numerical modeling of ﬂuvial strath-terrace formation in ... · gregory s. hancock* department of...

For permission to copy, contact [email protected] 2002 Geological Society of America 1131

GSA Bulletin; September 2002; v. 114; no. 9; p. 1131–1142; 11 figures.

Numerical modeling of fluvial strath-terrace formation inresponse to oscillating climate

Gregory S. Hancock*Department of Geology, College of William and Mary, Williamsburg, Virginia 23187, USA

Robert S. AndersonDepartment of Earth Sciences and Center for the Study of Imaging and Dynamics of the Earth, University of California,Santa Cruz, California 95064, USA

ABSTRACT

Many river systems in western NorthAmerica retain a fluvial strath-terrace rec-ord of discontinuous downcutting into bed-rock through the Quaternary. Their impor-tance lies in their use to interpret climaticevents in the headwaters and to determinelong-term incision rates. Terrace formationhas been ascribed to changes in sedimentsupply and/or water discharge produced bylate Quaternary climatic fluctuations. Weuse a one-dimensional channel-evolutionmodel to explore whether temporal varia-tions in sediment and water discharge cangenerate terrace sequences. The model in-cludes sediment transport, vertical bedrockerosion limited by alluvial cover, and lat-eral valley-wall erosion. We set limits onour modeling by using data collected fromthe terraced Wind River basin. Two typesof experiments were performed: constant-period sinusoidal input histories and variable-period inputs scaled by the marine d18Orecord. Our simulations indicate thatstrath-terrace formation requires inputvariability that produces a changing ratioof vertical to lateral erosion rates. Strathsare cut when the channel floor is protect-ed from erosion by sediment and areabandoned—and terraces formed—whenincision can resume following sediment-cover thinning. High sediment supply pro-motes wide valley floors that are abandonedas sediment supply decreases. In contrast,wide valleys are promoted by low effectivewater discharge and are abandoned as dis-charge increases. Widening of the valleyfloors that become terraces occurs over

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many thousands of years. The transitionfrom valley widening to downcutting andterrace creation occurs in response to sub-tle input changes affecting local divergenceof sediment-transport capacity. Formationof terraces lags by several thousand yearsthe input changes that cause theirformation.

Our results suggest that use of terraceages to set limits on the timing of a specificevent must be done with the knowledge thatthe system can take thousands of years torespond to a perturbation. The incision ratecalculated in the field from the lowest ter-race in these systems will likely be higherthan the rate calculated by using older ter-races, because the most recent fluvial re-sponse in the field is commonly downcut-ting associated with declining sedimentinput since the Last Glacial Maximum.This apparent increase in incision rates isobserved in many river systems and shouldnot necessarily be interpreted as a responseto an increase in rock-uplift rate.

Keywords: fluvial features, landscape evo-lution, modeling, river terraces, sedimentsupply.

INTRODUCTION

Fluvial strath-terrace sequences record riverincision through time and the timing of envi-ronmental perturbations to fluvial systems.Fluvial strath terraces are abandoned riverflood plains with thin mantles of alluviumoverlying a beveled bedrock platform (Fig. 1).Conceptually, strath-terrace formation requireschanges in the ratio of vertical incision to lat-eral planation in a downcutting stream. Gilbert(1877) postulated that as sediment load nears

stream-transport capacity, vertical erosionrates decrease, making lateral planation rela-tively more important. Flood plains are the re-sult of this lateral planation, and strath ter-races are vestiges of these flood plains,formed as river downcutting leads to flood-plain abandonment (Gilbert, 1877). Mackin(1937) proposed that incipient terraces formduring periods of river stability—i.e., grad-ed conditions—when lateral planation of val-ley walls produces wide valley floors. Gradedconditions occur when a river can transport allof the imposed sediment load and is achievedthrough a balance between sediment load andsize, discharge, and channel slope (Mackin,1937). According to Mackin (1937), the widthof the valley floor is related to the duration ofthe graded period. Both Gilbert (1877) andMackin (1937) suggested that the beveling ofthe bedrock strath and the deposition of thealluvial mantle occurred simultaneously andthat the channel geometry remained stablewhile the channel migrated across the valleyfloor. The beveled strath and thin alluvial cov-er distinguish strath from fill terraces, whichhave no underlying bedrock platform and canbe many tens of meters thick.

These hypotheses suggest that strath for-mation requires alternation between ‘‘graded’’periods of extensive lateral planation, and‘‘nongraded’’ periods of vertical incision. Riv-ers may obtain graded conditions when avail-able stream power equals the critical power,defined as the stream power necessary totransport enough sediment to maintain grade(Bull, 1979, 1990). Under these conditions in-cision rates (i.e., rates of vertical erosion) areat a minimum, allowing extensive valley wid-ening by lateral planation. Strath terraces arecreated when the available stream power be-comes greater than this critical power, leading

1132 Geological Society of America Bulletin, September 2002

HANCOCK and ANDERSON

Figure 1. Photograph looking downstream along the Wind River below Dubois, Wyoming.The width of the view represents ;1 km at the center. Above the active flood plain, threeterrace levels (ages: WR-1, ca. 20 ka; WR-3, ca. 120 ka; and WR-7, ca. 600 ka) mappedby Chadwick et al. (1997) can be seen along the course of the river.

to incision and flood-plain abandonment(Bull, 1979). In this conceptual model, switch-ing between these two states can occur by al-tering the available stream power (e.g., chang-ing discharge or slope), and/or the criticalpower (e.g., sediment load or size; Bull,1979). Available stream power may be in-creased, and hence, terraces may be created byincision, through basin-scale changes such asbase-level fall, hydrologic changes induced byclimate change, and/or increased tectonic ac-tivity (e.g., Merritts et al., 1994; Pazzaglia etal., 1998).

The well-preserved strath-terrace sequencesfound in many river systems of western NorthAmerica record discontinuous incision intobedrock throughout the late Quaternary (e.g.,Howard, 1970; Patton et al., 1991; Reheis etal., 1991; Ritter, 1967). Their widespread oc-currence implies response to a common re-gional forcing, and field data indicate climateoscillation between glacial and interglacialconditions, superimposed on long-term upliftand/or base-level lowering, may be responsi-ble. Many terrace deposits indicate transportof larger grain sizes by the terrace-formingrivers compared to their modern counterparts,suggesting greater peak discharges and/ormore effective headwater erosion during valley-widening periods than at present (Baker, 1974;Pierce and Scott, 1982). In addition, some ter-race deposits suggest that terrace-formingrivers were braided, unlike their modern me-andering counterparts (e.g., Baker, 1974; Jack-son et al., 1982; Ritter, 1967; Sinnock, 1981).Braided-stream conditions suggest greatersediment supply (particularly bedload) andlarger, more flashy channel flows (e.g., Bridge,

1993; Leopold and Wolman, 1957; Schummand Khan, 1972; Smith and Smith, 1984).

