The scaling and similarity of fluvial landscapes can reveal fundamental aspects of the physics driving their evolution. Here, we perform a dimensional analysis of the governing equation of a widely used landscape evolution model (LEM) that combines stream-power incision and linear diffusion laws. Our analysis assumes that length and height are conceptually distinct dimensions and uses characteristic scales that depend only on the model parameters (incision coefficient, diffusion coefficient, and uplift rate) rather than on the size of the domain or of landscape features. We use previously defined characteristic scales of length, height, and time, but, for the first time, we combine all three in a single analysis. Using these characteristic scales, we non-dimensionalize the LEM such that it includes only dimensionless variables and no parameters. This significantly simplifies the LEM by removing all parameter-related degrees of freedom. The only remaining degrees of freedom are in the boundary and initial conditions. Thus, for any given set of dimensionless boundary and initial conditions, all simulations, regardless of parameters, are just rescaled copies of each other, both in steady state and throughout their evolution. Therefore, the entire model parameter space can be explored by temporally and spatially rescaling a single simulation. This is orders of magnitude faster than performing multiple simulations to span multidimensional parameter spaces.

The characteristic scales of length, height and time are geomorphologically interpretable; they define relationships between topography and the relative strengths of landscape-forming processes. The characteristic height scale specifies how drainage areas and slopes must be related to curvatures for a landscape to be in steady state and leads to methods for defining valleys, estimating model parameters, and testing whether real topography follows the LEM. The characteristic length scale is roughly equal to the scale of the transition from diffusion-dominated to advection-dominated propagation of topographic perturbations (e.g., knickpoints). We introduce a modified definition of the landscape Péclet number, which quantifies the relative influence of advective versus diffusive propagation of perturbations. Our Péclet number definition can account for the scaling of basin length with basin area, which depends on topographic convergence versus divergence.

Hillslopes and river valleys are organized in striking patterns that appear to be repeated across landscapes and scales. Furthermore, within each landscape the transition from hillslopes to valleys seems to occur at a characteristic scale. These two properties have captivated scientists from the early days of geomorphology (e.g., Gilbert, 1877; Davis, 1892). Both properties are thought to be related to the scaling of processes that shape fluvial landscapes (e.g., Perron et al., 2008, 2012; Paola et al., 2009).

Scaling problems are often studied with the aid of dimensional analysis (e.g., Sonin, 2001; Bear and Cheng, 2010), which stems from Fourier's principle that all terms of physically meaningful equations should have consistent dimensions (Huntley, 1967). Dimensional analyses of landscape evolution models (LEMs) have been used to describe how the relative strengths of landscape-forming processes control properties of ridge and valley topography, such as drainage density (Willgoose et al., 1991), shapes of basins (Howard, 1994), the fluvial relief of mountains (Whipple and Tucker, 1999), topographic roughness (Simpson and Schlunegger, 2003), valley spacing (Perron et al., 2008, 2009), and drainage areas of first- and second-order valleys (Perron et al., 2012). The aforementioned studies used LEMs that, while differing in details, all assumed that fluvial landscapes are shaped by a combination of advective and diffusive erosion, with the former dominating valleys and the latter dominating hillslopes.

Here, we present a dimensional analysis of the governing equation of a simple, widely used LEM (see Eq. 1 below). Our work is based on two key premises. First, we define characteristic scales from the model parameters, rather than from extrinsic properties of the simulated landscape, such as the domain size or relief. Characteristic scales defined in this way are intrinsic to each landscape and its parameters (and thus to its underlying properties and to the strengths of the processes that shape it), and are independent of the initial and boundary conditions of any simulation. Second, in our approach we distinguish between the dimensions of horizontal length and vertical height. These two premises are not new (e.g., Willgoose et al., 1991; Whipple and Tucker, 1999; Perron et al., 2008; Robl et al., 2017), but here, for the first time, we apply them jointly to define and interpret all of the characteristic scales in this LEM. In so doing, we obtain three characteristic scales – a characteristic length, height, and time – that significantly simplify the model's governing equation. These characteristic scales are also geomorphologically interpretable, linking competition between processes to the scales of the features that emerge from these processes.

Our specific results directly apply only to the LEM that we dimensionally analyzed and to the landscapes that we can assume to be described by this LEM. However, our work illustrates an approach for the definition and interpretation of characteristic scales that could potentially be employed in dimensional analyses of other LEMs as well.

