Quantifying climatic and tectonic controls on hillslopes
| RESEARCH
Quantifying the climatic and tectonic controls on
hillslope steepness and erosion rate
Jon D. Pelletier1* and Craig Rasmussen2
1
DEPARTMENT OF GEOSCIENCES, UNIVERSITY OF ARIZONA, 1040 EAST FOURTH STREET, TUCSON, ARIZONA 85721-0077, USA
DEPARTMENT OF SOIL, WATER, AND ENVIRONMENTAL SCIENCE, UNIVERSITY OF ARIZONA, 1177 EAST FOURTH STREET, TUCSON, ARIZONA 85721-0038, USA
2
ABSTRACT
Hillslopes in humid regions are typically convex to concave in profile and have a relatively thick, continuous regolith cover. Conversely,
hillslopes in arid regions are typically cliff-dominated and have a relatively thin, discontinuous regolith cover. The difference between these
two end-member slope forms is classically attributed to climate, but climate, tectonics, and lithology all play a role. In this paper, we describe
a mathematical model for hillslope gradient and regolith thickness using basic climatic and tectonic input data for a given rock type. The
model first solves for the regolith thickness on a planar slope segment in topographic steady state using the soil production function and a
prescribed uplift/incision rate. The climatic and lithologic controls on soil production rates are quantified using an empirical energy-based
model for the physical weathering of bedrock. The slope gradient is then computed by balancing uplift/incision rates with sediment fluxes
calculated using a nonlinear depth- and slope-dependent sediment transport model. The model quantifies the ways in which, as aridity and
uplift/incision rates increase, regolith thicknesses decrease and hillslope gradients increase nonlinearly until a threshold condition is reached,
beyond which bare, cliff-dominated slopes form. The model can also be used to estimate long-term erosion rates and soil residence times
using basic input data for climate and regolith thickness. Model predictions for erosion rates closely match cosmogenically derived erosion
rates in granitic landscapes. This approach provides a better quantitative understanding of the climatic and tectonic controls on slope form,
and it provides a simple, widely applicable method for estimating long-term erosion rates and the thickness of regolith cover on hillslopes.
LITHOSPHERE; v. 1; no. 2; p. 73–80.
INTRODUCTION
It is a “textbook” observation in geomorphology that hillslopes in arid
regions tend to be cliff-dominated and have a thin, discontinuous regolith cover, while hillslopes in more humid regions tend to be convex to
concave in profile with a relatively thick, continuous regolith cover (Ritter et al., 2002) (Fig. 1). These differences in slope form are also associated with distinctly different modes of evolution: cliff-dominated, bareregolith slopes tend to evolve by lateral slope retreat, thus maintaining
steep slopes over time, even in the absence of uplift, while convex to
concave slopes with continuous regolith cover tend to evolve by slope
replacement, evolving to gentler slopes over time (Fig. 1). While the
formation of these two end-member slope types is classically attributed
to climate, uplift/incision rates and lithology must also play a role. In
arid climates where weathering rates are very low, for example, regolith
production can, nevertheless, keep pace with uplift rates if those rates
are sufficiently low. Rock type also plays a role in controlling regolith
cover and hillslope gradient, because rock types differ in their degree of
weatherability. The geologic conditions required for the formation of
bare, cliff-dominated slopes must, therefore, be a function of climate,
tectonics, and lithology. In addition, the precise mechanisms of climate
control on slope form (i.e., what are the respective roles of temperature
and precipitation?) are not well understood.
Recently, three major advances have been made that bear on the
link between slope form and climate and tectonics. First, Heimsath et
al. (1997, 1999) used in situ cosmogenic isotopes to constrain the soil
production function to be an exponential function of regolith thickness.
*E-mail: [email protected].
doi: 10.1130/L3.1
Heimsath et al.’s work directly links the rate of physical weathering of
bedrock with regolith thickness. Second, Roering and coworkers quantified the nonlinear dependence of sediment flux on slope gradient using
experimental, field, and modeling studies (e.g., Roering et al., 1999;
2001; Roering, 2004). Third, several recent studies have documented
the dependence of sediment flux on regolith thickness or depth (e.g.,
Gabet, 2000; Heimsath et al., 2005). By combining the nonlinear slopedependent transport model of Roering et al. with the depth-dependent
model of Gabet, Heimsath et al., and others, we get the following equation for unit sediment flux qs:
qs =
κhn S
1 − ( S / Sc )
2
,
(1)
where κ is a transport coefficient, hn is the regolith thickness normal to
the surface, S is the slope gradient, and Sc is the tangent of the angle of
stability. Equation 1 is not appropriate for very thick soils (i.e., more than
several meters) because the sediment flux cannot continue to increase
indefinitely as soil thickness increases (unless the hillslope is near the
angle of stability and capable of transporting the entire soil profile by
mass movement). More complex, depth-dependent models are available
that capture this depth-saturation effect (e.g., Roering, 2008). In a comparison of several end-member sediment transport models proposed in
the literature, Roering (2008) found that a depth- and slope-dependent
transport model similar to that of Equation 1 was the most accurate
model for quantifying sediment transport on the steep upland slopes of
the Oregon Coast Range.
