ESurfEarth Surface DynamicsESurfEarth Surf. Dynam.2196-632XCopernicus PublicationsGöttingen, Germany10.5194/esurf-5-807-2017Landscape evolution models using the stream power incision model show
unrealistic behavior when m/n equals 0.5KwangJeffrey S.jeffskwang@gmail.comhttps://orcid.org/0000-0002-3165-9700ParkerGaryhttps://orcid.org/0000-0001-5973-5296Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, USADepartment of Geology, University of Illinois at Urbana-Champaign, Urbana, IL, USAJeffrey S. Kwang (jeffskwang@gmail.com)6December2017548078208March201720March201717October20171November2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://esurf.copernicus.org/articles/5/807/2017/esurf-5-807-2017.htmlThe full text article is available as a PDF file from https://esurf.copernicus.org/articles/5/807/2017/esurf-5-807-2017.pdf
Landscape
evolution models often utilize the stream power incision model to simulate
river incision: E=KAmSn, where E is the vertical incision rate,
K is the erodibility constant, A is the upstream drainage area, S is
the channel gradient, and m and n are exponents. This simple but useful
law has been employed with an imposed rock uplift rate to gain insight into
steady-state landscapes. The most common choice of exponents satisfies
m/n= 0.5. Yet all models have limitations. Here, we show that
when hillslope diffusion (which operates only on small scales) is neglected,
the choice m/n= 0.5 yields a curiously unrealistic result: the
predicted landscape is invariant to horizontal stretching. That is, the
steady-state landscape for a 10 km2 horizontal domain can be stretched
so that it is identical to the corresponding landscape for a 1000 km2
domain.
Introduction
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, E, can be written as
E=KAmSn,
where K is the erodibility coefficient, A is the upslope drainage area,
S is the downstream slope, and m and n are exponents. This simple model
is thoroughly reviewed in Whipple and Tucker (1999) and Lague (2014), where
they hypothesize that m/n is between 0.35 and 0.60. This range is
consistent with results inferred from field work and map studies (Flint,
1974; Howard and Kerby, 1983; Tarboton et al., 1989, 1991; Willgoose et al.,
1990; Willgoose, 1994; Moglen and Bras, 1995; Snyder et al., 2000).
Furthermore, many researchers specifically suggest, or offer as a default,
the ratio m/n∼ 0.5 (Snyder et al., 2000; Banavar et al., 2001;
Hobley et al., 2017). The choice of this ratio is paramount in numerical
landscape evolution models (LEMs) that utilize SPIM, such as the
channel–hillslope integrated landscape development model, CHILD
(Tucker et al., 2001). The ratio m/n is also used to describe the
relationship between slope and drainage area in describing stream long
profiles (Flint, 1974). All models using SPIM, including studies on drainage
reorganization and stability (Willett et al., 2014), tectonic histories of
landscapes (Goren et al., 2014b; Fox et al., 2014), and persistent drainage
migration (Pelletier, 2004), involve the specification of this ratio. In
addition, the specific values of m and n are important (Tucker and
Whipple, 2002). Here, however, we focus on the ratio itself, and we show a
somewhat unexpected result: when m/n=0.5, SPIM-based LEMs exhibit
elevation solutions that are invariant to shape-preserving stretching of
horizontal domain. That is, except for the finest scales on which hillslope
diffusion becomes important, the model predicts the same solution for a
landscape with a total basin area of 10 km2 and one with a total basin
area of 1000 km2 under the constraint of identical horizontal basin
shape (e.g., square). The extremity of this result underscores a heretofore
unrecognized unrealistic aspect of SPIM.
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.
Motivation
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 m/n=0.5. Under this condition, SPIM alone will predict the same steady-state
relief for a 10 km2 domain as a 1000 km2 domain of the same
horizontal shape, as illustrated below. LEMs utilizing SPIM often sidestep
this problem with the use of a “hillslope diffusion” coefficient (e.g.,
Passalacqua et al., 2006), a useful but rather poorly constrained parameter
that lumps together a wide range of processes (Fernandes and Dietrich,
1997). Alternatively, the problem can be sidestepped with an externally
specified “hillslope critical length” (Goren et al., 2014a) that
essentially specifies the location of channel heads. For example, the model
simulations of Willett et al. (2014) employ the specific value of 500 m for
hillslope critical length in their characterization of tendencies for
drainage divide migration. The prediction of the hillslope diffusion
coefficient and the location of channels are outstanding problems in the
field of geomorphology (Montgomery and Dietrich, 1988). The intrinsic nature
of the SPIM model, however, is such that scale invariance persists for the
case m/n=0.5 on scales larger than a characteristic hillslope length scale,
whether it be externally specified or computed from a diffusion coefficient.
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.