On the basis of this evidence, valley wid-ening and strath formation are proposed to oc-cur during periods of maximum glaciation(e.g., Baker, 1974; Bull, 1991; Howard, 1970;Ritter, 1967; Sinnock, 1981), although glacierswithin the headwaters of a river system are notrequired (e.g., Pierce and Scott, 1982). Valleywidening and strath formation are initiated byincreased sediment supply and hydrologicchanges associated with a larger and later-melting snowpack, reduced evaporation andinfiltration capacity, extensive periglacial pro-cesses, and reduced vegetative cover duringcooler climate conditions (Pierce and Scott,1982). Incision and terrace formation by aban-donment of flood plains presumably occurduring interglacial periods or during the initialtransition from glacial to interglacial condi-tions (Sinnock, 1981). Church and Ryder(1972) and Jackson et al. (1982) have pro-posed that fluvial extraction of sediments cre-ated during glacial climates may continue foran extended time, called the paraglacial peri-od, after the transition toward interglacial con-ditions has begun. Renewed incision andstrath-terrace formation may therefore occurwell after this transition.

The age and height of paired strath-terracesequences provide the primary data for esti-mating the incision history and incision ratesin fluvial systems (e.g., Pazzaglia et al., 1998).In western North America, incision historiesdeduced from precisely dated strath-terrace se-quences are rare, but many of these sequencesindicate accelerating incision rates during theQuaternary (e.g., Hancock et al., 1999; Patton

et al., 1991; Reheis et al., 1991; Repka et al.,1997). The cause for this acceleration is notclear, but could indicate regional intensifica-tion of climate, base-level, and/or tectonicchange. Alternatively, Mills (2000) has pro-posed that apparent late Cenozoic accelerationof incision rates deduced from many terracesequences in the southeastern United States isan artifact of averaging over longer periods asterrace age increases. In our modeling, we at-tempt to test this hypothesis.

The timing, duration, and mechanisms ofstrath-terrace formation are difficult to infersolely from field observations because terracesequences represent incomplete records, aredifficult to date, and formed during fluvialconditions that differ from the present. Herewe explore strath-terrace formation in re-sponse to climate oscillation by using a nu-merical stream-profile simulation. This ap-proach differs significantly from previousmodeling efforts focused on the generation offluvial terraces (e.g., Boll et al., 1988; Veld-kamp, 1992; Veldkamp and Vermeulen, 1988)because we rely on physically based rules anda realistic set of stream inputs. First, we dis-cuss the model framework and rule set, theconstraints on initial and boundary conditions,and our modeling strategy. Second, we discussthe results and their implications for terraceformation and interpretation. Specifically, weexplore the following questions: (1) Why doterraces and terrace sequences form? (2) Howis terrace formation related in time to forcingmechanisms? (3) How long does it take toform a strath terrace? (4) What past climaticepisodes are most likely to leave a terrace re-cord? and (5) What implications do the an-swers have for interpreting fluvial terracesequences?

MODELING APPROACH

We use a one-dimensional finite-differencecode simulating the evolution of the long val-ley profile of a river in the face of temporallyvarying inputs of water and sediment. We in-corporate (1) sediment transport, (2) verticalincision of bedrock at the base of the activechannel, and (3) lateral planation of the valleywalls. Our incorporation of lateral planationdifferentiates our approach from previousstream-evolution models (see Howard et al.,1994) and allows us to investigate the condi-tions conducive to generation of strath terrac-es. Aside from lateral planation, our goals andphysical rules are similar to those of Tuckerand Slingerland (1997), but our focus is solelyon the channel response to climaticperturbations.

Geological Society of America Bulletin, September 2002 1133

NUMERICAL MODELING OF FLUVIAL STRATH-TERRACE FORMATION

Climatic forcing may be manifested in flu-vial systems by changes in discharge, sedi-ment load, and sediment size. Our model rulesmust therefore explicitly incorporate grain sizeand water discharge. We assume that transportof sediment is dictated by the local bed shearstress, and erosion of bedrock, either of thewall or of the bed, is related to the availableenergy per unit area of bed, or unit streampower (e.g., Whipple and Tucker, 1999). Theone-dimensional model tracks sediment thick-ness, bedrock erosion, and valley width atevenly spaced channel cells along a profile,analogous to channel reaches. We use a stag-gered grid to enhance model stability. Thethickness of the sediment, water discharge,channel elevation, and width of the channeland the valley floor are tracked at the centerof each channel cell. The physical quantitiesthat dictate fluxes between the cells, such asslope and shear stress, and the fluxes them-selves, are calculated at the edges of the cells.

Sediment Transport

Sediment mass within each cell is conserved,as expressed by the continuity equation:

dh 1 dQs5 2 , (1)dt w dx

where h is sediment thickness, t is time, Qs isvolumetric sediment-transport rate, x is thedownstream distance, and w is either valleywidth, w, or channel width, wc. If sedimentflux decreases in the downstream direction(i.e., dQs/dx is negative, which will cause sed-iment deposition), deposition is specified tooccur evenly over the active valley width, w.Here the active valley width is defined as theregion at nearly the same elevation as thechannel, and across this width the channel mi-grates. Hence, channel migration deposits sed-iment across the active valley floor. When sed-iment flux increases downstream (i.e., if dQs/dx is positive, which will cause sedimenterosion), sediment is removed only from theactive channel width, wc. Taken together, theserules allow sediment to be sequestered by de-position on wide valley floors and allow thesemantled valley floors to be preserved duringsubsequent channel incision.

We must therefore formulate a model al-gorithm for sediment discharge, and we fol-low the dominant-discharge (i.e., geomorphi-cally effective) strategy of Tucker andSlingerland (1997). Most sediment-transportequations suggest that the transport capacity,Qs, depends on the local shear stress, t,

t 5 rgHS, (2)

where r is water density, g is gravitational ac-celeration, H is water depth, and S is energyslope, here assumed equal to channel-bedslope. Equation 2 is valid for steady, uniformflow. Both the channel slope and the waterdepth must be known in order to calculate theshear stress and, hence, the sediment dis-charge. The channel slope, S, at each cell edgeis calculated by differencing the bed elevation,z, from the adjacent channel cells and dividingby the cell spacing, dx.

We calculate the flow depth, H, in a cellfrom the water discharge, Q. The effective dis-charge, Q, defined as the flow having the rightcombination of frequency and magnitude tocontrol channel form, is related to the drainagearea, A, through a power-law relationship(e.g., Dunne and Leopold, 1978),

cQ 5 bA , (3)

where b and c are empirical constants. Tem-poral variation in discharge is prescribed byvarying b and/or c. Channel width, wc, is cal-culated by using the hydraulic geometry re-lationship

ew 5 dQ ,c (4)

where d and e are again empirical constants.In our models, appropriate values for b, c, d,and e in equations 3 and 4 are taken from theWind River, Wyoming, and are typical ofthose obtained in numerous river systems (Le-opold and Maddock, 1953).