We perform a dimensional analysis of a simple, widely used model that
describes the evolution of landscapes under the influence of stream-power
incision, linear diffusion, and uplift, according to the governing equation

The incision term

The diffusion term

The term

If a landscape evolves to a condition in which, at every point, the three
terms of the right-hand side of Eq. (1) cancel each other out, then

Equation (1) is presumed to describe soil-mantled landscapes with sufficiently cohesive soils and gentle slopes, where we can assume that incision is detachment-limited and diffusion is linear (e.g., Perron et al., 2008). Other types of landscapes are shaped by processes that cannot be described by Eq. (1). For example, diffusive processes on steep soil-mantled hillslopes are better described by a nonlinear diffusion term (e.g., Roering et al., 1999, 2007).

Equation (1) assumes that all three processes act at all points of the landscape, without distinguishing between channels or hillslopes (e.g., Howard, 1994; Simpson and Schlunegger, 2003). All three terms are needed to model fluvial landscapes. Uplift fulfills the role of the source term, forcing the evolution of the landscape (e.g., Tucker and Hancock, 2010). Without it, the landscape would decay to a flat surface of zero elevation. The combination of the incision and diffusion terms is necessary for the emergence of ridges and valleys. Whereas the incision term amplifies topographic perturbations, setting in motion a positive feedback, the diffusion term dampens them, leading to a negative feedback (e.g., Smith and Bretherton, 1972; Perron et al., 2012). Both types of feedback are needed for the synthesis of surfaces with complex structures that resemble ridge and valley topography; therefore, the incision and diffusion terms of Eq. (1), or other terms with equivalent properties, represent the simplest combination of processes that can model landscapes characterized by ridge and valley topography (e.g., Smith and Bretherton, 1972; Howard, 1994). Because different points have different topographic properties (drainage area, slope, and curvature), the modeled incision and diffusion processes have different relative strengths across the landscape. Thus, even though distinct convergent (channel) and divergent (hillslope) landforms are not specified a priori, they can emerge from Eq. (1) (e.g., Howard, 1994), at scales that can be explored using dimensional analysis (e.g., Perron et al., 2008, 2009, 2012).

Our dimensional analysis of Eq. (1) begins by specifying the dimensions of
its various variables. We will rescale Eq. (1) in the horizontal direction
separately from the vertical direction so that we can study scaling of
lengths and reliefs separately (e.g., Dietrich and Montgomery, 1998).
Therefore, we must assume that the coordinates of points (

Because the dimensions of all variables of Eq. (1) can be expressed in terms
of L, H, and T, we can non-dimensionalize Eq. (1) using a characteristic
length

We define the characteristic length scale as

We introduce a characteristic height scale

The characteristic scales of length, height, and time

In Appendix A, we define analogous characteristic scales for an LEM with
generic exponents

Using the characteristic scales

Specifically, each differential (here using d

Substituting Eqs. (6), (8), (10), (12), and (14) into the governing equation
(Eq. 1) yields

Equation (16) includes only dimensionless variables and no parameters.
Because Eq. (16) has no parameters to be adjusted, for a given set of
boundary and initial conditions, it will have only one steady-state solution,
which will be arrived at via only one path of evolution. This implies that
all simulated landscapes with any parameters (but properly rescaled domains,
boundary conditions, and initial conditions) will evolve as rescaled replicas
of each other because they can all be reduced to Eq. (16) through rescaling
by the characteristic scales

Alternative dimensionless forms of Eq. (1) can reveal properties of the LEM
that are not revealed by Eq. (16). For example, Perron et al.'s (2008)
Eq. (19) was derived using the domain half width as a characteristic length
and the steady-state maximum relief as a characteristic height. In this way,
Perron et al.'s (2008) dimensionless equation includes information about the
domain size and the initial conditions (which influence the final relief;
e.g., Howard, 1994); therefore, that equation highlights the dependence of
its solutions on the domain size and the initial conditions. Perron et al.'s (2008)
Eq. (19) is equivalent to our Eq. (16) if one can express the domain
size and relief in terms of

Likewise, Eq. (16) does not reveal that flow-routing algorithms, and thus LEM
solutions, can be resolution-dependent if the channel width

The fact that Eq. (16) includes no parameters has an additional important
implication. One can use the factors that appear in front of terms of a
dimensionless equation to infer the relative importance of each term (e.g.,
Huntley, 1967). In the case of Eq. (16), all such factors are equal to 1,
which implies that none of the terms of this LEM (Eq. 1) is negligible
everywhere across a landscape. In other words, each term may be dominant at
some points of a landscape, depending on the local values of drainage area

We can rewrite Eq. (16) in a form that reveals what controls the relative
dominance of each process across a landscape. Specifically, the
dimensionless quantities of Eq. (16) are equal to the ratio of the
corresponding dimensional quantities over appropriate characteristic scales
(Eqs. 8, 10, 12, and 14). Therefore, we can rewrite Eq. (16) as

List of symbols (in the order they appear).