One-dimensional model: scale invariance and singularities
An LEM can be implemented using the following equation of mass conservation
for rock/regolith subject to uplift and denudation:
∂η/∂t=υ-E+D∇2η,
where η is the local landscape elevation, t is time, υ is the rock
uplift rate, and D is the hillslope diffusion coefficient. The term D∇2η accounts for hillslope diffusion (Somfai and Sander, 1997;
Banavar et al., 2001). The effect of diffusion is commonly neglected at
coarse-grained resolution (Somfai and Sander, 1997; Banavar et al., 2001;
Passalacqua et al., 2006), at which any resolved channels can be taken to be
fluvially dominated bedrock channels (Montgomery and
Foufoula-Georgiou,1993). In our analysis, we use Eq. (1) to specify the
incision term in Eq. (2). It should be noted that SPIM refers to the
incision in the direction normal to the bed, implying that there are both
horizontal and vertical components of incision. In much of the literature
using SPIM, however, the horizontal component is neglected in accordance
with the original formulation of Howard and Kerby (1983), and incision is
assumed to be purely vertical and downward. Here, we preserve this simplification
in order to better understand the overall behavior of SPIM. Last, in
correspondence with most 2-D implementations of SPIM within LEM, we neither
resolve channels nor compute their hydraulic geometry in our 2-D
implementation. The focus of this paper is the most simplified form (e.g.,
Eq. 1) of SPIM. This way we can analyze the most fundamental behavior of SPIM
itself.
Equation (2) characterizes landscape evolution in 2-D; i.e., elevation η=η (x,y), where x and y are horizontal coordinates. It is useful for
some purposes, however, to simplify Eq. (2) into a 1-D form. Neglecting
hillslope diffusion, the 1-D conservation equation is
∂η/∂t=υ-KAm-∂η/∂ln,
where l is the horizontal stream distance from the ridge, at which l= 0. It
should be noted that the negative sign appears in front of the term ∂η/∂l because ∂η/∂l is negative in the downstream
direction, so that streambed slope S=-∂η/∂l. In SPIM, slope S is assumed to be
positive. In order to solve Eq. (3), a relationship between A and l must be
established. Here, we assume a generalized form of Hack's Law (Hack, 1957):
A=Clh,
where C and h are positive values. Hack's Law assumes that upslope area
increases with lh. From empirical data, Hack found the exponent h to be
∼ 1.67 (Hack, 1957).
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, L1-D, was neither shown nor
discussed. Previous studies that use Eq. (4) (Whipple and Tucker, 1999;
Willett, 2010) involve nondimensionalization of both the horizontal and
vertical coordinates by the total horizontal length of the profile,
L1-D. As we show below, this step obscures the effect of the
horizontal scale on the relief of the profile. In our study, we
nondimensionalize the vertical coordinate, η, by a combination of
υ and the acceleration of gravity, g. Our nondimensionalization
of the coordinates is shown below.
η=υ2g-1η^t=υg-1t^l=L1-Dl^
Substituting Eqs. (4) and (5) into Eq. (3) results in the following
dimensionless conservation equation:
∂η^/∂t^=1-P1-D-nl^hm-∂η^/∂l^n,
where the dimensionless number P1-D, termed the 1-D Pillsbury number
herein for convenience, is given by the relation
P1-D=K-1/nC-m/nL1-D1-hm/nυ1/n-2g.
At steady state, Eq. (6) becomes
P1-D=l^hm/n-∂η^/∂l^.
From this equation, we see that as we approach the ridge, i.e., l^→0, the slope term -∂η^/∂l^ always approaches
infinity for positive values of h, m, and n.
The value of the 1-D Pillsbury number P1-D increases with stream
profile length L1-D when hm/n < 1, is invariant to
changes in L1-D when hm/n= 1, and decreases with
L1-D when hm/n > 1. This can be further illustrated
by integrating Eq. (8). To solve this first-order differential equation, we
need to specify a single boundary condition, shown below.
η^l^=1=0
This boundary condition sets the location and elevation of the outlet, where
flow is allowed to exit the system. Integrating Eq. (8) yields
η^=-P1-Dlnl^ifhm=n1-hm/n-1P1-D1-l^1-hm/nifhm≠n.
The steady-state profiles defined by Eq. (10) are shown in Fig. 1.
Inspecting Eq. (10), we see that elevation is infinite at the ridge (l= 0)
when hm/n≥1, and elevation is finite when hm/n < 1. In addition, when
hm/n= 1, P1-D is no longer dependent on the horizontal scale, L1-D, and
η^ is independent of the scale of the basin. Using the empirical
value from Hack's original work (1957), i.e., h= 1.67, the ratio m/n must
take the value 0.6 for scale invariance. This ratio is within the range
reported in the literature (Whipple and Tucker, 1999).
