SHORT RESEARCH Dip, Layer Spacing, and Incision Rate Controls on the Formation of Strike Valleys, Cuestas, and Cliffbands In

SHORT RESEARCH

Dip, layer spacing, and incision rate controls on the formation of strike valleys, cuestas, and cliffbands in heterogeneous stratigraphy

Dylan J. Ward DEPARTMENT OF GEOLOGY, UNIVERSITY OF CINCINNATI, CINCINNATI, OHIO 45221, USA

ABSTRACT

Landscapes developed over heterogeneous stratigraphy exhibit a spectrum of landforms from dramatic cliffbands to hogbacks, depending on the dip and spacing of the layers. In deeply incised landscapes, a single cliffband may consist of multiple resistant layers, whereas similar stratigraphy elsewhere is separated by strike valleys into individual cuesta benches or hogbacks. This paper presents a geometric analysis, informed by a numerical landscape model, to explain the conditions for development of a strike valley floored by erodible rocks. The results define a threshold incision rate below which strike valleys are more likely to form; this threshold incision rate is proportional to the stratigraphic spacing of cliff-forming layers and a trigonometric function of dip angle. The analysis also yields a time scale for the adjustment of structural landforms to changes in regional incision rate, which is a function of dip angle and the coupling between cliff retreat rate and escarpment height. In example landscapes of the Colorado Plateau, this time scale is likely much longer than that of documented variations of incision rates due to late Quaternary climate and land-use changes. The transitional state of escarpments in layered rock may therefore contain information about regional downcutting rates over time scales different from those recorded by the fluvial network. The utility of such features will require better understanding of the coupling between incision of a foot slope and the retreat rate of the cliff above in different kinds of rocks.

LITHOSPHERE; v. 11; no. 5; p. 697–707; GSA Data Repository Item 2019305 | Published online 2 August 2019 https://doi.org/10.1130/L1056.1

INTRODUCTION on landform is therefore complicated by multiple sources of variability: changes to dip, stratigraphic thickness, erodibility, and internal sources of Motivation transience in the downcutting rates (Darling and Whipple, 2015). Nonfluvial landforms that commonly develop on variably erodible In landscapes with uniform rocks and a steady rate of rock uplift or stratigraphy include cliffbands, cuestas, homoclinal ridges, hogbacks, base-level fall, topography evolves to a quasi-steady form that equalizes and flatirons. These “structurally controlled” landforms are typically clas- erosion rates spatially across the landscape (Gilbert, 1877; Tucker and sified based on somewhat arbitrary ranges of dip angles of the resistant Bras, 1998). Even where rocks differ spatially in erodibility, a quasi-steady rock capping the feature (Twidale and Campbell, 2005). There is not, topography can form under conditions of uniform base-level fall, where however, any unifying theory that relates stratigraphic and lithologic landforms on more resistant rocks evolve to have steeper slopes to attain properties under various rates of base-level fall to specific structurally the same erosion rates as those on weaker rocks (Gilbert, 1877; Hack, controlled landforms. It is my purpose here to explain a key morphologi- 1960). This relationship implies that, given appropriate information about cal distinction along this spectrum of landforms, which is the presence material and erosion processes, uplift rates and gradients thereof may be or absence of a strike valley floored by more erodible rocks. The result deduced from topography (Wobus et al., 2006). lends insight into the steady and transient states of landscapes developed In contrast, recent work has highlighted how the heterogeneity in rock on heterogeneous stratigraphy. strength can lead to the indefinite persistence of transient topography and to varying local erosion rates in landscapes experiencing steady base-level Colorado Plateau fall. For example, where fluvial networks cross dipping stratigraphy, lithologic knickpoints that form on harder units will migrate upstream, which Structurally controlled landscapes occur on sedimentary rocks world- may be updip or downdip. This process results in a local, internal base- wide. Prominent examples include the Valley and Ridge region of eastern level control that behaves differently from the external base level (Cook et North America; the Atlas Mountains of northern Africa; the Jura Moun- al., 2009; Berlin and Anderson, 2009; Forte et al., 2016; Perne et al., 2017). tains of Europe; the Arabian Desert and Negev Desert; the Flinders and Interpretation of erosion rates, or tectonics, in these landscapes based Macdonnell Ranges of Australia; and the Qinling and Dalou Mountains of China. Some of the most iconic landscapes of this type are found in Dylan Ward http://orcid.org/0000 -0002 -5741 -7830 the Colorado Plateau region of the United States.