Parameters and resulting characteristic scales of the three landscapes presented in Figs. 1–5. Landscape B, the baseline landscape, has parameters with typical values (e.g., Perron et al., 2008; Tucker, 2009; Clubb et al., 2016). We use the baseline landscape to demonstrate properties of height and length scales in Figs. 7–10.

The dimensionless form of the governing equation (Eq. 16) implies that landscapes with any parameters but with properly rescaled boundary and initial conditions (see immediately below what we term as “proper” rescaling) will evolve in such a way that snapshots of these landscapes at properly rescaled moments in time will be (horizontally and vertically) rescaled copies of each other. In other words, the evolution of these landscapes will obey temporal and geometric similarity. This, in turn, implies that such landscapes will reach geometrically similar steady states.

We consider domains, elevations, and time to be properly rescaled if they are
equivalent when normalized by the characteristic scales of length, height,
and time

We should point out that simulations of these two landscapes will reach
geometrically similar steady states only if we rescale the threshold

In the following subsections we use a numerical model to demonstrate the
temporal and geometric similarity of rescaled landscapes that is implied by
the dimensionless governing equation (Eq. 16). In addition, in Appendix B we
outline a simple analytical proof of this similarity property. That proof
suggests that rescaling works only if we rescale initial conditions
(elevations) by

Horizontal similarity of rescaled landscapes. Steady-state shaded
relief maps demonstrate the horizontal geometric similarity of three
landscapes with widely varying parameters but properly rescaled domains (see
Eq. 18 for definition of proper rescaling). We label axes in units of each
landscape's characteristic length

Vertical similarity of rescaled landscapes. Steady-state elevation
maps and transects demonstrate the vertical geometric similarity of the three
landscapes of Fig. 1. We color the maps by elevation with color scales that
are rescaled by each landscape's characteristic height

We used the Channel-Hillslope Integrated Landscape Development (CHILD) model (Tucker et al., 2001) to numerically demonstrate the similarity property revealed by the dimensionless governing equation (Eq. 16). We chose CHILD due to its wide use by the geomorphologic community and due to the fact that it uses triangular irregular networks (TINs), which avoid the geometric bias of regular grids (e.g., Braun and Sambridge, 1997). We selected CHILD modules and parameters such that CHILD would simulate Eq. (1). Specifically, we selected CHILD's detachment-limited incision module, with constant, uniform precipitation, along with linear diffusion and uniform uplift (see Tucker et al., 2001, and Tucker, 2010, for definitions of CHILD's assumptions, modules, and parameters). In Appendix C we present in more detail how we set up our CHILD simulations and how we retrieved our results from CHILD's output files.

Temporal similarity of evolving, rescaled landscapes. We compare the
evolution of the three landscapes of Fig. 1 using shaded relief maps drawn at
four properly rescaled moments in time (see Eq. 18d for definition of proper
rescaling of time). The comparison shows that, at rescaled moments in time,
the horizontal patterns of the landscapes are geometrically similar (and
geometrically identical in units of

Temporal, horizontal, and vertical similarity of evolving, rescaled
landscapes. We compare elevation maps of the three evolving landscapes of
Fig. 2, using the same snapshots as in Fig. 3 and the same layout (i.e.,
landscapes sorted by row, rescaled times sorted by column). Color maps are
rescaled by each landscape's characteristic height,

We ran simulations using multiple combinations of the model parameters

We applied these parameter combinations on rescaled copies of two random,
dimensional TINs. In Appendix C1.2 we describe how we prepared the rescaled
TINs. Domain size was 200

Simulation time step lengths were not explicitly rescaled. Rather, we defined
simulation time step lengths using Courant–Friedrichs–Lewy criteria (Refice
et al., 2012) as described in Appendix C (Eq. C3). As seen in Eq. (C4), it
turns out that the resulting time step lengths were in effect rescaled by

We ran simulations until they reached numerical steady states, which we defined using rescaled steady-state thresholds according to Eq. (19). We compared the resulting landscapes during their evolution and at their steady states.

