Equilibrium form of horizontally retreating, soil-mantled hillslopes:...

Equilibrium form of horizontally retreating, soil-mantledhillslopes: Model development and applicationto 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 asoil-mantled hillslope retreating into a level plain in response to a horizontally migratingbase level. This model applies to several scenarios that commonly arise in landscapes,including widening valleys, eroding channel banks, and retreating scarps. For a sedimenttransport 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 ratioof the sediment transport coefficient to the base level migration speed. For the case inwhich sediment flux increases nonlinearly with slope, the solution has a similar formthat converges to the linear case as L increases. We use a numerical model to explore theeffects of different base level geometries and find that the one-dimensional analyticalsolution 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 agroundwater sapping channel network in the Florida Panhandle. The model agrees closelywith hillslope profiles measured from airborne laser altimetry, and we use a predictedlog linear relationship between topographic slope and horizontal distance to estimate L forthe measured profiles. Mapping 1/L over channel tips throughout the landscape revealsthat adjacent channel networks may be growing at different rates and that south facingslopes experience more efficient hillslope transport.

Citation: Perron, J. T., and J. L. Hamon (2012), Equilibrium form of horizontally retreating, soil-mantled hillslopes: Modeldevelopment 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 conti-nental landscapes, but the vast majority of the land surfaceconsists of hillslopes. Hillslope form reflects the processesthat produce and transport sediment, the physical and chem-ical properties of the underlying material, and boundary con-ditions that induce relative changes in elevation. Hillslopetopography can therefore be a sensitive indicator of theprocesses that drive mass transport over Earth’s surface. Inaddition, because hillslopes respond to channels that formtheir base level, hillslope form can also record channel net-work 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]. Thisis a reasonable approximation in many scenarios involvingerosional processes driven by gravity, but there are alsosettings in which hillslopes experience dominantly horizon-

tal base level migration. Examples include bank erosionby rivers that migrate or widen faster than they incise verti-cally [e.g., Hooke, 1980; Lawler, 1993], retreating coasts,escarpments and cliffs [e.g., Gilbert, 1928; Koons, 1955;Anderson et al., 1999; Hanks, 2000], headward advanceof channel networks [e.g., Dunne, 1980], and the lateralexpansion of karst features. Situations such as these pre-sent opportunities to test the predictions of hillslope trans-port laws, constrain rates of sediment transport and hillslopedevelopment, and examine spatial trends within evolv-ing 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 slopesmigrate laterally while maintaining a constant form, andGilbert’s [1928] observations of scarp retreat driven bybase level migration challenged Davis’s [1899] notion ofthe inevitable relaxation of topography through slopedecline. King’s [1953] studies of escarpments in SouthAfrica involved models of lateral slope retreat that weresimilar to Penck’s. Later studies of bedrock slopes in aridenvironments discussed evidence of slope retreat, and pro-posed geometric models of landform development thatemphasized the role of stratified rock [Koons, 1955;Oberlander, 1977, 1989].

1Department of Earth, Atmospheric and Planetary Sciences,Massachusetts Institute of Technology, Cambridge, Massachusetts, USA.

Copyright 2012 by the American Geophysical Union.0148-0227/12/2011JF002139

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117, F01027, doi:10.1029/2011JF002139, 2012

F01027 1 of 18

http://dx.doi.org/10.1029/2011JF002139

[5] Some of the earliest quantitative models of landformdevelopment focused on hillslope form [e.g., Culling, 1960;Scheiddeger, 1961; Kirkby, 1971; Hirano, 1975; Ahnert,1976]. These studies sought to relate hillslope topographicprofiles to sediment transport expressions through conser-vation of mass. But most models that have incorporated baselevel effects have restricted their analyses to a base levelwith a fixed horizontal position. This includes both the well-known parabolic solution for steady state, sediment-mantledhillslopes evolving in response to a lowering base level witha sediment flux linearly proportional to the local topographicgradient [e.g., Culling, 1963; Kirkby, 1971; Hirano, 1975]and analogous solutions for transport laws in which sedi-ment 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 erosionat the hillslope scale [e.g., Mudd and Furbish, 2005; Stark,2010], and physically based expressions for hillslope formsproduced by horizontal base level migration have beenproposed 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 throughcomparisons with field sites. The goals of this paper areto derive expressions for the steady form of horizontallyretreating hillslopes subject to linear and nonlinear sedimenttransport laws, test the ability of these expressions to predicthillslope form in a field site where horizontal base levelmigration is known to occur, and demonstrate their utility foridentifying spatial patterns of channel network growthrecorded 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 thatadvances horizontally through an otherwise level plateau.In section 2, we derive one-dimensional analytical expres-sions for the steady state topographic profiles of slopeson which soil flux depends either linearly or nonlinearlyon the topographic gradient. In section 3, we compare theseexpressions with a numerical model, and show how a mea-sured hillslope profile can be used to estimate the ratio of thetransport coefficient to the base level migration speed, evenif the migrating base level takes the form of a point, such asan advancing channel tip, rather than a linear boundaryperpendicular to the transport direction. We then use theanalytical solutions in section 4 to estimate this ratio formany channel tips in a valley network formed by ground-water sapping in the Florida Panhandle, yielding a map thatreveals spatial trends in channel growth rates and hillslopetransport coefficients.

2. Analytical Model of a Retreating Hillslope

[8] We consider a one-dimensional hillslope that retreatsbecause of horizontal migration of a base level with fixedelevation (Figure 1). The coordinate system moves with thebase level at a horizontal speed v in the positive x direction,with the base level always located at x = 0, z = 0. The hill-slope is assumed to be retreating into a flat, level plain withan elevation z∞ that extends infinitely in the positive xdirection. The hillslope surface rises in the positive x direc-tion, approaching z∞ as x → ∞. Soil or sediment is trans-ported downslope with a volume flux per unit width q(x).

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 equalsthe total mass flux from upslope as the hillslope erodes intothe plain,

rsqðxÞ ¼ �ðz∞ � zÞ�rv; ð1Þ

where rs is soil or sediment bulk density and �r is the averagebulk density of the material in the plain between z and z∞.To derive an expression for an equilibrium profile, a transportlaw relating q to the topography is required. We consider twocases for soil-mantled hillslopes: one in which q is linearlyproportional to slope, and another in which q increases non-linearly with slope.

