We examine the influence of incision thresholds on topographic and scaling properties of landscapes that follow a landscape evolution model (LEM)
with terms for stream-power incision, linear diffusion, and uniform uplift. Our analysis uses three main tools. First, we examine the graphical behavior of theoretical relationships between curvature and the steepness index (which depends on drainage area and slope). These relationships plot
as straight lines for the case of steady-state landscapes that follow the LEM. These lines have slopes and intercepts that provide estimates of
landscape characteristic scales. Such lines can be viewed as counterparts of slope–area relationships, which follow power laws in detachment-limited landscapes but not in landscapes with diffusion. We illustrate the response of these curvature–steepness index lines to changes in the values of parameters. Second, we define a Péclet number that quantifies the competition between incision and diffusion, while taking the incision threshold into account. We examine how this Péclet number captures the influence of the incision threshold on the degree of landscape dissection. Third, we characterize the influence of the incision threshold using a ratio between it and the steepness index. This ratio is a dimensionless number in the case of the LEM that we use and reflects the fraction by which the incision rate is reduced due to the incision threshold; in this way, it quantifies the relative influence of the incision threshold across a landscape. These three tools can be used together to graphically illustrate how topography and process competition respond to incision thresholds.
Introduction
Processes that shape landscapes leave topographic signatures, which can often be visualized by plotting different topographic metrics against one
another. An example is the relationship between river gradient and drainage area, which has been used to analyze landscapes and river profiles, as well as to
diagnose the processes that shape them (e.g., Montgomery and Foufoula-Georgiou, 1993; Howard, 1994; Montgomery and Dietrich, 1994; Dietrich et al.,
2003). For example, the stream-power incision model predicts that if tectonics, climate, and rock properties are uniform, then bedrock rivers should
approach a steady state in which their gradient scales as a power law of drainage area (e.g., Tucker, 2004; Lague, 2014). This power-law scaling
implies that river gradient data should plot as a straight line against drainage area data on logarithmic axes. The properties of this line can give
estimates of properties of the landscape; e.g., its slope gives the concavity index (Whipple, 2004). Plotting synthetic topographic data from
landscape evolution models (LEMs) helps to illustrate the effects of different model formulations or parameterizations. For example, including a
threshold in the incision term of an LEM affects the resulting slope–area line (e.g., Tucker, 2004; Lague et al., 2005; Deal et al., 2018).
In the case of landscapes that are influenced by diffusion, topographic slope does not scale as a power function of drainage area (e.g., Howard,
1994). Thus, slope and area data from these landscapes do not plot as straight lines. In Theodoratos et al. (2018), we presented a counterpart
relationship for the case of landscapes produced by an LEM that includes linear diffusion (along with stream-power incision and uplift). This
relationship predicts that in steady state, curvature and the steepness index (which depends on drainage area and slope; e.g., Whipple, 2001) plot as
a straight line against each other on linear (i.e., non-logarithmic) axes. The slope and intercept of this line depend on characteristic scales of
length and height of the landscape, which in turn depend on the relative strengths of the processes that shape it. Thus, this relationship predicts a
link between topographic and scaling properties of landscapes that follow the LEM.
Here, we demonstrate an example of the explanatory power of plots of the curvature–steepness index relationship. Our example shows that these plots
can visualize topographic and scaling effects of incision thresholds. Incision thresholds can markedly influence erosion, as shown by numerous
studies. For instance, incision thresholds can influence the relationship between river gradient and the uplift rate (e.g., Snyder et al., 2003), the
dependence of long-term erosion rates on the average, variability, and duration of precipitation events (e.g., DiBiase and Whipple, 2011;
Scherler et al., 2017), and the dynamics of migrating knickpoints (e.g., Lague, 2014). Here, we are not further elaborating on the insights of these
studies. Instead, we focus on the effects of incision thresholds on the competition between incision and diffusion and on the topographic and scaling
properties of landscapes reflecting this competition. The topographic and scaling effects that we examine have been studied before (e.g., Montgomery
and Dietrich, 1992; Howard, 1994; Tucker, 2004; Perron et al., 2008). Here, however, we present a novel, purely graphical method to identify,
quantify, and interpret these effects based on the relationship between curvature and the steepness index.
In Theodoratos et al. (2018), we dimensionally analyzed a frequently used LEM with terms for uplift, linear diffusion, and stream-power incision
without an incision threshold. In Theodoratos and Kirchner (2020), we added an incision threshold to this LEM and dimensionally analyzed it. Here, we
summarize the definitions of characteristic scales and dimensionless numbers that emerged from the dimensional analyses of these two LEMs in
Sect. 2. Then, in Sect. 3, we show that these characteristic scales and dimensionless numbers have a geomorphologic meaning that can be expressed
graphically using plots of curvature vs. the steepness index. The graphical explanatory power of these plots is further highlighted by comparing plots
of LEMs with and without an incision threshold (Figs. 1 and 2).
Relationship between curvature and the steepness index in steady-state topography without an incision threshold. We plot a straight line defined by Eq. (10) or (11), which describes how curvature should be related to the steepness index if the landscape follows the LEM (Eq. 1) and is in steady state. This line is parameterized by the characteristic scales of length and height (lc and hc; Eqs. 3 an 4); its slope is 1/lc2, its horizontal-axis intercept is A|∇z|=hc, and its vertical-axis intercept is ∇2z=-κc (where κc is a characteristic curvature defined as hc/lc2; Eq. 8). These intercepts reveal topographic properties of special points in a landscape, namely the steady-state curvature of drainage divides and the steady-state steepness index of hillslope–valley transitions. The characteristic length lc quantifies the competition between knickpoint advection and diffusion and predicts how landscape dissection scales with the parameters. Thus, the slope of the curvature–steepness index line visually expresses how dissected a landscape is. Note that the line's slope can be represented either as K/D=1/lc2 units of curvature per 1 unit of steepness index or 1 unit of curvature per D/K=lc2 units of steepness index. For simplicity, we use the latter notation to express the slopes of the lines in Figs. 2–4.
Stream-power incision and linear diffusion LEMsGoverning equations
The LEM without an incision threshold follows the governing equation (e.g., Howard, 1994; Dietrich et al., 2003):
∂z∂t=-KA|∇z|+D∇2z+U.
This equation gives the rate of elevation change ∂z/∂t as the sum of three terms, namely (a) stream-power incision KA|∇z|, where K is the incision coefficient, A is drainage area, and |∇z| is topographic slope; (b) linear
diffusion D∇2z, where D is the diffusion coefficient and ∇2z is the Laplacian curvature; and (c) the uplift rate U. We
assume that Eq. (1) has base dimensions of horizontal length L, height H (which we treat as dimensionally distinct from L), and time T. All
quantities in Eq. (1) have dimensions that are combinations of L, H, and/or T, which we show in Table 1.
Descriptions and dimensions of the terms, variables, and parameters in the governing equations (Eqs. 1 and 2). Dimensions are expressed in terms of the model's fundamental dimensions of horizontal length L, vertical length (height) H, and time T.
SymbolDescriptionDimensions∂z/∂tTotal rate of elevation change at a point (x,y)HT-1Rates of elevation change due to -KA|∇z|stream-power incision (in Eq. 1)HT-1-K(A|∇z|-θ)threshold-limited stream-power incision (in Eq. 2)HT-1D∇2zlinear diffusionHT-1UupliftHT-1(x,y)Horizontal coordinatesLzElevationHtTimeTADrainage areaL2|∇z|Topographic slopeHL-1∇2zLaplacian curvatureHL-2KIncision coefficientT-1DDiffusion coefficientL2T-1UUplift rateHT-1θIncision thresholdHA|∇z|Steepness indexH
Note that the incision term KA|∇z| is a special case of the more general incision term KAm(|∇z|)n. As we explained in Theodoratos et al. (2018), dimensional analysis of an LEM with generic exponents m and n would lead to equivalent
results as the analysis of Eq. (1), but these results would be expressed with much more complicated mathematical formulas. Therefore, in Theodoratos
et al. (2018) we focused on the case of exponents m=0.5 and n=1, and we presented the results for generic m and n in an appendix. Likewise, in
the current study, the main presentation focuses on the case of m=0.5 and n=1, and in Appendix A we demonstrate that our graphical method is also
valid for the case of generic exponents m and n.
