Downloaded from gsabulletin.gsapubs.org on December 21, 2010 Geological Society of America Bulletin

Numerical modeling of fluvial strath-terrace formation in response to oscillating climate

Gregory S. Hancock and Robert S. Anderson

Geological Society of America Bulletin 2002;114;1131-1142 doi: 10.1130/0016-7606(2002)114<1131:NMOFST>2.0.CO;2

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Numerical modeling of ¯uvial strath-terrace formation in response to oscillating climate

Gregory S. Hancock* Department of Geology, College of William and Mary, Williamsburg, Virginia 23187, USA Robert S. Anderson Department 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 thousands of years. The transition stream-transport capacity, vertical erosion from valley widening to downcutting and rates decrease, making lateral planation rela- Many river systems in western North terrace creation occurs in response to sub- tively more important. Flood plains are the re- America retain a ¯uvial strath-terrace rec- tle input changes affecting local divergence sult of this lateral planation, and strath ter- ord of discontinuous downcutting into bed- of sediment-transport capacity. Formation races are vestiges of these ¯ood plains, rock through the Quaternary. Their impor- of terraces lags by several thousand years formed as river downcutting leads to ¯ood- tance lies in their use to interpret climatic the input changes that cause their plain abandonment (Gilbert, 1877). Mackin events in the headwaters and to determine formation. (1937) proposed that incipient terraces form long-term incision rates. Terrace formation Our results suggest that use of terrace during periods of river stabilityÐi.e., grad- has been ascribed to changes in sediment ages to set limits on the timing of a speci®c ed conditionsÐwhen lateral planation of val- supply and/or water discharge produced by event must be done with the knowledge that ley walls produces wide valley ¯oors. Graded late Quaternary climatic ¯uctuations. We the system can take thousands of years to conditions occur when a river can transport all use a one-dimensional channel-evolution respond to a perturbation. The incision rate of the imposed sediment load and is achieved model to explore whether temporal varia- calculated in the ®eld from the lowest ter- through a balance between sediment load and tions in sediment and water discharge can race in these systems will likely be higher size, discharge, and channel slope (Mackin, generate terrace sequences. The model in- than the rate calculated by using older ter- 1937). According to Mackin (1937), the width cludes sediment transport, vertical bedrock races, because the most recent ¯uvial re- of the valley ¯oor is related to the duration of erosion limited by alluvial cover, and lat- sponse in the ®eld is commonly downcut- the graded period. Both Gilbert (1877) and eral valley-wall erosion. We set limits on ting associated with declining sediment Mackin (1937) suggested that the beveling of our modeling by using data collected from input since the Last Glacial Maximum. the bedrock strath and the deposition of the the terraced Wind River basin. Two types This apparent increase in incision rates is alluvial mantle occurred simultaneously and of experiments were performed: constant- observed in many river systems and should that the channel geometry remained stable period sinusoidal input histories and variable- not necessarily be interpreted as a response while the channel migrated across the valley period inputs scaled by the marine ␦18O to an increase in rock-uplift rate. ¯oor. The beveled strath and thin alluvial cov- record. Our simulations indicate that er distinguish strath from ®ll terraces, which strath-terrace formation requires input Keywords: ¯uvial features, landscape evo- have no underlying bedrock platform and can variability that produces a changing ratio lution, modeling, river terraces, sediment be many tens of meters thick. of vertical to lateral erosion rates. Straths supply. These hypotheses suggest that strath for- are cut when the channel ¯oor is protect- mation requires alternation between ``graded'' ed from erosion by sediment and are INTRODUCTION periods of extensive lateral planation, and abandonedÐand terraces formedÐwhen ``nongraded'' periods of vertical incision. Riv- incision can resume following sediment- Fluvial strath-terrace sequences record river ers may obtain graded conditions when avail- cover thinning. High sediment supply pro- incision through time and the timing of envi- able stream power equals the critical power, motes wide valley ¯oors that are abandoned ronmental perturbations to ¯uvial systems. de®ned as the stream power necessary to as sediment supply decreases. In contrast, Fluvial strath terraces are abandoned river transport enough sediment to maintain grade wide valleys are promoted by low effective ¯ood plains with thin mantles of alluvium (Bull, 1979, 1990). Under these conditions in- water discharge and are abandoned as dis- overlying a beveled bedrock platform (Fig. 1). cision rates (i.e., rates of vertical erosion) are charge increases. Widening of the valley Conceptually, strath-terrace formation requires at a minimum, allowing extensive valley wid- ¯oors that become terraces occurs over changes in the ratio of vertical incision to lat- ening by lateral planation. Strath terraces are eral planation in a downcutting stream. Gilbert created when the available stream power be- *E-mail: [email protected]. (1877) postulated that as sediment load nears comes greater than this critical power, leading

GSA Bulletin; September 2002; v. 114; no. 9; p. 1131±1142; 11 ®gures.

For permission to copy, contact [email protected] ᭧ 2002 Geological Society of America 1131 Downloaded from gsabulletin.gsapubs.org on December 21, 2010 HANCOCK and ANDERSON

