Interpretation and downstream correlation of bedrock river terrace treads created from propagating knickpoints

Authors

  • N. J. Finnegan

    Corresponding author
    1. Earth and Planetary Sciences Department, University of California, Santa Cruz, California, USA
    • Corresponding author: N. J. Finnegan, Earth and Planetary Sciences Department, University of California, Earth and Marine Science Building, Santa Cruz, CA 95064, USA. (nfinnega@ucsc.edu)

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Abstract

[1] Derivation of an analytical model for the slope of a strath terrace created following an upstream propagating wave of incision reveals that for detachment-limited river incision the exponent on channel slope, n, governs terrace tread slope. Terrace elevations can increase upstream (n > 1), downstream (n < 1), or can be horizontal (n = 1). Numerical modeling confirms these results for temporally evolving knickpoint geometries and for incision due to sudden base level fall and an increase in rock uplift rate. Except in the case of bedrock river incision with a slope threshold only exceeded during knickpoint propagation, a terrace created from transient headward incision contains no information about the paleo-channel gradient. Gradients in rock uplift rate along channels potentially complicate the interpretation of terraces by altering the primary tilt on strath terraces. In particular, monotonic gradients in rock uplift can produce terrace treads that are apparently folded. Because gradients in rock uplift rate are common, where terraces are not longitudinally traceable, care is warranted in terrace correlation. In simple tectonic settings where terraces are longitudinally traceable, the slope of a strath terrace created from headward incision may provide a means of estimating the dependency of river incision rate on channel slope. Terraces in four locations argued to have formed from headward incision are parallel or close to parallel with the active channel, implying a slope exponent on river incision rate that is much greater than one or a threshold slope for incision that is only exceeded during knickpoint propagation.

1 Introduction and Motivation

[2] Bedrock strath terraces, abandoned river-worn bedrock surfaces veneered in gravel, provide the only means of reconstructing an incising bedrock river's position in time and space. Consequently, where terraces can be dated and/or correlated in space to reconstruct a long profile, they provide constraints on rock deformation [e.g., Lavé and Avouac, 2000; Molnar et al., 1994; Pazzaglia and Gardner, 1994; Rockwell et al., 1984; Shyu et al., 2006; Simoes et al., 2007], longitudinal patterns of bedrock incision [e.g., Lavé and Avouac, 2001; Pazzaglia and Brandon, 2001; Yanites et al., 2010b] and changes in river longitudinal profile slope with time [e.g., Merritts et al., 1994]. Terraces are therefore essential to tectonic geomorphology.

[3] However correlation of terraces in space to reconstruct a long profile is commonly a subjective exercise if terraces are not longitudinally continuous along a river [Merritts et al., 1994; Seidl and Dietrich, 1992]. To the extent that these correlations are uncertain, calculations of incision and/or deformation patterns based on reconstructed profiles are similarly uncertain. Ideally one would work only from dated terrace surfaces, but even then the possibility that terrace levels are time-transgressive [e.g., Weldon, 1986; Zaprowski et al., 2001] means that a model for downstream terrace age and gradient is still needed to connect dated sparse terrace fragments to reconstruct a profile. If strong rock uplift gradients are present along a river, correlation also requires structural information [e.g., Molnar et al., 1994].

[4] Mapping and dating of terrace levels reveals that in some settings bedrock terrace formation occurs synchronously in space and that terrace treads have close to the same slope as the active river channel [e.g., Personius et al., 1993; Wegmann and Pazzaglia, 2002]. Additionally, both historical and geological observations of rapid basin-wide changes in river planform [e.g., Eschner et al., 1983; Galster et al., 2008; Harvey, 2001; Liébault and Piégay, 2001; Meyer, 1995] and alluvial cover [e.g., Clark and Wilcock, 2000] lend support to the view that terrace creation can occur essentially simultaneously throughout a basin [Hancock and Anderson, 2002].

[5] On the other hand, observations and/or interpretations of terrace formation driven by upstream propagating waves of incision [e.g., Brocard et al., 2003; Crosby and Whipple, 2006; Gran et al., 2011; Jansen et al., 2011; Ward et al., 2005; Weldon, 1986; Yanites et al., 2010a; Zaprowski et al., 2001] require that terrace formation is time-transgressive in many other settings (Figure 1a). Pulses of incision can be triggered by relative sea level fall [e.g., Castillo et al., 2013; Pazzaglia and Gardner, 1993], sudden rock uplift [e.g., Cook et al., 2012; Yanites et al., 2010a], incision of channels downstream [e.g., Zaprowski et al., 2001], an increase in rock uplift rate [e.g., Whipple and Tucker, 1999; Whittaker et al., 2007] or an increase in erosional efficiency [e.g., Bull, 1990; Whipple et al., 1999].