Flow depth in a channel is given by

QH 5 , (5)

vwm

where v is mean flow velocity and wm is meanflow width. Assuming a roughly rectangularchannel, wm ø wc. The mean velocity may beestimated by using the Manning equation

2/3 1/2kH Sv 5 , (6)

n

where k is a constant and n is the Manningroughness coefficient. Furthermore, we haveassumed a wide, shallow channel such that wc

» H, allowing substitution of flow depth H forthe hydraulic radius.

By combining equations 3, 4, 5, and 6, weobtain the relationship between flow depth, H,and measurable quantities:

3/5cbA n23/10H 5 S . (7)1 2wc

Finally, by combining equations 2 and 7,bed shear stress may be predicted with the de-sired explicit dependence on the discharge:

3/5cbA3/5 7/10t 5 rg n S . (8)1 2wc

To predict the volumetric sediment flux, Qs,we use a variation of the Bagnold bedloadsediment-transport equation (see Slingerlandet al., 1994),

Bw1/2 1/2Q 5 (t 2 t )(t 2 t ), (9)s c c1/2(r 2 r)r gs

where B is a dimensionless constant (;10), tc

is the critical shear stress required to transporta given grain size, and rs is the density of thesediment grains. The tc for sediment entrain-ment is estimated by using the Shields param-eter, and this is the means by which depen-dence upon grain size is brought into theproblem. The sediment discharge, Qs, is de-termined by using the shear stress (equation8) calculated at the edge of each channel cell.The divergence of the sediment-transport ca-pacity, dQs/dx, may then be calculated at eachchannel cell center. Although equation 9 isformulated solely for bedload transport in al-luvial or transport-limited rivers, bedload islikely the most critical component affectingstream incision and terrace generation, as thebedload must be in transit before rock erosioncan take place. The simulated streams arelargely covered by alluvium, with only patchybedrock exposure.

Bedrock Erosion

Rules for predicting bedrock erosion at thereach scale in landscape-evolution models arebased on either shear stress or the stream pow-er per unit bed area, v,

rgQsv 5 , (10)

wc

where dz/dt is the vertical erosion rate (e.g.,Whipple and Tucker, 1999). These rules aretypically of the form

dzr5 K(X 2 X ) , (11)0dt

where X is either shear stress or stream power,X0 is a threshold shear stress or stream power,



r is an exponent, and K is a parameter relating‘‘excess’’ shear stress or stream power to ero-sion rate. This formulation neglects explicitconsideration of the role of the sediment-sup-ply rate in controlling erosion (Sklar and Die-trich, 2001). In this formulation, the value ofK is related to rock properties that dictate theamount of rock that may be eroded per unitshear stress or stream power. Values for K arenot well defined, but have been determined byerosion measurements in badland channels(Howard and Kerby, 1983) and by comparingpast and present channel profiles in the SierraNevada, California (Stock and Montgomery,1999). The method for choosing K for a par-ticular setting is not known, however. Our ap-proach is to select a K that produces geolog-ically reasonable erosion rates, to assume r 51 and X0 5 0, and to use stream power perunit bed area. Results obtained by using thestream power and shear stress rules do not dif-fer substantially, as shown theoretically byWhipple and Tucker (1999). We use separaterules that predict vertical rock erosion (i.e., in-cision) rates and horizontal rock erosion (i.e.,lateral planation) rates.

Rock-Incision RatesPotential vertical rock-erosion rates, (dz/

dt)p, is expressed by

dz5 K v, (12)v1 2dt

p

where Kv is a ‘‘hardness’’ parameter for ver-tical rock erosion. This equation predicts onlythe maximum potential erosion rate becauserock incision can take place only when sedi-ment does not protect that rock from erosion.Field observations suggest that bedrock ero-sion can occur while there is a finite meansediment thickness within a reach, as sedimentcover is often variable (e.g., Leopold andMaddock (1953). Sediment may be transport-ed and stored within a reach even while bed-rock erosion is occurring (Shepherd andSchumm, 1974; Howard and Kerby, 1983)and may be required for this erosion to takeplace as it provides the abrasive tools (e.g.,Hancock et al., 1998; Sklar and Dietrich,2001; Whipple et al., 2000). We do not incor-porate the possibility that there may be an op-timal sediment-supply rate, above and belowwhich erosion rates decline (Sklar and Die-trich, 2001).

To accommodate these observations, weadd to equation 12 a sediment-scour rule. Thisis designed to acknowledge that a finite butthin sediment cover can be scoured away inhigh-transport conditions, exposing the under-

lying rock to erosion. We assume that thedepth of sediment scour is related to the sed-iment-transport capacity, Qs, and the channelwidth, wc, and may not be uniform. We usethe potential sediment-transport rate, Qs, cal-culated by using equation 9, to estimate a sed-iment-scour depth scale, L,

sQsL 5 , (13)wc

where s is a constant with dimensions of timeper length, and wc is channel width.

It is reasonable that the probability ofscouring to bedrock increases as the scourdepth scale, L, increases, and decreases as thesediment thickness, h, increases. FollowingHoward (1998), we hypothesize that the prob-ability, F, for rock exposure and, hence, ofrock incision, decreases exponentially as sed-iment thickness increases:

2(h/L)F 5 e . (14)

Appropriately, the probability of bedrockerosion is therefore 1 if h 5 0 and approacheszero as h » L. To calculate the actual verticalbedrock-incision rate in a cell, dz/dt, we com-bine equation 12 and equation 14 to yield

dz5 FK v. (15)vdt

Although speculative, this formulation ac-commodates several field observations, in-cluding (1) the need to scour to the base ofthe sediment layer in order to erode rock, and(2) the likelihood that scour depths will varywithin the channel. This formulation also rec-ognizes that rock-erosion rates should increaseas Qs increases and decrease as the sedimentthickness increases.

Lateral Planation RatesAlthough the processes of erosion of river

banks in self-formed, alluvial river channelshave been the subject of extensive investiga-tion (e.g., Andrews, 1982), the widening ofbedrock valley walls by rivers has receivedmuch less attention. We know of only onestudy of lateral erosion, and this study sug-gests that lateral erosion rates, dw/dt, are re-lated to v1/2 (Suzuki, 1982). However, theprobability of valley-wall contact was not con-sidered, as we discuss subsequently.

To proceed, we make the assumption thatthe potential rate of valley-wall erosion isanalogous to vertical rock erosion and is re-lated to stream power per unit bed area,

dw5 K v (16)w1 2dt

p

applied to the channel margins, where (dw/dt)p

is the potential valley-widening (not channel-widening) rate and Kw is a susceptibility con-stant that relates stream power to wall erosion.We acknowledge that this critical componentof our model cannot be justified rigorously.Investigation of the valley-widening processesin rock-floored channels and of the controlson widening rates is sorely needed. However,lateral planation that far exceeds vertical in-cision over time is a key field observation thatmust be reproduced in this model. For ex-ample, in the Wind River basin (Fig. 1), ter-race widths require average lateral planationrates to be several orders of magnitude great-er than vertical rates. As discussed subse-quently, the model succeeds in reproducingthis observation.