et al., 1991; Reheis et al., 1991; Repka et al., 1997). The cause for this acceleration is not clear, but could indicate regional intensi®ca- tion of climate, base-level, and/or tectonic change. Alternatively, Mills (2000) has pro- posed that apparent late Cenozoic acceleration of incision rates deduced from many terrace sequences in the southeastern United States is an artifact of averaging over longer periods as terrace age increases. In our modeling, we at- tempt to test this hypothesis. The timing, duration, and mechanisms of strath-terrace formation are dif®cult to infer solely from ®eld observations because terrace sequences represent incomplete records, are Figure 1. Photograph looking downstream along the Wind River below Dubois, Wyoming. dif®cult to date, and formed during ¯uvial The width of the view represents ϳ1 km at the center. Above the active ¯ood plain, three conditions that differ from the present. Here terrace levels (ages: WR-1, ca. 20 ka; WR-3, ca. 120 ka; and WR-7, ca. 600 ka) mapped we explore strath-terrace formation in re- by Chadwick et al. (1997) can be seen along the course of the river. sponse to climate oscillation by using a nu- merical stream-pro®le simulation. This ap- proach differs signi®cantly from previous modeling efforts focused on the generation of to incision and ¯ood-plain abandonment 1993; Leopold and Wolman, 1957; Schumm ¯uvial terraces (e.g., Boll et al., 1988; Veld- (Bull, 1979). In this conceptual model, switch- and Khan, 1972; Smith and Smith, 1984). kamp, 1992; Veldkamp and Vermeulen, 1988) ing between these two states can occur by al- On the basis of this evidence, valley wid- because we rely on physically based rules and tering the available stream power (e.g., chang- ening and strath formation are proposed to oc- a realistic set of stream inputs. First, we dis- ing discharge or slope), and/or the critical cur during periods of maximum glaciation cuss the model framework and rule set, the power (e.g., sediment load or size; Bull, (e.g., Baker, 1974; Bull, 1991; Howard, 1970; constraints on initial and boundary conditions, 1979). Available stream power may be in- Ritter, 1967; Sinnock, 1981), although glaciers and our modeling strategy. Second, we discuss creased, and hence, terraces may be created by within the headwaters of a river system are not the results and their implications for terrace incision, through basin-scale changes such as required (e.g., Pierce and Scott, 1982). Valley formation and interpretation. Speci®cally, we base-level fall, hydrologic changes induced by widening and strath formation are initiated by explore the following questions: (1) Why do climate change, and/or increased tectonic ac- increased sediment supply and hydrologic terraces and terrace sequences form? (2) How tivity (e.g., Merritts et al., 1994; Pazzaglia et changes associated with a larger and later- is terrace formation related in time to forcing al., 1998). melting snowpack, reduced evaporation and mechanisms? (3) How long does it take to The well-preserved strath-terrace sequences in®ltration capacity, extensive periglacial pro- form a strath terrace? (4) What past climatic found in many river systems of western North cesses, and reduced vegetative cover during episodes are most likely to leave a terrace re- America record discontinuous incision into cooler climate conditions (Pierce and Scott, cord? and (5) What implications do the an- bedrock throughout the late Quaternary (e.g., 1982). Incision and terrace formation by aban- swers have for interpreting ¯uvial terrace Howard, 1970; Patton et al., 1991; Reheis et donment of ¯ood plains presumably occur sequences? al., 1991; Ritter, 1967). Their widespread oc- during interglacial periods or during the initial currence implies response to a common re- transition from glacial to interglacial condi- MODELING APPROACH gional forcing, and ®eld data indicate climate tions (Sinnock, 1981). Church and Ryder oscillation between glacial and interglacial (1972) and Jackson et al. (1982) have pro- We use a one-dimensional ®nite-difference conditions, superimposed on long-term uplift posed that ¯uvial extraction of sediments cre- code simulating the evolution of the long val- and/or base-level lowering, may be responsi- ated during glacial climates may continue for ley pro®le of a river in the face of temporally ble. Many terrace deposits indicate transport an extended time, called the paraglacial peri- varying inputs of water and sediment. We in- of larger grain sizes by the terrace-forming od, after the transition toward interglacial con- corporate (1) sediment transport, (2) vertical rivers compared to their modern counterparts, ditions has begun. Renewed incision and incision of bedrock at the base of the active suggesting greater peak discharges and/or strath-terrace formation may therefore occur channel, and (3) lateral planation of the valley more effective headwater erosion during valley- well after this transition. walls. Our incorporation of lateral planation widening periods than at present (Baker, 1974; The age and height of paired strath-terrace differentiates our approach from previous Pierce and Scott, 1982). In addition, some ter- sequences provide the primary data for esti- stream-evolution models (see Howard et al., race deposits suggest that terrace-forming mating the incision history and incision rates 1994) and allows us to investigate the condi- rivers were braided, unlike their modern me- in ¯uvial systems (e.g., Pazzaglia et al., 1998). tions conducive to generation of strath terrac- andering counterparts (e.g., Baker, 1974; Jack- In western North America, incision histories es. Aside from lateral planation, our goals and son et al., 1982; Ritter, 1967; Sinnock, 1981). deduced from precisely dated strath-terrace se- physical rules are similar to those of Tucker Braided-stream conditions suggest greater quences are rare, but many of these sequences and Slingerland (1997), but our focus is solely sediment supply (particularly bedload) and indicate accelerating incision rates during the on the channel response to climatic larger, more ¯ashy channel ¯ows (e.g., Bridge, Quaternary (e.g., Hancock et al., 1999; Patton perturbations.

1132 Geological Society of America Bulletin, September 2002 Downloaded from gsabulletin.gsapubs.org on December 21, 2010 NUMERICAL MODELING OF FLUVIAL STRATH-TERRACE FORMATION

Climatic forcing may be manifested in ¯u- ␶ϭ␳gHS, (2) bAc n 3/5 H SϪ3/10. (7) vial systems by changes in discharge, sedi- ϭ ΂΃wc ment load, and sediment size. Our model rules where ␳ is water density, g is gravitational ac- must therefore explicitly incorporate grain size celeration, H is water depth, and S is energy Finally, by combining equations 2 and 7, and water discharge. We assume that transport slope, here assumed equal to channel-bed bed shear stress may be predicted with the de- of sediment is dictated by the local bed shear slope. Equation 2 is valid for steady, uniform sired explicit dependence on the discharge: stress, and erosion of bedrock, either of the ¯ow. Both the channel slope and the water wall or of the bed, is related to the available depth must be known in order to calculate the bAc 3/5 gnS3/5 7/10. (8) energy per unit area of bed, or unit stream shear stress and, hence, the sediment dis- ␶ ϭ ␳ ΂΃wc power (e.g., Whipple and Tucker, 1999). The charge. The channel slope, S, at each cell edge one-dimensional model tracks sediment thick- is calculated by differencing the bed elevation, To predict the volumetric sediment ¯ux, Qs, ness, bedrock erosion, and valley width at z, from the adjacent channel cells and dividing we use a variation of the Bagnold bedload evenly spaced channel cells along a pro®le, by the cell spacing, dx. sediment-transport equation (see Slingerland analogous to channel reaches. We use a stag- We calculate the ¯ow depth, H, in a cell et al., 1994), gered grid to enhance model stability. The from the water discharge, Q. The effective dis- thickness of the sediment, water discharge, charge, Q, de®ned as the ¯ow having the right Bw channel elevation, and width of the channel Q ϭ (␶ Ϫ ␶ )(␶1/2Ϫ ␶ 1/2), (9) combination of frequency and magnitude to sc(␳ Ϫ ␳)␳1/2g c and the valley ¯oor are tracked at the center control channel form, is related to the drainage s of each channel cell. The physical quantities area, A, through a power-law relationship where B is a dimensionless constant (ϳ10), ␶ that dictate ¯uxes between the cells, such as c (e.g., Dunne and Leopold, 1978), is the critical shear stress required to transport slope and shear stress, and the ¯uxes them- a given grain size, and ␳ is the density of the selves, are calculated at the edges of the cells. s Q ϭ bAc, (3) sediment grains. The ␶c for sediment entrain- ment is estimated by using the Shields param- Sediment Transport where b and c are empirical constants. Tem- eter, and this is the means by which depen- poral variation in discharge is prescribed by dence upon grain size is brought into the Sediment mass within each cell is conserved, varying b and/or c. Channel width, wc, is cal- problem. The sediment discharge, Qs, is de- as expressed by the continuity equation: culated by using the hydraulic geometry re- termined by using the shear stress (equation lationship 8) calculated at the edge of each channel cell. dh 1 dQ The divergence of the sediment-transport ca- ϭϪ s, (1) e pacity, dQ /dx, may then be calculated at each dt w dx wc ϭ dQ , (4) s channel cell center. Although equation 9 is formulated solely for bedload transport in al- where h is sediment thickness, t is time, Q is where d and e are again empirical constants. s luvial or transport-limited rivers, bedload is volumetric sediment-transport rate, x is the In our models, appropriate values for b, c, d, likely the most critical component affecting downstream distance, and w is either valley and e in equations 3 and 4 are taken from the stream incision and terrace generation, as the width, w, or channel width, w . If sediment Wind River, Wyoming, and are typical of c bedload must be in transit before rock erosion ¯ux decreases in the downstream direction those obtained in numerous river systems (Le- can take place. The simulated streams are (i.e., dQ /dx is negative, which will cause sed- opold and Maddock, 1953). s largely covered by alluvium, with only patchy iment deposition), deposition is speci®ed to Flow depth in a channel is given by bedrock exposure. occur evenly over the active valley width, w. Here the active valley width is de®ned as the Q Bedrock Erosion region at nearly the same elevation as the H ϭ , (5) vwm channel, and across this width the channel mi- grates. Hence, channel migration deposits sed- Rules for predicting bedrock erosion at the where v is mean ¯ow velocity and w is mean iment across the active valley ¯oor. When sed- m reach scale in landscape-evolution models are ¯ow width. Assuming a roughly rectangular iment ¯ux increases downstream (i.e., if dQ / based on either shear stress or the stream pow- s channel, w ഠ w . The mean velocity may be dx is positive, which will cause sediment m c er per unit bed area, ␻, estimated by using the Manning equation erosion), sediment is removed only from the ␳gQs active channel width, wc. Taken together, these ␻ ϭ , (10) kH2/3 S 1/2 w rules allow sediment to be sequestered by de- v ϭ , (6) c position on wide valley ¯oors and allow these n mantled valley ¯oors to be preserved during where dz/dt is the vertical erosion rate (e.g., subsequent channel incision. where k is a constant and n is the Manning Whipple and Tucker, 1999). These rules are We must therefore formulate a model al- roughness coef®cient. Furthermore, we have typically of the form gorithm for sediment discharge, and we fol- assumed a wide, shallow channel such that wc low the dominant-discharge (i.e., geomorphi- » H, allowing substitution of ¯ow depth H for dz ϭ K(X Ϫ X )r , (11) cally effective) strategy of Tucker and the hydraulic radius. dt 0 Slingerland (1997). Most sediment-transport By combining equations 3, 4, 5, and 6, we equations suggest that the transport capacity, obtain the relationship between ¯ow depth, H, where X is either shear stress or stream power,

