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Keywords:

  • stream power;
  • fold kinematics;
  • Otago folds

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Simple Models of Fold Growth
  5. 3. Stream Power and Channel Erosion
  6. 4. Deformation Modeling
  7. 5. Rough Ridge
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Numerical Method
  11. Acknowledgments
  12. References
  13. Supporting Information

[1] As soon as a fold begins to grow above base level, it becomes subject to erosion. A numerical model of simultaneous fold growth and river erosion captures the early stages of channel formation and delineates the evolution of longitudinal channel profiles. This numerical exploration suggests that the patterns of channel incision into growing folds can be used to guide kinematic interpretations of fold growth. Four types of fold limb growth are modeled: limb lengthening through (1) the fold crest or (2) the fold toe, (3) limb rotation, and (4) curved fold limbs due to trishear folding. Initially, the pattern of channel incision is sensitive to the kinematics of surface deformation. Subsequently, more mature streams evolve toward similar incision patterns with deepest incision near the midpoint of the longitudinal channel profile. We compare these model predictions to Rough Ridge on New Zealand's South Island, where high-resolution topographic data reveal the multitude of channels incised into the flanks of these folds. Because the opposing limbs of these folds have followed distinct deformation pathways, the early sensitivity of channel incision to deformation pathway can be examined. Furthermore, the lateral propagation of the fold through time allows for an approximate space-for-time substitution, such that the evolution of these incision patterns can be examined from early through more mature stages. This analysis indicates that the west limb of Rough Ridge has grown by limb lengthening through the toe and the east limb has grown primarily by lengthening through the crest.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Simple Models of Fold Growth
  5. 3. Stream Power and Channel Erosion
  6. 4. Deformation Modeling
  7. 5. Rough Ridge
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Numerical Method
  11. Acknowledgments
  12. References
  13. Supporting Information

[2] Folded unconformities preserved either in growth strata or as fluvial terraces are commonly exploited as incremental strain markers to infer the kinematics of fold growth [Daëron et al., 2007; Hubert-Ferrari et al., 2007; Lavé and Avouac, 2000; Wilson et al., 2009], but such unconformities are not ubiquitously preserved, nor do they necessarily indicate a unique deformation pathway. Here we consider the incision pattern of consequent streams, those that form due to fold growth, as another indicator of fold growth history. Although the insights gained from the patterns of channel incision suffer from the same limitations of limited preservation and ambiguity, such insights are based on a different set of assumptions. For example, the assumption of nearly instantaneous abandonment of a wide channel cut surface to form a terrace is not required. Similarly, the assumption of isochrony for stratigraphic horizons used to delineate growth strata is unnecessary. River incision profiles can, therefore, provide an additional piece of information to answer questions of fold kinematics even in the presence of steady hydrologic and tectonic conditions.

2. Simple Models of Fold Growth

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Simple Models of Fold Growth
  5. 3. Stream Power and Channel Erosion
  6. 4. Deformation Modeling
  7. 5. Rough Ridge
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Numerical Method
  11. Acknowledgments
  12. References
  13. Supporting Information

[3] Imagine a tilted geomorphic surface that is 1 km long from crest to toe and has a constant dip of 10°. Some combination of three end-member kinematic histories (Figure 1) could produce such a structure: (1) while always maintaining a constant limb dip, the surface could have been laterally advected through a synclinal hinge at the toe of the fold; (2) the fold limb could have remained 1 km long for its entire history and rotated to achieve a dip of 10°, or (3) the surface could have been laterally advected through an anticlinal hinge at the fold crest while maintaining constant dip [Ahmadi et al., 2006; Wilson et al., 2009]. These geometric growth patterns can be associated with specific fault geometries that could lead to their formation (Figures 1a and 1b), namely, a fault bend fold [Suppe, 1983; Suppe et al., 1997] and a listric thrust fault [Amos et al., 2007; Erslev, 1986]. Given that other mechanisms such as detachment folding [Scharer et al., 2004] could produce similar surface deformation patterns, we consider pure lengthening and rotation of a fold limb, rather than explicitly including an underlying fault geometry. Real deformation processes likely involve some combination of rotation and lengthening, and they rarely result in perfectly planar surfaces. For this reason, we also consider a curved limb geometry (Figure 1c) that may form in trishear-style folding [Erslev, 1991; Hardy and Poblet, 2005; Zehnder and Allmendinger, 2000].

image

Figure 1. Schematic representations of end-member fold growth models. (a) Limb lengthening through the toe and crest illustrated here as the result of deformation above a fault bend fold. During early growth, hanging wall material moves through fold axes a and b which are stationary with respect to the footwall. (b) Limb rotation illustrated as deformation above a listric thrust. (c) Formation of a curved limb in the forelimb of a trishear fold.