To widen the valley, a channel must be incontact with its valley wall. Because our mod-el is one-dimensional, we do not track thechannel position explicitly. Instead, we as-sume that the channel position within a valleyis random if sampled over a long time periodin the absence of preferred positioning im-posed by tilting or other external forcing. Inthis case, the channel width, wc, and the valleywidth, w, determine the probability of valley-wall contact, W,

wcW 5 . (17)w

The probability increases as wc increases andas w decreases. The rate of valley-wall erosionis then determined by combining equations 16and 17:

dw5 WK v. (18)wdt

We note that valleys should widen rapidly atfirst and more slowly thereafter as the proba-bility of channel contact with the valley walldeclines.

This lateral erosion is divided between thetwo valley walls. In order to allow asymmetryin wall erosion, we specify a symmetry con-stant, a, where 0 # a # 1. Rates of valley-wall erosion are adw/dt and (1—a)dw/dt, onopposite valley walls. If a 5 0.5, there is nopreferred slip direction. As illustrated in Fig-ure 1, preferred slip across the valley is animportant factor in preserving terraces insome fluvial systems. In simulations presentedhere, we choose a 5 1, equivalent to forcing



channel migration in a constant direction andmaximizing terrace preservation.

MODELING STRATEGY

Our goal is to explore how a river respondsto variations in sediment and water fluxes(here referred to as inputs) driven by climateoscillation. We seek those conditions that leadto terrace formation. We discuss two experi-ments in which these inputs were varied: (1)constant period and amplitude variation of in-puts (the ‘‘constant-period experiments’’), and(2) irregular period and amplitude variationsscaled by the marine oxygen isotope record(the ‘‘variable-period experiments’’). Al-though we are interested in the general phe-nomenon of strath-terrace formation, we haveselected the Wind River, Wyoming, to providelimits for the model initial and boundary con-ditions and the magnitudes of discharge, sed-iment load, and sediment size. This choicedoes not limit the significance of the model tothis river system, but ensures that we selectgeologically reasonable values.

Initial and Boundary Conditions

All experiments start with a straight longi-tudinal channel profile with a slope S 5 0.01and a total length of 150 km; the spacing ofthe model cells is 1 km. The slope is equiv-alent to maximum slopes measured at the ki-lometer scale on the oldest terrace profiles inthe Wind River system. Boundary conditionsmust be specified at the upper and lower endsof the channel profile. The downstreamboundary is lowered at 0.15 mm per modelyear, the average incision rate obtained fromdated terraces at the lower end of the terracedreach of the Wind River near Riverton, Wy-oming (see Fig. 2 in Chadwick et al., 1997).All sediment reaching the lowest channel cellis allowed to escape downstream, preventingaggradation at the lower boundary. Sedimentis delivered only to the cell farthest upstream.The bedrock susceptibilities to erosion, Kv andKw, are set to 2 3 10–11 m/(W·s/m2) [5(m·W·s)–1] in all cells and in all simulations,implying uniform rock resistance. This valueyields geologically reasonable erosion ratesduring the simulations.

Water Discharge and Channel Geometry

The geomorphically effective—i.e., ‘‘dom-inant’’ (Tucker and Slingerland, 1997)—dis-charge is the relevant quantity for predictionof sediment transport and channel erosion(e.g., Wolman and Miller, 1960). The relation-

ship between flow frequency and magnitudethat produces the effective discharge for rockerosion in bedrock-floored rivers is notknown. We assume that the effective flowsequal the 2-yr-recurrence-interval peak dis-charge. Values for effective discharge weredetermined from modern gauging records onthe Wind River. This discharge increases withdrainage area, suggesting values of b 5 0.29and c 5 0.76 in equation 3. Channel widthsat various flows have been measured at manysites along the Wind River (Leopold and Wol-man, 1957; Smalley et al., 1994), yielding val-ues of d 5 3.0 and e 5 0.55 in equation 4.

Because effective discharge, Q, likelyvaries in response to changing climate condi-tions, Q is varied in our model. Paleohydrol-ogic studies in western North America suggestthat peak discharges in some terrace-formingstreams were much greater than at present(Baker, 1974; Pierce and Scott, 1982). In thesimulations, the effective discharge is in-creased by up to two times the modern effec-tive discharge, which is a reasonable estimatefor sustained, frequent flow events. In exper-iments in which water discharge is varied, thevalue of b, representing the effective basinrunoff in equation 3, is increased by up to twotimes the modern value (i.e., 0.29 , b ,0.58). The duration of effective discharge is10 days per model year, a choice based uponthe observed duration of the peak discharge ofthe Wind River. Although not considered here,changes in the duration of the effective partof a river hydrograph could also be importantin a fluvial system.

Sediment Load

The sediment load and grain size alsochange in response to climate oscillations anddictate the ability of the river to access anderode bedrock. The initial grain size, D, is tak-en to be equal to the mean grain size measuredin the modern Wind River (D50 ø 0.5 cm,Smalley et al., 1994). The grain size is as-sumed to be uniform along the model channel;transport of only this grain size is considered.The minimum sediment-load input is ;5000m3/yr, equal to the measured average annualbedload transported by the modern Wind Riv-er near its headwaters at Dubois (Leopold etal., 1960).

Hallet et al. (1996) have shown that exten-sive glacial cover can increase sediment yieldby up to an order of magnitude over nongla-ciated or slightly glaciated basins. On the ba-sis of this observation, the sediment-load in-put to the uppermost cell is increased by upto 10 times the minimum value. In the Wind

River system, mean grain sizes transported bythe modern river are similar to those foundwithin terrace deposits, whereas in other riversystems, mean and maximum grain sizes areseveral-fold larger in preserved terrace deposits(e.g., Baker, 1974; Pierce and Scott, 1982). Inthe experiments we report here, the grain sizeis kept constant. Experiments incorporatingvariations in the grain size used produce resultscomparable to sediment-supply variation.

Numerical Experiments

For the constant-period experiments, a fixedsinusoidal input variation with a period of 100k.y. is selected to mimic the dominant periodof climate cycling through glacial to intergla-cial conditions since ca. 800 ka. Maximumsediment input, sediment size, and water dis-charge are assumed to coincide with maxi-mum glaciation. We describe first a controlexperiment with steady inputs, followed bythree experiments: variable sediment load(variation by a factor of 10); variable dis-charge (variation by a factor of 2); and com-bined discharge and sediment-load variation.All experiments are run for 800 0000 modelyears. We discuss only the last 400 000 yr, bywhich time the channel response to the im-posed initial conditions is complete.

In the variable-period experiments, the ben-thic marine oxygen isotope record of Imbrieet al. (1984) is used to scale the inputs. Be-cause this record is a proxy for the extent ofcontinental glaciation, we assume that (1) themagnitude of climate change (e.g., glacier ex-tent) in our modeled river system parallels thecontinental ice volume, and (2) the relation-ship between climate change and sedimentsupply and water discharge is linear. Althoughthese assumptions are certainly not entirelyvalid, we are interested primarily in exploringthe river response to complex input variation.We consider two variable-period experiments:(1) variation of sediment supply by a factor of10 and (2) variation of water discharge by afactor of 2. In all cases, the peak in the inputis set to occur when the oxygen isotope curveindicates glacial conditions. In order to re-move any memory of the initial conditions,the ca. 800 ka record of Imbrie et al. (1984)is repeated once, producing experiments witha total duration of 1600 3 103 model years.