Qs, depends on the local shear stress, ␶, and measurable quantities: X0 is a threshold shear stress or stream power,

Geological Society of America Bulletin, September 2002 1133 Downloaded from gsabulletin.gsapubs.org on December 21, 2010 HANCOCK and ANDERSON r is an exponent, and K is a parameter relating lying rock to erosion. We assume that the dw ϭ K ␻ (16) ``excess'' shear stress or stream power to ero- depth of sediment scour is related to the sed- w ΂΃dt p sion rate. This formulation neglects explicit iment-transport capacity,Qs, and the channel consideration of the role of the sediment-sup- width, wc, and may not be uniform. We use applied to the channel margins, where (dw/dt)p ply rate in controlling erosion (Sklar and Die- the potential sediment-transport rate, Qs, cal- is the potential valley-widening (not channel- trich, 2001). In this formulation, the value of culated by using equation 9, to estimate a sed- widening) rate and Kw is a susceptibility con- K is related to rock properties that dictate the iment-scour depth scale, L, stant that relates stream power to wall erosion. amount of rock that may be eroded per unit We acknowledge that this critical component shear stress or stream power. Values for K are sQs of our model cannot be justi®ed rigorously. not well de®ned, but have been determined by L ϭ , (13) wc Investigation of the valley-widening processes erosion measurements in badland channels in rock-¯oored channels and of the controls (Howard and Kerby, 1983) and by comparing where s is a constant with dimensions of time on widening rates is sorely needed. However, past and present channel pro®les in the Sierra per length, and wc is channel width. lateral planation that far exceeds vertical in- Nevada, California (Stock and Montgomery, It is reasonable that the probability of cision over time is a key ®eld observation that 1999). The method for choosing K for a par- scouring to bedrock increases as the scour must be reproduced in this model. For ex- ticular setting is not known, however. Our ap- depth scale, L, increases, and decreases as the ample, in the Wind River basin (Fig. 1), ter- proach is to select a K that produces geolog- sediment thickness, h, increases. Following race widths require average lateral planation ically reasonable erosion rates, to assume r ϭ Howard (1998), we hypothesize that the prob- rates to be several orders of magnitude great- 1 and X0 ϭ 0, and to use stream power per ability, F, for rock exposure and, hence, of er than vertical rates. As discussed subse- unit bed area. Results obtained by using the rock incision, decreases exponentially as sed- quently, the model succeeds in reproducing stream power and shear stress rules do not dif- iment thickness increases: this observation. fer substantially, as shown theoretically by To widen the valley, a channel must be in Whipple and Tucker (1999). We use separate F ϭ eϪ(h/L). (14) contact with its valley wall. Because our mod- rules that predict vertical rock erosion (i.e., in- el is one-dimensional, we do not track the cision) rates and horizontal rock erosion (i.e., Appropriately, the probability of bedrock channel position explicitly. Instead, we as- lateral planation) rates. erosion is therefore 1 if h ϭ 0 and approaches sume that the channel position within a valley zero as h » L. To calculate the actual vertical is random if sampled over a long time period Rock-Incision Rates in the absence of preferred positioning im- Potential vertical rock-erosion rates, (dz/ bedrock-incision rate in a cell, dz/dt, we com- bine equation 12 and equation 14 to yield posed by tilting or other external forcing. In dt)p, is expressed by this case, the channel width, wc, and the valley width, w, determine the probability of valley- dz dz ϭ FKv␻. (15) wall contact, W, ϭ Kv␻, (12) dt ΂΃dt p w where K is a ``hardness'' parameter for ver- Although speculative, this formulation ac- W ϭ c . (17) v w tical rock erosion. This equation predicts only commodates several ®eld observations, in- the maximum potential erosion rate because cluding (1) the need to scour to the base of The probability increases as w increases and rock incision can take place only when sedi- the sediment layer in order to erode rock, and c ment does not protect that rock from erosion. (2) the likelihood that scour depths will vary as w decreases. The rate of valley-wall erosion Field observations suggest that bedrock ero- within the channel. This formulation also rec- is then determined by combining equations 16 sion can occur while there is a ®nite mean ognizes that rock-erosion rates should increase and 17: sediment thickness within a reach, as sediment as Qs increases and decrease as the sediment cover is often variable (e.g., Leopold and thickness increases. dw ϭ WKw␻. (18) Maddock (1953). Sediment may be transport- dt ed and stored within a reach even while bed- Lateral Planation Rates rock erosion is occurring (Shepherd and Although the processes of erosion of river We note that valleys should widen rapidly at Schumm, 1974; Howard and Kerby, 1983) banks in self-formed, alluvial river channels ®rst and more slowly thereafter as the proba- and may be required for this erosion to take have been the subject of extensive investiga- bility of channel contact with the valley wall place as it provides the abrasive tools (e.g., tion (e.g., Andrews, 1982), the widening of declines. Hancock et al., 1998; Sklar and Dietrich, bedrock valley walls by rivers has received This lateral erosion is divided between the 2001; Whipple et al., 2000). We do not incor- much less attention. We know of only one two valley walls. In order to allow asymmetry porate the possibility that there may be an op- study of lateral erosion, and this study sug- in wall erosion, we specify a symmetry con- timal sediment-supply rate, above and below gests that lateral erosion rates, dw/dt, are re- stant, ␣, where 0 Յ ␣ Յ 1. Rates of valley- which erosion rates decline (Sklar and Die- lated to ␻1/2 (Suzuki, 1982). However, the wall erosion are ␣dw/dt and (1Ð␣)dw/dt, on trich, 2001). probability of valley-wall contact was not con- opposite valley walls. If ␣ϭ0.5, there is no To accommodate these observations, we sidered, as we discuss subsequently. preferred slip direction. As illustrated in Fig- add to equation 12 a sediment-scour rule. This To proceed, we make the assumption that ure 1, preferred slip across the valley is an is designed to acknowledge that a ®nite but the potential rate of valley-wall erosion is important factor in preserving terraces in thin sediment cover can be scoured away in analogous to vertical rock erosion and is re- some ¯uvial systems. In simulations presented high-transport conditions, exposing the under- lated to stream power per unit bed area, here, we choose ␣ϭ1, equivalent to forcing