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[4] We use a numerical model to investigate how these deformation pathways can affect the pattern of incision by stream channels that form on a deforming fold limb. In all cases, we assume that the fold is fully emergent, such that its limbs rise above an unchanging local base level and are not being mantled with growth strata. This simplification of real, and possibly complex, fluctuations in base level is likely appropriate to the modeled spatial and temporal scale because (1) nascent channels on newly emergent structures may form with little connection to an established regional fluvial network and (2) channel head migration is not initially driven by headward erosion of existing channels.

3. Stream Power and Channel Erosion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Simple Models of Fold Growth
  5. 3. Stream Power and Channel Erosion
  6. 4. Deformation Modeling
  7. 5. Rough Ridge
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Numerical Method
  11. Acknowledgments
  12. References
  13. Supporting Information

[5] Stream incision into a deforming surface can be modeled according to a stream power formulation [Howard and Kerby, 1983] modified from the work of Oskin and Burbank [2007]. Erosion rate (E) is calculated as

  • equation image

where K is a constant (Table 1). The upstream drainage area is used as a proxy for stream discharge and is defined as Xp where X is the downstream distance from the drainage divide. In applying this model to nearly straight channels formed on the flanks of folds, we assume that downstream distance is equivalent to the perpendicular distance from the divide. The value of p describes the rate of downstream drainage area accumulation which may range between 1 for “pipe-like” drainages and 2 for “box-like” drainages [Oskin and Burbank, 2007]. We set p to 1.1 to represent the nearly pipe-like drainages that often form parallel to one another on folds for which limb dips are >5°. The channel slope is S, and the exponents θ and n describe the potentially nonlinear increase of erosion rate with respect to drainage area and slope. The exponents are assumed to be 0.5 and 1 in this simple model. Although this formulation is a simplistic description of the myriad processes that can cause bedrock channel erosion [Whipple et al., 2000], predictions of this model regarding patterns of channel incision are not heavily dependent on the details of the erosion rule. In general, any erosion rule that describes erosion rate as an increasing function of slope and drainage area or discharge would yield broadly similar predictions.

Table 1. Summary of Model Parameters
ModelaGeometric and Model CharacteristicsTrishear Fold Characteristics
  • a

    Model numbers are 1, lengthen through toe; 2, lengthen through crest; 3, rotating limb; 4, trishear.

1,2Slope of lengthening limbs (10°)Trishear fault dip (45°)
3Final slope of rotating limb (10°)Trishear apical angle (40°)
1,2Rate of limb lengthening (3 mm/yr)Rate of slip and tip propagation (0.5 mm/yr)
1,2,3,4Modeled time (5 × 105 yr)Initial depth of trishear fault tip (650 m)
1,2,3,4Stream power constant K (3 × 10−5)Type of trishear deformation field (Linear)

4. Deformation Modeling

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Simple Models of Fold Growth
  5. 3. Stream Power and Channel Erosion
  6. 4. Deformation Modeling
  7. 5. Rough Ridge
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Numerical Method
  11. Acknowledgments
  12. References
  13. Supporting Information

[6] Each simulation tracks the position of two markers within the 2-D model. One marker traces the preerosion surface, and a second traces the profile of the incising channel. Initially, these markers are colocated in a straight horizontal line representing an initially unincised and undeformed surface. To model limb lengthening through the toe, the deformation velocity field v(x, y) contains a kink at x = a1 representing the toe of the fold:

  • equation image
  • equation image

The rate of limb lengthening is D, and limb dip is equation image. The vectors equation image and equation image are unit vectors in the horizontal and vertical directions, respectively (Figure 2a).

image

Figure 2. Setup of each of the four kinematic models of fold limb deformation. Displacements from the initial condition (gray lines) to the deformed state (black lines) after a time Δt are shown with vector arrows. The horizontal and vertical unit vectors equation image and equation image are also depicted extending from the origin.

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[7] Limb lengthening through the crest is described similarly:

  • equation image
  • equation image

Here a2 represents the position of the fold crest (Figure 2b). Both of the equation sets used to describe limb lengthening dictate the surfaces experience an abrupt change in direction, but not magnitude, of displacement through the kink.

[8] The velocity of a rotating fold limb (Figure 2c) is the same as that of any rigid body rotation. The tangentially directed velocity v of points at a given distance r from the center of a body rotating with a given angular velocity equation image can be simply described by

  • equation image

Recasting this description of rotation in Cartesian coordinates allows rotation of limb with length L to be described by

  • equation image
  • equation image

This description of pure limb rotation approximates the deformation pattern above a listric thrust, but does not capture the axial surface migration that occurs in such structures [Amos et al., 2007].