RESULTS

Constant-Period Experiments

In the control experiment, the channel pro-file evolves toward a slightly concave-up



Figure 2. Successive channel profiles for every 100 k.y. model years in the control exper-iment, from initial conditions (t 5 0) to simulation end (t 5 800 ka). As in all of theconstant-period experiments, the channel profile achieves a nearly uniform incision rateover the last several 100 k.y. The downstream end of the modeled profile is at 0 km.

Figure 3. View of one side of the terraced-valley topography generated during the tenfoldsediment-supply-variation experiment. The view extends upstream from 122 to 128 kmalong the model profile. The view is looking upstream and illuminated from the left. Thevalley floor (i.e., the river) is on the right at 0 m distance across the valley (bottom axis).In this experiment, the channel was forced to slip across the valley in one direction only;hence, the missing valley side would be a vertical wall. Each numbered terrace level wasproduced during one sediment-supply input cycle. All are strath terraces, with alluviumthickness of less than or approximately equal to 1 m. The topographic profile extendingfrom A to B is shown in Figure 7.

shape. The slope in the lower half of the mod-el space declines monotonically through time,whereas that near the top of the profile steep-ens slightly (Fig. 2). Both vertical and lateralerosion rates at any point within the modelstream decline monotonically during the sim-ulation. By the end of the simulation, the pro-file has achieved a nearly uniform erosionrate. No terraces are produced in this simula-tion. All other constant-period simulationsproduced terrace sequences, as illustrated forthe sediment-load experiment in Figure 3. Thesix terrace levels in Figure 3 correspond to thefinal six input cycles completed. We discuss

the controls on terrace formation, timing, andappearance, by describing the response of onerepresentative channel cell 125 km above thedownstream end of the model profile.

Vertical bedrock-incision rates, dz/dt, andlateral planation rates, dw/dt, vary throughmodel time in all simulations in which inputsare varied (Fig. 4). The most rapid incisionoccurs when stream power and/or the likeli-hood of bedrock access is high. As a conse-quence, the vertical rock-incision rate is in-versely proportional to variations in sedimentsupply (Fig. 4A), because bedrock access andthe energy expended to transport sediment are

partially controlled by sediment supply. Incontrast, vertical rock-incision rates are di-rectly proportional to changes in discharge(Fig. 4B), because increasing discharge in-creases available stream power.

The ratio of vertical incision to lateral pla-nation rates, dz/dt:dw/dt, is also time depen-dent and is driven largely by variation in ver-tical incision rates (Fig. 5). Lateral planationrates are controlled almost entirely by theavailable stream power and are far less sen-sitive to variations in sediment supply than arevertical incision rates (Fig. 5). The valleywidth is controlled by the magnitude of thisratio and by the duration of lateral planation.Hence, wide valley floors are created duringlong periods when the ratio is low, when dw/dt k dz/dt. Because this ratio varies, valleywidth is also time dependent (Fig. 5). As en-visioned by Gilbert (1877) and Mackin(1937), variation in the relative importance oflateral planation rates and vertical incisionrates, here expressed as the ratio dz/dt:dw/dt,is therefore responsible for producing terracesequences. Valley floors are widened when theratio dz/dt:dw/dt is low for extended periods,and terraces form when these valley floors areabandoned during subsequent incision whenthe ratio dz/dt:dw/dt is high (Fig. 5, arrowsshow times of valley abandonment and terraceformation).

Lateral planation and therefore terrace cut-ting occurs when the sediment input forces dz/dt:dw/dt to be low. In the sediment-load ex-periments, vertical incision rates are lowestwhen the input of sediment is high, allowingextensive lateral planation (Fig. 5A). This re-lationship corresponds to times when the sed-iment cover is sufficient to reduce the likeli-hood of scour through this cover, preventingvertical rock erosion. Declining sediment loadleads to thinning of the sediment cover, allow-ing more rapid incision and abandonment ofwide valley floors to form strath terraces. Onthe other hand, valleys widen when effectivewater discharge is low and are abandoned asincreasing discharge results in more rapid ver-tical incision (Fig. 5B). Although the lateralplanation periods necessary to form terracesmay be related in time to maximum sedimentload or minimum water discharge, the actualtiming of terrace formation (arrows in Fig. 5)is not synchronous with these periods. Instead,in all simulations, the timing of actual terraceformation (valley-floor abandonment) typical-ly lags these variations in the forcing by up toseveral tens of thousands of years.

Because sediment supply and water dis-charge modulate vertical incision rates differ-ently, these inputs when combined compete to



Figure 4. Vertical (solid line) and lateral (dashed line) erosion-rate histories at km 125during constant-period variation of fluvial inputs (gray line). Simulations are (A) tenfoldsediment-supply variation, and (B) twofold dominant-discharge variation. Input variationproduces variable erosion rates and more significantly modulated vertical erosion ratesthan horizontal erosion rates. High vertical erosion rates are associated with periods mostconducive to enhancing rock accessibility (i.e., low sediment supply) and/or to producinghigh stream power (i.e., high dominant discharge). The cycle in the ratio of vertical tolateral erosion rates is responsible for creating terraces.

Figure 5. History of normalized sediment inputs (gray line), normalized valley-floor width(i.e., flood-plain width) at river level (dashed line), and the ratio of vertical to lateralerosion rates (solid black line) during two simulations: (A) tenfold sediment-supply vari-ation and (B) twofold dominant-discharge variation. Inputs and valley-floor widths arenormalized by dividing the value at each time by the maximum value during each simu-lation. The valley floor (i.e., the ‘‘flood plain’’) widens most significantly when verticalerosion rates are low, producing a ratio of vertical to lateral erosion that is small or zero.Flood plains are abandoned to form terraces when vertical erosion rates and the ratio ofvertical to lateral erosion rates increase, leading to renewed downcutting and narrowingof the active valley. Terraces are formed at the transition from valley-floor widening tonarrowing (arrows). Terrace formation does not occur at the times of either maximuminputs or minimum inputs, but instead significantly lags the timing of maximum (e.g.,sediment-supply maxima) or the minimum (e.g., water-discharge minimum) inputs. In thesimulations, all terraces generated are strath terraces.

control the timing of valley widening and ter-race formation (Fig. 6). For the chosen am-plitudes in this simulation, the sediment sup-ply dominates over discharge in modulatingthe timing and magnitude of incision rates, asdz/dt is inversely proportional to the sedimentsupply. We illustrate with one simulationwhere sediment supply varies by tenfold whileeffective discharge varies twofold. Extensivevalley widening occurs when sediment load ishighest, and terraces are formed as sedimentload decreases (Fig. 6). Valley widening ispromoted by high effective discharge, whenthe lateral planation rate reaches a maximum,while vertical incision rates are low becausesediment load is high and prevents access tothe bedrock (Fig. 6). For this particular choiceof geologically defensible input amplitudes,the variation of sediment load dictates the tim-ing of valley widening.