Two-dimensional model: scale invariance
One-dimensional analytical dimensionless solutions for elevation profiles at
steady-state equilibrium over a range of ratios hm/n (Hack's Law) = 0.7,
0.8, 0.9, 1.0, 1.1, and 1.2 and P1-D= 1.0.
In 2-D, the conservation equation using SPIM and neglecting hillslope
diffusion can be written as
∂η/∂t=υ-KAm∂η/∂x2+∂η/∂y2n/2.
To understand the behavior of Eq. (11) in response to scale, we need to use
a dimensionless formulation in a fashion similar to the previous 1-D
analysis. Here, L2-D denotes the horizontal length of the entire domain,
which is taken to be square for convenience. For the 2-D analysis, our
nondimensionalization is
η=υ2g-1η^t=υg-1t^A=L2-D2A^x=L2-Dx^y=L2-Dy^.
The form of Eq. (11) in which x, y, and A have been made dimensionless using
the definitions shown in Eq. (12) is
∂η^/∂t^=1-P2-D-nA^m∂η^/∂x^2+∂η^/∂y^2n/2,
where the dimensionless number P2-D, termed the 2-D Pillsbury number, is
given as
P2-D=K-1/nL2-D1-2m/nυ1/n-2g.
At steady state, Eq. (13) becomes
P2-D=A^m/n∂η^/∂x^2+∂η^/∂y^21/2.
The form of the parameter P2-D specified by Eq. (14) is similar to
the 1-D form, Eq. (7), but is different due to the different dimensionality.
The parameter, P2-D, scales with the relief of the landscape; as it
increases, the slope term on the RHS (right-hand side) of Eq. (15) also increases.
The value of P2-D increases with L2-D for
m/n < 0.5, remains constant with L2-D for
m/n= 0.5, and decreases with L2-D for
m/n > 0.5. For the ratio m/n= 0.5, the exponent to
which L2-D is raised in Eq. (14) becomes 0 and the relief of the
landscape becomes invariant to horizontal scale. When m/n= 0.5, the
same steady-state solution to Eq. (15) prevails regardless of the value of
L2-D. We note here that this scale invariance, which is the key
result of this paper, is intrinsic to the model itself and is not a function
of the discretization scheme used in implementing numerical solutions.
(a) Two-dimensional numerical landscapes at steady state
using a ratio of m/n= 0.5, n= 1.0,
υ= 4 mm yr-1,
K= 2.83 × 10-11 s-1,
M2= 1002 cells, and L2-D2= 125 and
2000 km2. For each case, the 2-D Pillsbury number was the same:
2.73 × 1021. (b) Results of panel (a)
expressed in terms of dimensionless horizontal scale. Each basin is made
dimensionless by its basin size, L2-D. (c) Nine 2-D
numerical simulations at dynamic equilibrium for three different values of
L2-D and three different values of m/n. The value of K has been
chosen to be different for each value of m/n for clarity in the figures.
From left to right, L2-D2= 5 × 102,
5 × 104, and 5 × 106 km2. To make the
relief of the landscapes comparable, the 2-D Pillsbury number,
P2-D, is set to 2.73 × 1021 for solutions of all
m/n ratios with L2-D2= 5 × 102 km2.
To achieve this for υ= 4 mm yr-1,
K= 2.10 × 10-10 m0.2 s-1,
2.83 × 10-11 s-1, and
3.82 × 10-12 m-0.2 s-1 for m/n= 0.4, 0.5,
and 0.6, respectively.
Our 2-D model was solved using the following boundary conditions:
ηy=0=0,∂η/∂yy=L2-D=0,ηx=0=ηx=L2-D.
The bottom (outlet) side of the domain presented in Fig. 2 is fixed at the
base level η= 0 m, corresponding to an open boundary where flow can
exit the system while satisfying Eq. (16). The top side of the domain is
designated as an impermeable boundary to flow, i.e., the drainage divide
satisfies Eq. (17). Periodic boundary conditions satisfying Eq. (18) are
applied at the left and right boundaries. Flow, slope, and drainage area are
determined using the D8 flow algorithm, where flow follows the route of
steepest descent (O'Callaghan and Mark, 1984). The initial condition is a
gently sloped plane oriented towards the outlet with small random elevation
perturbations.
For the results of Fig. 2, we use regular grids that contain 1002
cells. The number of cells is constant, regardless of the value of
L2-D. This is in contrast to holding cell size constant and instead
increasing the number of cells with L2-D. We argue that the former shows
the fundamental behavior of SPIM, while the latter obscures this behavior
due to the existence of slope and elevation singularities near the ridges in
the landscape. The next sections show this singular behavior in the 2-D
numerical model.
Figure 2a shows steady-state solutions for m/n= 0.5 and
two values of L2-D using the same initial condition. At each corresponding
grid cell between the two solutions, the slope, S, decreases as L2-D
increases. However, the relief structures of each landscape are identical.