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In all of these areas, the landforms are clearly controlled by the strati- the trunk streams due to drainage integration through the Grand Canyon. graphic spacing of resistant units. For example, resistant layers spaced This transience resulted in several sets of major (~200-m-relief) knick closely together commonly form a compound cliffband, wherein the foot zones along the river system, some of which may reflect a combination of slope below an upper cliff drains directly over the lower cliff (Fig. 1). base-level and lithological effects (Cook et al., 2009; Darling et al., 2012; Where resistant layers are spaced more widely, broad benches may form Bursztyn et al., 2015) or differential tectonic uplift rates (e.g., Crow et al., on the weaker rock between them, separating the cliffs and routing drain- 2014). Superimposed on these major, long-term incision events, there are age parallel to strike. Dip is another primary control: Steeply dipping 102 to 104 yr cut-and-fill cycles that are typically of 1 to 10 m in magnitude layers form homoclinal ridges or hogbacks separated by strike valleys, (Harvey et al., 2011; Pederson et al., 2013b; Sheehan and Ward, 2018). whereas more horizontal layers of the same rock may form compound These episodes of incision of different magnitudes and durations imply cliffs. The Laramide monoclines of the Colorado Plateau provide clas- that variability in the surrounding landforms may be due to their current sic examples of this stratigraphic and dip control on the topography of states of response to the variability in downcutting, with their specific escarpments (Fig. 1). morphology conditioned by the bedrock template. Because the field-measurable terms of interlayer spacing and structural dip contribute to the variability of landforms, the relationship between Reference Case: Coal Cliffs Cuesta, Utah, USA landform and base-level fall is not as straightforward as in uniform sub- strates (Forte et al., 2016). In the example of the Colorado Plateau, the As noted already, steeply dipping cap rocks that form hogbacks are cliff and cuesta landforms developed during a time of significant docu- commonly separated by strike valleys, but it is less common for strike val- mented spatial and temporal variability in downcutting rates (Darling et leys to form among subhorizontal rocks. An example of this less common al., 2012; Jochems and Pederson, 2015). The most prominent transience case is the Coal Cliffs cuesta (Fig. 2), located on the western flank of the in the fluvial networks is related to deep, late Cenozoic entrenchment of San Rafael Swell in central Utah. The San Rafael Swell is a Laramide-aged

mesa A compound cliff strike valley hogbacks backscarp strike valley (bench) compound cliffs cuesta cuesta

Dip: steeper subhorizontal B

compound cliffs transverse stream valleys compoun d strike cliff s & hogback

strike valley

N38.32º, W111.04º View to the north

Figure 1. (A) Schematic cross section of an upwarp such as the San Rafael Swell, illustrating dip and stratigraphic control on structural landforms. (B) Strike valley on the Colorado Plateau between Hanksville, Utah, and Capitol Reef National Park. The near-horizontal stratigraphy of North Caineville Mesa in the background is ringed by compound cliffs containing several cliff-forming layers. Closely spaced sandstone units form a compound cliffband on the east side of the strike valley, which is developed in a thicker shale unit with a moderate dip. As the dip steepens to the west (left side of the image), more closely spaced resistant strata form hogbacks separated by narrow strike valleys. The strike valleys drain to a tributary of the Fremont River, which acts here as a transverse stream, crossing the layers perpendicular to strike.

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A B

Figure 2. (A) Location and topography of the Coal Cliffs cuesta on the western, shallowly dipping flank of the San Rafael Swell, Utah (UT). R— River. (B) Oblique air photo of the Coal Cliffs cuesta and strike valley. Below the capping Ferron Sandstone Member, there is a steep foot slope of Tununk shale, which tapers to a low-relief strike valley above the Dakota Formation sandstone. The strike valley drains through the Dakota layer at multiple pour points along strike (white circles on panel A). Largest talus boulders are ~10 m in diameter. (C) Strike valley is widest to the north where these pour points are farther from the upper (Ferron) cliffline and disappears to the south as the pour points encroach on the foot slope of the upper cliff. This change appears to be related to the deeper entrenchment of Muddy Creek (Ck) and its tributaries where they cross the escarpment at the south end of the image.