Following Perron et al. (2008), we can add an incision threshold to the LEM by recasting the incision term as K(A|∇z|-θ), where θ is the incision threshold. This formulation assumes that the incision rate KA|∇z| is reduced
everywhere by the constant quantity Kθ. The LEM examined here is based on the assumption that sediment transport is detachment-limited. Thus,
it does not include deposition, and negative incision rates would not be meaningful. Therefore, the incision term is set to zero where the term K(A|∇z|-θ) would be negative, i.e., where A|∇z|≤θ, and the governing equation
becomes
∂z∂t=D∇2z+U,A|∇z|≤θ-K(A|∇z|-θ)+D∇2z+U,A|∇z|>θ.
The incision threshold θ has the same dimensions as A|∇z|, i.e., dimensions of H.
Equation (2) assumes that precipitation rates are constant in time and uniform in space, and it incorporates climatic effects into the incision
coefficient K. Other LEMs use stochastic precipitation to drive their incision terms (e.g., Tucker, 2004, Whipple, 2004; Lague et al., 2005; DiBiase
and Whipple, 2011; Deal et al., 2018). The incision thresholds of these LEMs define limiting values of shear stress or stream power, below which no
incision occurs. At any given location in the landscape, these limiting values might be exceeded during some stochastic events and not exceeded during
other events, depending on their intensities. By contrast, in the case of the LEM that we examine, the assumption of constant and uniform
precipitation implies that any given combination of drainage area A and slope |∇z| would lead to the same value of stream power
(or shear stress) for any storm event (as all events would be equal), and this value of stream power would either be above or below the incision
threshold. In this idealized case, defining a topographic threshold based on A|∇z| is exactly equivalent to defining a
threshold of stream power (or shear stress).
We acknowledge that the LEMs with stochastic precipitation allow much more realistic integration of incision rates over time compared to the LEM that
we examine here. Therefore, these LEMs are more appropriate for studying the influence of incision thresholds on erosion rates compared to the LEM
that we use. However, our study has a different focus. Our study focuses on how the incision threshold θ influences topographic and scaling
properties of landscapes and on how this influence can be graphically expressed with curvature–steepness index lines. For these tasks, the
simplified formulation of the incision term of Eq. (2) is more practical. It may be possible in future work to extend this approach to include
incision thresholds driven by stochastic precipitation.
Summary of definitions and formulas used in this study.
DescriptionDefinitionEquationCharacteristic lengthlc=D/K(3)Characteristic heighthc=U/K(4)Characteristic timetc=1/K(5)Characteristic areaAc=lc2=D/K(6)Characteristic gradientGc=hc/lc=U/DK(7)Characteristic curvatureκc=hc/lc2=U/D(8)Incision-threshold numberNθ=Kθ/U(9)Curvature–steepness index relationship, without θ∇2z=(K/D)A|∇z|-(U/D)(10)Curvature–steepness index relationship, without θ∇2z=(1/lc2)A|∇z|-κc(11)Curvature–steepness index relationship, with θ∇2z=-κc,A|∇z|≤θ∇2z=(1/lc2)A|∇z|-(1+Nθ)κc,A|∇z|>θ.(12)Fraction of incision rate reduction1-Φ=1,A|∇z|≤θθA|∇z|,A|∇z|>θ.(22)Threshold factorΦ=0,A|∇z|≤θ1-θA|∇z|,A|∇z|>θ.NAFlow path lengthl: the distance along flow paths from a point to the farthest ridgeNADiffusion timescaletD=l2D(13)Kinematic wave celerity, without θc=KANAIncision timescale, without θtI=l/c=l/(KA)(14)Péclet number, without θPeθ=0=tD/tI=Allc2=AAcllc(15)Kinematic wave celerity, with θc=0,A|∇z|≤θKA-Kθ/|∇z|,A|∇z|>θ.(17)Incision timescale, with θtI=l/c=+∞,A|∇z|≤θl/KA-Kθ/|∇z|,A|∇z|>θ.(18)Péclet number, with θPe=0,A|∇z|≤θAAcllc-NθllcGc|∇z|,A|∇z|>θ.(20)Péclet number, with θPe=0,A|∇z|≤θ1-θA|∇z|Allc2,A|∇z|>θ.(23)
NA: not available.
Characteristic scales
The two governing equations (Eqs. 1 and 2) can be non-dimensionalized using characteristic scales of length, height, and time (lc,
hc, and tc) defined as (Theodoratos et al., 2018; Theodoratos and Kirchner, 2020)
3lc=D/K,4hc=U/K,5tc=1/K.
Effects of incision threshold on steady-state topography as reflected in the curvature–steepness index line. We show curvature–steepness index lines of landscapes with and without an incision threshold using black and gray, respectively (see Eqs. 11 and 12). The gray line in this figure is identical to the line in Fig. 1. Adding an incision threshold to the LEM changes the resulting steady-state topography, as indicated by the differences between the gray and black lines. The black line consists of two segments. The horizontal segment corresponds to points where incision is fully suppressed by the threshold. This horizontal segment is at ∇2z=-κc, which is the vertical-axis intercept of the gray line. This shows that the hilltop curvature (the most negative curvature value in the landscape) spreads to points on hillslopes beyond drainage divides. Thus, hillslopes become more convex due to the threshold. The inclined segment of the black line is parallel to the gray line and at a horizontal distance θ to its right. Thus, the horizontal-axis intercept is increased from hc to hc+θ due to the threshold; i.e., the hillslope–valley transition occurs at a larger steepness index value. This increase corresponds to a larger drainage area A and/or steeper slope |∇z|, both of which are consistent with the steepening of landscapes and the decrease in their drainage density by the incision threshold. In the case of the LEM that includes an incision threshold, the degree of landscape dissection is expressed by the length scale 1+Nθlc (see Eq. 21), where Nθ is a dimensionless incision-threshold number (Eq. 9). The square of this length scale is (1+Nθ)lc2, which is equal to (hc+θ)/κc, the ratio of the two intercepts of the black line. The quantity (1+Nθ)lc2 is the reciprocal of the slope of the black dashed line that connects the two intercepts. Thus, by comparing the slope of this auxiliary line and of the gray curvature–steepness index line, we can graphically express the effect of the incision threshold on landscape dissection, as shown by the white arrow.
We summarize these and other definitions for this presentation in Table 2. The characteristic scales lc, hc, and
tc can be viewed as intrinsic properties of a landscape in the sense that they depend exclusively on the values of the parameters K,
D, and U and not on extensive properties of the landscape such as the size of its domain or its maximum relief. We present geomorphologic
interpretations of these characteristic scales in Sect. 3.
By combining lc, hc, and tc we can define additional characteristic scales (Theodoratos et al., 2018). For
example, given that drainage areas A have dimensions of L2, we can define a characteristic area Ac
as the square of the characteristic length:
Ac=lc2=D/K.
Likewise, we can define a characteristic gradient Gc as
Gc=hc/lc=U/DK
and a characteristic curvature κc as
κc=hc/lc2=U/D.