By relief structure, we are describing the elevation value at each
corresponding grid cell in the two steady-state solutions. This is confirmed
by nondimensionalizing the horizontal scale of landscape without adjusting
the vertical scale (Fig. 2b). Using the same numerical methods and the
parameters from Fig. 2a, the results of a similar analysis using different
ratios m/n= 0.4, 0.5, and 0.6 are shown in Fig. 2c.
In Fig. 2c, the case of scale invariance can be seen when m/n= 0.5. For m/n= 0.4, the relief of the entire landscape increases with
increasing L2-D, and for m/n= 0.6, the relief decreases with increasing
L2-D. When m/n≠0.5, the landscapes do not exhibit scale invariance.
However, the overall planform drainage network structure shows resemblance
across scales. That is, the location of the major streams and rivers in the
numerical grid are similarly organized. It should be noted that the
landscapes are not identical. When the landscapes are shown in dimensional
space, as shown in Fig. 2a, the landscapes appear to be quite different. In
the case of Fig. 2b, however, the smaller landscape can be stretched
horizontally to be precisely identical to the large one. The drainage
network structure described above persists in each simulation due to the
imprinting of the initial condition, which always consists of the same
randomized perturbations.
Nine 2-D numerical simulations at steady state for three different
values of M2 and three different values of m/n. In this figure, the
horizontal scale is kept constant: L2-D= 10 km for all
solutions. The value of K has been chosen to be different for each value of
m/n for clarity in the figures. From left to right, the number of cells
M2= 402, 802, and 1602. To make the relief of the
landscapes comparable, the 2-D Pillsbury number, P2-D, is set to
3.10 × 1023 for solutions of all m/n ratios with
L2-D= 10 km. To achieve this for
υ= 1 mm yr-1,
K= 6.31 × 10-12 m0.2 s-1,
1.00 × 10-12 s-1, and
1.58 × 10-13 m-0.2 s-1 for m/n= 0.4, 0.5,
and 0.6, respectively. Relief increases with the number of cells because the
ridge singularity is resolved at finer resolution.
Two-dimensional model: quasi-theoretical analysis of singular behavior
Like the 1-D model of Eq. (8), the 2-D model, Eq. (15), has slope, S,
approaching infinity as area, A, approaches 0 at steady state. In contrast
to the 1-D model, however, general steady-state solutions for elevation in
the 2-D model, Eq. (15), cannot be determined analytically. However, the
ratio m/n for which elevation singularities occur can be determined by
analyzing the behavior of the 2-D numerical model in close proximity to a
ridge. Here, we first develop a quasi-theoretical treatment to study
near-ridge behavior, and we then use it to infer singular behavior in the
numerical model. Converting the coordinate system from Cartesian to a system
that follows the stream-wise direction, we rewrite Eq. (11) as
∂η/∂t=υ-KAm-∂η/∂sn,
where s is the distance along the path of steepest descent away from the
ridge. From dimensional considerations, A [L2] must scale with s2
[L2] near the ridge (s= 0), and therefore,
A=βs2ass→0,
where β is the scaling factor. For this analysis, our
nondimensionalization is
η=υ2g-1η^t=υg-1t^s=LRs^,
where LR is the horizontal ridge scale. Near the ridge, Eq. (19) can be
nondimensionalized into
∂η^/∂t^=1-PR-ns^2m-∂η^/∂s^n,
where PR is another dimensionless Pillsbury number, here denoted as
PR=K-1/nβ-m/nL1-D1-2m/nυ1/n-2g.
At steady state (∂η/∂t= 0), Eq. (22) becomes
PR=s^2m/n-∂η^/∂s^.
From Eq. (24), we see that at the ridge (s^=0), there is
a singularity in slope, i.e., the slope (-∂η^/∂s^) goes to
infinity. Integration of Eq. (24) using the downstream boundary condition,
η^s^=1=0, allows for the delineation of
the conditions for elevation singularities in the 2-D model. The profile is
given as
η^=-PRlns^if2m=n1-2m/n-1PR1-s^1-2m/nif2m≠n.
Instead of the elevation singularity occurring when hm/n≥1 as seen in the
1-D model, Eq. (10), this analysis for the 2-D model shows an elevation
singularity at the ridge when m/n≥0.5.
Nine 2-D numerical simulations at steady state for three different
values of M2 and three different values of m/n. In this figure, the
grid size, L2-D/M= 125 m, is used in all the solutions. The
value of K has been chosen to be different for each value of m/n for
clarity in the figures. From left to right, the number of cells
M2= 402, 802, and 1602. To make the relief of the
landscapes comparable, the 2-D Pillsbury number, P2-D, is set to
3.10 × 1023 for solutions of all m/n ratios with
L2-D= 10 km. To achieve this for
υ= 1 mm yr-1,
K= 6.31 × 10-12 m0.2 s-1,
1.00 × 10-12 s-1, and
1.58 × 10-13 m-0.2 s-1 for m/n= 0.4, 0.5,
and 0.6, respectively. Like Fig. 3, relief increases with the number of cells
because the ridge singularity is resolved at a finer resolution.
