Equilibrium Form of Horizontally Retreating, Soil-Mantled
Hillslopes: Model Development and Application to a
Groundwater Sapping Landscape
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Perron, J. Taylor, and Jennifer L. Hamon. “Equilibrium Form of
Horizontally Retreating, Soil-mantled Hillslopes: Model
Development and Application to a Groundwater Sapping
Landscape.” Journal of Geophysical Research 117.F1 (2012).
©2012. American Geophysical Union
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http://dx.doi.org/10.1029/2011jf002139
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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117, F01027, doi:10.1029/2011JF002139, 2012
Equilibrium form of horizontally retreating, soil-mantled
hillslopes: Model development and application
to a groundwater sapping landscape
J. Taylor Perron1 and Jennifer L. Hamon1
Received 30 June 2011; revised 23 January 2012; accepted 24 January 2012; published 20 March 2012.
[1] We present analytical solutions for the steady state topographic profile of a
soil-mantled hillslope retreating into a level plain in response to a horizontally migrating
base level. This model applies to several scenarios that commonly arise in landscapes,
including widening valleys, eroding channel banks, and retreating scarps. For a sediment
transport law in which sediment flux is linearly proportional to the topographic slope,
the steady state profile is exponential, with an e-folding length, L, proportional to the ratio
of the sediment transport coefficient to the base level migration speed. For the case in
which sediment flux increases nonlinearly with slope, the solution has a similar form
that converges to the linear case as L increases. We use a numerical model to explore the
effects of different base level geometries and find that the one-dimensional analytical
solution is a close approximation for the hillslope profile above an advancing channel tip.
We then compare the analytical model with hillslope profiles above the tips of a
groundwater sapping channel network in the Florida Panhandle. The model agrees closely
with hillslope profiles measured from airborne laser altimetry, and we use a predicted
log linear relationship between topographic slope and horizontal distance to estimate L for
the measured profiles. Mapping 1/L over channel tips throughout the landscape reveals
that adjacent channel networks may be growing at different rates and that south facing
slopes experience more efficient hillslope transport.
Citation: Perron, J. T., and J. L. Hamon (2012), Equilibrium form of horizontally retreating, soil-mantled hillslopes: Model
development and application to a groundwater sapping landscape, J. Geophys. Res., 117, F01027, doi:10.1029/2011JF002139.
1. Introduction
[2] Channel networks drive the evolution of most continental landscapes, but the vast majority of the land surface
consists of hillslopes. Hillslope form reflects the processes
that produce and transport sediment, the physical and chemical properties of the underlying material, and boundary conditions that induce relative changes in elevation. Hillslope
topography can therefore be a sensitive indicator of the
processes that drive mass transport over Earth’s surface. In
addition, because hillslopes respond to channels that form
their base level, hillslope form can also record channel network development.
[3] Most studies exploring these relationships have
focused on vertical rates of base level change [e.g., Kirkby,
1971; Hirano, 1975; Fernandes and Dietrich, 1997]. This
is a reasonable approximation in many scenarios involving
erosional processes driven by gravity, but there are also
settings in which hillslopes experience dominantly horizon1
Department of Earth, Atmospheric and Planetary Sciences,
Massachusetts Institute of Technology, Cambridge, Massachusetts, USA.
Copyright 2012 by the American Geophysical Union.
0148-0227/12/2011JF002139
tal base level migration. Examples include bank erosion
by rivers that migrate or widen faster than they incise vertically [e.g., Hooke, 1980; Lawler, 1993], retreating coasts,
escarpments and cliffs [e.g., Gilbert, 1928; Koons, 1955;
Anderson et al., 1999; Hanks, 2000], headward advance
of channel networks [e.g., Dunne, 1980], and the lateral
expansion of karst features. Situations such as these present opportunities to test the predictions of hillslope transport laws, constrain rates of sediment transport and hillslope
development, and examine spatial trends within evolving landscapes.
[4] Horizontally retreating slopes figured prominently in
some early studies of landscape evolution. Penck’s [1924]
conceptual model of parallel slope retreat, in which slopes
migrate laterally while maintaining a constant form, and
Gilbert’s [1928] observations of scarp retreat driven by
base level migration challenged Davis’s [1899] notion of
the inevitable relaxation of topography through slope
decline. King’s [1953] studies of escarpments in South
Africa involved models of lateral slope retreat that were
similar to Penck’s. Later studies of bedrock slopes in arid
environments discussed evidence of slope retreat, and proposed geometric models of landform development that
emphasized the role of stratified rock [Koons, 1955;
Oberlander, 1977, 1989].
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PERRON AND HAMON: RETREATING HILLSLOPES
[5] Some of the earliest quantitative models of landform
development focused on hillslope form [e.g., Culling, 1960;
Scheiddeger, 1961; Kirkby, 1971; Hirano, 1975; Ahnert,
1976]. These studies sought to relate hillslope topographic
profiles to sediment transport expressions through conservation of mass. But most models that have incorporated base
level effects have restricted their analyses to a base level
with a fixed horizontal position. This includes both the wellknown parabolic solution for steady state, sediment-mantled
hillslopes evolving in response to a lowering base level with
a sediment flux linearly proportional to the local topographic
gradient [e.g., Culling, 1963; Kirkby, 1971; Hirano, 1975]
and analogous solutions for transport laws in which sediment flux increases nonlinearly with slope [Roering et al.,
2007; Perron, 2011].
[6] A few studies have emphasized the importance of
considering both vertical and lateral components of erosion
at the hillslope scale [e.g., Mudd and Furbish, 2005; Stark,
2010], and physically based expressions for hillslope forms
produced by horizontal base level migration have been
proposed for linear sediment transport laws [e.g., Hanks,
2000]. Yet, unlike the case of vertical base level change,
these expressions have not been widely tested through
comparisons with field sites. The goals of this paper are
to derive expressions for the steady form of horizontally
retreating hillslopes subject to linear and nonlinear sediment
transport laws, test the ability of these expressions to predict
hillslope form in a field site where horizontal base level
migration is known to occur, and demonstrate their utility for
identifying spatial patterns of channel network growth
recorded in the surrounding hillslopes.
[7] In the sections that follow, we consider the case of
soil-mantled slopes evolving in response to a base level that
advances horizontally through an otherwise level plateau.
In section 2, we derive one-dimensional analytical expressions for the steady state topographic profiles of slopes
on which soil flux depends either linearly or nonlinearly
on the topographic gradient. In section 3, we compare these
expressions with a numerical model, and show how a measured hillslope profile can be used to estimate the ratio of the
transport coefficient to the base level migration speed, even
if the migrating base level takes the form of a point, such as
an advancing channel tip, rather than a linear boundary
perpendicular to the transport direction. We then use the
analytical solutions in section 4 to estimate this ratio for
many channel tips in a valley network formed by groundwater sapping in the Florida Panhandle, yielding a map that
reveals spatial trends in channel growth rates and hillslope
transport coefficients.
2. Analytical Model of a Retreating Hillslope
[8] We consider a one-dimensional hillslope that retreats
because of horizontal migration of a base level with fixed
elevation (Figure 1). The coordinate system moves with the
base level at a horizontal speed v in the positive x direction,
with the base level always located at x = 0, z = 0. The hillslope is assumed to be retreating into a flat, level plain with
an elevation z∞ that extends infinitely in the positive x
direction. The hillslope surface rises in the positive x direction, approaching z∞ as x → ∞. Soil or sediment is transported downslope with a volume flux per unit width q(x).
F01027
We seek an equilibrium topographic profile, such that z = z(x),
independent of time. To maintain an equilibrium profile,
conservation of mass requires that the mass flux at x equals
the total mass flux from upslope as the hillslope erodes into
the plain,
rs qðxÞ ¼ ðz∞ zÞ
rv;
ð1Þ
is the average
where rs is soil or sediment bulk density and r
bulk density of the material in the plain between z and z∞.