[9] The trishear deformation field is considerably more complicated and is subdivided into the rigid hanging wall, the zone of active trishear, and the rigid footwall. Within the zone of active trishear, particle velocities are greatest at the upper boundary with the hanging wall and decrease linearly toward the lower boundary with the footwall (Figure 2d). The mathematical descriptions of these velocity fields are published elsewhere [Hardy and Poblet, 2005; Zehnder and Allmendinger, 2000; Allmendinger, 1998]. Trishear model parameters and parameters for the planar limb models are included in Table 1. We solve for stream channel and terrace surface position through time using an explicit first-order finite difference scheme (Appendix A).

4.1. Predictions of Combined Fold and Erosion Models

[10] Longitudinal river profiles are often used as indicators of deformation [Kirby and Whipple, 2001; Stock et al., 2004; Wobus et al., 2006]. Here, the difference in elevation between the channel and the surface into which it is incised, an incision profile, is used instead. When simultaneous erosion and deformation are modeled, a characteristic incision profile emerges depending on the deformation pathway. If the surface is deformed via lengthening such that new material is added to the limb through the fold toe, the channel is initially most deeply incised near its midpoint (Figures 3a, 4a, and 5a). The downstream channel reaches have both a greater upstream catchment area and a steep initial slope in the newly incorporated lowest part of the limb that together lead to the fastest instantaneous incision rates, but the lower reaches have not been elevated above local base level for as much time as upstream reaches [Ahmadi et al., 2006]. This mode of channel evolution is mathematically identical to the progressive exposure of a tilted, but static, surface as a less resistant cover is progressively stripped away such that base level is continuously lowered [Oskin and Burbank, 2007].

image

Figure 3. (a-d) Results of numerical model with simultaneous surface (black lines) deformation and channel (gray lines) incision from early (T1) to later (T5) stages of development. Open circles indicate points of deepest incision. (e) Longitudinal profiles developed at T5 for each of the three models with planar fold limbs. Though successive surface profiles overlap in the limb lengthening scenarios (Figures 3a and 3b insets), they are separated in this illustration for visual clarity.

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image

Figure 4. Depth of incision calculated as the difference between the channel elevation and preerosion surface at each of the five stages of development shown in Figure 3.

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image

Figure 5. Modeled incision profiles plotted in terms of relative incision depth (incision depth/maximum incision depth at a given time) and relative downstream distance (distance from crest/limb length). Gray curve in Figure 5a represents instantaneous incision rate along the channel at the end of the model run.

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[11] In contrast, a channel that forms on a planar, but progressively rotating, surface is most deeply incised toward its lower end where both drainage area and stream power are greatest (Figures 3c, 4c, and 5c). Because erosion rate is calculated as an increasing function of slope and drainage area (equation (1)), the predicted incision pattern into a fold that is formed by advection of a previously horizontal surface into the fold limb through the crest is similar to that predicted for a rotating limb (Figures 3b, 4b, and 5b). Limb lengthening through the fold crest increases the drainage area contributing to channel erosion.

[12] If only the longitudinal river profiles are considered (Figure 3e), important differences among the fold growth scenarios remain. Limb rotation and lengthening through the crest produce river profiles that are more concave near the toe of the fold than the profile produced by limb lengthening through the toe. This contrast suggests that even in the absence of a preerosion surface, some information regarding fold growth history may be encoded in river profiles.

[13] In the case of a curved fold limb (Figures 3d, 4d, and 5d), the concave-up shape of the surface dictates that the surface slope is much greater near the fold crest than near the toe. As a result, incision is narrowly focused closer to the fold crest than in the previous three models.

4.2. Evolution of Incision Profiles Through Time

[14] Incision profiles represent the cumulative effect of channel erosion. As a consequence, the channel reach experiencing the highest instantaneous erosion rates may not be the most deeply incised reach if some other reach has experienced moderate erosion rates for a longer time. This concept is illustrated in the time evolution of channels on a fold limb that lengthens through the fold hinge at its toe (Figure 5a). Although the predicted instantaneous incision rate will always be fastest at the toe (because it has the greatest upstream catchment area), the point of deepest incision is located well upstream of the channel toe, because the hillslope just upstream of the toe has only recently begun to incise. The exact location of deepest incision varies slightly through time depending on the sensitivity of erosion rate to the increasing length of the channel as parameterized by the values of p and θ (equation (1)). Even though the shape of normalized incision profiles (Figure 5a) remains similar throughout model development, the channel does not reach a steady state in which incision and uplift are balanced.

