1998 tucker

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WATER RESOURCES RESEARCH, VOL. 34, NO. 10, PAGES 2751-2764, OCTOBER 1998

Hillslope processes, rainage density, and landscapemorphology

Gregory E. Tucker and Rafael L. Bras

Department of Civil and EnvironmentalEngineering,Massachusettsnstitute of Technology,Cambridge

Abstract. Catchmentmorphology nd drainagedensityare strongly nfluencedby

hillslopeprocesses. he consequencesf severaldifferent hillslopeprocessaws are

explored n a seriesof experimentswith a numericalmodel of drainagebasinevolution.

Five different modelsare considered,ncludinga simplediffusive-advectiverocess

transition,a runoff generation hreshold,an erosion hreshold,and two typesof threshold-

activated andsliding. hese different hillslopeprocesses lter both the visualappearance

of the landscape nd the predicted elationshipbetweenslopeand contributingarea. On

the basisof the different threshold heories,we derive expressionsor the relationships

betweendrainagedensityand environmental actorssuchas rainfall, relief, and mean

erosion ate. These relationships ary dependingon the dominanthillslope hreshold. n

particular, he signof the predicted elationshipbetweendrainagedensityand relief is

positive n semiarid, ow-relief andscapes nd negative n humid landscapes ominatedby

a saturation hresholdand/or in high-relief andscapes ominatedby simple hreshold

landsliding.

1. Introduction

The structure of catchment opographydepends o a large

extent on the interactionbetween hillslope and channelpro-

cesses.n recognitionof this, a number of quantitativemodels

have been developed in recent years to explore three-

dimensional drainage basin structure and evolution. These

modelshave made it possible o simulate he long-term mor-

phologicconsequencesf interactinghillslopeand fluvial pro-

cesses e.g.,Kirkby, 1986;Ahnert, 1987; Willgoose t al., 1991;

Howard, 1994;Tucker nd Slingerland, 997], o investigatehe

scalingpropertiesof three-dimensionalerrain [e.g., Rigon et

al., 1992],and to explore andscape volution n the contextof

large-scaleectonics e.g.,Lifton and Chase,1992;Beaumontet

al., 1992;Tuckerand Slingerland, 994].To date, however,most

models have relied on a highly simplified representationof

hillslopeprocesses. ittle attentionhasbeen paid, for example,

to the role of factorssuchas landsliding, ariable runoff gen-

eration, and vegetationcover n the contextof a catchmentas

opposed o a hillslopeprofile. In this paper, we address ome

of these ssues heoreticallyby modeling he influenceof dif-

ferent typesof hillslope hresholdon catchmentmorphology

and examining he implications or drainage density.

One of the most basicpropertiesof a landscape s its degree

of dissection, ften expressedn termsof drainagedensity.The

transition rom straightor convexhillslopes o concave alley

forms s widely understood o representa transition n process

dominance, but the nature of that transition has been debated.

Gilbert 1909] argued hat convex-concaveormsreflect a grad-

ual transition n processdominance rom creep to wash with

increasing istance rom a drainagedivide.Gilbert's model was

quantified n terms of a linear stabilityanalysis y Smith and

Bretherton 1972] and is based on the view that valleys orm

where flow convergence auses ill or gully excavation y run-

off erosion o outpace nfilling by diffusiveprocesses uch as

rain splash. he instabilitymodel hasbeen extended o include

Copyright1998 by the American GeophysicalUnion.

Paper number 98WR01474.

0043-1397/98/98 WR-01474509.00

finite-scale ffects Loewenherz, 991] and more generalpro-

cess aws Kirkby,1987, 1993;D. Tarboton,unpublishedmanu-

script, 1994].

An alternativeview is that valley and channel ormation are

controlledby geomorphic hresholds. numberof researchers

have argued, or example, hat hillslope-valley ransitionsoc-

cur where a threshold or runoff erosion s regularlyexceeded

during large storms Horton, 1945; Montgomery nd Dietrich,

1989;Willgooset al., 1991]. Data on channelhead ocations n

watersheds n the western United States show a correspon-

dencebetweenvalleygradientand channelhead sourcearea, a

finding hat appears o support he thresholdmodel [Montgom-

eryand Dietrich,1989;Dietrichet al., 1992], houghperhapsnot

uniquely. In fact, the threshold and instability theories need

not be mutually exclusive; ather, the two models constitute

end-member ases, nd any given andscapemaybe threshold-

dominatedor instability-dominated, ependingon the climate,

relief, geology,and stageof evolution Kirkby,1993, 1994].

A threshold or runoff erosion s only one of severaldiffer-

ent process hresholdshat may influence andscapemorphol-

ogy and drainage density. Drainage density and landscape

structure may alternatively be controlled, for example, by

thresholds or runoff generation [e.g., Kirkby, 1980; Ijjdsz-

Vdsquez t al., 1992;Dietrichet al., 1993] or by thresholds f

slope stability [e.g., Montgomery nd Dietrich, 1989; Howard,

1994]. Drainage density n particular may be controlled to

varyingdegreesby any of these hresholds, nd each different

thresholdmay producea different functional elationshipbe-

tween drainagedensityand factors elated to climate,geology,

and relief. For example, the slope models of Kirkby [1980,

1993] predict that under a humid climate, drainage density

shoulddecreasewith increasing elief, while under a semiarid

climate, drainage density might be independent of relief.

Howard [1997] presentsa detachment-limitedmodel in which

the relationshipbetween drainage densityand mean erosion

rate dependson (1) the dominanthillslope ransportprocess

(creep or landsliding) nd (2) the presenceor absence f a

threshold for runoff erosion.

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2752 TUCKER AND BRAS: HILLSLOPE PROCESSES, DRAINAGE DENSITY, AND TOPOGRAPHY

In this paper we developa theoreticaland numerical rame-

work to explore two related questions:

1. How do different typesof hillslope hreshold nfluence

the three-dimensionalmorphologyof the landscape,both vi-

suallyand in terms of the slope-areastatistic?

2. How might these different thresholds hange he func-

tional relationshipbetween drainagedensityand factorssuch

as rainfall, erosion rate, and relief?

The theory is implemented numerically n the form of a

simulation model of drainage basin evolution. We show that

each type of thresholdprocessmplies a different relationship

between drainagedensity expressedn terms of valley head

sourcearea) and external actors elated o tectonics, limate,

and material properties.Our emphasiss on valley densityas a

measureof landscape issection,ather than on the more vari-

able property of stream network density;however, the two

conceptsare closely elated, as was discussed y Montgomery

and Dietrich [1994a].As a startingpoint we consider he simple

case n whichhillslope-valleyopography esults rom compe-

tition between low-driven rosionand creep-typediffusive)

hillslope transport, as envisioned by Smith and Bretherton

[1972]. We then consideralternativecases nvolving our dif-

ferent typesof threshold: 1) a runoffgenerationhreshold, 2)

a simple opographicallyasedslopestability hreshold,3) a

pore pressure-dependent lope stability hreshold hat incor-

porates he effectsof flow convergence,nd (4) a threshold or

erosion by surfacerunoff.