To derive an expression for an equilibrium profile, a transport
law relating q to the topography is required. We consider two
cases for soil-mantled hillslopes: one in which q is linearly
proportional to slope, and another in which q increases nonlinearly with slope.
2.1. Linear Transport Law
[9] On soil-mantled hillslopes with low to moderate gradients, it has been proposed from simple arguments [Culling,
1960, 1963, 1965] and demonstrated through field measurements [Monaghan et al., 1992; McKean et al., 1993;
Small et al., 1999] that soil volume flux per unit width is
linearly proportional to, and opposite in direction from, the
topographic gradient,
qðxÞ ¼ D
dz
;
dx
ð2Þ
where D is a transport coefficient. Although recent studies
suggest that equation (2) may be at best an approximation
for the true pattern of mass transport [Heimsath et al., 2005;
Furbish et al., 2009; Foufoula-Georgiou et al., 2010; Tucker
and Bradley, 2010], numerous studies of hillslope evolution
and topography have shown it to be a useful approximation.
Substituting equation (2) into equation (1) and solving for
dz/dx yields an expression for slope as a function of elevation above the base level,
v
r
dz
¼
ðz∞ zÞ:
dx rs D
ð3Þ
, rs and D are
Separating variables and assuming that r
independent of x and z yields
Z
v
r
dz
¼
z∞ z r s D
Z
dx;
ð4Þ
which we integrate to obtain
lnðz∞ zÞ ¼
r v
x þ C;
rs D
ð5Þ
where C is an integration constant. The boundary condition
z(0) = 0 gives C = ln z∞. Using this value and solving
equation (5) for the normalized steady state elevation profile
z/z∞ gives
z
¼ 1 ex=L ;
z∞
ð6Þ
where the length scale L ¼ ðrs DÞ=ð
r vÞ. Hillslopes for which
L is small (rapid retreat or small transport coefficient) have
steep slopes that rapidly approach z∞, whereas hillslopes for
which L is large (slow retreat or large transport coefficient)
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PERRON AND HAMON: RETREATING HILLSLOPES
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flux q increases nonlinearly with the topographic gradient.
Several expressions have been proposed, but the most
commonly used is that proposed by Andrews and Bucknam
[1987] and Roering et al. [1999], in which | q| → ∞ as S
approaches a critical slope Sc,
qðxÞ ¼
dz
K dx
dz
2 ;
1 j dx j=Sc
ð10Þ
with Sc 1. Substituting equation (10) into equation (1)
yields
Figure 1. Schematic diagram of a hillslope retreating into a
level plain with elevation z∞ because of horizontal migration
of a base level (point b) at a speed v. The coordinate system
moves with the base level, such that point b is always at the
origin. At steady state, the point at (x, z) must convey a volume flux per unit width, q, that is proportional to (z∞ z)v.
have gentler slopes that gradually approach z∞. Equation (6)
is analogous to solutions proposed for depositional landforms such as prograding deltas [Kenyon and Turcotte,
1985] and foreland basins adjacent to thrust belts [Pelletier,
2007], and is identical to the analytical solution of Hanks
[2000, equation (25)] for the steady form of a retreating escarpment.
[10] When comparing this predicted topographic profile to
a measured profile, it is desirable to avoid estimating z∞
directly. The model profile only approaches z∞ far from the
base level, and in natural topography, the assumption of a
smooth, nearly level surface will usually break down much
closer to the steep portion of the profile. There are two
simple approaches for determining z∞ indirectly that also
provide estimates of L. First, defining S = dz/dx, equation (3)
can be written
z∞ z
S¼ ;
L L
ð7Þ
which predicts a linear relationship between slope and elevation, the slope of the relationship being 1=L ¼ ð
rvÞ=ðrs DÞ.
Second, differentiating equation (6) with respect to x yields
S¼
z∞ x=L
;
e
L
2 !
v
r
dz
dz
1
¼
ðz∞ zÞ 1 :
dx rs K
dx Sc2
A solution to equation (11), which gives a dimensionless
steady state hillslope profile, is
z
LSc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼1
W ðFÞð2 þ W ðFÞÞ;
z∞
2z∞
z∞ x
;
L L
f ¼ W ðfÞeW ðfÞ ;
F¼
1
4a
2x
exp
1
;
L
LSc 2 L
2.2. Nonlinear Transport Law
[11] If hillslope gradients are sufficiently steep, mechanical arguments [Andrews and Bucknam, 1987; Roering et al.,
1999], laboratory experiments [Roering et al., 2001], and
field observations [Anderson, 1994; Pierce and Colman,
1986; Roering et al., 1999; Gabet, 2000] indicate that the
ð14Þ
with
2 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13
2
2
LSc2 4 @
2z∞
2z∞ A5
A
ln L
1þ
a¼
1 ⋅ exp@ 1 þ
:
4
LSc
LSc
ð15Þ
[12] As with the linear transport law, we seek a way of
comparing the profile predicted by the nonlinear transport
law with a measured topographic profile that does not
require a direct estimate of z∞. The simplest approach is to
differentiate equation (11) with respect to x, yielding
2
2
S
ð
S=S
Þ
1
c
d z
1
¼
:
dx2
L ðS=Sc Þ2 þ 1
2
which predicts a linear relationship between the logarithm of
slope and horizontal distance, with the slope of the relationship
again being 1/L. Once L is known, z∞ can be determined from
the intercept of equation (7) or (9).
ð13Þ
and the quantity F is
ð8Þ
ð9Þ
ð12Þ
where L ¼ ðrs KÞ=ð
r vÞ, W is the Lambert W function,
defined by
which can be cast as a linear equation,
ln S ¼ ln
ð11Þ
ð16Þ
This predicts a linear relationship between the second
derivative of hillslope elevation and the quantity involving
slope on the right-hand side, with the slope of the relationship being 1/L. Equation (16) is more flexible than
equation (7) or (9) because it allows for the potentially
nonlinear character of the transport law, but it has the disadvantages that it requires measurements of concavity,
which are typically more uncertain than measurements of
slope, and requires an estimate of Sc. Note that as S/Sc → 0,
equation (16) reduces to the simple form
3 of 18
d2z
S
¼ ;
dx2
L
ð17Þ
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PERRON AND HAMON: RETREATING HILLSLOPES
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Figure 2. Steady state profiles of retreating hillslopes with z∞ = 20 m and Sc = 1 for different values of L
in (a) dimensionless coordinates and (b) dimensional coordinates. In Figure 2a, the solution for the linear
transport law (equation (6), solid black line) is the same for all L. The solution for the nonlinear transport
law (equation (12), dashed gray lines) converges to the solution for the linear law as L increases.
which is the same result obtained for the linear transport law
by differentiating equation (8).
2.3. Predicted Hillslope Forms
[13] Figure 2 compares steady state hillslope profiles for
the linear and nonlinear transport laws. In dimensionless
form (Figure 2a and equation (6)), the shape of the solution
for the linear law is independent of L. In contrast, the shape
of the solution for the nonlinear law depends on L (and on z∞
and Sc). For small L, which would correspond to a rapidly
retreating base level or a small soil transport coefficient, the
profile approaches an angle of repose slope, with a nearly
straight lower section with a gradient slightly less than Sc,
and a narrow concave-down section near z = z∞. For large L,
which would correspond to a slowly retreating base level
or a large soil transport coefficient, the profile approaches
the solution for the linear transport law, as implied by
a comparison of equations (3) and (11), which converge as
S/Sc → 0. The reason for the convergence of the solutions is
more apparent in dimensional form (Figure 2b): both solutions have gentler slopes for larger L, so the nonlinear
transport effects are less important. This transition between
hillslope forms predicted by the linear and nonlinear transport laws is qualitatively similar to the case of vertical
base level change analyzed by Roering et al. [2007] and
Perron [2011].