[15] Deformation either by lengthening through the crest or by rotation results in channels that are initially most deeply incised near the toe of the fold (Figures 5b and 5c). Continued deformation does not expose any new increment of the deformed surface at the fold toe to erosion. Furthermore, the modest structural relief near the toe with respect to a fixed base level limits the depth of incision that is possible on the lower part of the fold limb. Thus, the point of deepest incision migrates upstream once the downstream reaches have incised down to their base level of erosion.

[16] In our realization of the trishear model, the fault tip approaches the surface and, therefore, leads to a progressively narrower zone of deformation that is focused near the midpoint of the fold limb (Figure 3d). Channel reaches near the toe of the fold never become deeply incised because both the surface slope is gentle relative to upstream reaches and relief is low. Because a multitude of folding patterns (including convex fold limbs) could be generated with different trishear model parameters, this result will not apply to every trishear fold.

[17] In general, the modeled channels do not reach a steady state in which incision rates ubiquitously balance vertical uplift rates. In the case of limb rotation, for example, the portion of the channel nearest the toe of the fold does approach such a balance, but the upstream reaches do not (Figure 6). The combination of relatively low uplift rates near the hinge of the rotating limb and large accumulation area allow the channel to incise at pace with uplift in its downstream reaches.

image

Figure 6. The approach to a steady state in the limb rotation simulation. In the lower half of the channel (solid lines), the average incision rate (gray solid line) has nearly reached the average uplift rate (black solid line). In the upper half of the channel (dotted lines), the average incision rate (gray dotted line) remains far below the average uplift rate (black dotted line).

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[18] The differences among incision profiles produced in this model (Figure 5) indicate that under the right conditions, stream incision patterns may be used to constrain fold kinematics. In order for this method to be applied, we assume that instantaneous erosion rates are a simple increasing function of slope and drainage area. Also, remnants of the preerosion surface must allow for its reconstruction across incised drainages. Natural examples of folds that retain a well-defined preerosion surface include the Otago folds of New Zealand [Jackson et al., 1996], Wheeler Ridge in California [Burbank et al., 1996; Keller et al., 1999; Mueller and Suppe, 1997], and the Kyrgyz and Gory Baybeiche Ranges in central Asia's Tien Shan [Bullen, 1999; Oskin and Burbank, 2007]. Each of these sites could provide a natural laboratory for testing the predictions of our incision profile model; we focus on one of the Otago folds.

5. Rough Ridge

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Simple Models of Fold Growth
  5. 3. Stream Power and Channel Erosion
  6. 4. Deformation Modeling
  7. 5. Rough Ridge
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Numerical Method
  11. Acknowledgments
  12. References
  13. Supporting Information

[19] Rough Ridge on New Zealand's South Island is a northward plunging anticline formed over a blind, west dipping thrust fault [Jackson et al., 1996]. At least three smaller-scale, nearly parallel folds merge to form the topographic expression of Rough Ridge (Figure 7). These smaller constituent folds are North Rough Ridge, Rough Ridge, and South Rough Ridge. Previous studies [Bennett et al., 2006; Craw et al., 2007; Jackson et al., 2002; Yeats, 1987; Youngson et al., 2005] have helped to determine the relative and absolute timing of fold growth, as well as the structures that underlie the growing folds. The relatively resistant Otago schist has preserved the form of the anticline. In contrast, a thin Tertiary sedimentary sequence has generally been stripped from the folds since their Pliocene, or later, initiation [Bennett et al., 2006; Hutton et al., 1875; Youngson et al., 1998].

image

Figure 7. (a) Oblique view toward the northwest of Rough Ridge from TOPSAR DEM. (b) Simplified map of the structures that form Rough Ridge on New Zealand's South Island. The shaded relief represents our reconstruction of the preerosion surface of Rough Ridge. Dashed boxes designate regions covered by topographic profiles in Figure 10.

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[20] Many subparallel channels etch both the eastern and western flanks of the fold. On the commonly preserved wide interfluves, the presence of numerous 10 m high tors suggests uniform lowering of the fold surface by slow erosion (∼0.01 mm/yr) over the past 1 My [Bennett et al., 2006]. The geometry of folding is, therefore, well represented by the undissected surface.

