsubscript{sn-q}} is analogous to Erosion Index
(EI) used by Finlayson et al., (2002) provided that \emph{m/n}
[?]\selectlanguage{greek}\emph{θ\textsubscript{ref}}\selectlanguage{english} , such that EI =
(\emph{k\textsubscript{sn-q}} )\emph{\textsuperscript{n}} . Also,
as\emph{k\textsubscript{sn}} is the slope of \selectlanguage{greek}χ-\selectlanguage{english}transformed river
profiles in \selectlanguage{greek}χ-\selectlanguage{english}elevation space, if \selectlanguage{greek}χ \selectlanguage{english}is redefined to include
precipitation to estimate discharge, slopes of \selectlanguage{greek}χ-\selectlanguage{english}transformed profiles
would instead represent \emph{k\textsubscript{sn-q}} (Royden \& Perron,
2013; Yang et al., 2015). To the extent that the SPM captures the
influence of discharge on erosional efficiency, it predicts that
along-stream variations in \emph{k\textsubscript{sn-q}} should scale
with local erosion rate, precisely as it does for
\emph{k\textsubscript{sn}} where\emph{K} is spatially uniform.
Hereafter, we use\emph{k\textsubscript{sn}} and
\emph{k\textsubscript{sn-q}} to refer to upstream averaged values,
consistent with their common usage in catchment-mean erosion rates
analyses, unless we specifically state that they represent local values.

Millennial-scale catchment-averaged erosion rates measured, for example,
using cosmogenically-derived \textsuperscript{10}Be found in quartz in
alluvial sediment (e.g. Bierman \& Steig, 1996; Brown et al., 1995;
Granger et al., 1996), seek to quantify erosion rates at the river basin
scale. At steady state, spatially averaged erosion rate, local incision
rate, and rock uplift rate are equivalent; however, during periods of
transient adjustment these values differ, complicating interpretations
(Willenbring et al., 2013; Wobus et al., 2006). To make our results more
portable to studies of natural landscapes, we calculate the spatially
averaged erosion rate (\emph{E\textsubscript{avg}} ) in addition to the
instantaneous vertical incision rate (\emph{E} ):

\({E_{\text{avg}}}_{j}=\frac{\sum_{x_{h}}^{j}\left(E_{j}\ \bullet\left(A_{j}-A_{j-1}\right)\right)}{A_{j}},\)(5)

where \emph{j} corresponds to a downstream node of the profile,
and\emph{x\textsubscript{h}} is the channel head.

\textbf{3. Longitudinal Profiles (1-D) 3.1 Influence of Longitudinal
Rainfall Gradients on River Profiles at Steady State}

Where rainfall is spatially uniform, topographic metrics (e.g., fluvial
relief, channel steepness) at steady state are expected to vary
inversely and monotonically with mean rainfall (Figure 1a). However,
spatially variable rainfall patterns complicate these expectations, as
shown in Figure 1b, where comparisons between rivers that experience
different rainfall patterns instead result in positive relationships
between these topographic metrics and mean rainfall. This reversal
reflects limitations of using spatially averaged metrics where climate
is spatially variable (e.g., in most mountain landscapes).

Systematic longitudinal variations in rainfall require that upstream
average rainfall values change systematically downstream, which
similarly affects erosional efficiency (\emph{K} ), and thus equilibrium
channel slope (Equation 1). Where such spatial variations exist, mean
values of rainfall and \emph{k\textsubscript{sn}} therefore depend on
where they are measured. In contrast, where equation 1 holds and\emph{m}
/\emph{n} = \selectlanguage{greek}\emph{θ\textsubscript{ref}}\selectlanguage{english} ,\emph{k\textsubscript{sn-q}} is
independent of changes in mean rainfall (Figure 1). Comparison of SPM
equations for \emph{k\textsubscript{sn}}and \emph{k\textsubscript{sn-q}}
at steady state (\emph{E} = \emph{U} ) further clarifies this
difference:

\(k_{\text{sn}}=\ \left(\frac{U}{K_{p}{\overset{\overline{}}{P}}^{m}}\right)^{1/n}\), (6a)\(k_{sn-q}=\ \left(\frac{U}{K_{p}}\right)^{1/n}\). (6b)

For spatially uniform rock uplift rate (\emph{U} )
and\emph{K\textsubscript{p}} , steady state fluvial relief (\emph{R} )
is proportional to the upstream integrated discharge (Han et al., 2015;
Roe et al., 2003; Royden \& Perron, 2013). Integrating equation 1c from
base level (\emph{x\textsubscript{b}} ) upstream to the channel head
(\emph{x\textsubscript{h}} ), it can be shown that:

\(R=\ \left(\frac{U}{K_{p}}\right)^{1/n}\int_{x_{b}}^{x_{h}}{Q^{-m/n}\text{\ dx}}\). (7)

This demonstrates clearly how fluvial relief depends on the cumulative
effect of discharge and implies that fluvial relief does not necessarily
scale monotonically with discharge or rainfall measured at any single
position, or averaged along any segment of a profile, except under the
special condition where rainfall is spatially uniform (Gasparini \&
Whipple, 2014; Han et al., 2015). This is an important result,
particularly for understanding the topographic evolution of mountain
landscapes because it suggests that considering how rainfall patterns,
specifically, have changed with time is critical to predicting responses
to changes in climate. For instance, shifts toward `wetter' climates may
support topographic growth, contrary to expectations and even in the
absence of any change in tectonics, depending on where rainfall is
concentrated, or vice versa.

\textbf{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\emph{.}

Changes in discharge at a given location following a temporal change in
rainfall pattern reflect changes upstream average rainfall conditions.
(Hereafter we use subscripts \emph{i} and \emph{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 (\emph{Q\textsubscript{f}}
\textgreater{}\emph{Q\textsubscript{i}} ) but decrease in downstream
locations (\emph{Q\textsubscript{f}} \textless{}
\emph{Q\textsubscript{i}} ), or vice versa.

We refer to the position of such a reversal (e.g. from increasing to
decreasing discharge or vice versa) as \emph{x\textsubscript{sc}} . At
this position discharge remains constant, and thus equilibrium river
slope does not change following a temporal change in rainfall pattern
(at \emph{x} = \emph{x\textsubscript{sc}} , \emph{Q\textsubscript{f}}
=\emph{Q\textsubscript{i}} and \emph{S\textsubscript{f}}
=\emph{S\textsubscript{i}} ). 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 \emph{x\textsubscript{sc}} initial and final steady state
profiles begin to converge (see Figure 2).
Thus,\emph{x\textsubscript{sc}} 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 and\emph{K\textsubscript{p}}
, the maximum difference in elevation along the profile between initial
and final steady states, \selectlanguage{greek}Δ\selectlanguage{english}\emph{z\textsubscript{Sc}} , 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 \emph{x\textsubscript{zc}} (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, \emph{x\textsubscript{zc}} 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 \emph{x\textsubscript{zc}} are
those that lead to positive relationships between spatially averaged
mean rainfall and fluvial relief (Figure 1b).

\textbf{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 \emph{U} , \emph{K\textsubscript{p}} ,
\emph{m} , and \emph{n} provided \emph{m} /\emph{n} is unchanged.
Channel length has a negligible influence for channels longer than a few
kilometers where\emph{A} \textgreater{}\textgreater{}
\emph{A\textsubscript{c}} , and only minor influence for different
\emph{m} /\emph{n} ratios (\textasciitilde{}0.4-0.6). The \emph{h}
exponent in Hack's Law (Equation 2) can influence field boundaries, as
indicated in Figure 3a; however, the effect is minor for typical
\emph{h} values (1.67 [?]\emph{h} [?] 2; e.g., Rigon et al., 1996). Contours
of\emph{Q\textsubscript{f} /Q\textsubscript{i}} illustrate the extent to
which a given rainfall gradient represents a net wetter
(\emph{Q\textsubscript{f}} /\emph{Q\textsubscript{i}} \textgreater{} 1)
or drier (\emph{Q\textsubscript{f}}
/\emph{Q\textsubscript{i}}\textless{} 1) condition (Figure 3b). Contours
of\emph{R\textsubscript{f} /R\textsubscript{i}} describe the extent to
which fluvial relief increases
(\emph{R\textsubscript{f}}/\emph{R\textsubscript{i}} \textgreater{} 1)
or decreases (\emph{R\textsubscript{f}} /\emph{R\textsubscript{i}}
\textless{} 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 \emph{x\textsubscript{sc}} , but
not\emph{x\textsubscript{zc}} \textsubscript{}). The mode of transient
adjustment is variable in space and time upstream
of\emph{x\textsubscript{sc}} (variably \emph{E} \textgreater{} \emph{U}
or\emph{E} \textless{} \emph{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. \emph{x\textsubscript{sc}} 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 \emph{x\textsubscript{sc}} and \emph{x\textsubscript{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
from\emph{x\textsubscript{sc}} . 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
at\emph{x\textsubscript{sc}} 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 \emph{x\textsubscript{sc}} 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 (\emph{R\textsubscript{f}} /\emph{R\textsubscript{i}} always
\textless{} 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.,\emph{K\textsubscript{p}} , \emph{U, h} , \emph{L} ). We find that
a temporal change in rainfall pattern alone may support as much as
\textasciitilde{}10\textsuperscript{2}--10\textsuperscript{3} 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
\textasciitilde{}10\textsuperscript{1}--10\textsuperscript{2} 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.