upwarp, ~60 km across, with a steeply dipping (20°–80°) eastern flank Climate-driven incision cycles have resulted in short-term erosion and a shallowly dipping (2°–10°) western flank. The stratigraphy of the rates of 1–3 mm/yr in the vicinity of the Coal Cliffs (Sheehan and Ward, western flank is composed of Triassic through Cretaceous terrestrial and 2018). The escarpment is crosscut by multiple tributaries to the San Rafael marine sedimentary rocks (Cashion, 1973). The sandstones tend to act River and Dirty Devil River, which join the main-stem Green River and as cliff-forming units, and the weak shale and mudstone interbeds form Colorado River, respectively, ~100 km downstream. Near its junction with foot slopes and strike valleys. the San Rafael River, the Green River has been incising at 0.45 mm/yr The Coal Cliffs escarpment is developed in the Cretaceous Mancos over the last 100 k.y. (Pederson et al., 2013a). Upstream, in Desolation Formation. The 10–20-m-thick Ferron Sandstone Member of the Man- Canyon, the Green River has incised more slowly, at ~0.04 mm/yr, over cos Formation acts as the cap rock for the Coal Cliffs (Fig. 2B). Below, the last 1.8 m.y. (Darling et al., 2012). The faster rates downstream pos- the ~100-m-thick Tununk Shale Member forms a bench or strike valley, sibly represent a transient acceleration in downcutting rates within the ranging from a few hundreds of meters to just over 1 km wide atop the Green-Colorado network (Cook et al., 2009; Roberts et al., 2012; Pederson Dakota Formation sandstone, which itself forms the cap of a lower cliff- et al., 2013a), but it is not clear how far into the San Rafael Swell such band. Above the Ferron member, there is the Blue Gate Shale Member of a signal has propagated. Considering these factors, rates of longer-term the Mancos Formation, which is several hundred meters thick and forms fluvial incision in the Coal Cliffs are likely on the order of 0.1–0.5 mm/yr. a kilometers-wide strike valley between the Coal Cliffs and the base of The Coal Cliffs represent a clear end-member expression of low dip the Mesaverde Group sandstones at the edge of the Wasatch Plateau to angles, thin resistant and thick weak layers, and well-developed strike the west. These layers dip consistently 3°–4° to the west. valleys that drain to escarpment-crossing transverse streams, some of

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which have a catchment area above the Ferron sandstone and some of layers, the structural dip, and the regional downcutting rate to the pres- which do not. ence or absence of a strike valley, under conditions of steady base-level lowering and steady topographic relief. I then discuss some salient aspects Pour Points and Benches of the transient evolution of these systems as base-level lowering rates change through time. The upper segments of any escarpment that exposes two resistant units must ultimately drain across the lower resistant unit. On a classic hogback, ANALYTICAL MODEL or a cuesta like the Coal Cliffs, the foot slope of the upper cliffband drains to a bench underlain by the more erodible rock. Channels on the bench Model Geometry drain parallel to strike, until reaching a larger transverse stream or break in the lower cap rock. Following the Coal Cliffs example, consider an escarpment exposing

Where a bench or strike valley is present, the width over which it two resistant units separated by a weaker unit of thickness Hi, and dipping is floored by more erodible rocks is correlated with the distance from upstream relative to larger transverse streams that cross the escarpment the upper cliffline to the nearest pour point through the lower cap rock approximately perpendicular to strike (Fig. 3). This analysis seeks the (Fig. 2C). Unsurprisingly, the farther the pour point lies outboard of the conditions of stratigraphy, dip angle α, and base-level lowering rate (ζ) upper cliffline, the wider is the area in which the more erodible rocks are under which, upon reaching steady topographic relief, the transverse exposed in the strike valley. Where the pour point distance becomes less stream’s pour point through the lower cap rock is aligned along strike with than the length of the steep foot slope of the upper cliffband, the strike the top of the upper cliff (Fig. 3D). Geometrically, this occurs when the

valley disappears (Fig. 2A), and the cliffs transform to a compound form. steady-state upper cliff height Hs is equal to Hv, defined as the apparent

Drainage along strike becomes limited, and small channels of the foot thickness of the weak rock interlayer in the vertical direction (i.e., Hv =

slope of the upper cliff drain directly over the edge of the next cliff below. Hi/cos α). Hs < Hv is the bench-forming case (Figs. 3A and 3C), and Hs >

In well-developed compound cliffs, the pour points through the lower cliff Hv favors drainage of the upper cliff over the lower cliff and a compound are positioned within the bounds of the upper cliffline, as in the classical cliff (Figs. 3B and 3E). “rule of V’s.” Similarly, where transverse streams are superimposed on a For simplicity, and based on the Coal Cliffs example, the transverse

compound cliff form, they cross the lower cap rock at pour points that are streams are assumed to have significant upstream area (>>Abench). They inboard of the upper rim of the escarpment. In turn, the pour point posi- therefore maintain low slopes (here taken to be zero slope), even where tions are related to the relative entrenchment of the local drainage network. crossing the resistant layers, and respond quasi-instantaneously to base- I posit that this geometry of pour point location dictates the form of level change. This assumption can be relaxed if necessary with a geometric the escarpment by controlling the route by which base-level changes are correction to account for the gradient of the transverse stream. communicated to the upper portion of the escarpment. Next, I derive the The rate of change of the upper cliff’s height H can be written: conditions under which the lower cap rock pour point on an escarpment- dH crossing stream would be aligned along strike with the forefront of the =ζ− R tanα, (1) upper cliffband. The goal is to relate the stratigraphic spacing of resistant dt

A Map C Profile R(Hs) ζ ζ R tan a = Hs( )< Hv pour point α Hv Hs ζ<ζ Hi c pour Bench point strike valley present