The exponential soil production function implies that bedrock lowering is a maximum for bare bedrock slopes and decreases exponentially
For permission to copy, contact [email protected] | © 2009 Geological Society of America
73
PELLETIER and RASMUSSEN
A
arid
C
D
B
humid
arid
humid
P0
∂b
∂t
P0 e-h/h0(exponential)
humped
0.9P0 h
h0
0
2
1
3
4
h/h0
Figure 1. Comparison of slope form and evolution in arid and humid climates. Photographs are shown for hillslopes characteristic of
relatively arid (A) and humid (B) climates, from Arizona and Bolivia, respectively. (C) Schematic diagram illustrating the typical evolution
of arid slopes, which tend to evolve primarily by lateral slope retreat, and humid slopes, which tend to evolve by slope replacement. (D)
Plot of the rate of change of bedrock lowering ( | ∂b/∂t | ) as a function of normalized regolith thickness (h/h0) for the exponential and
humped production functions.
with increasing regolith thickness. This exponential relationship has been
inferred from in situ cosmogenic isotope measurements in upland soil profiles (Heimsath et al., 1997, 1999). Conceptually, this relationship reflects
the fact that regolith cover acts as a buffer for the underlying bedrock, protecting it from the diurnal temperature changes and infiltrating runoff that
drive subsurface weathering. The exponential soil production function may
not capture the full complexity of regolith production, however. As regolith
thickness decreases below a critical value in arid and semiarid climates, the
landscape may be unable to store enough water to promote weathering or
support plant life. Therefore, in some environments, weathering rates may
actually increase as regolith thickness increases (for relatively thin regolith cover), leading to a “humped” production function (Fig. 1D) (Ahnert,
1977; Cox, 1980; Dietrich et al., 1995; Anderson and Humphrey, 1989;
Furbish and Fagherazzi, 2001; Anderson, 2002; Furbish, 2003; Minasny
and McBratney, 1999, 2006). Recent cosmogenic radionuclide data from
granitic landscapes in Australia provide at least preliminary support for a
humped production function (Heimsath et al., 2006). Also, depending on
whether tors are included or separated from analyses of soil production data
sets, earlier cosmogenic analyses may also be consistent with a humped production function. Tors are small bedrock hills characterized by thin regolith
cover and relatively low weathering rates. Because tors are associated with
locally more resistant bedrock, Heimsath et al. (1997, 1999) excluded tors
from their analysis of the soil production function. Strudley et al. (2006),
however, argued that tor formation, although triggered by locally resistant
bedrock, also involves a feedback process between slope gradient, regolith
cover, and erosion that they associated with a humped production function.
In this paper, we consider both the exponential and humped production
functions in our analyses, comparing and contrasting the model predictions
for these two possible end-member forms of the soil production function.
74
MODEL DESCRIPTION AND RESULTS
Hillslopes in upland landscapes are composed of a system of two interacting surfaces: the topographic surface, with elevations given by z(x,y),
and the underlying bedrock surface or weathering front, given by b(x,y).
The difference between these two surfaces is the soil or regolith thickness,
h(x,y). The topographic and weathering-front surfaces are strongly coupled because the shape of the topography controls erosion and deposition,
which, in turn, changes the values of h(x,y) (e.g., Furbish and Fagherazzi,
2001). The values of h(x,y), in turn, control weathering rates. In the following analysis, we will assume topographic steady state at the hillslope
scale, i.e., that bedrock uplift and soil erosion rates are equal after the
effect of the relative densities of rock and soil are taken into account. The
appropriateness of this assumption and the robustness of the model predictions with respect to this assumption will be addressed in the discussion
section. If topographic steady state is assumed, the rate of bedrock lowering must equal the rate of bedrock uplift:
P0e − h h0 = U ,
(2)
assuming the exponential form of the soil production function, and
P0e − h h0 = U
if
h > 0.6h0
h
(3)
= U if h ≤ 0.6h0,
h0
assuming the humped form of the soil production function plotted in
Figure 1, where P0 is the maximum bedrock lowering rate, h0 is a characteristic regolith depth, and U is the uplift/incision rate. Equations 2 and
0.9 P0
www.gsapubs.org | Volume 1 | Number 2 | LITHOSPHERE
Quantifying climatic and tectonic controls on hillslopes
3 assume that regolith production is controlled by the regolith thickness
measured vertically. It is more reasonable to assume that regolith production is controlled by the regolith thickness measured normal to the
surface, since it is that latter thickness that buffers the bedrock from the
elements (Heimsath et al., 2001). This “normal-directed” soil production function has the effect of introducing a cos θ term (where θ is the
hillslope angle) into the analysis. We will consider that effect later in this
section. The humped production function represented by Equation 3 is a
linear function of h for small h that rises to a maximum value of approximately one-half the maximum rate of the exponential form and then
matches the exponential form for larger values of h. This specific form
of the humped production function is not unique, and other forms that
increase from a low value at h = 0 to an exponential form at large values
of h are possible. We refer to U as the “uplift/incision” rate because it is
the rate of channel incision into bedrock at the base of the slope, not the
regional uplift rate, that controls hillslope response to uplift. Local incision rates may be higher or lower than regional uplift rates, depending
on how the channel network responds to regional uplift.