2. Landscape Evolution Model

Computermodelingprovides ne of the few ways o study

the long-term morphologicconsequences f short-term pro-

cess aws and to conductcontrolled experiments n landscape

evolution.The model used n this study s a variant of GOLEM

[Tuckerand Slingerland, 996, 1997] and is basedon the prin-

cipleof continuityof sedimentmasswithin the landscape.Mass

continuity mplies that the rate of changeof elevation of a

point on the landscapes proportional o the local divergence

of sediment flux,

O (Oqsxqs•l

D-= - 55-x oy '

wherez is elevation, is time, U is uplift or base evel owering

rate,andqsxandqsyare sedimentluxesn thex andy direc-

tions, respectively.The erosion term can be divided into a

"wash" erm, representing article erosionby surface unoff,

and a generichillslopeerosion erm H(x, y, t), in whichcase

(1) may be rewritten

Oz OQs

ot = u •xx+ H(x,y, t), (2)

where Qs is the time-averaged lux of water-borne sediment

and x is a vector oriented along the directionof surface low.

In the numericalmodel, a continuousopographic urface s

approximated s a discrete attice of cells.Water and sediment

are routed across his surfaceby assuminghat water entering

each cell (either from upslopeor from direct precipitation)

flowsdownhill n the steepest irection owardone of the eight

surroundingneighborcells. n a landscapemantled by easily

detachable egolith, he water-bornesediment lux Q s is equal

to the total sediment ransport capacity referred to as the

transport-limited ase),whereas f the surfacematerialsare

resistant o detachment e.g., bedrock,coarseor cohesive ed-

iment,or a thickly egetated urface),Qs will generally e less

than he sedimentransport apacitythe detachment-limited

case). For purposesof this study, we assumea transport-

limited condition n the sense hat sediment s always rans-

portedat its capacity or a givenprocess; owever, he capacity

itself may be zero below a given detachment hreshold.

The sediment ransport ate per unit width by flowingwater,

qs, is modeled as a power function of slope S and specific

surfacewater dischargeq:

qs= ks(ktqS• - Oc)•, (3)

where ks, kt, or, j•, and •/are constantsand Oc s an erosion

threshold.A number of commonlyused sediment ransport

formulas anbe cast n thisgeneral orm. This equationmaybe

expressedn terms of total sediment ransport ate in a chan-

nel, Qs, and total surfacewater dischargeQ by assuminghat

the width of channelized low is related to discharge ccording

to [Leopold nd Maddock, 1953;Yalin, 1992]

W = kwQ% (4)

where 60 s typically --0.5 for alluvial streams that value is

assumed enceforth).We assumehat the width-dischargee-

lationship lsoapplies o overlandand rill flow, or which here

is some experimentalevidence Parsons t al., 1994]. Making

this substitution,

Qs= ks2Q'ø(kt2QO-'ø)St3Oc), (5)

where ks2 = kskw and kt2 = ktk,7, . If Oc = 0, this reduces

to

Qs= kfQmsn, (6)

wherekf = ks k •2 m = to + ( a - a ooy andn = /33,.

Sediment ransport s modeledusing 6) in all but one of the

experimentsdescribedbelow, with m = n = 2.

The simulationdomain consists f a 48-by-48cell grid that

representsan idealized square drainage basin. This domain

size s large enough o producehillslope-valleyopography ut

small enough o allow for reasonable omputation imes. The

initial condition s a nearly flat surfaceseededwith a small

randomperturbation n the elevationof each cell. The bound-

ary condition or the model s a single ixedoutlet n one corner

(Figure 1). The model s drivenby a uniform effective unoff

rate P and base evel lowering at the outlet at a rate U. The

runoff rate represents n effectivegeomorphicaverage,and

would not necessarily e equivalent to mean annual runoff.

The total surface-plus-subsurfaceunoff Qt(x, y, t) is assumed

to be proportional o contributing reaA,

Qt(x, y, t) = PA (x, y, t). (7)

In all but one of the casesconsideredbelow, all flow is assumed

to travel assurface unoff Q. In one experiment, he total flow

Q t is apportionedbetweensurfaceand subsurfacelow accord-

ing to a topographically ased runoff generationcriterion.

Note that in contrast o the linear relationshipn (7), it could

be argued hat long-termeffectivedischargemight nstead ary

less han linearlywith drainagearea (as is often the case or

floodsof a given ecurrencenterval).Were we to account or

such an effect, the only changewould be a reduction n the

effective alue of "m" in the expressionsor valleygradientand

sourcearea developedbelow.

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TUCKER AND BRAS: HILLSLOPE PROCESSES, DRAINAGE DENSITY, AND TOPOGRAPHY 2753

DRAINAGE AREA

(Logarithmic scale)

ZERO ':'"'"'----•"'""'""••••:.-• MAX.

Figure 1. Simulationn which hillslope-valleyopography volves hroughsimplecompetition etween

creep nd unoff rosion0c = 0). Eachmodel ixel s 40 m by40 m; thecatchmentize s •-3.6km .

Howard [1994] describes detachment-limitedmodel of

drainage asinevolution hat is similar o the presentmodel n

that it also represents unoff erosionas a power function of

slope and discharge. n fact, althoughwe assume ransport-

limited conditions as long as process hresholds re exceed-

ed), most of the findings eported n this paper would also

apply with minor modifications o Howard"sdetachment-

limited theory.

3. Simulations With Varying Hillslope Processes

In each of the simulations described below, the model is run

with a constant ate of baselevel owering at the outlet until a

Table 1. Model Parameter Values

Run Description Parameter* Value

1 no thresholds P 1 m yr 1

2 simplehresholdandsliding

3 saturation threshold

U 0.1 mmyr •

k•e 10 8 yr m 3

kd 10 2 m2 yr •

0c 0

•x 40 m

Sc 0.466 (25ø)

T l0 s m3 yr •

(cellwidth)1

0.8 (38.7

l0 s

5OO

1/2

2/3

3

3

4 pore pressure-.ependent tan qb

landsliding

5 erosion threshold

[p•Tb pwP

oc

kt2

*Unlessnoted,parametersor runs2-5 are the sameas hose or run

1

balancebetweenuplift and erosion s reached.This dynamic

equilibrium condition certainly does not apply to all land-

scapes, ut it has he advantage f providinga consistent asis

for comparison etweendifferent models.The simulations re

summarized in Table 1.