The numerical results confirmed the rescaling properties of the dimensionless
governing equation (Eq. 16). Simulations which were run on rescaled versions
of the same random TIN evolved similarly in space and time. Specifically, at
time steps rescaled by

Temporal and vertical similarity of evolving, rescaled landscapes. Transects of the three evolving landscapes of Fig. 2 corroborate the vertical component of their temporal and geometric similarity. We use the same snapshots as in Figs. 3 and 4 with the same layout (i.e., landscapes sorted by row, rescaled times sorted by column). The rescaled transects are identical along each column, demonstrating the exact temporal and geometric similarity of the rescaled landscapes. Transects pass through the highest peak of the steady-state landscapes and are marked on the maps of Fig. 4 with thick black lines. The fourth column shows steady-state transects, i.e., those of Fig. 2.

In Figs. 1 and 2 we present steady-state results for three landscapes (our
baseline and two alternatives), and in Figs. 3–5 we present transient
results during their evolution. These figures illustrate the geometric and
temporal similarity of these landscapes. The parameter combinations of these
three landscapes are a subset of all the combinations that we used; their
values can be seen in Table 2. We are presenting these specific combinations
for demonstration purposes as they lead to wide ranges of

In Fig. 1 we show steady-state shaded relief maps, and in Fig. 2 we show
elevation maps and transects. In both figures,

In the shaded relief maps of Fig. 1, the spatial pattern of ridges and
valleys is identical across the three landscapes, illustrating their
horizontal geometric similarity, although their shaded relief contrast
varies, reflecting their different characteristic gradients

The geometric similarity of the three steady-state landscapes is exact, not
just visually convincing. Our domain rescaling procedure does not affect the
point IDs of the TIN vertices, so we can directly compare corresponding
points using their IDs. In these simulated landscapes, the maximum absolute
difference in dimensionless elevations

Figures 3–5 show shaded relief maps, elevation maps, and transects for four
snapshots in time during the evolution of the three landscapes. Each column
shows snapshots of the three landscapes that correspond to the same moment in
dimensionless time (but different moments in dimensional time), with time
increasing from left to right. Each row shows the evolution of one landscape,
with one set of model parameters. As in Figs. 1 and 2, lengths and
elevations on axes and color bars in units of kilometers or meters are shown in bold, and
the corresponding values in units of

As in the case of steady-state landscapes, the temporal and geometric
similarity of the three evolving landscapes is exact, not just visually
convincing. Throughout the evolution of the three landscapes, the maximum
absolute difference in dimensionless elevations

The geometric similarity of rescaled landscapes implies that all horizontal
coordinates (

The characteristic scales of length, height, and time

As an additional example, the ratio of the characteristic length

The temporal and geometric similarity of our rescaled simulations (Sect. 3.1)
implies that we can explore the entire

Exploring a parameter space by rescaling can be orders of magnitude more
efficient than running multiple simulations for multiple parameter
combinations. For example, consider a numerical experiment exploring 10
values for each of the three parameters

Inferring the results of a simulation by rescaling the results of another
simulation assumes that the sizes of the simulation domains are equal in
units of characteristic length

However, if we vary the model parameters such that the features of interest are no longer small with respect to the domain, then these features will be influenced by boundary effects and may not be able to express their intrinsic shapes or behaviors, i.e., the shapes or behaviors that they would have if they were small relative to the domain. On the other hand, if we vary the parameters such that features of interest are not sufficiently large with respect to the resolution, then these features may be influenced by resolution effects because they may be insufficiently resolved. In both of these cases, we can no longer reasonably assume that we can study the features of interest with a rescaling approach.

To be able to assess which combinations of domain sizes and resolutions and
model parameters

In general, one should consider all the specifications of simulations not
only in units of meters and years but also in units of

The values of the characteristic scales depend on the relative magnitudes of
the model parameters

The incision, diffusion, and uplift terms of the governing equation (Eq. 1)
give the rates of change of elevation due to the respective processes. We can
scale these rates using the characteristic time

The definitions of the characteristic height (

In geometrically similar landscapes, the incision and diffusion heights of
corresponding points will be rescaled by the characteristic height

Note that Eq. (23a) shows that, if we define a dimensionless incision height

Schematic illustration of height scales. Profiling the incision,
diffusion, and characteristic height scales

We can express the governing equation Eq. (1) in terms of the incision,
diffusion, and characteristic height scales

First, we focus on points that have zero curvature

Second, we focus on drainage divides, where the drainage area

Equations (27) and (28) refer to special points where incision or diffusion
is zero. These are special cases of a general steady-state property of