The final width of a terrace is dictated bythe product of the lateral planation rate with achannel-residence time at a particular eleva-tion. We define the channel-residence time tobe the time spent by the channel between suc-cessive elevation contours, here taken to be 1m. The residence time in lateral planation pe-riods over one simulation (that with 103 sed-iment-supply variation) varies from ;15 k.y.to ;60 k.y., i.e., ;15% to ;60% of each 100k.y. cycle (Fig. 7). We note that despite thisvariation, the terraces produced during theseperiods are roughly the same width. As down-cutting progresses, an increasing duration oflateral planation (low dz/dt:dw/dt) is offset bya decrease in the instantaneous lateral plana-tion rates caused by the decline in channelslope (Fig. 7). The likelihood of wall contact(equation 17) decreases as terraces widen andalso acts to limit terrace width.

Variable-Period Experiments

In the variable-period experiments, all sim-ulations once again produce terrace sequences(Fig. 8), and the relationship between inputsand the timing and mechanisms of terrace for-mation is similar to that relationship for theconstant-period simulations (Fig. 9). However,the irregular input variation produces an irreg-ular channel-incision history in which thestream is unable to incise for much of eachsimulation (Fig. 9). Periods of lateral plana-tion (low dz/dt:dw/dt) account for ;50%–90%of the total simulation time, implying that thetypical mode of operation for the modelstream system involves extended periods oflateral planation and long residence time onthe flood plain interrupted by short periods ofrapid incision (Fig. 10).



Figure 6. Ratio of vertical to lateral erosion rates (solid black line) and normalized activevalley width (dashed black line) at km 125 during constant-period variation of both sed-iment supply (dashed gray line) and effective discharge (gray line). As shown in Figure5, the timing of maximum vertical incision produced by these inputs is not in phase.Therefore, these inputs compete to set the timing of maximum vertical incision and, hence,terrace formation. The arrows mark the timing of valley abandonment and terrace for-mation. Although there is a continuum of possible combinations of these two inputs, withthe field-defined amplitudes chosen for this simulation, the sediment-supply fluctuationsdominate.

Figure 7. The topographic profile along A–B of the terraced valley illustrated in Figure 3(thick line) and channel-residence time (see definition in text) as a function of elevationduring downcutting in this simulation (thin line). Terraces are related to periods of lateralplanation during long channel residence within a narrow elevation range. Residence timeat individual terrace levels reaches up to many tens of thousands of years, indicating thatthe river spent most of the simulation forming terraces.

Unlike the constant-period experiments, notall significant input fluctuations form terraces.The terrace sequences preserved include fourto six well-defined terrace levels as well asseveral less distinct levels (Figs. 8 and 10).The widest terraces form in response to thelongest periods of high sediment load or lowwater discharge (Fig. 9). However, neither ter-race width nor preservation perfectly recordsthe duration and occurrence of climate events.A distinct terrace is not formed by the short-lived (;20 k.y.) fluctuation analogous to themarine isotope stage 4 event (ca. 75 ka, Fig.9). Instead, evidence for this fluctuation is lostwithin the last terrace formed, created at theend of a lateral planation period lasting ;80k.y. and spanning from isotope stage 5 to the

end of isotope stage 2 (Fig. 9). Within thesame simulation, three fluctuations betweenca. 200 and 250 ka that are comparable to iso-tope stage 4 do leave three small but distinctterraces, one of which forms terrace level 3 inFigure 10.

The number of terraces preserved in a sim-ulation depends upon the extent to which thechannel migrates in a preferred direction orslips sideways in its valley. In the simulationswe have presented, we forced a constant mi-gration direction, and seven to eight terracelevels are preserved. In the absence of this im-posed unidirectional migration, many terracesare cannibalized during later lateral planationperiods. With no migration, at most one ortwo terrace levels are preserved; for example,

in the sediment-variation experiments, the sur-viving terraces have model ages of ca. 20 andca. 420 ka. In the absence of a preferred mi-gration direction, terrace preservation requiresthat successively younger terrace levels benarrower, reflecting a shorter duration of lat-eral planation and/or slower lateral planationrates. In this sense, the preservation of terraceswithin sequences is analogous to glacial mo-raine survival (Gibbons et al., 1984); if a rivercannot migrate primarily in a single direction,the terrace sequence will include only thoseterraces that are able to survive later lateralplanation events. Situations that promote uni-directional channel migration include repeatedriver diversion (e.g., Chadwick et al., 1997)and tectonic tilting (e.g., Reheis, 1985).

Although base-level lowering in these sim-ulations is constant, the average vertical channel-incision rate between successive terrace levelsvaries widely (Fig. 11). In the sediment-supply-variation experiment, rates of channelincision generally increase toward the end ofthe simulation; most dramatically, the averageincision rate calculated from the age and ele-vation of the last terrace created is severaltimes higher than any other calculated incisionrate (though this increased rate seems to bemore apparent than real; see discussion andFig. 11). We reiterate that these results arefrom the last 800 k.y. of the total 1600 k.y.model run, and the response to initial condi-tions was complete before this period.

DISCUSSION

In our simulations, the conditions associat-ed with terrace formation support many of theexisting conceptual models for terrace devel-opment. A prerequisite for terrace formationis variability in the ratio of vertical incisionrates to lateral planation rates, as suggested byGilbert (1877) and Mackin (1937), amongothers. Wide valleys form during periods oflateral planation when bedrock exposure islimited and/or available stream power is low(i.e., the stream power nearly equals the crit-ical power), resulting in low vertical channel-incision rates. These conditions are promotedby relatively high sediment load and/or loweffective discharge. Vertical incision promotesabandonment of these valleys, producing ter-races, when these factors reverse. As suggest-ed in the introduction, sediment load and grainsize are thought to have increased in manywestern North American streams under colderclimates. Our model results therefore supportthe general hypothesis that valleys are wid-ened during glacial times, when sediment pro-duction in river headwaters is generally high,



Figure 8. View of one side of the terraced-valley topography generated during the tenfoldsediment-supply-variation experiment scaled by the oxygen isotope record. The view isupstream, extends from 122 to 128 km along the model profile, and is illuminated fromthe left. The valley floor (i.e., the river) is on the right at 0 m distance across the valley(bottom axis). In this experiment, the channel was forced to slip across the valley in onedirection only; hence, the missing valley side would be a vertical wall. All terraces arestrath terraces, with alluvium thickness of less than or approximately equal to 1 m. Incontrast to sinusoidal variation (Fig. 3), the irregular variation of sediment supply pro-duces terraces of variable width. The most extensive terraces (1, 2, 5, 6, and 7) are pro-duced by lateral planation during prolonged channel residence at each terrace elevation(see Fig. 10). The topographic profile extending from A to B is shown in Figure 10.

