Evidence for Nonlinear, Diffusive Sediment Transport on Hillslopes and Implications for Landscape Morphology Joshuaj

WATER RESOURCES RESEARCH, VOL. 35, NO. 3, PAGES 853-870, MARCH 1999

Evidence for nonlinear, diffusive sediment transport on hillslopes and implications for landscape morphology JoshuaJ. Roering, JamesW. Kirchner, and William E. Dietrich Department of Geologyand Geophysics,University of California, Berkeley

Abstract. Steep, soil-mantledhillslopes evolve through the downslopemovement of soil, driven largelyby slope-dependenttransport processes. Most landscapeevolution models representhillslope transport by linear diffusion,in which rates of sedimenttransport are proportionalto slope,such that equilibriumhillslopes should have constantcurvature betweendivides and channels.On many soil-mantledhillslopes, however, curvature appearsto vary systematically,such that slopesare typicallyconvex near the divide and becomeincreasingly planar downslope.This suggeststhat linear diffusionis not an adequatemodel to describethe entire morphologyof soil-mantledhillslopes. Here we showthat the interactionbetween local disturbances(such as rainsplashand biogenic activity)and frictional and gravitationalforces results in a diffusivetransport law that dependsnonlinearly on hillslopegradient. Our proposedtransport law (1) approximates linear diffusionat low gradientsand (2) indicatesthat sedimentflux increasesrapidly as gradient approachesa critical value. We calibratedand tested this transportlaw using high-resolutiontopographic data from the Oregon CoastRange. These data, obtainedby airborne laser altimetry, allow us to characterizehillslope morphology at •2 m scale.At five smallbasins in our studyarea, hillslopecurvature approaches zero with increasing gradient,consistent with our proposednonlinear diffusive transport law. Hillslope gradientstend to clusternear valuesfor which sedimentflux increasesrapidly with slope, suchthat large changesin erosionrate will correspondto small changesin gradient. Therefore averagehillslope gradient is unlikely to be a reliable indicator of rates of tectonicforcing or baselevellowering. Where hillslopeerosion is dominatedby nonlinear diffusion,rates of tectonicforcing will be more reliably reflected in hillslopecurvature near the divide rather than averagehillslope gradient.

1. Introduction of subsurfaceflow and transient inputs of rainfall. Because shallowlandslides typically scour the soil mantle, thus ampli- The morphology of hillslopesreflects the erosional pro- fying the topographicconvergence that may contributeto their cessesthat shape them. In steep, soil-mantled landscapes, initiation, they tend to inciseterrain rather than diffuse mate- ridge and valley topography(Figure 1) is formed by the inter- rial across it. action between two general types of mass-wastingprocesses: This paper exploresthe mechanicsof diffusiveprocesses and hillslopediffusion and landsliding.Diffusive processes, such as their influenceon hillslopemorphology. In soil-mantledland- rainsplash,tree throw, and animal burrowing,detach and mo- scapes,slopes tend to steepenwith distancedownslope from bilize sediment,moving soil graduallydownslope [Dietrich et the drainage divide [Gilbert, 1909]. On sufficientlylong hill- al., 1987;Black and Montgomery,1991; Heimsath et al., 1997]. slopes,slope anglestend to convergetoward a limiting gradi- These disturbance-drivenprocesses have been termed diffu- ent, before shallowingat the transitionto the valley network sive becausethe resultingsediment flux is thought to be pri- [Strahler,1950; Penck, 1953; Schumm, 1956; Howard, 1994a,b]. marily slope-dependent.Additional diffusiveprocesses include Many soil-mantledhillslopes are not only convexin profile but the cyclicwetting and dryingof soils,as well asfreeze/thaw and also convexin planform, which createsdivergent pathways of shrink/swellcycles [Carson and Kirkby,1972]. Soil-mantled hill- downslopesediment transport [Hack and Goodlett,1960]. Hill- slopesare also shapedby shallowlandslides, which commonly slopeswith profile and planform convexityoccur in diverse begin in topographicallyconvergent areas and may travel long climatic and tectonic regimes,suggesting that diffusive sedi- distancesthrough the low-orderchannel network, scouring and ment transportprocesses control hillslope morphology in many depositingsediments along their runout path [e.g., Pierson, different settings. 1977; Sidle, 1984;Johnson and Sitar, 1990;Hungr, 1995;Benda Followingthe qualitativeobservations of Davis [1892], Gil- and Dunne, 1997;Iverson et al., 1997]. Small shallowlandslides bert [1909] usedthe followingconceptual model to suggestthat can also initiate on steep, planar sideslopes,coming to rest convexhillslopes result from slope-dependenttransport pro- after travelinga short distance.Landslides are often triggered cesses.On a one-dimensionalhillslope that erodes at a con- by elevated pore pressuresin the shallow subsurface,and stantrate, sedimentflux mustbe proportionalto distancefrom therefore are sensitiveto topographicallydriven convergence the divide.If gravityis the primaryforce impellingsediment Copyright1999 by the American GeophysicalUnion. downhill,the hillslopemust become steeper with distancefrom Paper number i998WR900090. the dividein order to transportthe requiredsediment flux, and 0043-1397/99/1998WR900090 $09.00 a convex form results.

853 854 ROERING ET AL.: EVIDENCE FOR NONLINEAR, DIFFUSIVE SEDIMENT TRANSPORT

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Figure 1. Aerial photographof steep, soil-mantledlandscape, Oregon Coast Range. Note the steep and relatively uniform gradient hillslopes.

Culling [1960, 1963] and Hirano [1968] modeled hillslope This relationshipindicates that all else being equal, equilib- evolutionthrough an analogywith Fick's law of diffusion(in rium hillslopesthat erode by linear diffusionshould have con- which fluxesare proportionalto chemicalgradients). In the stant curvature. hillslopeanalogy, sediment flux (L 3/L/T) Ftsis proportional to Recently, geomorphicsimulation models have been devel- the topographicgradient Vz: oped to explorehow tectonicactivity, climate change, and land use affect landscapeevolution [e.g., Nash, 1984;Andrews and •s = Kiln•7z (1) Hanks, 1985;Hanks and Schwartz,1987; Koons, 1989; I4qllgoose whereKli , is lineardiffusivity (L2/T) andz is elevation(L). et al., 1991; Rosenbloom and Anderson, 1994; Rinaldo et al., Field estimatesof downslopesediment flux on low to moderate 1995;Arrowsmith et al., 1996; Kooi and Beaumont, 1996; Braun gradient hillslopesare consistentwith (1) over both short and Sambridge,1997; Tucker and Slingerland,1997]. These [Schumm, 1967] and long [McKean et al., 1993] timescales. modelssimulate hillslope erosion as a linear diffusiveprocess, Sediment flux and landscapelowering rates can be related but the morphologyof most soil-mantled hillslopesis incon- throughthe continuityequation sistent with the linear diffusion law. Soil-mantled hillslopes typicallydo not exhibit constantcurvature; instead, curvature Oz --Os-•-: [Js•7ø•sqLOreo (2a) tends to approach zero as slopessteepen toward a limiting angle (Figure 2). Extreme examplescan be found in steep, where Or and Ps are the bulk densitiesof rock and sediment, high-relief mountainous terrain, where many hillslopes are respectively(M/L3), Oz/Ot is the rateof changein the land nearly planar, with marked convexityonly near divides.This surfaceelevation (L/T), and C O is the rate of rock uplift zone of convexityis broader on soil-mantled hillslopes,but (L/T). If the rate of surfaceerosion is approximatelybalanced there is nonethelessa pronounceddecrease in convexitydown- by rock uplift (i.e., dynamicequilibrium, as positedby Gilbert slopewith slopesbecoming nearly planar far from the divide. [1909] and Hack [1960]), Oz/Ot --- 0 and (2a) becomes This hillslopeform conflictswith constant-curvature(i.e., par- abolic)slope profiles predicted by the linear diffusivetransport --Oreo= Os•7ø •s (2b) law (Figure 2). What processescould causethis downslope We can combine(1) and (2b) to obtainan expressionrelating decreasein convexity,and how can such processesbe mod- the ratio of the landscapeerosion rate (which equalsthe rock eled? uplift rate Co) and diffusivity,C o/Klin,and hillslopecurvature To simulatethe evolutionof nearly planar hillslopes,recent •72Z- landscapeevolution models hypothesizethat, on steep slopes, sedimenttransport increases nonlinearly with gradient[Kirkby, prCo 1984, 1985; Andersonand Humphrey, 1989; Anderson, 1994; --psK,i-- • = V2z (3) Howard, 1994a,b, 1997].These models postulate that sediment ROERING ET AL.' EVIDENCE FOR NONLINEAR, DIFFUSIVE SEDIMENT TRANSPORT 855

