3.1. Numerical Modeling Framework
 The geomorphic evolution of the central Rockies is controlled primarily by fluvial incision patterns that respond to the tectonic and climatic drivers described above. To explore quantitatively the implications of these proposed forcing mechanisms, we develop and apply a numerical model of regional stream incision for the eastern slope of the central Rockies and compare its predictions with two documented, broad-scale patterns of geomorphic change: Quaternary fluvial incision rates [Dethier, 2001] and total erosion since the middle to late Tertiary [McMillan et al., 2006]. While glacial erosion has impacted the highest elevations in the landscape in the Quaternary, we focus on downstream areas and timescales longer than glacial-interglacial cycles. Therefore we do not attempt to model headwater glaciers and their associated erosional, sedimentological and hydrological effects.
 The model is structured to incorporate several important aspects of the landscape and its geologic history.
 1. The streams cross dramatically different lithologies as they flow from the crystalline ranges into the highly erodible sedimentary basins (Figure 1).
 2. The ranges and basins reside in close proximity, and therefore exhumation of one range-basin pair may drive isostatic uplift that influences exhumation of nearby pairs (e.g., exhumation of the Bighorn basin may result in isostatic uplift of that basin and nearby basins such as the Wind River basin).
 3. The onset of late Cenozoic exhumation varied across the region but was ongoing in most places by 3–4 Ma [McMillan et al., 2006]. Prior to this time, we assume that streams maintained steady profiles across the adjacent basins while sculpting subsummit surfaces in most ranges.
 4. Stream profiles within and adjacent to the crystalline cores of the ranges have knickpoints that indicate that this portion of the landscape has not attained a steady state morphology [Anderson et al., 2006; Safran et al., 2005; Zaprowski et al., 2001]. Therefore we do not assume that erosion rates are in equilibrium with rock uplift rates.
 5. The streams have incised into bedrock and therefore can be considered bedrock streams over million year timescales.
 We use a quasi-two-dimensional, finite difference model (Table 1) of bedrock streams in which stream incision is a function of specific stream power ω, the rate of energy generated per unit area of channel bed by the loss of potential energy as water flows downhill [e.g., Bagnold, 1977; Burbank and Anderson, 2001; Hancock and Anderson, 2002; Howard and Kerby, 1983]. Each model experiment begins at 5 Ma, allowing the onset of incision in most basins to begin by 3 Ma. The two-dimensional topology of the model space is based on the modern landscape, with model stream channels located in their modern orientation as extracted from the GTOPO-30 digital elevation model (Figure 1a). As the streams are kept in a fixed location, incision calculations are one-dimensional. We represent the plan view geometry of major basins in the model space as circles and calculate erosion as if it occurred as a uniform “slab” of sediment (i.e., a disc load) [Lambeck, 1988; Wessel and Keating, 1994].
Table 1. Model Equations and Parameters
|Model||Equation or Value|
|Channel elevation change, m/yr||dz/dt = Ut + Uf − ɛ|
|Basin elevation change, m/yr||dz/dt = Ut + Uf − 0.75ɛ|
|Channel incision, m/yr||ɛ = k(ω − ωc)|
|Specific stream power, N/s m||ω = ρwgQS/W|
|Tectonic uplift, m/yr||Ut = “Yellowstone” + “Rio Grande Rift”|
|Flexural uplift, m/yr||Uf = analytical solution for disc loads|
|Drainage area, km2||A = 0.712x1.785|
|Upstream channel distance, km||x = distances extracted from GTOPO-30|
|Discharge, m3/s||Q = PA|
|Channel width, m||W = 3.0Q0.55|
|Stream incision time step||100 years|
|Rock uplift time step||0.1 Myr|
|Model duration||5 Myr|
|Precipitation P||0.1 m/yr|
|Gravitational acceleration g||9.81 m/s2|
|Water density ρw||1000 kg/m3|
|Rock erodibility in basins kb||1.5 × 10−12 m s2/kg|
|Rock erodibility in ranges kr||5 × 10−14 m s2/kg|
|Threshold stream power ωc||25 N/s m|
|Crust density ρc||2600 kg/m3|
|Mantle density ρm||3300 kg/m3|
|Flexural rigidity D||1024 N m|