R(Hv) R tan a = ζ D Figure 3. (A–B) Schematic map H (ζ) = H transverse streams H s v and (C–E) cross-section views of Cuesta v Hi Hs a cuesta escarpment (top) and a compound cliff (bottom). Terms ζ=ζ α pour point c and variables are defined in the B ~aligned with salients in upper clifftop text. Downcutting rate increases top to bottom. transitional pour R(H ) point v Increasing incision rate Increasing R tan a < ζ E ζ) Hs( > Hv Hv Hi H pour s point α transverse stream ζ>ζ c no strike valley Compound cliffband

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where H is the instantaneous height of the upper cliff, ζ is the base-level A Contours of incision rate at form crossover (mm/yr) 300 5 lowering rate, assumed to be applied synchronously along the entire trans- 0.5

0.1

verse stream, R is the horizontal retreat rate of the upper escarpment, and 2 α is the dip angle. Assuming steady relief has been attained, dH/dt = 0, and 250 0.05 1

20 0.2

0.02 10

ζ= α. (2) 0.01 R tan 200 5 0.5

0.1 Unless the retreat rate R is a function of escarpment height, there can be 0.001 2

0.05 (m)

no steady-relief condition. This analysis is predicated on the assumption i 150 1

H 0.2

20 that such a condition is attainable and therefore that such a relationship 10 0.02

exists. There is not any universal theory for the form of R(…), but there 0.01 100 0.5 is some empirical and model support for taller escarpments experienc- 5 0.1

0.005 2

ing larger retreat rates (cf. Howard and Selby, 2009; Haviv et al., 2010), 0.001 0.2 1 particularly where the retreat is driven by undermining of the resistant 50 0.05 0.5 0.02 0.1 20 layer by downcutting of the foot slope (e.g., Koons, 1955; Howard, 1995). 10 0.01 0.05 0.2 An increase in retreat rate with height is furthermore consistent with the 0.005 0.02 0.050.1 0 0.001 0.001 scaling of hillslope relief with uplift/incision rate in linear and nonlinear- 0 10 20 30 40 50 60 70 80 90 , degrees diffusive settings (Roering et al., 2001). The simplest such form of R(H) = 0.1 mm/yr = 1 mm/yr is linear with proportionality coefficient c1 [1/T]: 2 300 1 B 0.5

2 1 H* 0.5 H*

Rc= 1H. (3) 250

The c1 term here encapsulates the mechanics of the backwasting process,

scaling the creation of relief by the channel network of the foot slope to 5 200 2 cuesta 0.5 1 1

2 strike the size and time distribution of rockfall failure events. 0.5

compound benches

For an escarpment as illustrated in Figure 3, H cannot physically (m)

i valleys 150 cliffs exceed Hv, but we may still consider that there is a theoretical steady height H for the upper cliff. Inserting Equation 3 into Equation 2, and solving for cuesta / strike 2

height, which per the assumptions of Equation 2 must be the steady-state 100 benches 1 0.5 5 compound1 0.5 valleys height H , we obtain: 2 s ζ cliffs H = . (4) / s α 50 c1 tan 2 1 0.5 2 5 1 Where Hs = Hv, the pour point through the lower cliff is aligned along 0.5 5 2 1 0.5 strike with the salient edge of the upper cliff (Fig. 3D). The system is at 0 5 1 0 10 20 30 40 50 60 70 80 90 the point of transition from bench to compound cliff, as streams flowing , degrees along the bench must cross the lower cap rock before reaching the more competent transverse stream. Figure 4. Relationship between incision rate and form for different dip −6 –1 angles and layer spacing, using a value of c1 = 3.5 × 10 yr (see “Model Results” subsection of “Numerical Model” section in text). (A) Contours ζ H i = . (5) of incision rate, ζ , where the form crossover occurs for a particular dip c α α c 1 tancos and layer spacing. Rates lower than this threshold permit development of

Defining the incision rate that results in this configuration asζ c, and strike valleys. (B) Contours of H* for ζ = 0.1 mm/yr and 1 mm/yr. rearranging Equation 5, we obtain:

tanα ζ=cicH1 . (6) cosα H* > 1 represents conditions likely to form a compound cliff. The ratio

Therefore, for a given layer spacing Hi and dip angle α, whether a weak- H* is linearly dependent on the collection of terms: floored bench or strike valley develops depends on a threshold incision ζ rate that increases linearly with interlayer spacing and is proportional to , (8) tan α / cos α (Fig. 4A). cH1 i which is the dimensionless ratio between the base-level lowering rate and Dimensionless Height the cliff retreat rate at a cliff height equal to the interlayer spacing. The interlayer spacing here takes the role of a scale height for the behavior Nondimensionalization can reveal the key relationships between terms of the escarpment against the regional downcutting rate. that govern the overall behavior of a system. Equation 4 can be divided

by Hv as a reference length, yielding a dimensionless number H*: Analytical Results H ζ cosα H * ≡=s . (7) Equations 6 and 7 posit a nonlinear relationship for resistant layer α Hcv 1H i tan spacing, dip angle, and incision rate that dictates whether the landscape H* < 1 means that the equilibrium cliff height is less than the vertical will evolve toward a compound cliffband or cuesta/hogback form as it apparent thickness of the interbed, and therefore a cuesta bench may form. approaches equilibrium relief (Fig. 4). At low dip angles, the apparent