[21] The depth of incision into the opposing limbs of these folds (Figure 8) is measured from a TOPSAR digital elevation model (DEM) with 25 m resolution. First, a preerosion surface is reconstructed over the fold. To define this surface, the DEM is inverted, and hydrologic flow-routing algorithms are used to identify the inverted ridges. A preerosion surface is then interpolated through the highest points of the interfluves (Figure 7). This topographic envelope reveals that the western limb of Rough Ridge dips at 8 degrees. The northeast and southeast limbs dip at 12 and 14 degrees, respectively. Although the height of the North Rough Ridge and Rough Ridge fold crests both grow by 300 m along the first 10 km of their length, the crestal elevation of North Rough Ridge is approximately constant beyond 10 km. This reconstructed surface forms the reference frame against which incision is measured.

image

Figure 8. Map of incision depth below the hypothetical preerosion surface. Black dots indicate which channels are plotted in Figure 9.

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[22] Unlike channel patterns on the ridge flanks, deeply incised channels do not dissect the relatively undeformed plains surrounding Rough Ridge. The absence of a deeply incised regional fluvial network suggests that the incised streams draining the flanks of Rough Ridge have formed in response to local uplift rather than a drop in regional base level. This interpretation is consistent with the model assumption of fixed base level at the fold toe. Based on the incision map, however, regional aggradation and rise in base level at a rate slower than the growth of Rough Ridge cannot be dismissed.

[23] The position of deepest incision along 42 minor channels on the west limb does not vary systematically along the length of the fold and is localized near the midpoint of the fold limb on average (Figure 9). The study area on the eastern limb encompasses three north plunging fold tips. The drainages etched into the most southerly fold tip have headwaters that extend well above the crest of the fold and onto the larger structure of Rough Ridge (Figure 8). Because these channels are likely antecedent to the growth of the third fold tip, we do not consider them in this analysis. In contrast, most of the drainages on the two northern fold tips end at the fold crest, are roughly parallel, and have few tributary junctions. Near the northern tip of each fold, streams are most deeply incised near the base of the fold limb. The point of deepest incision moves systematically upstream in channels situated farther south of each fold tip (Figure 9).

image

Figure 9. The relative downstream position of deepest incision for each of the channels indicated in Figure 8. Channels from the west and east limb are shown in the left and right plots, respectively, and are arranged according to the distance of the channel away from the northern tip of north Rough Ridge. Channels on the eastern limbs are separated according to the fold on which they are formed. Gray lines show the best linear fit through each subset of the data.

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[24] Exposure ages measured through the accumulation of in situ cosmogenic nuclides along the northern tip of South Rough Ridge indicate that this fold tip has grown by northward propagation during Quaternary times at a rate of 1–2 mm/yr [Bennett et al., 2006; Jackson et al., 2002]. Deflected antecedent drainages provide additional evidence for the northward propagation of the folds [Jackson et al., 1996]. This growth pattern permits the observed spatial pattern of channel incision along the length of the fold to be recast in terms of channel evolution through time. Jackson et al. [2002] argue that lateral propagation of the South Rough Ridge was episodic rather than constant. Furthermore, if lateral propagation of the fold had been steady and continuous, the height of the fold crest would likely grow steadily along strike [Childs et al., 2003; Manighetti et al., 2001]. Instead, elevation profiles across the preerosion surface (Figure 10) indicate that crest height of North Rough Ridge initially increases south of its northern tip and then remains nearly constant. Thus, we do not expect that distance from the fold tip is strictly proportional to the age of initial uplift, only that channels formed at greater distances from the fold tip are relatively older and more mature than those near the fold tip.

image

Figure 10. Topographic profiles measured across the preerosion surface of three Rough Ridge fold limbs. See Figure 7 for locations. In all cases, limb lengthening is responsible for most fold growth. The two profiles nearest to the fold tip on the northeast and northwest limbs indicate some limb rotation in addition to lengthening.

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5.1. Modeling Regional Aggradation and Deformation

[25] Up to this point, the modeling of coupled erosion and deformation has assumed a base level fixed at the toe of the fold. Nonetheless, the Rough Ridge anticline could have formed in an aggradational setting such that the toes of the fold limbs were steadily covered by sediment. Through time, the structure would still grow above the surrounding plains as long as the rate of tectonic uplift exceeded the rate of aggradation. The three models of planar limb growth through lengthening and rotation show little sensitivity to aggradation at a rate equivalent to half of the relative uplift rate (Figures 11a11c). This insensitivity is hardly surprising given that both the planar shape of the limbs and underlying kinematics remain unchanged.

image

Figure 11. (a-d) Comparisons between original models of planar limb growth (gray curves) and models with aggradation at 50% of the tectonic uplift rate (black curves). (e) Surface and channel profiles developed on the trishear forelimb model with 50% aggradation.