 Specific stream power is proportional to the product of the effective stream discharge Q (the volume of water available per unit time) and the local slope S (the proportion of the distance traveled that is downward), and is inversely proportional to channel width W:
where ρw is the density of water, and g is gravitational acceleration. A fraction of the excess specific stream power (specific stream power above a threshold value ωc) is available to drive incision of the streambed:
where z is elevation of the channel bed and k is the proportionality coefficient that accounts for factors such as the efficiency of the flow in delivering energy to the bed and the susceptibility of the bed to erosion. We employ a rule in which incision is linearly related to stream power, and we allow a finite threshold (i.e., ωc ≠ 0). Because sinks and flat areas of the digital elevation map prevent accurate extraction of drainage areas over such a large region, we use a power law relationship to express drainage area A as a function of upstream channel distance x, constrained by the drainage areas of USGS stream gages in the central Rockies. At tributary junctions, the upstream distance for each stream is added, resulting in a stepped increase in drainage area. We use power law relationships to express Q and W as functions of A [Hancock and Anderson, 2002; Roe et al., 2002; Snyder et al., 2000].
 We use Equation (1) to calculate stream erosion because it explicitly incorporates effective discharge, which can be adjusted in climate change scenarios. However, we note that our incision rule is equivalent to excess shear stress rules for stream incision, in which dz/dt = KAmSn (summarized by Tucker and Whipple ), where m = 0.45 and n = 1 and K incorporates ρw, g, and coefficients of the power law equations Q = f1(A) and W = f2(A). The concavity index, θ = m/n = 0.45, is well within the 0.11–1 range of indices reported for a variety of drainage basins [Tucker and Whipple, 2002]. As a first-order approximation, we hold concavity steady spatially and temporally, although analysis of digital elevation models of the Great Plains indicates that modern concavity varies spatially [Zaprowski et al., 2005]. We use two values for k, 1.5 × 10−12 m s2/kg for highly erodible basin fill and 5 × 10−14 m s2/kg for resistant crystalline bedrock, the same values used by Anderson et al.  in modeling incision of Boulder Creek, which drains a portion of the Front Range. The parameters we employ are similar to the lithology coefficients used in excess shear stress incision models (converting k to K, our basin value is K = 2.25 × 10−5 m0.1/yr; Whipple and Tucker  use K = 2.00 × 10−5 m0.1/yr when n = 1), and the lithology contrast is well within the four-orders-of-magnitude range of K values reported by Stock and Montgomery .
3.2. Initial and Boundary Conditions
 The detailed morphology of the Laramide stream network prior to the onset of late Cenozoic exhumation is poorly known. Stream captures have rearranged several drainages [e.g., Mackin, 1937; Reheis et al., 1991; Ritter, 1972; Wayne et al., 1991; Zaprowski et al., 2001], and initial stream gradients are known only in localized areas [McMillan et al., 2002]. The eastern slope of the central Rockies has undergone long-wavelength rock subsidence producing tilt down to the west during the Laramide orogeny and subsequent rock uplift with tilt down to the east since the mid-Tertiary [Mitrovica et al., 1989]. During this time interval, the landscape evolved from one characterized by deposition and relief reduction to one characterized by incision and relief enhancement. Presumably, during active subsidence and during initial recovery from the subsidence, low regional stream gradients prevailed and streams could not incise into bedrock except perhaps locally and temporarily within the crystalline ranges. Long-term persistence of such conditions would have promoted lateral planation that carved subsummit surfaces within the ranges. The ubiquity of these surfaces suggests that the stream network was below the threshold for incision for many millions of years [Epis and Chapin, 1975; Mears, 1993]. Long-term sediment deposition and rock uplift eventually produced regional gradients that were sufficiently great that the stream network must have been at the threshold for incision. Any subsequent rock uplift and/or climate change would have caused stream power to exceed the incision threshold.