Two-dimensional model: numerical analysis of singular behavior
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 M2 and three values of
m/n, i.e., 0.4, 0.5, and 0.6. In both figures, the number of cells is
quadrupled from column to column. The leftmost column contains 402 cells, the middle column contains 802 cells, and the rightmost column
contains 1602 cells. Figure 3 shows simulations where the horizontal
length scale, L2-D, is held constant in all simulations. By increasing
the number of cells, the grid size decreases. In all cases of m/n, the maximum
relief increases with the number of cells. However, our quasi-theoretical
analysis predicted the absence of an elevation singularity at the ridge for
m/n < 0.5. To illustrate this point, we take a different approach,
shown later in this section.
Figure 4 contains simulations where grid size is held constant at 125 m.
Here, the horizontal length scale, L2-D, increases with the number of
cells. In Fig. 4, the leftmost column contains 402 cells with
L2-D= 5 km, the middle column contains 802 cells with L2-D= 10 km, and the rightmost column contains
1602 cells with L2-D= 20 km. Regardless of the m/n ratio and whether L2-D or grid size is kept
constant, the maximum relief of the landscape increases as the number of
cells increases. Relief increases in both sets of simulations because with
more grid cells, we are numerically sampling closer to ridges, and by
sampling closer to ridges, we are resolving the ridge singularity on a finer
scale. We emphasize, however, that the issue of dependence of the solution
on grid size is separate from the issue of scale invariance for m/n= 0.5,
the latter result being deduced from the governing equation itself (Eq. 15)
before any discretization is implemented and illustrated in Fig. 2c.
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 m/n, i.e., 0.4, 0.5, and 0.6.
The largest simulation is for L2-D2=106 km2;
simulations were also performed at progressively 1 order
of magnitude less in area down to L2-D2=10-6 km2. The number of grid cells, M2, is held constant at
252. In each simulation, then, the closest distance to the ridge that
can be resolved is one grid cell, given by
Δli=107-i/2/25[km]i=1,2…13.
From each of the simulations, we construct two synthetic river profiles: one
that intersects the highest point of the basin divide (high profile) and one
that intersects the lowest point of the basin divide (low profile). The
choice of these two elevations was made so as to bracket the possible range
of behavior; analogous results would be obtained from starting points along
the basin divide at intermediate elevations. We use these synthetic profiles
to characterize whether or not the numerical model is tending toward a
singularity near ridges. We do this because the numerical model itself
cannot directly capture singular behavior. We outline the details of the
methodology for the high profile only, as the case of the low profile
involves a transparent extension.
(a) Construction of the synthetic profile,
ηS(l). The opaque points represent the synthetic profile, and
the transparent points represent the untranslated profiles. The green points
represent the profile for i= 1, blue represents i= 2, and red
represents i= 3. After η13(l) has been utilized in
ηS(l), the synthetic profile is complete.
(b) One-dimensional steady-state equilibrium analytical solutions
fitted to 2-D numerical results using P1-D. Each m/n ratio
contains two profiles: one generated from a flow path from the highest point
on the ridge corresponding to the basin divide (HP) and one from the lowest
point on the basin divide (LP). The circles (HP) and crosses (LP) represent
the 2-D model data, and the red (HP) and blue (LP) lines represent the 1-D
analytical model. For each m/n ratio, υ= 3 mm yr-1,
M2= 252 cells, n= 1.0, and
L2-D2= 10-6 to 106 km2. (I) Using
K= 5.00 × 10-12 m0.2 s-1, m/n= 0.4
(2-D), and hm/n= 0.8 (1-D),
P1-D= 6.45 × 1021 (LP) and
P1-D= 7.89 × 1021 (HP). (II) Using
K= 2.83 × 10-11 s-1, m/n= 0.5 (2-D), and
hm/n= 1.0 (1-D), P1-D= 5.79 × 1021 (LP)
and P1-D= 6.47 × 1021 (HP). (III) Using
K= 3.82 × 10-12 m-0.2 s-1, m/n= 0.6
(2-D), and hm/n= 1.2 (1-D),
P1-D= 2.13 × 1023 (LP) and
P1-D= 2.15 × 1023 (HP).