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[26] The trishear forelimb is characterized by a long, gently sloping toe that steepens near the crest of the fold. With aggradation at half of the uplift rate, the gentle toe of the trishear forelimb is buried so that only the steeper, updip portion of the fold is exposed to channel incision (Figure 11e). Through time, the exposed portion of the forelimb steepens so that deformation is approximately rotational. Indeed, the channel incision pattern is marked by upstream migration of the point of deepest incision similar to limb rotation models (Figure 11d). This upstream migration stands in sharp contrast to the downlimb migration of the zone of deepest incision that is predicted in the absence of aggradation (Figure 5d).

5.2. Incision Profiles With Different Model Parameters

[27] We use the Rough Ridge DEM to measure one of the stream power parameters (equation (1)). The value of p describes the increase in drainage area (A) with along-stream distance from the divide, where

  • equation image

[28] On a logarithmic scale, a plot of drainage area against distance from the drainage divide (Figure 12) indicates that the value of 1.1 used in our model accurately describes the rate of drainage accumulation in the upper 1 km of the channels. When data from the entire length of the fold limbs are considered, 1.3 is a more appropriate value. An increase in p enhances erosion of the downstream reaches of modeled channels, because larger p predicts more rapid growth in upstream catchment area and, therefore, in discharge per distance downstream (Figure 13). This more focused erosion at the toe of the fold leads to more rapid upstream migration of the point of deepest incision (Figures 13b and 13c) in the models of rotation and lengthening through the crest. Similarly, the downstream migration of deepest incision that is observed in the trishear model is enhanced (Figure 12d).

image

Figure 12. DEM-derived drainage area and streamwise distance from the drainage divide for each of the channels in Figure 8. Measurements from the upper 1 km of channels (light gray points) are fit by a power law with exponent of 1.1. Measurements from the entire length of the limbs (dark points) are better fit with an exponent of 1.3.

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image

Figure 13. Comparison of relative incision depth between model runs with p = 1.1 and p = 1.3 (black curves). Dashed curves show early (light gray) and late (dark gray) incision profiles from Figure 5. Increasing the sensitivity of erosion rate to downstream distance by increasing p leads to greater migration of the point of deepest incision (Figures 3b3d). Dashed arrow highlights the point of deepest incision in original model run. Compare with solid arrow that indicates the analogous point when p = 1.3.

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[29] Increasing p without adjusting any other model parameter leads to faster erosion rates and deeper incision. In order to examine the sensitivity of incision profiles to changes in n (the exponent on both slope and area: equation (1)), both n and the proportionality constant K are varied so that the final depth of incision remains near 50 m in all model runs (Figure 14). Because the sensitivity of incision rate to slope is governed by n and because limb rotation produces a range of surface slopes without changing drainage area, the limb rotation model is used to explore this aspect of model sensitivity. Although changes in the shape of predicted longitudinal profiles occur as n is varied, the upstream migration of deepest incision occurs with little variation in all models.

image

Figure 14. (a-c) Comparison of normalized incision profiles developed during limb rotation with varying values of K and n. K is varied so that the final depth of incision is approximately 50 m in all models. (d) Comparison of longitudinal profiles developed in each model.

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[30] Longitudinal river profiles developed with different deformation styles, but with identical stream incision parameters (Figure 3e), clearly differ from each other. Nonetheless, the form of longitudinal profiles developed with identical deformation, but with different incision parameters (Figure 14d), is almost as variable as profiles generated with different deformation styles. Despite the dependence of longitudinal profile shape on incision parameters, these parameters are unlikely to differ among channels lying near one another unless underlying bedrock characteristics are changing, too. In general, a change in model parameters that increases incision rates at the toe of the fold leads to more rapid upstream migration of the point of deepest incision in models of rotation or lengthening through the crest.

6. Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Simple Models of Fold Growth
  5. 3. Stream Power and Channel Erosion
  6. 4. Deformation Modeling
  7. 5. Rough Ridge
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Numerical Method
  11. Acknowledgments
  12. References
  13. Supporting Information

[31] Slip on a buried fault is interpreted to be responsible for the formation of the Rough Ridge anticline [Jackson et al., 1996]. Can the observed pattern of channel incision provide additional information on the kinematics of fold growth?