3.1. Process Competition

The long-term average rate of sediment transport by soil

creepand relatedprocessessuchas rain splash) s commonly

assumedo be proportional o hillslopegradient, eading o the

well-knownhillslopediffusionequation

H(x, y, t) = [Oz/Ot]cr kaV2z, (8)

where the subscript"cr" refers to creep erosion. Figure 1

shows simulated'landscapeormed by the simultaneous c-

tion of creep equation 8)) and surface low erosion equation

(5)), assumingc = 0 (Table 1). This s arguablyhe simplest

model one could magine hat produceshillslope-valleyopog-

raphy,and it is a case hat hasbeen consideredn somedetail

in previouswork [e.g., Kirkby, 1986; Willgoose t al., .1990;

Tarboton t al., 19.92; oward, 1994].The closest atural analog

would be a semiarid, ow- to moderate-relief andscapewith

predominantlyHortonian overland flow (spatially uniform

runoff generation),sparse egetation,and loosesurfacesoils.

Hillslope-valley ransitionsn this casenaturally end to occur

at those points where the rate of runoff-driven gullying out-

pacesgully nfillingby diffusiveprocessesFigure 1).

For a one-dimensionalillslope rofile, he point of transi-

tion from diffusion-dominated to runoff-dominated erosion

can be analyticallydefined as the point at which the two pro-

cesses re equally effective n transportingsediment [e.g.,

Howard, 1997]. At the point along the slopewhere both pro-

cesses re equallyeffective,eachprocessmust ransporthalf of

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10

1

lO

2

1003 ................................. 710 10 10 10

Contributing rea (rn2)

Figure 2. Plot of surface lope in the downstreamirection)

versus ontributing rea for the simulation ictured n Figure 1.

the total sediment lux,which or a steady tateprofile s equal

to the erosion rate U times the distance rom the divide,x. For

creep ransport equation 8)), this mplies hat

U

• Uxt kdSt or St= • xt, (9)

wherex t is the distance o the transitionpoint and S is the

slopeat that point.For runofferosion equation 6)), assuming

that runoff occurs as channelized flow in rills with the total

width of flow described y (4), that Q = Pxb (b is unit width),

and that 0c - 0,

• Ux=kfpmxmS•r St--2kfpm t1-m)/n.10)

Equating he two slopes, he distance o the processransition

point is givenby

( ) 1-n)/(m+n-1)

-- kf-1/(m+n-1)p-m/(m+n-1)k•/(m+n-1).11)

Note that this includes he casem = 2, n = 1 considered y

Kirkby1986],orwhichxt ckd/kf)/2.Equation11)may e

usedas an approximationor valley-head ourcearea in irreg-

ular topography Moglenand Bras, 1994; Howard, 1997] by

replacing he slope distancex t with upstreamcontributing

source area As:

( ) 1-n)/(m+n-1)

soc •-l/(m+n-1)pm/(m+n-1) k•/(m+n-1).

(12)

This expression houldbe considered nly an approximation

becauset neglectshe effectsof slopedivergence nd/orcon-

vergence whichwould end to alter the relative mportance f

creep ransport),nonlocal diffusive" ransportprocesseshat

might dampensurface nstabilities t shortwavelengthsLoe-

wenherz, 991],and the influenceof microtopographyn flow

aggregationDunneet al., 1995].What the analysis oespro-

vide, however, s an indicationof the relativedegreeof depen-

dence f source reaon the controllingarametersf, P, U,

andk a. The predicteddependenciesor the process-transition

model may then be comparedwith modelsbasedon different

typesof process hreshold.

Interestingly, he sign of the relationshipbetween source

area and the erosion ate U (equal to the uplift rate for a

steady tatebasin)depends n the slopeparametern. As long

as runoff-driven ediment ransportdependsmore strongly n

gradient than does diffusive ransport i.e., n > 1), source

area will tend to decreasewith increasing plift rate, and vice

versa.Many sediment ransport ormulas mplyn ranging rom

1 and 2 [e.g.,Howard, 1980;Moore and Burch, 1986;Willgoose

et al., 1991; Tuckerand Slingerland, 996], in which case he

theorypredicts weak nverse elationship etweenAs and U.

By comparison, he detachment-limitedmodel of Howard

[1997] predictsa positive elationshipbetweenerosion ate

and sourcearea. This reflects he assumptionhat the runoff-

driven erosionrate E is linearly proportional o bed shear

stress% which mplies oughly --- Sø'7.However,he de-

tachment-limitedmodelwouldalsopredictan inverse elation-

shipbetweensourcearea and erosion ate if erosion ate were

instead ssumedo be proportionalo •' raised o a powergreater

thanabout /2 (for example,3would ive oughly ---S2).

Figure 2 shows plot of slopeversus ontributing rea for

the simulation ictured n Figure 1. The slope-area lot shows

a characteristic turnover that reflects the transition from con-

vex hillslopes o concavevalleys.Similar trends have been

observed n some slope-areadata derived rom digital eleva-

tion data [e.g.,Turboton t al., 1991;Montgomerynd Foufoula-

Georgiou, 993; 14qllgoose,994], thoughmany data setsshow

a more complexpattern [e.g., jjdsz-Vdsqueznd Bras, 1995;

Tucker,1996].

3.2. Threshold Landsliding

In higher-reliefsettings,hillslopedenudation s typically

dominatedby landsliding, ith landslides ommonly riggered

onlywhen a threshold f soilor rock strength s exceeded. his

thresholddependences not well described y the linear dif-

fusionequation. n the next simulation, hallow egolith and-

sliding s modeledusinga numericalapproachbasedon the

thresholdconcept.To describe he combinedeffectsof soil

creepand shallowandsliding,he hillslope edimentlux term

may be written

qhs= kdS + oo S > Sc,

qhs= kdS + 0 otherwise

(whereSc is a critical alueor threshold radient hat depends

on materialstrength),which s related o the hillslope rosion

term in (2) by continuity f mass,

(Oqhsqhsh 13b)

(x, , ) =- • + Oy '

Equations 13) are implementedn the modelby applying n

algorithm hat triggers "landslide" t anypoint on the model

grid where S > Sc. Each such andslide emovesust enough

material to reduce he slope o the thresholdvalue Sc. The

material released rom each grid cell cascades ownslope

across he grid until it reachesa locationwhere S < Sc, at

which point it is deposited.This process epeats teratively

during each model time step until no oversteepened oints

remain. The net behaviorof this algorithm s similar o that of

models that represent andslidingusing nonlinear diffusion

[Kirkby, 987;Anderson nd Humphrey, 990;Howard,1994].

Figure3 shows simulationn whichhillslope rosion ccurs

by a combination f soil creep and threshold andsliding. e-

cause he thresholdslope s assumedo be constant cross he

landscape,he hillsidesend to be planarand do not develop

concavehollows.Diffusive creep dominateson the convex

ridge tops,where curvature s high. A natural analog o this

model might be an arid or semiarid,moderate- o high-relief

terrain with a thin regolith cover,suchas the badlands ormed

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Figure 3. Simulation n which hillslopesare dominated by simple threshold andsliding.

on MancosShale on the Colorado Plateau [Howard, 1997]. n

sucha setting, he thin regolithwould tend to saturatequickly

during arge storms,so that the effectivestrengthof material

on side slopeswould vary relatively ittle with topography.