1134 Geological Society of America Bulletin, September 2002 Downloaded from gsabulletin.gsapubs.org on December 21, 2010 NUMERICAL MODELING OF FLUVIAL STRATH-TERRACE FORMATION channel migration in a constant direction and ship between ¯ow frequency and magnitude River system, mean grain sizes transported by maximizing terrace preservation. that produces the effective discharge for rock the modern river are similar to those found erosion in bedrock-¯oored rivers is not within terrace deposits, whereas in other river MODELING STRATEGY known. We assume that the effective ¯ows systems, mean and maximum grain sizes are equal the 2-yr-recurrence-interval peak dis- several-fold larger in preserved terrace deposits Our goal is to explore how a river responds charge. Values for effective discharge were (e.g., Baker, 1974; Pierce and Scott, 1982). In to variations in sediment and water ¯uxes determined from modern gauging records on the experiments we report here, the grain size (here referred to as inputs) driven by climate the Wind River. This discharge increases with is kept constant. Experiments incorporating oscillation. We seek those conditions that lead drainage area, suggesting values of b ϭ 0.29 variations in the grain size used produce results to terrace formation. We discuss two experi- and c ϭ 0.76 in equation 3. Channel widths comparable to sediment-supply variation. ments in which these inputs were varied: (1) at various ¯ows have been measured at many constant period and amplitude variation of in- sites along the Wind River (Leopold and Wol- Numerical Experiments puts (the ``constant-period experiments''), and man, 1957; Smalley et al., 1994), yielding val- (2) irregular period and amplitude variations ues of d ϭ 3.0 and e ϭ 0.55 in equation 4. For the constant-period experiments, a ®xed scaled by the marine oxygen isotope record Because effective discharge, Q, likely sinusoidal input variation with a period of 100 (the ``variable-period experiments''). Al- varies in response to changing climate condi- k.y. is selected to mimic the dominant period though we are interested in the general phe- tions, Q is varied in our model. Paleohydrol- of climate cycling through glacial to intergla- nomenon of strath-terrace formation, we have ogic studies in western North America suggest cial conditions since ca. 800 ka. Maximum selected the Wind River, Wyoming, to provide that peak discharges in some terrace-forming sediment input, sediment size, and water dis- limits for the model initial and boundary con- streams were much greater than at present charge are assumed to coincide with maxi- ditions and the magnitudes of discharge, sed- (Baker, 1974; Pierce and Scott, 1982). In the mum glaciation. We describe ®rst a control iment load, and sediment size. This choice simulations, the effective discharge is in- experiment with steady inputs, followed by does not limit the signi®cance of the model to creased by up to two times the modern effec- three experiments: variable sediment load this river system, but ensures that we select tive discharge, which is a reasonable estimate (variation by a factor of 10); variable dis- geologically reasonable values. for sustained, frequent ¯ow events. In exper- charge (variation by a factor of 2); and com- iments in which water discharge is varied, the bined discharge and sediment-load variation. Initial and Boundary Conditions value of b, representing the effective basin All experiments are run for 800 0000 model runoff in equation 3, is increased by up to two years. We discuss only the last 400 000 yr, by All experiments start with a straight longi- times the modern value (i.e., 0.29 Ͻ b Ͻ which time the channel response to the im- tudinal channel pro®le with a slope S ϭ 0.01 0.58). The duration of effective discharge is posed initial conditions is complete. and a total length of 150 km; the spacing of 10 days per model year, a choice based upon In the variable-period experiments, the ben- the model cells is 1 km. The slope is equiv- the observed duration of the peak discharge of thic marine oxygen isotope record of Imbrie alent to maximum slopes measured at the ki- the Wind River. Although not considered here, et al. (1984) is used to scale the inputs. Be- lometer scale on the oldest terrace pro®les in changes in the duration of the effective part cause this record is a proxy for the extent of the Wind River system. Boundary conditions of a river hydrograph could also be important continental glaciation, we assume that (1) the must be speci®ed at the upper and lower ends in a ¯uvial system. magnitude of climate change (e.g., glacier ex- of the channel pro®le. The downstream tent) in our modeled river system parallels the boundary is lowered at 0.15 mm per model Sediment Load continental ice volume, and (2) the relation- year, the average incision rate obtained from ship between climate change and sediment dated terraces at the lower end of the terraced The sediment load and grain size also supply and water discharge is linear. Although reach of the Wind River near Riverton, Wy- change in response to climate oscillations and these assumptions are certainly not entirely oming (see Fig. 2 in Chadwick et al., 1997). dictate the ability of the river to access and valid, we are interested primarily in exploring All sediment reaching the lowest channel cell erode bedrock. The initial grain size, D, is tak- the river response to complex input variation. is allowed to escape downstream, preventing en to be equal to the mean grain size measured We consider two variable-period experiments: aggradation at the lower boundary. Sediment in the modern Wind River (D50 ഠ 0.5 cm, (1) variation of sediment supply by a factor of is delivered only to the cell farthest upstream. Smalley et al., 1994). The grain size is as- 10 and (2) variation of water discharge by a

The bedrock susceptibilities to erosion, Kv and sumed to be uniform along the model channel; factor of 2. In all cases, the peak in the input ±11 2 Kw, are set to 2 ϫ 10 m/(W´s/m )[ϭ transport of only this grain size is considered. is set to occur when the oxygen isotope curve (m´W´s)±1] in all cells and in all simulations, The minimum sediment-load input is ϳ5000 indicates glacial conditions. In order to re- implying uniform rock resistance. This value m3/yr, equal to the measured average annual move any memory of the initial conditions, yields geologically reasonable erosion rates bedload transported by the modern Wind Riv- the ca. 800 ka record of Imbrie et al. (1984) during the simulations. er near its headwaters at Dubois (Leopold et is repeated once, producing experiments with al., 1960). a total duration of 1600 ϫ 103 model years. Water Discharge and Channel Geometry Hallet et al. (1996) have shown that exten- sive glacial cover can increase sediment yield RESULTS The geomorphically effectiveÐi.e., ``dom- by up to an order of magnitude over nongla- inant'' (Tucker and Slingerland, 1997)Ðdis- ciated or slightly glaciated basins. On the ba- Constant-Period Experiments charge is the relevant quantity for prediction sis of this observation, the sediment-load in- of sediment transport and channel erosion put to the uppermost cell is increased by up In the control experiment, the channel pro- (e.g., Wolman and Miller, 1960). The relation- to 10 times the minimum value. In the Wind ®le evolves toward a slightly concave-up