3. Numerical Model
3.1. Model Description
[14] To test whether the analytical solutions in section 2
can provide a useful description of natural hillslopes that
deviate from a strictly one-dimensional form, we created a
two-dimensional numerical model of a retreating hillslope.
The model solves the equation
rs
∂z
∂z
v ;
þ r ⋅ ðrs qÞ ¼ r
∂t
∂x
ð18Þ
where q, the vector flux, is given by a two-dimensional
version of either the linear transport law, equation (2), or the
nonlinear transport law, equation (10). The model coordinate
system moves with the hillslope’s base level, and therefore
the advection term on the right-hand side of equation (18),
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PERRON AND HAMON: RETREATING HILLSLOPES
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Figure 3. Steady state numerical solutions for a hillslope retreating in response to (a) a migrating linear
base level and (b) an advancing channel tip. Both solutions use the nonlinear transport law with L = 10 m,
z∞ = 20 m, Sc = 1, Dx, Dy = 5 m, and Dt = 100 years. (c and d) Profiles through the numerical solutions
(black points) compared with the one-dimensional analytical solution (equation (12), black line).
which causes the solution to shift in the negative x direction
at a speed v, is equivalent to a base level that migrates in the
positive x direction in a fixed reference frame. The elevation
of the positive x boundary is fixed at z∞, which assumes that
it is sufficiently far from the hillslope’s base level that z ≈ z∞.
We use the implicit method of Perron [2011] to solve
equation (18) forward in time until the topography reaches
a static steady state.
3.2. Comparison of Numerical and Analytical Solutions
[15] We investigated two cases. In the first case, the y
boundaries are periodic, and a base level with a fixed elevation of zero covers half of the grid. This produces a onedimensional hillslope that does not vary in the y direction
(Figure 3a). A profile through this solution in the x direction
is identical to the one-dimensional analytical solutions in
section 2 (Figure 3c). In the second case, the y boundaries
have fixed elevations of z∞, the negative x boundary is free,
and the base level consists of a channel tip with a fixed
elevation of zero extending into the grid in the x direction
from the negative x boundary. This case is intended to simulate the topography surrounding a horizontally advancing
channel tip, and the solution consists of concave-down
hillslopes that wrap around the channel tip (Figure 3b).
Because of this convergent topography, points near the channel tip must convey a larger flux at steady state than in the onedimensional case, and the hillslope profile in Figure 3b is
therefore steeper and more curved than the profile in Figure 3a.
Despite this difference, a profile through the two-dimensional
topography deviates only slightly from the one-dimensional
analytical solution (Figure 3d). The steady state solutions for
both cases are independent of the initial conditions.
[16] To investigate the effect of contour curvature on
L values inferred from analysis of topographic profiles,
we calculated steady state numerical solutions using the
boundary conditions in Figure 3b for a range of L, and used
the expressions derived in section 2 to determine apparent
values of L from profiles through the numerical solutions.
All numerical calculations used the nonlinear transport
r ¼ 1 , K = 0.01 m2/yr, z∞ = 20 m, Sc = 1,
law with rs =
and Dx, Dy = 5 m. The speed v ranged from 0.1 mm/yr to
1 mm/yr, such that 10 m ≤ L ≤ 100 m, a range that encompasses most of the hillslopes at the study site investigated
in section 4.
[17] We extracted a topographic profile from each
numerical solution at the location shown in Figure 3b, and
approximated dz/dx and d2z/dx2 with second-order finite
differences. We then used iteratively reweighted least
squares regression to fit the relationships in equations (9)
and (16) to the model solution, and determined L from the
regression slopes. Figure 4 compares the L values inferred
from the regression with the actual values. Because the
analytical solutions neglect the effect of convergent topography illustrated in Figure 3, both systematically underestimate L. However, both expressions provide estimates of L
that are within 13% of the true value over the range we
tested, and within 2% for L = 10 m. Moreover, because the
solutions for the two transport laws are similar for L ≳ 10 m
(Figure 2), the regression based on the linear transport law
estimates L with accuracy comparable to the regression
based on the nonlinear law (and even slightly better, because
the linear law predicts a steeper, more curved profile that
mimics the two-dimensional effect). For sites in which
L falls in the range tested here, it is therefore possible to
obtain good estimates of L from hillslope profiles by using
equation (9), which avoids the potentially noisy measurements of profile curvature required for equation (16). While
the specific errors plotted in Figure 4 do not apply to all of
5 of 18
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PERRON AND HAMON: RETREATING HILLSLOPES
Figure 4. Comparison of L values inferred from regression
analysis of topographic profiles through numerical solutions
like that in Figure 3b. Values marked with circles were
determined from equation (16) for the nonlinear transport
law, and those marked with crosses were determined from
equation (9) for the linear transport law.
our results, the magnitudes should be similar, and our
interpretations in sections 4 and 5 are based largely on relative measures of L, which are not significantly influenced
by these systematic errors.
4. Application to Sapping Valley Networks
of the Florida Panhandle
[18] To test the analytical model, and to use the associated
predictions to examine spatial variability in v or K, we
sought a landscape with a large number of measurable hillslope profiles subject to boundary conditions similar to those
assumed in section 2: a base level migrating horizontally at a
constant rate into a level plain. We also sought a site with
minimal spatial variability in transport processes or the
mechanical characteristics of the substrate, such that hillslope form is likely to be a sensitive indicator of rates of
sediment transport and base level migration. We selected a
site in the Florida Panhandle where groundwater sapping
channel networks have created hundreds of hillslopes that
satisfy most of these criteria.
4.1. Site Description
[19] The Western Highlands of the Florida Panhandle are
built from the highly permeable Plio-Pleistocene sands of
the Citronelle Formation [Sellards and Gunter, 1918]. The
highlands surface rises a few tens of meters above sea level,
slopes gently toward the Gulf Coast, and is locally very
planar. In some locations adjacent to rivers or water bodies,
the subdued highlands topography is interrupted by steepsided valley networks containing perennial, spring-fed
streams. The valley networks typically have straight main
F01027
stems and short tributaries with nearly orthogonal junction
angles. Tributary valleys terminate abruptly upstream in
steep, roughly semicircular headwalls known colloquially as
“steepheads” [Means, 1981]. The near absence of channel
incision upslope of the valley heads and the alignment of
major valleys with the average direction of groundwater
flow led early investigators to conclude that the valley networks were incised by groundwater sapping at the spring
sites. Sapping valleys are thought to form through a positive
feedback in which the focusing of flow toward a spring leads
to accelerated erosion where the spring emerges from the
ground, which in turn advances the channel tip and causes
the groundwater flow to converge more strongly [Dunne,
1980; Howard and McLane, 1988; Howard, 1988]. Early
studies suggested that sapping valleys in Florida may also
have been influenced by low-permeability clay beds in the
Citronelle Formation that direct groundwater flow and
enhance this focusing effect, or by indurated layers near the
surface that inhibit erosion [Sellards and Gunter, 1918], but
subsequent studies have found no evidence of such layers
[Schumm et al., 1995; Abrams et al., 2009].