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vertical thickness of weak units between the cap rocks is only slightly approaching a steady height (Fig. 5). More rapid uplift rates resulted in exaggerated, so the product of this vertical thickness and the retreat con- taller escarpments. Exhumation of the backscarp occurred more rapidly

stant c1 is outstripped by even fairly slow incision rates. Thus, for a given with higher uplift rates, requiring longer model runs and longer domains

value of c1, low dip angles require low erosion rates for benches to form, to capture the growth phase of increasing relief and approach to steady and even widely spaced resistant layers will tend to form a compound relief that are relevant for these experiments. cliff. Conversely, vertical dips result in an infinite apparent thickness of The instantaneous rate of retreat during the growth phase was found by the weak interbed, and so the cap-rock layers must be separated by a strike extracting the average y position and height of the clifftop through each valley under any incision rate (in that case, parallel hogbacks). model run (Fig. 5B). Retreat rates were noisy, due to the stochastic, cel- lular nature of the rockfall process, but systematically increased through NUMERICAL MODEL model time as the escarpment grew taller. Dividing the instantaneous retreat rate by the instantaneous escarpment height revealed a constant Model Description proportionality, as was asserted heuristically in Equation 3. This propor- −6 –1 tionality constant took on a value (c1 ≈ 3.5 × 10 yr ) that did not vary The simplicity of the previous solution relies on the assumption that between runs with different uplift rates. Sensitivity testing showed that

R = c1H. To determine the functional form of this relationship for a set of this constant is not strongly affected by dip angle, cap-rock thickness, or common erosion rules, I applied a two-dimensional (2-D) finite-difference the fluvial erodibility of either rock type. Instead, it is an approximately landscape evolution model derived from that of Ward et al. (2011). The linear function of the specified gradient of initiation for the rockfall pro- model simulates stochastic failure of a hard-capped cliff within a detach- cess in the model (see Data Repository item). In other words, it dictates ment-limited fluvial landscape carved into more erodible rock. Hillslopes how steep the upper extent of the drainage network must become to cause evolve according to a linear-diffusion scheme (Tucker and Bras, 1998), rockfall, effectively coupling the stream network and the retreat rate via and bedrock incision is based on unit stream power (Whipple and Tucker, the cliff height. 1999). Stochastic rockfall from the cliffband is simulated by instantaneously reduction of the height of a selected cliff-edge cap-rock grid cell Transient Approach to Steady Height to that of its lowest downslope neighbor (following Howard, 1995). The equivalent volume of the cell is added to a model layer that tracks rockfall The approach to a steady escarpment height following either emer- debris thickness and is distributed across the landscape below this cell gence of the cliff or a change in base-level lowering rate, given Equations by relaxing the surface of the debris layer to the angle of repose. Debris 1 and 3, has an exponential secular-equilibrium form: is continuously redistributed to the angle of repose as the surrounding ζ landscape evolves, or until it erodes away. Further details of the model , (9) Ht()= −−H 01(1 exp(−αct tan )) + H 0 1 α implementation are included in the GSA Data Repository material. c1 tan

A single cap rock was configured among uniformly weaker rocks where H0 is the initial height of the escarpment. This can be seen in the within a long model domain oriented parallel to the dip direction. Model model results between the times of cliff emergence and backscarp dissec- –1 stratigraphy dipped 3.4° in the +y direction, based on the example of the tion (Fig. 5B). The e-folding time scale defined byτ = (c1 tan α) dictates Coal Cliffs (Fig. 2). The resistant layer was 20 m thick and 200× more the time needed for the cliff height and retreat rate to respond to a new resistant to erosion than the surrounding rock. The cliff was initially set uplift/incision rate. In the model configuration used here, this time scale to fail by the rockfall process when the upper edge of the cliff reached a is 4.8 m.y., so nearly full adjustment requires on the order of 10 m.y. In slope exceeding 60° (in the model, corresponding to a particular elevation rocks with steeper dips, this response time is reduced as 1/tan α.