 In the modeling scenarios we present here, we assume that the streams at 5 Ma were on the cusp of beginning to incise, with ω = ωc. Because initial conditions are set such that starting stream incision rates are 0 m/Myr, incision is driven only by rock uplift or climate change imposed during the model run; this was confirmed by a run in which no tectonic or climatic forcing was imposed, and indeed no incision occurred. Therefore the model results can be considered sensitivity tests to the imposed forcing. There are no published values of ωc for the central Rockies, despite the recognized importance of erosional thresholds [e.g., Snyder et al., 2003]. McMillan et al.  calculate that the preexhumation slope of the ancestral Platte River system in the western Great Plains was ∼0.001. Setting S = 0.001 and using reasonable discharge and channel width values for the Platte River, Q = 100 m3/s and W = 40 m, we find that ωc = 25 kg/s3. Sensitivity studies in which ωc values were varied over several orders of magnitude indicated that calculated stream incision in this model is not sensitive to ωc, although the initial stream gradient is.
 We do not attempt to reproduce stream captures and drainage reconfigurations that have occurred in the central Rockies since 5 Ma. While changes in the plan view morphology of the stream network have affected local erosion rates, it is unlikely that these changes have caused regional incision. Stream capture and/or migration may drive incision if there is a corresponding increase in channel gradient. The expected incision is Δz = SΔx, where S is the steady state channel gradient and Δx is the change in channel length generated by the capture/migration. For a stream system with a gradient of 0.001, a 200–1200 km decrease in channel length would be required to drive 200–1200 m of incision, as suggested for the amount of erosion in the large Laramide basins. Sufficiently large changes in channel lengths have not been proposed.
 We dictate that no erosion occurs at the downstream node of the modeled drainage network, consistent with minimal late Cenozoic erosion rates in the eastern Great Plains [Dethier, 2001; McMillan et al., 2006]. Because erosion at this downstream boundary is prevented from occurring, the boundary does not generate incision upstream.
3.3. Short-Wavelength, Tectonically Driven Rock Uplift
 In the southern part of the model space, short-wavelength rock uplift is based on calculated tilting of the Ogallala Formation. Leonard  presents profiles of tilting to the east and north. We reproduce this pattern of tilting by multiplying two Gaussian curves to capture decreasing uplift to the north and east, choosing parameters to match the uplift profiles presented by Leonard :
where A1 is the maximum amplitude of the uplift, 600 m; Dn and De are the distances north of the southern model boundary and east of the western model boundary, respectively; and σn and σe are the length scales over which the uplift decreases to 1/e of its maximum, 500 km and 800 km, respectively (Figure 1b). Rock uplift tapers both to the east and north, becoming negligible at the eastern edge of the model space and decreasing to only ∼200 m at 41°N (the Colorado-Wyoming border) [Leonard, 2002]. We assume northward tapering of 100 m per degree of latitude suggested by Leonard  continues; therefore rock uplift becomes negligible at ∼43°N. Uplift in the southwest part of the model space possibly overestimates true uplift, but because there are no modeled streams in this region, the model results are unaffected by the imposed uplift in this region. We assume that the uplift has occurred steadily since 5 Ma.
 Short-wavelength rock uplift in the northwest part of the model space is based on the observed Yellowstone topographic anomaly [Smith and Braile, 1993], a radially symmetrical bulge that can be approximated by
where dz is the amplitude of the anomaly, A2 is the amplitude of the center of the hot spot, 700 m; Dr is the radial distance from the hot spot center; and σr is the radial length scale over which the topographic anomaly decreases, 250 km. The center of the hot spot, migrates at 4.5 cm/yr with a bearing of 56°; surface uplift associated with the migrating hot spot at time i is Δz = dzi − dzi−1 (Figure 1b). We assume that the corresponding rock uplift is equal to the surface uplift, although this is not well constrained in the geologic record. A remnant deposit of the White River unit near the southern border of Yellowstone National Park indicates that erosion here has been <100 m [Love et al., 1976; McMillan et al., 2006]. We calculate hot spot migration in 0.1 Myr time steps, with rock uplift imposed at the beginning of each time step.