2.1. Linear Transport Law

[9] On soil-mantled hillslopes with low to moderate gra-dients, it has been proposed from simple arguments [Culling,1960, 1963, 1965] and demonstrated through field mea-surements [Monaghan et al., 1992; McKean et al., 1993;Small et al., 1999] that soil volume flux per unit width islinearly proportional to, and opposite in direction from, thetopographic gradient,

qðxÞ ¼ �Ddz

dx; ð2Þ

where D is a transport coefficient. Although recent studiessuggest that equation (2) may be at best an approximationfor the true pattern of mass transport [Heimsath et al., 2005;Furbish et al., 2009; Foufoula-Georgiou et al., 2010; Tuckerand Bradley, 2010], numerous studies of hillslope evolutionand topography have shown it to be a useful approximation.Substituting equation (2) into equation (1) and solving fordz/dx yields an expression for slope as a function of eleva-tion above the base level,

dz

dx¼ �r

rs

v

Dðz∞ � zÞ: ð3Þ

Separating variables and assuming that �r, rs and D areindependent of x and z yieldsZ

dz

z∞ � z¼ �r

rs

v

D

Zdx; ð4Þ

which we integrate to obtain

�lnðz∞ � zÞ ¼ �rrs

v

Dxþ C; ð5Þ

where C is an integration constant. The boundary conditionz(0) = 0 gives C = �ln z∞. Using this value and solvingequation (5) for the normalized steady state elevation profilez/z∞ gives

z

z∞¼ 1� e�x=L; ð6Þ

where the length scale L ¼ ðrsDÞ=ð�rvÞ. Hillslopes for whichL is small (rapid retreat or small transport coefficient) havesteep slopes that rapidly approach z∞, whereas hillslopes forwhich L is large (slow retreat or large transport coefficient)

PERRON AND HAMON: RETREATING HILLSLOPES F01027F01027

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have gentler slopes that gradually approach z∞. Equation (6)is analogous to solutions proposed for depositional land-forms 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 retreat-ing 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 thebase level, and in natural topography, the assumption of asmooth, nearly level surface will usually break down muchcloser to the steep portion of the profile. There are twosimple approaches for determining z∞ indirectly that alsoprovide estimates of L. First, defining S = dz/dx, equation (3)can be written

S ¼ z∞L� z

L; ð7Þ

which predicts a linear relationship between slope and eleva-tion, the slope of the relationship being�1=L ¼ �ð�rvÞ=ðrsDÞ.Second, differentiating equation (6) with respect to x yields

S ¼ z∞Le�x=L; ð8Þ

which can be cast as a linear equation,

lnS ¼ lnz∞L� x

L; ð9Þ

which predicts a linear relationship between the logarithm ofslope and horizontal distance, with the slope of the relationshipagain being�1/L. Once L is known, z∞ can be determined fromthe intercept of equation (7) or (9).

2.2. Nonlinear Transport Law

[11] If hillslope gradients are sufficiently steep, mechani-cal arguments [Andrews and Bucknam, 1987; Roering et al.,1999], laboratory experiments [Roering et al., 2001], andfield observations [Anderson, 1994; Pierce and Colman,1986; Roering et al., 1999; Gabet, 2000] indicate that the

flux q increases nonlinearly with the topographic gradient.Several expressions have been proposed, but the mostcommonly used is that proposed by Andrews and Bucknam[1987] and Roering et al. [1999], in which |q | → ∞ as Sapproaches a critical slope Sc,

qðxÞ ¼ �K dzdx

1� j dzdx j=Sc� �2 ; ð10Þ

with Sc � 1. Substituting equation (10) into equation (1)yields

dz

dx¼ �r

rs

v

Kðz∞ � zÞ 1� dz

dx

� �2 1

Sc2

!: ð11Þ

A solution to equation (11), which gives a dimensionlesssteady state hillslope profile, is

z

z∞¼ 1� LSc

2z∞

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiW ðFÞ 2þW ðFÞð Þ

p; ð12Þ

where L ¼ ðrsKÞ=ð�rvÞ, W is the Lambert W function,defined by

f ¼ W ðfÞeW ðfÞ; ð13Þ

and the quantity F is

F ¼ 1

Lexp

4aLSc

2 �2x

L� 1

� �; ð14Þ

with

a ¼ LSc2

4ln L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2z∞

LSc

� �2s

� 1

0@

1A ⋅ exp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2z∞

LSc

� �2s0

@1A

24

35:

ð15Þ

[12] As with the linear transport law, we seek a way ofcomparing the profile predicted by the nonlinear transportlaw with a measured topographic profile that does notrequire a direct estimate of z∞. The simplest approach is todifferentiate equation (11) with respect to x, yielding

d2z

dx2¼ � 1

L

S S=Scð Þ2 � 1� �2S=Scð Þ2 þ 1

: ð16Þ

This predicts a linear relationship between the secondderivative of hillslope elevation and the quantity involvingslope on the right-hand side, with the slope of the relation-ship being �1/L. Equation (16) is more flexible thanequation (7) or (9) because it allows for the potentiallynonlinear character of the transport law, but it has the dis-advantages that it requires measurements of concavity,which are typically more uncertain than measurements ofslope, and requires an estimate of Sc. Note that as S/Sc → 0,equation (16) reduces to the simple form

d2z

dx2¼ � S

L; ð17Þ

Figure 1. Schematic diagram of a hillslope retreating into alevel plain with elevation z∞ because of horizontal migrationof a base level (point b) at a speed v. The coordinate systemmoves with the base level, such that point b is always at theorigin. At steady state, the point at (x, z) must convey a vol-ume flux per unit width, q, that is proportional to (z∞ � z)v.


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which is the same result obtained for the linear transport lawby differentiating equation (8).

2.3. Predicted Hillslope Forms

[13] Figure 2 compares steady state hillslope profiles forthe linear and nonlinear transport laws. In dimensionlessform (Figure 2a and equation (6)), the shape of the solutionfor the linear law is independent of L. In contrast, the shapeof the solution for the nonlinear law depends on L (and on z∞and Sc). For small L, which would correspond to a rapidlyretreating base level or a small soil transport coefficient, theprofile approaches an angle of repose slope, with a nearlystraight 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 levelor a large soil transport coefficient, the profile approachesthe solution for the linear transport law, as implied bya comparison of equations (3) and (11), which converge asS/Sc → 0. The reason for the convergence of the solutions ismore apparent in dimensional form (Figure 2b): both solu-tions have gentler slopes for larger L, so the nonlineartransport effects are less important. This transition between

hillslope forms predicted by the linear and nonlinear trans-port laws is qualitatively similar to the case of verticalbase level change analyzed by Roering et al. [2007] andPerron [2011].

3. Numerical Model

3.1. Model Description

[14] To test whether the analytical solutions in section 2can provide a useful description of natural hillslopes thatdeviate from a strictly one-dimensional form, we created atwo-dimensional numerical model of a retreating hillslope.The model solves the equation

rs∂z∂t

þr ⋅ rsqð Þ ¼ �rv∂z∂x

; ð18Þ

where q, the vector flux, is given by a two-dimensionalversion of either the linear transport law, equation (2), or thenonlinear transport law, equation (10). The model coordinatesystem moves with the hillslope’s base level, and thereforethe advection term on the right-hand side of equation (18),

Figure 2. Steady state profiles of retreating hillslopes with z∞ = 20 m and Sc = 1 for different values of Lin (a) dimensionless coordinates and (b) dimensional coordinates. In Figure 2a, the solution for the lineartransport law (equation (6), solid black line) is the same for all L. The solution for the nonlinear transportlaw (equation (12), dashed gray lines) converges to the solution for the linear law as L increases.


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which causes the solution to shift in the negative x directionat a speed v, is equivalent to a base level that migrates in thepositive x direction in a fixed reference frame. The elevationof the positive x boundary is fixed at z∞, which assumes thatit is sufficiently far from the hillslope’s base level that z ≈ z∞.We use the implicit method of Perron [2011] to solveequation (18) forward in time until the topography reachesa static steady state.