\textbf{4. Transient Catchment Response to Changes in Rainfall Patterns}

Many of the ideas laid out above for 1-dimensional profiles intuitively
transfer to understanding how signals related to changing rainfall
patterns propagate through a drainage network, albeit with some
additional considerations. It is important to remember that trunk
(transverse) rivers control base level for tributaries. Complex
responses, like those described above where both the magnitude and mode
of transient adjustment vary along the trunk profile in space and time,
necessarily result in varying boundary conditions for tributaries. In
addition, tributary responses to these variable base level signals are
modulated by the rainfall history experienced by a given tributary,
which is always different from the trunk. Finally, adjustments migrate
upstream at a finite rate, so there is a time lag between a change in
rainfall pattern and arrival of the associated base level signals from
the trunk adjustment to a given tributary. The duration of this lag, as
well as local rainfall conditions within a tributary catchment, are a
function of its position.

Following Riihimaki et al. (2007), we use a quasi-two-dimensional model
to explore catchment response. We abstract river basin topology to
comprise a single one-dimensional trunk profile and 51 regularly spaced
(1 km spacing) one-dimensional tributary profiles. Tributary outlets are
fixed to the elevation of the trunk profile at their confluence.
Discharge does not pass from tributaries to the trunk river as drainage
area along the trunk and each tributary follows Hack's Law (equation 2)
independently. This means that there is not two-dimensional hydrological
coupling between trunk and tributary rivers through discharge, but
transient signals are communicated through variations in tributary base
level. This approach allows closely spaced tributaries with identical
characteristics to isolate the influence of spatial variations in
rainfall within the larger catchment that would not be possible to the
same degree with 2-dimensional modelling approaches.

Although our model setup is abstract, it allows us to compare expected
patterns of erosion and channel response along the trunk and within
small trunk-stream tributaries in a way that is portable to natural
river networks. Because drainage area increases along the modeled trunk
river following Hack's Law, independent of the distribution of modeled
tributaries, at each point along the trunk river discharge accumulation
approximates that of a typical drainage basin aligned along the
orographic rainfall gradient. In this light, the trellis configuration
of tributaries we model is representative of a subset of small,
approximately trellised trunk-stream tributaries that typically exist
within more complex river network structures and are often targeted for
sampling of detrital sediment (e.g., Ouimet et al., 2009). These small
tributaries also do not contribute significantly to downstream increases
in drainage area (or discharge) of larger rivers into which they drain,
meaning discharge accumulation along large rivers is largely decoupled
from the hydrology of small tributaries. Therefore, the simplifications
we make to the hydrology are also generally consistent with conditions
created by targeting small tributaries situated along large rivers in a
trellis-like fashion.

In the following, we explore the transient response of modelled river
basins to four representative climate change scenarios: (1) a spatially
uniform decrease in rainfall, (2) a spatially uniform increase in
rainfall, (3) a shift from spatially uniform rainfall to a bottom-heavy
rainfall gradient, and (4) a shift from spatially uniform rainfall to a
top-heavy rainfall gradient. The initial condition for all models is
steady state with spatially uniform rainfall. Imposed changes in
rainfall evolve at a linear rate over the first 10 kyr of model time at
all points along each channel. Tributaries are modelled with spatially
uniform rainfall set by their position along the trunk profile, and
individual tributaries experience spatially uniform changes in rainfall
as rainfall gradients evolve. This simplification is consistent with the
notion that, due to their smaller areal extent and orientation,
tributaries set within mountain-belt scale orographic rainfall patterns
generally experience relatively uniform rainfall. As we show in section
5.1, more realistic scenarios, such as the intensification or relaxation
of an existing orographic rainfall gradient, produce analogous behavior
to these simple scenarios. The simplicity of the idealized scenarios
described here makes them especially effective for developing intuition
about response characteristics in general.

\textbf{4.1 Case 1: Spatially Uniform Decrease in Rainfall}

The modelled catchment response to a spatially uniform decrease in
rainfall is characterized by adjustment to uniformly
higher\emph{k\textsubscript{sn}} and an increase in steady state fluvial
relief. For this model run, rainfall is decreased from 2 to 1 m/yr,
causing discharge to decrease similarly by 50\% (Movie S1).

Erosion rate decreases across the entire channel network following the
change in rainfall, driven by a decrease in erosional efficiency
directly proportional in magnitude (i.e., 50\%; recall \emph{m} = 1).
The resulting disequilibrium, with \emph{E} \textless{} \emph{U} ,
drives surface uplift and upstream migration of a convex-up slope-break
knickpoint, and eventually a \textasciitilde{}40\% increase in fluvial
relief.

In \emph{k\textsubscript{sn}--E\textsubscript{avg}} space (Figure 4a),
the decrease in erosional efficiency driven by the change rainfall
causes the trunk river and tributary network to shift uniformly onto a
different erosional efficiency curve that describes the expected
relationship between \emph{k\textsubscript{sn}} and erosion rate for a
given erosional efficiency at steady state. Specifically, both shift
from \emph{K=2} \selectlanguage{ngerman}·\emph{K\textsubscript{p}} to
\emph{K=K\textsubscript{p}}(\emph{K\textsubscript{f}} =
0.5·\emph{K\textsubscript{i}} ). Following this initial shift, the trunk
river and individual tributaries approximately follow this new curve
(\emph{K=K\textsubscript{p}} ) during adjustment toward higher erosion
rates and \emph{k\textsubscript{sn}} to return to steady state, although
in detail they deviate slightly. This deviation is a result of averaging
segments above and below the migrating knickpoint into mean
(upstream-averaged)\emph{k\textsubscript{sn}} and erosion rate values.
Insofar as the relationship between channel steepness and erosion rate
implies a given erosional efficiency, these deviations imply an apparent
erosional efficiency different, albeit minor in this case, from the
modelled value.

In \emph{k\textsubscript{sn-q}} --\emph{E\textsubscript{avg}} space
(Figure 4b), both the initial and final steady state conditions plot in
the same location on the \emph{K=K\textsubscript{p}} curve. Here, the
initial decrease in erosional efficiency causes the trunk and
tributaries to shift uniformly along this curve to
lower\emph{k\textsubscript{sn-q}} and erosion rate rather than shifting
onto a different curve. During adjustment both generally follow this
curve to return to steady state, minor transient deviations as seen
for\emph{k\textsubscript{sn}} notwithstanding.

\textbf{4.2 Case 2: Spatially Uniform Increase in Rainfall}

The modelled catchment response to a spatially uniform increase in
rainfall is characterized by adjustment to uniformly
lower\emph{k\textsubscript{sn}} and a decrease in steady state fluvial
relief. To compare responses, we invert the change in rainfall from the
previous example, and rainfall is increased from 1 to 2 m/yr, resulting
in a 100\% increase in discharge (Movie S2).

We observe a broadly symmetrical response to Case 1, where the twofold
increase in rainfall leads to an initial twofold increase in erosion
rate (\emph{E} \textgreater{} \emph{U} ) and a \textasciitilde{}30\%
decrease in steady state fluvial relief. The transient knickpoint is
concave-up in this case and it broadens as it migrates upstream, as is
expected for a concave-up knickpoints where \emph{n} \textgreater{} 1
(Royden \& Perron, 2013). The signal of transient adjustment
communicated to tributaries is consequently protracted, making
adjustments more diffuse.

Responses reflected in\emph{k\textsubscript{sn}}
--\emph{E\textsubscript{avg}} and\emph{k\textsubscript{sn-q}}
--\emph{E\textsubscript{avg}} relationships also mirror Case 1. As is
characteristic for \emph{k\textsubscript{sn}} , the initial change in
erosional efficiency causes the trunk and tributary network to shift
onto a different steady state erosional efficiency curve; in this case,
from \emph{K} =\emph{K\textsubscript{p}} to \emph{K} =
2·\emph{K\textsubscript{p}} , and they generally follow this curve
during adjustment (Figure 4a). Minor deviations from this curve exhibit
a convex-up pattern (inverted from Case 1) due to the opposite
knickpoint shape. Meanwhile in\emph{k\textsubscript{sn-q}}
--\emph{E\textsubscript{avg}} space (Figure 4b), the initial and final
equilibrium conditions for both the trunk and tributary network plot in
the same location, as in Case 1. The change in rainfall again causes a
shift only along the \emph{K=K\textsubscript{p}}curve, but to uniformly
higher \emph{k\textsubscript{sn-q}} and erosion rate in this case, and
they generally follow this curve during adjustment back to steady state.