25 discussionof Howard [1997]). These sedimenttransport mod- OregonCoast Range els have not been calibrated or directly tested against field data. 20 • •'••.•:•• linearmodel (eqn. 1) In this contributionwe derive a simple, theoretical expres- - sion that describeshow sediment transport on soil-mantled N15 ß nonlinearmodel(eqn.8) hillslopesvaries with slope angle. In contrastto previousstudies,which have assumedthat disturbance-driventransport pro- 0•10 cessesmay be represented by linear diffusion, our analysis suggeststhat diffusive transport'varieslinearly with slope at 5 low gradientsbut increasesnonlinearly as slope approachesa critical value. We test and calibrate our proposed transport 0 law, using a unique high-resolutiontopographic data set ob- 1.0 tained by airborne laser altimetry. At our studysite, hillslope B morphologyis consistentwith our proposednonlinear diffusion law. Our resultsindicate that on soil-mantledhillslopes, small changesin slope angle may lead to disproportionate "• 0.6 - changesin erosion rate.

c- - =5 0.4 - 2. Theory: Nonlinear Diffusive E:h - Sediment Transport 0.2- On soil-mantledhillslopes, disturbances (such as tree throw, animalburrowing, rainsplash, and wet/dry cycles) mobilize sed- 0.0 - iment, allowing it to be transporteddownslope. The balance 0.00 between local frictional and gravitationalforces acts to dissi- pate the energysupplied by disturbances[e.g., van Burkalow, -0.01 - 1945;Jaeger and Nagel, 1992] and may control how sediment flux varies with hillslope angle. Here we present a derivation -0.02 - showingthat the combined effects of disturbances,friction, and gravityproduce a nonlinearrelationship between sediment -0.03 - flux and hillslopegradient. First, we write a general statement -0.04 - for sedimentflux (L3/L/T) as V -0.05 F:/s= •- • (4)

-0.06 where V/A is the volume of mobile sediment per unit area 0 10 20 30 40 alongthe slope(L 3/L2) and• isthe velocity of itsmovement horizontaldistance (m) downslope(L/T). We assumethat over geomorphictime- Figure 2. (a) Elevation,(b) gradient,and (c) curvaturealong scales,disturbance processes expend energy at a given rate a profile for (1) a theoreticalhillslope modeled with linear (i.e., supplypower) in detachingand mobilizingsediment. Us- diffusion(equation (1)), shownby thicklines, (2) a hillslopein ing the physicalrelationship P = F•; where P is work done in the OregonCoast Range (OCR), shownwith graypoints, and transportingsoil per unit time by disturbingagents and F is the (3) a theoreticalhillslope modeled with our proposednonlin- net force resistingtransport, we can substitutefor velocity in ear model(equation (8)), shownby a thin line. The theoretical (4) and expresssediment flux as a functionof power per unit hillslopesare calculatedassuming constant erosion. For both area P/A and dissipativeforce per unit volume F/V: modelsimulations the diffusivityequals 0.003 m2/yr, the constanterosion rate Co equals0.075 mm/yr, and Pr/Psequals 2.0. P/A For the nonlinear hillslope,S c = 1.2. Els= F/V (5)

In the absenceof evidencesuggesting a directionalpreference flux combinestwo distinctprocess types: (1) diffusiveprocesses to disturbanceprocesses, we assumethat geomorphicdistur- (e.g., biogenicactivity, rainsplash, soil creep,and solifluction) bancessupply power isotropicallyon hillslopes,such that equal typicallyrepresented with equation(1) and (2) slope failure power is availableto move sedimentin all directions.The net processes(e.g., soil slipsand slumps)which occur more fre- downslopesediment flux then becomesthe differencebetween quentlyon steepslopes and may be influencedby a threshold the fluxesin the downslopeand upslopedirections, according to slope angle.According to these models,at low slopes,sediment fluxincreases approximately linearly with gradient,but as slopesapproach a critical value, slope failure processesbe- comemore prevalent,and smallincreases in hillslopegradient qs= F/VJdown-F/VJup=• - (F/V)dow n- (F/V)up/ (6) causelarge increasesin sedimentflux. At or just below this critical gradient,sediment flux is effectivelyinfinite, suchthat The upslope and downslopefluxes differ becausethe down- steeperslopes cannot be maintained.As a result,nearly planar slopedissipative force (F/•dow n is lessthan the upslopedis- hillslopesmay evolve, limited by the criticalhillslope angle (see sipativeforce (F/•u p.Downslope transport is resisted by fric- 856 ROERING ET AL.: EVIDENCE FOR NONLINEAR, DIFFUSIVE SEDIMENT TRANSPORT

(A) nonlinear model (eqn. 8)

•,,•,.--• K (B)linear(eqn.1)mødel gradient,critical '•Sc gradient Figure 3. Theoreticalrelationships between sediment flux and gradient.(curve a) Nonlinear transportlaw (equation(8)); (line b) linear diffusionlaw (equation(1)). The criticalgradient Sc is the gradientat whichflux becomesinfinite for the nonlinear transport law. tionalforce (F/V)f and aidedby gravitationalforce (F/V)a, by Howard [1994a,b, 1997] in that sedimentflux increasesin a whereasupslope transport is resistedby both friction and grav- nearly linear fashionat low hillslopegradients (which is sup- ity, thus (F/[/)Sown -- (F/[/)f - (F/De , and (F/V)up = ported by a field study [McKeanet al., 1993]) and increases (F/V)f + (F/V)a. Forthis formulation we assume that friction rapidly as the gradientapproaches a criticalvalue (Figure 3) is the only componentof shear strengthin the soil. The fric- (note:the transportlaw presentedby Howard [1994a,equation tional force per unit volumecalculated along the slopeis tZps# (6)] was misprinted;see Howard [1997, equation (2)] for the cos 0, and the downslopecomponent of the gravitationalforce correctform). In our proposedmodel the rate of sedimentflux per unit volumeis Ps# sin 0, where Ix is the effectivecoefficient becomesinfinite at the critical hillslopegradient S•. This at- of friction, Psis the bulk densityof sediment,# is gravitational tribute of the model is consistentwith the conceptof threshold force, and 0 is slope angle. The net downslopesediment flux hillslopes;when slope anglesapproach the critical value, in- calculatedin the horizontalplane then becomes creasesin the rate of downcuttingdo not significantlysteepen slopes,but insteadlead to higher sedimentfluxes. q_s=•- P ( (tzpsgCOS cos0-psgsin0)0 We distinguishthe criticalgradient parameter S• contained in this model from the commonlyused "threshold"slope or gradientSt. Assumingthat erosiondoes not becomeweather- - (tzpsgcoscos 0+ 0 Psg sin 0) ) (7) ing-limited,slope angles equal to S• shouldnot be observable in the field becauseflux is infinite for such slopesand they where the cos0 term in the numeratorprojects the along-slope would rapidly decline.In contrast,the thresholdslope angle is forcesinto the horizontalplane. Equation (7) can be simpli- fied.By substitutingK = (P/A)(2/gtjC2ps),Vz = tan 0, andSc generally equated with the angle of repose and thus is the --- Ix, we obtain a theoreticalexpression for how sedimentflux maximumangle observed in the field [e.g.,Strahler, 1950; Car- varieswith hillslopegradient: son and Petley,1970; Young, 1972; Burbank et al., 1996]. Our formulation is not sensitiveto the assumptionof iso- KVz tropic power expenditureby diffusiveagents. On steepslopes, _ (iVzl/Sc)= (8) where power may be preferentially suppliedin the downhill whereK is diffusivity(L2/T) andS• is the criticalhillslope direction (i.e., anisotropically),the resultingsediment flux is similar to that predictedby assumingisotropic power expen- gradient.In our proposedmodel, diffusivityvaries linearly with the power per unit area supplied by disturbanceprocesses. diture becausethe net flux is dominated by the downslope DiffusivityK alsovaries inversely to the squareof the effective component. coefficientof friction, Ix, which• follows the intuition that sed- Our model doesnot explicitlyinclude small soil slips(which iment mobilization and transport will vary with the shear travel short distancesover divergentor planar hillslopes)as strengthof the soil. Thus, all else being equal, sedimentswith part of its conceptual framework, but it may capture their more frictional resistance will have lower diffusivities. Andrews diffusivebehavior over long timescales.However, our trans- and Bucknam [1987] derived a model effectivelyequivalent to port law does not addresslarger landslidesthat (1) are con- (8), usinga frameworkin whichballistic particles are projected trolled by elevatedpore pressuresdue to topographicallyin- along a hillslope[Hanks and Andrews,1989]. duced convergentsubsurface flow, (2) tend to travel long The behaviorof (8) is similarto the transportlaw proposed distancesthrough low-order channelnetworks, (3) scourcol- ROERING ET AL.: EVIDENCE FOR NONLINEAR, DIFFUSIVE SEDIMENT TRANSPORT 857