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partially controlled by sediment supply. In contrast, 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 planation rates are controlled almost entirely by the available stream power and are far less sen- sitive to variations in sediment supply than are vertical incision rates (Fig. 5). The valley width is controlled by the magnitude of this ratio and by the duration of lateral planation. Hence, wide valley ¯oors are created during long periods when the ratio is low, when dw/ Figure 2. Successive channel pro®les for every 100 k.y. model years in the control exper- dt k dz/dt. Because this ratio varies, valley ka). As in all of the 800 ؍ to simulation end (t (0 ؍ iment, from initial conditions (t width is also time dependent (Fig. 5). As en- constant-period experiments, the channel pro®le achieves a nearly uniform incision rate visioned by Gilbert (1877) and Mackin over the last several 100 k.y. The downstream end of the modeled pro®le is at 0 km. (1937), variation in the relative importance of lateral planation rates and vertical incision shape. The slope in the lower half of the mod- the controls on terrace formation, timing, and rates, here expressed as the ratio dz/dt:dw/dt, el space declines monotonically through time, appearance, by describing the response of one is therefore responsible for producing terrace whereas that near the top of the pro®le steep- representative channel cell 125 km above the sequences. Valley ¯oors are widened when the ens slightly (Fig. 2). Both vertical and lateral downstream end of the model pro®le. ratio dz/dt:dw/dt is low for extended periods, erosion rates at any point within the model Vertical bedrock-incision rates, dz/dt, and and terraces form when these valley ¯oors are stream decline monotonically during the sim- lateral planation rates, dw/dt, vary through abandoned during subsequent incision when ulation. By the end of the simulation, the pro- model time in all simulations in which inputs the ratio dz/dt:dw/dt is high (Fig. 5, arrows ®le has achieved a nearly uniform erosion are varied (Fig. 4). The most rapid incision show times of valley abandonment and terrace rate. No terraces are produced in this simula- occurs when stream power and/or the likeli- formation). tion. All other constant-period simulations hood of bedrock access is high. As a conse- Lateral planation and therefore terrace cut- produced terrace sequences, as illustrated for quence, the vertical rock-incision rate is in- ting occurs when the sediment input forces dz/ the sediment-load experiment in Figure 3. The versely proportional to variations in sediment dt:dw/dt to be low. In the sediment-load ex- six terrace levels in Figure 3 correspond to the supply (Fig. 4A), because bedrock access and periments, vertical incision rates are lowest ®nal six input cycles completed. We discuss the energy expended to transport sediment are when the input of sediment is high, allowing extensive lateral planation (Fig. 5A). This re- lationship corresponds to times when the sed- iment cover is suf®cient to reduce the likeli- hood of scour through this cover, preventing vertical rock erosion. Declining sediment load leads to thinning of the sediment cover, allow- ing more rapid incision and abandonment of wide valley ¯oors to form strath terraces. On the other hand, valleys widen when effective water discharge is low and are abandoned as increasing discharge results in more rapid ver- tical incision (Fig. 5B). Although the lateral planation periods necessary to form terraces may be related in time to maximum sediment load or minimum water discharge, the actual Figure 3. View of one side of the terraced-valley topography generated during the tenfold timing of terrace formation (arrows in Fig. 5) sediment-supply-variation experiment. The view extends upstream from 122 to 128 km is not synchronous with these periods. Instead, along the model pro®le. The view is looking upstream and illuminated from the left. The in all simulations, the timing of actual terrace valley ¯oor (i.e., the river) is on the right at 0 m distance across the valley (bottom axis). formation (valley-¯oor abandonment) typical- In this experiment, the channel was forced to slip across the valley in one direction only; ly lags these variations in the forcing by up to hence, the missing valley side would be a vertical wall. Each numbered terrace level was several tens of thousands of years. produced during one sediment-supply input cycle. All are strath terraces, with alluvium Because sediment supply and water dis- thickness of less than or approximately equal to 1 m. The topographic pro®le extending charge modulate vertical incision rates differ- from A to B is shown in Figure 7. ently, these inputs when combined compete to

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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 modulating the timing and magnitude of incision rates, as dz/dt is inversely proportional to the sediment supply. We illustrate with one simulation where sediment supply varies by tenfold while effective discharge varies twofold. Extensive valley widening occurs when sediment load is highest, and terraces are formed as sediment load decreases (Fig. 6). Valley widening is promoted by high effective discharge, when the lateral planation rate reaches a maximum, while vertical incision rates are low because sediment load is high and prevents access to Figure 4. Vertical (solid line) and lateral (dashed line) erosion-rate histories at km 125 the bedrock (Fig. 6). For this particular choice during constant-period variation of ¯uvial inputs (gray line). Simulations are (A) tenfold of geologically defensible input amplitudes, sediment-supply variation, and (B) twofold dominant-discharge variation. Input variation the variation of sediment load dictates the tim- produces variable erosion rates and more signi®cantly modulated vertical erosion rates ing of valley widening. than horizontal erosion rates. High vertical erosion rates are associated with periods most The ®nal width of a terrace is dictated by conducive to enhancing rock accessibility (i.e., low sediment supply) and/or to producing the product of the lateral planation rate with a high stream power (i.e., high dominant discharge). The cycle in the ratio of vertical to channel-residence time at a particular eleva- lateral erosion rates is responsible for creating terraces. tion. We de®ne the channel-residence time to be the time spent by the channel between suc- cessive elevation contours, here taken to be 1 m. The residence time in lateral planation pe- riods over one simulation (that with 10ϫ sed- iment-supply variation) varies from ϳ15 k.y. to ϳ60 k.y., i.e., ϳ15% to ϳ60% of each 100 k.y. cycle (Fig. 7). We note that despite this variation, the terraces produced during these periods are roughly the same width. As down- cutting progresses, an increasing duration of lateral planation (low dz/dt:dw/dt) is offset by a decrease in the instantaneous lateral plana- tion rates caused by the decline in channel slope (Fig. 7). The likelihood of wall contact (equation 17) decreases as terraces widen and also 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 inputs Figure 5. History of normalized sediment inputs (gray line), normalized valley-¯oor width and the timing and mechanisms of terrace for- (i.e., ¯ood-plain width) at river level (dashed line), and the ratio of vertical to lateral mation is similar to that relationship for the erosion rates (solid black line) during two simulations: (A) tenfold sediment-supply vari- constant-period simulations (Fig. 9). However, ation and (B) twofold dominant-discharge variation. Inputs and valley-¯oor widths are the irregular input variation produces an irreg- normalized by dividing the value at each time by the maximum value during each simu- ular channel-incision history in which the lation. The valley ¯oor (i.e., the ``¯ood plain'') widens most signi®cantly when vertical stream is unable to incise for much of each erosion rates are low, producing a ratio of vertical to lateral erosion that is small or zero. simulation (Fig. 9). Periods of lateral plana- Flood plains are abandoned to form terraces when vertical erosion rates and the ratio of tion (low dz/dt:dw/dt) account for ϳ50%±90% vertical to lateral erosion rates increase, leading to renewed downcutting and narrowing of the total simulation time, implying that the of the active valley. Terraces are formed at the transition from valley-¯oor widening to typical mode of operation for the model narrowing (arrows). Terrace formation does not occur at the times of either maximum stream system involves extended periods of inputs or minimum inputs, but instead signi®cantly lags the timing of maximum (e.g., lateral planation and long residence time on sediment-supply maxima) or the minimum (e.g., water-discharge minimum) inputs. In the the ¯ood plain interrupted by short periods of simulations, all terraces generated are strath terraces. rapid incision (Fig. 10).