The 13 simulations result in 13 elevation profiles ηi, where
i= 1, 2... 13, each extending from Δli (i.e., one grid point
from the divide) to a downstream value lDi that is somewhat
larger than the value 103-(i/2) km (because the down-channel path of
steepest descent does not follow a straight line). We assemble a synthetic
channel profile, ηS(l), from these as follows. The first leg
of ηS(l) is identical to η1(l) and extends from
l=Δl1 to lD1. We extend the synthetic
profile by translating the second profile upward until its elevation at its
downstream point lD2 matches with
ηS(lD2), as shown in Fig. 5a. The profile ηS(l) now extends from Δl2 to lD1.
As shown in Fig. 5a, we repeat this process until all 13 profiles have been
used to assemble the synthetic profile, which now extends from Δl13 to lD1.
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, P1-D, as a fitting
parameter. To account for the difference in dimensionality, the 1-D
steady-state profiles with hm/n= 0.8, 1.0, and 1.2 are fitted to the
2-D data for m/n= 0.4, 0.5, and 0.6, respectively. The scatter in the
synthetic profile is due to the randomness in the pathway, as dictated by the
initial conditions.
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 m/n < 0.5 converge to finite values as l
approaches 0 and infinity for m/n≥0.5. While these results do not
constitute analytical proof of this asymptotic behavior, they provide
compelling evidence for it.
Scale behavior in other landscape evolution models
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 E is given as
E=KGA1-Qs/QtQs/W,
where KGA is the abrasion coefficient, Qs is the bed load sediment flux,
W is the channel width, and Qt is the bed load transport capacity. Gasparini et
al. (2007) use the following relation for Qt:
Qt=KtAmtSnt,
where Kt is a transport constant and mt and nt are exponents. At
steady state, the total sediment flux at any point in the landscape must
equal the production rate of sediment due to rock uplift:
Qs=KBAυ,
where KB is the fraction of sediment produced that contributes to
bed load (the remainder being moved out of the system as wash load). For
channel width, they use a relation of the form
W=kwQb,
where Q is the water flow discharge, kw is the hydraulic geometry
constant, b is the hydraulic geometry exponent (e.g., Finnegan et al., 2005).
The value of b has been found to vary between 0.3 and 0.5 for bedrock
rivers (Whipple, 2004); Gasparini et al. (2007) use b= 0.5 in their
model. They also estimate discharge as an effective precipitation rate,
kq, multiplied by a drainage area to the power of c, where c≤1:
Q=kqAc.
The resulting relation for steady-state slope is
S=∂η/∂x2+∂η/∂y21/2=KBKt-1υA1-mt1/nt1-kqbkwKB-1KGA-1Abc-1-1/nt.
Using the nondimensionalization terms from Eq. (12), we nondimensionalize Eq. (32)
to
∂η^/∂x^2+∂η^/∂y^21/2=PG1A^1/nt-mt/nt1-PG2A^bc-1-1/nt,
where PG1 and PG2 are two dimensionless Pillsbury numbers:
PG1=υ-2gKBKt-1υL2-D2-2mt+nt1/nt,PG2=KB-1KGA-1L2-D2bc-2.
Horizontal-scale invariance results only when both dimensionless numbers are
independent of the horizontal length scale, L2-D. Gasparini et al. (2007)
use mt= 1.5 and nt= 1.0. This parameter does indeed make the
exponent 2 - 2mt+nt, equal to 0, so that PG1 is
independent of L2-D. The parameter PG2 is invariant to the horizontal
scale when the product of b and c is equal to 1. However, realistic values of
b are between 0.3 and 0.5 (Whipple, 2004) and the value of c is less than or equal
to 1. This means that the maximum value of bc is 0.5. It follows that
PG2 is not independent of the horizontal scale and that the model of
Gasparini et al. (2007) does not satisfy horizontal-scale invariance.
Sensitivity of relief to hillslope length and profile length
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 hm/n≥1, ridge elevation is infinite, thus formally implying
infinite relief. This problem has been sidestepped by introducing a critical
hillslope length lc, upstream of which it is assumed that there
is no channel (e.g., Goren et al., 2014a). This point may be thought of as
loosely corresponding to the channel–hillslope transition in the slope–area
relation discussed by Montgomery and Dietrich (1988, 1992). Here, then, we
let the hillslope zone cover the range 0≤l≤lc, where
lc is an appropriately small fraction of profile length
L1-D. Modifying Eq. (10) accordingly, we can determine the total
relief, R, of the channel profile as follows:
R^=-P1-Dlnl^cifhm=n1-hm/n-1P1-D1-l^c1-hm/nifhm≠n,
where
R=υ2g-1R^lc=L1-Dl^c.
We remind the reader that according to Eq. (7),
P1-D∼L1-D1-hm/n.
We now consider the scale-invariant case, hm/n= 1, and inquire as to how the
relief of the basin might change. Increasing L1-D does not increase
relief because the parameter P1-D∼L1-D0. It is
thus seen from Eqs. (36) and (37) that relief can be increased only by
decreasing l^c. But from Eq. (36), R^→∞ as
l^c→0. It follows that relief is extremely sensitive to the
choice of l^c. Based on our previous analysis, we expect that this
result carries over to the case m/n= 0.5 for the 2-D model.