Figure 9. History of normalized inputs (gray line), normalized valley-floor width (i.e.,flood-plain width) at river level (solid line), and ratio of vertical to lateral erosion rates(dashed black line) during the tenfold sediment-supply-variation experiment. Inputs andvalley-floor widths are normalized as in Figure 5. Arrows indicate time of terrace for-mation, and the mechanism for terrace formation was as described in Figure 5. Theirregular variation of inputs produces a more complex terrace sequence (Fig. 8), andterraces are not formed from all significant input variations. During the general increasein sediment supply between full interglacial (i.e., lowest sediment supply) and full glacialconditions (i.e., peak sediment supply), minor fluctuations superimposed on the largertrend do not produce a terrace. For instance, in the growth of valley width between modeltime ca. 120 ka and ca. 20 ka, an increase and subsequent decrease in sediment supplyat model time ca. 75 ka (isotope stage 4) produces no terrace in spite of a relatively largeamplitude sediment-supply change. In western North America, preservation of isotopestage 4 terraces is rare, despite a readvance of alpine glaciers at this time.

and terraces are formed by renewed incisioninto these valley floors during the transition tointerglacial conditions.

Regardless of which input fluctuates, ter-race formation lags significantly behind thetiming of forcing. This lag arises because thefluvial system takes a finite period of time topass the sediment accumulated during periodsof low water discharge and high sediment in-put. Once the sediment cover thins at a site,bedrock becomes more accessible to erosion,allowing the incision rate there to increase.This model observation is consistent with theidea of a paraglacial recovery period (e.g.,Church and Ryder (1972), during which chan-nels adjust to the sediment loads and sizessupplied by glacial advances. If relevant to thefield, this result implies that ages obtainedfrom terraces may significantly lag the timingof maximum glaciation or cooling that drovethe formation of the terrace.

If our numerical results are relevant for realriver systems, the observation of long lateralplanation periods interrupted by brief and rap-id incision in the model suggests that inter-pretation of terrace age is more complex thanpreviously recognized. If terrace surfaces areconstructed during tens of thousands of years(Fig. 10), it is likely that the deposition ofalluvium and cutting of the strath surface arediachronous in the cross-valley direction on anotherwise continuous terrace surface. An ab-solute age obtained by dating material incor-porated in a terrace surface could reflect anytime between the initiation of strath cuttingand the timing of strath abandonment. Anynumerical age should therefore be considereda maximum estimate for the timing of terraceformation, unless geologic evidence suggestsotherwise. Multiple samples from the sameterrace surface would help set limits on theresidence time on the strath surface and wouldplace bounds on the extent of diachroneitywithin the terrace material.

In field settings, the heights of dated strathterraces above modern river level are used fre-quently to estimate mean incision rates (e.g.,Burbank et al., 1996; Chadwick et al., 1997;Repka et al., 1997). Our simulations suggestthat channels spend a large fraction of time inlateral planation, and mean incision rates ob-tained by using terrace ages are many timeslower than the maximum instantaneous inci-sion rates within the period of averaging. Inthe sediment-supply simulation, for instance,calculated mean incision rates are 1.5–7 timeslower than the maximum instantaneous inci-sion rates occurring during river incision (Fig.9). Because average incision rates may incor-porate periods of little to no vertical incision



Figure 10. The topographic profile along A–B of the terraced valley illustrated in Figure8 (thick line) and channel-residence time (see definition in text) as a function of elevationduring downcutting in this simulation (thin line). Terraces are related to periods of lateralplanation during long channel residence within a narrow elevation range. As in the con-stant-period experiments, channel-residence time at individual terrace levels reaches upto many tens of thousands of years, indicating that the river spent most of the simulationtime forming terraces. The widest terraces are produced by the longest residence times,and incision between terrace levels occurs over relatively short time intervals.

of unknown duration, comparison of mean in-cision rates obtained in two different riversduring the same time period—or within thesame river system over differing time peri-ods—neglects the actual duration of incision.Drawing conclusions about the controls onrock-incision rates by comparing downcuttingrates obtained from dated terraces should bedone cautiously in light of our findings. In ad-dition, variation in rock-incision rate is an in-evitable consequence of climatically inducedvariation in the sediment and water inputs tothe fluvial system. Changes in the mean inci-sion rates through time as deduced from a se-ries of terrace ages therefore do not necessar-ily imply long-term changes in uplift rates, butcan instead simply result from variations inthe sequencing and duration of input events.A stream may incise rapidly not only becauseit has high stream power and/or is flowingacross weak rock, but also because the chan-nel is able to transport the supplied sedimentto provide sufficient access to the bed.

Many dated terrace sequences in westernNorth America and elsewhere indicate accel-eration of incision rates toward the present(e.g., Chadwick et al., 1997; Patton et al.,1991; Reheis et al., 1991). Our simulationssuggest that this apparent acceleration maysimply reflect the particular time at whichthese systems are sampled. To illustrate, in ourexperiments where sediment supply moststrongly controls vertical erosion rates, as wesuspect is the real case, the model stream isincising actively at the end of the simulationperiod (e.g., Figs. 4 and 9). As a consequence,

the average channel-incision rate calculatedfrom the height and abandonment age of theyoungest terrace is much higher than incisionrates calculated between older terrace levels(compare incision rates for terraces, Fig. 11).The increase in the average incision rate arisessolely because the simulation ends when sed-iment input is low and when incision rates areat their highest, rather than from changes intectonic or climatic forcing. The apparent ac-celeration of incision rates in several westernNorth American river systems may occur forthis reason, as the highest measured rates areobtained from the youngest dated terrace.

Because strath terraces imply long periodsof lateral planation and nearly constant riverelevation, they are inferred to reflect a time ofbalance between sediment supply and dis-charge such that neither net aggradation nordegradation occurs (i.e., dQs/dx ø 0 and dz/dtø 0). This state is called a graded river (Bull,1990; Mackin, 1937). Although lateral plana-tion implies nearly constant elevation, oursimulations suggest that these graded periodsshould not be taken to indicate steady forcing.All of the experiments produce terraces, oftenby abrupt onset of downcutting, even thoughthe input is being changed continuously andeven gradually (e.g., Figs. 4 and 9). The valley-widening periods are characterized by slightlypositive divergences in sediment-transport ca-pacity (dQs/dx . 0), where x increases down-stream, reflecting a balance within a channelreach wherein all sediment delivered into areach and the sediment generated by bedrockerosion are transported downstream. This bal-

ance keeps the sediment thickness on a form-ing strath relatively constant. The transition todowncutting occurs as dQs/dx increases, be-coming more positive, allowing net sedimentremoval from the channel and enhanced ac-cess to the bedrock. Conversely, an increasein sediment supply, for instance, would pushdQs/dx toward negative values (i.e., aggrada-tion), switching the river from strath forma-tion to valley filling. Fill terraces can be gen-erated in such circumstances, which would bean interesting subject for a future modelingeffort. A threshold exists between valley filland strath-terrace formation that is dictated bythe divergence in sediment-transport capacity,rather than implying distinctly different inputsand river response to those inputs. As a con-sequence, two identical river systems withonly slight differences in the amplitude of flu-vial inputs may produce two different recordsof the same event: a strath terrace in one anda fill terrace in the other.