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in the sediment-variation experiments, the sur- viving terraces have model ages of ca. 20 and ca. 420 ka. In the absence of a preferred mi- gration direction, terrace preservation requires that successively younger terrace levels be narrower, re¯ecting a shorter duration of lat- eral planation and/or slower lateral planation rates. In this sense, the preservation of terraces within sequences is analogous to glacial mo- raine survival (Gibbons et al., 1984); if a river Figure 6. Ratio of vertical to lateral erosion rates (solid black line) and normalized active cannot migrate primarily in a single direction, valley width (dashed black line) at km 125 during constant-period variation of both sed- the terrace sequence will include only those iment supply (dashed gray line) and effective discharge (gray line). As shown in Figure terraces that are able to survive later lateral 5, the timing of maximum vertical incision produced by these inputs is not in phase. planation events. Situations that promote uni- Therefore, these inputs compete to set the timing of maximum vertical incision and, hence, directional channel migration include repeated terrace formation. The arrows mark the timing of valley abandonment and terrace for- river diversion (e.g., Chadwick et al., 1997) mation. Although there is a continuum of possible combinations of these two inputs, with and tectonic tilting (e.g., Reheis, 1985). the ®eld-de®ned amplitudes chosen for this simulation, the sediment-supply ¯uctuations Although base-level lowering in these sim- dominate. ulations is constant, the average vertical channel- incision rate between successive terrace levels varies widely (Fig. 11). In the sediment- supply-variation experiment, rates of channel incision generally increase toward the end of the simulation; most dramatically, the average incision rate calculated from the age and ele- vation of the last terrace created is several times higher than any other calculated incision rate (though this increased rate seems to be more apparent than real; see discussion and Fig. 11). We reiterate that these results are from 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 Figure 7. The topographic pro®le along A±B of the terraced valley illustrated in Figure 3 (thick line) and channel-residence time (see de®nition in text) as a function of elevation In our simulations, the conditions associat- during downcutting in this simulation (thin line). Terraces are related to periods of lateral ed with terrace formation support many of the planation during long channel residence within a narrow elevation range. Residence time existing conceptual models for terrace devel- at individual terrace levels reaches up to many tens of thousands of years, indicating that opment. A prerequisite for terrace formation the river spent most of the simulation forming terraces. is variability in the ratio of vertical incision rates to lateral planation rates, as suggested by Gilbert (1877) and Mackin (1937), among Unlike the constant-period experiments, not end of isotope stage 2 (Fig. 9). Within the others. Wide valleys form during periods of all signi®cant input ¯uctuations form terraces. same simulation, three ¯uctuations between lateral planation when bedrock exposure is The terrace sequences preserved include four ca. 200 and 250 ka that are comparable to iso- limited and/or available stream power is low to six well-de®ned terrace levels as well as tope stage 4 do leave three small but distinct (i.e., the stream power nearly equals the crit- several less distinct levels (Figs. 8 and 10). terraces, one of which forms terrace level 3 in ical power), resulting in low vertical channel- The widest terraces form in response to the Figure 10. incision rates. These conditions are promoted longest periods of high sediment load or low The number of terraces preserved in a sim- by relatively high sediment load and/or low water discharge (Fig. 9). However, neither ter- ulation depends upon the extent to which the effective discharge. Vertical incision promotes race width nor preservation perfectly records channel migrates in a preferred direction or abandonment of these valleys, producing ter- the duration and occurrence of climate events. slips sideways in its valley. In the simulations races, when these factors reverse. As suggest- A distinct terrace is not formed by the short- we have presented, we forced a constant mi- ed in the introduction, sediment load and grain lived (ϳ20 k.y.) ¯uctuation analogous to the gration direction, and seven to eight terrace size are thought to have increased in many marine isotope stage 4 event (ca. 75 ka, Fig. levels are preserved. In the absence of this im- western North American streams under colder 9). Instead, evidence for this ¯uctuation is lost posed unidirectional migration, many terraces climates. Our model results therefore support within the last terrace formed, created at the are cannibalized during later lateral planation the general hypothesis that valleys are wid- end of a lateral planation period lasting ϳ80 periods. With no migration, at most one or ened during glacial times, when sediment pro- k.y. and spanning from isotope stage 5 to the two terrace levels are preserved; for example, duction in river headwaters is generally high,