Incision-threshold number Nθ
In Theodoratos and Kirchner (2020), we derived a dimensionless number, whose definition and interpretation we summarize here. Dimensional analysis of
the governing equation with an incision threshold θ (Eq. 2) yielded the dimensionless grouping of parameters Kθ/U. Specifically, all
terms in Eq. (2) give rates of elevation change and have dimensions of HT-1. Therefore, to non-dimensionalize Eq. (2), we divided all of
its terms by the uplift rate U. The quantity Kθ, which is included in the incision term in Eq. (2) and gives the reduction in the rate of
incision due to the threshold, also has dimensions of HT-1. Therefore, dividing the incision term in Eq. (2) by U yielded the
dimensionless ratio Kθ/U. We defined this dimensionless ratio as an incision-threshold number Nθ:
Nθ=Kθ/U.
This analysis led to a dimensionless version of Eq. (2) that includes only one parameter: the incision-threshold number Nθ. This implies
that Nθ is a control on the topography of landscapes that follow Eq. (2). Specifically, model landscapes that have equal incision-threshold
numbers Nθ can be set up such that they follow geometrically similar evolutions. Model landscapes that have different Nθ cannot
evolve geometrically similarly, and their topographies differ in ways that depend on their Nθ values. Simulation results illustrating these
points are presented in Theodoratos and Kirchner (2020).
We proposed two interpretations of the incision-threshold number Nθ in Theodoratos and Kirchner (2020). First, Nθ is defined as
the incision rate reduction Kθ relative to the uplift rate U (Eq. 9). The uplift rate U can be viewed as a characteristic rate of
elevation change because it is equal to the ratio of the characteristic height to the characteristic time, i.e., U=hc/tc
(Eqs. 4 and 5). Consequently, Nθ is a normalized incision rate reduction with respect to U. Second, if we rearrange Eq. (9) as Nθ=θ/(U/K), then we can interpret Nθ as giving the magnitude of θ relative to the parameter ratio U/K. Thus, the
definition of Nθ shows that incision thresholds from different landscapes should not be compared to each other according to their own
values but instead according to their values relative to the ratio U/K of each landscape.
Graphical illustrations of topographic and scaling effects of the incision thresholdDefining a steady-state topographic relationship between the steepness index and curvature
In Theodoratos et al. (2018), we presented a relationship that describes the steady-state topography of landscapes that evolve according to
Eq. (1). Specifically, if we set ∂z/∂t=0 and we solve the governing equation for curvature ∇2z, we obtain
∂z/∂t=0:∇2z=(K/D)A|∇z|-(U/D).
The quantity A|∇z| is equal to the steepness index (defined as Am/n|∇z| for drainage area and slope
exponents m and n; e.g., Whipple, 2001). For this reason, we refer to Eq. (10) as the curvature–steepness index relationship.
In a coordinate system in which the steepness index (A|∇z|) and curvature (∇2z) are plotted on the horizontal
and vertical axes, respectively, Eq. (10) plots as a straight line (for example, see Fig. 1, which we describe in more detail further
below). Equation (10) is a testable, quantitative prediction; if a landscape is in steady state and has evolved according to Eq. (1), then curvature
should plot as a straight line against the steepness index. Furthermore, this line can give estimates of the parameters K, D, and U because its
slope is K/D, and its intercepts are ∇2z=-U/D and A|∇z|=U/K. While we have not validated this
prediction with data, Eq. (10) is a rearranged version of Eq. (5) in Perron et al. (2009), which has been successfully tested with real-world
landscape data and has been used to estimate model parameters. Testing Eqs. (10), (11), and (12) as well as Figs. 1 and 2, which are described further
below, would be a reasonable next step after the current study.
Characteristic scales and the curvature–steepness index relationship
If we substitute the characteristic scales lc and κc for the parameter ratios K/D and U/D, then the
curvature–steepness index relationship (Eq. 10) becomes
∂z/∂t=0:∇2z=(1/lc2)A|∇z|-κc.
As this equation shows, an interpretation of lc and hc is that they control steady-state topography. Specifically, for a
landscape to be in steady state, drainage area A, topographic slope |∇z|, and curvature ∇2z must obey Eq. (11),
which is parameterized by the characteristic scales lc and κc or, equivalently, by lc and
hc because κc=hc/lc2 (Eq. 8). We can graphically illustrate the control of
lc and hc on the topography by plotting the curvature–steepness index line described by Eq. (11). As Fig. 1 shows, the
properties of such a line are controlled by lc and hc; specifically, its slope is 1/lc2, and its
intercepts are ∇2z=-κc=-hc/lc2 and A|∇z|=hc. Note
that the slope of this line can be represented either as K/D=1/lc2 units of curvature per 1 unit of steepness index or 1 unit of
curvature per D/K=lc2 units of steepness index. For simplicity, we use the latter notation to express the slopes of the
curvature–steepness index lines in Figs. 2–4.
Graphical illustration, using curvature–steepness index lines, of how parameters influence landscape properties. The three plots show how curvature–steepness index lines respond to increases in the uplift rate U, incision coefficient K, and diffusion coefficient D. (a) An increase in U parallel-shifts the line to the right and downward. This makes the vertical-axis intercept smaller (more negative) and the horizontal-axis intercept bigger, showing that ridges become more convex and that gradients become steeper (i.e., relief becomes higher). The line's slope remains 1/lc2, indicating that the scale of dissection does not change. (b) An increase in K rotates the line counterclockwise around the vertical-axis intercept. This makes the horizontal-axis intercept smaller and the line's slope bigger, showing that gradients become gentler (i.e., relief becomes lower) and that the landscape becomes more dissected (i.e., the scales of dissection become smaller). (c) An increase in D rotates the line clockwise around the horizontal-axis intercept. This moves the vertical-axis intercept closer to zero and decreases the line's slope, showing that ridges become less convex and that the landscape becomes less dissected (i.e., the scales of dissection become larger).
Graphical illustration, using curvature–steepness index lines, of how an incision threshold and parameters control landscape properties. The four plots show how curvature–steepness index lines respond to increases in the incision threshold θ, uplift rate U, incision coefficient K, and diffusion coefficient D. (a) An increase in the incision threshold θ parallel-shifts the line to the right. Note the difference with the shift in (b), which is to the right and downward. The shift in (a) makes the horizontal segment of the line longer and the horizontal-axis intercept bigger, showing that the zones of maximum convexity become wider, gradients become steeper, and drainage areas of valley heads become larger. The curvature value of the horizontal segments and the line's slope remain unchanged. The increase in the horizontal-axis intercept changes its ratio to the vertical-axis intercept, which expresses the degree of landscape dissection as explained in Sect. 3.3. This ratio can be visualized by the dashed auxiliary lines that connect the horizontal- and vertical-axis intercepts, and the change in the value of the ratio can be visualized by the rotation of these lines. (b) An increase in the uplift rate U shifts the line to the right and downward. This makes the vertical-axis intercept smaller (more negative) and the horizontal-axis intercept bigger, showing that ridges become more convex and that gradients become steeper (i.e., relief becomes higher). The changes in these two intercepts are not proportional and thus their ratio changes. This change can be visualized in the inset, where we plot the auxiliary lines such that they share the same starting point. This shows that the degree of landscape dissection changes when U is increased, whereas it did not change in the case of the LEM without an incision threshold (see Fig. 3). (c) An increase in the incision coefficient K rotates the inclined segment of the line counterclockwise around its intersection with the horizontal segment. The horizontal segment remains unchanged. Thus, the horizontal-axis intercept becomes smaller, which shows that gradients become gentler and the landscape becomes more dissected. (d) An increase in the diffusion coefficient D rotates the inclined segment line clockwise around the horizontal-axis intercept. This moves the horizontal segment and the vertical-axis intercept closer to zero and changes the ratio of the two intercepts, showing that ridges become less convex and that the landscape becomes less dissected.