\textbf{4.3 Case 3: Spatially Uniform to Bottom-heavy}

In this scenario, we model the catchment response to a change in
rainfall pattern from spatially uniform 1.5 m/yr to a gradient that
decreases upstream from 4 to 0.5 m/yr, resulting in a complex transient
response. Mean rainfall increases by \textasciitilde{}80\%, which
remarkably also drives a 30\% increase in fluvial relief that contrasts
with the 25\% decrease expected for a spatially uniform increase in
rainfall of the same magnitude (Figure 3b; Movie S3).

\emph{4.3.1 Case 3: Trunk Response}

In this case, because the change in rainfall is spatially variable along
the trunk river, the initial change in erosion rate is also variable.
The trunk river experiences an approximately 80\% increase in erosion
rate at the outlet (\emph{E \textgreater{} U} ), and a decrease of 67\%
in the headwaters (\emph{E \textless{} U} ), corresponding to the change
in upstream average rainfall along its length.
At\emph{x\textsubscript{sc}} (located \textasciitilde{}27 km upstream
from the outlet), \emph{Q\textsubscript{f}} = \emph{Q\textsubscript{i}}
,\emph{S\textsubscript{f}} = \emph{S\textsubscript{i}} , and so
immediately following the change in rainfall \emph{E} = \emph{U} .
Enhanced incision at the outlet produces a concave-up knickpoint;
however, as this knickpoint migrates upstream it progressively sharpens
and eventually evolves into an oversteepened convex-up knickpoint,
contrasting with expectations for the increase in rainfall (e.g\emph{.}
, Case 2). Oversteepening is a consequence of the upstream decrease in
erosional efficiency driven by the rainfall gradient that is exacerbated
by, but does not depend on, differing modes of adjustment upstream and
downstream of \emph{x\textsubscript{sc}} related to the complex
response. This is analogous to knickpoint behavior described by Forte et
al. (2016) and Darling et al. (2020) where modelled lithologic contacts
demarcate similar relative variations in erosional efficiency (i.e. hard
rocks over soft rocks). Lastly, we note that everywhere upstream
of\emph{x\textsubscript{sc}} over-adjusts during the transient response,
which is a characteristic of complex responses in general, leading to
variable modes of adjustment in time and space. The overadjustment we
observe is essentially the whiplash response described by Gasparini et
al. (2006, 2007), but notably results here without sediment flux. This
continuous evolution of the trunk knickpoint has important consequences
for signals passed to tributaries.

\emph{4.3.2 Case 3: Tributary Response}

Tributary responses to the change in rainfall pattern depend largely on
their position. Individual tributaries experience changes in erosional
efficiency proportional to their change in rainfall. Changes to both
quantities are always different from those of the trunk river at their
confluence (Figure 2a inset). Additionally, tributaries also respond to
changing boundary conditions related to adjustment of the trunk river.
These signals are often conflicting. For example , enhanced incision
along the trunk river downstream from \emph{x\textsubscript{sc}} causes
tributaries there to experience a relative increase in the rate of
base-level fall. Alone, this should promote steepening, but higher
erosional efficiency (higher rainfall) counteracts steepening. The net
effect of this competition plays out differently as a function of
tributary position as 1) the discrepancy between the local rainfall
conditions experienced by tributaries and the upstream averaged rainfall
experienced by the trunk profile narrows upstream (Figure 2a inset), 2)
the transient base level signal (i.e., trunk knickpoint) changes shape
as it sweeps upstream, and 3) the duration of transient adjustment
increases upstream.

\emph{4.3.3 Case 3: k\textsubscript{sn}--E\textsubscript{avg} and
k\textsubscript{sn-q}--E\textsubscript{avg}}

Plots of \emph{k\textsubscript{sn}} --\emph{E\textsubscript{avg}}
and\emph{k\textsubscript{sn-q}--E\textsubscript{avg}} clarify some
additional features of the transient response (Figure 5). The trunk
profile exhibits higher \emph{k\textsubscript{sn}}
and\emph{k\textsubscript{sn-q}} values and spans a relatively narrower
range in erosion rates than tributaries during transient adjustment,
reflecting the fundamentally different ways they experience the modelled
rainfall gradient. This behavior illustrates that the network of
tributaries (isolated catchments that individually experience relatively
uniform rainfall but collectively span a range of conditions) inherently
incorporates a more direct signal of the change in rainfall patterns
compared to the trunk, which averages upstream rainfall variations. This
is consistent with findings by Han et al. (2015) for steady state
landscapes exposed to orographic rainfall and is important for designing
an effective sampling strategy in the field -- discussed further in
section 5.3.

In \emph{k\textsubscript{sn}} --\emph{E\textsubscript{avg}} space
(Figures 5a, 5b), shifts onto different erosional efficiency curves
occur as in Cases 1 \& 2, but here the spatial rainfall variability
causes different positions along the trunk and individual tributaries
shift by different amounts. The trunk profile response spans
from\emph{K} = 0.5·\emph{K\textsubscript{p}} to
\textasciitilde{}1.8·\emph{K\textsubscript{p}} , while the network of
tributaries spans from \emph{K} = 0.5·\emph{K\textsubscript{p}} to
4·\emph{K\textsubscript{p}} . To first order, points representing a
given location along the trunk or a given tributary move along these
curves reflecting local erosional efficiency during adjustment toward
steady state as in Cases 1 \& 2, but again deviate in detail.
Trajectories are more complex in this case because of the interplay
between changes in erosional efficiency and baselevel variations in
modulating tributary channel steepness and the over-adjustments
mentioned previously (\emph{c.f.} , Figures 4 \& 5). Finally, we note an
interesting feature where the range of erosional efficiency values that
correspond to upstream mean \emph{k\textsubscript{sn}} at steady state
for the trunk profile (\emph{K} = 0.5·\emph{K\textsubscript{p}} to
\textasciitilde{}1.8·\emph{K\textsubscript{p}} ) are lower than that
implied by mean rainfall (i.e., \emph{K} =
\textasciitilde{}2.7·\emph{K\textsubscript{p}} ; Figures 1b, 5a) --
discussed further in section 5.2.

In \emph{k\textsubscript{sn-q}} --\emph{E\textsubscript{avg}} space
(Figures 5c, 5d), differences between trunk and tributary responses
and\emph{k\textsubscript{sn}} vs. \emph{k\textsubscript{sn-q}} are
readily apparent. Initial and final steady state conditions plot in the
same position, as is characteristic of \emph{k\textsubscript{sn-q}}
where uplift rate is constant. Following the change in rainfall, the
trunk profile expands slightly obliquely to the
\emph{K=K\textsubscript{p}}curve, where downstream locations are
systematically shifted toward higher erosional efficiency. This shift
reflects systematic slope adjustments that must occur along the trunk to
bring it into equilibrium with the non-uniform rainfall pattern and
decreases with time as these adjustments take place. Apart from this
minor shift, different positions along the trunk profile generally
follow the \emph{K=K\textsubscript{p}}curve during adjustment back to
steady state. In tributaries, on the other hand, the change in rainfall
causes expansion precisely along the\emph{K=K\textsubscript{p}} curve
because they experience no along-stream variations in rainfall.
Tributaries again generally evolve along the \emph{K=K\textsubscript{p}}
curve toward steady state (as with cases 1 and 2). Transient
morphological adjustments affect this trajectory in detail and
deviations, which affect apparent erosional efficiency, are more
significant in drier tributaries near the headwaters. Overadjustment is
also evident for both the trunk and tributaries in this space, but
because transient evolution is generally along the steady state curve,
it does not significantly affect the apparent erosional efficiency. As a
final note, dispersion around the steady state erosional efficiency
curve in\emph{k\textsubscript{sn-q}} --\emph{E\textsubscript{avg}} space
is minor over the duration of the transient adjustment compared to
dispersion in\emph{k\textsubscript{sn}} --\emph{E\textsubscript{avg}}
space -- we expand on implications from this point in section 5.3.

\textbf{4.4 Case 4: Spatially Uniform to Top-heavy}

In this final case, we the model the catchment response to a change in
rainfall pattern from spatially uniform 1.5 m/yr to a gradient that
increases upstream from 0.25 to 2.25 m/yr, which also results in a
complex transient response. Mean rainfall decreases by 33\% and fluvial
relief also decreases by 10\% (Figure 3; Movie S4), which contrasts with
the 22\% increase in fluvial relief expected for a spatially uniform
decrease in rainfall of the same magnitude.

