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.

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 (

The LEM without an incision threshold follows the governing equation (e.g., Howard, 1994; Dietrich et al., 2003):

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.

Note that the incision term

Following Perron et al. (2008), we can add an incision threshold to the LEM by recasting the incision term as

Equation (2) assumes that precipitation rates are constant in time and uniform in space, and it incorporates climatic effects into the incision
coefficient

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

Summary of definitions and formulas used in this study.

NA: not available.

The two governing equations (Eqs. 1 and 2) can be non-dimensionalized using characteristic scales of length, height, and time (

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

We summarize these and other definitions for this presentation in Table 2. The characteristic scales

By combining

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

We proposed two interpretations of the incision-threshold number

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

In a coordinate system in which the steepness index (

If we substitute the characteristic scales

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

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

Likewise, the curvature–steepness index relationship that corresponds to the LEM with an incision threshold

Equations (11) and (12) and Figs. 1 and 2 show that the characteristic scales

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

Second, the curvature–steepness index line without an incision threshold (Fig. 1, Eq. 11) has a horizontal-axis intercept of

In Theodoratos et al. (2018), we derived an interpretation of the characteristic length

The Péclet number is defined (e.g., Perron et al., 2008) as the ratio of a diffusion timescale

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):

The conditions

Adding the threshold

Note that the diffusion timescale

Note that the Péclet number definition by Perron et al. (2008) also includes a reduction that depends on

Using Eq. (20) we see that the conditions

Equation (21) shows that, in the case of a landscape that includes an incision threshold

The length scales

It should be noted that the characteristic length

As we show in Figs. 3 and 4, differences in the properties of landscapes with different 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

Figure 4 illustrates in four panels how curvature–steepness index lines respond to increases in the value of either the incision threshold

In Fig. 4a we illustrate an increase in

In Fig. 4b we show that an increase in the uplift rate

In Fig. 4c we illustrate the response of the curvature–steepness index line to an increase in the incision coefficient

Thus far, we have examined how the influence of the incision threshold

We can quantify the relative influence of the threshold

We can associate the fractional reduction in incision rate with the Tucker (2004) threshold factor

How the relative influence of the incision threshold changes across a landscape. We plot the quantity

Comparison of the relative influence of incision thresholds with different magnitudes. We present curves of the quantity

Control of the incision-threshold number

In Fig. 5 we plot

To indicate how different parts of the

With Fig. 6, we examine how the value of the incision-threshold number

Figure 7 shows maps of the quantity

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

The fractional reduction in incision rate

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

For the first graphical method, we plot steady-state relationships between curvature

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

In Sect. 3.3, we examine in more detail the effects of the incision threshold

The second graphical method consists of plots of the dimensionless fraction

The two dimensionless numbers examined here,

The three dimensionless numbers, Pe,

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

For generic exponents

Given that the steepness index is defined as

Setting

When plotted in axes of

Note that the characteristic scales of length and height (

Likewise, when plotted in axes of

Finally, for generic exponents

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

We simulated the four landscapes on triangular irregular networks (TINs) with a total extent of 7.5

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

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.

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.

The authors declare that they have no conflict of interest.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This study was made possible by financial support from ETH Zurich. We thank Eric Deal for helpful discussions.

This paper was edited by Paola Passalacqua and reviewed by Fiona Clubb and Philippe Steer.