Likewise, the curvature–steepness index relationship that corresponds to the LEM with an incision threshold θ is controlled by the
characteristic scales lc and κc. This relationship, however, is also controlled by the incision-threshold number
Nθ. To derive this relationship, we set ∂z/∂t=0 in Eq. (2), and we solve it for ∇2z. When we do this
for the second subdomain (where A|∇z|>θ), we encounter the ratio Kθ/D. This ratio can be rewritten as
Kθ/D=(KθU)/(UD)=Nθκc (Eqs. 8 and 9). Thus, we obtain the curvature–steepness index relationship:
∂z/∂t=0:∇2z=-κc,A|∇z|≤θ∇2z=(1/lc2)A|∇z|-(1+Nθ)κc,A|∇z|>θ.
We plot this equation in Fig. 2 in black, and for comparison we also plot the curvature–steepness index line without an incision threshold (Eq. 11) in
gray. The black line consists of two segments that correspond to the two subdomains in Eqs. (2) and (12). The first segment is horizontal and
describes a uniform steady-state curvature value of ∇2z=-κc for points with A|∇z|≤θ, where incision is fully suppressed by the threshold and only diffusion and uplift operate. The second segment is inclined and corresponds to points
with A|∇z|>θ where all three processes operate.
Equations (11) and (12) and Figs. 1 and 2 show that the characteristic scales lc, hc, and κc describe
the steady-state topography at points of special interest (see also Theodoratos et al., 2018). Furthermore, some effects of incision thresholds on
landscape properties can be visualized by comparing the curvature–steepness index lines with and without an incision threshold (black and gray lines
in Fig. 2).
First, the vertical-axis intercept of the curvature–steepness index line without an incision threshold (Fig. 1, Eq. 11) corresponds to ridges and
drainage divides, which have A=0 and/or |∇z|=0, i.e., A|∇z|=0. This intercept shows that the
steady-state curvature of ridges and drainage divides is ∇2z=-κc=-U/D (see also Roering et al., 2007; Perron
et al., 2009). Note that -κc is the most negative value of curvature. The horizontal segment of the black line in Fig. 2 (described
by the first subdomain in Eq. 12) expresses the fact that, in landscapes with an incision threshold θ, the points with A|∇z|≤θ have the same steady-state curvature as ridges and drainage divides, i.e., the most negative value of curvature. This shows that
adding an incision threshold to the LEM results in more convex hillslopes (e.g., Howard, 1994; Theodoratos and Kirchner, 2020).
Second, the curvature–steepness index line without an incision threshold (Fig. 1, Eq. 11) has a horizontal-axis intercept of A|∇z|=hc. This intercept corresponds to points with curvature ∇2z=0, which can be viewed as defining the transition
between hillslopes and valleys (e.g., Howard, 1994). Thus, points with a steepness index equal to the characteristic height hc can be used
to map hillslope–valley transitions (Theodoratos et al., 2018). Adding an incision threshold θ to the LEM makes landscapes steeper and
decreases the drainage density; i.e., it makes first-order basins bigger (e.g., Montgomery and Dietrich, 1992; Howard, 1994; Perron et al., 2008). These
two effects lead to steeper gradients |∇z| and larger drainage areas A at hillslope–valley transitions. Specifically, as Fig. 2
shows, the horizontal-axis intercept increases from A|∇z|=hc (gray line) to A|∇z|=hc+θ (black line).
Quantifying and visualizing the effect of the incision threshold on the scales of landscape dissection
In Theodoratos et al. (2018), we derived an interpretation of the characteristic length lc by analyzing the competition between the
advection and diffusion of elevation perturbations (e.g., knickpoints), which gives rise to ridges and valleys and controls their characteristic
sizes (e.g., Smith and Bretherton, 1972; Howard, 1994; Perron et al., 2008). Following Perron et al. (2008, 2009, 2012), we quantified the relative
strength of advection vs. diffusion using a Péclet number (Pe). The definition of our Péclet number differs somewhat from that of Perron
et al. Specifically, our definition includes a length scale l that we termed flow path length and that we defined as the distance along flow
paths from a given point to the farthest ridge.
The Péclet number is defined (e.g., Perron et al., 2008) as the ratio of a diffusion timescale tD to an incision timescale tI, each of
which gives a measure of the strength of the respective process. Specifically, a diffusion timescale can be defined as (e.g., Perron et al., 2008)
tD=l2D.
This timescale characterizes diffusive propagation over a distance l. In Theodoratos et al. (2018), to define tI, we first calculated
the celerity c that corresponds to the incision term in Eq. (1), which is a kinematic wave term (e.g., Whipple and Tucker, 1999). This celerity is
equal to c=KA. Perturbations can be assumed to be advected at this celerity (e.g., Berlin and Anderson, 2007; Perron et al., 2008). Lague
(2014) has criticized this assumption because it does not take into account the effects of knickpoints on hydraulics (e.g., on stream width) and their
feedbacks on the rate of knickpoint propagation, especially in the presence of incision thresholds. While we acknowledge this limitation, we
nonetheless assume that the rate of knickpoint advection is equal to the celerity c in Eq. (17) because our current focus is on interpreting the
characteristic scales lc, hc, and tc that pertain to Eqs. (1) and (2), which do not describe hydraulics
explicitly. Therefore, in Theodoratos et al. (2018), we defined the incision timescale tI as the ratio of the flow path length l,
which characterizes the location of points within drainage basins, to the celerity c, which characterizes the strength of advection:
tI=l/c=lKA.
Note that small values of tI and tD correspond to strong advection and diffusion, respectively.
We can quantify the relative strengths of advection and diffusion using the ratio of the respective timescales, which defines the Péclet number
(Theodoratos et al., 2018):
Pe=tD/tI=Allc2=AAcllc.
Diffusive propagation is stronger at points with a Péclet number smaller than 1, and advective propagation is stronger where the Péclet number is
larger than 1. Where the Péclet number is roughly equal to 1, diffusion and advection will be roughly equal (when measured by tD
and tI). Equation (15) shows that if a point's flow path length l is roughly equal to the characteristic length lc and
its drainage area A is roughly equal to the characteristic area Ac, then its Péclet number will be roughly equal to 1, i.e.,
l≈lc,A≈Ac≈lc2⟹Pe≈1.
Note that if the incision term has a slope exponent n≠1, then the condition |∇z|≈Gc must be included along
with l≈lc and A≈lc2 for the Péclet number to be Pe≈1.
The conditions A≈lc2 and l≈lc (Eq. 16) are not the only combination of A and l that give
Pe≈1, but they are significant because they lead to an interpretation of lc. Specifically, these conditions show that
advective propagation, which promotes valley dissection, is dominant at points farther than lc from the ridge and with a drainage area
greater than lc2. Therefore, in Theodoratos et al. (2018) we interpreted the characteristic length lc as giving a
measure of the smallest scales of dissection. This interpretation does not imply that valley heads are exactly 1 lc away from ridges or
that they have drainage areas exactly equal to 1 lc2. Rather, it implies that flow path lengths and drainage areas of valley heads
are of a similar order of magnitude as lc and lc2, respectively. Furthermore, it implies that valley heads in different
landscapes have l and A that scale with lc and lc2, respectively.
Adding the threshold θ to the incision term of the LEM changes the kinematic wave celerity c and thus the incision timescale
tI and the Péclet number (Pe). Specifically, the celerity becomes
c=0,A|∇z|≤θKA-Kθ/|∇z|,A|∇z|>θ,
and thus the incision timescale tI becomes
tI=l/c=+∞,A|∇z|≤θlKA-Kθ/|∇z|,A|∇z|>θ.