3.2. Comparison of Numerical and Analytical Solutions

[15] We investigated two cases. In the first case, the yboundaries are periodic, and a base level with a fixed ele-vation of zero covers half of the grid. This produces a one-dimensional hillslope that does not vary in the y direction(Figure 3a). A profile through this solution in the x directionis identical to the one-dimensional analytical solutions insection 2 (Figure 3c). In the second case, the y boundarieshave fixed elevations of z∞, the negative x boundary is free,and the base level consists of a channel tip with a fixedelevation of zero extending into the grid in the x directionfrom the negative x boundary. This case is intended to sim-ulate the topography surrounding a horizontally advancingchannel tip, and the solution consists of concave-downhillslopes that wrap around the channel tip (Figure 3b).Because of this convergent topography, points near the chan-nel tip must convey a larger flux at steady state than in the one-dimensional case, and the hillslope profile in Figure 3b istherefore steeper and more curved than the profile in Figure 3a.Despite this difference, a profile through the two-dimensionaltopography deviates only slightly from the one-dimensionalanalytical solution (Figure 3d). The steady state solutions forboth 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 theboundary conditions in Figure 3b for a range of L, and usedthe expressions derived in section 2 to determine apparentvalues of L from profiles through the numerical solutions.All numerical calculations used the nonlinear transportlaw with rs=�r ¼ 1 , K = 0.01 m2/yr, z∞ = 20 m, Sc = 1,and Dx, Dy = 5 m. The speed v ranged from 0.1 mm/yr to1 mm/yr, such that 10 m ≤ L ≤ 100 m, a range that encom-passes most of the hillslopes at the study site investigatedin section 4.[17] We extracted a topographic profile from each

numerical solution at the location shown in Figure 3b, andapproximated dz/dx and d2z/dx2 with second-order finitedifferences. We then used iteratively reweighted leastsquares regression to fit the relationships in equations (9)and (16) to the model solution, and determined L from theregression slopes. Figure 4 compares the L values inferredfrom the regression with the actual values. Because theanalytical solutions neglect the effect of convergent topog-raphy illustrated in Figure 3, both systematically underesti-mate L. However, both expressions provide estimates of Lthat are within 13% of the true value over the range wetested, and within 2% for L = 10 m. Moreover, because thesolutions for the two transport laws are similar for L ≳ 10 m(Figure 2), the regression based on the linear transport lawestimates L with accuracy comparable to the regressionbased on the nonlinear law (and even slightly better, becausethe linear law predicts a steeper, more curved profile thatmimics the two-dimensional effect). For sites in whichL falls in the range tested here, it is therefore possible toobtain good estimates of L from hillslope profiles by usingequation (9), which avoids the potentially noisy measure-ments of profile curvature required for equation (16). Whilethe specific errors plotted in Figure 4 do not apply to all of

Figure 3. Steady state numerical solutions for a hillslope retreating in response to (a) a migrating linearbase 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).


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our results, the magnitudes should be similar, and ourinterpretations in sections 4 and 5 are based largely on rel-ative measures of L, which are not significantly influencedby these systematic errors.

4. Application to Sapping Valley Networksof the Florida Panhandle

[18] To test the analytical model, and to use the associatedpredictions to examine spatial variability in v or K, wesought a landscape with a large number of measurable hill-slope profiles subject to boundary conditions similar to thoseassumed in section 2: a base level migrating horizontally at aconstant rate into a level plain. We also sought a site withminimal spatial variability in transport processes or themechanical characteristics of the substrate, such that hill-slope form is likely to be a sensitive indicator of rates ofsediment transport and base level migration. We selected asite in the Florida Panhandle where groundwater sappingchannel networks have created hundreds of hillslopes thatsatisfy most of these criteria.

4.1. Site Description

[19] The Western Highlands of the Florida Panhandle arebuilt from the highly permeable Plio-Pleistocene sands ofthe Citronelle Formation [Sellards and Gunter, 1918]. Thehighlands surface rises a few tens of meters above sea level,slopes gently toward the Gulf Coast, and is locally veryplanar. In some locations adjacent to rivers or water bodies,the subdued highlands topography is interrupted by steep-sided valley networks containing perennial, spring-fedstreams. The valley networks typically have straight main

stems and short tributaries with nearly orthogonal junctionangles. Tributary valleys terminate abruptly upstream insteep, roughly semicircular headwalls known colloquially as“steepheads” [Means, 1981]. The near absence of channelincision upslope of the valley heads and the alignment ofmajor valleys with the average direction of groundwaterflow led early investigators to conclude that the valley net-works were incised by groundwater sapping at the springsites. Sapping valleys are thought to form through a positivefeedback in which the focusing of flow toward a spring leadsto accelerated erosion where the spring emerges from theground, which in turn advances the channel tip and causesthe groundwater flow to converge more strongly [Dunne,1980; Howard and McLane, 1988; Howard, 1988]. Earlystudies suggested that sapping valleys in Florida may alsohave been influenced by low-permeability clay beds in theCitronelle Formation that direct groundwater flow andenhance this focusing effect, or by indurated layers near thesurface that inhibit erosion [Sellards and Gunter, 1918], butsubsequent 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 Rivernear Bristol, Florida (Figure 5). Several lines of evidence,including surveys of groundwater table elevations [Abramset al., 2009; Petroff et al., 2011], analyses of valley longi-tudinal 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 chan-nel bifurcation angles [Petroff, 2011] support the inferencethat the valley networks were formed by dominantly hori-zontal migration of groundwater sapping sites through thesandy bluffs. As the springs at the tips of the channel net-work have advanced through the nearly level surface thatsits approximately 50 m above the Apalachicola River,they have created boundary conditions very similar to thescenario in Figure 1. The case of perennial, spring-fedchannels bounded by highly permeable slopes that experi-ence little overland flow is also consistent with the sharpboundary between hillslope and fluvial domains assumed insection 2. The many valley heads across the site thereforeprovide an opportunity to test the analytical solutions pre-sented in section 2, and to obtain a snapshot of the evolu-tion of the valley network by measuring relative rates ofhillslope retreat.[21] Although the level surface of the bluffs has been

clear-cut, the sapping valleys remain forested, with conifersand a few hardwoods dominating the higher elevations ofthe valley walls, and abundant magnolia, beech, and ever-green shrubs in the wetter, more densely vegetated lowerelevations [Means, 1981]. Soil transport appears to occurthrough a combination of bioturbation, small slumps andraveling events, and, less commonly, shallow landslides.Hillslopes at valley heads are concave down and typicallygrade smoothly into the level plain (Figure 6), which isqualitatively consistent with the analytical model.

4.2. Topographic Measurements and Determinationof L

[22] To enable a quantitative comparison, we measuredhillslope elevation profiles above channel tips throughout

Figure 4. Comparison of L values inferred from regressionanalysis of topographic profiles through numerical solutionslike that in Figure 3b. Values marked with circles weredetermined from equation (16) for the nonlinear transportlaw, and those marked with crosses were determined fromequation (9) for the linear transport law.