Equations 2 and 3 can be solved for the regolith thickness h along a planar slope segment in topographic steady state given values for P0, U, and
h0. For the exponential production function, regolith thickness is obtained
by solving Equation 2 for h to obtain
⎛P ⎞
h = h0 ln ⎜ 0 ⎟
⎝U⎠
if
h=0
if
P0
≥1
U
P0
< 1.
U
⎛P ⎞
h = h0 ln ⎜ 0 ⎟
⎝U⎠
h U
or h = 0
0.9 P0
h=0
⎧⎪ 1 ⎡⎛ MAT − 21.5 ⎞ 2 ⎛ MAP − 4412 ⎞ 2 ⎤ ⎫⎪
EEMT = 347,134 exp ⎨− ⎢⎜
+
⎟⎠ ⎥ ⎬.
⎝ −10.1 ⎟⎠ ⎜⎝
1704
⎦ ⎭⎪
⎩⎪ 2 ⎣
if
P0
≥2
U
if
P0
< 2.
U
(5)
h ρs h
,
=
E ρ bU
(6)
where E is the rate of soil erosion, and ρb and ρs are the densities of the
bedrock and soil, respectively.
Climate and rock type enter into the model through their control on the
maximum rate of bedrock lowering, P0 (or P0/2 in the humped model).
Recently, Rasmussen et al. (2005) and Rasmussen and Tabor (2007) proposed a new variable, called “effective energy and mass transfer” (EEMT),
for quantifying the role of climate in pedogenic processes. Unlike previous attempts to empirically relate rates of pedogenesis to climatic indices
such as mean annual temperature and precipitation, the EEMT variable is
explicitly based on a conceptual model of energy input and flow through
the soil profile, including thermal (temperature) and material (water and
biomass) forms of energy. These authors demonstrated the effectiveness
of the EEMT framework by documenting significant correlations between
EEMT and soil order, regolith thickness, and a variety of pedogenic
indices using data from study sites over a broad range of climates and
rock types. Plotting these data versus mean annual temperature (MAT)
LITHOSPHERE | Volume 1 | Number 2 | www.gsapubs.org
(7)
Biotic controls on EEMT are implicitly included in Equation 7 because
they are empirical functions of MAP and MAT. Rasmussen and Tabor
(2007) found that regolith thickness on stable (low gradient) slopes
increased exponentially with EEMT, suggesting that rates of regolith production also increase exponentially with EEMT. This hypothesis suggests
a relationship of the form
P0 = ae b⋅EEMT,
(8)
where a (units of m/k.y.) and b (units of m2/[kJ yr]) are empirical coefficients. In order to constrain the parameters a and b for granitic landscapes,
we used published data from Riebe et al. (2004) (Figs. 2A and 2B). Physical erosion rates measured by Riebe (E) were used to estimate the rate of
peak regolith production using the equation
(4)
With the humped production function, two solutions for h are possible.
The solution that applies depends on whether uplift rates are increasing
or decreasing over time. We will return to this point later in this section.
Given a solution for regolith thickness (h), the soil residence time can
be computed as
Tr =
and mean annual precipitation (MAP), in contrast, yielded no significant
correlations. As such, the EEMT variable appears to provide an effective
quantitative measure of the controlling influence of climate and lithology
on rates of regolith production. EEMT may be computed using an empirical function of MAT and MAP (Rasmussen and Tabor, 2007):
P0 =
If P0/U < 1, the model predicts that a bare bedrock slope will form. For
the humped production function, solving Equation 3 for h gives
| RESEARCH
ρs hn / h0
,
Ee
ρb
(9)
assuming the exponential production function, where hn is the soil thickness normal to the surface, and
P0 =
ρs hn / h0
Ee
ρb
P0 =
h0 ρs
E if
0.9hn ρ b
if
hn > 0.6h0
hn ≤ 0.6h0,
(10)
assuming the humped production function, with h0 = 0.5 m. The resulting estimates for P0 are plotted in Figures 2A and 2B as a function of
EEMT on logarithmic scales, assuming ρb/ρs = 1.35. The data points plotted in Figures 2A and 2B include all of the data published in Riebe et
al. (2004) for which regolith thicknesses were measured. The continuous
curve plotted in Figures 2A and 2B corresponds to Equation 8 with bestfit coefficients a = 0.037 m/k.y. and b = 0.00003 m2/(kJ yr). The two data
points plotted with open circles represent outliers in this and subsequent
analyses of the data set from Riebe et al. (2004). Potential limitations of
the model with respect to these outlying points will be discussed at the
end of this section. The coefficients a and b vary with rock type, and the
variation in these parameters represents the control lithology has on slope
and soil thickness in the model. In this paper, we focus on model results
for granitic landscapes only, because it is in these landscapes where regolith production rates are currently most well constrained. Rasmussen and
Tabor (2007), however, provided preliminary values of a and b for basalt
and andesite as well. Future studies that apply the model to those rock
types, therefore, may use the preliminary values provided by Rasmussen
and Tabor (2007).