difference between cells as a proxy for the height of the vertical portion Following Equation 4, at steady state, τ = Hs/ζ. This relationship allows

of the cliff; the effects of this threshold are discussed below). The angle an estimate in the field ofτ , and therefore also c1, in the rare circumstances of repose of rockfall debris was 25°. To isolate the intrinsic relationship where steady-state relief can be assumed. For example, if the ~100 m between height and backwasting rate, rockfall debris was assigned a high height of the Ferron escarpment is assumed to be in equilibrium with a erodibility, such that its effect on cliff retreat was minor and identical for long-term average downcutting rate of 0.1 mm/yr at the Coal Cliffs site, −5 –1 all experiments. Harder debris would create an armoring effect on the foot then c1 = 1.4 × 10 yr , and τ = ~1 m.y. Even if the escarpment there is slope, which is a potentially important factor in the height–retreat rate rela- equilibrated to a much faster long-term mean downcutting rate of 1 mm/ −4 –1 tionship (Ward et al., 2011; Glade et al., 2017). The strike-parallel model yr, then c1 = 1.4 × 10 yr , and τ = ~100 k.y. Although this escarpment boundaries were open, with periodic boundaries on the transverse edges. has not been shown to be in relief steady state, these example calcula- I imposed a range of rock uplift rates (10−5 to 10−4 m/yr) and allowed the tions suggest that the landform would be expected to equilibrate over a cap rock to be exhumed, develop an escarpment, and retreat until retreat time scale longer than that of late Pleistocene–Holocene environmental rates became quasi-steady. changes in downcutting rate.

Model Results: Retreat Rate Proportional to Height MODEL RUNS WITH MULTIPLE CAP ROCKS AND TRANSVERSE STREAMS Once the cap rock was exhumed, the escarpment height increased over the duration of each model run until exhumation of the downdip edge of Model Setup the cap rock resulted in rapid erosion of the entire landform via dissection of the backscarp. During the increasing-relief phase, the height of The same numerical model was used to examine the transient response the escarpment edge increased most rapidly at first and then slowed until of simulated landscapes with more than one cap rock that are near the

1 GSA Data Repository Item 2019305, Numerical model description and Movies SM1–SM3, is available at http://www.geosociety.org/datarepository/2019 or on request from [email protected].

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A ζ = 1x10-5 m/yr ζ = 2x10-5 m/yr

) )

ζ = 5x10-5 m/yr ζ = 10x10-5 m/yr

) )

B 6 10 6

(1/yr) 6 1 1 4 10

2 10 6

c = H / R 0 1.5 1.0

(mm/yr) 0.5

R Retreat rate rate Retreat 0 Backscarp dissection reaches cliff face 400 Cliff emergence

(m) 200

H height f f Cli 0 ζ = 5x10-5 m/yr ζ = 10x10-5 m/yr

5 ζ = 2x10-5 m/yr

position

(km)

ζ = 1x10-5 m/yr f Cli 0 5 10 15 20 Model time (m.y.)

Figure 5. (A) Snapshots from model runs under different uplift rates. (B) Dividing instantaneous cliff retreat rate by height reveals a linear proportionality that is independent of uplift rate. Retreat rates are smoothed by a 10 point running average to reduce noise caused by the rockfall process.