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the valley networks. An airborne laser altimetry map of theBristol site was produced by the National Center for Air-borne Laser Mapping (NCALM). The raw point cloud wasfiltered to remove laser returns from vegetation and griddedto a horizontal point spacing of 1 m. We used elevation andslope maps to estimate the locations of spring sapping sitesat the heads of valleys. A spring site was identified as abreak in slope at the base of a headwall. From each spring,we drew a linear transect extending upslope in a directionparallel to the valley until the elevations reached a nearlyconstant value, and then interpolated elevations along eachprofile at a spacing of approximately 1 m. We inspected eachprofile and removed concave-up portions at the downslopeend, which can result from small errors in our estimate of thespring location, and portions at the upslope end where thesurface sloped away from the spring, which were usuallyassociated with human modification of the topography or

deviations of the plain from a perfectly level surface. Thecoordinates of the first point in the profile were set to x = 0,z = 0, and the other coordinates were measured relative tothis point. The profiles were then examined for obvioussigns that the hillslopes were not in a steady state, such asmajor breaks in slope, inflections in curvature, or largebumps or dips. Profiles containing such features were dis-carded. After this screening, 201 profiles remained foranalysis, including most of the major valley heads visible inthe laser altimetry as well as many smaller valleys thatextend only a short distance from the main valleys. Loca-tions, directions, and lengths of the transects are listed inTable 1.[23] The expressions derived in section 2 allow us to

obtain an estimate of L, and therefore K/v, for each channeltip. We used equation (9) to determine the best fit value of L.Although equation (9) is strictly only valid for the linear

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 rectanglemarks the area shown in Figure 9. The Apalachicola River, at the far left, flows south. Image is fromthe National Agriculture Imagery Program.


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Table 1. Hillslope Profile Data

Profile Eastinga (m) Northinga (m) Length (m) Directionb (deg) Lc (m) Estimatedd R2

1 693413 3372435 126.4 356.1 3.9 � 0.4 0.892 695120 3374711 39.8 0.0 5.6 � 0.2 0.983 694644 3375406 112.5 325.8 6.8 � 0.5 0.914 696138 3374388 111.4 272.7 7.1 � 0.4 0.925 698157 3373164 80.7 1.6 7.5 � 0.7 0.866 694068 3374435 66.4 276.0 8.0 � 0.6 0.877 694528 3374268 80.4 291.7 8.1 � 0.4 0.948 694349 3375375 119.3 286.8 8.2 � 0.4 0.919 695179 3372476 62.8 160.0 10.4 � 0.6 0.9410 694541 3375370 154.5 269.3 10.5 � 0.8 0.8511 697169 3374982 114.3 248.7 10.5 � 1.0 0.7812 694108 3374144 126.3 276.3 10.8 � 0.7 0.8813 695418 3374423 112.6 258.8 10.8 � 0.5 0.9314 696985 3375821 176.7 221.8 10.9 � 0.5 0.9315 694040 3375361 127.3 268.3 11.2 � 0.9 0.7916 695635 3375100 118.8 169.1 11.4 � 1.0 0.8317 695566 3372569 57.2 3.8 11.4 � 0.9 0.8418 693798 3375513 121.9 267.4 11.5 � 0.4 0.9519 695210 3374715 38.7 183.3 11.5 � 1.0 0.8820 694491 3375394 124.6 262.5 11.6 � 0.6 0.9321 694872 3374330 118.5 268.8 11.7 � 0.5 0.9322 694295 3375394 112.5 203.8 12.0 � 0.7 0.8823 696927 3374693 105.6 311.1 12.0 � 0.8 0.9124 694804 3375549 96.2 302.3 12.3 � 0.5 0.9225 694015 3375891 71.0 265.6 12.4 � 1.6 0.6426 698106 3374153 75.5 260.5 12.5 � 0.9 0.8627 697051 3375802 168.7 294.7 12.5 � 1.2 0.7828 695327 3374477 104.5 180.0 12.6 � 0.4 0.9529 696871 3374756 81.7 206.5 12.7 � 0.6 0.9230 695082 3374730 49.9 221.4 12.7 � 0.7 0.9431 695202 3372547 87.1 137.1 12.7 � 0.6 0.9132 695720 3375199 117.6 32.4 12.8 � 0.9 0.8633 695594 3374415 113.2 254.9 12.9 � 0.8 0.8634 697719 3374768 155.6 249.2 13.1 � 0.8 0.8735 698856 3374145 109.3 325.9 13.3 � 1.2 0.8136 698787 3374131 85.8 269.1 13.5 � 0.6 0.9037 693788 3375839 49.8 190.4 14.0 � 0.9 0.9038 696354 3375109 141.9 128.7 14.2 � 0.6 0.9339 693557 3375313 117.6 289.7 14.2 � 0.6 0.9240 696989 3375299 87.1 131.7 14.3 � 1.1 0.8341 695027 3373018 141.2 16.9 14.5 � 0.9 0.8642 696839 3375847 145.8 256.0 14.6 � 0.5 0.9543 695644 3375168 115.1 155.1 14.7 � 0.6 0.9244 695251 3372494 74.0 49.2 14.7 � 1.0 0.9545 696812 3374824 113.2 225.0 14.8 � 0.6 0.9246 696111 3374999 156.8 89.5 14.8 � 1.0 0.8247 696301 3373825 117.7 215.0 14.9 � 0.7 0.9548 693922 3375443 97.1 218.2 15.0 � 0.5 0.9449 696477 3375225 165.1 168.8 15.0 � 0.7 0.9250 695557 3372417 77.0 14.7 15.2 � 1.1 0.7951 695348 3373397 71.0 260.2 15.4 � 1.2 0.8452 695561 3372629 103.8 358.6 15.6 � 1.6 0.6853 697194 3374020 150.4 127.7 15.6 � 1.2 0.8254 694993 3372948 117.6 238.4 15.8 � 1.0 0.8655 694928 3373052 120.2 202.5 16.0 � 0.9 0.8856 695768 3372977 107.5 309.2 16.3 � 1.7 0.7657 695239 3375761 96.5 237.3 16.6 � 2.3 0.5758 695456 3372660 108.7 154.1 16.7 � 0.7 0.9159 694646 3374983 107.4 163.1 16.9 � 0.8 0.8760 694154 3373054 78.8 287.8 17.0 � 0.8 0.9361 696361 3374018 85.8 139.3 17.0 � 1.3 0.8062 693510 3372132 150.6 47.2 17.2 � 0.8 0.9063 695314 3373416 85.9 229.7 17.5 � 0.7 0.9464 694016 3373706 121.8 188.0 17.7 � 1.0 0.8565 695191 3375097 95.0 13.7 17.9 � 0.9 0.9066 693799 3372739 102.4 230.0 17.9 � 1.3 0.7867 694148 3374452 170.6 292.1 18.0 � 0.7 0.9268 694997 3374386 157.4 261.1 18.1 � 0.9 0.8769 693795 3375804 58.1 180.0 18.2 � 1.0 0.9070 695010 3373141 115.9 0.1 18.3 � 2.1 0.6471 694435 3374306 111.8 270.0 18.5 � 0.9 0.8772 693758 3372767 101.8 202.4 18.6 � 1.0 0.87


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Table 1. (continued)