Figures 3A and 3B plot model predictions for regolith thickness h given
by Equations 3 and 4 as a function of EEMT and uplift/incision rate for
both the exponential (Figs. 3A and 3B) and humped production functions
(Figs. 3C and 3D), assuming a = 0.037 m/k.y., b = 0.00003 m2/(kJ yr), and
h0 = 0.5 m. For the exponential production function, regolith thickness
75
PELLETIER and RASMUSSEN
humped
exponential
104
P0
(m/k.y.)
102
A
104
P0
(m/k.y.)
102
100
100
10-2
10-2
3,000
0.25
B
10,000
3,000
30,000 100,000 300,000
0.25
0.2
0.2
Ep 0.15
(m/k.y.)
Ep 0.15
(m/k.y.)
0.1
0.1
0.05
0.05
0.05
0.1
0.15
100,000 300,000
30,000
EEMT (kJ/(m2yr))
C
0.0
0.0
10,000
EEMT (kJ/(m2yr))
0.2
D
0.0
0.0
0.1
0.05
Em (m/k.y.)
0.15
0.2
Em (m/k.y.)
Figure 2. Calibration and tests of the model using Riebe et al. (2004) data set. (A–B) Plots of maximum regolith production rate (P0) and a function of effective energy and mass transfer (EEMT) assuming (A) exponential and (B) humped
production functions. Circles represent P0 values calculated from measurements of Riebe et al. (2004), while the curve
corresponds to Equation 8 using a = 0.037 m/k.y. and b = 0.00003 m2/(kJ yr). (C–D) Plots of the predicted erosion rate
as a function of the measured rate assuming (C) exponential and (D) humped production functions. The straight lines
correspond to perfect agreement between predicted and measured values. Open circles represent the FR6 and FR8 sites
where the model-predicted erosion rates are several times higher than measured erosion rates.
increases linearly with EEMT and decreases exponentially with uplift/
incision rate until the threshold condition P0/U < 1 is met, beyond which,
bare slopes are formed. For example, assuming an EEMT of 50,000 kJ/
(m2 yr) (i.e., a moist, temperate climate) and an uplift/incision rate of
0.01 m/k.y., Figure 3A predicts a regolith thickness of 1.4 m. For the
humped production function, the threshold condition for bare slopes to
form requires lower values for U and/or higher values for P0 compared
to the exponential case, i.e., P0/U < 2, meaning that bare slopes will form
under a wider range of climatic and uplift/incision values compared to the
exponential case. Also, in the humped production case, there is a “left”
and a “right” branch of solutions corresponding to relatively thin and thick
regolith, respectively. The “thin” solutions are plotted in Figure 3B, while
the “thick” solutions are the same as those of the exponential production
function plotted in Figure 3A for P0/U < 2 (otherwise h = 0). The solution
that is applicable depends on whether the value of P0/U is increasing or
decreasing with time. If a hillslope is initially in a state of tectonic quiescence (i.e., thick soils have had time to form) and the hillslope experiences
steadily increasing uplift rates, regolith thickness will decrease from its
initially high value along the right branch of the humped production function until uplift outpaces production and a bare slope forms. Alternatively,
if a hillslope begins as a bare slope (e.g., formed during a period of high
uplift rates) and uplift rates steadily decrease, the regolith thickness will
increase along the left branch until equilibrium is achieved with the new
uplift rate. Regolith thicknesses are much lower along the left branch compared to the right branch for similar uplift rates. As such, if a hillslope is
governed by a humped production function, regolith thicknesses will be
76
lower for uplift rates decreasing below the threshold condition for bare,
cliff-dominated slopes to form compared to regolith thicknesses in conditions of increasing uplift rates. This behavior suggests a type of “hysteresis,” in which thin soils persist on a landscape subject to decreasing
uplift rates, while thick soils persist on a landscape subject to increasing
uplift rates. In this example, we described the hysteresis effect in terms
of increasing or decreasing uplift rates, but climate change also exerts an
influence through its effect on P0. Hence, climate change could thicken or
thin the regolith cover hysteretically in a similar manner.
The slope gradient of a hillslope can be estimated using the solutions
for regolith thickness plotted in Figures 3A and 3B together with the
relationship
κhn S
1 − ( S / Sc )
2
= EL =
ρb
UL.
ρs
(11)
Equation 11 applies the sediment continuity equation to the base of a
slope of horizontal length L with a basal slope gradient S and basal regolith
thickness normal to the surface hn, undergoing constant erosion at a rate E.
Equation 11 is general and does not place any limitations on the variations
in S and hn upslope. However, when Equation 11 is used together with
input values for hn obtained by solving Equations 9 and 10 (which assume
a planar slope segment in topographic steady state), then the predictions of
Equation 11 are exactly applicable only to planar slopes and thus provide
only approximate slope gradients for more general hillslope shapes. Equation 11 is a quadratic equation with solution
www.gsapubs.org | Volume 1 | Number 2 | LITHOSPHERE
Quantifying climatic and tectonic controls on hillslopes
4
0.3
exponential
A
| RESEARCH
humped (left branch)
B
0.2
3
h
(m)
h
(m)
2
0.1
1
0
10-3
10-2
10-1
U (m/k.y.)