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threshold erosion rate expected from the analytical treatment above. A sec- form should result on approach to a state of steady relief: Closely spaced ond, 10-m-thick cap rock was added, 30 m below the 20-m-thick cap rock resistant layers, shallow dip angles, and rapid downcutting rates each favor of the single-layer runs. The dip-parallel boundaries were changed to open, a compound cliff morphology. If background downcutting rates remain fixed-elevation boundaries and are equivalent to the transverse streams. slower than a threshold defined by Equation 6, cuestas or hogbacks sepa- Here, I report the results of simulations given two uplift histories that rated by strike valleys will form so long as there are streams cutting the illustrate the key behavior. In both cases, there was an equilibration period escarpment that connect the strike valley to base level. in which uplift rate was steady at 0.1 mm/yr for 1.8 million model years, Equilibration time scales to reach these states following a change in until a near-steady topography was reached over a homogeneous substrate. incision rate are geologically significant and may be long enough that This was followed by 0.2 m.y. of zero uplift to allow the initial relief to the spatially variable dip of the beds, further changes to downcutting rate, decay somewhat before the cliff was exhumed, preventing the existing or dissection of the escarpment via the backscarp would be encountered topography from draining over the upper cliff as it emerged. before steady state is reached (Howard and Selby, 2009). In the numeri- Following the 2 m.y. equilibration period, the first scenario maintained cal model runs, the e-folding time scale for adjustment was 4.8 m.y.; this a constant 0.008 mm/yr until 10 m.y. and then increased to 0.03 mm/yr was the outcome of an arbitrary parameterization. For highly erodible for the remainder of the model run (Fig. 6A). These values lie near to escarpments in the Colorado Plateau, such as the Coal Cliffs, conservative but on either side of the H* = 1 line for this stratigraphic geometry. The estimates of the response time scale are ~100 k.y., but it is more likely on second uplift scenario consisted of a series of brief pulses of 0.1 mm/yr the order of 1 m.y. Different landforms may therefore be in different states uplift spaced irregularly (Fig. 6E). The pulses resulted in long-term uplift of adjustment to the same changes in incision rate. For example, because rates that averaged 0.03 mm/yr for the next 4.4 m.y. of the model run and of the dip dependence of the response time scale, portions of the steep 0.008 mm/yr for the final 3.8 m.y. This time series more closely idealizes limb of the San Rafael Swell dipping at 25° would be expected to attain the episodic, transient forcing documented on the Colorado Plateau, and steady relief ~7× faster than the shallowly dipping limb at 4° dip (Eq. 9). it allows examination of the system response over different time scales. The response of the escarpment form locally affects fluvial incision rates. Because separation between cuesta cliffbands expands and contracts Model Results: Transient Behavior with Two Cap Rocks during transients in the incision rate, the stream networks experience a fluctuating configuration of upstream area (see Movies SM1 and SM2 in Simple Step Change in Uplift Rates the Data Repository). These locally driven fluctuations of incision rate In this scenario, during the slow uplift (H* < 1) phase, a bench of con- would be further modulated by the dip control on local incision rates stant width formed between the two layers. When the uplift rate increased described by Berlin and Anderson (2009) and Forte et al. (2016). (H* > 1), there was a period of transient adjustment that lasted ~5 million Nonetheless, the transient numerical model results indicate that the model years, as first the upper cliff increased in height until the lower cap dominant form of the escarpment reflects the downcutting rate averaged rock was exposed beneath it, and then the bench of weak rock disappeared. over the equilibration time scale, including any superimposed shorter As the height of the upper cliff above the lower became limited by the periods of more rapid or less rapid incision. In the context of the Colorado vertical projection of the interlayer thickness, its backwearing rate also Plateau, this means that escarpment landforms with longer equilibration became limited. This allowed the lower cliff to “catch up,” combining time scales likely reflect the long-term regional rock uplift rates, or long- the two cliffs into a compound cliff (Figs. 6B–6D). term downcutting trends due to river integration over the late Cenozoic, Episodic Uplift while faster-responding landforms may change substantially in response to Throughout the model run, the cliffs began to separate to form a bench hydrologically driven cycles of downcutting related to glacial-interglacial during the intervals between uplift pulses and then converged again when climate variability. uplift resumed. During the portion of the model run with the faster average uplift rate (H* > 1), the cliffband generally behaved as a compound Retreat Rate as a Function of Height: Role of Lithology and cliff. In this phase, drainage from the upper foot slope over the lower cliff Debris Armoring enhanced lower cliff retreat, creating alcoves and bringing the two cliffs closer during uplift pulses. These relationships between downcutting rate and landform are pri- During the phase with the slower average uplift rate (H* < 1), the marily due to large-scale geometry, and they are surprisingly independent pulses of uplift were too infrequent to maintain convergence of the two of erosion process and the detailed lithology of the rocks involved. As cliffs, and a persistent cuesta bench formed. Erosion transients reached the long as cliff retreat rate is a monotonic function of escarpment height, upper cliff via small scarps that propagated from the transverse drainages the systematics should be grossly similar to those described here. In along strike on the erodible rock of the bench. (Equivalent features are the model framework presented here, the coupling is assumed to be lin-

observed in the field; see Sheehan and Ward, 2018.) The upper cliff at the early proportional to the constant c1, which is validated by the numerical midpoint between the two transverse streams was the last area to respond to model (given its own set of erosion rules and corresponding assumptions). each uplift pulse, and salients formed in the cliffband there (Figs. 6F–6H). A nonlinear, but monotonic relationship (e.g., R ~ Hp, where p is a real number) would result in a different steady height for a given erosion DISCUSSION rate, and therefore a different threshold incision rate for form transition, but the same general framework relating dip, spacing, and incision rate Expected Steady Form and Landform Adjustment Time Scales would persist. If retreat rate is a superlinear function of height (p > 1), the landscape adjustment time scale would be shorter than in the linear The analytical and numerical model results presented here describe case; in the case of a sublinear relationship (p < 1), it would be longer. how a spectrum of landforms can result from a common base-level forc- This emphasizes the importance of understanding the coupling between ing due to the nonlinear interplay among dip, stratigraphic bed thickness, backwearing rate and escarpment height in a given setting. and incision rate (cf. the “multiple modes of adjustment” described by For this analysis, I explicitly assumed that the retreat of the cap-rock Phillips, 2003). The analysis presented in Figure 4 shows what escarpment edge at the top of a cliff is controlled by erosion of the weaker rocks