73 694902 3373761 88.5 154.2 18.8 � 1.2 0.7974 694778 3374417 144.1 265.9 18.8 � 0.7 0.9475 697632 3374033 161.4 358.5 18.8 � 1.9 0.7076 695978 3375009 72.1 105.5 18.9 � 1.1 0.8777 697263 3375024 74.3 329.9 18.9 � 1.3 0.8278 696294 3374595 79.4 19.9 18.9 � 1.5 0.7779 695678 3374493 119.8 345.1 19.0 � 1.0 0.8680 696431 3374761 116.1 279.5 19.3 � 1.3 0.8181 693702 3374429 85.4 273.5 19.7 � 0.8 0.9382 695378 3375682 136.8 310.8 19.8 � 2.0 0.6783 695570 3373064 128.1 183.2 20.0 � 1.4 0.7884 695100 3374707 95.3 256.8 20.0 � 1.3 0.8085 695981 3375002 135.0 106.0 20.0 � 0.7 0.9386 695889 3375819 199.2 272.6 20.0 � 2.7 0.6787 695222 3374702 52.8 275.7 20.5 � 1.2 0.8888 695914 3374497 109.3 220.2 20.5 � 0.9 0.8789 695678 3374497 99.1 37.6 20.5 � 1.0 0.8990 694019 3374874 119.9 310.8 20.9 � 1.2 0.8391 695577 3372707 79.6 86.5 21.0 � 1.7 0.7492 694570 3374304 90.8 322.5 21.1 � 0.8 0.9293 698322 3374226 165.3 276.4 21.1 � 1.3 0.8194 696529 3373944 107.7 358.8 21.2 � 1.0 0.8895 695757 3375866 200.9 185.7 21.2 � 1.8 0.7996 694305 3374302 94.1 272.1 21.3 � 1.4 0.7697 695616 3372690 80.0 5.2 21.3 � 2.1 0.6598 697473 3375122 87.8 253.6 21.8 � 1.2 0.8599 696924 3372927 108.0 273.6 22.3 � 1.6 0.76100 696458 3374788 122.8 344.3 22.3 � 1.1 0.87101 694184 3374452 64.2 289.0 22.4 � 0.8 0.95102 695382 3373387 127.4 279.8 22.5 � 1.3 0.90103 697321 3375967 114.5 189.5 22.5 � 1.1 0.87104 695492 3372696 95.8 114.4 22.6 � 1.0 0.87105 693658 3374855 54.2 141.1 22.7 � 1.0 0.94106 697096 3375838 87.5 331.7 22.8 � 1.0 0.90107 694040 3374219 80.2 204.3 22.8 � 0.9 0.91108 694528 3373081 117.4 306.6 22.9 � 1.1 0.88109 695543 3374720 65.2 274.3 22.9 � 1.6 0.82110 694615 3373240 109.1 305.1 23.0 � 0.8 0.94111 693855 3372760 70.4 264.1 23.0 � 1.1 0.90112 693739 3372815 93.1 211.2 23.1 � 1.0 0.87113 693857 3374414 45.8 188.8 23.2 � 1.1 0.92114 695769 3373883 85.5 175.3 23.2 � 0.9 0.92115 693751 3374438 105.0 275.7 23.4 � 1.2 0.81116 694955 3372405 92.2 173.9 23.6 � 2.1 0.71117 694260 3375742 89.0 179.8 23.8 � 2.4 0.65118 693794 3374785 130.3 325.9 24.1 � 0.7 0.94119 697731 3374846 148.9 168.1 24.4 � 1.2 0.86120 695575 3373283 124.2 187.9 24.9 � 1.5 0.81121 693874 3374345 91.3 219.2 25.2 � 1.2 0.87122 697127 3375107 116.5 213.9 25.6 � 1.0 0.90123 695015 3374714 90.1 278.9 26.1 � 0.9 0.92124 695645 3374791 96.8 283.8 26.1 � 1.0 0.92125 694367 3375926 85.7 276.1 26.2 � 1.2 0.90126 695886 3373977 124.9 85.6 26.3 � 0.7 0.95127 693941 3374276 105.1 241.3 26.4 � 1.6 0.77128 694242 3375835 76.3 180.0 26.5 � 2.2 0.76129 694866 3373431 50.6 87.2 26.7 � 2.5 0.78130 696850 3373883 150.6 116.5 26.7 � 1.0 0.88131 694385 3372832 91.9 87.0 26.8 � 1.5 0.86132 693746 3375004 102.3 156.8 26.9 � 0.7 0.96133 697521 3374042 88.0 95.8 26.9 � 1.4 0.86134 694300 3375528 102.5 173.0 27.3 � 1.4 0.83135 695708 3374802 100.1 283.0 27.4 � 1.0 0.91136 695519 3372707 103.6 85.2 27.4 � 1.4 0.84137 694927 3374707 94.8 282.5 27.5 � 1.8 0.82138 695970 3373089 158.5 352.8 27.8 � 1.6 0.82139 694984 3373890 163.3 93.4 28.4 � 1.5 0.85140 693693 3374686 61.0 178.6 28.5 � 1.4 0.91141 693970 3372641 123.1 239.4 28.7 � 1.3 0.84142 695511 3373884 100.0 352.0 28.9 � 1.0 0.91143 695132 3374705 101.3 275.9 29.0 � 1.7 0.85144 695052 3375161 84.1 82.9 29.2 � 1.3 0.87


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transport law, the exercise in Figure 4 demonstrates that thisapproach can yield good estimates of L even for profiles inwhich nonlinear effects are significant. Moreover, prelimi-nary analyses not presented here indicated that the curvaturevalues required to apply equation (16), which is valid for thenonlinear transport law, were sufficiently noisy that they

introduced a larger source of uncertainty in our estimates ofL 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 eachpoint, except profile endpoints, with second-order finitedifference approximations, and 1/L was determined from the

Table 1. (continued)