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• 0 500 l•00m FiBure 4. Shadedrelief m•p of study•m•. Topographicd•t• were obtainedwith •irbome l•se[ •lfimet• •t •n •vemge sp•cingof 2.3 m. The sm•11b•sins •n•lyzed he[e •[• outlined (MRI: •, MR2 = 2, MR3 = 3, MR4 = 4, •nd MRS = S). luvium and sedimentsand exposebedrock, and (4) tend to along sideslopes,ridges, and nosestransport sediment to un- inciseinto hillslopesrather than diffusesediment across them. channeledvalleys, where it accumulatesuntil it is removed by In the following sections,we describeour study site in the landsliding.The probability of landslidingincreases with soil Oregon Coast Range, test the proposed transport law, and depth; thus the rate of sediment supply from adjacent hill- describe two techniquesfor calibratingK and Sc. We also slopescontrols the timescaleof cyclicsediment accumulation compareour proposedtransport law with the linear diffusion and evacuationin hollows [Reneauet al., 1989]. In some lo- law and explore its implicationsfor hillslopeevolution. cales, inactive deep-seatedlandslides dominate the topography, having transformed ridge/valley sequences into low- gradient, relatively undissected hillslopes. These massive 3. Study Site: Oregon Coast Range landslidescommonly have a downslopeorientation coincident 3.1. Description with the downdipdirection of the local bedrock,and appear to We testedand calibratedour proposedtransport law at five be concentrated in areas with high dip magnitudes of the smallwatersheds within a 2 km2 area of the centralOregon underlyingbedrock [Roeringet al., 1996]. The transportlaw we CoastRange (OCR), near Coos Bay, Oregon (Figure 4) (see proposehere appliesto purely slope-dependenttransport pro- Montgomeryet al. [1997] for location map). We selectedthis cesseson divergentand planar hillslopes,but doesnot encom- study area because erosion rates are well-documented for passdebris flow processesor deep-seatedlandsliding. smalland large spatialscales. Also, at this site, high-resolution The uniformityof highlydissected topography in the Oregon topographicdata accuratelyreveal the morphologicsignature Coast Range suggeststhat locally, erosion rates may be spa- of diffusiveprocesses. The OCR is a humid, forested,moun- tially uniform. Using data from studiesof long-term hillslope tainouslandscape. Its central and southernregions are under- erosionand short-term,basinwide sediment yield, Reneauand lain by a thick sectionof Eoceneturbidites mapped as the Tyee Dietrich [1991] argued that denudation rates (approximately Formation [Baldwin, 1956; Snavelyet al., 1964; Lovell, 1969; 0.06-0.1 mm/yr) in the central Oregon Coast Range do not Chan and Dott, 1983;Heller et al., 1985]. The Tyee (or some- vary significantlywith spatial scale, such that the landscape times referred to as the Flournoy Formation) has been com- may be in approximate equilibrium. These denudation rates pressedinto a seriesof low-amplitude,north-northeast striking are also consistent with rates of exfoliation in colluvial bedrock folds that rarely exhibit dip anglesgreater than 20ø [Baldwin, hollows [Reneauand Dietrich, 1991]. Subsequentstudies have 1956].The Oregon CoastRange is situatedabove a subduction estimatedrock uplift rates as rangingfrom 0.03 to 0.23 mm/yr zone and has experienceduplift over the last 20-30 Myr [Orr in the central Oregon Coast Range [Kelseyand Bockheim, et al., 1992]. 1994;Kelsey et al., 1994]. By dating strath terracesalong many The topographyof the Oregon CoastRange is variable,but Oregon Coast Range rivers,Personius [1995] estimatedsimilar significantportions are characterizedby steep,highly dissected uplift rates of 0.1-0.2 mm/yr. More recently,cosmogenic ra- terrain with distinct ridge and valley topography(Figure 4) dionuclideshave been applied to estimatelong-term rates of [Dietrichand Dunne, 1978;Montgomery, 1991; Montgomery and erosionand soil production,which are 0.1 _+0.03 mm/yr in a Dietrich, 1992]. Thin soils typicallymantle ridges, and thick small basin within our study site [Heimsathet al., 1996]. The colluvial depositsfill unchanneledvalleys at the uppermost similarity in estimated rates of erosion and uplift suggests extent of the channel network. Debris flows originating in approximate equilibrium conditions.However, variability in unchanneledvalleys may control the dissectionof low-order tectonic and climatic processesmay affect rates of channel channelnetworks, as fluvial erosionis infrequentin theseareas incision, thereby altering the boundary conditionsthat drive [Dietrichand Dunne, 1978; Swansonet al., 1982; Reneauand hillslope erosion. Fernandesand Dietrich [1997] studied the Dietrich, 1990; Seidl and Dietrich, 1992]. Diffusive processes time required for diffusion-dominatedhillslopes to establish 858 ROERING ET AL.: EVIDENCE FOR NONLINEAR, DIFFUSIVE SEDIMENT TRANSPORT