Note that the diffusion timescale tD is not affected by the incision threshold. Thus, we can use Eqs. (13) and (18) to define a
Péclet number (Pe) for the LEM with an incision threshold θ (Eq. 2), specifically
Pe=tD/tI=0,A|∇z|≤θAl-(θl/|∇z|)lc2,A|∇z|>θ.
It can be shown that Eq. (19) can be rewritten as
Pe=tD/tI=0,A|∇z|≤θAAcllc-NθllcGc|∇z|,A|∇z|>θ,
where Nθ is the incision-threshold number (Eq. 9). Equation (20) shows that adding an incision threshold θ to the LEM reduces the
Péclet number relative to the Péclet number for the LEM without a threshold (Eq. 15). This agrees with the fact that the threshold weakens the
incision term. More specifically, the Péclet number for the LEM with θ is reduced by the quantity Nθ(l/lc)(Gc/|∇z|).
Note that the Péclet number definition by Perron et al. (2008) also includes a reduction that depends on Nθ (denoted as
θ′ in Perron et al., 2008). The two definitions differ in that ours includes the product Al (where A is the drainage area
and l is the flow path length), whereas the Perron et al. definition includes only a length scale (squared). By including Al, our definition
can account for the scaling of A with l, which depends on the convergence or divergence of topography. The implications of this property of our
Péclet number are discussed in Sect. 4.2.3 of Theodoratos et al. (2018).
Using Eq. (20) we see that the conditions l≈lc, A≈lc2, and |∇z|≈Gc, which lead to a Péclet number roughly equal to 1 for the case without an incision threshold (Eq. 16), will lead to Pe≈1-Nθ<1 when θ is included. The fact that the difference between Pe and 1 is equal to Nθ suggests that we could obtain the
value Pe≈1 by adjusting the values of l, A, and |∇z| such that they depend on Nθ. Indeed, we
observe that
l≈1+Nθlc,A≈(1+Nθ)lc2,|∇z|≈1+NθGc⟹Pe≈1.
Note that 1+Nθlc is larger than lc, which agrees with observations that incision thresholds reduce
landscape dissection (e.g., Montgomery and Dietrich, 1992; Howard, 1994; Perron et al., 2008).
Equation (21) shows that, in the case of a landscape that includes an incision threshold θ, the smallest scales of dissection are not
characterized by the characteristic length lc on its own, but rather jointly by lc and the incision-threshold number
Nθ through the quantity 1+Nθlc. Consequently, the presence of θ changes the dependence of the scales
of dissection on the LEM parameters. Without an incision threshold, the scales of landscape dissection depend on lc, which depends on
the incision and diffusion coefficients K and D (Eq. 3). On the other hand, when θ is included in the LEM, the scales of dissection depend
on 1+Nθlc, which depends on K and D, but also on the uplift rate U and the incision threshold θ. We
illustrate an example of the dependence on U in Fig. 4b.
The length scales lc and 1+Nθlc can be expressed graphically by the horizontal- and vertical-axis
intercepts of curvature–steepness index lines, specifically by the ratio of these intercepts (or, more precisely, by the ratio of their absolute
values). This ratio is equal to hc/κc=lc2 in the case without an incision threshold (see Fig. 1) and
equal to (hc+θ)/κc=(1+Nθ)lc2 in the case that includes the incision
threshold θ (see Fig. 2). Note that the first ratio is equal to the inverse of the slope of the curvature–steepness index line, which is 1/lc2. On the other hand, the second ratio is not the inverse of this slope, which remains 1/lc2 when the
threshold θ is included. Instead, it is the inverse of the slope of an auxiliary line connecting the two intercepts. In Fig. 2, we show this
auxiliary line as a black dashed line. The effect of the incision threshold on valley dissection can be visualized graphically by comparing
the slope of the curvature–steepness index line against the slope of the black dashed auxiliary line. We show this comparison as a thick white
arrow.
It should be noted that the characteristic length lc depends only on K and D when the slope exponent is n=1. However, for
other values of n, lc will also depend on the uplift rate U; in this more general case, lc=(K-1DnU1-n)1/(n+2m) (see Appendix A in Theodoratos et al., 2018). Therefore, the degree of landscape dissection in general is not independent
of U. Specifically, an increase in U leads to a decrease in landscape dissection for n>1 and to an increase in landscape dissection for n<1,
which agrees with previous observations of the dependence of drainage density on the uplift rate (e.g., Clubb et al., 2016). Interestingly, as
revealed by the current study, if an incision threshold is included, the degree of landscape dissection depends on U even for n=1.
How the curvature–steepness index line responds to parameter value changes
As we show in Figs. 3 and 4, differences in the properties of landscapes with different parameters K, D, U, and θ can be graphically
summarized by curvature–steepness index lines because the slopes and intercepts of these lines depend on the characteristic scales lc,
hc, and κc, as well as on the incision-threshold number Nθ, which in turn depend on the parameters.
Figure 3 shows curvature–steepness index lines without incision thresholds. It consists of three panels, each showing how the lines respond to an
increase in one of the three parameters U, K, and D. In Fig. 3a, an increase in the uplift rate U shifts the curvature–steepness index line
downward and to the right without changing its slope. This illustrates that the characteristic height and curvature (hc and
κc), which control the intercepts of the line, are proportional to U (Eqs. 4 and 8), while the characteristic length
lc, which controls the line's slope, is independent of U (Eq. 3). The parallel shift of the line corresponds to more convex ridges (so
that diffusion can keep up with uplift), steeper gradients (so that incision can keep up with uplift), and unchanged landscape
dissection. Analogously, Fig. 3b shows that an increase in the incision coefficient K leads to a counterclockwise rotation of the line around the
vertical-axis intercept, which corresponds to a more dissected landscape (smaller lc), milder gradients (smaller hc), and
unchanged ridge convexity (unchanged κc). Finally, in Fig. 3c, an increase in the diffusion coefficient D results in a clockwise
rotation of the line around the horizontal-axis intercept. This corresponds to a smoother landscape with less dissection (larger lc),
less convex ridges (smaller κc), and an unchanged steepness index at hillslope–valley transitions (unchanged hc).
Figure 4 illustrates in four panels how curvature–steepness index lines respond to increases in the value of either the incision threshold θ
or one of the parameters U, K, and D. As a reminder, a curvature–steepness index line with an incision threshold consists of two segments, a
horizontal and an inclined. Note that, as we explain in the previous subsection (Sect. 3.3), a curvature–steepness index line that includes an
incision threshold does not express landscape dissection through the slope of its inclined segment, which depends only on the characteristic length
lc, but rather through the ratio of the horizontal- and vertical-axis intercepts, which is equal to 1+Nθlc. This ratio can be graphically illustrated by the slope of an auxiliary line that connects the two intercepts, such as the dashed
black line in Fig. 2. In each panel of Fig. 4, we show two dashed black auxiliary lines to illustrate how the ratio of intercepts responds to the
parameter changes.
In Fig. 4a we illustrate an increase in θ. The steepness index A|∇z| must reach a greater value before exceeding
the increased θ, and thus the horizontal segment of the curvature–steepness index line becomes longer. The vertical position of this segment
(along with the vertical-axis intercept) does not change because the characteristic curvature κc does not depend on θ. The
slope of the curvature–steepness index line also does not change because lc does not depend on θ. Thus, the increase in
θ parallel-shifts the inclined segment of the line to the right. Consequently, the horizontal-axis intercept increases, which expresses the
steepening of gradients and the decrease in landscape dissection. The decrease in dissection is also expressed by the fact that the ratio of the
horizontal- to the vertical-axis intercept increases, as shown by the clockwise rotation of the dashed auxiliary line.