[20] We focused on a cluster of sapping valley networks
incised into bluffs on the east side of the Apalachicola River
near Bristol, Florida (Figure 5). Several lines of evidence,
including surveys of groundwater table elevations [Abrams
et al., 2009; Petroff et al., 2011], analyses of valley longitudinal profiles [Devauchelle et al., 2011], cross sections
[Lobkovsky et al., 2007], and head shapes [Petroff et al.,
2011], comparisons with laboratory experiments [Howard,
1988; Lobkovsky et al., 2007], and measurements of channel bifurcation angles [Petroff, 2011] support the inference
that the valley networks were formed by dominantly horizontal migration of groundwater sapping sites through the
sandy bluffs. As the springs at the tips of the channel network have advanced through the nearly level surface that
sits approximately 50 m above the Apalachicola River,
they have created boundary conditions very similar to the
scenario in Figure 1. The case of perennial, spring-fed
channels bounded by highly permeable slopes that experience little overland flow is also consistent with the sharp
boundary between hillslope and fluvial domains assumed in
section 2. The many valley heads across the site therefore
provide an opportunity to test the analytical solutions presented in section 2, and to obtain a snapshot of the evolution of the valley network by measuring relative rates of
hillslope retreat.
[21] Although the level surface of the bluffs has been
clear-cut, the sapping valleys remain forested, with conifers
and a few hardwoods dominating the higher elevations of
the valley walls, and abundant magnolia, beech, and evergreen shrubs in the wetter, more densely vegetated lower
elevations [Means, 1981]. Soil transport appears to occur
through a combination of bioturbation, small slumps and
raveling events, and, less commonly, shallow landslides.
Hillslopes at valley heads are concave down and typically
grade smoothly into the level plain (Figure 6), which is
qualitatively consistent with the analytical model.
4.2. Topographic Measurements and Determination
of L
[22] To enable a quantitative comparison, we measured
hillslope elevation profiles above channel tips throughout
6 of 18
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PERRON AND HAMON: RETREATING HILLSLOPES
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Figure 5. Aerial image of groundwater sapping valley networks in the Apalachicola Bluffs near Bristol,
Florida. Sapping valleys are easily identified by denser, darker green vegetation. The black rectangle
marks the area shown in Figure 9. The Apalachicola River, at the far left, flows south. Image is from
the National Agriculture Imagery Program.
the valley networks. An airborne laser altimetry map of the
Bristol site was produced by the National Center for Airborne Laser Mapping (NCALM). The raw point cloud was
filtered to remove laser returns from vegetation and gridded
to a horizontal point spacing of 1 m. We used elevation and
slope maps to estimate the locations of spring sapping sites
at the heads of valleys. A spring site was identified as a
break in slope at the base of a headwall. From each spring,
we drew a linear transect extending upslope in a direction
parallel to the valley until the elevations reached a nearly
constant value, and then interpolated elevations along each
profile at a spacing of approximately 1 m. We inspected each
profile and removed concave-up portions at the downslope
end, which can result from small errors in our estimate of the
spring location, and portions at the upslope end where the
surface sloped away from the spring, which were usually
associated with human modification of the topography or
deviations of the plain from a perfectly level surface. The
coordinates of the first point in the profile were set to x = 0,
z = 0, and the other coordinates were measured relative to
this point. The profiles were then examined for obvious
signs that the hillslopes were not in a steady state, such as
major breaks in slope, inflections in curvature, or large
bumps or dips. Profiles containing such features were discarded. After this screening, 201 profiles remained for
analysis, including most of the major valley heads visible in
the laser altimetry as well as many smaller valleys that
extend only a short distance from the main valleys. Locations, directions, and lengths of the transects are listed in
Table 1.
[23] The expressions derived in section 2 allow us to
obtain an estimate of L, and therefore K/v, for each channel
tip. We used equation (9) to determine the best fit value of L.
Although equation (9) is strictly only valid for the linear
7 of 18
PERRON AND HAMON: RETREATING HILLSLOPES
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Table 1. Hillslope Profile Data
Profile
Eastinga (m)
Northinga (m)
Length (m)
Directionb (deg)
Lc (m)
Estimatedd R2
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
693413
695120
694644
696138
698157
694068
694528
694349
695179
694541
697169
694108
695418
696985
694040
695635
695566
693798
695210
694491
694872
694295
696927
694804
694015
698106
697051
695327
696871
695082
695202
695720
695594
697719
698856
698787
693788
696354
693557
696989
695027
696839
695644
695251
696812
696111
696301
693922
696477
695557
695348
695561
697194
694993
694928
695768
695239
695456
694646
694154
696361
693510
695314
694016
695191
693799
694148
694997
693795
695010
694435
693758
3372435
3374711
3375406
3374388
3373164
3374435
3374268
3375375
3372476
3375370
3374982
3374144
3374423
3375821
3375361
3375100
3372569
3375513
3374715
3375394
3374330
3375394
3374693
3375549
3375891
3374153
3375802
3374477
3374756
3374730
3372547
3375199
3374415
3374768
3374145
3374131
3375839
3375109
3375313
3375299
3373018
3375847
3375168
3372494
3374824
3374999
3373825
3375443
3375225
3372417
3373397
3372629
3374020
3372948
3373052
3372977
3375761
3372660
3374983
3373054
3374018
3372132
3373416
3373706
3375097
3372739
3374452
3374386
3375804
3373141
3374306
3372767
126.4
39.8
112.5
111.4
80.7
66.4
80.4
119.3
62.8
154.5
114.3
126.3
112.6
176.7
127.3
118.8
57.2
121.9
38.7
124.6
118.5
112.5
105.6
96.2
71.0
75.5
168.7
104.5
81.7
49.9
87.1
117.6
113.2
155.6
109.3
85.8
49.8
141.9
117.6
87.1
141.2
145.8
115.1
74.0
113.2
156.8
117.7
97.1
165.1
77.0
71.0
103.8
150.4
117.6
120.2
107.5
96.5
108.7
107.4
78.8
85.8
150.6
85.9
121.8
95.0
102.4
170.6
157.4
58.1
115.9
111.8
101.8
356.1
0.0
325.8
272.7
1.6
276.0
291.7
286.8
160.0
269.3
248.7
276.3
258.8
221.8
268.3
169.1
3.8
267.4
183.3
262.5
268.8
203.8
311.1
302.3
265.6
260.5
294.7
180.0
206.5
221.4
137.1
32.4
254.9
249.2
325.9
269.1
190.4
128.7
289.7
131.7
16.9
256.0
155.1
49.2
225.0
89.5
215.0
218.2
168.8
14.7
260.2
358.6
127.7
238.4
202.5
309.2
