The ability of erosional processes to incise into a topographic surface can be limited by a threshold. Incision thresholds affect the topography of landscapes and their scaling properties and can introduce nonlinear relations between climate and erosion with notable implications for long-term landscape evolution. Despite their potential importance, incision thresholds are often omitted from the incision terms of landscape evolution models (LEMs) to simplify analyses. Here, we present theoretical and numerical results from a dimensional analysis of an LEM that includes terms for threshold-limited stream-power incision, linear diffusion, and uplift. The LEM is parameterized by four parameters (incision coefficient and incision threshold, diffusion coefficient, and uplift rate). The LEM's governing equation can be greatly simplified by recasting it in a dimensionless form that depends on only one dimensionless parameter, the
incision-threshold number

In the uppermost parts of drainage basins, water is not flowing over the ground surface or is flowing too weakly to incise into it. At least two kinds of limits must typically be overcome for erosion by flowing water to begin. First, sufficient drainage area must be accumulated for overland flow to start; second, this flow must exert sufficient shear stress on the surface to overcome the mechanical resistance of rocks or soils and thus mobilize sediment (e.g., Perron, 2017).

Channel-incision terms in landscape evolution models (LEMs) often capture
both of these limits by including an incision threshold below which no
incision occurs. For instance, if

In addition, incision thresholds can have notable consequences on the relationship between climate and long-term incision rates as described, for example, by Snyder et al. (2003), Tucker (2004), Lague et al. (2005), Perron (2017), and Deal et al. (2018). Specifically, incision thresholds stop smaller events from eroding the surface. In many wet climates, the total annual streamflow is high, but small, frequent events tend to contribute most of this total; in contrast, in many dry climates, a larger fraction of the total annual streamflow tends to be contributed by rare but intense events (e.g., Rossi et al., 2016). Therefore, a sufficiently high incision threshold could render ineffective a larger fraction of the total precipitation in wetter climates than in drier climates. This behavior can lead to a nonlinear dependence of long-term erosion rates on average precipitation; it can even lead to the counterintuitive observation that, in some cases, larger average precipitation corresponds to smaller long-term erosion rates (e.g., DiBiase and Whipple, 2011).

Furthermore, incision thresholds can play a role in setting the smallest
scales of valley dissection, which are among the fundamental scales that
characterize landscapes. For instance, Horton (1945) suggested that valley
dissection stops because further dissection would lead to hillslopes that
are too short to yield flow that can erode the surface. Montgomery and
Dietrich (1992) found that thresholds of the topographic quantity

In Theodoratos et al. (2018), we performed a scaling analysis of an incision–diffusion LEM that did not include an incision threshold. In the present study, we add an incision threshold to that LEM and examine how our analysis needs to be modified to account for this threshold. More specifically, in Theodoratos et al. (2018), we dimensionally analyzed an LEM that includes three parameters – an incision coefficient, a diffusion coefficient, and an uplift rate. For that analysis, we used three characteristic scales (of length, height, and time) that are defined in terms of the three parameters of the LEM. As we explained in detail in Theodoratos et al. (2018), because the characteristic scales depend on the model parameters and because there are three parameters and three characteristic scales, the LEM can be greatly simplified by being recast in a dimensionless form that has no parameters.

Adding an incision threshold to the LEM that we analyzed in Theodoratos et al. (2018) increases the number of its parameters to four (see Eq. 1 below). This leads to the question of whether the LEM with incision threshold can be dimensionally analyzed using the same three characteristic scales that we used to dimensionally analyze the LEM without incision threshold (Theodoratos et al., 2018). Here, we hypothesize that these three scales are reasonable choices even after adding an incision threshold to the LEM, and we test this hypothesis by applying these scales and examining the resulting rescaled equations.

We study an LEM described by the governing equation (e.g., Howard, 1994;
Dietrich et al., 2003):

The stream-power incision term

Dimensions of the variables and parameters of the LEM.

Equation (1) is defined piecewise on two subdomains. The first subdomain,
where

We test whether the characteristic scales defined in Theodoratos et al. (2018) are reasonable choices to analyze the LEM that includes an incision threshold

Given that all the terms of Eq. (1) have dimensions in H, L, and T, we can dimensionally analyze Eq. (1) using characteristic scales of length, height, and time. In Eqs. (3)–(8) below, we summarize the dimensional analysis of Theodoratos et al. (2018) as necessary background for the new analysis presented here. The dimensional analysis of Theodoratos et al. (2018) is based on a characteristic length that is defined as

To non-dimensionalize the rate of elevation change in the left-hand side of
Eq. (1) and the topographic properties in the right-hand side of Eq. (1)
(drainage area

If we divide all of the terms of the governing equation (Eq. 1) by the
uplift rate

Summary of characteristic scale definitions and of derivation of dimensionless quantities.