At steadystate the process ransition rom threshold ands-

liding o fluvial erosion n the model occurs t pointswhere the

equilibrium slope for runoff erosion ust equals the stability

thresholdSc. Upstream of this location, luvial erosionalone

cannot maintain a gradient that is gentler than the threshold

gradient,and as a result, the slopesare insteaddominatedby

landsliding. he equilibrium luvial gradientcan be obtained

by noting hat the sediment ransport ate (equation 6)) must

equal the erosion ate U times the contributing rea. Substi-

tuting equation 7) for discharge,he equilibriumslope-area

relationship s

Seq A O-m)/n, (14)

where the exponent erm (1 - m)/n describesongitudinal

profile concavity nd is equal to the slopeon a log-logplot of

streamgradientversusdrainagearea [Willgooset al., 1991]. f

Sc is spatiallyuniform, the transition rom fluvial erosion to

landslidingccurswhereSeq = Sc. The source rea at this

transitionpoint is given by

kS)/(m-)

Z•s S• l-m). (15)

Sediment ransport heoryand data suggestm • [ 1, 2]. Thus,

unlike the creep-dominated ase, he theory predictsa strong

positive elationshipbetweenerosion ate and sourcearea. A

similar result was also found by Howard [1997] for a detach-

ment-limitedmodel. The positiveU - As relationship eflects

the fact that a higher erosionrate implies higher relief; with

higher relief a greater areal proportionof a catchmentwill be

susceptibleo slope ailure.

Figure 4a showsa slope-areaplot from the simulationpic-

tured in Figure 3. The landslide-dominated illslopes re rep-

resentedby a seriesof pointswith a uniform slopeangle,equal

to the stability hreshold.The transitionbetween he landslid-

ing and fluvial domains s abrupt, reflecting he assumption f

a spatiallyuniform Sc. In reality, one would expectsignificant

(a) 10

o10

Threshold slope

..

-2

1003 .......................... 6 ....... 7

10 10 10 10

ContributingArea (m2)

(b) 10 ...........................

16;03 104 105 106 107

Contributing rea (rn )

Figure 4. Slopeversuscontributingarea for simple andslid-

ing simulations.a) SpatiallyuniformSc. (b) Spatiallyuncor-

relatedrandomvariation n Sc (dashed inesshow he rangeof

variation n Sc).

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SURFACE FLOW RATE

(Logarithmic scale)

ZERO ................•' •••••'••• ......... MAX.

white = unsaturated

Figure 5. Effects of saturation-excessunoff on simulatedequilibrium andscape.

scatter about this line, due to spatial variations n material

strengthand variations n the time elapsedsince he last slide.

Figure 4b shows he slope-areaplot from a simulation n which

Sc varies n spaceas an uncorrelated andom ield. The mean

trend through he scatterof pointsslopes ownward,ndicating

that on average,hillslopepoints with a larger upstreamcon-

tributingarea tend to havea slightly ower gradient.The down-

ward trend reflects the mixed influence of points that are

dominatedby landsliding low Sc) and those hat are domi-

natedby runoff erosion high Sc).

3.3. Runoff Production by Saturation Overland Flow

One of the important limiting assumptionsn the previous

cases and in many models) is the assumptionhat runoff

production s uniform across basin.Suchan approximations

reasonable or arid catchments nd/or catchmentswith imper-

meablesoils,where nfiltration-excessverland low is the pri-

mary runoff generationmechanism. n most humid and semi-

humidenvironments, owever, unoff s generated rimarilyby

saturation excess and return flow and thus varies as a function

of topography nd soil thicknessDunne,1978].There is both

theoreticaland field evidence o suggesthat saturation hresh-

olds may influencedrainagebasinmorphology. irkby [1980]

andMontgomery nd Dietrich 1989]hypothesizedhat a topo-

graphicallydependentsaturation hresholdcan mposea limit

to drainagedensity.This problemwasexplorednumerically y

Ijjdsz-Vdsquezt al. [1992], who found that introducing sat-

uration threshold in a model of basin evolution altered the

predictedcatchment ypsometric urve. n the next simulation

we explore he long-termgeomorphic onsequencesf a topo-

graphicallydependentsaturation hreshold n a steadystate

catchment nd examine ts implications or drainagedensity.

For subsurfacelow parallel to the groundsurface, he sat-

urated subsurfacelow rate can be represented s the product

of soil transmissivity and surfaceslopeS,

Qss TS, (16)

where transmissivitys the depth ntegral of hydraulicconduc-

tivity [Bevenand Kirkby, 1979; O'Loughlin,1986]. For steady

state runoff the total flow is the product of contributingarea

and he effective ainfallrate (equation 7)). The overland low

component s the total flow minus the amount that travels n

the shallow subsurface,

Q=PA - TS, if PA > TS, (17)

Q = 0, otherwise.

Saturation occurs where

A/S -> T/P, (18)

with P now representingainfall rate minus osseso evapora-

tion and deeper drainage.When this runoff threshold s incor-

porated nto the landscape volutionmodel, the model pre-

dicts an abrupt hillslope-valleyransition Figure 5). In this

case, he catchmentmorphologys shapedby the interactionof

three rather than two topographicallydependentprocesses.

Given a high enough value of T/P, the model landscape

evolves oward a state n which saturationoccursonly within

the valleynetwork,with an abruptbreak n topography ccur-

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ring where return flow emergesat valley heads. The abrupt

hillslope-valleyransition s partly a result of the dependence

of subsurfacelow on topography.On hillslopes he increase n

gradient away rom drainagedividesalso ncreases roundwa-

ter flow capacity, hereby nhibitingsaturation.Ultimately, the

convergence f flow lines at some point downstream orces

saturation.The emergence f surface low at these ocations s

associated ith a large increase n erosiveenergy,which pro-

ducesan abruptreduction n slope.The abruptnature of valley

headscan be seen as a distinctoffset on a plot of slopeversus

contributing rea (Figure6a). Points o the left of the offsetare

unsaturated nd thereforecontrolledsolelyby creep erosion.

Points to the right are dominatedby fluvial erosion.

The abrupt hillslope-valleyransition n Figures5 and 6a is

alsoof coursepartly the result of usinga single ainfall rate to

approximatea natural sequenceof stormsof varying size and

intensity. This assumptioncan be relaxed by introducing a

stochastic omponent o rainfall in the model. Figure 6b shows

a slope-areaplot from a simulation in which the model is

driven by a seriesof random storm events.Rainfall intensity,

duration, and interstorm arrival time are chosen at random

from an exponentialdistribution or each storm [Eagleson,

1978; Tuckerand Bras, 1997]. The main effect of rainfall vari-

ability s to blur the otherwise brupt hillslope-valleyransition

and remove the sharp offset between hillslope and channel

points (Figure 6b). Notably, the transitionbetweenhillslope

points thosedominatedby diffusive ransport)and the fluvial

points thosedominatedby runoff erosion) n Figure 6b still

occursclose o the threshold or saturationby the mean rain-

fall event. Abrupt basal concavitysimilar to that in Figure 5

was also a feature of the hillslopeprofile modelsexploredby

Kirkby 1993],which ncorporated ariability n rainstormsize.