237.3
154.1
163.1
287.8
139.3
47.2
229.7
188.0
13.7
230.0
292.1
261.1
180.0
0.1
270.0
202.4
3.9 0.4
5.6 0.2
6.8 0.5
7.1 0.4
7.5 0.7
8.0 0.6
8.1 0.4
8.2 0.4
10.4 0.6
10.5 0.8
10.5 1.0
10.8 0.7
10.8 0.5
10.9 0.5
11.2 0.9
11.4 1.0
11.4 0.9
11.5 0.4
11.5 1.0
11.6 0.6
11.7 0.5
12.0 0.7
12.0 0.8
12.3 0.5
12.4 1.6
12.5 0.9
12.5 1.2
12.6 0.4
12.7 0.6
12.7 0.7
12.7 0.6
12.8 0.9
12.9 0.8
13.1 0.8
13.3 1.2
13.5 0.6
14.0 0.9
14.2 0.6
14.2 0.6
14.3 1.1
14.5 0.9
14.6 0.5
14.7 0.6
14.7 1.0
14.8 0.6
14.8 1.0
14.9 0.7
15.0 0.5
15.0 0.7
15.2 1.1
15.4 1.2
15.6 1.6
15.6 1.2
15.8 1.0
16.0 0.9
16.3 1.7
16.6 2.3
16.7 0.7
16.9 0.8
17.0 0.8
17.0 1.3
17.2 0.8
17.5 0.7
17.7 1.0
17.9 0.9
17.9 1.3
18.0 0.7
18.1 0.9
18.2 1.0
18.3 2.1
18.5 0.9
18.6 1.0
0.89
0.98
0.91
0.92
0.86
0.87
0.94
0.91
0.94
0.85
0.78
0.88
0.93
0.93
0.79
0.83
0.84
0.95
0.88
0.93
0.93
0.88
0.91
0.92
0.64
0.86
0.78
0.95
0.92
0.94
0.91
0.86
0.86
0.87
0.81
0.90
0.90
0.93
0.92
0.83
0.86
0.95
0.92
0.95
0.92
0.82
0.95
0.94
0.92
0.79
0.84
0.68
0.82
0.86
0.88
0.76
0.57
0.91
0.87
0.93
0.80
0.90
0.94
0.85
0.90
0.78
0.92
0.87
0.90
0.64
0.87
0.87
8 of 18
PERRON AND HAMON: RETREATING HILLSLOPES
F01027
F01027
Table 1. (continued)
Profile
Eastinga (m)
Northinga (m)
Length (m)
Directionb (deg)
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
694902
694778
697632
695978
697263
696294
695678
696431
693702
695378
695570
695100
695981
695889
695222
695914
695678
694019
695577
694570
698322
696529
695757
694305
695616
697473
696924
696458
694184
695382
697321
695492
693658
697096
694040
694528
695543
694615
693855
693739
693857
695769
693751
694955
694260
693794
697731
695575
693874
697127
695015
695645
694367
695886
693941
694242
694866
696850
694385
693746
697521
694300
695708
695519
694927
695970
694984
693693
693970
695511
695132
695052
3373761
3374417
3374033
3375009
3375024
3374595
3374493
3374761
3374429
3375682
3373064
3374707
3375002
3375819
3374702
3374497
3374497
3374874
3372707
3374304
3374226
3373944
3375866
3374302
3372690
3375122
3372927
3374788
3374452
3373387
3375967
3372696
3374855
3375838
3374219
3373081
3374720
3373240
3372760
3372815
3374414
3373883
3374438
3372405
3375742
3374785
3374846
3373283
3374345
3375107
3374714
3374791
3375926
3373977
3374276
3375835
3373431
3373883
3372832
3375004
3374042
3375528
3374802
3372707
3374707
3373089
3373890
3374686
3372641
3373884
3374705
3375161
88.5
144.1
161.4
72.1
74.3
79.4
119.8
116.1
85.4
136.8
128.1
95.3
135.0
199.2
52.8
109.3
99.1
119.9
79.6
90.8
165.3
107.7
200.9
94.1
80.0
87.8
108.0
122.8
64.2
127.4
114.5
95.8
54.2
87.5
80.2
117.4
65.2
109.1
70.4
93.1
45.8
85.5
105.0
92.2
89.0
130.3
148.9
124.2
91.3
116.5
90.1
96.8
85.7
124.9
105.1
76.3
50.6
150.6
91.9
102.3
88.0
102.5
100.1
103.6
94.8
158.5
163.3
61.0
123.1
100.0
101.3
84.1
154.2
265.9
358.5
105.5
329.9
19.9
345.1
279.5
273.5
310.8
183.2
256.8
106.0
272.6
275.7
220.2
37.6
310.8
86.5
322.5
276.4
358.8
185.7
272.1
5.2
253.6
273.6
344.3
289.0
279.8
189.5
114.4
141.1
331.7
204.3
306.6
274.3
305.1
264.1
211.2
188.8
175.3
275.7
173.9
179.8
325.9
168.1
187.9
219.2
213.9
278.9
283.8
276.1
85.6
241.3
180.0
87.2
116.5
87.0
156.8
95.8
173.0
283.0
85.2
282.5
352.8
93.4
178.6
239.4
352.0
275.9
82.9
9 of 18
Lc (m)
18.8 18.8 18.8 18.9 18.9 18.9 19.0 19.3 19.7 19.8 20.0 20.0 20.0 20.0 20.5 20.5 20.5 20.9 21.0 21.1 21.1 21.2 21.2 21.3 21.3 21.8 22.3 22.3 22.4 22.5 22.5 22.6 22.7 22.8 22.8 22.9 22.9 23.0 23.0 23.1 23.2 23.2 23.4 23.6 23.8 24.1 24.4 24.9 25.2 25.6 26.1 26.1 26.2 26.3 26.4 26.5 26.7 26.7 26.8 26.9 26.9 27.3 27.4 27.4 27.5 27.8 28.4 28.5 28.7 28.9 29.0 29.2 1.2
0.7
1.9
1.1
1.3
1.5
1.0
1.3
0.8
2.0
1.4
1.3
0.7
2.7
1.2
0.9
1.0
1.2
1.7
0.8
1.3
1.0
1.8
1.4
2.1
1.2
1.6
1.1
0.8
1.3
1.1
1.0
1.0
1.0
0.9
1.1
1.6
0.8
1.1
1.0
1.1
0.9
1.2
2.1
2.4
0.7
1.2
1.5
1.2
1.0
0.9
1.0
1.2
0.7
1.6
2.2
2.5
1.0
1.5
0.7
1.4
1.4
1.0
1.4
1.8
1.6
1.5
1.4
1.3
1.0
1.7
1.3
Estimatedd R2
0.79
0.94
0.70
0.87
0.82
0.77
0.86
0.81
0.93
0.67
0.78
0.80
0.93
0.67
0.88
0.87
0.89
0.83
0.74
0.92
0.81
0.88
0.79
0.76
0.65
0.85
0.76
0.87
0.95
0.90
0.87
0.87
0.94
0.90
0.91
0.88
0.82
0.94
0.90
0.87
0.92
0.92
0.81
0.71
0.65
0.94
0.86
0.81
0.87
0.90
0.92
0.92
0.90
0.95
0.77
0.76
0.78
0.88
0.86
0.96
0.86
0.83
0.91
0.84
0.82
0.82
0.85
0.91
0.84
0.91
0.85
0.87
PERRON AND HAMON: RETREATING HILLSLOPES
F01027
F01027
Table 1. (continued)
Profile
Eastinga (m)
Northinga (m)
Length (m)
Directionb (deg)
Lc (m)
Estimatedd R2
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
694085
696346
695153
693725
694111
695423
697213
698063
697786
693888
694806
694455
694993
694170
693785
695414
698574
696449
694307
697909
695036
693963
697369
694473
696119
697203
694955
697326
695255
693679
694079
695911
693897
694303
695276
696762
698217
694662
694282
696086
694054
697102
694058
693823
697811
698688
698728
696499
693960
695958
695287
693555
694174
696852
696415
694552
698172
3375586
3373895
3373352
3374952
3374195
3374592
3375354
3374277
3374798
3372699
3375130
3372557
3372948
3374967
3375079
3373969
3374245
3374029
3375579
3374926
3374393
3375055
3375458
3374609
3373497
3373642
3372577
3375434
3373873
3374741
3372477
3373928
3375764
3374736
3374905
3375111
3374284
3374543
3374701
3374848
3375658
3375291
3375659
3372839
3375059
3374271
3374274
3375017
3372786
3373398
3374905
3375466
3374600
3373361
3373224
3374772
3374932
98.3
147.8
112.5
114.1
67.2
109.1
95.7
124.3
106.1
94.0
70.8
154.4
122.7
63.9
120.5
126.8
157.7
107.3
100.5
97.3
108.6
94.6
107.5
48.6
96.2
175.7
106.2
86.4
213.3
92.4
166.2
150.8
97.9
93.0
102.7
103.0
81.6
100.0
122.9
79.0
117.6
97.9
98.9
101.0
123.6
97.4
71.7
95.8
128.7
116.4
93.4
118.0
128.8
133.5
126.5
123.1
102.2
25.0
183.1
326.9
158.5
5.3
194.9
133.9
180.7
327.8
228.2
123.1
0.0
280.2
1.1
88.8
80.4
254.2
66.6
173.7
298.2
338.4
53.9
90.0
94.1
276.9
315.0
115.4
118.7
128.5
176.9
189.3
11.9
15.1
149.5
124.4
30.5
288.8
277.0
153.2
276.4
82.3
99.8
48.6
33.2
190.3
258.5
262.2
299.4
45.8
323.2
75.8
24.7
91.6
24.9
247.0
98.4
248.0
29.6 2.0
29.7 0.8
30.3 1.1
30.5 1.0
30.5 1.8
31.0 0.8
31.3 1.4
31.4 1.6
32.0 2.1
32.5 1.4
32.5 2.3
32.8 4.7
33.4 3.1
33.4 1.4
33.5 1.2
33.6 2.1
33.8 1.7
33.9 2.2
34.1 1.8
34.2 2.4
34.5 1.4
35.2 1.8
35.3 2.3
36.2 2.8
36.7 1.6
36.7 2.9
36.9 4.0
38.1 2.7
38.3 2.7
39.3 1.4
39.5 2.3
39.6 1.4
41.3 1.9
44.0 3.1
45.8 3.7
46.6 5.3
47.1 4.2
47.7 2.6
49.0 2.8
49.0 4.6
51.8 3.8
52.2 3.1
52.8 4.8
54.3 2.5
55.9 3.0
58.1 3.2
58.4 4.5
62.4 4.1
64.1 3.9
68.8 5.9
70.8 6.0
71.3 4.3
73.0 5.2
73.6 5.0
75.9 8.3
107.3 7.5
149.6 26.1
0.76
0.92
0.89
0.95
0.85
0.95
0.87
0.88
0.73
0.90
0.79
0.51
0.75
0.93
0.91
0.81
0.80
0.77
0.83
0.74
0.87
0.84
0.73
0.82
0.86
0.68
0.54
0.73
0.75
0.91
0.80
0.87
0.84
0.74
0.67
0.55
0.70
0.79
0.80
0.66
0.71
0.77
0.63
0.85
0.75
0.80
0.78
0.75
0.72
0.60
0.64
0.71
0.66
0.67
0.48
0.64
0.28
a
UTM zone 16N.