Bringing together the dimensionless terms derived above, we obtain a
dimensionless form of the governing equation (Eq. 1):

The LEM without incision threshold, which we studied in Theodoratos et al. (2018), has a dimensionless form that does not include any parameters (see Eq. 16 in Theodoratos et al., 2018). Having no parameters to be adjusted, the dimensionless form has a single solution for any given combination of boundary and initial conditions. This implies that landscapes with any parameters but with the same boundary and initial conditions (when
normalized by the characteristic scales

In contrast, the dimensionless form of the LEM with an incision threshold,
Eq. (10), includes one parameter, the incision-threshold number

The elimination of three out of four parameter-related degrees of freedom
from the LEM (from the four parameters

In this section, we numerically demonstrate that landscapes that follow
Eq. (1) but have different parameters will evolve geometrically similarly if
they have equal incision-threshold numbers

Values of parameters (

In this context, geometric similarity is defined in the following way. Let
the first landscape have characteristic scales

We perform numerical simulations using the Channel-Hillslope Integrated Landscape Development (CHILD) model (Tucker et al., 2001). Below, we briefly explain how we set up the simulations, and in Appendix A we present formulas that relate the parameters of CHILD to the parameters of the governing equation (Eq. 1). We refer readers to Theodoratos et al. (2018) for more details about setting up numerical simulations that follow geometric similarity (Sect. 3.1.1 and Appendix C) and about the theory behind such simulations (Appendix B).

For our similarity analysis, we simulate nine landscapes, each having a
different combination of the parameters

Note that the incision threshold values

To obtain domains and initial conditions that are equivalent when normalized
by the characteristic scales of length and height

Horizontal geometric similarity of landscapes with equal
incision-threshold numbers

Normalizing the initial conditions is necessary for landscapes to evolve
geometrically similarly and to reach geometrically similar steady states.
Specifically, landscapes can be geometrically similar at some time step if
they were geometrically similar at the previous time step. By extension,
landscapes must start from geometrically similar initial conditions. Note
that evolving landscapes must be compared at times that are normalized by
each landscape's characteristic timescale. For example, if two landscapes
have characteristic timescales of

Note that landscapes can reach geometrically similar steady states only if
the criteria that define the steady state are normalized by appropriate
characteristic scales, as explained in Sect. 3 of Theodoratos et al. (2018).
In the present study, for instance, we assume that a simulation reaches its
steady state when the absolute rate of elevation change

Horizontal and vertical geometric similarity of landscapes
with equal incision-threshold numbers

The nine simulated landscapes are all geometrically similar to each other, both during their evolution and in steady state. In Figs. 1–3, we graphically demonstrate that our simulated landscapes reach geometrically similar steady states. Specifically, we illustrate shaded relief maps in Fig. 1, elevation maps in Fig. 2, and maps of the extents of the zones of zero incision in Fig. 3. In the present study, we illustrate only steady-state results. For examples of graphical demonstrations of geometric similarity during landscape evolution, we refer readers to Figs. 3–5 of Theodoratos et al. (2018). For clarity, we present maps of only four out of the nine landscapes, specifically, of landscapes A–D in Table 3. However, all nine landscapes evolve geometrically similarly.

Horizontal geometric similarity of zones of zero incision. Red regions show the Voronoi polygons of points with

In Figs. 1–3, the four landscapes are arranged in a

In the shaded relief maps of Fig. 1, ridges and valleys form identical
plan-view patterns across the four landscapes, illustrating their horizontal
geometric similarity. Note that the characteristic scales of length and
height

In the elevation maps of Fig. 2, the spatial pattern of colors is identical
across the four landscapes. This shows that the four landscapes are
geometrically similar both horizontally and vertically, because the color
scales are rescaled by

In Fig. 3, we map the zones of zero incision of the four landscapes. To
illustrate these zones, we find the Voronoi polygons associated with points
for which

The landscapes in Figs. 1–3 do not just visually appear to be geometrically
similar. They are in fact geometrically similar. To test this quantitatively, we normalize the elevations

In this subsection, we demonstrate that landscapes with different
incision-threshold numbers

For these simulations, too, we use CHILD, as described in Appendix A. We
perform nine simulations with incision-threshold numbers

Incision-threshold numbers

As we mentioned in the Introduction (Sect. 1), the inclusion of incision
thresholds in LEMs leads to increasing topographic slopes, decreasing
drainage densities, and more convex hillslopes (e.g., Howard, 1994; Tucker
and Bras, 1998; Perron et al., 2008). In Figs. 4–10, we illustrate these
topographic effects using steady-state results of the nine simulations
defined above (Sect. 3.2.1; Table 4). More specifically, we present shaded
relief maps (Figs. 4 and 5), maps of elevation

Overview of influence of value of incision-threshold number

Influence of incision-threshold number

Increase in relief as the incision-threshold number

Expanding extent of zones of zero incision as the incision-threshold number

Deeper and sparser valleys and wider hillslopes, in
landscapes with higher incision-threshold numbers

Steepening of profiles as the incision-threshold number

Equivalence of elevations that are normalized by

We observe that landscapes become steeper as

Furthermore, we observe that landscapes become less dissected and appear
smoother in plan view as