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Figure 5. Theoreticalrelationships between curvature and gradientgenerated by three generalizeddiffusive sedimenttransport laws (assuming a constanterosion rate Co). (line a) A linear diffusionlaw giveshillslopes with constantcurvature; (line b) a linear diffusionlaw coupledwith a slopethreshold St, suchthat steeper slopeangles are instantaneouslyreduced, generates hillslopes with constantcurvature below the thresholdand planar hillslopesfor slopesat the threshold;and (curvec) a nonlineardiffusion law (equation(8)) generates hillslopeswith curvature that approacheszero with increasinggradient. Curvature becomes zero when gradient equalsthe critical value Sc. approximateequilibrium following a change in the channel from the groundsurface, as well as the stochasticnature of tree incision rate. According to their analysis,hillslopes similar to throw pits and local lithologicvariability. thosein our studyarea would require 150,000years to reach approximateequilibrium after a doublingof the channelinci- sionrate (exactequilibrium is approachedasymptotically). By 4. Characterizing Sediment Transport contrast,the OCR has been uplifting for the last 20 million Laws With Topographic Derivatives years [Orr et al., 1992]. 4.1. Method These observationssuggest that portions of the Oregon CoastRange may approximatean equilibriumlandscape, such We usedplots of hillslopecurvature as a functionof gradient that over time the terrain erodes at approximatelythe same (e.g., Figures5 and 6) to test diffusivetransport laws against rate and rock uplift balancesdenudation. In other words, the the hillslope morphologyof our study site. Most landscape topographyof the region may be consideredrelatively time- evolutionmodels employ one of three generalrelationships for independent(see Howard [1988] for discussion).Despite these how sedimentflux varieswith slope.Each of thesethree gen- observationssupporting equilibrium, we observelocal features, eral transport laws generatesa distinct relationshipbetween suchas ancientdeep-seated landslides, that may indicatelocal hillslopegradient and curvaturefor equilibriumhillslopes (Fig- erosionaldisequilibrium. In performingthe analysesdescribed ure 5). First, the linear diffusivetransport law (equation (1)) below, we focus on five small watersheds with well-defined impliesthat on equilibriumhillslopes, curvature does not vary divides and relatively uniform ridge and valley morphology with gradient (see equation (3)). Second,some studieshave becausethese areas may be more likely to approximate an suggestedthat hillslopesdenude by linear diffusion unless a equilibriumlandscape. thresholdangle St is attained [e.g.,Ahnert,1976;Avouac, 1993; Tuckerand Slingerland,1994; Arrowsmith et al., 1996]. Upon 3.2. High-Resolution Topographic Data: Airborne Laser reachingthe thresholdangle, hillslopesexperience an imme- Altimetry diate downsloperedistribution of sediment(i.e., landslide). In our study area, high-resolutiontopographic data were Equilibrium hillslopes modeled with such a transport law obtainedusing airborne laser altimetry(Airborne Laser Map- would exhibit constant curvature for gradients below the ping, Inc.), which can accuratelycharacterize fine-scale topo- thresholdand be planar (zero curvature)for gradientsat or graphicfeatures over large areas[e.g., Garvin, 1994;Ritchie et near the threshold. Thus plots of curvature versus gradient al., 1994;Armstrong et al., 1996; Ridgwayet al., 1997]. Topo- would approximatea stepfunction at the thresholdslope angle graphicdata pointswere collectedwith an averagespacing of (seeFigure 5). Finally, nonlineartransport laws (e.g., equation 2.3 m and a vertical resolutionof approximately0.2-0.3 m. In (8)) imply that hillslopecurvature should be roughlyconstant addition to depictingfundamental geomorphic features, such at low gradients, convergingcontinuously toward zero with asridges and unchanneledvalleys, and anthropogenicfeatures, increasinggradient (see Figure 5). Curvature equalszero for suchas loggingroads and landings,the topographicdata pro- gradientsequal to the criticalvalue Sc. Using the relationship vide a detailed descriptionof meter-scaleirregularities on hill- between hillslope gradient and curvature, we can assessthe slopes(see Figure 4). The expressionof small-scaleroughness validity of the three generalizedsediment transport laws. presumablyreflects data errors in distinguishingvegetation Traditionally, hillslope profiles have been used to analyze 860 ROERING ET AL.: EVIDENCE FOR NONLINEAR, DIFFUSIVE SEDIMENT TRANSPORT

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Figure6. (a-e) Plot of gradientagainst curvature for our studybasins (MR1-MR5) and (f) moving downslopealong individual hillslopes in MR1-MR4. Opendiamonds indicate binned' average values of curvature.Low-gradient terrain tends to beconvex (highly negative curvature), whereas steep terrain tends to benearly planar (near zero curvature). The relationship isconsistent with our nonlinear transport law (see Figure5), suchthat curvature decreases continuously with gradient and approaches zero for steepslopes. the morphologicexpression of sedimenttransport laws [e.g., from the coefficientsof the bestfit equations.Our analysisis Davis, 1892;Penck, 1953;King, 1963;Kirkby, 1971;Howard, basedon laseraltimetry-derived topographic data that cap- 1994a,b], aswe showin Figure2. Profilesare a simpleand turesmeter-scale variability in the topography.Topographic appealingmethod for describingslope morphology, but they dataobtained from digitalelevation models with a largegrid cannotaccount for planform(i.e., contour)curvature. In our spacing(e.g., 30 m or larger)will not accuratelydepict small- studysite, as well as in manysoil-mantled landscapes, hillslope scalevariations in hillslopemorphology that are indicativeof segmentswith negligibleplanform curvatureare rare, con- most diffusiveprocesses. foundingour abilityto characterizehillslope morphology with profiles.Alternatively, plots of gradientagainst curvature (cal- 4.2. Results culatedin both the x andy directions)allow us to directly Plate 1 depictsa smallcatchment (MR1, whichis approxi- comparehillslope form to modelsof hillslopeerosion. Using mately52,000 m 2) with characteristic ridges and valleys in our this approach,we can test thesemodels across large areas, studysite near CoosBay, Oregon.The spatialdistribution of ratherthan restrictingour analysisto selectedhillslope pro- gradient(IVzl) andcurvature (calculated as the Laplacian files. operatorV2z) delineatesthe structureof ridgesand valleys. We estimatedtopographic gradient and curvature by fitting Ridges(which are narrow,divergent hilltops that extendover two-dimensional,second-degree polynomials to patchesof lo- longdistances) and noses (which are divergenthillslopes with cal topographicdata points (300-500 m 2) at evenlyspaced unchanneledvalleys on either side) tend to have divergent intervalsthroughout the studybasins. We useda weightedleast (highlynegative V2z) axesand high gradient, nearly planar squaresroutine (which gives greater weight to pointsnear the (nearzero V2z) sideslopes.Noses alternate with convergent center of each patch) and calculatedgradient and curvature (highlypositive V2z) unchanneled valleys. Curvature appears ROERING ET AL.: EVIDENCE FOR NONLINEAR, DIFFUSIVE SEDIMENT TRANSPORT 861 to decreasegradually as one movesin the downslopedirection 1/2 from hilltops. RMSE = - (Co -- E,) 2 (•0) To quantifyhillslope morphology and assessthe sediment /l = transportlaws discussedabove, we plotted gradient against curvaturefor hillslopes(which we defined as areas with V2z < where n is the number of pointsin the landscapefor which we 0) in the five smallbasins depicted in Figure 4. Low-gradient model the erosion rate and E i is the modeled erosion rate at portions of hillslopestend to be highly divergent,whereas the ith point in the landscape(calculated with equation(9)). steepsideslopes typically have near zero values of curvature.In The vertical resolution of topographicdata may affect our each of our studybasins, we observea gradualtrend toward results.To estimatehow uncertaintyin the elevation(i.e., z) zero curvaturewith increasinggradient (Figures 6a-6e), which values affects our calibrated parameters, we performed a indicatesthat the topographyis consistentwith our nonlinear Monte Carlo simulation.Specifically, we randomlyvaried the diffusivetransport law (equation (8)). Moving downslope elevation values within their range of uncertainty and recali- alongindividual hillslopes in thesebasins, we alsoobserve the brated the model parametersfrom this "synthetic"data set. By trend of decreasingconvexity with increasinggradient (Figure performing many iterations of this process,we estimatedthe 6f). More generally,we observethat (1) curvatureis not con- probability distribution and standard error of the calibrated stantand (2) an abruptthreshold slope angle does not separate model parameters. low-gradient,divergent slopes from steep,planar slopes.Thus the morphologicsignature depicted in Figure 6 is fundamen- 5.2. Results tally inconsistentwith both the linear diffusion law and the We performed the calibration on five small basinsin our linear diffusion/slopethreshold law. study area (see Figure 4). For catchmentMR1 we plotted RMSE (normalizedby Co) as a function of K and S• and found the minimum value of RMSE/Co (approximately0.4) 5. Model Calibration: Divergence whenK = 0.0032 _+0.0009 m2/yrand Sc- 1.25 _+0.10 of Sediment Flux (Figure 7). Calibratedparameters for all five studybasins are shown in Table 1. Values for K range from 0.0031 to 0.0045 5.1. Method m2/yrwith an averageof 0.0036m2/yr, and S• rangesfrom 1.2 Our calibrationprocedure finds the valuesof K and Sc that to 1.35with an averagevalue of 1.27. Thus the variability of the are most consistentwith the topographicform of our study parameterestimates between our five smallbasins is consistent area and its measured long-term erosion rate. We calculate with our estimate of the parameter uncertaintywithin each individual basin. erosion rate E as the divergenceof sedimenttransport, E -- V ßF/s. Substituting (8), we obtainthe followingexpression that The shape of the surface defining RMSE/Co in Figure 7, relatesthe rate of landscapeerosion to model parameters(K which dipssteeply in either directionparallel to the K axisand and So) and derivativesof the local topographicfield: dipsless steeply parallel to the S• axis,indicates that the model calibration is more sensitiveto variability in K than in S•. DiffusivityK is linearly related to the spatiallyconstant landscapeerosion rate, suchthat any adjustmentsin Co give pro- E = iOr/Ps-g[1 -- (IvzV2z/Sc) 2 portional changesin K. Recalibrating the model parameters in our Monte Carlo simulation,we observedthat our resultsare relativelyinsensi- [(0z)0z(oz,oz ,ozoz tive to uncertaintyin the topographicdata. For example,our + - tlzlc)5 (9) calculationsshow that the standarderror in K due to topo- 2 +2[¾oxoxoygraphicuncertainty is approximately0.0001 m2/yr and for S• it From (9) and assumedvalues for K and S•, we can calculate is approximately0.02. Thus topographicuncertainty is a small the local erosion rates of individual points on the landscape part of the total uncertaintyin estimatesof K and So. from their topographicderivatives. If the landscapeis in steady state, then each point on the landscapeshould be eroding at the same rate. Our calibration procedure searchesfor the 6. Model Calibration: Area Slope valuesof K and S• that make thesemodeled erosionrates as 6.1. Method uniform as possibleacross the landscape,and as consistentas possiblewith independentlyderived estimates of the long-term We can also calibrate the model parametersby modeling averageerosion rate Co. We limited our analysisto divergent sedimentfluxes using (8) and comparingthem to the sediment or nearlyplanar hillslopes (i.e., V2z < 0) becauseour sedi- fluxes required to maintain a spatiallyuniform erosion rate. ment transportmodel (equation(8)) doesnot addressmass- This methodtests our proposedmodel more directly,although wastingprocesses associated with convergenttopography. it doesnot definethe estimatedparameters as preciselyas the The calibratedor bestfit parametersare thosethat minimize divergencemethod. In an equilibriumlandscape the volumet- the differencebetween modeled erosion rates (equation(9)) ric sedimentflux per unit length acrosseach point is and the assumedlong-term erosionrate Co. For the calibra- Or a qs=Co (11) tion we useda constantlandscape erosion rate Co of 0.1 _+0.03 psb mm/yr [Heimsathet al., 1996] and Pr/Ps= 2.0 _+0.2 [Reneau and Dietrich, 1991]. We plot the root mean square error where a/b is drainagearea per unit contourlength. We plot (RMSE) betweenCo and modelederosion rates for a rangeof massflux (equation(11)) as a functionof hillslopegradient valuesof K and S•, and choosethe parametersthat give the and estimateK and S• by fitting theoreticalflux curves(equa- lowest value of RMSE: tion (8)) to the data points. 862 ROERING ET AL.' EVIDENCE FOR NONLINEAR, DIFFUSIVE SEDIMENT TRANSPORT