Measured counterclockwise from east.
c
Mean standard error.
d
The correlation coefficient is undefined for robust regression. The estimated R2 listed here is less than or equal to the value that would be obtained from
ordinary least squares regression.
b
transport law, the exercise in Figure 4 demonstrates that this
approach can yield good estimates of L even for profiles in
which nonlinear effects are significant. Moreover, preliminary analyses not presented here indicated that the curvature
values required to apply equation (16), which is valid for the
nonlinear transport law, were sufficiently noisy that they
introduced a larger source of uncertainty in our estimates of
L than the use of equation (9).
[24] We used the procedure described in section 3.2 to
determine L for each profile. Slopes were calculated at each
point, except profile endpoints, with second-order finite
difference approximations, and 1/L was determined from the
10 of 18
F01027
PERRON AND HAMON: RETREATING HILLSLOPES
Figure 6. Photographs of hillslopes above sapping channel
tips in the Apalachicola Bluffs near Bristol, Florida.
(a) Sandy slopes more than 100 m from the channel are concave down and slope gently toward the channel. Timber
farming has altered the vegetation cover. (b) Slopes closer
to the channel are steeper. Note people for scale on the right
in the middle distance of both images.
slope of an iteratively reweighted least squares regression of
ln S against x. The standard error of the regression slope
provided an estimate of the uncertainty in 1/L. Steep sections
with nearly constant slope at the downslope ends of some of
the profiles suggested significant nonlinear transport effects,
and were excluded from the regressions. Sections with
nearly constant slopes of approximately zero at the upslope
ends of some profiles suggested that those portions of the
plain have not responded significantly to the propagating channel tips, and were also excluded. Our approach for
estimating L is conceptually similar to that of Abrams et al.
[2009], who used the radius of curvature of the valley rim as
an order-of-magnitude estimate of K/v. The expressions
derived here demonstrate the basis for the ratio they obtained
by dimensional analysis.
[25] L values for the 201 analyzed profiles are listed
in Table 1, and the frequency distribution is plotted in
Figure 7. L is lognormally distributed, with a mode of
=rs ¼ 1 , a reasonable approxiapproximately 20 m. For r
mation for the sands of the Citronelle Formation, and a
typical transport coefficient of K = 0.01 m2/yr for weakly
cohesive sediment in a humid environment [Nash, 1980;
F01027
Hanks et al., 1984; Rosenbloom and Anderson, 1994;
Fernandes and Dietrich, 1997; Small et al., 1999; Hanks,
2000], this would correspond to a modal channel tip velocity of 0.5 mm/yr, with 95% of the velocities faster than
0.17 mm/yr and 95% slower than 0.95 mm/yr. Given the
one-dimensional approximation used in our model, it is
likely that our calculations underestimate L, and therefore
overestimate channel tip velocities. However, the analysis in
section 3 suggests that this systematic error is unlikely to
exceed 10% (Figure 4). Our estimates of L may also be
biased by channel tips that are slowing down or speeding up,
resulting in hillslope profiles that are not representative of
the present-day migration speed. However, the observation
that most measured profiles follow the predicted log linear
steady state relationship between slope and horizontal distance (see section 4.3) suggests that this bias is not substantial. Another way to assess the potential for nonsteady
state profiles is to compare the response time of the hillslope profiles to the timescale for channel network growth.
Using the values of K = 0.01 m2/yr and v = 0.5 mm/yr
estimated above, the diffusion time for a hillslope with
z∞ = 20 m is z2∞/K = 40 kyr, whereas the time required to
grow a tributary with a length ‘ = 500 m (Figure 5) is ‘/v = 1
Myr. Hillslopes that experience nonlinear transport due to
fast-moving tips have an even larger effective diffusivity,
and a shorter diffusion time. We therefore expect that hillslopes respond rapidly compared with the growth of the
channel network, and that hillslope profiles will be close to a
steady state even for channel tips that are gradually slowing down or speeding up. As noted above, profiles that
contained clear deviations from the predicted steady state
form were not included in our analysis.
[26] Abrams et al. [2009] used a growth model for the
channel network based on a steady state groundwater flow
field to estimate channel tip velocities. Their reconstruction
suggests that tip velocities averaged over the entire 1 Myr
history of the channel network were as fast as 5.3 mm/yr,
but that channel tips have slowed to an average of about
0.5 mm/yr over the past 10 kyr as channel tips have
approached one another and competition for groundwater
Figure 7. Frequency distribution of L inferred from the
201 hillslope profiles in Figure 9.
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Figure 8. Comparison of analytical solutions with topographic profiles upslope of channel tips. (a) Dimensionless
elevation profiles (compare with Figure 2a). Profile 130 is
compared with the solution for the linear transport law
(equation (6), solid line), and profile 3 is compared with
the solution for the nonlinear law (equation (12) with Sc = 1.2
and z∞ = 16.5 m, dashed line). Locations and statistics for
the profiles are listed in Table 1. (b and c) Plots of ln S
against horizontal distance, with horizontal scales chosen
such that each point appears at the same horizontal position
as it does in Figure 8a. Points with negative slope are not
shown. Regression lines, which ignore constant-slope sections at the ends of the profiles, are used to determine L from
equation (9).
has intensified. Abrams et al. [2009] used a slightly higher
transport coefficient of 0.02 m2/yr to arrive at their estimates. This K value would increase our modal channel tip
velocity to 1 mm/yr. Thus, our estimates of channel tip
velocities based on hillslope profiles are slightly faster than
the estimates of Abrams et al. [2009] and an estimate of “an
inch or two per century” for other Florida sapping channels
cited by Schumm et al. [1995], but are still consistent with
these previous estimates to within a factor of two.