In the absenceof other significant hresholds, alley-head

source rea n this model s controlledby the saturation hresh-

old when T/P > ka/U (if ka/U is large, the minimum source

area will be imposedby diffusive ransport,as in run 1). For

this case, he sourcearea can be approximatedas that area for

which he equilibriumchannelgradient s equal to the gradient

requiredor saturationo occur Seq --Ssat).Theslope t the

pointof saturations givenby (18). Settingt equal o Seq

(equation 14)),

As = A 1-m)/n

,

(19)

or

rn/(m n- 1)U1/(m n- 1)

As - kl/(m+n_l)pm+n/(m+n_l).20)

Equation 20) describeshe point of intersection etween he

saturation curve for the mean storm and the fluvial scaling

curveon a slope-area lot (Figure 6) and predicts hat source

area for a saturation-limitedcatchmentshould be positively

correlated with soil transmissivity nd erosion rate, and in-

versely correlated with mean rainfall intensity and material

erodibility.The inversedependence n erosion ate U, which

was alsoobserved y Kirkby 1993] n a one-dimensional lope

model, implies that drainagedensityshoulddecreasewith in-

creasing elief in humid climates.The physicalexplanation s

that all elsebeing equal, a higher erosion ate impliessteeper

slopes,which in turn implies an increasedsubsurface low

capacityand decreasedoverland flow production. However,

higher erosion ateswould also ikely be associated ith thin-

1

(a) 10

10

10

2

lO

lO

(b)

10

10

,'

• o ogarithmicinnedverage

3 4 5 6 7

10 10 10 10

Contributing rea (rn2)

ß ,0

10 10

Contributingrea (rn )

Figure 6. Plots of slope versuscontributingarea for simula-

tionsdrivenby saturation-excessunoff production. a) Deter-

ministicmodel.Note the offsetat the point of saturation. b)

Model driven by discretestorm eventswith random, exponen-

tially distributed ntensity,duration, and interstormperiod.

ner soils, and thus reduced transmissivity. trictly speaking,

therefore, the erosionrate-sourcearea relationshipshould

only apply to cases n which most subsurfacelow is carried n

the uppermostpart of the soil column, n which caseT would

be relatively nsensitive o soil depth.

3.4. Pore Pressure-ActivatedLandsliding

The topographicallyasedsubsurfacelow model equations

(16) and (17)) has alsobeen used to model the influenceof

variations n water table height on landslidesusceptibility.

Montgomery nd Dietrich [1994b]developed model of pore

pressure-inducedhallow andsliding y combininghe infinite

slopestabilitymodel or shallow oilswith equation 16), under

the assumption f steady tate low (equation 7)). The result-

ing criterion predictsslope nstabilityat locationswhere

--->---- 1 sin0, (21)

b pwP tan

where 0 is the slopeangle, b s the angleof internal riction,b

is contour width, and Ps and Pw are the densitiesof soil and

water, respectively.Equation (21) applies only to slopes

steeper han the maximumstableangle or fully saturated oils,

which s givenby [Montgomerynd Dietrich,1994b]

tan 0-> [(Os- Pw)/Ps]an (b • (1/2) tan

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DRAINAGE AREA

(Logarithmic scale)

ZERO -:-:•• .... MAX.

Figure 7. Simulation n which landsliding s modeled using the pore pressurestability criterion of Mont-

gomery nd Dietrich 1994b].Elevatedpore pressure roundpointsof flow convergenceeads o the formation

of hollowsby landslideerosion.

To model the effectsof pore pressure-inducedandsliding

on catchmentmorphology, 21) is used as the slopestability

criterion n the landscape volutionmodel (Figure 7). At each

iteration, the critical drainageareaA c is computedaccording

to (21). If A > A c, a landslide s initiated. Landslidedebris

then cascades ownslopeuntil reachinga locationwhere S <

(1/2)tan 45 the maximum tablegradient or saturated oil).

The simulated andscape Figure 7) is characterized y the

formation of landslide-dominated hollows and an overall in-

crease n valley density elative to the previousexperiments.

Unlike the uniform hreshold ase Figure 3), landsliding ere

is restricted o the hollowsand lower slopes,where the pore

pressure ends o be high. This model might be analogouso

moderate-relief, soil-mantled andscapes uch as the Marin

County region studiedby Montgomery nd Dietrich [1989]. A

slope-areaplot from the simulation Figure 8a) reflects he

shape of the slope stability envelope,and the distribution s

similar to that predictedby the form instabilitymodel of D.

Tarboton unpublishedmanuscript, 994). The slope-areadi-

agramcanbe divided nto four regions from right to left): (1)

the fluvialscaling rend, (2) a flat region hat represents atu-

rated andsliding,3) a second ownward caling rend, repre-

senting partially saturated landsliding,and (4) a level or

slightly ncreasingrend that reflectsa combination f diffusion

and unsaturated andsliding.Notably, these scalingzonesre-

semble he zones dentified n digital elevationmodel (DEM)

data by Montgomery nd Foufoula-Georgiou1993] and Ijjtisz-

Vtisquez nd Bras [1995]. Spatialvariations n soil transmissiv-

ity (Figure8b) and n tan 45 Figure8c) introduce catter n the

slope-area elationship ut do not significantly lter the mean

trend.

The landslide-dominated hollows in the simulation shown in

Figure 7 developbecause he convergence f subsurfacelow

around initial perturbations educes he stability threshold,

therebyacceleratinghe local erosion ate and attractingmore

flow. In that sense, he modified andslidingmodel exhibitsa

type of form instabilitysimilar o that considered y Smithand

Bretherton 1972]. However, the landslidingmodel does not

predict form instability under all circumstances. ather, the

tendency oward form instabilitydependson the parameters

governing runoff generation and erosion. Figure 9 illustrates

three possibleoutcomes or a hypotheticalcatchment. The

figure depicts he equilibriumslope-area elationship or run-

off erosion RS) and diffusion DS), alongwith thresholdsor

saturation SAT) and slopestability SLP). Points n the sim-

ulation model will tend to follow one of the two process qui-

librium lines, exceptwhere they are limited by the slope sta-

bility threshold.Form instabilitywill tend to occur when the

equilibriumgradient or the rate-limitingprocess runoff ero-

sion, andsliding, r diffusion)decreases ith increasing on-

tributing area [Tarbotonet al., 1992]. By this criterion, runoff

erosion line RS) is an unstableprocess, hile diffusion line

DS) is a stabilizing rocess. ore pressure-drivenandsliding

can act as either a neutral processor as an unstable one,

dependingon the catchmentproperties and position n the

landscape. he processs neutral where the thresholdgradient

is constant reflectedby a horizontal ine in Figure 9) and is

unstablewhere the thresholdgradient decreaseswith increas-

ing contributing rea (the curvingportion of the stabilityen-

velope n Figure 9).