145 694085 3375586 98.3 25.0 29.6 � 2.0 0.76146 696346 3373895 147.8 183.1 29.7 � 0.8 0.92147 695153 3373352 112.5 326.9 30.3 � 1.1 0.89148 693725 3374952 114.1 158.5 30.5 � 1.0 0.95149 694111 3374195 67.2 5.3 30.5 � 1.8 0.85150 695423 3374592 109.1 194.9 31.0 � 0.8 0.95151 697213 3375354 95.7 133.9 31.3 � 1.4 0.87152 698063 3374277 124.3 180.7 31.4 � 1.6 0.88153 697786 3374798 106.1 327.8 32.0 � 2.1 0.73154 693888 3372699 94.0 228.2 32.5 � 1.4 0.90155 694806 3375130 70.8 123.1 32.5 � 2.3 0.79156 694455 3372557 154.4 0.0 32.8 � 4.7 0.51157 694993 3372948 122.7 280.2 33.4 � 3.1 0.75158 694170 3374967 63.9 1.1 33.4 � 1.4 0.93159 693785 3375079 120.5 88.8 33.5 � 1.2 0.91160 695414 3373969 126.8 80.4 33.6 � 2.1 0.81161 698574 3374245 157.7 254.2 33.8 � 1.7 0.80162 696449 3374029 107.3 66.6 33.9 � 2.2 0.77163 694307 3375579 100.5 173.7 34.1 � 1.8 0.83164 697909 3374926 97.3 298.2 34.2 � 2.4 0.74165 695036 3374393 108.6 338.4 34.5 � 1.4 0.87166 693963 3375055 94.6 53.9 35.2 � 1.8 0.84167 697369 3375458 107.5 90.0 35.3 � 2.3 0.73168 694473 3374609 48.6 94.1 36.2 � 2.8 0.82169 696119 3373497 96.2 276.9 36.7 � 1.6 0.86170 697203 3373642 175.7 315.0 36.7 � 2.9 0.68171 694955 3372577 106.2 115.4 36.9 � 4.0 0.54172 697326 3375434 86.4 118.7 38.1 � 2.7 0.73173 695255 3373873 213.3 128.5 38.3 � 2.7 0.75174 693679 3374741 92.4 176.9 39.3 � 1.4 0.91175 694079 3372477 166.2 189.3 39.5 � 2.3 0.80176 695911 3373928 150.8 11.9 39.6 � 1.4 0.87177 693897 3375764 97.9 15.1 41.3 � 1.9 0.84178 694303 3374736 93.0 149.5 44.0 � 3.1 0.74179 695276 3374905 102.7 124.4 45.8 � 3.7 0.67180 696762 3375111 103.0 30.5 46.6 � 5.3 0.55181 698217 3374284 81.6 288.8 47.1 � 4.2 0.70182 694662 3374543 100.0 277.0 47.7 � 2.6 0.79183 694282 3374701 122.9 153.2 49.0 � 2.8 0.80184 696086 3374848 79.0 276.4 49.0 � 4.6 0.66185 694054 3375658 117.6 82.3 51.8 � 3.8 0.71186 697102 3375291 97.9 99.8 52.2 � 3.1 0.77187 694058 3375659 98.9 48.6 52.8 � 4.8 0.63188 693823 3372839 101.0 33.2 54.3 � 2.5 0.85189 697811 3375059 123.6 190.3 55.9 � 3.0 0.75190 698688 3374271 97.4 258.5 58.1 � 3.2 0.80191 698728 3374274 71.7 262.2 58.4 � 4.5 0.78192 696499 3375017 95.8 299.4 62.4 � 4.1 0.75193 693960 3372786 128.7 45.8 64.1 � 3.9 0.72194 695958 3373398 116.4 323.2 68.8 � 5.9 0.60195 695287 3374905 93.4 75.8 70.8 � 6.0 0.64196 693555 3375466 118.0 24.7 71.3 � 4.3 0.71197 694174 3374600 128.8 91.6 73.0 � 5.2 0.66198 696852 3373361 133.5 24.9 73.6 � 5.0 0.67199 696415 3373224 126.5 247.0 75.9 � 8.3 0.48200 694552 3374772 123.1 98.4 107.3 � 7.5 0.64201 698172 3374932 102.2 248.0 149.6 � 26.1 0.28

aUTM zone 16N.bMeasured counterclockwise from east.cMean � standard error.dThe 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.


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slope of an iteratively reweighted least squares regression of�ln S against x. The standard error of the regression slopeprovided an estimate of the uncertainty in 1/L. Steep sectionswith nearly constant slope at the downslope ends of some ofthe profiles suggested significant nonlinear transport effects,and were excluded from the regressions. Sections withnearly constant slopes of approximately zero at the upslopeends of some profiles suggested that those portions of theplain have not responded significantly to the propagat-ing channel tips, and were also excluded. Our approach forestimating L is conceptually similar to that of Abrams et al.[2009], who used the radius of curvature of the valley rim asan order-of-magnitude estimate of K/v. The expressionsderived here demonstrate the basis for the ratio they obtainedby dimensional analysis.[25] L values for the 201 analyzed profiles are listed

in Table 1, and the frequency distribution is plotted inFigure 7. L is lognormally distributed, with a mode ofapproximately 20 m. For �r=rs ¼ 1, a reasonable approxi-mation for the sands of the Citronelle Formation, and atypical transport coefficient of K = 0.01 m2/yr for weaklycohesive sediment in a humid environment [Nash, 1980;

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 veloc-ity of 0.5 mm/yr, with 95% of the velocities faster than0.17 mm/yr and 95% slower than 0.95 mm/yr. Given theone-dimensional approximation used in our model, it islikely that our calculations underestimate L, and thereforeoverestimate channel tip velocities. However, the analysis insection 3 suggests that this systematic error is unlikely toexceed �10% (Figure 4). Our estimates of L may also bebiased by channel tips that are slowing down or speeding up,resulting in hillslope profiles that are not representative ofthe present-day migration speed. However, the observationthat most measured profiles follow the predicted log linearsteady state relationship between slope and horizontal dis-tance (see section 4.3) suggests that this bias is not sub-stantial. Another way to assess the potential for nonsteadystate profiles is to compare the response time of the hill-slope profiles to the timescale for channel network growth.Using the values of K = 0.01 m2/yr and v = 0.5 mm/yrestimated above, the diffusion time for a hillslope withz∞ = 20 m is z∞

2 /K = 40 kyr, whereas the time required togrow a tributary with a length ‘ = 500 m (Figure 5) is ‘/v = 1Myr. Hillslopes that experience nonlinear transport due tofast-moving tips have an even larger effective diffusivity,and a shorter diffusion time. We therefore expect that hill-slopes respond rapidly compared with the growth of thechannel network, and that hillslope profiles will be close to asteady state even for channel tips that are gradually slow-ing down or speeding up. As noted above, profiles thatcontained clear deviations from the predicted steady stateform 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 flowfield to estimate channel tip velocities. Their reconstructionsuggests that tip velocities averaged over the entire �1 Myrhistory of the channel network were as fast as 5.3 mm/yr,but that channel tips have slowed to an average of about0.5 mm/yr over the past 10 kyr as channel tips haveapproached one another and competition for groundwater

Figure 6. Photographs of hillslopes above sapping channeltips in the Apalachicola Bluffs near Bristol, Florida.(a) Sandy slopes more than 100 m from the channel are con-cave down and slope gently toward the channel. Timberfarming has altered the vegetation cover. (b) Slopes closerto the channel are steeper. Note people for scale on the rightin the middle distance of both images.

Figure 7. Frequency distribution of L inferred from the201 hillslope profiles in Figure 9.


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has intensified. Abrams et al. [2009] used a slightly highertransport coefficient of 0.02 m2/yr to arrive at their esti-mates. This K value would increase our modal channel tipvelocity to 1 mm/yr. Thus, our estimates of channel tipvelocities based on hillslope profiles are slightly faster thanthe estimates of Abrams et al. [2009] and an estimate of “aninch or two per century” for other Florida sapping channelscited by Schumm et al. [1995], but are still consistent withthese previous estimates to within a factor of two.