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A E Step-change base-level forcing for transient model runs Episodic base-level forcing for transient model runs cumulative uplift (m) cumulative uplift (m) 1 1 m/yr) 400 m/yr) 300 -4 equili- -4 equilibration bration period 200 0.5 period 200 0.5 100

0 0 0 0 rock uplift rate (10 0 5 10 15 rock uplift rate (10 0 5 10 model time (m.y.) model time (m.y.) B F alcoving where bench t=10 m.y. t=5 m.y. drains H* < 1 H* > 1 over lower cliff

bench drains to transverse stream without crossing lower cap rock C t=13 m.y. incipient G H* > 1 alcove where t=6.5 m.y. bench drains H* > 1 over lower cliff

D t=14 m.y. H H* > 1 t=10 m.y. H* < 1

small escarpments propagate along bench and capture area that drains over the lower cliff

Figure 6. (A) Uplift history driving the two cap-rock model with a step change in uplift rates at 10 m.y. (B) During the low-uplift period, H* < 1, and a strike valley forms. (C) After the uplift rate increases, the bench must drain across resistant rock to reach a transverse stream, and it begins to be routed over the lower cliff. (D) Following a transition period, a compound cliff has formed (the short length of this cliff segment is due to the close spacing of the lateral boundaries in this model run). See also Movie SM1 in the Data Repository. (E) Uplift history driving the two cap-rock model with episodic uplift and a decrease in average uplift rate at ~6 m.y. (F–H) Snapshots from the numerical model run showing the transition from compound cliff to cuesta bench as mean erosion rate declines, with transverse escarpments migrating along the bench. See also Movies SM2 and SM3 in the Data Repository.

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beneath, and not the properties of the cap rock itself. The presence of threshold erosion rate below which a strike valley will emerge increases lithologically different cap rocks with different backwasting mechanics linearly with interlayer spacing and is proportional to the tangent over the might invalidate these assumptions or alter the form of the relationship cosine of the layer dip α. Therefore, the threshold incision rate decreases between height and backwearing rate. For example, on the Colorado rapidly with increasing dip over low dip angles and becomes less sensi- Plateau, the backwearing process of cliffs in massive Jurassic eolianites tive to dip at high dip angles. without exposed basal shales may be dominated by rock mass strength The approach to steady-state relief has an e-folding time scale that and fracture patterns, and not by basal undermining (e.g., Kimber et al., scales with 1/tan α, and this is possibly several million years given rea-

2002; Howard and Selby, 2009). sonable values for the height–retreat rate coupling coefficient c1. This An additional lithologic effect on the height–retreat rate coupling is due response time scale can be found as the ratio of escarpment height to to the presence of debris shed from the hard cap rock onto the foot slope. downcutting rate in a setting known to be in relief steady state. However, Depending on the longevity, size distribution, and thickness of this talus, given the range of dips and rock types found on the Colorado Plateau, it it may create a significant negative feedback on downcutting of the foot is expected that various cliff and cuesta landforms are in different states slope, and therefore on the retreat rate, due to channel-armoring (Bryan, of response to documented episodes of downcutting. 1940; Ward et al., 2011) and modification of hillslope sediment fluxes The most difficult term to constrain for a particular cliffband is the (Glade et al., 2017, 2019). As the talus may vary significantly depending coupling between escarpment height and backwearing rate. This coupling on the character of the cap rock from which it is shed, and its thickness and may vary widely depending on the mechanics of erosion and may not residence time may be independently modulated by climate and vegeta- everywhere (or anywhere) be linear in form. This strongly motivates a tion (Bull, 1991; Gutiérrez et al., 1998), it represents a significant factor physical approach, via understanding undermining mechanics and debris in the relationship between height and retreat rate. dynamics, as well as further chronometric work to determine rates of cliff retreat in response to documented base-level changes. Role of Transverse Streams ACKNOWLEDGMENTS The role of transverse streams in both the analytical treatment and This work was supported by U.S. Army Research Office grant 67195-EV-YIP. Chris Sheehan, Missy Eppes, Brian Yanites, and Bob Anderson are gratefully acknowledged for insightful numerical model is to provide a path of communication of base-level discussions. Critical comments by two anonymous reviewers greatly improved the presenta- change into the region between two exposed cap-rock layers. If the bench tion. Model code is available via the Community Surface Modeling Dynamics System (https:// or strike valley cannot respond quickly to a base-level change, then the csdms.colorado.edu). response of the upper cliffband is delayed as well, favoring loss of the bench to the retreating lower escarpment. It is important to the forma- REFERENCES CITED Berlin, M.M., and Anderson, R.S., 2009, Steepened channels upstream of knickpoints: Con- tion of a strike valley that the transverse stream is erosionally efficient trols on relict landscape response: Journal of Geophysical Research–Earth Surface, v. 114, relative to the other streams that drain the upper foot slopes. 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