[32] Substituting the duration of fold growth for distance from the northern tip of the Rough Ridge anticline allows for comparison between our model predictions of channel incision patterns through time (Figures 3, 4, and 5) and the incision patterns observed along the ridge (Figure 9). On the western flank of Rough Ridge, this ergodic substitution indicates that the point of deepest incision has remained near the channel midpoint throughout fold evolution (Figure 9). Thus, south of the fold tip and irrespective of age and position, the channels incising into the west limb of Rough Ridge indicate deformation via limb lengthening. This pattern alone cannot, however, conclusively determine the fold growth history, because the incision pattern predicted for such an actively growing fold limb is indistinguishable from the incision pattern that would form on a static, previously tilted horizon from which a less resistant cover sequence was progressively stripped away as base level was lowered [Ramsey et al., 2008]. Four observations argue against such a stripping mechanism that would lead to sequential base level lowering. First, the thickness of Tertiary sediments (a few tens of meters [Youngson et al., 1998]) deposited across this region is far smaller than the height of the fold limb (∼500 m). Second, the presence of tors of similar height across the upper flanks and crest of the fold suggests an equal duration of exposure, as opposed to progressive stripping and exposure. Third, cosmogenic exposure ages along the fold flanks show no systematic younging toward lower parts of fold limbs [Youngson et al., 2005], as would be expected if cover rocks were progressively removed. Fourth, if the observed incision patterns are due to this stripping mechanism rather than active deformation, we expect that the east limb would exhibit the same incision pattern, but it does not.

[33] The incision patterns on two fold tips of the eastern flank indicate that initial channel incision is greatest near the fold toe, but migrates upstream with time as the fold matures, as shown by the progression of channels from north to south (Figure 9). This observation is consistent with two of the modeled end-member deformation pathways: either limb lengthening through the fold crest or limb rotation. Elevation profiles across the preerosion surface (Figure 10) address this ambiguity. Successive, equally spaced profiles from each of the three fold limbs are used to represent the evolution of the fold through time. These profiles indicate that all of the limbs have grown primarily through limb lengthening, although the tip zone along the NE limb shows significant rotational steepening during early stages of growth. Neither topographic profiles nor channel incision patterns alone can constrain Rough Ridge's pattern of growth. Taken together the two lines of evidence show that the western limb has grown primarily by lengthening through the toe, and the eastern limbs have grown primarily by lengthening through the fold crest with a secondary amount of limb rotation.

[34] In summary, deepest channel incision that remains near the channel midpoint indicates either limb lengthening through the toe or top-down stripping of a cover sequence that causes sequential base level lowering. This ambiguity may be resolved by comparison between opposing fold limbs, where base level lowering by stripping should produce a symmetrical pattern. Deepest incision near the fold toe may indicate either limb lengthening through the crest or limb rotation. Examination of topographic profiles across the reconstructed preerosion surface of the fold (Figure 10) can then be used to determine whether lengthening or rotation dominates. Strongly concave fold limbs, such as those predicted from trishear models (Figures 1c and 3d), would also be apparent in topographic profiles. Because the surface slope increases near the fold crest of concave limbs, channel reaches near the fold crest typically are most deeply incised, despite having relatively small catchment areas.

[35] To be consistent with these findings, a description of the kinematic history of Rough Ridge must include lengthening of the west limb throughout fold growth, early steepening of the east limbs, and later growth of the east limbs by lengthening through the crest. Additionally, drainage asymmetry and patterns of defeated drainages surrounding Rough Ridge indicate that North Rough Ridge formed first, followed by Rough Ridge, and then South Rough Ridge [Jackson et al., 1996]. These geomorphic constraints suggest the following kinematic history. First, North Rough Ridge began to form above a blind west dipping reverse fault (Figure 15). During this earliest stage of growth, the west limb would have lengthened through the toe and remained nearly planar. The east limb may have steepened via trishear-style propagation of the fault tip toward the surface (Figure 15a). Given the 8° dip of the west limb, this initial period of fold growth was likely accompanied by at least an 8° increase in the dip of the fault at depth [Chen et al., 2007]. The transition from early steepening of the east limb to lengthening through the toe may be related to a decrease in the dip of the fault as it approached the surface (Figure 15b). Continued slip on this fault would result in lengthening of the west limb through the toe and lengthening of the east limb through the crest. Later growth of the same fault system toward the surface could cause similar steepening and later lengthening of the eastern limb of the younger Rough Ridge anticline.

image

Figure 15. One combination of fault dips and fault propagation patterns that is compatible with the fold growth history derived from channel incision modeling. The software program 2DMove was used to produce this diagram only.