4.3. Comparison With Analytical Models

[27] Having estimated L for each profile, we calculated thecorresponding analytical solution from equation (6) andcompared it with the measured profile. For profiles with

nearly straight sections at the downslope end, we also cal-culated the analytical solution for the nonlinear transportlaw, using the value of z∞ determined from the regressionand a value of Sc = 1.2, which we estimated from the dis-tribution of measured slopes. Figure 8a shows two end-member profiles: profile 130, with L = 26.7 � 1.0, which iswell described by the solution for the linear law, and pro-file 3, with L = 6.8 � 0.5, one of the shortest measuredL values, which is best described by the solution for thenonlinear law. Both profiles are very similar to the anal-ytical solutions. When the profiles are plotted in dimen-sionless coordinates, the effect of L on the profile shapenoted earlier in Figure 2a is apparent: profiles with longerL approach z/z∞ = 1 at smaller x/L, whereas profiles withshorter L have a straight section where S ≈ Sc that causes amore gradual approach to z/z∞ = 1. In dimensional coordi-nates (not shown in Figure 8), the profile with shorter L issteeper, and approaches z∞ at a shorter distance, as in thedashed profiles in Figure 2b.[28] In addition to the elevation profiles, another test of

whether the analytical solutions are a good description of themeasured profiles is the goodness of fit of the regression lineused to determine 1/L. Figures 8b and 8c show the fits forprofiles 130 and 3, respectively. Both profiles contain pointsthat deviate from the predicted trend at one or both ends,as noted in section 4.2, and at slight irregularities in theprofiles, but both contain a section that is reasonably welldescribed by a linear trend in �ln S versus x space. Table 1lists estimated correlation coefficients, R2, for the linear fits.Although the example profiles shown in Figure 8 are amongthe “cleanest” matches to the analytical solutions, in thesense that they contain few topographic irregularities, thetable shows that most of the other profiles have comparablegoodness of fit.

5. Discussion

5.1. Analysis of Spatial Trends in Hillslope Form

[29] The model proposed for sapping channel networkdevelopment, in which channel growth and deflection of thegroundwater flow field are linked though a positive feedback[Dunne, 1980; Howard, 1988; Abrams et al., 2009], impliesthat transient rates of channel growth may vary considerablyacross a groundwater sapping landscape. Our hillsloperetreat model predicts that such variations in channelmigration rates should be recorded in the morphology ofhillslopes above channel heads. To search for such trends,we examined the spatial distribution of the L values deter-mined in section 4.2. Figure 9 plots the quantity

1

L¼ �r

rs

v

Kð19Þ

over a shaded relief map of the sapping channel network.Provided that �r=rs is nearly constant across the site, a rea-sonable assumption for the uniform sands of the CitronelleFormation, larger values of 1/L indicate either rapidlyadvancing 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 of1/L. For example, Figure 10a compares distributions of 1/L

Figure 8. Comparison of analytical solutions with topo-graphic profiles upslope of channel tips. (a) Dimensionlesselevation profiles (compare with Figure 2a). Profile 130 iscompared with the solution for the linear transport law(equation (6), solid line), and profile 3 is compared withthe solution for the nonlinear law (equation (12) with Sc = 1.2and z∞ = 16.5 m, dashed line). Locations and statistics forthe profiles are listed in Table 1. (b and c) Plots of �ln Sagainst horizontal distance, with horizontal scales chosensuch that each point appears at the same horizontal positionas it does in Figure 8a. Points with negative slope are notshown. Regression lines, which ignore constant-slope sec-tions at the ends of the profiles, are used to determine L fromequation (9).


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Figure

9.Map

of1/Lforhillslop

esabov

echanneltips.Black

linesmarkthelocation

sof

measuredprofiles

andpo

intin

theinferred

directionof

channelp

ropagation

.Pointsmarkthechanneltipsandhave

areasprop

ortion

alto

1/Landcolors

prop

ortion

alto

log 1

01/L.


13 of 18

for the two networks that lie completely within the laseraltimetry coverage. The northern network has a more log-normal distribution, with numerous hillslopes with largevalues of 1/L, whereas the southern network has a distribu-tion skewed toward smaller 1/L. It is possible that variationsin the characteristics of the sand cause a gradient in K acrossthe site, but we observed no evidence of such differences inthe field. The more likely explanation is that the northernnetwork is growing faster, perhaps because the southernnetwork developed earlier and its outermost channel tipsslowed sooner. The fact that the northern network alsoappears to have a higher density of active channel tips,as indicated by the larger number of measurable profilesdespite the smaller overall area of the network (Figures 9and 10), also supports this idea.[31] The second and more surprising trend revealed by

Figure 9 is that tips growing southward (which form

hillslopes that face northward) have larger average 1/L.A comparison of distributions of 1/L between north facingand south facing hillslopes confirms this observation(Figure 10b). This could indicate either that channels growsouthward faster than they grow northward, or that northfacing hillslopes have less efficient hillslope transport thansouth facing hillslopes. The former explanation is lessplausible. Although water availability appears to have asignificant effect on channel tip propagation rates [Abramset al., 2009; Petroff et al., 2011], it seems unlikely that dif-ferences in evaporation on the steep slopes immediatelyupslope of springs would have a large enough impact onspring discharge to slow northward migrating channels,because springs are fed by deeper flow from a larger areathat is mostly flat. None of the recent studies of sappingvalley networks in the Florida Panhandle have reportedstructural heterogeneities in the Citronelle Formation thatwould have driven asymmetric tributary growth. The orien-tation of the main valleys appears to be controlled by theoverall direction of groundwater flow, which is toward theApalachicola River and its tributaries (Figure 5). Moreover,the valley networks lack the asymmetry of tributary lengthsone would expect to see if southern tributaries have beengrowing faster for a prolonged interval.[32] The more plausible explanation for the asymmetry is

that K varies with slope aspect. Many studies have presentedevidence that microclimates produced by differences in solarradiation can influence the long-term efficiencies of ero-sional 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 strongmicroclimates often have asymmetric topography [e.g.,Bass, 1929; Emery, 1947; Dohrenwend, 1978]. In particular,Pierce and Colman [1986] documented faster regolith creeprates on equator-facing terrace scarps on alluvial fans thanon pole-facing scarps, the same sense of asymmetry in Kimplied by our measurements. In the Apalachicola Bluffs,we observed pronounced aspect-related differences in veg-etation in some parts of the landscape, with south facingslopes dominated by an open canopy of conifers, and northfacing slopes covered by deciduous trees with denserundergrowth. It is not certain how this difference in vege-tation would affect soil transport rates, but the likely role ofbioturbation and the potential inhibition of mass wasting byroot cohesion suggest that there could be important effects[Dietrich and Perron, 2006]. A difference in K is also con-sistent with the asymmetric cross sections of east-westtrending valleys, which typically have gentler northernside slopes and steeper southern side slopes (Figure 11).Although valley walls may experience some retreat becauseof minor seepage after they are initially created by anadvancing channel tip, the dominant effect appears to begradual relaxation of the initial walls, as recorded by theprogressively gentler slopes on both sides of the valleysas they approach their junction with the Apalachicolafloodplain (Figure 11). The asymmetric cross sections aretherefore likely to be the product of different transport effi-ciencies rather than different seepage-driven retreat rates.

5.2. Broader Applications of Hillslope Retreat Models

[33] Although the example of sapping channels is specificto a certain landscape, the model for retreating hillslopes

Figure 10. (a) Histograms of 1/L for hillslope profiles inthe two complete valley networks in the study area (seeFigures 5 and 9). The modes of the two distributions are sim-ilar, but the northern valley network has more hillslopes withvery large 1/L. (b) Histograms of 1/L for north facing andsouth facing hillslope profiles in the study area. North facingslopes have a larger mean value of log10 (1/L) than south fac-ing slopes (one-tailed t test, p = 0.00025).