1.0
100
50
100
150
200
0.0
10-3
EEMT
(x10 kJ/(m2 yr))
1.0
C
10-1
100
50
150
200
EEMT
(x10 kJ/(m2 yr))
3
D
0.8
0.8
S/Sc
S/Sc
0.6
0.6
0.4
0.4
0.2
0.2
0
10-3
10-2
U (m/k.y.)
3
100
10-2
10-1
U (m/k.y.)
100
50
100 150
200
EEMT
3
(x10 kJ/(m2 yr))
0
10-3
10-2
10-1
U (m/k.y.)
100
50
100 150
200
EEMT
(x103 kJ/(m2 yr))
Figure 3. (A–B) Plots of model-predicted values for regolith thickness h as a function of uplift/incision rate (U) and effective
energy and mass transfer (EEMT). (C–D) Plots of normalized slope gradient (S/Sc) as a function of uplift/incision rate (U)
and EEMT. (A, C) Exponential production function. For the humped production function, B and D represent solutions for
the “left” (thin) branch of solutions, while the right branch of solutions match those of the exponential function plotted
in A and C.
2
S
1 ⎛ ρ κh S ⎞ 1 ⎛ ρ s κhn Sc ⎞
= − ⎜ s n c⎟ +
+ 4.
Sc
2 ⎝ ρ b UL ⎠ 2 ⎜⎝ ρ b UL ⎟⎠
(12)
Equation 12 is plotted in Figures 3C and 3D for the exponential and
humped production functions, assuming the same parameters as in Figures 3A and 3B and κ = 0.1 m/k.y., L = 50 m, and Sc = 1. Figures 3C and
3D illustrate that slope gradients are inversely related to regolith thickness. For all climatic and tectonic conditions for which bare slopes are
predicted (i.e., h = 0), the model predicts slope gradients that are equal to
the threshold of stability (i.e., S/Sc = 1). However, although the maximum
value for S/Sc is 1 in the model, actual maximum slope gradients may
be much larger because topographic steady state is impossible to achieve
above the threshold condition for bare slopes. As such, the slope gradi-
LITHOSPHERE | Volume 1 | Number 2 | www.gsapubs.org
ent will continually increase in such cases as long as the uplift/incision
rate outpaces soil production. Figure 3C illustrates that the model prediction that slope gradient increases with increasing uplift/incision rate and
decreasing EEMT values until bare, cliff-dominated slopes form. If the
humped production function is assumed, there are two solutions for slope
gradient corresponding to the two solutions of regolith thickness plotted
in Figures 3A and 3B. Figure 3D shows that, in cases where thin regolith
forms on the left branch of the humped production function, the slope will
compensate with very steep slopes over a wide range of climates and uplift
rates. This result underlines the fact that bare, cliff-dominated slopes can
persist under a wide range of conditions once they form in a given landscape (if the humped production function applies). This model behavior
may help explain the persistence of cliff-dominated slopes even in areas
not undergoing active tectonic uplift (e.g., Monument Valley and other
77
PELLETIER and RASMUSSEN
cliff-dominated landscapes of the Colorado Plateau that are far enough
from the Colorado River to have been unaffected by late Cenozoic incision). Qualitatively, the ways in which the slope gradients predicted by
the model vary with climate and uplift rate do not depend on the specific
values of κ, L, and Sc. Quantitatively, predicted slope values decrease as
the ratio κSc /L increases. The threshold for cliff-dominated slopes to form,
however, given by P0 /U < 1 and P0 /U < 2 for the exponential and humped
production functions, respectively, does not depend on κ, L, or Sc.
Equations 2 and 3 relate regolith production to the regolith thickness
in the vertical direction. As noted at the beginning of this section, it is
more appropriate to relate these variables in a coordinate system normal
to the surface (Heimsath et al., 2001). If we adopt this approach, then
Equations 2 and 3 become
P0 1 + S 2 e − h ( h0
1+ S 2 )
=U =
ρs
E,
ρb
(13)
assuming the exponential form of the soil production function, and
P0 1 + S 2 e
0.9 P0
(
− h h0 1+ S 2
) = U = ρs E
ρb
ρ
h
=U = s E
ρb
h0
if
if
h > 0.6h0 1 + S 2
h ≤ 0.6h0 1 + S 2 ,
(14)
assuming the humped form. If slope gradients are known, the analytic
solutions to Equations 13 and 14 are
⎛P
⎞
h = h0 1 + S 2 ln ⎜ 0 1 + S 2 ⎟
⎝U
⎠
if
h=0
if
P0
1+ S2 ≥ 1
U
P0
1 + S 2 < 1,
U
(15)
and
⎛P
⎞
h = h0 1 + S 2 ln ⎜ 0 1 + S 2 ⎟
⎝U
⎠
or h =
h0 U
0.9 P0
h=0
if
P0
1+ S2 ≥ 2
U
if
P0
1 + S 2 < 2 . (16)
U
Equations 11 and either 13 or 14 form a set of two coupled equations for
two unknowns that cannot be solved analytically. The simplest numerical method for solving these equations is to use the analytic solutions
for Equations 12 and either 15 or 16 to iteratively converge on the solution. In this method, Equation 15 or 16 is used to estimate h assuming
S = 0. This estimate for h is then used in Equation 12 to calculate a
value for S that is then used in Equation 15 or 16 to refine the value of
h. This process can be repeated and converges to the correct solution
in all cases. Figure 4 illustrates the numerical solutions for h and S
obtained in this way for the exponential case. For S/Sc < 0.4, the analytic
solutions predicted by Equations 3, 4, and 12 for the original, vertically directed production model differ from the numerical solutions for
the revised, normally directed production model by less than 1%. For
steeper slopes, however, the revised model predicts significantly thicker
regolith for the same values of U and EEMT compared to the original
model. Slope gradients predicted by the revised model are not significantly different from those of the original model as a function of uplift/
incision rate and EEMT.