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7. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Simple Models of Fold Growth
  5. 3. Stream Power and Channel Erosion
  6. 4. Deformation Modeling
  7. 5. Rough Ridge
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Numerical Method
  11. Acknowledgments
  12. References
  13. Supporting Information

[36] Although both growth strata and deformed geomorphic surfaces can define a history of fold growth and underpin analyses of fold kinematics, such data are commonly unavailable. In contrast, almost all emergent folds are ornamented with minor channels that have etched their limbs during fold growth. Here we show that the pattern of incision by these channels is sensitive to the kinematics of fold growth. Fold growth by limb lengthening through the fold toe leads to the formation of channels that are most deeply incised near their midpoint: a pattern that shows little change during continued fold growth. In contrast, either limb lengthening through the crest or limb rotation leads to the formation of channels that are initially most deeply incised near their downstream end. During subsequent fold growth, this point of deepest incision migrates systematically upstream. Thus, on nascent or propagating folds, a spatial pattern in the position of deepest incision on the fold flanks can be used to deduce the kinematic style of deformation.

[37] Channel incision patterns must be measured relative to a preerosion surface. Constructing such a surface through remnant topography introduces additional uncertainty into measurements of channel incision. Furthermore, descriptions of channel incision that are governed only by drainage area and channel slope may be too simplistic in some regions or at larger spatial scales where changes in sediment load [Johnson et al., 2009; Sklar and Dietrich, 1998, 2004; Yanites and Tucker, 2010] and precipitation and snowmelt patterns [Bookhagen and Burbank, 2010] could dominate.

[38] The incision patterns of nearly 80 channels into the flanks of Rough Ridge indicate that channels formed during fold growth, rather than incising into a static, previously folded surface as a cover sequence was stripped away. A combination of channel incision patterns with profiles of fold topography suggests that the western limb grew by lengthening through the fold toe, whereas the eastern limb grew primarily via initial rotation followed by lengthening through the fold crest.

Appendix A:: Numerical Method

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Simple Models of Fold Growth
  5. 3. Stream Power and Channel Erosion
  6. 4. Deformation Modeling
  7. 5. Rough Ridge
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Numerical Method
  11. Acknowledgments
  12. References
  13. Supporting Information

[39] The four coupled erosion and deformation models are solved numerically by dividing both the terrace and channel marker lines into 4 m long cells, and calculating erosion and deformation rates at 50 year intervals throughout model development. This computation routine performs the following operations at each time step: (1) calculate the tectonic displacements of each cell of the channel and surface markers according to the model velocity description (equations (2)(4)); (2) apply the vertical and horizontal displacements to both the channel and surface cells; (3) calculate the slope of the newly displaced channel based on the difference in position between a given cell and the cell immediately downstream; (4) calculate the number of upstream cells that contribute to each channel cell (We do not consider the possible contribution of cells situated beyond the upper limit of the tilted fold limb.); (5) calculate the erosion rate at each cell (equation (1)) based on the local slope and distance downstream from steps 3 and 4; (6) apply the calculated erosion vertically to the channel cells.

[40] The displaced and eroded channel, as well as the displaced surface, is then used in the next loop of calculations. The horizontal channel reach downstream of the fold toe defines the local base level. In order to insure that the channel cell situated immediately downstream of the fold toe does not incise below base level, this calculation scheme does not allow the channel slope upstream of a given cell to contribute to the erosion rate calculation (step 3).

[41] The spatial and temporal resolution of the model allows for efficient computation on a personal computer. In order to test the numerical robustness of this procedure, we compute a series of model realizations in which the spatial and temporal step size is decreased by a factor of 2 and 4. Incision depth, the model result of primary interest, changes by <2% and <3% when doubling and quadrupling model resolution.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Simple Models of Fold Growth
  5. 3. Stream Power and Channel Erosion
  6. 4. Deformation Modeling
  7. 5. Rough Ridge
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Numerical Method
  11. Acknowledgments
  12. References
  13. Supporting Information

[42] This work benefited greatly from discussion with Bodo Bookhagen. MATLAB code from geomorphtools.org facilitated analysis of Rough Ridge streams. We also thank George Hilley and an anonymous reviewer whose thoughtful reviews led to substantial improvements. We wish to acknowledge NASA for providing TOPSAR data.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Simple Models of Fold Growth
  5. 3. Stream Power and Channel Erosion
  6. 4. Deformation Modeling
  7. 5. Rough Ridge
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Numerical Method
  11. Acknowledgments
  12. References
  13. Supporting Information

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Simple Models of Fold Growth
  5. 3. Stream Power and Channel Erosion
  6. 4. Deformation Modeling
  7. 5. Rough Ridge
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Numerical Method
  11. Acknowledgments
  12. References
  13. Supporting Information
FilenameFormatSizeDescription
jgrb16584-sup-0001-t01.txtplain text document1KTab-delimited Table 1.

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