Figure 6 schematically illustrates how the incision, diffusion, and
characteristic height scales

Substituting the definitions of

In this sense, we can interpret the characteristic height

Steady-state relationship between drainage areas, slopes, and
curvatures, parameterized by characteristic length and height scales

For drainage area and slope exponents

Given that Eq. (30) can be rewritten as

Equations (30) and (31), the relationships that constrain steady-state
topography, are testable predictions. They imply that we can test whether the
governing equation Eq. (1) describes a given, presumably steady-state,
real-world landscape by plotting estimates of the product

Steady-state valley networks visualized by incision height

Equation (31) can be used to estimate model parameters. Specifically, we can
estimate

Another equation that could hypothetically be used to estimate model
parameters is Eq. (27), which is reminiscent of the steepness-index formula

Prediction of ridge migration by differences of incision height

Valleys have been defined as areas where the quantity

Equations (26) and (31) can be combined as

Given that one is a linear function of the other, are there practical reasons
to prefer

Equation (33) holds only in steady state; its counterpart in transient states
would be

Perron et al. (2008, 2009, 2012) expressed the competition between diffusion,
which smooths landscapes, and incision, which dissects them, in terms of a
Péclet number

To examine the properties of

The incision term of Eq. (1) has the form of a kinematic wave and advects
perturbations at a rate equal to the kinematic wave celerity

The diffusion term of Eq. (1) smooths elevation differences by redistributing
them over an expanding region of neighboring points. For example, an
elevation difference that is initially concentrated at a single point will
evolve as a Gaussian function, centered around this point and with a standard
deviation that grows proportionally to

Following Perron et al. (2008, 2009, 2012), we quantify the relative
horizontal influence of incision versus diffusion across a landscape by the
ratio of the diffusion time

The transition between the regimes of horizontally dominant diffusion and
incision can be assumed to occur where the incision and diffusion timescales are roughly equal, i.e., where

Steady-state valley networks visualized by Péclet number

To calculate values of the Péclet number Pe across a landscape, we must
specify what the length scale

Figure 10 shows how the Péclet number Pe varies across a landscape. We
calculated Pe according to Eq. (36), assuming that

Just as the incision and diffusion heights

In this subsection, we discuss the differences between the Péclet number
defined in this study and the Péclet number defined by Perron et
al. (2008, 2009, 2012). Our definition (Eq. 36) includes both the drainage
area

To determine how our Péclet number scales with the flow path length

Dependence of drainage area on flow path length and convergence or
divergence of topography. Three points that have equal flow path lengths can
have very different drainage areas. Elevation contours (gray lines) reveal
that point P

Figure 11 schematically illustrates how the scaling relationship between flow
path length

Given that the Péclet number Pe is proportional to

Perron et al.'s (2008, 2009, 2012) Péclet number scales as

In this study, we perform a dimensional analysis of an LEM that includes terms
for stream-power incision, linear diffusion, and uplift (Eq. 1; e.g., Howard,
1994; Dietrich et al., 2003). The governing equation that we analyze in the
main text (Eq. 1) includes the relatively simple incision term

Our dimensional analysis is based on two key premises. First, we assume that
the dimensions of length and height are conceptually distinct. Second, we use
only intrinsic characteristic scales, i.e., scales that depend only on the
parameters of the model (the incision coefficient

First, rescaling the governing equation (Eq. 1) by

Second,

The temporal and geometric similarity of rescaled landscapes implies that we
can explore all combinations of the model parameters

Such a modeling approach assumes that simulation domains are rescaled, but this is not always physically realistic (e.g., if a domain represents an island, changing model parameters may change the sizes of ridges and valleys on the island, but not the size of the island itself). Nonetheless, as we explain in Sect. 3.2.2, landscape features that are sufficiently small with respect to the domain size may locally remain statistically similar, even if the landscapes globally are not similar. Therefore, the rescaling approach may offer insights into how such features depend on model parameters, even if the domain is not rescaled.