Thus far, we have assumed uplift rate to be a prescribed input to the
model. The model framework can also be inverted to estimate erosion
rates by solving Equations 13 and 14 using site-specific measurements
for h and S and empirical data for P0 and h0. In contrast to the previous
results in this paper, this approach does not require topographic steady
state and hence can be expected to be more generally applicable. Figures 5A and 5B plot the erosion rate as a function of regolith thickness
5
A
1.0
B
4
0.8
S/Sc
h 3
(m)
0.6
2
0.4
1
0
10-3
0.2
10-2
10-1
U (m/k.y.)
100
50
100
150
200
EEMT
3
(x10 kJ/(m2 yr))
0
10-3
10-2
10-1
U (m/k.y.)
100
50
100
150
200
EEMT
(x10 kJ/(m2 yr))
3
Figure 4. (A) Plot of model-predicted values for regolith thickness (h) as a function of uplift/incision rate (U) and effective
energy and mass transfer (EEMT) for the revised exponential production model based on thickness normal to the surface.
(B) Plot of normalized slope gradient (S/Sc) as a function of uplift/incision rate (U) and EEMT for the revised model.
78
www.gsapubs.org | Volume 1 | Number 2 | LITHOSPHERE
Quantifying climatic and tectonic controls on hillslopes
rates at sites FR6 and FR8. Persistent snow cover affects both the accuracy of cosmogenically derived erosion rates, and it serves to buffer the
subsurface bedrock from diurnal temperature changes and, hence, may
decrease erosion rates relative to model predictions. The empirical function for EEMT (i.e., Eq. 7) does not account for seasonality and timing
of precipitation inputs because it is based on mean annual temperature
and precipitation only. As such, higher MAP values drive higher weathering rates regardless of precipitation type, i.e., rain or snow. In areas of
persistent snow cover, such as the low-relief, high-elevation zones of the
Sierra Nevada, the empirical function in Equation 7 may be of limited
accuracy because higher MAP values in such cases may equate to lower
weathering rates resulting from the buffering effect provided by the persistent snow layer. The original formulation of EEMT (Rasmussen et al.,
2005) is based on monthly data and mitigates this problem to an extent
because precipitation that falls during below-freezing temperature conditions does not contribute to weathering.
normal to the surface (hn) and EEMT for the exponential and humped
production functions, respectively. The corresponding soil residence
times, calculated using Equation 6, are plotted in Figures 5C and 5D.
The surfaces plotted in Figures 5C and 5D dip down and away from the
reader, so that the thickest soils have the largest residence times. As a
test of the model predictions for erosion rate, Figures 2C and 2D illustrate the predicted erosion rates versus the measured rates using the data
of Riebe et al. (2004), assuming exponential and humped production
functions, respectively. The straight line in this figure represents an exact
match between the predicted and measured values. Figure 2C shows that
the model does an excellent job of predicting erosion rates using both
the exponential and humped production functions, except for two sites
(FR6 and FR8) on the low-relief, high-elevation plateau of the Sierra
Nevada (shown as open circles), where predicted rates are several times
higher than actual rates. Persistent snow cover is one reason why the predicted erosion rates are substantially higher than the measured erosion
102
102
A
exponential
101
E
(m/k.y.)
| RESEARCH
B
humped
101
100
E
(m/k.y.)100
10-1
10-2
10-1
10-3
10-2
10-4
10-5
0.2
1.0
2.0
3.0
hn (m)
106
4.0
C
50
100 150
200
10-3
0.2
EEMT
(x103 kJ/(m2 yr))
0.5
1.0
1.5
hn (m)
104
2.0
50
100
150
200
EEMT
(x10 kJ/(m2 yr))
3
D
105
103
104
Tr
(k.y.) 103
Tr 102
(k.y.)
101
102
101
100
100
10-1
200
10-1
0.2
1.0
2.0
h (m)
3.0
4.0
50
100
150
EEMT
3
(x10 kJ/(m2 yr))
10-2
0.2
0.5
1.0
h (m)
1.5
2.0
50
100
150 200
EEMT
(x10 kJ/(m2 yr))
3
Figure 5. (A–B) Plots of model-predicted values for erosion rate (E) as a function of regolith thickness normal to the surface (hn) and
effective energy and mass transfer (EEMT), and (C–D) soil residence time (Tr) (C, D), as a function of regolith thickness measured
vertically (h) and EEMT. (A, C) Exponential production function. (B, D) Humped production function.