\emph{4.4.1 Case 4: Trunk Response}

In this case, the trunk river experiences a 33\% initial decrease in
erosional efficiency and erosion rate at the outlet, and a 50\% increase
in the headwaters following the change in rainfall pattern (Figure 6).
This range is significantly narrower than the range of variations in
rainfall because of the buffering effect that concentrating rainfall in
the headwaters has on downstream changes in discharge (e.g., Figure 2b
inset). Like Case 3, because this scenario also exhibits a complex
transient response, the transition between decreases in erosion rate
downstream and increases upstream is initially at
position\emph{x\textsubscript{sc}} . Unlike Case 3, knickpoint shape
does not invert during transient adjustment and is always convex-up.
Interestingly, combined with observations from Case 3, this suggests
that complex responses generally are more likely to exhibit convex-up
knickpoints. Retention of the knickpoint shape is accommodated by
over-adjustment upstream of \emph{x\textsubscript{sc}} , specifically by
progressively more rapid upstream adjustment (higher erosional
efficiency) toward gentler slopes that outpaces downstream adjustment.
By the time that the migrating trunk knickpoint reaches positions
upstream of \emph{x\textsubscript{sc}} , the profile has incised below
its equilibrium elevation and is shallower than its equilibrium slope.
Thus, the profile is forced to uplift and steepen to reach steady state,
preserving the convex-up knickpoint shape (Movie S4).

\emph{4.4.2 Case 4 Tributary Response}

Complexities in the trunk response again dramatically affect the
tributary responses, and generally mirror complexities discussed in Case
3. Tributaries again respond to variable and commonly conflicting
signals. In this case, tributaries downstream
from\emph{x\textsubscript{sc}} initially respond to a decrease in
rainfall (lower \emph{} erosional efficiency) by steepening. However,
decreases in erosional efficiency along the trunk profile drives
steepening and surface uplift, forcing tributaries to respond to a
conflicting signal of base-level rise (Movie S4). Additionally,
tributaries in downstream locations are relatively much drier and adjust
relatively slowly compared to upstream locations. Interestingly, the
trunk profile adjusts and communicates transient base-level signals
upstream to wetter tributaries relatively quickly compared to the
adjustment timescale of these dry tributaries, which follows as the
trunk river has much higher erosional efficiency from rainfall
concentrated in its headwaters. The initially counterintuitive result of
this is that central portions of the catchment are the first to achieve
steady state (i.e., both the trunk profile and tributaries achieve the
new steady state), followed by the headwaters, and lastly tributaries
near the outlet where the transient signal originated, which contrasts
with the common expectation of adjustment proceeding in an upstream
fashion.

\emph{4.3.3 Case 4: k\textsubscript{sn}--E\textsubscript{avg} and
k\textsubscript{sn-q}--E\textsubscript{avg}}

Evolution of\emph{k\textsubscript{sn}}
\textsubscript{}--\emph{E\textsubscript{avg}}and
\emph{k\textsubscript{sn-q}} --\emph{E\textsubscript{avg}}relationships
during the transient response are generally similar to Case 3, but with
a few exceptions. First, the disparity between final steady state
conditions for the trunk profile and network of tributaries is
significantly greater than Case 3 (Figures 6a, 6b), which reflects the
extent to which the rainfall pattern buffers variations in erosional
efficiency along the trunk. Also, in\emph{k\textsubscript{sn-q}}
--\emph{E\textsubscript{avg}} space, apparent erosional efficiency is
more strongly affected during transient adjustment in the tributary
network compared to the trunk profile (transient deviations from the
\emph{K=K\textsubscript{p}} curve), but also compared to Case 3. Large
deviations are again restricted to drier tributary catchments, and
comparison to Case 3 expresses that changes in rainfall have a
non-linear effect (Equations 1). That said, even these stronger effects
on apparent erosional efficiency in\emph{k\textsubscript{sn-q}}
--\emph{E\textsubscript{avg}} space are still minor compared to
representing any equivalent time in the\emph{k\textsubscript{sn}}
\textsubscript{}--\emph{E\textsubscript{avg}}relationship with a
spatially and temporally uniform erosional efficiency value. Finally,
also like Case 3, the range of erosional efficiency values for the trunk
profile (\emph{K} = \textasciitilde{}1.8·\emph{K\textsubscript{p}} to
2.25·\emph{K\textsubscript{p}} ) are different, but in this case higher
than is implied by mean rainfall (i.e., \emph{K} =
\textasciitilde{}\emph{K\textsubscript{p}} ; Figures 1b, 6a) --
discussed further in section 5.2.

\textbf{5. Discussion}

The central themes we have explored so far are how the spatial rainfall
pattern influences the channel profile morphology, and how temporal
changes in rainfall pattern affect erosion rates and profile morphology
during periods of transient adjustment. We have detailed the expected
response to along-stream variations in erosional efficiency caused by
spatial rainfall gradients according to the SPM and have shown how the
transient response to a change in rainfall pattern is fundamentally
different from a spatially uniform change in rainfall. A change in
rainfall pattern will always result in spatially variable changes of
erosion rates that also change with time during the transient response.
In some circumstances a given location may, over time, experience both
elevated and reduced erosion rates (and channel steepness values)
relative to equilibrium in response to a single change in rainfall. It
is important to keep in mind, however, that the nature of transient
response depends strongly on the initial conditions at the time of the
change in rainfall pattern. Therefore, there is a complex relationship
between the transient response at any given location or time and both
the change in mean rainfall and the final rainfall pattern. In the
following discussion, we focus on highlighting some implications for the
different expectations that follow from changes in rainfall pattern and
discussing examples where conventional expectations based on spatially
uniform changes in rainfall can potentially lead researchers astray.
Where possible, we attempt to identify additional information or
strategies that may be leveraged by future studies.

\textbf{5.1 Revising Expectations for Erosional and Morphological
Responses to Changing Climate} \emph{5.1.1 Relative Nature of Erosional
Response}

To this point, our choice of a steady state initial condition with
spatially uniform rainfall has been convenient, as have been the terms
top-heavy and bottom-heavy to describe typical orographic rainfall
patterns. While idealized, this provides an intuitive starting point for
understanding how more complicated -- but almost certainly more
realistic -- climate change scenarios might play out. Recall, according
to the SPM, transient climate-driven changes in erosion rate are
dictated by a \emph{relative} change in discharge. Where discharge is
increased, erosion rates increase in response and river gradient
declines toward a new equilibrium steepness; thus, a river subjected to
an increase in discharge can be considered locally, if transiently,
oversteepened relative to equilibrium, and vice versa. As we have shown,
because discharge generally accumulates non-linearly downstream within a
river basin, a change in rainfall pattern can create circumstances where
the relative change in discharge inverts along the river length -- at
position \emph{x\textsubscript{sc}} -- producing a complex transient
response (Figures 2 \& 3). This implies that the river is simultaneously
oversteepened and understeepened on either side of
position\emph{x\textsubscript{sc}} . These transient states dictate
whether erosion rates initially increase or decrease following the
change in rainfall, respectively, not whether the new rainfall pattern
is itself top-heavy or bottom-heavy, and the positions of these
transient states shift throughout adjustment.

The nature of landscape response to relative changes in discharge
implies, for instance, that relaxation of a bottom-heavy rainfall
gradient can cause a complex transient response resembling a change from
uniform to top-heavy rainfall patterns. That is, a weaker bottom-heavy
gradient is relatively top-heavy compared to an extreme bottom-heavy
gradient; similarly, a gentler top-heavy gradient is relatively
bottom-heavy compared to an extreme top-heavy gradient, and vice versa
(Figure 7). Thus, for example, in the case of a change in climate that
causes an extreme bottom-heavy rainfall gradient to become less
bottom-heavy \emph{and} results in a complex transient response (e.g.,
Figure 7a), rainfall and erosion rate are expected to increase in the
headwaters of the catchment and decrease near the outlet as seen for
Case 4 (uniform to top-heavy). This response is not consistent with
expectations for any uniform increase or decrease in rainfall, even if
such a shift accurately reflects the change in mean rainfall. Therefore,
neither the final rainfall pattern alone (i.e., modern observed pattern)
nor accurate inference about the relative change in mean rainfall
(wetter or drier) necessarily allow a robust prediction of changes in
erosion rate within a catchment following a change in climate where
rainfall patterns have changed significantly.

Interestingly, changes in climate do not need to involve extreme changes
in rainfall patterns (e.g., reversal from top-heavy to bottom-heavy), or
to occur over short timescales to drive complex transient responses.
Indeed, even subtle changes in rainfall pattern potentially driven by
minor, commonly occurring variations temperature and atmospheric
conditions (e.g., Mutz et al., 2018; Roe et al., 2003; Siler \& Roe,
2014), may induce complex responses and significantly, if temporarily,
alter the spatial pattern of erosion in a catchment (Figures 3 \& 7).
Indeed, such climate changes may have occurred in the Peruvian Andes and
eastern-central Himalaya in the transition from Pliocene to Pleistocene
climates, the latter represented by Last Glacial Maximum conditions
(LGM; Figure 7c \& 7d). Even if rivers in each of these ranges were in a
transient state during Pliocene time, any adjustment toward equilibrium
with the Pliocene rainfall pattern that occurred would then be in
disequilibrium with the Pleistocene (LGM) rainfall pattern, and would
have driven a complex response.