1.1

1 o.ooo 0.003 0.006 0.009 o.o12 o.o15 K (m2/yr) Figure 7. Contour plot of RMSE/Co as a functionof K and Sc for basinMR1. The surfacedefines the misfit, accordingto equation (10), betweenmodeled erosionrates and the spatiallyconstant rate (0.1 mm/yr) for different parametervalues. The lowestvalue of RMSE/Co, asshown by the graylines, corresponds to the best fit parameters(K = 0.0032 m2/yr,and Sc = 1.25). Contourswith extremely high values of RMSE/Co were omitted for clarity.

Analogousto Gilbert's [1909] conceptualmodel of a one- [1997]). In this calculationthe total area drainingout of each dimensionalhillslope, (11) statesthat massflux must increase grid cell is divided between its downslopeneighbors (both with drainagearea on equilibriumhillslopes. If sedimenttrans- adjacentand diagonal)in proportionto the localgradient. This port variesprimarily with gradient (as we hypothesizefor di- algorithm produces a spatial distribution of a/b that avoids vergent and planar hillslopes),points with higher a/b should obviousgrid artifactsthat would be createdif the maximumfall alsohave steeperhillslope gradients. To restrictthis analysisto line were used to calculatedrainage area. hillslopes,we limited our calculationsto pointsof the landscapewith negativecurvature. Generally, these points have a/b 6.2. Results values below a particular threshold, such that higher values On hillslopesin catchmentMR1, a/b increaseswith gradi- tend to be associatedwith convergenttopography (e.g., hol- ent, which is consistentwith the equilibrium assumption,such lows and channels)[Tarboton et al., 1991;Dietrich et al., 1992; that points with larger drainage areas must transmit higher Montgomeryand Foufoula-Georgiou,1993]. To calculatea/b, massfluxes (Figure 8). However,for convergentparts of the we used a grid-based analysisand applied a multidirection, landscape,a/b variesinversely with gradient;these points are weighted-areaalgorithm that usesthe grid node spacingas b part of the valley network and thus are not includedin the (similar to Costa-Cabraland Burges [1994] and Tarboton model calibration. In catchment MR1 we observe that flux (calculatedwith equation(11) for pointsin the landscapewith Table 1. Calibrated ParametersUsing Divergence negativecurvature) increases nearly linearlywith gradientfor Calibration Method in Study Basins low slopesand increasesrapidly on steeperslopes (Figure 9). To calibrate the model parameters,we fit two model flux Site K, m2/yr Sc curves(equation (8)) to the data, suchthat over 98% of the

MR1 0.0032 1.25 data pointsare enclosedby the two curves.These curves define MR2 0.0031 1.20 a range of model parameters;K varies between 0.0015 and MR3 0.0045 1.35 0.0045m2/yr, and S c variesbetween 1.0 and 1.4. These param- MR4 0.0032 1.35 eters define the bounds of values consistentwith our study MR5 0.0039 1.22 basin and are similar to parameter values estimatedwith the Average 0.0036 _+0.0016 1.27 _+0.16 divergencemethod (K = 0.0032 m2/yr,and Sc = 1.25). For verylow gradients(<0.2), massflux estimatesexceed those pre- ROERING ET AL.' EVIDENCE FOR NONLINEAR, DIFFUSIVE SEDIMENT TRANSPORT 863

10000 MR1 convergent divergent

000

100

• %ø0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 gradient Figure 8. Semilogplot showingthe relationshipbetween drainage area per unit contourlength (a/b) and gradientfor basinMR1 (seePlate 1). Blackdots represent terrain with divergentcurvature (which we classify ashillslopes), and gray dots represent convergent terrain. Values are calculated from gridded topography with a 4 m spacing,thus the smallestpossible value of a/b equals4 m. dictedby the model becausecells in our simulationhave a finite valuesof K and S•, we plotted model curvesagainst the hill- size(4 x 4 m) and smallerdrainage areas cannot be computed. slopeprofile depicted in Figure2 (whichhad negligibleplanform curvature)to demonstratethat nonlineardiffusive transport is more appropriate for representing the observed 7. Modeling Hillslope Morphology hillslopemorphology than linear diffusion. To analyzehow our proposedtransport law influenceshill- For verylow values of PrCo/PsK (<0.01 m-X), the modeled slopemorphology, we calculatedhow erosionrate affectsgra- hillslopehas relatively constant curvature and approximatesa dient and curvaturealong an equilibriummodel hillslope.Spe- hillslope modeled with linear diffusion. In other words, when cifically,we solved(9) in one dimensionassuming a constant erosionrates are low relativeto diffusivity,hillslope gradients erosionrate Co and usingthe calibratedvalue of Sc = 1.25. will not approachthe critical slope S• excepton very long We used this simple, one-dimensionalanalysis because two- hillslopes.With increasingerosion rate (or decreasingdiffusiv- dimensionalhillslope evolution modeling requires the specifi- ity), the point along the hillslopeat which curvaturedeviates cationand calibrationof transportlaws for valley-formingpro- from a constantvalue moves closer to the divide.For extremely cessessuch as debris flows and fluvial erosion (those endeavors highvalues of PrCo/PsK, slopeangles increase rapidly downhill exceedthe scopeof this contribution).Figure 10 depictsgra- of the divide and approacha thresholdvalue suchthat lower dient (Figure 10a) and curvature(Figure 10b) as a functionof sectionsof the hillslope are steep and nearly planar. More distancefrom the divide (L) for a rangeof prCo/PsK. Figure generally,with increasingerosion rate, decreasingdiffusivity, 10 showsthat for slopegradients greater than roughly0.5 So, and increasinghillslope length, more of the hillslopewill be large increasesin erosionrate can be accommodatedby small closeto the critical angle. increasesin hillslopegradient. For example,25 m downslope of the divide, only a 25% increasein gradient (from 0.75 to 0.97) is required to accommodatea doubling of the erosion 8. Observedand Modeled Hillslope Morphology rate(from prCo/Ps K equals0.05 to 0.10m-l). With increasing To explorehow nonlinear diffusion may affect hillslope mor- distancefrom the divide, hillslopegradient becomesincreas- phology,we plottedthe frequencydistribution of hillslopegra- ingly uniform, and both slope and curvaturebecome increas- dient in our studycatchments, and comparedit to the slope ingly insensitiveto changesin erosion rate. Using calibrated dependenceof sedimentflux (as estimated from equation(8)). 864 ROERING ET AL.: EVIDENCE FOR NONLINEAR, DIFFUSIVE SEDIMENT TRANSPORT

