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

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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 ..":".... .*',/½'5½"' = 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 previousstud- ies,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 diffu- sion law. Our resultsindicate that on soil-mantledhillslopes, small changesin slope angle may lead
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