However, if landscapes are not rescaled and model parameters are varied too
widely, then boundary or resolution effects may arise. These boundary and
resolution effects will be minimized if the domain size is much larger than

Third,

In steady-state landscapes, the characteristic height

Equation (30) is a testable prediction that can discriminate between
landscapes that are in steady state and follow the governing equation
(Eq. 1), and those that do not. Furthermore, it can be rearranged to estimate

The characteristic scales of length and time

Our definition of the Péclet number differs from that in Perron et al. (2008,
2009, 2012) in that ours includes both the length scale

To summarize,

We perform a dimensional analysis of an LEM with an incision term with generic
drainage area and slope exponents

The variables and parameters of Eq. (A1) have the following dimensions.
Coordinates of points (

All terms of Eq. (A1) have dimensions that can be expressed in terms of L, H,
and T. Therefore, we can non-dimensionalize Eq. (A1) using a characteristic
length

We assume that

We can find the exponents

The solution of this system is

Therefore, the characteristic length can be defined as

Note the exponents of the parameters

For

For

Therefore, eliminating

Equation (1) satisfies both

Based on the definitions of

We define a characteristic horizontal velocity as

We have shown that the characteristic scales of the governing equation in the main text, Eq. (1), are consistent with the characteristic scales of Eq. (A1). Below we confirm that all results and interpretations that refer to Eq. (1) and were presented in the main text are also consistent with Eq. (A1).

We define dimensionless variables according to Eqs. (6, 8, 10, 12, and 14)
(using the characteristic scales defined by Eqs. A8–A13) and substitute
into Eq. (A1):

We define an incision height as the erosion due to incision during one unit
of

The incision height

We define a diffusion height as the elevation change due to diffusion per
unit of

Multiplying Eq. (A1) by

At points of zero curvature (

At drainage divides (

Substituting the definition of

The celerity of the incision term is

The diffusion timescale of Eq. (A1) is the same as in the case of Eq. (1);
i.e., according to Eq. (35), it is

Therefore, the Péclet number is defined as

The condition

We outline an analytical proof of the rescaling property implied by the dimensionless governing equation (Eq. 16), namely that landscapes with any parameters will evolve temporally and geometrically similarly and will reach geometrically similar steady states provided that we properly rescale their boundary and initial conditions. Equation (18a–d) defines what we term as proper rescaling.

A simple way to demonstrate this property is to explore the necessary
conditions for the dimensionless governing equation (Eq. 16) to describe two
different landscapes and then show that the same conditions lead to
a temporally and geometrically similar evolution of these two landscapes. Let
a landscape have parameters

First, we derive rescaling relationships between characteristic scales of the
two landscapes as functions of

Second, we derive relationships between coordinates of the landscapes and
between moments in time during their evolution. We substitute Eq. (B1a)
into Eq. (18a) and obtain

Third, one can show that drainage areas, time derivatives, and differential
operators of the two landscapes will be related according to

Fourth, we can retrieve the following dimensional governing equation for the
second landscape if we start from the dimensionless Eq. (16) and use the
rescaling formulas of Eq. (18):

Fifth, we can now show that rescaling according to Eqs. (B2a–h) additionally
leads to temporal and geometric similarity. Specifically, we can show that if
the two landscapes are geometrically similar (i.e., they obey Eq. B2d) at any
moments in time

Therefore, we can conclude that if the two landscapes have geometrically similar initial conditions, then they will evolve temporally and geometrically similarly and will reach geometrically similar steady states.

To simulate the governing equation (Eq. 1) with CHILD we used the
detachment-limited module; constant, uniform, and continuous precipitation;
zero infiltration; hydraulic geometry scaling exponents

For this choice of exponents, the rate of elevation change due to incision
is calculated by CHILD from the following equations (in CHILD notation):

Equating the incision term of Eq. (1) to

We varied the values of

We defined combinations of the parameters

We synthesized two random TINs by randomly perturbing a deterministic TIN generated by the geometry definition module of MATLAB's PDE toolbox. We used MATLAB to better control the rescaling procedure that we describe below. We exported the rescaled TINs of simulations as text files. Using the coordinates of TIN points included in these files, CHILD calculated the corresponding Delaunay triangulation using its own, built-in modules. We set all four domain boundaries to be open (see Tucker, 2010).

To prepare rescaled TINs, we followed the following procedure. First, we
synthesized two random TINs on rectangular domains with

We defined the time step length

CHILD allows the definition of a single time step length, which is used
throughout the entire simulation. However, since

We assume that

CHILD produces output files with various variables. Relevant to our model are
those with data of elevation

Using the triangle and edge output data, we define the Voronoi polygon
associated with a point

Our study used numerical data produced by the CHILD model (Tucker et al., 2001). All CHILD input files needed to reproduce these numerical 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, HS, and JWK interpreted them. NT drafted the paper, and NT, HS, 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. We thank Taylor Perron and Sean Willett for helpful discussions and Greg Tucker and an anonymous referee for their feedback, which helped improve our paper. Edited by: Jean Braun Reviewed by: Gregory Tucker and one anonymous referee