As transient adjustments proceed relatively more rapidly where rainfall
is more concentrated (i.e., erosional efficiency is higher), changes in
rainfall pattern have the potential to produce spatially distinct
effects different from what would be expected from considering uniform
changes in mean climate. Transient adjustments may therefore be
relatively enhanced or underdeveloped in different locations within the
same catchment, or adjustment to quasi-equilibrium may be essentially
complete in some locations while others reflect only an incipient
response to the climate change. We noted an example of this behavior in
Case 4, where low-elevation dry tributary catchments preserve transient
conditions the longest, contrasting with the notion that headwater
catchments should be the last to equilibrate. Similarly segregated
conditions occur in Case 3, where adjustment to quasi-equilibrium is
essentially complete in wet low-elevation catchments long before the
migrating trunk knickpoint even reaches drier high-elevation catchments.
Because such complex, climate change-driven landscape adjustments are
not reasonably captured by a conceptual framework based on spatially
uniform changes in rainfall (e.g., compare Cases 3 \& 4 to Cases 1 \&
2), apparent inconsistencies between expectations and observations have
the potential to give a false impression about the primary forcing(s)
controlling erosion rates.

Additionally, if large-scale changes in rainfall patterns like we model
develop incrementally over long timescales (e.g., millions of years),
they could still result in complex transient responses.
Greenhouse-icehouse transitions and orogenic growth are among many
geologically significant events that may cause temporally distinct,
sustained, and dramatic changes to climate and/or circulation patterns
where complex responses could arise (Mutz et al., 2018; Poulsen et al.,
2010; Roe et al., 2003; Zachos et al., 2001), If, for example, the
bottom-heavy gradient in Case 3 instead develops over several million
years, regardless any added complexity to the general trajectory of this
change in rainfall pattern, the result is that it supports a 30\%
increase fluvial relief despite also increasing total rainfall by
\textasciitilde{}80\%, and channel steepness patterns fundamentally
change as the catchment adjusts. Gradual changes in rainfall patterns
cause morphological adjustments to become more diffuse, and induce
relatively smaller transient changes in erosion rate than abrupt
changes. However, the spatial pattern of erosion is still significantly
affected (i.e., in excess of a factor or two from steady state) so long
as the timescale over which the rainfall pattern evolves does not far
exceed that of the catchment adjustment timescale, which may be several
million years for large river basins (Roe et al., 2003; Whipple, 2001).
As such, the general characteristics of the classes of transient
behavior following changes in rainfall pattern toward
\emph{relatively}bottom-heavy or top-heavy conditions remain intact,
even for long-term transient responses.

\emph{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., \emph{k\textsubscript{sn}} ) 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 (\emph{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.

\textbf{5.2 Recognizing the Influence of Climate on Topography and
Erosion Rates} \emph{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 (\emph{K} ) at steady
state (Equation 6), and as shown by curves in Figures 4-6. Because the
role of climate is encapsulated in \emph{K} , it is important to
remember that a uniform \emph{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
like\emph{k\textsubscript{sn}} should relate to rainfall in a simple way
generally relies on an assumption of a spatially uniform \emph{K} value.
Rainfall gradients systematically affect this expectation, where
bottom-heavy gradients result in higher \emph{k\textsubscript{sn}}(lower
apparent erosional efficiency), while top-heavy gradients result in
lower \emph{k\textsubscript{sn}} (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, \emph{k\textsubscript{sn}} 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 \emph{k\textsubscript{sn}} 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.,
mean\emph{k\textsubscript{sn}} , 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 \emph{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.,
using\emph{k\textsubscript{sn-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.

\emph{5.2.2 Misleading Transient Signals}

Spatial patterns in erosion rate are commonly used to inform tectonic
models and to infer rock uplift rates in mountain landscapes (e.g.,
Adams et al., 2020; Godard et al., 2014; Kober et al., 2015; Morell et
al., 2015; Safran et al., 2005; Scherler et al., 2014). However, we have
shown that changes in rainfall patterns can drive long-lived and complex
spatial patterns of erosion that differ from expectations for uniform
changes in rainfall and thus may not be readily recognized and
interpreted. We have also shown that that ongoing transient adjustment
may not be obviously expressed in landscape morphology (especially for
catchment-mean metrics) under some circumstances. If these caveats are
not considered, subtly expressed transient spatial variations in erosion
rate may be mistaken as representing quasi-steady state spatial
variations in uplift rate (i.e., \emph{E} = \emph{U} ). At once this
would give a false impression both about the spatial pattern of uplift
and the importance of past climate changes on a landscape's evolution,
with direct implications for understanding connections among climate,
surface processes, and tectonics. Determining whether there are
circumstances in which spatial patterns of erosion and topography
produced by changes in rainfall patterns that can be misleading enough
to confound interpretations about factors controlling landscape
evolution is critically important.

During the early transient adjustment in Case 3 (transition to a
bottom-heavy rainfall pattern) there is a clear example of how such
confusion may occur (Figure 10). Recall, in this case, early transient
adjustment produces a concave-up knickpoint along the trunk profile but
as it migrates upstream the shape evolves. This creates a broad
adjustment zone. Over the first \textasciitilde{}500 kyr, quasi-steady
state adjustment proceeds \textasciitilde{}60\% upstream along the
trunk, but the broad adjustment zone means most tributaries along this
length experience a protracted signal of base-level changes related to
trunk adjustment. Because these tributaries all also experience a net
increase in rainfall, knickpoints associated with local adjustment of
the trunk river (Stage 2) tend to relax as they work upstream making
them more diffuse. This protracted competition between local rainfall
and spatio-temporally variable rates of base-level fall, generally
results in diffuse concave to broad convexo-concave adjustment zones in
tributaries (e.g., Figure 8a; Movie S5). Broad adjustment zones,
particularly concave-up adjustment zones, are inherently subtle and this
can inhibit their recognition. This problem may be further compounded by
the influence of sediment flux in natural settings (Brocard \& van der
Beek, 2006; Whipple \& Tucker, 2002). Indeed, even in our idealized
model (i.e., no sediment influence), along-stream variations in trunk
and tributary local \emph{k\textsubscript{sn}} variation is diffuse
(Figure 10c). Based on a lack of significant knickpoints that might
indicate transient adjustment and the several-fold spatial variation in
erosion rate, one might reasonably interpret relationships depicted in
Figure 10a reflect a quasi-steady-state landscape adjusted to a spatial
gradient in uplift rate. In the absence of known surface breaking
structures that might accommodate this gradient in uplift, blind
structures may be inferred, with potential implications for tectonic
models. The apparent viability of this interpretation is supported by
the SPM if rock uplift rate is assumed to match the observed pattern of
catchment averaged erosion rates in tributaries (steady-state
conditions) as illustrated in Figure 10b. Figure 10b shows that the
predicted steady-state upstream-averaged
\emph{k\textsubscript{sn}}pattern along the trunk river and mean
\emph{k\textsubscript{sn}} values exhibited by the tributary network is
essentially identical to the transient pattern in Figure 10a. Moreover,
even in detail, there are only subtle differences in the along-stream
pattern of local\emph{k\textsubscript{sn}} between the two scenarios
(Figure 10c). Thus, in this instance, \emph{k\textsubscript{sn}}
patterns and erosion rates that actually record a complex transient
response to a change in rainfall pattern could reasonably be mistaken
for a steady state landscape adjusted to a spatial gradient in uplift.

Although subtle variations in \emph{k\textsubscript{sn}} values might
give a misleading impression that a landscape is in quasi-steady-state,
the spatial pattern of \emph{k\textsubscript{sn-q}} unambiguously
suggests along-stream variations in erosion rate exist along both the
trunk river and tributaries. \emph{k\textsubscript{sn-q}} also exhibits
a coherent pattern of downstream adjustment that could readily be
interpreted as a transient signal sweeping upstream through the
catchment that, significantly, is inconsistent with a steady state
landscape adjusted to the spatial gradient in uplift shown in Figure 10b
(Figure 10c). This example shows the potential usefulness
of\emph{k\textsubscript{sn-q}} , both as a diagnostic tool for detecting
ongoing transient adjustment to changes in rainfall patterns
where\emph{k\textsubscript{sn}} may be misleading and for resolving
the relative influences of tectonics and climate.

Finally, we emphasize that our intention is not to suggest all, or any
specific examples, where spatial patterns of rock uplift are inferred
from erosion rates and channel steepness patterns are incorrect. Rather,
our intention here is to highlight the extent to which confusion may be
possible under the right circumstances, and how explicitly accounting
for rainfall patterns can be a step toward addressing these challenges.