LITHOSPHERE | Volume 1 | Number 2 | www.gsapubs.org
79
PELLETIER and RASMUSSEN
DISCUSSION
REFERENCES CITED
The framework of this paper relies upon the assumption of topographic
steady state at the hillslope scale. Given that topographic steady state is
difficult to prove and may also be rarely achieved in nature, it is appropriate to ask how robust the model predictions are with respect to this
assumption. Topographic steady state assumes that uplift and erosion are
in precise balance, implying that elevation values do not change over time.
If, however, uplift and erosion rates are in approximate balance (e.g., values of E and ρbU/ρs are within 20% of each other), then the model predictions can be expected to apply with comparable accuracy. Therefore,
topographic steady state need not apply precisely in order for the model
to provide useful predictions. For example, even in a landscape with no
active tectonic uplift, isostatic adjustment generates rock uplift rates that
are equal to ~80% of erosion rates. Therefore, an approximate balance
between uplift and erosion rates can be expected to hold even in landscapes that are not subject to active tectonic uplift.
In the arid and semiarid regions of the southwestern United States,
cliff formation is especially prevalent in the relatively undeformed
sedimentary rocks of the Colorado Plateau. In order to understand that
prevalence within the context of this paper, it is important to note that the
erosion of bedded or banded rocks on cliff-dominated slopes is a special
case because the erosion of one stratum may depend on the erodibility
of strata exposed above it, because weaker strata cannot erode faster than
more slowly eroding, resistant cap rock units above them. For this reason, cliff formation is more common in layered strata because the effective resistance to erosion of the entire exposed sequence is controlled by
the most resistant units within that sequence. In the model of this paper,
the role of rock type is quantified using lithologically dependent coefficients linking regolith production rate to climate. As more quantitative
data become available on the values of these parameters for different
rock types and as the model becomes more broadly applied, it will be
important to remember that in bedded and banded rocks, the erodibility
of one rock type can exert a rate-limiting effect on the erosion of rocks
exposed below it.
The method used in this paper provides a quantitative tool for estimating the regolith cover and slope gradient based on readily available indices
for climate (i.e., MAT and MAP) and tectonics (i.e., uplift/incision rate).
Roering et al. (2007) took a broadly similar approach, utilizing the nonlinear slope–dependent transport model to derive a generally applicable analytic relationship between erosion rate and hillslope relief. This approach
by Roering et al. provides a very useful theoretical basis for quantitatively
linking slope gradient and erosion rates. Here, we extend the results of
Roering et al. by including both transport-limited (regolith covered) and
weathering-limited (bare) slopes (and the transition between them) and
the effects of climate and lithology on erosion rates and slope form.
Ahnert, F., 1977, Some comments on quantitative formulation of geomorphological processes
in a theoretical model: Earth Surface Processes and Landforms, v. 2, p. 191–201.
Anderson, R.S., 2002, Modeling the tor-dotted crests, bedrock edges, and parabolic profiles
of high alpine surfaces of the Wind River Range, Wyoming: Geomorphology, v. 46,
p. 35–58, doi: 10.1016/S0169-555X(02)00053-3.
Anderson, R.S., and Humphrey, N.F., 1989, Interaction of weathering and transport processes in the evolution of arid landscapes, in Cross, T.A., ed., Quantitative Dynamic
Stratigraphy: Englewood Cliffs, New Jersey, Prentice Hall, p. 349–361.
Cox, N.J., 1980, On the relationship between bedrock lowering and regolith thickness: Earth
Surface Processes, v. 5, p. 271–274.
Dietrich, W.E., Reiss, R., Hsu, M.-L., and Montgomery, D.R., 1995, A process-based model for
colluvial soil depth and shallow landsliding using digital elevation data: Hydrological
Processes, v. 9, p. 383–400, doi: 10.1002/hyp.3360090311.
Furbish, D.J., 2003, Using the dynamically coupled behavior of land surface geometry and
soil thickness in developing and testing hillslope evolution models, in Wilcock, P.R.,
and Iverson, R.M., eds., Predictions in Geomorphology: Washington, D.C., American
Geophysical Union, p. 169–181.
Furbish, D.J., and Fagherazzi, S., 2001, Stability of creeping soil and implications for hillslope
evolution: Water Resources Research, v. 37, p. 2607–2618, doi: 10.1029/2001WR000239.
Gabet, E.J., 2000, Gopher bioturbation: Field evidence for nonlinear hillslope diffusion: Earth Surface Processes and Landforms, v. 25, p. 1419–1428, doi:
10.1002/1096-9837(200012)25:13<1419::AID-ESP148>3.0.CO;2-1.
Heimsath, A.M., Dietrich, W.E., Nishiizumi, K., and Finkel, R.C., 1997, The soil production function and landscape equilibrium: Nature, v. 388, p. 358–361, doi: 10.1038/41056.
Heimsath, A.M., Dietrich, W.E., Nishiizumi, K., and Finkel, R.C., 1999, Cosmogenic nuclides,
topography, and the spatial variation of soil depth: Geomorphology, v. 27, p. 151–172,
doi: 10.1016/S0169-555X(98)00095-6.