In Fig. 4b we show that an increase in the uplift rate U parallel-shifts the curvature–steepness index line downward and to the right. Furthermore,
the horizontal- and vertical-axis intercepts move to the right and downward, respectively (κc and hc are proportional
to U), and the slope of the inclined segment remains unchanged (lc does not depend on U). As we explain in Sect. 3.3, the value
of U affects the value of 1+Nθlc, which expresses the scales of landscape dissection. Specifically, the increase
in U leads to a decrease in Nθ. This reflects the fact that θ becomes less important relative to the increased U. Thus, the
decrease in dissection due to the threshold is somewhat moderated by the increase in U. This moderation is graphically illustrated by the slopes of
auxiliary lines connecting the intercepts of the curvature–steepness index lines. These auxiliary lines do not intersect, and thus their slopes
cannot be readily compared visually. Therefore, we plot them again in an inset such that they start from the same point. In this way, we can see that
the increase in U leads to a counterclockwise rotation of the auxiliary lines, which expresses the increase in dissection.
In Fig. 4c we illustrate the response of the curvature–steepness index line to an increase in the incision coefficient K. The horizontal segment of
the line remains unchanged and the inclined segment is rotated counterclockwise around the point of transition between the two segments. Likewise, the
dashed auxiliary line connecting the horizontal- and vertical-axis intercepts is rotated counterclockwise. These responses express the fact that dissection is
decreased and that gradients become milder when K is increased. Finally, in panel (d), we show that increasing the diffusion coefficient D leads
to a clockwise rotation of the inclined segment of the line around its horizontal-axis intercept, which remains unchanged. The rotation results in
moving the horizontal segment up and in rotating the dashed auxiliary line clockwise. These changes express the reduction in landscape dissection and
the reduction in the convexity of ridges and hillslopes.
Quantifying how the influence of the incision threshold varies within a landscape
Thus far, we have examined how the influence of the incision threshold θ varies between different landscapes with different parameters using
the incision-threshold number Nθ (Eq. 9). This number is constant for any given landscape if the parameters of the landscape are
constant. Now, we turn our attention to how the influence of the incision threshold θ varies within a given landscape.
We can quantify the relative influence of the threshold θ on the rate of incision using the fraction θ/(A|∇z|). This fraction is equal to Kθ, which is the reduction in the incision rate due to the threshold, divided by KA|∇z|, which is the
incision rate if there were no threshold. Therefore, θ/(A|∇z|) shows by what fraction the incision rate is
reduced due to the threshold. Where A|∇z|=θ, the fraction θ/(A|∇z|) is equal
to 1, which agrees with the incision rate being reduced by 100 % (i.e., being reduced to zero). At points with A|∇z|
smaller than θ, calculating the fraction θ/(A|∇z|) would not be meaningful; instead, because the threshold
completely suppresses incision under these conditions, we assign a value of 1 to the fractional reduction in incision rate.
We can associate the fractional reduction in incision rate with the Tucker (2004) threshold factor Φ. Tucker (2004) defined Φ to quantify the
fraction of precipitation events leading to shear stress above a threshold value, i.e., the fraction of events that lead to erosion. Tucker (2004)
used Φ to express the incision term of his LEM as KAmbSnbΦ. In the case of the LEM examined here
(Eq. 2), following Tucker's notation, we can express the incision term as KA|∇z|Φ, where the threshold factor
Φ is equal to 1-θ/(A|∇z|) for A|∇z|>θ and to 0 for A|∇z|≤θ. Thus, the quantity 1-Φ is equal to the fractional reduction in incision rate, i.e.,
1-Φ=1,A|∇z|≤θθA|∇z|,A|∇z|>θ.
Consequently, in what follows we denote the fractional reduction in incision rate as 1-Φ. We illustrate the properties of the quantity 1-Φ
with plots and maps in Figs. 5–7.
How the relative influence of the incision threshold changes across a landscape. We plot the quantity 1-Φ vs. the steepness index A|∇z|, where Φ is a threshold factor that can subsume the incision threshold θ (definitions in Eq. 22 and Sect. 4). The quantity 1-Φ is dimensionless and expresses the fractional reduction in incision rate due to the incision threshold θ. The 1-Φ curve presented here shows how this fraction varies across the landscape. The value 1-Φ=1 corresponds to points where incision is fully suppressed by the threshold, i.e., where the incision rate is reduced by 100 %. Thus, the horizontal segment of the 1-Φ curve corresponds to the horizontal segment of the curvature–steepness index line in Fig. 2. At points with A|∇z| much larger than θ (the far right of the plot), the incision rate is reduced by a very small fraction, and the 1-Φ curve asymptotically approaches 0. The black dot on the 1-Φ curve corresponds to the steepness index value hc+θ, which corresponds to hillslope–valley transitions, as shown in Fig. 2. The position of the black dot on the curve helps us visualize how large the incision rate reduction fraction is across different regimes of a landscape. This position depends on the characteristic height hc, θ, and the dimensionless ratio θ/hc. We define this ratio as the incision-threshold number Nθ (Eq. 9).
Comparison of the relative influence of incision thresholds with different magnitudes. We present curves of the quantity 1-Φ vs. the steepness index A|∇z| for four different values of Nθ. The black dots show the values of A|∇z| and 1-Φ that correspond to hillslope–valley transitions. The curve with Nθ=0.2, the smallest of the four values of Nθ, starts with a short horizontal segment and then descends steeply and approaches 0 rapidly. Furthermore, its black dot corresponds to the value 1-Φ=0.2/1.2=1/6. By contrast, the curve with the largest value, Nθ=2, starts with a long horizontal segment, descends gradually, approaches 0 slowly, and has a black dot with 1-Φ=1/3=0.333. These differences show that as Nθ increases, (a) incision is fully suppressed by the threshold in bigger portions of hillslopes, (b) the steepness index must reach greater values for the influence of the threshold to start becoming negligible, and (c) the hillslope–valley transition occurs at larger values of 1-Φ; i.e., the threshold has a strong influence not only on hillslopes, but also on the valley network.
Control of the incision-threshold number Nθ on the spatial distribution of the fractional reduction in incision rate. We map the quantity 1-Φ across four steady-state simulated landscapes with different values of Nθ. We use the same values of Nθ as in Fig. 6. Details about the setup and parameters of these simulations are presented in Appendix B. Lighter colors correspond to larger values of 1-Φ, i.e., to stronger influence of the incision threshold. The spatial distribution of 1-Φ follows the dendritic pattern of the valley network. As Nθ increases, the maps become lighter; i.e., areas with a strong influence of the incision threshold become more widespread both on hillslopes and in valleys.
In Fig. 5 we plot 1-Φ vs. the steepness index A|∇z| according to Eq. (22). The curve consists of two parts. The
first is a horizontal segment that describes the value 1-Φ=1 and corresponds to points with A|∇z|≤θ, where
incision is fully suppressed by the threshold. The second part corresponds to points with A|∇z|>θ, forming part of a
hyperbola that asymptotically approaches 0. This asymptotic approach expresses the fact that, at points with a steepness index A|∇z| much larger than θ, the incision threshold has a very small relative influence on the incision rate.
To indicate how different parts of the 1-Φ curve in Fig. 5 correspond to different regimes of a landscape, we identify the point that
corresponds to hillslope–valley transitions. As explained in Sect. 3.2, hillslope–valley transitions can be defined as points with zero curvature
and therefore with a steady-state steepness index of A|∇z|=hc+θ. Consequently, the fractional
reduction in incision rates θ/(A|∇z|) at these points is θ/(hc+θ). We can rewrite this
value in terms of the incision-threshold number Nθ as Nθ/(1+Nθ). Thus, in Fig. 5, hillslope–valley transitions
correspond to the point with coordinates (A|∇z|,1-Φ)=(hc+θ,Nθ/(1+Nθ)), which we mark with a black dot. The part of the curve above and to the left of this dot corresponds to hillslopes, and the part below and to the
right corresponds to the valley network.