We next provide an example illustrating the dependence of relief on
hillslope length and profile length when hm/n≠1. Specifically, we
consider the case hm/n= 0.9, with a dimensionless hillslope length
l^c= 0.01. According to Eq. (36), a halving of l^c to
0.005 increases the relief by 11.4 %. In order to achieve the same
increase in relief by changing profile length L1-D while holding
l^c constant, L1-D would have to be increased by 196 %. It is
thus seen that relief of the channel profile can be more sensitive to a
relative change in dimensionless critical channel length than it is to a
relative change in horizontal scale.
Discussion and conclusion
Our 1-D analytical solutions, Eq. (10) and Fig. 1, characterize the scale
behavior of 1-D SPIM, with horizontal-scale invariance satisfied when
hm/n= 1.0. Our 2-D numerical solutions shown in Fig. 2 illustrate
our analytical result that 2-D SPIM shows horizontal-scale invariance when
m/n= 0.5. That is, 2-D models using SPIM with m/n= 0.5 show the same relief
structure regardless of the horizontal scale. This scale invariance has been
previously demonstrated for neither the 1-D nor the 2-D SPIM model. Our result
calls into question the common usage of the ratio m/n= 0.5 in landscape
evolution models (Gasparini et al., 2006). For example, the Python-based
landscape modeling environment, Landlab (Hobley et al., 2017) offers a
default m/n ratio of 0.5. Our result also motivates further investigation as
to why analysis of field data commonly yields values of m/n∼ 0.5 (e.g., Snyder et al., 2000). It should be noted that local empirical
measurements indicating m/n= 0.5 do not necessarily mean m/n=0.5 should be
used as a universal ratio in SPIM. Gasparini and Brandon (2011) used
multiple incision laws, other than SPIM, to simulate steady-state
landscapes and were able to fit E, A, and S in Eq. (1) to find empirical
values of m′ and n′ (prime denotes an empirical value). They found that the
ratio of m′/n′ was sensitive to the incision model's parameters as well as the
rock uplift pattern in each landscape. This implies that both m′ and n′ have
dependency on landscapes properties and are not universal from landscape to
landscape.
In addition to the horizontal-scale invariant case m/n= 0.5 for the
2-D SPIM model, we also emphasize the relationship between the steady-state
landscape relief and horizontal when m/n≠0.5. Equations (14) and (15)
and the results in Fig. 2c show that the relief structure of the landscape
scales with P2-D. Within P2-D, the horizontal length
scale term is L2-D1-2m/n. For the m/n ratio range 0.35 to 0.6
(Whipple and Tucker, 1999), the corresponding exponent range in the
horizontal length scale term is -0.2 to 0.3. This means that over the
stated range of m/n, the relief structure has a weak dependence on the
horizontal length scale. For m/n < 0.5, relief weakly increases
with horizontal scale. For m/n > 0.5, relief weakly but
unrealistically decreases with horizontal scale. The underlying physics of
channel and hillslope processes that might dictate such behavior are, at
present, unhelpfully opaque. In natural systems, larger landscapes would
yield longer rivers. Since elevation monotonically increases with upstream
distance, one would expect relief to increase with horizontal scale. The
results of SPIM, where m/n≥0.5, clearly contradict this intuitive understanding.
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
hm/n ratios (1-D) and all m/n ratios (2-D). For hm/n≥1 (1-D) and m/n≥0.5
(2-D), the models exhibit singular behavior in elevation at ridges as well.
When relief is limited by a hillslope length lc, elevation and slope do
indeed reach finite values at the channel heads, but the effects of the
singularity persist. For example, for the case hm/n≥ 1 in the 1-D model,
relief approaches infinity as hillslope length approaches 0. Our analysis
of ridge singularities in SPIM shows that the choice of hillslope
parameterization plays a key role in determining the relief of natural
landscapes.
Numerical solutions of the 2-D model indicate that it cannot be
grid-invariant for m/n≥0.5. In the absence of hillslope diffusion,
ridges reach infinite elevation as grid size becomes vanishingly small. This
result underlines the critical role of hillslope diffusion in obtaining
meaningful results from the 2-D model. Field estimates of hillslope diffusion
have been obtained on the hillslope scale, but there are unanswered
questions about their application to large-scale models (Fernandes and
Dietrich, 1997). Our results suggest that for the ratio m/n < 0.5,
there are steady-state grid-invariant solutions. However, the grid size
below which grid invariance is realized may be so small, e.g., sub-meter
scale, that the validity of Eq. (1) is called into question. Issues with
SPIM when used on large scales include the following. Studies commonly
neglect the effect of hillslope diffusion when the scale of the grid is
larger than the hillslope scale (Somfai and Sander, 1997; Banavar et al.,
2001; Passalacqua et al., 2006). On coarse-grained scales, increasing the
size of the numerical domain, while keeping the number of cells constant,
will result in the behavior shown in Fig. 2. In Fig. 4 we see that adding
more cells to compensate for the increase in size of the domain, such that
the grid size remains constant, produces heavily biased (i.e., ever more
singular) behavior near the ridges.