The results point to several future modelingexperiments. First, sediment should be deliv-ered at tributary junctions as well as at thehead of the trunk channel. Tributary inputsmay influence the timing and magnitude oflateral planation in the down-valley direction,perhaps producing diachroneity along terraceprofiles. Second, the quantity of sediment in-put and the ease with which the valley can bewidened control the ability of the river to wid-en and form strath terraces, as opposed to ag-grading to form fill terraces. Although wehave focused on production of strath sequenc-es here, as are seen in the Wind River systemfrom which we deduced the parameter setused, it would also be interesting to explorethe conditions leading to fill-terrace formation.Preliminary simulations suggest that less eas-ily eroded valley walls promote the formationof significant fill terraces. Finally, although wehave maintained a steady base-level fall here,a goal should be exploring how nonsteadybase-level fall, with superimposed variationsin river inputs, influences the resulting terracesequence.

CONCLUSIONS

Starting with the conceptual hypotheses ofGilbert (1877) and Mackin (1937), we havecreated a physically based model that suc-ceeds in producing distinct fluvial strath-terrace sequences in response to time-dependentinputs to the fluvial system. We have selecteda physically based rule set and have set limitson inputs from available field evidence. Ournumerical experiments indicate that strath ter-races can be created through climatic modu-



Figure 11. Graph of height of model terraces above the channel (as labeled on left y-axis)vs. terrace age produced by the tenfold sediment-supply simulation scaled by the oxygenisotope curve (sediment-supply-variation history shown by gray curve). The average in-cision rates between terrace levels calculated from terrace age and height are shown withbars (as labeled on the right y-axis). There is a general trend of increasing average incisionrates toward the present (i.e., end of the simulation). A similar trend has been noted forseveral terrace sequences in western North America (see text for references), often basedon terraces of ages similar to model terraces 1, 2, and 7 in this graph. Although suchacceleration may be ascribed to increasing uplift rates and/or increasing climate intensityin the field, our results suggest this may only be an apparent acceleration given the uniquetime period that we are observing. In our simulation, this apparent acceleration occurs inspite of a constant tectonic forcing (i.e., base-level lowering). The high rate associated withthe most recent terrace (level 1) is obtained because, since abandonment of this terracelevel, the channel has been largely in a period of downcutting. Incision rates calculatedbetween older terrace levels include long periods of little to no downcutting; hence, averagerates tend to be lower solely because of the time sampled. Between the time of formationof terrace 7 and terrace 5 in our simulation, the stream spends an unusually long timeforming terraces (i.e., little vertical incision) relative to the remainder of the simulation.Incorporating this period into the calculation of average incision between terrace level 7and any terrace created after terrace 5 (like terrace 2) tends to produce a relatively lowincision rate. Thus, our simulations explain why incision rates appear to increase, but donot require either increased uplift or enhanced climate severity.

lation of sediment load and water dischargedelivered to a fluvial system. This modulationcauses rates of lateral planation and river in-cision to vary through time, and this variationfosters creation of wide valley floors that areabandoned to form terraces during periods ofmore rapid incision. Terrace development andriver response are strongly dependent on theamplitude of input fluctuations with time, andterrace width, maximum incision rates, andvalley floor residence time increase as the in-put amplitude increases. Terrace width is con-trolled largely by the river’s residence time ata particular elevation, and the width of cli-matically generated terraces is indicative ofthe duration and/or magnitude of climaticchange.

Several aspects of model behavior, as yetundocumented in field settings, provide new

ideas for investigation and interpretation ofterrace sequences. The periods of lateral pla-nation required to form the valley floors thatbecome terraces may be very long. In our sim-ulations, these surfaces were occupied formany tens of thousands of (model) years anda significant fraction of the overall river his-tory. Deposition of gravel across and down aparticular terrace tread is likely diachronous,with a potentially large variation in the timingof deposition. Given this variability, assigninga single numeric age to a terrace surfaceshould be done with caution and with refer-ence to the time within the terrace cycle (e.g.,valley abandonment) to which the age corre-sponds. Although these periods of lateral pla-nation with low incision rates are representa-tive of the classic graded-river condition,achievement of grade in our simulations does

not require that inputs be steady. Continuouslateral planation occurs in our simulationseven though inputs are irregular and non-steady. Terraces should not be interpreted toreflect a period of steady forcing, but insteada period of delicate balance achieved even asthe external forcing on the fluvial system con-tinues to change. The graded condition in themodel fluvial systems reflects a delicate bal-ance between incision and aggradation; de-struction of this balance occurs in response tomodest changes in fluvial inputs.

The simulations also indicate that the tim-ing of terrace formation significantly lags theinput cycles driving terrace formation. Thisobservation is consistent with field documen-tation of a paraglacial period that follows ma-jor climate shifts during the transition fromglacial to interglacial conditions. As in thefield, our model fluvial systems require timeto respond and adjust to changing inputs, par-ticularly during the transition from periods oflateral planation to more rapid incision. Ter-race formation occurs well after the onset ofthe input changes forcing their creation. Datesfrom terraces in the field should be expectedto lag the onset of climate changes that affectfluvial inputs, such as the transition from gla-cial advance to retreat.

The incision-rate histories obtained fromterrace sequences in extant fluvial systemsmay be distorted by current river incision.Many fluvial systems are now actively down-cutting following an extended period of lateralplanation during the last glaciation. The youn-gest terrace in these fluvial systems is oftenrelated to the Last Glacial Maximum, and insome cases abandonment of the Last GlacialMaximum valley floor may not yet be com-plete (e.g., Chadwick et al., 1997). Because ofthis active downcutting, incision rates ob-tained from the age and height of a Last Gla-cial Maximum terrace in these systems are av-eraged over a period of continual incision.Incision rates from older terraces, on the otherhand, are averaged over both prolonged peri-ods of lateral planation (i.e., incision ratesnear 0) and incision periods. As a conse-quence, incision rates calculated from themost recent terrace level would be high evenif the long-term average incision rate is notchanging. This apparent acceleration of inci-sion rate has been observed in many fluvialsystems, but should not necessarily be inter-preted to indicate fluvial response to changesin external forcing, such as increasing rock-uplift rates.

ACKNOWLEDGMENTS

This research was supported in part by grantsfrom the National Science Foundation (EAR-



9417798 and EAR-9803837) and a grant from theGeological Society of America (Fahnestock Awardto Hancock). We thank Andy Fisher, Justin Reven-augh, Eric Small, Ellen Wohl, John Gosse, and DonEasterbrook for careful reviews of an earlier versionof the manuscript. The manuscript benefited greatlyfrom excellent reviews by Frank Pazzaglia, GregTucker, and Kelin Whipple. We thank as well OliverChadwick for sharing insights into the Wind Riversystem. This is University of California, Santa Cruz,Center for the Study of Imaging and Dynamics ofthe Earth contribution number 420.

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