If the saturation hresholdratio T/P is small (becauseof

impermeable oils, or example) Figure9a), the steeper lopes

will tend to be uniformlysaturatedduring storms, n which case

the solution educes o the uniform Sc case as in the simula-

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tion in Figures and 4). If T/P is arge Figure9b), sideslopes

will tend o have ow pore pressuresnd hereforewill tend not

to fail until they approach he stability hreshold or dry soil

(tan 4•). Hollowsmay form where sufficient low accumulates

to reduce he slope stability hresholdand/or generateocca-

sionalwasherosionduring arge storms thesehollowswould

correspondo the point labeled"valleyheads" n Figure 9b).

The tendency or landsliding o excavatevalleys would be

greatest, owever, n the intermediate ase Figure 9c). Here

diffusiongivesway to landsliding long he curvingportion of

the slope stability curve, which representspositions n the

landscapewhere the landslide hresholddependsstronglyon

drainagearea. The simulation n Figures7 and 8 corresponds

to this hird case.By comparison,he parameters erived rom

DEM databyDietrichet al. [1992]seem o fall between he two

casesdepicted n Figures9b and 9c.

3.5. Erosion Threshold

Horton [1945] hypothesized hat an unchanneledhillside

representsa "belt of no erosion"within which overland flow

strength s below a thresholdnecessaryor erosion.Montgom-

eryandDietrich 1989, 1992]andDietrichet al. [1993]presented

data on channelhead locationsand terrain morphology hat

are consistentwith this hypothesis. he conceptof a runoff

erosion hresholdhas been explored n severalmodels Will-

gooseet al., 1991; Howard, 1994; Kirkby, 1994; Rinaldo et al.,

1995;Tucker nd Slingerland,997]. n run 5 (Figure 10), the

threshold erm in equation 5) is retained.The existence f a

(a) 10 ................................

"- ' "_'"'",'•.'t;,.........

• .

03 "•'"•'x.

101

103

(b)

1oø

LU

0310.1

(c)

..

10 10 10 10

10ø

LU

• 10 1

03

øJ,.g:ø%,,

'11;-I'

logarithmicinned verage

10 1O 10 10

103

%.

o = logarithmic innedaverage o

10 10 1;6 10

DRAINAGE AREA (sq. m)

Figure 8. Slope versus contributing area for simulations

drivenby pore pressure-sensitiveandsliding.a) Spatially ni-

form T and Sc. (b) Spatiallyuncorrelated andomvariation n

Sc. (c) Spatiallyuncorrelated andomvariation n T.

(a)

o

o

Valley heads

log( drainage area )

(b)

SLP / / •,5' heads

log( drainagearea )

(c)

Valley • •.o /

heads-• ,•'.

log( drainagearea )

SLP = slopestabilityhreshold

RS = runoff rosion quilibriumlope

DS = hillslope iffusionquilibriumlope

SAT = saturation threshold

Figure 9. Effect of slope stability envelope on slope-area

distribution.Valley forms are predicted to correspond o

pointsof form instability,which occurwhere the equilibrium

slopedecreases ith increasing rainage rea. (Note that by

definition he SAT line alwaysntersectshe break n the slope

stabilitycurve).

runofferosion hresholdn the model Figure10) lendsgreater

curvature to the hillslope forms, which are now controlled

solely by creep transport. Introduction of a threshold also

changeshe predicted elationship etween alley-head ource

area and parameters such as erosion rate [Kirkby, 1994;

Howard,1997].Using he samederivation hat led to equation

(12), the source rea for the threshold ase s givenby

kf m]A[(1-m)/Y]-/3kt--• -½+/3)=aa

(22)

where/5= a(1 - co). hiscomplicated xpressionescribeshe

generalcase n whichvalley ormationbeginswhere (1) the

thresholds exceeded nd (2) there s sufficient rosive ower

that diffusive rocesseso not inhibit ncision.A simplerend-

membercase s that in whichvalley ormationbeginsclose o

the point at which the erosion threshold s exceeded.This

might be the case, or example, n landscapes ith a resistant

vegetation mat [Dietrichet al., 1993; Kirkby, 1994; Howard,

1996].The source rea at the pointwhere he equilibrium ill-

slope radient ecomesuststeep noughhatkt2P•AaS3 - 0c

(equation 5)) is givenby

k •./(/5/3) 1/(15+/3)

As= •/(•+/3)/(•+U/3/(•+/3) (23)

P /3) ,

8/16/2019 1998 Tucker

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Figure 10. Simulated andscapewith a finite runoff erosion hreshold.

where •5and/3 are typically --1/3 and ---2/3,respectively e.g.,

Tuckerand Slingerland, 997]. This would be appropriate or

large thresholds,when the thresholddominatesover stability

considerationsKirkby,1993].Note that 0c representshe ero-

sional resistance of the unchanneled surface rather than that of

sedimentgrainswithin an established hannel,whichmight be

lower. This sourcearea expressionmplies hat sourcearea in

a threshold-limited atchmeritwill tend to vary positivelywith

hill. lope iffusivitya function f climate ndmaterial)and 0c

(likely a function of vegetationdensity)and to vary inversely

with erosion rate and effective rainfall rate. The threshold

control on valley-head ocations s illustrated n Figure 11.

It is interesting to note that when 0c > 0, the model no

longer predictsa simplepower law relationshipbetween slope

and drainagearea for an equilibrium iver network though n

practice, he slope-area elation might still closely esemblea

power aw). With the inclusionof a threshold erm, the equi-

librium slope-area elationship or the drainagenetwork is

S -- A l-m)/7t_,7A 15 (24)

where the prime on 0'c indicates that it now represents a

threshold for sediment entrainment within an established

channel. If the threshold term is small, the solution reduces to

S crAt(1-m)/'•l equation14)),whereasf the hresholderm

dominates, crA-("(•-ø'))/•. These wo end-member ases

might correspond o fine-bed alluvial and threshold gravel

channels, espectively Howard, 1987]. Both casesdescribea

power-lawrelationshipbetween slope and contributingarea,

but generally with different exponents.On the other hand,

recent modelingwork with multiple grain sizes Gaspariniet

al., 1997] suggestshat the dependenceon mean grain size

implied by (24) may not apply to streamswith bimodal bed

sediment mixtures.

4. Discussion

An important mplicationof the modelswe have analyzed s

that the relationship etween drainage ensity nd factorssuch

as climate, geology,and relief will vary in a predictableway

dependingon the nature of geomorphicprocesses perating

on hillslopes.We have quantified his relationship n terms of

valley-headsourcearea rather than directly n terms of drain-

age densityD•, but the two concepts re closely elated.Mo-

glenet al. [1998] analyzed he relationshipbetweensourcearea

and D• in DEM data for sevenbasins n the United States.For

each data set, they varied the sourcearea used to extract the

channelnetwork and compared t with the resultingdrainage

density.The results showeda power law relationship,D• •

A j-•, with • • 0.5, aswould be expectedgiven he dimensions

: I IIq'

::::.".,.,,,

0 , "%

1(52 ...................................

