Landscape
evolution models often utilize the stream power incision model to simulate
river incision:

The stream power incision model (SPIM) (e.g., Howard, 1994; Howard et al.,
1994) is a commonly used physically based model for bedrock incision. The
incision rate,

In this paper, we perform a scaling analysis of SPIM. First, we use a 1-D model to analytically derive steady-state river profiles, to illustrate the problem of scale invariance, and to delineate conditions for which elevation singularities occur at the ridge. Then, using a 2-D numerical model, we demonstrate the effects of horizontal scale on the steady-state relief of landscapes and infer the conditions for which elevation singularities occur at ridges.

SPIM is a simple model that has been used to gain considerable insight into landscape evolution. Previous studies using SPIM have shown how landscapes respond to tectonic and climate forcing (e.g., Howard, 1994; Howard et al., 1994). Yet like most simple models, SPIM is in some sense an oversimplification. Here, we demonstrate this by showing that it satisfies a curiously unrealistic scale invariance relation. By demonstrating this limitation, we hope to motivate the formulation of models that overcomes it.

The fundamental limitation on SPIM becomes apparent when the ratio

The existence of scale invariance exemplifies an unrealistic aspect of SPIM, which we believe to be associated with its omission of natural processes, such as abrasion due to sediment transport. Gilbert (1877) theorized two roles that sediment moving as bed load could play in bedrock incision: the first as an abrasive agent that incises the bed via collisions and the second as a protector that inhibits collisions of bed load on the bed. These observations have been implemented quantitatively by many modelers (e.g., Sklar and Dietrich, 2001, 2004, 2006; Lamb et al., 2008; Zhang et al., 2015), some of whom have implemented them in LEMs (e.g., Gasparini et al., 2006, 2007). Egholm et al. (2013) have directly compared landscape models using SPIM on the one hand and models using a saltation–abrasion model on the other hand. Here, we shed light on an unrealistic behavior of SPIM with the goal of motivating the landscape evolution community to develop more advanced treatments that better capture the underlying physics. A further goal is to emphasize the importance of scaling and nondimensionalization in characterizing LEMs.

An LEM can be implemented using the following equation of mass conservation
for rock/regolith subject to uplift and denudation:

Equation (2) characterizes landscape evolution in 2-D; i.e., elevation

Previous researchers have presented 1-D analytical solutions for elevation
profiles (Chase, 1992; Beaumont et al., 1992, Anderson, 1994; Kooi and
Beaumont, 1994, 1996; Tucker and Slingerland, 1994; Densmore et al., 1998;
Willett, 1999, 2010; Whipple and Tucker, 1999). In their solutions, the
effect of the horizontal scale, which in the 1-D model we define as the total
length of the stream profile,

The value of the 1-D Pillsbury number

One-dimensional analytical dimensionless solutions for elevation profiles at
steady-state equilibrium over a range of ratios

In 2-D, the conservation equation using SPIM and neglecting hillslope
diffusion can be written as

Our 2-D model was solved using the following boundary conditions:

For the results of Fig. 2, we use regular grids that contain 100

Figure 2a shows steady-state solutions for

In Fig. 2c, the case of scale invariance can be seen when

Nine 2-D numerical simulations at steady state for three different
values of

Like the 1-D model of Eq. (8), the 2-D model, Eq. (15), has slope,

Nine 2-D numerical simulations at steady state for three different
values of

In Figs. 3 and 4 we present results which serve to distinguish the
fundamental behavior of SPIM from the numerical behavior associated with a varying density of discretization. Figures 3 and 4 each show nine steady-state simulations, each using three values of

Figure 4 contains simulations where grid size is held constant at 125 m.
Here, the horizontal length scale,

Our quasi-theoretical analysis infers the conditions for singular behavior in the 2-D model. If elevation singularities exist, the model will not satisfy grid invariance, causing the relief between the ridge and outlet to increase indefinitely as grid size decreases. In contrast, in simulations where singularities do not exist, the relief between the ridge and outlet can be expected to converge as the grid size decreases. In both cases, understanding ridge behavior in the 2-D model requires studying solution behavior as grid size approaches 0.

We do this by extracting river profiles from 13 landscape simulations of
different scales for each of three values of

The 13 simulations result in 13 elevation profiles

This procedure results in a high synthetic profile encompassing all 13 profiles (circles) and in a low synthetic profile (crosses) (Fig. 5b). One-dimensional
analytical solutions, Eq. (10), are then fitted to the profiles of the 2-D
simulations using the 1-D Pillsbury number,

Figure 5b shows good fit between the 2-D results and the corresponding 1-D
steady-state profiles. This allows us to make inferences concerning
asymptotic behavior at a ridge. The analytical curves for elevation that
best fit the 2-D data for

We offer here an example of a landscape model that does not necessarily
satisfy horizontal-scale invariance, i.e., that of Gasparini et al. (2007).
They incorporate the formulation of Sklar and Dietrich (2004) for bedrock
abrasion due to wear in their model. The rate of erosion

In the river profiles of Figs. 1 and 5b, we see that a sizable proportion of
the relief is confined to the headwaters, i.e., near a ridge. In our 1-D
model, for

We next provide an example illustrating the dependence of relief on
hillslope length and profile length when

Our 1-D analytical solutions, Eq. (10) and Fig. 1, characterize the scale
behavior of 1-D SPIM, with horizontal-scale invariance satisfied when

In addition to the horizontal-scale invariant case

Our work neglects the effect of hillslope diffusion because our intent is to
study the behavior of SPIM itself. Without hillslope diffusion, SPIM causes
singular behavior at ridges in both the 1-D and 2-D formulation. Indeed, both
the 1-D and 2-D models exhibit singularities in slope at ridges for all

Numerical solutions of the 2-D model indicate that it cannot be
grid-invariant for

Our analysis illustrates that SPIM has two important limitations:
(a) unrealistic scale invariance when

The datasets used in this paper are available at

The authors declare that they have no conflict of interest.

This material is based upon work supported by the US Army Research Office under grant no. W911NF-12-R-0012 and by the National Science Foundation Graduate Research fellowship under grant no. DGE-1144245. Edited by: Jean Braun Reviewed by: two anonymous referees