0.015

0.012

0.009

0.006 eq. 8: K=0.0045; Sc=l.0

0.003

eq. 8: K=0.0015; Sc=1.4

0.000 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 gradient

Figure 9. Relationshipbetween sedimentflux and gradient in basin MR1. Data points indicate flux computedwith equation(11), whichis calculatedfor parcelsof the landscapewith divergentcurvature. Two model curves,which are computedwith equation(8), are plotted aroundthe data points,such that over 98% of the datapoints are enclosed by the twocurves. The dashedcurve has K = 0.0015 m2/yr,and S c = 1.4; the solid graycurve has K = 0.0045 m2/yr,and Sc - 1.0. Fluxdoes not approach zero (or fit withinthe model curves) becausea finite grid cell size limits the smallestpossible value of a/b.

In our study basins,ridges and noses typically bisect steep musthave zero slopeat the divide,whereas divides in our study sideslopes;on these sideslopes,angles approach and some- basinsare typicallyinclined alongtheir axes;thus slopesin the timesexceed 45 ø . The scaleof the highlyconvex region is of the studybasins tend to be steeper.In addition,some hillslopes in order of 10 m, and the average hillslope length is approxi- our study basinshave significantplanform curvature,which mately 35 m; thus a large fraction of the hillslope area lies furtherdiminishes the areawith low gradient(i.e., ridges).We outside the highly convex region and instead is steep and also compared the cumulative distribution of curvature for nearly planar. We plotted our calibrated sedimenttransport MR1-MR5 and the theoreticalhillslope (Figure 12). These law (K = 0.0032 m2/yr,and Sc = 1.25) andthe frequency distributionsillustrate that the majority of hillslope area has distributionof gradient for hillslopesin MR1-MR5 (Figure curvature between 0 and -0.04. 11a). In our studybasins, hillslope gradients tend to cluster near valuesfor which sedimentflux is a highly nonlinear function of slope(for example,in MR1, mean hillslopegradient is 9. Modeling Erosion Rates: Linear and 0.80, and 50% of the terrain has gradientsbetween 0.74 and Nonlinear Diffusive Transport Laws 0.92). Sedimentflux increasesrapidly with gradient near the Toevaluate how our proposed transport law compares with criticalslope; this feedback results in a limit to slopesteepness. linear diffusionfor modelingerosion rates at our studysite, we To test the observedclustering of gradientsnear the non- calculatedthe frequencydistribution of erosionrate usingboth linearity in massflux, we comparedthe gradient distributions laws. In both cases,we calculated erosion rates at evenly for MR1-MR5 (seeFigure 11a) with the gradientdistribution spaced points on divergent terrain in basin MR1. For the for an equilibrium,one-dimensional, model hillslope(Figure nonlinearcase we used (9) (with the calibratedparameters 11b). This simple,one-dimensional analysis captures how our K = 0.0032 m2/yr,and S c = 1.25) andplotted the frequency transport law influencesthe distributionof hillslopegradient distributionof the calculatederosion rates (Figure 13). For the and curvature.We modeledthe equilibriumhillslope with the linear modelwe used(10) to calibrate(1) and foundthe value followingparameters: K = 0.0032 m2/yr,S c = 1.25, Co = of Knn that minimizes the misfit between modeled erosion 0.1 mm/yr, 9,-/9s = 2.0, and hillslopelength is equal to 35 m. rates and the spatially constantvalue Co = 0.1 mm/yr. We Hillslope gradientsfor the studybasins and model hillslopeare usedthe bestfit Knn(approximately 0.0053 m2/yr) to model similarly clusteredbetween 0.7 and 1.0, although the model erosionrates with (1) and (2a) and plotted the frequency hillslopehas more area with lower gradients,between 0 and distributionof calculatederosion rates (Figure 13). 0.6. This discrepancyarises because our theoretical hillslope From this analysiswe observethat neither transport law ROERING ET AL.: EVIDENCE FOR NONLINEAR, DIFFUSIVE SEDIMENT TRANSPORT 865

0.10 I I I I 10. Discussion A)Gradient Our analysis yields a nonlinear diffusivetransport law that is consistentwith the tendencyof hillslopesin our studyarea to 0.08- becomemore planar as they steepen.Our transportlaw be- haves similarly to those proposed by Kirkby [1984, 1985], Anderson [1994], and Howard [1994a, b, 1997] in that flux increasescontinuously with gradient and increasesrapidly as • 0.06 gradientsapproach a critical value. Although these models 'F ' have sometimesbeen termed "landslide"laws, we refer to (8) as a nonlinear diffusive transport law becauseour proposed c• c• 0.04- model doesnot encompasslarge landslides or debrisflows and doesnot requirethem to producethe observednonlinear slope dependenceof sedimentflux. Transportlaws that attempt to representdebris flows should include a mechanismto account 0.02- for the area dependencyof pore pressuredevelopment, such that convergentareas experience more frequent soil saturation •0.05• - and thus have a higher probability of generatinga shallow

0.00 i i i landslide[Tucker and Bras, 1998]. Our proposedtransport law 0.10 I I differs from those discussedabove because it suggeststhat ature purelydiffusive processes, such as biogenic activity, generate a nonlinear relationship between hillslope gradient and sediment flux. Kirkby [1984, 1985],Anderson [1994], and Howard O.O& [1994a, b, 1997] suggestthat a processtransition occurson ß steeperslopes, such that slopefailure processesbecome more prevalent and cause a rapid increasein flux with gradient. 0.06- Becausethe critical gradient term Sc imposesa condition of •-. E chronic soil instabilityin our model, we suggestthat it may v effectivelyencompass small soil slipsand slumpsthat do not travel far, althoughthese processes are not explicitlyincluded 0.04- in the theoretical formulation. Sedimenttransport rates predicted with our proposedmodel are similar to those obtained by field studies.Reneau and