10 lC• 10 14

Contributingrea rn )

Figure 11, Slope-area plot from model run with erosion

threshold shown n Figure 10).

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Table 2. Predicted aluesof K n the RelationDd ocR K,Assuming d oc1/A /2

Case

K (• = 1/2, K (•: 1/3, • (•: 2/3,

/3-- 2/3, /3 = 2/3, /3 = .2/3,

3' = 3)* 3' = 3)? 3' = 3/2)$

Creep, uniform runoff generation,Oc = 0

Creep, uniform runoff generation,{}Oc> 0

Creep, saturation-excessunoff productionll

Landsliding, patiallyuniform Sc

1/3 2/5 0

4/7 2/3 1/2

(-1/3) (-2/5) (-2/3)

-1 -2 -1

*Correspondso rn = 2, n = 2 when 0c = 0 (this paper,Kirkby 1987] and Willgooset al. [1991]).

?Correspondso rn = 3/2, n = 2 when Oc = 0 [e.g.,Howard, 1994].

$Correspondso the rn - 3/2, n = 1, caseconsidered y Kirkby [1993] with an additionalconstraint

on flow width (equation 4)).

{}Applicable nlywhen the threshold tself s not the biggest ontributor o relief (when he threshold

term in equation 24) is smallrelative o the other term).

IIAssumes is independentof relief.

of length and area. For the sake of argument,we will assume

thatDd oc •-ø's.Assuminglarger alueof '1,such sDd

A•- • [MontgomeryndDietrich, 989],would mplya stronger

relationshipbetween drainagedensityand the various actors

that control sourcearea, but would not alter the sign or the

relative magnitudeof the dependenceon different factors.

4.1. Drainage Density and Relief

Of particular interest s the relationshipbetween drainage

densityD,• and topographic elief R. Topographic elief is not

one of the governingariablesn our theoretical odel or

source area, but it can be deduced from three of those vari-

ables,U, P, and kf. There existmanydifferent elief-based

measures, uchas otal relief (maximumelevationabovecatch-

ment outlet), relief ratio (total relief divided by catchment

length), and local relief (maximumelevationdifference n an

analysiswindowof specified ize). n a large catchment,most

of the reliefwill arise rom the elevation hange long he main

streambetween he outlet and headwaters.Recognizing his,

we can define a "drainage network relief" R,• as the total

elevationifferenceetweenheoutlet nda poin{ longhe

main streamat a distance L upstreamof the outlet, where L

is the total length of the main stream and p is a suitable

proportion e.g.,90%). The main streammaybe definedas he

longeststream n the basin [Hack, 1957] or as the streamwith

the largestcontributing rea at any givenconfluence. he

statistichas severaladvantages:t is relativelyeasy o measure

in a DEM; it can be readilyobtained rom theoreticalmodels;

and because treamprofilesare typically ogarithmic, t does

not depend stronglyon catchmentsize. On the other hand,

because f the assumptionhat mostof the relief occurswithin

the luvial etwork,heR,•statisticrovidesnapproximation

of total relief only at spatialscales hat are significantlyarger

than typicalhillslope-valleywavelengths.

Drainage network relief R d can be derived rom the model

as follows.Assuminga typicaldrainagebasingeometry, t can

be shown hat (14) impliesa power aw relationship etween

relief and steadystate erosion ate, U,

e oc kp m)n•l, (25)

with a dependence n basin size only if the slope-areaexpo-

nent (1 - m)/n is significantly ifferent from 0.5. Among

other things, his relationshipprovidesa meansof comparing

the theoreticalmodelwith empiricalstudies f denudation ate

as a function of relief [Schummand Hadley, 1961; Schumm,

1963;Ahnert, 1970; Summerfield nd Hulton, 1994]. Equation

(25) may be substituted nto the source-areaexpressions

(equations12), (t5), (20), and (23)) to solve or source rea as

a function of relief. Table 2 listsvaluesof the parameter K in

the power aw relationD,• ocR,• for typicalvaluesof m, n, 3,

/3, and3,.Note hatn isderived nder heassumptionhatk,•,

kf, P, andotherparametersreprimarily ontrolled yclimate

and/or material propertiesand therefore do not vary system-

aticallywith R,•. For most of the parameters, his s probablya

reasonableassumption.Soil transmissivity , however,might

be expected o varywith relief owing o associated ariations n

regolith hickness. or this reason, he saturation-excessunoff

modelshouldbe considered pplicable nly o landscapes ith

thick soils (which generallyexcludesmountainous errain)

and/or soils in which most subsurface runoff travels in the

upper ew centimeters. or such andscapeshe model predicts

an inversecorrelationbetween elief and drainagedensity,as

compared with landscapesdominated by Horton overland

flow, in which case he model prescribeshe opposite elation-

ship. This finding is consistentwith Kirkby's 1987] stability

analysis, which predicted a positive relationship between

sourcearea and valley gradient or humid (saturationdomi-

nated) settings nd an inverse elationshipor semiarid infil-

tration excess-dominated)ettings for the casen = 2).

In the first case n Table 2 (creep-dominated,Hortonian

runoff with a negligible hreshold),drainage density s inde-

pendent of relief only when n •- 1. Thus an observation hat

drainagedensitydoes vary with relief could not by itself be

taken as evidence or thresholdbehaviorwithout ndependent

knowledgeof the climate and of the physical elationshipbe-

tween slopeand sediment ransportefficiency.

Notably, although he theory predictsa positivecorrelation

between D,• and R,• for Hortonian, creep-dominated and-

scapes,t also predictsa strong nversecorrelation or land-

slide-dominated nes. Such a difference appears o be sup-

ported by observations.n low- to moderate-relief settings,

moststudies howa positivecorrelationbetweenD,• and relief.

Schumm 1956] found a positivecorrelationbetweenD,• and

the relief ratio in a humid badlands.Montgomery nd Dietrich

[1989, 1994a] eporteda positivecorrelationbetweendrainage

densityand valley gradient n stream-headhollows n the west-

ern United States. A positive correlation between drainage

density nd slopehasalsobeenobservedn experimentalwork

[Schumm t al., 1987]. n contrast o the data from thesemostly

(thoughnot exclusively)ow- to moderate-reliefcases,Oguchi

[1997] found an inverserelationshipbetween relief and the

densityof deepvalleys n mountainouserrain n Japan. Ogu-

8/16/2019 1998 Tucker

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Slope/area trend for WE38 catchment

1.000 - ' '

o.oo -+

0.010

0.001

101 102 103 104 105 106 107

Drainage Area (square meters)

Figure 12. Slope versuscontributingarea obtained from a

5-m-resolution igital elevationmodel (DEM) of the WE-38

watershed,centralPennsylvania, .S.A. Each point represents

an averageof 100 consecutive EM pixels,arranged n order

by contributing rea.

chi also analyzed different methods for measuringD a and

showed hat the results depend on the criteria used. If only

well-definedvalley orms are identified as drainages, here is a

clear inverseD a-R relationship,whereas f all contour nflec-

tions are considered, the relationship disappears. In the

present contextwe are concernedwith D a as a measureof

landscape issection, o the first method s more meaningful.)