Heimsath, A.M., Dietrich, W.E., Nishiizumi, K., and Finkel, R.C., 2001, Stochastic processes
of soil production and transport: Erosion rates, topographic variation and cosmogenic
nuclides in the Oregon Coast Range: Earth Surface Processes and Landforms, v. 26,
p. 531–552, doi: 10.1002/esp.209.
Heimsath, A.M., Furbish, D.J., and Dietrich, W.E., 2005, The illusion of diffusion: Field evidence for depth-dependent sediment transport: Geology, v. 33, p. 949–952, doi: 10.1130/
G21868.1.
Heimsath, A.M., Chappell, J., Finkel, R.C., Fifield, K., and Alimanovic, A., 2006, Escarpment
erosion and landscape evolution in southeastern Australia, in Willett, S.D., Hovius, N.,
Brandon, M.T., and Fisher, D.M., eds.,Tectonics, Climate, and Landscape Evolution: Geological Society of America Special Paper 398, p. 173–190, doi: 10.1130/2006.2398(10).
Minasny, B., and McBratney, A.B., 1999, A rudimentary mechanistic model for soil production and landscape development: Geoderma, v. 90, no. 1–2, p. 3–21, doi: 10.1016/S00167061(98)00115-3.
Minasny, B., and McBratney, A.B., 2006, Mechanistic soil-landscape modelling as an
approach to developing pedogenetic classifications: Geoderma, v. 133, no. 1–2, p. 138–
149, doi: 10.1016/j.geoderma.2006.03.042.
Rasmussen, C., and Tabor, N., 2007, Applying a quantitative pedogenic energy model across
a range of environmental gradients: Soil Science Society of America Journal, v. 71,
no. 6, p. 1719–1729, doi: 10.2136/sssaj2007.0051.
Rasmussen, C., Southard, R.J., and Horwath, W.R., 2005, Modeling energy inputs to predict
pedogenic environments using regional environmental databases: Soil Science Society of America Journal, v. 69, no. 4, p. 1266–1274, doi: 10.2136/sssaj2003.0283.
Riebe, C.S., Kirchner, J.W., and Finkel, R.C., 2004, Erosional and climatic effects on long-term
chemical weathering rates in granitic landscapes spanning diverse climate regimes: Earth
and Planetary Science Letters, v. 224, no. 3–4, p. 547–562, doi: 10.1016/j.epsl.2004.05.019.
Ritter, D.F., Kochel, R.C., and Miller, J.R., 2002, Process Geomorphology: Long Grove, Illinois,
Waveland Press, 560 p.
Roering, J.J., 2004, Soil creep and convex-upward velocity profiles: Theoretical and experimental investigation of disturbance-driven sediment transport on hillslopes: Earth Surface Processes and Landforms, v. 29, p. 1597–1612, doi: 10.1002/esp.1112.
Roering, J.J., 2008, How well can hillslope evolution models “explain” topography? Simulating soil production and transport using high-resolution topographic data: Geological
Society of America Bulletin, v. 120, p. 1248–1262, doi: 10.1130/B26283.1.
Roering, J.J., Kirchner, J.W., and Dietrich, W.E., 1999, Evidence for nonlinear, diffusive
sediment transport on hillslopes and implications for landscape morphology: Water
Resources Research, v. 35, p. 853–870, doi: 10.1029/1998WR900090.
Roering, J.J., Kirchner, J.W., and Dietrich, W.E., 2001, Hillslope evolution by nonlinear, slopedependent transport: Steady-state morphology and equilibrium adjustment timescales:
Journal of Geophysical Research, v. 106, p. 16,499–16,513, doi: 10.1029/2001JB000323.
Roering, J.J., Perron, J.T., and Kirchner, J.W., 2007, Functional relationships between denudation and hillslope form and relief: Earth and Planetary Science Letters, v. 264, p. 245–
258, doi: 10.1016/j.epsl.2007.09.035.
Strudley, M.W., Murray, A.B., and Haff, P.K., 2006, Regolith-thickness instability and the
formation of tors in arid environments: Journal of Geophysical Research, v. 111, doi:
10.1029/2005JF000405.
CONCLUSIONS
In this paper, we presented a new method for estimating regolith thickness, slope gradient, and erosion rates as a function of basic tectonic and
climatic input data. The model provides a more complete understanding
of the controlling factors that influence the formation of cliff-dominated
versus convex to concave slopes. Model results highlight the dynamic
complexity inherent in landscapes governed by a humped soil production function. Application of the model framework is currently limited
by the relative paucity of data on regolith thickness and long-term erosion rates. The model framework would therefore greatly benefit from
additional calibration and test data, particularly in humid climates, areas
of persistent snow cover, and in study sites with nongranitic lithologies.
80
MANUSCRIPT RECEIVED 14 JULY 2008
REVISED MANUSCRIPT RECEIVED 21 OCTOBER 2008
MANUSCRIPT ACCEPTED 29 OCTOBER 2008
Printed in the USA
www.gsapubs.org | Volume 1 | Number 2 | LITHOSPHERE