We observe that as

Moreover, we observe that as

Consequently, hillslopes become more convex as

Finally, we observe that the widening of the zones of zero incision
eventually leads to a qualitative change in the operation of the laws of the
LEM across the landscapes. Specifically, the zones of zero incision almost
entirely occupy the hillslopes of the landscape with

With the above observations in mind, we can explain the observation that
landscapes become steeper as

In an alternative interpretation, one could potentially view the quantity

Equation (11) suggests that the quantity

We can test this hypothesis using the profiles of Fig. 9, since they belong
to landscapes that have different incision-threshold numbers

In Fig. 10, we observe that the elevations of the normalized profiles deviate systematically from one another. Specifically, we observe that, whereas the reliefs of the original (un-normalized) profiles grow as

Lengths, reliefs, and mean slopes of profiles along the longest
flow paths of landscapes with different incision-threshold numbers

Unlike the LEM studied here, other LEMs, such as those of Tucker (2004) or Deal et al. (2018), do not define zones of zero incision, i.e., areas where incision never operates, because those LEMs define incision terms based on conceptually different temporal averaging of rainfall events, in comparison to the LEM examined here.

Specifically, those other LEMs derive long-term incision rates by integrating stochastic rainfall over time, assuming that incision occurs when the shear stress (or, equivalently, the stream power) exceeds a threshold value. Given that the value of shear stress depends on discharge and slope, points with different drainage areas or slopes will experience different shear stress values during any given event. Therefore, any given combination of drainage area and slope corresponds to a critical rainfall intensity that is sufficient to generate a shear stress that equals the threshold shear stress. Long-term incision rates can be derived by integrating over the rainfall events that exceed this critical rainfall intensity. This approach implies that, in theory, any point with nonzero drainage area and slope can experience incision with a nonzero probability (provided that rainfall can theoretically become sufficiently intense). Therefore, in LEMs that follow this approach, zero-incision zones are not defined. Note, however, that in those LEMs one can define zones of probability of exceedance of the critical rainfall intensity, i.e., of probability of incision.

In contrast, the LEM studied here assumes constant, uniform rainfall, which
leads to constant stream power for any given combination of drainage area
and slope (i.e., for any given value of

In this study, we have examined whether the characteristic scales of length,
height, and time (

Dimensional analysis can ensure that a set of characteristic scales is
dimensionally consistent and can provide the number of degrees of freedom
that can be eliminated from a model (e.g., Buckingham, 1914), but it cannot
show a priori which characteristic scales should be used. For example, in
the case of Eq. (1), if we assume that length L and height H are distinct
dimensions, then together with time T they form a group of three dimensions, and dimensional analysis will show that any manipulation of Eq. (1) can eliminate at most 3 degrees of freedom. Because the characteristic scales

A future study could examine the utility of characteristic scales that are defined to depend on model parameters (Eqs. 3–5) in the case of parameters that vary in space or time. In such a case, the characteristic scales would also vary. We expect that if the parameter variation is gradual and follows a systematic pattern (e.g., the differential uplift across a fold described by Kirby and Whipple, 2001), then the resulting variable characteristic scales could be useful. For example, designing a lab-scale sandbox landscape that models differential uplift might benefit from these nonuniform characteristic scales. However, if the parameters were randomly heterogeneous, or they varied greatly over distances much smaller than typical landscape units, then the resulting “characteristic” scales might not be characteristic of any landscape properties, and thus they might lose their explanatory power.

In this study, we perform a dimensional analysis of an LEM that includes
terms describing stream-power incision, linear diffusion, and uplift
(Eq. 1). The LEM assumes that incision is limited by a threshold,
specifically, that there is no incision at points with drainage area

Our dimensional analysis is based on characteristic scales of length, height, and time (

In Sect. 2.3, using the characteristic scales

The fact that the incision-threshold number

In Sect. 3.2.2, we explore how these different

Finally, we derive a quantitative prediction of the increase in relief with
the incision-threshold numbers

To implement the governing equation of the LEM (Eq. 1) with CHILD, we use
CHILD's detachment-limited module, and we set the parameter DETACHMENT_LAW equal to 0. Furthermore, we use constant, uniform, and continuous precipitation, define infiltration to be 0, and set the hydraulic geometry scaling exponents

For this choice of exponents, CHILD uses the following equations to calculate the rate of elevation change due to incision (in CHILD notation):

Equating

In Eqs. (A2) and (A3), we assume constant values of

All data used in this study were synthesized using the CHILD model (Tucker et al., 2001). The input files needed to reproduce this data are available from the corresponding author upon request.

NT derived analytical results, and NT and JWK interpreted them. NT designed, performed, and analyzed numerical simulations, 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.

This study was made possible by financial support from ETH Zurich.

This paper was edited by Jean Braun and reviewed by Wolfgang Schwanghart, Eric Deal, and one anonymous referee.