0.02- Dietrich [1991] estimatedcolluvial transport rates by defining ---0.015 sideslopepathways of sedimenttransport into hollowsand measuringthe amount of colluvium deposited above dated organicmaterial. The averagecolluvial transport rate for nine 0.00 0 1'0 2'0 3•0 4'0 50 sitesin the southernOregon Coast Range is 0.0032 ___0.0023 L (m) m2/yr.Most of the sideslopesin their studyarea have a relatively uniform slope angle, and the averagevalue was mea- Figure 10. Contour plot of hillslope (a) gradient and (b) sured for each site. To test (8) against the transport rates curvatureas a function of the ratio of erosionrate to diffusivity measuredby Reneauand Dietrich [1991], we calculatedsedi- (Co/K) and hillslopelength L for an equilibriumhillslope ment flux (or colluvial transport rate) using our calibrated modeledin one dimensionwith equation(9) (Sc = 1.25). model and the averagegradient of sideslopesin their study (approximately0.7). Our model predictsa colluvialtransport rate of approximately0.0036 m2/yr, which is similarto their generatesa uniform erosionrate. In fact, both modelsindicate averagevalue of 0.0032m2/yr. Using their averagecolluvial that local erosion rates vary between 0 and over 0.16 mm/yr. transportrate and the averagesideslope gradient, we calcu- Despitethis variability, our nonlinearmodel produces a cluster lated an averagelinear diffusivityK•i n which is approximately of values around the assumed constant erosion rate of 0.1 0.0048m2/yr. This value is verysimilar to the calibratedvalue mm/yr. The linear model generatesmore low erosion rates of Klin (0.0053m2/yr) that we estimatedusing the equilibrium becauseerosion is proportionalto curvatureand our studysite assumptionand topographicdata. Estimatedvalues of K•in are is dominatedby nearly planar hillslopes.For the nonlinear larger than estimatesof K becauseof the nonlinearityin our model, over 70% of the studysite has a modeled erosionrate proposedflux relationship(equation (8)). For a givendiffusiv- between 0.05 and 0.15 mm/yr, whereasfor the linear model, ity our proposedtransport law predictshigher sediment fluxes less than 40% of the terrain has a modeled erosion rate in that for gradientsnear the criticalvalue than doesthe linear diffu- range.This result suggeststhat the proposednonlinear trans- sion law. We suggestthat steep, soil-mantledlandscapes may port law would better preservethe current topographythan be appropriatelymodeled with a nonlineardiffusive transport would the linear law. In other words, denudationmodeled by law because(1) the topographyof our studysite is consistent the linear model would substantiallyalter hillslopemorphol- with a nonlineardiffusive transport law and (2) our calibrated ogy,rounding sideslopes and producinga wider distributionof model generatesrates of massflux similarto thosemeasured in gradient.The nonlinearmodel is more consistentwith steady the field. state erosiongiven the observedhillslope morphology. Our topographicanalyses (Figures 6 and 9) and erosionrate 866 ROERING ET AL.: EVIDENCE FOR NONLINEAR, DIFFUSIVE SEDIMENT TRANSPORT

J i 0.018A) Study basins / 0.015 • --, sedimentflux • fraction of area-MR1 I

• fraction of area-MR2 !

• fraction of area-MR3 / •>'0.012 fraction of area-MR4 / x• 0.009 •------fraction of area-MR5 /

• 0.006

0.003

0.000 j f

0.018 ' • ' • ' J i B) Theoretical hillslope I 0 015 .-,-. -,-- sedimentflux I fraction of area _ t 0.012 - I / 0.009 -

0.006 -

0.003 -

0.000 0.0 0.2 0.4 0.6 0.8 1.0 1.2 gradient

Figure 11. Calibratedrelationship between flux and gradient(shown by dashedcurve, with K - 0.0032 m2/yr,and S c = 1.25) andthe frequencydistribution of hillslopegradients (shown with solid curves) for (a) our studybasins and (b) a one-dimensional,equilibrium hillslope modeled with equation(8) (K = 0.0032 m2/yr,and Sc = 1.25). Gradientdistributions reveal a clusteringbetween 0.7 and 1.0. calculations(Figure 13) suggestthat the morphologyof our representthe total shear strengthof the soil, although the study basins is consistentwith an approximateequilibrium model formulation does not explicitly account for shear landscapeeroding by nonlinear diffusiveprocesses. Although strengthfrom sourcesother than granular friction. Our esti- hillslope convexitytends to decreasewith gradient (as our mate of Sc is somewhatscale-dependent because slopes may model predicts),we observesignificant variance in this trend, be even steeper locally (e.g., within a pit/mound feature). which may indicate that some locales are eroding faster or Nonetheless, field measurementsof slope angle generally slower than our assumedconstant value Co. Similarly, we agreewith calculationsfrom the digital topographicdata, and observea tendencyfor modelederosion rates to clusteraround rarely did we observeslope anglesexceeding 450-50 ø in the our assumedconstant value (Co = 0.1 mm/yr),but portionsof field. basin MR1 have modeled erosion rates that are not within 50% Our analysisassumes that soil depth doesnot affect the rate of Co. These observationssuggest that local deviationsfrom of sedimenttransport on hillslopes.Ahnert [1967, 1976] hy- equilibrium may exist in our study basins.These deviations pothesizedthat sedimentflux may vary nonlinearlywith soil may reflecthillslope response to changingboundary conditions depth,such that equilibriumhillslopes may be planarwith soil or the influenceof variable rock properties. depth increasingdownslope. He suggestedthat shear stresses Through our model calibration, we estimated the critical generatedat the base of a soil column may causethe soil to gradientto be approximately1.25 (or 51ø).This value exceeds deformor flow [e.g.,Fleming and Johnson,1975]. In our study the internal friction angle of most soils,suggesting that cohe- area we observean easily distinguishableboundary between sion from roots or other sourcescontributes shear strengthto colluvium,which is typicallycoarse-grained and rich in organic soils in our study area. We suggestthat Sc may effectively material, and weathered bedrock or saprolite. On ridges, ROERING ET AL.: EVIDENCE FOR NONLINEAR, DIFFUSIVE SEDIMENT TRANSPORT 867

assumptionis consistentwith our field observations.If erosion • m m theoreticalhillslope ratesexceed rates of soilproduction, hillslopes may be stripped of their soil mantle, and different erosionalprocesses may • • MR3 becomedominant, such as rock fall, rock toppling,and other formsof bedrocklandsliding [e.g., Howard and Selby,1994]. • 60 • MR4 Heimsathet al. [1997] suggestthat biogenicprocesses control the rate of soilproduction, which decreases exponentially with 40 soildepth. Climatic and biogenic variability, whether natural or anthropogenic,may influencethe soilproduction function defined by Heimsathe! al. [1997],although these relationships 20 have been unexplored. Previousfield and experimentalstudies have reportedre- 0 • sultsthat are consistentwith a nonlineardiffusion transport 0.00 -0.02 -0.04 -0.06 -0.08 -0.10 law. Strahler[1950, p. 673] analyzedhillslopes in southern planar curvature(m -1) divergent Californiaand found a uniformityof slopeangle that he at- Figure 12. Plot of cumulativepercentage of divergentterrain tributedto "a prevailingcondition of form-equilibrium."Li- for our studybasins and for a one-dimensional,equilibrium thology,climate, soil, vegetation,and channellocation were hillslopemodeled with equation(8). The majorityof curvature usedto distinguishslopes in differentstages of development. valuesfor both the studysite and the theoreticalhillslope are In his analysis,Strahler [1950] noted that hillslopeswith chan- between -0.04 and 0.0. nelsactively incising at theirbase tended to exhibitthe steepest slopeangles, which tended to clusteraround a value similar to the angleof repose.Furthermore, gentler slopes were associ- noses,and sideslopes,soil depth rangesbetween about 0.2 and ated with creep processesand were distinct from those con- 1.0 m and does not covarywith curvature,slope angle, or trolledby a limitingslope angle. Several hillslope profiles from distancedownslope (K. M. Schmidt,manuscript in prepara- Strahler[1950] showed distinctly convex hilltops with increas- tion, 1998;A.M. Heimsathet al., manuscriptin preparation, inglyplanar slopes downhill, such that creepand landsliding 1998), suggestingthat sedimentflux doesnot vary systemati- may combineto generatethe observedhillslope morphology. callywith soil depth. The stochasticnature of diffusivepro- Experimentalstudies of rainsplashare consistentwith our cesses,such as tree throw and mammalburrowing, leads to nonlinear model. For a given rainfall rate, measurementsof largevariability in local soil depth.We arguethat soil depth rainsplashcreep, which is definedas sedimenttransport by maynot affectsediment flux in our studyarea becausethe soils raindropimpact, demonstrate a nonlinearincrease in creep are coarse;frictional resistancewould prevent significant rate with increasinggradient [Moeyersons, 1975]. In addition, shearingor flowing.Instead, soils are transportedby periodic Moeyersons[1975] reported that flux ratesare linearlyrelated disturbances, which overcome the frictional resistance of the to rainfall rate. This findingis consistentwith our proposed colluviumand the shearstrength provided by root cohesion. theoreticalmodel in that poweris linearlyrelated to sediment Our proposedmodel appliesonly to the transportof soil, flux, if we considerrainfall rate as a measureof the power thus diffusivetransport rates are limited by soil production responsiblefor mobilizingand transportingsediment (see ratesif bedrockslopes emerge. Our topographicanalysis and equation(8)). In an experimentalstudy of sedimenttransport modelcalibration procedures assume that ratesof soilproduc- by rainsplash,Mosley [1973] quantifiedflux as a function of tion are sufficientto maintainsoil-mantled hillslopes, and this gradient,although he did not report resultsfor slopesgreater