4.3. Comparison With Analytical Models
[27] Having estimated L for each profile, we calculated the
corresponding analytical solution from equation (6) and
compared it with the measured profile. For profiles with
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nearly straight sections at the downslope end, we also calculated the analytical solution for the nonlinear transport
law, using the value of z∞ determined from the regression
and a value of Sc = 1.2, which we estimated from the distribution of measured slopes. Figure 8a shows two endmember profiles: profile 130, with L = 26.7 1.0, which is
well described by the solution for the linear law, and profile 3, with L = 6.8 0.5, one of the shortest measured
L values, which is best described by the solution for the
nonlinear law. Both profiles are very similar to the analytical solutions. When the profiles are plotted in dimensionless coordinates, the effect of L on the profile shape
noted earlier in Figure 2a is apparent: profiles with longer
L approach z/z∞ = 1 at smaller x/L, whereas profiles with
shorter L have a straight section where S ≈ Sc that causes a
more gradual approach to z/z∞ = 1. In dimensional coordinates (not shown in Figure 8), the profile with shorter L is
steeper, and approaches z∞ at a shorter distance, as in the
dashed profiles in Figure 2b.
[28] In addition to the elevation profiles, another test of
whether the analytical solutions are a good description of the
measured profiles is the goodness of fit of the regression line
used to determine 1/L. Figures 8b and 8c show the fits for
profiles 130 and 3, respectively. Both profiles contain points
that deviate from the predicted trend at one or both ends,
as noted in section 4.2, and at slight irregularities in the
profiles, but both contain a section that is reasonably well
described by a linear trend in ln S versus x space. Table 1
lists estimated correlation coefficients, R2, for the linear fits.
Although the example profiles shown in Figure 8 are among
the “cleanest” matches to the analytical solutions, in the
sense that they contain few topographic irregularities, the
table shows that most of the other profiles have comparable
goodness of fit.
5. Discussion
5.1. Analysis of Spatial Trends in Hillslope Form
[29] The model proposed for sapping channel network
development, in which channel growth and deflection of the
groundwater flow field are linked though a positive feedback
[Dunne, 1980; Howard, 1988; Abrams et al., 2009], implies
that transient rates of channel growth may vary considerably
across a groundwater sapping landscape. Our hillslope
retreat model predicts that such variations in channel
migration rates should be recorded in the morphology of
hillslopes above channel heads. To search for such trends,
we examined the spatial distribution of the L values determined in section 4.2. Figure 9 plots the quantity
v
r
1
¼
L rs K
ð19Þ
over a shaded relief map of the sapping channel network.
=rs is nearly constant across the site, a reaProvided that r
sonable assumption for the uniform sands of the Citronelle
Formation, larger values of 1/L indicate either rapidly
advancing channel tips (fast v), or less efficient soil transport
(small K).
[30] Two main trends are apparent in Figure 9. First,
adjacent valley networks can have different distributions of
1/L. For example, Figure 10a compares distributions of 1/L
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Figure 9. Map of 1/L for hillslopes above channel tips. Black lines mark the locations of measured profiles and point in
the inferred direction of channel propagation. Points mark the channel tips and have areas proportional to 1/L and colors
proportional to log10 1/L.
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PERRON AND HAMON: RETREATING HILLSLOPES
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PERRON AND HAMON: RETREATING HILLSLOPES
Figure 10. (a) Histograms of 1/L for hillslope profiles in
the two complete valley networks in the study area (see
Figures 5 and 9). The modes of the two distributions are similar, but the northern valley network has more hillslopes with
very large 1/L. (b) Histograms of 1/L for north facing and
south facing hillslope profiles in the study area. North facing
slopes have a larger mean value of log10 (1/L) than south facing slopes (one-tailed t test, p = 0.00025).
for the two networks that lie completely within the laser
altimetry coverage. The northern network has a more lognormal distribution, with numerous hillslopes with large
values of 1/L, whereas the southern network has a distribution skewed toward smaller 1/L. It is possible that variations
in the characteristics of the sand cause a gradient in K across
the site, but we observed no evidence of such differences in
the field. The more likely explanation is that the northern
network is growing faster, perhaps because the southern
network developed earlier and its outermost channel tips
slowed sooner. The fact that the northern network also
appears to have a higher density of active channel tips,
as indicated by the larger number of measurable profiles
despite the smaller overall area of the network (Figures 9
and 10), also supports this idea.
[31] The second and more surprising trend revealed by
Figure 9 is that tips growing southward (which form
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hillslopes that face northward) have larger average 1/L.
A comparison of distributions of 1/L between north facing
and south facing hillslopes confirms this observation
(Figure 10b). This could indicate either that channels grow
southward faster than they grow northward, or that north
facing hillslopes have less efficient hillslope transport than
south facing hillslopes. The former explanation is less
plausible. Although water availability appears to have a
significant effect on channel tip propagation rates [Abrams
et al., 2009; Petroff et al., 2011], it seems unlikely that differences in evaporation on the steep slopes immediately
upslope of springs would have a large enough impact on
spring discharge to slow northward migrating channels,
because springs are fed by deeper flow from a larger area
that is mostly flat. None of the recent studies of sapping
valley networks in the Florida Panhandle have reported
structural heterogeneities in the Citronelle Formation that
would have driven asymmetric tributary growth. The orientation of the main valleys appears to be controlled by the
overall direction of groundwater flow, which is toward the
Apalachicola River and its tributaries (Figure 5). Moreover,
the valley networks lack the asymmetry of tributary lengths
one would expect to see if southern tributaries have been
growing faster for a prolonged interval.
[32] The more plausible explanation for the asymmetry is
that K varies with slope aspect. Many studies have presented
evidence that microclimates produced by differences in solar
radiation can influence the long-term efficiencies of erosional processes [e.g., Kane, 1970; Pierce and Colman,
1986; Burnett et al., 2008; Istanbulluoglu et al., 2008;
Yetemen et al., 2010], such that landscapes with strong
microclimates often have asymmetric topography [e.g.,
Bass, 1929; Emery, 1947; Dohrenwend, 1978]. In particular,
Pierce and Colman [1986] documented faster regolith creep
rates on equator-facing terrace scarps on alluvial fans than
on pole-facing scarps, the same sense of asymmetry in K
implied by our measurements. In the Apalachicola Bluffs,
we observed pronounced aspect-related differences in vegetation in some parts of the landscape, with south facing
slopes dominated by an open canopy of conifers, and north
facing slopes covered by deciduous trees with denser
undergrowth. It is not certain how this difference in vegetation would affect soil transport rates, but the likely role of
bioturbation and the potential inhibition of mass wasting by
root cohesion suggest that there could be important effects
[Dietrich and Perron, 2006]. A difference in K is also consistent with the asymmetric cross sections of east-west
trending valleys, which typically have gentler northern
side slopes and steeper southern side slopes (Figure 11).
Although valley walls may experience some retreat because
of minor seepage after they are initially created by an
advancing channel tip, the dominant effect appears to be
gradual relaxation of the initial walls, as recorded by the
progressively gentler slopes on both sides of the valleys
as they approach their junction with the Apalachicola
floodplain (Figure 11). The asymmetric cross sections are
therefore likely to be the product of different transport efficiencies rather than different seepage-driven retreat rates.
5.2. Broader Applications of Hillslope Retreat Models
[33] Although the example of sapping channels is specific
to a certain landscape, the model for retreating hillslopes
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PERRON AND HAMON: RETREATING HILLSLOPES
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Figure 11. (a) Shaded relief map showing locations of transects across the two main valleys. (b) Elevation profiles along transects in Figure 11a.
presented here has broader applicability. In principle, it can
describe any sediment-mantled slope bounding a nearly
level surface in which sediment that reaches the base of the
slope is removed. This can include plateaus being dissected
by growing drainage networks, valley walls or channel
banks, and scarps of various scales.