\textbf{5.3 Toward Detecting the Influence of Climate on Topography and
Erosion Rates}

Remote analysis of channel steepness patterns can provide a preliminary
means to detect whether the climate may be influencing river profiles in
a landscape, and whether transient adjustment to a change in climate may
be ongoing. For landscapes well-described by the SPM, discrepancies
between rainfall conditions experienced by trunk and tributary basins
should cause systematic differences in trunk and
tributary\emph{k\textsubscript{sn}} (Figures 5\&6; Gasparini \& Whipple,
2014), but not in \emph{k\textsubscript{sn-q}} . As illustrated in
Figure 10c, the expected contrast in trunk and tributary
\emph{k\textsubscript{sn}}is most developed at steady state, but it
begins manifesting immediately during transient adjustment as
differences in local\emph{k\textsubscript{sn}} at confluences that
migrate upstream (Figure 10c, Movie S5). In contrast, precise agreement
between trunk and tributary \emph{k\textsubscript{sn-q}} is expected at
steady state, and it is approximately maintained even during periods of
adjustment to a change in rainfall pattern. Agreement between trunk and
tributary\emph{k\textsubscript{sn-q}} that weakens upstream from
confluences may also be an important indication of ongoing transient
adjustment that may be difficult detect from the
\emph{k\textsubscript{sn}} pattern alone (Figure 10c, e.g., as discussed
in section 5.2.2). Therefore, comparison of \emph{k\textsubscript{sn}}
and \emph{k\textsubscript{sn-q}} patterns may be a useful way to
diagnose and further assess the potential extent of influence by
rainfall gradients, provided the pattern of discharge accumulation can
be reasonably estimated (e.g., using high spatial resolution satellite
rainfall or nested stream gauge data).

Following topographic analysis, our results have additional implications
for designing effective catchment-averaged erosion rate sampling
strategies. Broadly, sampling strategies can be grouped into two
classes: nested and distributed, described below. Nested, or
hierarchical, sampling strategies comprise multiple samples from the
same river basin where some or all samples are collected along the trunk
river, and where averages typically integrate over very large
(10\textsuperscript{2}--10\textsuperscript{5} km\textsuperscript{2})
drainage areas (e.g., Abbühl et al., 2010; Henck et al., 2011; Portenga
et al., 2015; Reinhardt et al., 2007; Safran et al., 2005; Whittaker et
al., 2007; Willenbring et al., 2013; Wittmann et al., 2016). In
principle, such strategies can allow researchers to assess
reproducibility of individual measurements, test sediment mixing models,
and sub-divide basins into different sectors to identify along-stream
variations in erosion rates. Given the complex along-stream patterns of
erosion rate we observe in response to changing rainfall patterns and
widely disparate responses between trunk and tributary profiles,
however, our results suggest that caution is warranted in interpreting
patterns of erosion rates collected in a nested fashion. In particular,
this includes datasets that compile measurements from along trunk rivers
and tributary catchments, but also those that compare samples from
different large catchments that experience different mean climates and
rainfall patterns, even at quasi-steady-state, because how each reflects
and experiences variations in rainfall may be fundamentally different.
Furthermore, because we find that along-stream variations in erosion
rate due to changes in rainfall pattern are characteristically muted
along the trunk profile (e.g., Figure 10a), nested strategies may not be
appropriate in many settings, especially if the goal is to measure the
influence of climate on fluvial incision.

The other widely used strategy, which we refer to as a distributed
sampling strategy, targets single samples from catchments that are
distributed across a landscape or mountain front, and typically --
though certainly not always -- comprises relatively smaller
(10\textsuperscript{0}-10\textsuperscript{2} km\textsuperscript{2})
catchments (e.g., Adams et al., 2020; Binnie et al., 2008; Carretier et
al., 2013; DiBiase et al., 2010; Godard et al., 2014; Morell et al.,
2015; Ouimet et al., 2009; Scherler et al., 2014). This type of strategy
generally allows more freedom to carefully select preferable catchments
(relatively uniform channel steepness, rainfall, lithology), with the
limitation that spatial variations in erosion at the sub-catchment scale
are not resolvable with single measurements. In contrast to muted
spatial variations in erosion noted for nested strategies, our model
results suggest distributed strategies inherently record a more direct
signal of climatic influences than nested approaches, consistent with
previous findings by Han et al. (2015). However, if inter-catchment
variations in rainfall and therefore erosional efficiency are not
accounted for, or transient conditions are not recognized, this
sensitivity may cause significant dispersion or distortion in measured
relationships among landscape metrics, as shown
for\emph{k\textsubscript{sn}} --\emph{E\textsubscript{avg}}
relationships (Figures 5b \& 6b), and/or lead to misleading spatial
patterns in erosion rate and \emph{k\textsubscript{sn}} (Figure 10).

Lastly, we note that transient adjustments in response to changes in
rainfall pattern do not significantly affect apparent erosional
efficiency in \emph{k\textsubscript{sn-q}}
--\emph{E\textsubscript{avg}}relationships where no variations in rock
properties exist, even in response to the dramatic shifts in rainfall
patterns that we model, regardless of sampling strategy. Indeed,
transient deviations from expected steady state relationships modelled
in any location are generally well within the analytical uncertainty of
measured catchment-average erosion rates from natural landscapes
(Figures 5 and 6). Exceptions to this appear to be limited to scenarios
where landscapes experience a shift toward arid climates. Nevertheless,
transient, spatially variable patterns of erosion caused by changes in
rainfall pattern are reflected in \emph{k\textsubscript{sn-q}} patterns
with good accuracy in our model (Figure 10c, Movie S3-S5). As
such,\emph{k\textsubscript{sn-q}} may be used to recognize ongoing
adjustment to changes in climate where \emph{k\textsubscript{sn}} can be
ambiguous. We suggest that \emph{k\textsubscript{sn-q}} , or a different
metric that encompasses the spatial distribution of rainfall (runoff),
may be vital component for future efforts to detect climate's influence
on and from topography and erosion rates in mountain landscapes where
rainfall is inherently spatially variable.

\textbf{5.4 Caveats and Limitations}

Several important caveats should be kept in mind when evaluating the
results and subsequent discussion presented here. First, and foremost,
our modelling efforts explicitly assume that landscapes are inherently
sensitive to climate (through rainfall) in a manner described by the
SPM. While the intuitive support for such sensitivity is strong, and
evidence from natural landscapes is mounting that broadly support
predictions of the SPM (e.g., Adams, et al., 2020; Harel et al., 2016;
Lague, 2014), there remains a large amount of uncertainty about the
strength of the sensitivity to climate.

Following the core assumption that the SPM is broadly applicable, we
note that our model setup is very simple. We assume that all
precipitation is rainfall, all rainfall is converted directly to runoff,
and we impose constant rainfall gradients that act precisely along the
trunk stream and basin axis and that span the entire length of our
modelled river basin. While some river basins set within mountain-belt
scale orographic precipitation patterns may indeed experience rainfall
patterns consistent with this simple geometry, we note that non-linear,
and even non-monotonic, rainfall gradients are common for large river
basins or those characterized by high local relief (Roe, 2005). As noted
previously, however, the framework we have developed translates well to
these more complex scenarios. In addition, we model tributary catchments
that are uniform in size that experience uniform rainfall. Preliminary
model investigations suggest that although allowing tributary sizes and
rainfall to vary causes some dispersion in the relationships illustrated
here, the overall behavior we describe remains the dominant signal.
Nevertheless, both aspects warrant further investigation.

Lastly, we emphasize that we use a simple version of the SPM where,
among other simplifications, we treat erosion exclusively as detachment
limited and do not explicitly model erosional thresholds. While assuming
continuum detachment-limited conditions is common, particularly for
describing erosion in mountain settings, it may not always be
appropriate even in these settings to describe transient behavior
(Lague, 2014; Whipple \& Tucker, 2002). Related to this, our treatment
of \emph{K\textsubscript{p}} , while incrementally more complex than
spatially uniform erosional efficiency, remains highly simplified. We do
not explore the myriad of other factors that control erosional
efficiency (\emph{K} ), and the likelihood that rainfall, or more
broadly climate, will influence rock erodibility
(\textasciitilde{}\emph{K\textsubscript{p}} ) and size distribution of
sediment delivered to rivers (Ferrier et al., 2013; Murphy et al., 2016;
Neely \& DiBiase, 2020; Riebe et al., 2015; Sklar et al., 2017).