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and terraces are formed by renewed incision into these valley ¯oors during the transition to interglacial conditions. Regardless of which input ¯uctuates, ter- race formation lags signi®cantly behind the timing of forcing. This lag arises because the ¯uvial system takes a ®nite period of time to pass the sediment accumulated during periods of 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 the idea of a paraglacial recovery period (e.g., Figure 8. View of one side of the terraced-valley topography generated during the tenfold Church and Ryder (1972), during which chan- sediment-supply-variation experiment scaled by the oxygen isotope record. The view is nels adjust to the sediment loads and sizes upstream, extends from 122 to 128 km along the model pro®le, and is illuminated from supplied by glacial advances. If relevant to the the left. The valley ¯oor (i.e., the river) is on the right at 0 m distance across the valley ®eld, this result implies that ages obtained (bottom axis). In this experiment, the channel was forced to slip across the valley in one from terraces may signi®cantly lag the timing direction only; hence, the missing valley side would be a vertical wall. All terraces are of maximum glaciation or cooling that drove strath terraces, with alluvium thickness of less than or approximately equal to 1 m. In the formation of the terrace. contrast to sinusoidal variation (Fig. 3), the irregular variation of sediment supply pro- If our numerical results are relevant for real duces terraces of variable width. The most extensive terraces (1, 2, 5, 6, and 7) are pro- river systems, the observation of long lateral duced by lateral planation during prolonged channel residence at each terrace elevation planation periods interrupted by brief and rap- (see Fig. 10). The topographic pro®le extending from A to B is shown in Figure 10. id incision in the model suggests that inter- pretation of terrace age is more complex than previously recognized. If terrace surfaces are constructed during tens of thousands of years (Fig. 10), it is likely that the deposition of alluvium and cutting of the strath surface are diachronous in the cross-valley direction on an otherwise continuous terrace surface. An ab- solute age obtained by dating material incor- porated in a terrace surface could re¯ect any time between the initiation of strath cutting and the timing of strath abandonment. Any numerical age should therefore be considered a maximum estimate for the timing of terrace formation, unless geologic evidence suggests otherwise. Multiple samples from the same terrace surface would help set limits on the residence time on the strath surface and would place bounds on the extent of diachroneity within the terrace material. In ®eld settings, the heights of dated strath terraces above modern river level are used fre- Figure 9. History of normalized inputs (gray line), normalized valley-¯oor width (i.e., quently to estimate mean incision rates (e.g., ¯ood-plain width) at river level (solid line), and ratio of vertical to lateral erosion rates Burbank et al., 1996; Chadwick et al., 1997; (dashed black line) during the tenfold sediment-supply-variation experiment. Inputs and Repka et al., 1997). Our simulations suggest valley-¯oor widths are normalized as in Figure 5. Arrows indicate time of terrace for- that channels spend a large fraction of time in mation, and the mechanism for terrace formation was as described in Figure 5. The lateral planation, and mean incision rates ob- irregular variation of inputs produces a more complex terrace sequence (Fig. 8), and tained by using terrace ages are many times terraces are not formed from all signi®cant input variations. During the general increase lower than the maximum instantaneous inci- in sediment supply between full interglacial (i.e., lowest sediment supply) and full glacial sion rates within the period of averaging. In conditions (i.e., peak sediment supply), minor ¯uctuations superimposed on the larger the sediment-supply simulation, for instance, trend do not produce a terrace. For instance, in the growth of valley width between model calculated mean incision rates are 1.5±7 times time ca. 120 ka and ca. 20 ka, an increase and subsequent decrease in sediment supply lower than the maximum instantaneous inci- at model time ca. 75 ka (isotope stage 4) produces no terrace in spite of a relatively large sion rates occurring during river incision (Fig. amplitude sediment-supply change. In western North America, preservation of isotope 9). Because average incision rates may incor- stage 4 terraces is rare, despite a readvance of alpine glaciers at this time. porate periods of little to no vertical incision

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ance keeps the sediment thickness on a form- ing strath relatively constant. The transition to

downcutting occurs as dQs/dx increases, be- coming more positive, allowing net sediment removal from the channel and enhanced ac- cess to the bedrock. Conversely, an increase in sediment supply, for instance, would push

dQs/dx toward negative values (i.e., aggrada- tion), switching the river from strath forma- tion to valley ®lling. Fill terraces can be gen- erated in such circumstances, which would be an interesting subject for a future modeling effort. A threshold exists between valley ®ll and strath-terrace formation that is dictated by Figure 10. The topographic pro®le along A±B of the terraced valley illustrated in Figure the divergence in sediment-transport capacity, 8 (thick line) and channel-residence time (see de®nition in text) as a function of elevation rather than implying distinctly different inputs during downcutting in this simulation (thin line). Terraces are related to periods of lateral and river response to those inputs. As a con- planation during long channel residence within a narrow elevation range. As in the con- sequence, two identical river systems with stant-period experiments, channel-residence time at individual terrace levels reaches up only slight differences in the amplitude of ¯u- to many tens of thousands of years, indicating that the river spent most of the simulation vial inputs may produce two different records time forming terraces. The widest terraces are produced by the longest residence times, of the same event: a strath terrace in one and and incision between terrace levels occurs over relatively short time intervals. a ®ll terrace in the other. The results point to several future modeling experiments. First, sediment should be deliv- of unknown duration, comparison of mean in- the average channel-incision rate calculated ered at tributary junctions as well as at the cision rates obtained in two different rivers from the height and abandonment age of the head of the trunk channel. Tributary inputs during the same time periodÐor within the youngest terrace is much higher than incision may in¯uence the timing and magnitude of same river system over differing time peri- rates calculated between older terrace levels lateral planation in the down-valley direction, odsÐneglects the actual duration of incision. (compare incision rates for terraces, Fig. 11). perhaps producing diachroneity along terrace Drawing conclusions about the controls on The increase in the average incision rate arises pro®les. Second, the quantity of sediment in- rock-incision rates by comparing downcutting solely because the simulation ends when sed- put and the ease with which the valley can be rates obtained from dated terraces should be iment input is low and when incision rates are widened control the ability of the river to wid- done cautiously in light of our ®ndings. In ad- at their highest, rather than from changes in en and form strath terraces, as opposed to ag- dition, variation in rock-incision rate is an in- tectonic or climatic forcing. The apparent ac- grading to form ®ll terraces. Although we evitable consequence of climatically induced celeration of incision rates in several western have focused on production of strath sequenc- variation in the sediment and water inputs to North American river systems may occur for es here, as are seen in the Wind River system the ¯uvial system. Changes in the mean inci- this reason, as the highest measured rates are from which we deduced the parameter set sion rates through time as deduced from a se- obtained from the youngest dated terrace. used, it would also be interesting to explore ries of terrace ages therefore do not necessar- Because strath terraces imply long periods the conditions leading to ®ll-terrace formation. ily imply long-term changes in uplift rates, but of lateral planation and nearly constant river Preliminary simulations suggest that less eas- can instead simply result from variations in elevation, they are inferred to re¯ect a time of ily eroded valley walls promote the formation the sequencing and duration of input events. balance between sediment supply and dis- of signi®cant ®ll terraces. Finally, although we A stream may incise rapidly not only because charge such that neither net aggradation nor have maintained a steady base-level fall here, it has high stream power and/or is ¯owing degradation occurs (i.e., dQs/dx ഠ 0 and dz/dt a goal should be exploring how nonsteady across weak rock, but also because the chan- ഠ 0). This state is called a graded river (Bull, base-level fall, with superimposed variations nel is able to transport the supplied sediment 1990; Mackin, 1937). Although lateral plana- in river inputs, in¯uences the resulting terrace to provide suf®cient access to the bed. tion implies nearly constant elevation, our sequence. Many dated terrace sequences in western simulations suggest that these graded periods North America and elsewhere indicate accel- should not be taken to indicate steady forcing. CONCLUSIONS eration of incision rates toward the present All of the experiments produce terraces, often (e.g., Chadwick et al., 1997; Patton et al., by abrupt onset of downcutting, even though Starting with the conceptual hypotheses of 1991; Reheis et al., 1991). Our simulations the input is being changed continuously and Gilbert (1877) and Mackin (1937), we have suggest that this apparent acceleration may even gradually (e.g., Figs. 4 and 9). The valley- created a physically based model that suc- simply re¯ect the particular time at which widening periods are characterized by slightly ceeds in producing distinct ¯uvial strath- these systems are sampled. To illustrate, in our positive divergences in sediment-transport ca- terrace sequences in response to time-dependent experiments where sediment supply most pacity (dQs/dx Ͼ 0), where x increases down- inputs to the ¯uvial system. We have selected strongly controls vertical erosion rates, as we stream, re¯ecting a balance within a channel a physically based rule set and have set limits suspect is the real case, the model stream is reach wherein all sediment delivered into a on inputs from available ®eld evidence. Our incising actively at the end of the simulation reach and the sediment generated by bedrock numerical experiments indicate that strath ter- period (e.g., Figs. 4 and 9). As a consequence, erosion are transported downstream. This bal- races can be created through climatic modu-