The model resultssuggesthat the varyingbehaviorof Da with

respect o R may reflect differencesn process ominanceon

hillslopes.Given that landsliding s active n the basinsstudied

by Oguchi 1997] and colleagues, e suggesthat their findings

reflect an increasingareal proportion of planar, landslide-

dominatedhillslopes n steeper errain. This interpretation s

also consistentwith the study of Howard [1997], in which he

documented a change from a positive to a negative R-Da

relationshipwith increasing elief in badland andscapes n the

Colorado Plateau, U.S.A.

4.2. Drainage Density and Climate

Each of the modelswe have considered redictsa positive

correlation between D d and the rainfall parameter P. The

predictedsensitivitys greatest or landslide-dominated atch-

ments and for catchments in which channel network extent is

limited by a saturation hreshold.However, he predictedsen-

sitivity o P doesnot account or changesn other factors hat

are also ikely to be correlatedwith climate.Most studies f the

relationshipbetween climate and D d show either an inverse

relation [Melton, 1958] or a humped curve in which D a in-

creaseswith mean annualprecipitationunder semiarid o arid

climates,and decreases nder more humid climates Gregory

and Gardiner,1975;Moglenet al., 1998].Moglenet al. [1998]

emphasizehe importanceof vegetationas a limiting actoron

D a. In the context of this model, vegetationwould serve to

increaseboth surface esistanceOcand transmissivity , both

of which end to decrease a (equations 23) and (23)). There

is alsoevidence hat creepdiffusivity a maybe correlatedwith

climate. ield estimatesange rom -2 x 10 4 m2/yr n the

aridAravaValley Enzel tal., 1996] o -10 -2 m2/yrn humid

to subhumid ettings n the United States Nash, 1980;Hanks

et al., 1984;Rosenbloom ndAnderson,1994].Higher diffusiv-

ity under a humid climate would tend to decreaseD d, and

converselyor an arid climate. ncreasedweatheringrates un-

der a humid climatewould also end to increase egolith thick-

ness, hereby increasingT and decreasingD a. All of these

factorsmay contribute o counterbalancinghe positiveeffects

of increased ainfall on Dd. In addition,changesn precipita-

tion variabilitywould be likely to influencedrainagedensity

[Tucker ndSlingerland,997]. heparametersf andP lump

information about both mean rainfall and rainfall variability,

so that P is not necessarily quivalent o mean annual rainfall

or runoff. Given the importanceof large, relatively nfrequent

events n shaping andscapes,t is possible hat the geomor-

phicallyeffective unoff rate (if indeed here is sucha thing)

actually ncreasesunder an arid climate with highly variable

precipitation e.g.,Knox, 1983].

4.3. Process "Fingerprints" in Slope-Area Data

Each of the different processes xploredhere has an impact

on the slope-area elationship,suggestinghe possibility hat

slope-areadata may be used o discriminate etweendifferent

geomorphicprocess egimes Tarbotonet al., 1992;Dietrich et

al., 1993; Montgomery nd Dietrich, 1994] as well as to test

theories of slope and landscapedevelopment [e.g., Kirkby,

1993].n eachof themodels onsidered,illslope.process(es)

introduceone or more inflectionpoints n the mean slope-area

trend. Such inflectionshave been observed n many data sets

[Tarbotonet al., 1991; Montgomery nd Foufoula-Geourgiou,

1993;Wiltgoose,994; jjdsz-Vdsqueznd Bras, 1995],and their

significance as been debated. jjdsz-Vdsqueznd Bras [1995]

recognized hat a commonpattern in many (thoughnot all)

data setswas the presenceof two apparentlyparallel scaling

trends,with the larger one representing he main valley net-

work (Figure 12). Of the variousmodels, he slope-arearend

producedby the landslide-dominated imulationsmost closely

reproduces his pattern. Montgomery nd Foufoula-Georgiou

[1993]alsonoted the presence f multiple nflectionsn slope-

area data, thoughwith somewhatdifferentpatterns han those

in Figure 12 and in the data of Ijjdsz-Vdsqueznd Bras [1995],

and they interpreted the larger-area nflection as a transition

from debris flow-dominated channels to alluvial channels. The

data of Figure 12 are from a moderate-relief catchment n

Pennsylvania n which landsliding s not today a significant

process.However, t is possible hat the landslide-like signa-

ture" reflectsmassmovementby solifluctionunder a perigla-

cial climateduring he last glaciation.Alternatively, he pattern

may be controlledby low-gradient, ow-area pixelswithin the

largervalleys.Further simulationmodelingmay help elucidate

the origin of these ypesof pattern in slope-areadata, though

at present his approach s hamperedby a lack of DEM data of

sufficient esolution o properlymap andscape tructure t the

hillslopescale.

5. Conclusions

The modelswe have explored n this paper providea frame-

work for understanding he impact of different hillslopepro-

cesses nd process hresholdson the structureof terrain. A

numberof important ssues emain unanswered,ncluding he

role of vegetation,whichhasa clear mpacton drainagedensity

and sedimentyield, and the role of short-termvariability in

rainfall and runoff. Our analysis as considered nly equilib-

rium landscapes, nd it shouldbe emphasized hat thesemod-

els apply o landscapeshat have had sufficient ime to adjust

8/16/2019 1998 Tucker

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to prevailing onditionsbase evel owering ate, rainfall,etc),

but do not necessarily escribewhat might happen for exam-

ple to drainage density) in the short term in response o

changes n environmentalconditions.Such issuesmight be

addressedhrougha combination f analysis f high-resolution

DEM data, dating of geomorphic eatures in specific and-

scapes, nd modelingof transient andscape tatesand process

interactions.

Acknowledgments. This researchwas supportedby the U.S. Army

ConstructionEngineering Research Laboratory (DACA 88-95-R-

0020) and the Army ResearchOffice (DAAH 04-95-1-0181).This

work hasbenefited rom discussionsith Ying Fan, Nicole Gasparini,

Stephen Lancaster, Jeff Niemann, Daniele Veneziano, and Kelin

Whipple, and from thoughtful eviewsby Alan Howard, Glenn Mo-

glen, David Montgomery,and David Tarboton.

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R. L. Brasand G. E. Tucker correspondinguthor),Departmentof

Civil and EnvironmentalEngineering,Room 48-108, Massachusetts

Institute of Technology,Cambridge,MA 02139. (e-mail: gtucker@

mit.edu)

(ReceivedOctober27, 1997;revisedApril 22, 1998;

acceptedMay 1, 1998.)

1998 tucker

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