With Fig. 6, we examine how the value of the incision-threshold number Nθ of a landscape controls the relationship between the quantity
1-Φ and the steepness index A|∇z|. Specifically, in Fig. 6 we show curves of 1-Φ vs. A|∇z| for four landscapes with incision-threshold numbers Nθ equal to 0.2, 0.4, 1, and 2. The landscapes are assumed to have equal
parameters K, D, and U and therefore to have equal characteristic scales. The curves with greater values of Nθ also have greater
incision thresholds θ, and thus they have longer horizontal segments. Furthermore, the curves with greater values of Nθ go towards
zero more slowly. On each curve, we show the hillslope–valley transition using a black dot. The value of the quantity 1-Φ corresponding to each
dot becomes larger as Nθ increases. Thus, in landscapes with smaller Nθ, the incision rate is reduced by large fractions only on
the hillslopes, and in valleys it is reduced by small fractions. By contrast, in landscapes with greater Nθ, incision can be reduced by
large fractions both on hillslopes and in valleys.
Figure 7 shows maps of the quantity 1-Φ across four steady-state landscapes. We simulated these landscapes with the CHILD (Channel–Hillslope Integrated Landscape Development) numerical model
(Tucker et al., 2001). Details about these simulations and additional results are presented
in Theodoratos and Kirchner (2020). Here, we provide brief information about the parameters and setup of these simulations in Appendix B. To
illustrate how the spatial distribution of 1-Φ depends on the incision-threshold number Nθ, we ran four simulations with Nθ values of 0.2, 0.4, 1, and 2, i.e., the same Nθ values as in Fig. 6. The pixels of the four maps are colored according to their values
of 1-Φ using a grayscale that ranges from white to black. Lighter colors correspond to larger values of 1-Φ, i.e., to a stronger influence
of the incision threshold. As expected, lighter colors appear near ridges and on hillslopes where the incision threshold has a stronger influence.
The patterns in Fig. 7 reflect the spatial distribution of drainage area and slope because the incision threshold in Eq. (2) is defined as a
topographic threshold. However, maps of the quantity 1-Φ would be useful for other formulations of the incision threshold as well. For example,
the Tucker (2004) formulation of the incision threshold assumed stochastic precipitation. Tucker quantified the influence of this incision threshold
using the threshold factor Φ, which ranges between 0 and 1 (and on which our quantity 1-Φ is based, as mentioned above). Therefore, the
quantity 1-Φ could be calculated for the case of the Tucker (2004) LEM, and maps of this quantity would visualize how the influence of the
incision threshold is spatially distributed across landscapes.
The fractional reduction in incision rate 1-Φ and the threshold factor Φ can be used to simplify the definition of the Péclet
number (Pe). Specifically, we can rearrange the definition of Pe (Eq. 19) such that it includes the fraction θ/(A|∇z|):
Pe=0,A|∇z|≤θ1-θA|∇z|AAcllc,A|∇z|>θ.
If this equation is combined with the definition of 1-Φ (Eq. 22), then we can rewrite the definition of Pe in compact form as
Pe=Φ⋅Peθ=0,
where Peθ=0 is the Péclet number for the LEM without an incision threshold (see Eq. 15). Equations (23) and (24) reveal that the
influence of the incision threshold on the Péclet number varies across the landscape. Specifically, larger values of Pe, which correspond to
larger values of the steepness index A|∇z|, are less sensitive to the incision threshold.
Summary and conclusions
We present graphical methods that summarize topographic and scaling properties of landscapes following a simple stream-power incision and linear
diffusion LEM (Eq. 1) and that illustrate the effects of adding an incision threshold θ (Eq. 2). Our results referring to the LEM without
an incision threshold (Eq. 1) have been presented before (Theodoratos et al., 2018), but we show them here again to contrast them against those referring
to the LEM with the threshold θ (Eq. 2). The two LEMs (Eqs. 1 and 2) assume that the incision term has drainage area and slope exponents
m=0.5 and n=1 because this combination significantly simplifies the mathematical derivations. However, as we show in Appendix A, our results are
also valid for generic exponents m and n.
For the first graphical method, we plot steady-state relationships between curvature ∇2z and the steepness index A|∇z| (Eqs. 10, 11, and 12), which we obtain from the governing equations (Eqs. 1 and 2). These relationships can be viewed as counterparts of
the relationship between topographic slope and drainage area, which is typically assumed to follow a power law in detachment-limited landscapes but
not in landscapes that are also influenced by hillslope diffusion. These relationships plot as straight lines (Figs. 1 and 2), whose properties (slope
and intercepts) depend on the incision threshold θ and on the characteristic scales of the landscape, which in turn depend on the parameters of
the LEM, i.e., on the incision coefficient K, the diffusion coefficient D, and the uplift rate U. (Eqs. 3, 4, and 8). A reasonable follow-up
study would be to validate these results against real-world landscapes and specifically explore whether curvature and steepness index data from
real landscapes would plot against each other as straight lines.
With Fig. 2, we show that curvature–steepness index lines can graphically illustrate the effects of incision thresholds on landscapes. Specifically, the
ways in which curvature–steepness index lines with and without a threshold differ from each other illustrate that thresholds make hillslopes more
convex, make gradients steeper, and reduce the drainage density. These effects have been presented elsewhere (e.g., Montgomery and Dietrich, 1992;
Howard, 1994; Perron et al., 2008), but the curvature–steepness index lines offer new ways to visualize them graphically. In Figs. 3 and 4, we
illustrate the dependence of these properties on the parameters K, D, U, and θ by showing how curvature–steepness index lines respond to
increases in these parameters, one at a time. These figures demonstrate an advantage of curvature–steepness index lines: the topographic effects of
model parameter changes are expressed as simple shifts and rotations of these lines.
In Sect. 3.3, we examine in more detail the effects of the incision threshold θ on drainage density and the scales of landscape dissection, as well as
how these effects can be visualized with curvature–steepness index lines. We assume that dissection is controlled by the competition between the
advection and diffusion of perturbations (e.g., Smith and Bretherton, 1972; Howard, 1994; Perron et al., 2008), and thus we examine the effects of
θ using a Péclet number (Pe) (Eqs. 15 and 19; see also Perron et al., 2008, 2012; Theodoratos et al., 2018). For the LEM that does not include
an incision threshold, we found in Theodoratos et al. (2018) that the characteristic length lc characterizes the smallest scales of
dissection. Note that the slope of curvature–steepness index lines is 1/lc2; therefore, this slope graphically expresses the
scales of the dissection of landscapes without incision thresholds. Adding the incision threshold θ, we find that the smallest scales of dissection
are characterized by the length scale 1+Nθlc, where Nθ is a dimensionless incision-threshold number defined
as Nθ=Kθ/U (Eq. 9). This length scale is longer than lc, which expresses the fact that incision thresholds reduce
the drainage density. The square of this length scale is (1+Nθ)lc2 and is equal to the ratio between the horizontal- and
vertical-axis intercepts of the curvature–steepness index line. As we show in Fig. 2, an auxiliary line connecting these two intercepts would have a
slope of 1/((1+Nθ)lc2). Thus, we can graphically visualize the effect of the incision threshold on landscape
dissection by comparing the slope of this auxiliary line with the slope of the curvature–steepness index line.