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not require that inputs be steady. Continuous lateral planation occurs in our simulations even though inputs are irregular and non- steady. Terraces should not be interpreted to re¯ect a period of steady forcing, but instead a period of delicate balance achieved even as the external forcing on the ¯uvial system con- tinues to change. The graded condition in the model ¯uvial systems re¯ects a delicate bal- ance between incision and aggradation; de- struction of this balance occurs in response to modest changes in ¯uvial inputs. The simulations also indicate that the tim- ing of terrace formation signi®cantly lags the input cycles driving terrace formation. This observation is consistent with ®eld documen- tation of a paraglacial period that follows ma- Figure 11. Graph of height of model terraces above the channel (as labeled on left y-axis) jor climate shifts during the transition from vs. terrace age produced by the tenfold sediment-supply simulation scaled by the oxygen glacial to interglacial conditions. As in the isotope curve (sediment-supply-variation history shown by gray curve). The average in- ®eld, our model ¯uvial systems require time cision rates between terrace levels calculated from terrace age and height are shown with to respond and adjust to changing inputs, par- ticularly during the transition from periods of bars (as labeled on the right y-axis). There is a general trend of increasing average incision lateral planation to more rapid incision. Ter- rates toward the present (i.e., end of the simulation). A similar trend has been noted for race formation occurs well after the onset of several terrace sequences in western North America (see text for references), often based the input changes forcing their creation. Dates on terraces of ages similar to model terraces 1, 2, and 7 in this graph. Although such from terraces in the ®eld should be expected acceleration may be ascribed to increasing uplift rates and/or increasing climate intensity to lag the onset of climate changes that affect in the ®eld, our results suggest this may only be an apparent acceleration given the unique ¯uvial inputs, such as the transition from gla- time period that we are observing. In our simulation, this apparent acceleration occurs in cial advance to retreat. spite of a constant tectonic forcing (i.e., base-level lowering). The high rate associated with The incision-rate histories obtained from the most recent terrace (level 1) is obtained because, since abandonment of this terrace terrace sequences in extant ¯uvial systems level, the channel has been largely in a period of downcutting. Incision rates calculated may be distorted by current river incision. between older terrace levels include long periods of little to no downcutting; hence, average Many ¯uvial systems are now actively down- rates tend to be lower solely because of the time sampled. Between the time of formation cutting following an extended period of lateral of terrace 7 and terrace 5 in our simulation, the stream spends an unusually long time planation during the last glaciation. The youn- forming terraces (i.e., little vertical incision) relative to the remainder of the simulation. gest terrace in these ¯uvial systems is often Incorporating this period into the calculation of average incision between terrace level 7 related to the Last Glacial Maximum, and in and any terrace created after terrace 5 (like terrace 2) tends to produce a relatively low some cases abandonment of the Last Glacial incision rate. Thus, our simulations explain why incision rates appear to increase, but do Maximum valley ¯oor may not yet be com- not require either increased uplift or enhanced climate severity. plete (e.g., Chadwick et al., 1997). Because of this active downcutting, incision rates ob- tained from the age and height of a Last Gla- lation of sediment load and water discharge ideas for investigation and interpretation of cial Maximum terrace in these systems are av- delivered to a ¯uvial system. This modulation terrace sequences. The periods of lateral pla- eraged over a period of continual incision. causes rates of lateral planation and river in- nation required to form the valley ¯oors that Incision rates from older terraces, on the other cision to vary through time, and this variation become terraces may be very long. In our sim- hand, are averaged over both prolonged peri- fosters creation of wide valley ¯oors that are ulations, these surfaces were occupied for ods of lateral planation (i.e., incision rates abandoned to form terraces during periods of many tens of thousands of (model) years and near 0) and incision periods. As a conse- more rapid incision. Terrace development and a signi®cant fraction of the overall river his- quence, incision rates calculated from the river response are strongly dependent on the tory. Deposition of gravel across and down a most recent terrace level would be high even amplitude of input ¯uctuations with time, and particular terrace tread is likely diachronous, if the long-term average incision rate is not terrace width, maximum incision rates, and with a potentially large variation in the timing changing. This apparent acceleration of inci- valley ¯oor residence time increase as the in- of deposition. Given this variability, assigning sion rate has been observed in many ¯uvial put amplitude increases. Terrace width is con- a single numeric age to a terrace surface systems, but should not necessarily be inter- trolled largely by the river's residence time at should be done with caution and with refer- preted to indicate ¯uvial response to changes a particular elevation, and the width of cli- ence to the time within the terrace cycle (e.g., in external forcing, such as increasing rock- matically generated terraces is indicative of valley abandonment) to which the age corre- uplift rates. the duration and/or magnitude of climatic sponds. Although these periods of lateral pla- change. nation with low incision rates are representa- ACKNOWLEDGMENTS Several aspects of model behavior, as yet tive of the classic graded-river condition, This research was supported in part by grants undocumented in ®eld settings, provide new achievement of grade in our simulations does from the National Science Foundation (EAR-

Geological Society of America Bulletin, September 2002 1141 Downloaded from gsabulletin.gsapubs.org on December 21, 2010 HANCOCK and ANDERSON

9417798 and EAR-9803837) and a grant from the [Ph.D. thesis]: Baltimore, Maryland, Johns Hopkins sin, in Morrison, R.B., ed., Quaternary nonglacial ge- Geological Society of America (Fahnestock Award University, 361 p. ology: Conterminous US: Boulder, Geological Society to Hancock). We thank Andy Fisher, Justin Reven- Howard, A.D., 1998, Long pro®le development of bedrock of America, Geology of North America, v. K-2, augh, Eric Small, Ellen Wohl, John Gosse, and Don channels: Interaction of weathering, mass wasting, bed p. 407±440. erosion, and sediment transport, in Tinkler, K.J., and Repka, J.L., Anderson, R.S., and Finkel, R.C., 1997, Cos- Easterbrook for careful reviews of an earlier version Wohl, E.E., eds., Rivers over rock: Fluvial processes mogenic dating of ¯uvial terraces, Fremont River, of the manuscript. The manuscript bene®ted greatly in bedrock channels: Washington, D.C., American Utah: Earth and Planetary Science Letters, v. 152, from excellent reviews by Frank Pazzaglia, Greg Geophysical Union, p. 297±319. p. 59±73. Tucker, and Kelin Whipple. We thank as well Oliver Howard, A.D., and Kerby, G., 1983, Channel changes in Ritter, D.F., 1967, Terrace development along the front of Chadwick for sharing insights into the Wind River badlands: Geological Society of America Bulletin, the Beartooth Mountains, southern Montana: Geolog- system. This is University of California, Santa Cruz, v. 94, p. 739±752. ical Society of America Bulletin, v. 78, p. 467±484. 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1142 Geological Society of America Bulletin, September 2002