[34] We have demonstrated several applications of the
model. First, it can describe the equilibrium shapes of
retreating slopes (Figure 8a), and it provides a dimensionless
framework that allows comparisons among slope forms with
different absolute scales (Figures 2a and 8a). Second, it
predicts relationships among horizontal distance, elevation,
slope, and curvature that should occur at steady state
(section 2). The linear trends in Figures 8b and 8c
demonstrate one of these relationships. Third, these linear
relationships can be used to infer the ratio of the soil transport coefficient to the base level retreat speed via the length
scale L. If one of these two parameters can be constrained
independently, the hillslope profile can be used to infer the
other. Finally, we have demonstrated how variability in
hillslope form throughout a landscape can reveal spatial
trends in relative rates of landscape evolution, even if the
absolute rates are unknown. In the case of the sapping valley
networks of the Apalachicola Bluffs, the hillslope profiles
above channel tips act as relative speedometers for valley
network growth. The resulting map (Figure 9) provides
a snapshot of transient channel network evolution, and
also highlights variability in the hillslope response to this
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PERRON AND HAMON: RETREATING HILLSLOPES
F01027
that would be required to distinguish between these proposed formation mechanisms, particularly since valley networks with sapping-like characteristics have been shown
to form through other mechanisms [Lamb et al., 2006,
2007, 2008].
[36] To determine whether it is possible to distinguish
hillslopes driven by horizontal and vertical base level motion
solely on the basis of morphology, we compared the steady
state solutions derived in section 2 with a numerical model
of transient hillslope evolution in response to vertical channel incision into a level plateau. Beginning with a level
surface, the left boundary was lowered at a fixed rate,
and the elevations of the other points on the grid evolved
according to
∂z
∂q
¼ ;
∂t
∂x
Figure 12. Comparison of steady state solutions for horizontally retreating hillslopes (solid black lines) with transient numerical solutions for hillslopes responding to
vertical base level lowering (gray circles) for (a) the linear
soil transport law, equation (2), and (b) the nonlinear soil
transport law, equation (10). The dashed line is the initial
condition for the numerical model and z∞ for the analytical
solutions. Parameters used in the numerical model were
K = 0.01 m2/yr, Dx = 1 m, Dt = 10 years, and a boundary
lowering rate of 0.1 mm/yr, with profiles shown at t = 25,
50, 75, and 100 kyr (Figure 12a), and K = 0.01 m2/yr, Sc = 1,
Dx = 1 m, Dt = 10 years, and a boundary lowering rate of
1 mm/yr, with profiles shown at t = 2.5, 5, 7.5, and 10 kyr
(Figure 12b). The solid line in Figure 12a is equation (6)
with L = 25 m, and the solid line in Figure 12b is
equation (12) with L = 6 m.
forcing, such as possible microclimatic control of soil
transport coefficients.
[35] Another potential application that we have not
explored in this paper is determining whether a hillslope has
developed in response to horizontal or vertical base level
change. For example, an analysis of hillslope profiles like
that in section 4 might be a way to test whether a valley
network has been incised by vertical incision of stream
profiles or by approximately horizontal propagation of
channel tips by groundwater sapping. Such an approach
would be particularly useful in planetary settings, where
observations of erosional mechanisms are generally not
available. It has been proposed, for example, that some of
the fluvial networks on Saturn’s moon Titan, which appear
to have short tributaries with orthogonal junction angles,
formed through sapping erosion driven by subsurface flow
of liquid hydrocarbons [Tomasko et al., 2005; Soderblom
et al., 2007; Jaumann et al., 2010], whereas adjacent fluvial networks have characteristics more consistent with
surface incision driven by channelized flow [Perron et al.,
2006]. Although no topographic maps suitable for measuring high-resolution hillslope profiles currently exist for
Titan, it is useful to consider the topographic measurements
ð20Þ
where q, the volume flux per unit width, is given by either
the linear transport law, equation (2), or the nonlinear
transport law, equation (10). Equation (20) was solved using
a forward time, centered space (FTCS) finite difference
method, producing the transient hillslope profiles in
Figure 12. For each of the final profiles, we chose z∞ to be
the total lowering of the base level below the initial plateau,
and identified a value of L for which the steady state
analytical solution for a horizontally retreating hillslope
(equation (6) or (12)) closely matched the numerical solution. The comparison in Figure 12 demonstrates that the
transient profiles driven by vertical base level lowering and
the steady state solutions for horizontal retreat are difficult to
distinguish from one another for both the linear and nonlinear transport laws. In principle, the two could be distinguished by subtle differences in slope and the second
derivative, because the vertically lowering profiles do not
follow the relationships in equations (7), (8), (16) and (17).
In practice, these differences are sufficiently subtle that they
would be difficult to resolve with field data. It is possible
that the two-dimensional form of hillslopes responding to
vertical lowering of a channel head may be more easily
distinguished from hillslopes responding to a horizontally
advancing channel head, but such a comparison is beyond
the scope of this paper.
[37] Given this difficulty, a more practical way to distinguish valleys formed by horizontally advancing channel
networks from those formed by vertical incision would be to
compare downstream trends in valley cross-sectional form.
Valleys formed by horizontally advancing channel networks
should have nearly uniform depth but progressive relaxation
of valley sidewalls with downstream distance (Figure 11),
because the valley floor experiences little erosion after the
passage of advancing channel heads. In contrast, valleys
formed by vertical incision should deepen more substantially
downstream, because locations further downstream generally receive more fluid discharge and erode faster.
6. Conclusions
[38] We have presented analytical solutions for the equilibrium topographic profiles of hillslopes retreating into a
level plain in response to a horizontally migrating base level.
The profiles have an exponential form, with the hillslope
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PERRON AND HAMON: RETREATING HILLSLOPES
grading up to the elevation of the plain at a rate described by
an e-folding length L, which is proportional to the ratio of
the soil diffusivity to the base level migration speed. The
shape of the profile differs if soil transport rate increases
nonlinearly with the topographic gradient, but the solutions
converge as L increases. By transforming the analytical
solutions into linear relationships among distance, elevation,
slope and curvature, it is possible to infer L from regression
analyses of measured hillslope profiles. We compared the
analytical solutions with a numerical model of a retreating
hillslope, and found that it is possible to estimate L to within
10% of the true value even if the migrating base level
creates a two-dimensional hillslope form, such as a convergent hollow above an advancing channel tip. The growth of
groundwater sapping channel networks in the Florida Panhandle has created many hillslopes with this form, and we
used our analytical model to infer L for the hillslopes above
201 channel tips in an area adjacent to the Apalachicola
River that has been surveyed by airborne laser altimetry. The
measured profiles closely match the profiles predicted by
the analytical model, and illustrate the transition between the
forms predicted by the linear and nonlinear transport laws.
By combining the measured L values with a typical soil
diffusivity, we estimate a modal channel growth rate of
0.5 mm/yr, consistent with, but slightly faster than, previous
estimates. A map of 1/L for all the surveyed hillslopes
reveals that adjacent channel networks appear to be growing
at different rates, and that south facing slopes experience
more efficient soil transport. Beyond this specific example,
the hillslope retreat model should apply to any retreating,
sediment-mantled slope on the edge of a level surface where
material reaching the base of the slope is removed.
[40] Acknowledgments. We thank D. Rothman, A. Petroff, D. Abrams,
A. Lobkovsky, and O. Devauchelle for the invitation to participate in fieldwork
at the Apalachicola Bluffs and Ravines Preserve and for sharing the laser
altimetry map, which was acquired under contract by the National Center for
Airborne Laser Mapping (NCALM). We also thank the Nature Conservancy
for granting access to the site. This study was supported by the Massachusetts
Institute of Technology and NSF award EAR-0951672 to J.T.P.
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