3.5. Climate Change
 Ideally, a quantitative assessment of how late Cenozoic climate change has affected stream processes would include reconstructions of flood probability distributions along each stream [Lague et al., 2005; Molnar, 2001; Molnar et al., 2006; Snyder et al., 2003; Tucker and Bras, 1998]. However, this is an unrealistic expectation of the geologic record, particularly given the extensive erosion that has removed much of the late Cenozoic sedimentary record in the Rockies. We therefore attempt to keep our climate change scenario as simple as possible to illustrate the sensitivity of landscapes to basic changes in climate. Our implementation of climate change does not encompass all of the possible ways in which climate could have affected late Cenozoic erosion rates. We use a simple forcing in which streams gradually become more erosive through time, accomplished by linearly increasing effective discharge over the model runs. This pattern mimics the gradual increase in global oxygen isotopes since 5 Ma [Zachos et al., 2001] (Figure 2b). We present the results of several scenarios in which the final discharge increase is 2–6 times the initial discharge.
3.6. Isostatically Driven Rock Uplift
 We employ an analytical solution to calculate two-dimensional isostatic uplift due to removal of disc loads [Lambeck, 1988; Wessel and Keating, 1994] (Figure 3). We use a uniform flexural wavelength l of ∼100 km, corresponding to a flexural rigidity D of 1024 N m [Angevine and Flanagan, 1987; Lowry and Smith, 1995]. For Young's modulus E = 100 GPa and Poisson's ratio ν = 0.25, the effective lithospheric thickness Te ≈ 48 km. We assume a mantle density of 3300 kg/m3, a basin fill density of 2600 kg/m3, and an air density of 1.22 kg/m3. We perform this calculation every 0.1 Myr, sufficiently long for the landscape to achieve the full isostatic response to the unloading (of order 10 kyr [Adams et al., 1999]) and for significant, but still small, amounts of rock uplift to occur (<10 m per time step for all experiments).
Figure 3. (a) Schematic exhumation of a sedimentary basin and the modeled disc load approximation used for isostatic uplift calculations, shown in one dimension. The basin floor elevation lowers from the initial elevation (solid black line) to the final elevation (dashed gray line) in response to river incision in the center of the basin. We treat the material exhumed as a uniform slab of sediment, with a thickness 75% of the river incision (solid gray line). We expect the greatest error in this approximation at the basin edges. (b) Plot of the expected uplift generated by removing a disc with a radius of 100 km and thickness of 100 m from a sedimentary basin in the Laramide region. A maximum of 67.3 m of rock uplift occurs at the center of the exhumed basin; 33.5 m of uplift is generated along the basin periphery. For basins with smaller radii the amount of calculated uplift is smaller.
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 The thickness of each sediment slab removed depends on the mean lowering of the basin floor, which is a function of the rate of trunk stream incision (which we explicitly calculate), the response of tributaries to this incision, and the drainage density within the basin. The detailed relationship between trunk stream incision and mean basin exhumation is complicated in any basin. In the central Rockies, we make a first-order approximation that the basin floor experiences a mean erosion rate that is 75% of the trunk-stream incision rate. Our choice of the 75% value acknowledges that the basin floors are not perfectly flat, that tributaries do not cover the entire basin, but that the volume of uneroded basin fill is small relative to the volume of eroded sediment, as can be seen in the modern basins of the region. We expect the greatest error at the basin edges, where the approximation of circular basins does not capture more complex basin shapes and where tributary response times are longest [Riihimaki et al., 2006]. However, given the high flexural rigidity of the region, modeled stream incision rates and amounts are insensitive to the simplified basin geometries.
 We do not calculate rock uplift due to erosion of canyons within ranges. Small and Anderson [1998a] argue that this rock uplift was likely small enough to be balanced by slow bedrock weathering rates. Because the volume of material eroded from basins was much greater than the volume eroded from the adjacent ranges, the regional isostatic uplift should be driven primarily by uplift in response to basin erosion [Small and Anderson, 1998b].