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presented here has broader applicability. In principle, it candescribe any sediment-mantled slope bounding a nearlylevel surface in which sediment that reaches the base of theslope is removed. This can include plateaus being dissectedby growing drainage networks, valley walls or channelbanks, and scarps of various scales.[34] We have demonstrated several applications of the

model. First, it can describe the equilibrium shapes ofretreating slopes (Figure 8a), and it provides a dimensionlessframework that allows comparisons among slope forms withdifferent absolute scales (Figures 2a and 8a). Second, itpredicts 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 linearrelationships can be used to infer the ratio of the soil trans-port coefficient to the base level retreat speed via the lengthscale L. If one of these two parameters can be constrainedindependently, the hillslope profile can be used to infer theother. Finally, we have demonstrated how variability inhillslope form throughout a landscape can reveal spatialtrends in relative rates of landscape evolution, even if theabsolute rates are unknown. In the case of the sapping valleynetworks of the Apalachicola Bluffs, the hillslope profilesabove channel tips act as relative speedometers for valleynetwork growth. The resulting map (Figure 9) providesa snapshot of transient channel network evolution, andalso highlights variability in the hillslope response to this

Figure 11. (a) Shaded relief map showing locations of transects across the two main valleys. (b) Eleva-tion profiles along transects in Figure 11a.


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forcing, such as possible microclimatic control of soiltransport coefficients.[35] Another potential application that we have not

explored in this paper is determining whether a hillslope hasdeveloped in response to horizontal or vertical base levelchange. For example, an analysis of hillslope profiles likethat in section 4 might be a way to test whether a valleynetwork has been incised by vertical incision of streamprofiles or by approximately horizontal propagation ofchannel tips by groundwater sapping. Such an approachwould be particularly useful in planetary settings, whereobservations of erosional mechanisms are generally notavailable. It has been proposed, for example, that some ofthe fluvial networks on Saturn’s moon Titan, which appearto have short tributaries with orthogonal junction angles,formed through sapping erosion driven by subsurface flowof liquid hydrocarbons [Tomasko et al., 2005; Soderblomet al., 2007; Jaumann et al., 2010], whereas adjacent flu-vial networks have characteristics more consistent withsurface incision driven by channelized flow [Perron et al.,2006]. Although no topographic maps suitable for measur-ing high-resolution hillslope profiles currently exist forTitan, it is useful to consider the topographic measurements

that would be required to distinguish between these pro-posed formation mechanisms, particularly since valley net-works with sapping-like characteristics have been shownto 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 motionsolely on the basis of morphology, we compared the steadystate solutions derived in section 2 with a numerical modelof transient hillslope evolution in response to vertical chan-nel incision into a level plateau. Beginning with a levelsurface, the left boundary was lowered at a fixed rate,and the elevations of the other points on the grid evolvedaccording to

∂z∂t

¼ � ∂q∂x

; ð20Þ

where q, the volume flux per unit width, is given by eitherthe linear transport law, equation (2), or the nonlineartransport law, equation (10). Equation (20) was solved usinga forward time, centered space (FTCS) finite differencemethod, producing the transient hillslope profiles inFigure 12. For each of the final profiles, we chose z∞ to bethe total lowering of the base level below the initial plateau,and identified a value of L for which the steady stateanalytical solution for a horizontally retreating hillslope(equation (6) or (12)) closely matched the numerical solu-tion. The comparison in Figure 12 demonstrates that thetransient profiles driven by vertical base level lowering andthe steady state solutions for horizontal retreat are difficult todistinguish from one another for both the linear and non-linear transport laws. In principle, the two could be distin-guished by subtle differences in slope and the secondderivative, because the vertically lowering profiles do notfollow the relationships in equations (7), (8), (16) and (17).In practice, these differences are sufficiently subtle that theywould be difficult to resolve with field data. It is possiblethat the two-dimensional form of hillslopes responding tovertical lowering of a channel head may be more easilydistinguished from hillslopes responding to a horizontallyadvancing channel head, but such a comparison is beyondthe scope of this paper.[37] Given this difficulty, a more practical way to distin-

guish valleys formed by horizontally advancing channelnetworks from those formed by vertical incision would be tocompare downstream trends in valley cross-sectional form.Valleys formed by horizontally advancing channel networksshould have nearly uniform depth but progressive relaxationof valley sidewalls with downstream distance (Figure 11),because the valley floor experiences little erosion after thepassage of advancing channel heads. In contrast, valleysformed by vertical incision should deepen more substantiallydownstream, because locations further downstream gener-ally receive more fluid discharge and erode faster.

6. Conclusions

[38] We have presented analytical solutions for the equi-librium topographic profiles of hillslopes retreating into alevel plain in response to a horizontally migrating base level.The profiles have an exponential form, with the hillslope

Figure 12. Comparison of steady state solutions for hori-zontally retreating hillslopes (solid black lines) with tran-sient numerical solutions for hillslopes responding tovertical base level lowering (gray circles) for (a) the linearsoil transport law, equation (2), and (b) the nonlinear soiltransport law, equation (10). The dashed line is the initialcondition for the numerical model and z∞ for the analyticalsolutions. Parameters used in the numerical model wereK = 0.01 m2/yr, Dx = 1 m, Dt = 10 years, and a boundarylowering 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 of1 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 isequation (12) with L = 6 m.


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grading up to the elevation of the plain at a rate described byan e-folding length L, which is proportional to the ratio ofthe soil diffusivity to the base level migration speed. Theshape of the profile differs if soil transport rate increasesnonlinearly with the topographic gradient, but the solutionsconverge as L increases. By transforming the analyticalsolutions into linear relationships among distance, elevation,slope and curvature, it is possible to infer L from regressionanalyses of measured hillslope profiles. We compared theanalytical solutions with a numerical model of a retreatinghillslope, and found that it is possible to estimate L to within�10% of the true value even if the migrating base levelcreates a two-dimensional hillslope form, such as a conver-gent hollow above an advancing channel tip. The growth ofgroundwater sapping channel networks in the Florida Pan-handle has created many hillslopes with this form, and weused our analytical model to infer L for the hillslopes above201 channel tips in an area adjacent to the ApalachicolaRiver that has been surveyed by airborne laser altimetry. Themeasured profiles closely match the profiles predicted bythe analytical model, and illustrate the transition between theforms predicted by the linear and nonlinear transport laws.By combining the measured L values with a typical soildiffusivity, we estimate a modal channel growth rate of0.5 mm/yr, consistent with, but slightly faster than, previousestimates. A map of 1/L for all the surveyed hillslopesreveals that adjacent channel networks appear to be growingat different rates, and that south facing slopes experiencemore 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 wherematerial reaching the base of the slope is removed.

[40] Acknowledgments. We thankD. Rothman, A. Petroff, D. Abrams,A. Lobkovsky, and O. Devauchelle for the invitation to participate in fieldworkat the Apalachicola Bluffs and Ravines Preserve and for sharing the laseraltimetry map, which was acquired under contract by the National Center forAirborne Laser Mapping (NCALM). We also thank the Nature Conservancyfor granting access to the site. This study was supported by the MassachusettsInstitute of Technology and NSF award EAR-0951672 to J.T.P.

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