Our analysis illustrates that SPIM has two important limitations:
(a) unrealistic scale invariance when m/n takes the commonly used value
0.5, so that a 10 km2 basin has identical relief to a 1000 km2
basin, and (b) singular behavior near the ridges for m/n≥0.5 that makes
maximum relief entirely and unrealistically dependent on grid size. SPIM has
been shown to be of considerable use in the study of the general behavior of
landscapes (e.g., Howard, 1994; Howard et al., 1994). We believe, however,
that the time has come to move on to more sophisticated models. While
scientific questions remain that can be answered with the stream power
incision model, there are many more questions that require a more advanced
formulation (e.g., Gasparini et al., 2007; Crosby et al., 2007; Egholm et
al., 2013). The development of alternative, more physically based models for
incision (e.g., Sklar and Dietrich, 2004; Lague, 2014; Zhang et al., 2015)
and their application to landscape evolution (e.g., Davy and Lague, 2009;
Gasparini et al., 2006, 2007) offer exciting prospects for the future.
The datasets used in this paper are available at
10.13012/B2IDB-7434833_V2 (Kwang, 2017). The exact version of the code
used by the authors can be requested by contacting Jeffrey Kwang at
jeffskwang@gmail.com. However, open-source modeling toolkits, such as Landlab
(see http://landlab.github.io/#/), can also be used to reproduce our
results.
NotationAupslope drainage area (L2)bexponent defining relation between channel width and flow discharge (Gasparini et al., 2007) (–)Bprofile width (L)cexponent defining relation between flow discharge and drainage area (Gasparini et al., 2007) (–)CHack's law constant (L2-h)Dhillslope diffusion coefficient (L2/T)Elocal erosion rate (L/T)gacceleration of gravity (L/T2)hHack's law exponent (–)Kerodibility coefficient (L(1-2m)/T)KBfraction of sediment produced that contributes to bed load (Gasparini et al., 2007) (–)KGAconstant defining relation for the general abrasion model (Gasparini et al., 2007) (L-1)kqeffective precipitation rate (Gasparini et al., 2007) (L(3-2c)/T)Ktconstant defining relation for bed load transport capacity (Gasparini et al., 2007) (L(3-2mt)/T)kwconstant defining relationship between channel width and flow discharge (Gasparini et al., 2007) (L(1-3b)Tb)iindex denoting the profile, 1, 2... 13 (–)lhorizontal distance from the ridge in the 1-D profile (L)l^dimensionless horizontal distance from the ridge in the 1-D profile, l/L1-D (–)lccritical hillslope length (L)l^cdimensionless critical hillslope length, lc/L1-D (–)lDitotal length of profile, i (L)lihorizontal distance from the ridge of profile, i (L)L1-Dhorizontal length scale, profile length (L)L2-Dhorizontal length scale, basin size (L)LRhorizontal length scale, ridge (L)mexponent above A in SPIM (–)mtexponent above A in sediment transport capacity equation (Gasparini et al., 2007) (–)M2number of numerical cells (cells2)nexponent above S in SPIM (–)ntexponent above S in sediment transport capacity equation (Gasparini et al., 2007) (–)P1-DPillsbury number for the 1-D analysis (–)P2-DPillsbury number for the 2-D analysis (–)PG1first Pillsbury number for the Gasparini et at. (2007) analysis (–)PG2second Pillsbury number for the Gasparini et at. (2007) analysis (–)PRPillsbury number for the 2-D ridge analysis (–)Qsbed load sediment flux (L3/T)Qtbed load transport capacity (L3/T)Rtotal relief of the channel profile (L)R^dimensionless total relief, Rg/υ2 (–)sdistance from the ridge (L)s^dimensionless distance from the ridge, s/LR (–)Sstream gradient (–)ttime (T)t^dimensionless time, tg/υ (–)Wchannel width (L)xhorizontal coordinate orthogonal to y (L)x^dimensionless horizontal coordinate, x/L2-D (–)yhorizontal coordinate orthogonal to x (L)y^dimensionless horizontal coordinate, y/L2-D (–)βridge scaling constant (–)Δligrid size for profile, i (L)ηelevation (L)η^dimensionless elevation, ηg/υ2 (–)ηielevation of profile, i (L)ηSelevation of synthetic profile (L)υuplift rate (L/T)
The authors declare that they have no conflict of
interest.
Acknowledgements
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
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