\textbf{6. Summary and Conclusions}

In this study, we use a simple form of the SPM to explore how spatial
rainfall gradients influence both river profiles and expectations about
how fluvial landscapes should respond to changes in climate. Notably,
because changes in climate in mountain settings involve changes in
orographic precipitation patterns, advancing understanding about how
changes in climate influence the rivers and topography of mountain
landscapes requires a more nuanced, spatially variable, treatment of
climate at the catchment scale beyond broadly characterized changes in
mean climate. We show that spatially varying rainfall conditions
experienced by rivers occupying different landscape positions,
specifically trunk and tributary rivers, result in fundamentally
different expressions of a given rainfall pattern, and that they respond
to changes in rainfall pattern in fundamentally different ways. Further,
we show how complex transient responses may arise, and that even modest
variations in rainfall patterns can significantly affect spatial
patterns of erosion and topographic adjustments that directly contrast
with expectations for uniform changes in rainfall. In particular,
changes in rainfall pattern characteristically result in multi-stage
responses with adjustment zones that can be both spatially extensive and
subtly expressed, thus difficult to recognize. Interestingly, these
finding suggest more broadly that catchments of different sizes, shapes,
and locations set within regional orographic precipitation patterns may
have unique, and substantially variable, rainfall histories and
responses to changes in climate. These complications raise important
questions about how best to interpret spatial variations in erosion
rates and their relationship with topography. Finally, we discuss how
catchment-scale variations in rainfall generally complicate
relationships between conventional topographic metrics (e.g.,
mean\emph{k\textsubscript{sn}} ) and erosion rates, even at steady
state, and implications for empirical tests of the SPM in natural
landscapes. More precise metrics like \emph{k\textsubscript{sn-q}} that
leverage ever-increasing resolution of rainfall datasets to better
account for the spatial distribution of rainfall -- specifically its
effect on discharge and runoff -- may be a significant step toward
overcoming these challenges and may prove vital for future studies
seeking to quantify interactions between climate, tectonics, and erosion
in mountain landscapes.

\textbf{Acknowledgments}

This manuscript was inspired by and benefitted from many thoughtful
discussions with present and former students of the earth surface
processes group at ASU. We also thank Simon Mudd and two anonymous
reviewers for insightful comments that improved the clarity and quality
of the science presented. Funding for this project was provided by the
Tectonics program at NSF (EAR-1842065 to K.X.W.). Supplementary movies
are attached with this submission and accessible in open-access form
through ESSOAR along with a preprint version of the manuscript. The
model used for this analysis is described in detail in main text. No
data were used, nor created for this research.

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\textbf{Figures and Tables}\selectlanguage{english}
\begin{longtable}[]{@{}lll@{}}
\toprule
\textbf{Table 1.} Model Parameters & \textbf{Table 1.} Model Parameters
&\tabularnewline
\midrule
\endhead
Parameters & Value & Units\tabularnewline
\emph{U} & 5.0 x 10\textsuperscript{-4} &
m\selectlanguage{ngerman}·yr\textsuperscript{-1}\tabularnewline
\emph{K\textsubscript{p}} & 2.5 x 10\textsuperscript{-9} &
m\selectlanguage{ngerman}\textsuperscript{-1·}yr\textsuperscript{-1}\tabularnewline
\emph{m} & 1 &\tabularnewline
\emph{n} & 2 &\tabularnewline
\emph{k\textsubscript{a}} & 6.69 &\tabularnewline
\emph{h} & 1.67 &\tabularnewline
\emph{L} & &\tabularnewline
\emph{Trunk} & 50 & km\tabularnewline
\emph{Tributaries} & 5 & km\tabularnewline
\emph{A} & &\tabularnewline
\emph{Trunk} & 472 & km\textsuperscript{2}\tabularnewline
\emph{Tributaries} & 11 & km\textsuperscript{2}\tabularnewline
\emph{A\textsubscript{c}} & 1 & km\textsuperscript{2}\tabularnewline
Node spacing & 25 & m\tabularnewline
Time step & 50 & yr\tabularnewline
Tributary Spacing & 1 & km\tabularnewline
\bottomrule
\end{longtable}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}

\end{center}
\end{figure}

\textbf{Figure 1} . Influence of rainfall on channel profile form and
topographic metrics at steady state. a) Top row -- left panel shows
river profiles adjusted to different amounts of spatially uniform
rainfall. Corresponding discharge accumulation curves are shown in the
middle panel. Right panel shows the relationship between mean rainfall
(defined as \emph{Q} /\emph{A} ) calculated at the outlet, with fluvial
relief (\emph{R} ), \emph{k\textsubscript{sn}} ,
and\emph{k\textsubscript{sn-q}} for a fixed uniform rock uplift rate.
Note, at steady state fluvial relief and \emph{k\textsubscript{sn}} are
inversely related to mean rainfall, while \emph{k\textsubscript{sn-q}}is
independent of changes in rainfall. b) Bottom row -- all panels as in
(a) but for a comparable suite of steady-state profiles adjusted to
spatially uniform rainfall, a decreasing upstream rainfall
(bottom-heavy) pattern, and an increasing upstream (top-heavy) pattern.
Note intersections that occur between profiles in the left panel and
middle panels, and that fluvial relief and \emph{k\textsubscript{sn}}are
positively related to mean rainfall in these cases (right panel),
inverted from (a). All other model parameters are equal (see Table 1).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}

\end{center}
\end{figure}

\textbf{Figure 2} . Schematic illustration of key aspects of complex
transient responses -- defined in section 3.3 -- that can result from
changes to a relatively bottom-heavy (a) or a top-heavy rainfall
gradient (b) -- rainfall shown in inset. Horizontal grey bars highlight
resulting net change in fluvial relief (\selectlanguage{greek}Δ\selectlanguage{english}\emph{R} ) and maximum
elevation change (\selectlanguage{greek}Δ\selectlanguage{english}\emph{z\textsubscript{sc}} \textsubscript{}) along
the center of the profile. Colored circles mark positions
of\emph{x\textsubscript{sc}} (yellow) and \emph{x\textsubscript{zc}}
(red) that also demarcate segments of net divergent and convergent
transient adjustments -- see text for definitions and discussion. Net
modes of transient adjustment are indicated along top axis. Note
reversal at position \emph{x\textsubscript{zc}} (net incision
downstream, net uplift upstream); however, during transience, adjustment
upstream of\emph{x\textsubscript{sc}} is variable with time. Inset shows
both local rainfall rate (\emph{P} , blue) and upstream averaged
rainfall (\(\overline{P}\), black). \emph{P} = \(\overline{P}\) for
spatially uniform rainfall; only \emph{P} is depicted for initial
rainfall. Positions of \emph{x\textsubscript{sc}}
and\emph{x\textsubscript{zc}} \textsubscript{} in inset correspond to
those along the profiles in the main figure. Note along-stream offset in
position where \emph{P} and \(\overline{P}\) exceed the initial
rainfall rate, the latter corresponding to\emph{x\textsubscript{sc}}
\textsubscript{}. Also, note that the difference between \emph{P} and
\(\overline{P}\) increases systematically downstream.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}

\end{center}
\end{figure}

\textbf{Figure 3} . Sensitivity analysis of a river profile to different
spatial gradients in rainfall. a) Defines general domains of behavior
described in the sensitivity analysis shown in (b) -- see text for
descriptions of different fields. Black star in (a) marks the initial
condition; note, initial conditions in panels (a) and (b) are different.
Top-heavy and bottom-heavy domains are separated by the 1:1 line. The
primary influence of different \emph{h} values in Hack's Law is on the
slope of \emph{Q\textsubscript{f} /Q\textsubscript{i}} , as indicated.
b) Same as (a) but where the initial profile is adjusted to spatially
uniform \emph{P} = 1.5 m/yr, and grey fields in (a) are colored by
magnitude of \selectlanguage{greek}Δ\selectlanguage{english}\emph{z\textsubscript{sc}} \textsubscript{} (e.g., Figure
2). White stars show rainfall gradient scenarios explored in section 3
-- Cases 3 \& 4. Contours of
\emph{Q\textsubscript{f}/Q\textsubscript{i}} are shown in grey dashed
lines, and contours of\emph{R\textsubscript{f} /R\textsubscript{i}} in
black dotted lines.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}

\end{center}
\end{figure}

\textbf{Figure 4} . Representative time slice of transient evolution for
Case 1 (uniform decrease in rainfall) and Case 2 (uniform increase in
rainfall) through (a)\emph{k\textsubscript{sn}}
-\emph{E\textsubscript{avg}} and (b)\emph{k\textsubscript{sn-q}}
-\emph{E\textsubscript{avg}} parameter spaces. Case 1 is shown in red
shaded domain (transient decrease in erosion rate, \emph{E} \textless{}
\emph{U} ), and Case 2 is in blue shaded domain (transient increase in
erosion rate, \emph{E}\textgreater{} \emph{U} ). Curves expressing the
expected steady state relationship between channel steepness and erosion
rate with different uniform erosional efficiencies are shown; \emph{K =
K\textsubscript{p}} , is shown in black solid line, and grey lines show
different values of\emph{K} as multiples of \emph{K\textsubscript{p}} .
Vertical dashed lines show percent deviations from steady state erosion
rate. Initial and final conditions shown with yellow circles, as
labelled. Model data from different positions along the trunk profile
and individual tributaries shown with red and blue data points,
respectively. Trunk data shown are only profile nodes at tributary
junctions -- all trunk profile datapoints have a corresponding tributary
datapoint at the same position. Trunk profile data show upstream mean
values from their respective position, and tributary data show mean
values for the tributary profile measured at the junction with the trunk
profile.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image5/image5}