0.12

0.10 P• Nonlinear(equation8)

• 0.08

o c 0.06 o

'- 0.04

0.02

0.00

0.00 0.05 0.10 0.15 0.20 erosionrate (mrn/yr) Figure 13. Frequencydistribution of modelederosion rates in basinMR1 for calibratedlinear (shown by thindashed curve, equation (1)) andnonlinear (thick gray curve, equation (8)) transportlaws. Both transport lawswere calibratedassuming a constanterosion rate of 0.1 mm/yr(equation (10)). For the nonlinear transportlaw, K equals0.0032 m2/yr, and Sc = 1.25,whereas for the linear diffusion law, Kun = 0.0053m2/yr. 868 ROERING ET AL.: EVIDENCE FOR NONLINEAR, DIFFUSIVE SEDIMENT TRANSPORT than 25ø because"rolling and slidingof grains"became prev- 1994; Summerfieldand Nulton, 1994;Aalto and Dunne, 1996; alent. Such behavior may be consistentwith our proposed Grangeret al., 1996]. Near the critical slope,large changesin transport law in that rapid increasesin flux occur on steep erosionrates are associatedwith only slight changesin hillslopes. slopegradient, so the topographicsignature of tectonicforcing Diffusion models depend on measuresof topographiccur- will be small. Although changesin erosion rate have only a vature, which depend on the resolutionof the underlyingto- small effect on averagegradient, they will causeproportional pographicdata. Many studiesthat use linear diffusion(equa- changesin hillslopecurvature near hilltops and divides(see tion (1)) to explorelarge-scale landscape evolution are based Figure 10). Thus the topographicsignature of tectonicforcing on topographicdata with coarseresolution (30-1000 m) and may more clearly be found in hilltop curvaturerather than in assumethat "scaled-up"transport laws can appropriatelyrep- sideslopegradient. If the diffusivityK can be estimatedinde- resentgeomorphic processes at scaleslarger than their scaleof pendently,hilltop curvaturecan be usedas a quantitativemea- occurrence,as discussedby Andersonand Humphrey [1989] sureof erosionrate. Becauseslopes become more planar with and Koons[1989]. In the Oregon Coast Range, diffusivepro- distancefrom the divide, curvaturecan be more preciselyes- cessesoperate over length scalesof only a few metersor less, timated(and will be more clearlyrelated to erosionrate) close as is typical of soil-mantledlandscapes. The analysesreported to dividesand hilltops. here are possiblebecause of the availabilityof high-resolution topographicdata. 11. Conclusion Alternative approacheshave been proposedfor simulating rapid increasesin flux at high gradients.For example,Martin In steep, soil-mantledlandscapes, the interaction between and Church [1997] used historicallandslide data to define diffusiveprocesses and shallow landslidingis reflected in al- different diffusivitiesfor high and low gradienthillslopes. The ternating sequencesof ridges and unchanneledvalleys. Hill- high-gradient diffusivity represents "rapid, episodic mass slopesediment transport processes influence not only the evo- movements."Also, many models of landscapeevolution and lution of hillslope morphology,but also the rates of hollow fault scarpevolution couple a linear diffusionlaw and a thresh- infilling and evacuationby debrisflows, the rates of sediment old slope angle, such that higher anglesare instantaneously flux into channel networks, and the distribution of soil on reduced to the thresholdvalue [e.g., Ahnert, 1976; Avouac, hillslopes.We modeled diffusive sediment transport and its 1993;Tucker and Slingerland,1994; Arrowsmith et al., 1996].As implicationsfor hillslope morphology.Our conclusionsfrom opposedto these coupled models, nonlinear transport laws this analysisare as follows: predict that a rapid (but continuous)increase in sediment 1. Diffusive sediment transport has traditionally been transport can be used to approximate the development of modeledas a linear functionof gradient(equation (1)). Our thresholdslopes [Howard, 1997]. Our topographicanalysis (see theoreticalanalysis shows that diffusiveprocesses should pro- Figure 6) demonstratesthat a continuoustransport function ducea nonlinearrelationship between sediment flux and slope may be more appropriatethan slope thresholdsfor modeling (equation(8)). This analysisyields a nonlineardiffusive trans- the erosion of steep, soil-mantledhillslopes. port law in which sediment flux increasesnearly linearly for Previousstudies propose that hillslopegradient is influenced shallowgradients, but increasesrapidly as gradientapproaches by rates of valley incision and may reflect thresholdhillslope a critical slopeangle (Figure 3). processes.Through their analysisof river incision rates and 2. At our studysite, hillslopestend to becomeincreasingly digital topographicdata along a swathof terrain in the north- planar with increasinggradient, consistentwith our proposed western Himalayas, Burbank et al. [1996] concludethat the nonlineartransport law (Plate 1 and Figure 6). Hillslopemor- averageangle of hillslopesis steepand independentof erosion phologyat our studysite is inconsistentwith linear diffusion. rate, such that landslidingallows for hillslopesto adjust effi- 3. We calibrated our nonlinear transport law using high- ciently to river downcutting.Schmidt and Montgomery[1995] resolutiontopographic data for our studysite (Figure 4). This suggestthat bedrock landslidingresults from a combinationof calibrationassumes that the local terrain usedin our analysisis slope and local relief, suchthat the maximumhillslope angle in approximateerosional steady state. Our calibratedparam- varies with the elevation difference between ridges and valley eters yield sedimenttransport rates that are consistentwith bottoms. Howard [1997] provided insight on threshold- field estimates obtained by measuring colluvial deposition controlled erosional regimes through a landscapeevolution ratesin hollows.Topographic modeling of erosionrates shows model. In exploringrelationships between erosionrate, relief, a clusteringaround the estimated constantvalue. However, and drainagedensity, he states"in steepterrain, erosionrates modelederosion rates for 30% of our studyarea differ by more may be sufficientlyhigh that slopegradient is generallyclose to than 50% of the assumedconstant value, suggestingthat the failure conditions... slope gradient then becomesessentially equilibrium assumptionmay only be approximatelymet. independentof erosionrate or locationon the slope"[Howard, 4. With increasingerosion rate, decreasingdiffusivity, and 1997, p. 215]. In our studybasins, hillslope gradients are suf- increasinghillslope length, our proposedmodel predictsthat ficientlyclose to the criticalvalue to generatethe behaviorthat hillslopesshould become steeper and increasinglyplanar. Howard [1997] describes.The mean gradient for these hill- 5. In our studysite, hillslopegradients tend to clusternear slopesis approximately0.80 (or 65% of the criticalslope), and valuesfor which sedimentflux is a highlynonlinear function of over 80% of gradientsexceed 0.6. Slight increasesin slope gradient(0.7-0.9), suchthat slope anglesmay be limited by angle should generate large increasesin massflux, and slope large increasesin massflux at high gradients.Because slight anglesmay be limited by this feedback;for a given channel variationsin hillslopegradient may correspondto a large vari- spacing,local relief therefore may be largely independentof ationsin erosionrate, slopeangle will not be a sensitiveindi- erosion rate. cator of tectonicforcing. The signatureof tectonicforcing will Our analysissuggests that ratesof uplift or erosionmay not be more reliably manifestedin the topographiccurvature of be easilydiscerned from hillslopesteepness [e.g., Kelsey et al., hilltops and divides. ROERING ET AL.: EVIDENCE FOR NONLINEAR, DIFFUSIVE SEDIMENT TRANSPORT 869

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