The second graphical method consists of plots of the dimensionless fraction θ/A|∇z|, which gives the fraction by
which the incision rate is reduced due to the threshold (see the governing equation, Eq. 2). We found that this fraction is equal to 1-Φ
(Eq. 22), where Φ is a threshold factor (see Tucker, 2004) that subsumes the effect of the incision threshold θ on the incision term of
the LEM (see Eqs. 2 and 22). Thus, we denote the fractional reduction in the incision rate as 1-Φ. In Figs. 5–7, we present plots and maps of
1-Φ that illustrate how the relative influence of incision thresholds will vary across a given landscape and how the variation of this relative
influence depends on the landscape's incision-threshold number Nθ.
The two dimensionless numbers examined here, Nθ and 1-Φ, quantify the relative influence of the incision threshold θ: the
first with respect to the parameters K and U and the second with respect to the steepness index. Thus, Nθ quantifies how θ
affects different landscapes with different parameters, and 1-Φ quantifies how the influence of θ varies across different points of a
given landscape. We find that the definition of the Péclet number (Pe) can be rewritten in two equivalent forms (Eqs. 20 and 24), which reveal how
Pe depends on Nθ and on Φ, respectively.
The three dimensionless numbers, Pe, Nθ, and Φ, along with the characteristic scales lc, hc, and
tc, provide a thorough characterization of landscapes that follow the governing equation (Eq. 2). Furthermore, plots of the
curvature–steepness index relationship offer a straightforward way to graphically express the geomorphologic meaning of these dimensionless numbers
and characteristic scales. Even though the specific definitions of these quantities refer only to the LEMs examined here (Eqs. 1 and 2), the approach
that underpins our graphical methods is more generally applicable. For example, an LEM with an incision threshold and stochastic precipitation would have
a different governing equation than Eq. (2) and thus a different curvature–steepness index relationship than Eq. (12) and Fig. 2. However,
curvature and the steepness index would still be reasonable axes for plotting data from such an LEM if it included diffusion. Likewise, the quantity
1-Φ would follow a different formula than Eq. (22), but maps of this quantity would be useful in visualizing spatial patterns of the influence
of the incision threshold across a landscape. Consequently, our graphical methods could potentially be helpful for the analysis of a broader range of
models than those examined here.
Curvature–steepness index lines for generic drainage area and slope exponents m and n
In this Appendix, we demonstrate that our graphical method remains valid for the case of LEMs with incision terms that have generic drainage area and
slope exponents m and n, respectively.
For generic exponents m and n, the governing equations Eqs. (1) and (2) respectively become
∂z∂t=-KAm(|∇z|)n+D∇2z+U
and
∂z∂t=D∇2z+U,Am(|∇z|)n≤θ-K(Am(|∇z|)n-θ)+D∇2z+U,Am(|∇z|)n>θ.
Given that the steepness index is defined as ks=Am/n|∇z| (e.g., Whipple, 2001), the quantity Am(|∇z|)n in the above equations is equal to the steepness index raised to the power n, i.e., Am(|∇z|)n=ksn, and
the incision threshold θ is a threshold of the quantity ksn.
Setting ∂z/∂t=0 in Eqs. (A1) and (A2), we can derive the corresponding steady-state relationships between curvature and the
steepness index:
∇2z=(K/D)ksn-(U/D)
and
∇2z=-(U/D),ksn≤θ∇2z=(K/D)ksn-(1+Nθ)(U/D),ksn>θ,
where Nθ is the incision-threshold number, still defined as Nθ=Kθ/U.
When plotted in axes of ∇2z and ksn, Eq. (A3) has the same basic properties as Eq. (11), which is the curvature–steepness index relationship with m=0.5, n=1, and θ=0. Specifically, Eq. (A3) plots as a straight line with a slope equal to K/D, a vertical-axis
intercept equal to -U/D, and a horizontal-axis intercept equal to U/K. Consequently, for generic exponents m and n, changes in the values
of the parameters K, D, and U are still expressed graphically as shifts and rotations of the curvature–steepness index line, as seen in Fig. 3
for the case of m=0.5 and n=1.
Note that the characteristic scales of length and height (lc and hc) are not equal to D/K and U/K for generic
exponents m and n. Rather, they are defined by the more complicated formulas
lc=(K-1DnU1-n)1/(n+2m)
and
hc=(K-2Dn-2mU2-n+2m)1/(n+2m),
whose derivation can be seen in Appendix A of Theodoratos et al. (2018). However, the parameter ratios K/D and U/K still express the relative
strengths of incision vs. diffusion and incision vs. uplift. By contrast, note that the parameter ratio U/D remains equal to the characteristic
curvature κc, which expresses the relative strength of diffusion vs. uplift. Consequently, the shifts and rotations of the
curvature–steepness index line still express changes in scaling and topographic properties of landscapes, such as changes in the curvature of ridges, the
degree of dissection, and gradients at hillslope–valley transitions.
Likewise, when plotted in axes of ∇2z and ksn, Eq. (A4) has the same properties as Eq. (12): the
curvature–steepness index relationship with an incision threshold and with m=0.5 and n=1. Specifically, Eq. (A4) plots as a line with two segments:
a horizontal segment at ∇2z=-U/D for ksn≤θ and an inclined segment with slope equal to K/D and
horizontal-axis intercept equal to (U/K)+θ for ksn>θ. This line, too, responds to changes in the parameters θ, K,
D, and U, with shifts and rotations equivalent to the shifts and rotations shown in Fig. 4 for the case of m=0.5 and n=1.
Finally, for generic exponents m and n, the fractional reduction in incision rate 1-Φ is defined as
1-Φ=1,ksn≤θθ/ksn,ksn>θ,
which plots as shown in Figs. 5 and 6 but in axes of ksn.
Setup of numerical simulations
We prepared the maps in Fig. 7 with results from numerical simulations that we performed using the CHILD model, originally for the work discussed in
Theodoratos and Kirchner (2020). In that work, we present much more information about these simulations and their results. Here, we briefly summarize
the model setup and parameterization.
All four landscapes in Fig. 7 have an incision coefficient K= 2 × 10-6a-1, diffusion coefficient
D= 0.5 × 10-2m2a-1, and uplift rate U= 0.5 × 10-4ma-1. Each landscape's
incision threshold θ depends on the value of its incision-threshold number Nθ according to θ=Nθ⋅(U/K) (see
Eq. 9), where U/K= 25 m for all landscapes. Therefore, the landscapes have the incision thresholds seen in Table B1.
We simulated the four landscapes on triangular irregular networks (TINs) with a total extent of 7.5 km× 11.25 km and an average
TIN edge length of 20 m, which resulted in around 250 000 TIN points. Each map in Fig. 7 shows a part of the TIN,
specifically a rectangular region with a size of 5 km× 4 km centered around the largest drainage basin of the corresponding
landscape.
Details about the implementation of the governing equation (Eq. 2) in CHILD (Tucker et al., 2001) can be found in Theodoratos et al. (2018) and in
Theodoratos and Kirchner (2020).
Incision-threshold numbers Nθ and corresponding incision thresholds θ of the four landscapes illustrated in Fig. 7. The parameter ratio U/K is equal to 25 m for all landscapes.
The data presented here were synthesized using the CHILD model (Tucker et al., 2001). The input files needed to reproduce them are available from the corresponding author upon request.
Author contributions
NT derived and analyzed the theoretical, numerical, and graphical results, and NT and JWK interpreted them. NT drafted the paper, and NT and JWK edited it.
Competing interests
The authors declare that they have no conflict of interest.
Disclaimer
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Acknowledgements
This study was made possible by financial support from ETH Zurich. We thank Eric Deal for helpful discussions.
Review statement
This paper was edited by Paola Passalacqua and reviewed by Fiona Clubb and Philippe Steer.
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