\end{center}
\end{figure}

\textbf{Figure 5} . Transient evolution of the trunk profile (a \& c)
and network of tributaries (b \& d) in Case 3 (spatially uniform to
bottom-heavy) through\emph{k\textsubscript{sn}}
-\emph{E\textsubscript{avg}} (a \& b) and\emph{k\textsubscript{sn-q}}
-\emph{E\textsubscript{avg}} (b \& d) parameter spaces. Layout of
individual panels is like Figure 4. Six representative time slices are
shown; t\textsubscript{1} shows time at 10 kyr model time (i.e., first
timestep following change in rainfall), t\textsubscript{6} is at final
steady state. Time intervals are not equal in duration but
t\textsubscript{1}-t\textsubscript{6} are the same timestep for all
panels. Note range in \emph{k\textsubscript{sn}} values and erosional
efficiency values spanned at final steady state in panels (a) and (b).
Also, note the relatively muted responses by the trunk profile relative
to the network of tributaries in both\emph{k\textsubscript{sn}} and
\emph{k\textsubscript{sn-q}} . Star in panel (a) shows expected
erosional efficiency calculated from equation (1b) based on the final
mean rainfall of 2.68 m/yr (2.68\selectlanguage{ngerman}·\emph{K\textsubscript{p}} ) at steady
state, which corresponds to a predicted mean \emph{k\textsubscript{sn}}
of \textasciitilde{}275. Note offset between this value and range of
erosional efficiency and\emph{k\textsubscript{sn}} values exhibited at
t\textsubscript{6}.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image6/image6}

\end{center}
\end{figure}

\textbf{Figure 6} . Transient evolution of the trunk profile (a \& c)
and network of tributaries (b \& d) in Case 4 (spatially uniform to
top-heavy) through\emph{k\textsubscript{sn}}
-\emph{E\textsubscript{avg}} (a \& b) and\emph{k\textsubscript{sn-q}}
-\emph{E\textsubscript{avg}} (c \& d) parameter spaces; figure layout is
same as Figure 5. Like Case 3, the trunk response is muted relative to
the network of tributaries, but the disparity is more extreme in this
case. Star in panel (a) shows \emph{K}calculated from equation (1b)
based on the final mean rainfall of 1 m/yr (\emph{K} =
\emph{K\textsubscript{p}} ) at steady state, which corresponds to a
predicted mean \emph{k\textsubscript{sn}} of \textasciitilde{}450.
Again, note offset between this value and range of erosional efficiency
and \emph{k\textsubscript{sn}} values exhibited at t\textsubscript{6}.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image7/image7}

\end{center}
\end{figure}

\textbf{Figure 7} . Sensitivity analysis where initial conditions use
final rainfall patterns from Case 3 (a) and Case 4 (b), and examples
from the eastern flank of the Peruvian Andes (c) and eastern-central
Himalaya (d). Panel layouts are the same as in Figure 3b. Triangular
fields in demarcated in black dashes in (a) and (b) bound rainfall
gradients with the same pattern as the initial condition (i.e.,
bottom-heavy or top-heavy), but cause complex transient responses
similar to those expected for reversals in polarity of the rainfall
pattern as the rainfall becomes \emph{relatively} bottom- or top-heavy
-- see section 5.1 for further discussion. Panels (c \& d) show examples
of climate change scenarios from model simulations reported in
supplementary tables by Mutz et al. (2018), for a hypothetical
transverse river with parameters in Table 1. Initial condition is set in
the Pliocene (PLIO), and representative evolution of the orographic
rainfall pattern through to the Last Glacial Maximum (LGM) is plotted,
illustrating that climate changes that are predicted to trigger complex
responses are likely not uncommon.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image8/image8}

\end{center}
\end{figure}

\textbf{Figure 8.} Representative examples of multi-stage tributary
response in Cases 3 (a) \& 4 (b). Note, upstream positions always
experience relatively longer initial adjustment stages than downstream
positions, regardless of rainfall gradient. Grey shaded and white
regions show where initial and final steady states (SS) and different
adjustment stages are represented along the profile lengths, as
indicated across top axis. a) Shows a tributary located in a relatively
downstream position, 15 km from trunk outlet and downstream
from\emph{x\textsubscript{sc}} after 300 kyr of transient adjustment in
Case 3, where the initial adjustment stage is relatively short-lived.
Despite this, protracted adjustment creates a several-kilometer-wide
adjustment zone with relatively smooth but notable along-stream
variations in\emph{k\textsubscript{sn}} . This pattern is subtle on the
longitudinal profile. b) Shows a tributary located in an upstream
position, 45 km from the trunk outlet and upstream from
\emph{x\textsubscript{sc}} after 1.725 Myr of transient adjustment in
Case 4, where the initial adjustment stage is long-lived. The initial
adjustment stage reshapes the entire profile prior to adjustment of the
trunk profile, but no transient knickpoints or other obvious topographic
indicators of such significant modification are present, with smoothly
varying local\emph{k\textsubscript{sn}} values within 10\% of the mean.
Also, note overadjustment means that the increase in channel steepness
required to reach steady state contrasts with net reduction in channel
steepness during transient adjustment.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image9/image9}

\end{center}
\end{figure}

\textbf{Figure 9.} Plot illustrating how rainfall gradients
systematically influence correlation between mean rainfall
and\emph{k\textsubscript{sn}} at steady state, and consequences for
relationship between channel steepness and erosion rate (inset). A
common symbology is used in both the main figure and the inset. Expected
SPM relationship based on uniform climate (uniform erosional
efficiency,\emph{K} ) is shown in solid back line. Grey long-dashed
lines represent changes in \emph{K} =\emph{K\textsubscript{p}}
\(\overset{\overline{}}{P}\)\emph{\textsuperscript{m}} . Circles colored by mean
rainfall reflect scenarios shown in Figure 1 -- uniform rainfall of 1,
1.5, and 2 m/yr, and bottom- and top-heavy gradients (also used in Cases
3 \& 4). While \emph{K\textsubscript{p}}is equivalent for all scenarios,
rainfall gradients cause dispersion (apparent differences in erosional
efficiency) from the expected relationship (solid black line) equivalent
to differences in\emph{K\textsubscript{p}} of approximately a factor of
two (inset). For instance, the bottom-heavy case where mean rainfall is
\textasciitilde{}2.7 m/yr exhibits an apparent \textasciitilde{}50\%
reduction in erosional efficiency based on
mean\emph{k\textsubscript{sn}} (plots on\emph{K}
[?]0.5\emph{K\textsubscript{p}} curve in main figure) and thus plots where
one would expect a catchment that experiences half as much rainfall to
plot in the inset. Dotted line in main figure illustrates, conceptually,
how this dispersion could distort inferred relationships between mean
rainfall and channel steepness (weaker and quasi-linear in this case),
even at steady state and in systems where the SPM is a complete
description of the controls on channel profile form. Where uplift rates
are unknown, this systematic bias in apparent erosional efficiency due
to rainfall gradients is a potential source of dispersion in
relationships between \emph{k\textsubscript{sn}}  and erosion rate
(inset).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image10/image10}

\end{center}
\end{figure}



\textbf{Figure 10} . Illustration of potentially misleading spatial
patterns that arise among channel steepness, rainfall, and erosion rates
during transient adjustment to a change in rainfall pattern,
particularly when erosion rate and morphometrics are averaged at
catchment scale -- see section 5.1.2 for detailed discussion. a) Shows
time slice at 0.5 Myr into transient adjustment of Case 3
(t\textsubscript{2} in Figure 5). For clarity, a subsample of ten data
points is shown for each model dataset; connecting dashed line is
populated from full model. Thick colored bands (red = trunk, blue =
tributaries) show final steady state
upstream-averaged\emph{k\textsubscript{sn}} pattern toward which the
modelled catchment is adjusting. Position of trunk knickpoint, defined
as upstream extent of quasi-equilibrium adjustment, is shown by red
solid line, and zones describing morphological characteristics of
tributary catchments are shown across the top axis (compare both with
panel c). b) Modelled steady state \emph{k\textsubscript{sn}} and trunk
erosion rate pattern of hypothetical catchment experiencing a
bottom-heavy rainfall gradient and a spatial gradient in uplift rate
that matches the catchment-averaged erosion rate pattern recorded by the
tributary network in a). c) Shows planform development of
local\emph{k\textsubscript{sn}} , \emph{k\textsubscript{sn-q}} , and
erosion rates at initial and final steady states, a transient time slice
from Case 3, and steady state pattern from (b). Downstream is to the
left. Transient time slice and tributaries are the same as in panels (a
\& b). Note that trunk and tributary profiles are not illustrated to
scale (Table 1).

\selectlanguage{english}
\FloatBarrier
\end{document}