Figure 1.

(a) Schematic illustration of the process of strath terrace creation from upstream knickpoint propagation. (b) Derivation sketch for calculation of knickpoint trajectory in a vertically incising channel.

[6] Paired terrace formation can occur in association with such transient incision if the wave of incision is accompanied by changes in either planform geometry [e.g., Yanites et al., 2010a] or channel width [e.g., Whittaker et al., 2007] as it sweeps upstream. Alternatively, unpaired terrace formation can occur along the inside of active bedrock meanders if the migrating knickpoint increases the vertical incision rate relative to the lateral migration rate [Finnegan and Dietrich, 2011]. Although longitudinally continuous paired terraces are likely only to persist until the next episode of strath planation [e.g., Yanites et al., 2010a], terrace fragments from multiple paired terrace formation events are commonly observed in strath forming rivers because of incomplete erosional removal of previous terraces [e.g., Molnar et al., 1994]. In contrast, because unpaired terraces form on the inside of growing bedrock meander bends [e.g., Finnegan and Dietrich, 2011; Personius, 1993], they are less likely to be destroyed by subsequent terrace formation events. Thus regardless of the specific processes of terrace formation, which are beyond the scope of this paper, terraces are commonly preserved in settings with transient incision.

[7] Unfortunately, in most cases it is difficult to determine the mechanism(s) responsible for terrace formation based on observables in the field. Nonetheless knowledge of whether a terrace is time-transgressive is essential before attempting to correlate terraces along a river. Here my goal is to explore what governs the slope of a strath terrace in the situation described above in which terrace abandonment is triggered by a wave of incision that propagates through a catchment (Figure 1a). The aim here is to provide a physically based framework for interpreting, dating and correlating fluvial strath terraces in the field. Below a simple analytical model is derived for the slope of a strath terrace formed following an upstream propagating wave of incision. This analytical model reveals that the dependency of the river incision process on channel slope ultimately governs the slope of the terrace tread. More complex scenarios are then explored with a numerical model to determine the general applicability of the analytical approach. Finally, there is a discussion of the implications of these results for interpreting and correlating terraces in the field as well for constraining the slope dependency of river incision into bedrock.

[8] The goal here is to explore how different assumptions about knickpoint propagation influence strath terrace slope. Consequently, this analysis is predicated on the assumption that the details of the knickpoint retreat processes, which are not treated mechanistically here, are ultimately not necessary to reveal the consequences of knickpoint retreat on strath terrace slope at the watershed scale.

2 Analytical Derivation of Terrace Tread Slope

[9] If a terrace is created as a knickpoint propagates upstream (Figure 1a), then the terrace tread slope is defined by the trajectory of the knickpoint crest as it moves upstream [Culling, 1957] (Figure 1b). If the channel only incises via knickpoint propagation (in other words, the incision rate is zero in the upstream and downstream sections of the channel), then the knickpoint will propagate upstream along the slope of the channel (Figure 1b) [Culling, 1957]. In this case, the trajectory of the knickpoint in space will exactly mimic the channel slope and the resulting terrace will have exactly the same slope as the channel [Culling, 1957]. However, in the less restrictive case in which incision can occur above and below the knickpoint, as well as at the knickpoint, it is straightforward to see that the slope of the knickpoint trajectory in space will be less than the slope of the channel [Culling, 1957] (Figure 1b). This is because incision of the entire channel network (i.e., not just where there are knickpoints) occurs as the knickpoint propagates upstream. Consequently, a knickpoint is at a lower elevation in the upstream reaches of a channel than it otherwise would if the channel were not lowering [Culling, 1957; Finnegan and Dietrich, 2011]. The trajectory of the knickpoint crest in space is thus a function of the propagation rate of the knickpoint and the incision rate of the channel upstream of the knickpoint (Figure 1b). Specifically, the geometric relationships in Figure 1b imply that terrace tread slope, St, is given by:

display math(1)

where Sr is the slope of the channel upstream of the knickpoint, I is the vertical incision rate upstream of the knickpoint and C is the upstream propagation rate of the knickpoint (Figure 1b). Equation (1) is derived from geometry and alone requires no assumptions about specific processes of incision. To arrive at a prediction for terrace tread slope, below the stream-power erosion model is used to make substitutions for I and C in equation (1). However, it is worth emphasizing that equation (1) can accommodate other vertical incision and knickpoint retreat models.

[10] To model the process of terrace creation from upstream propagating incision, here the stream-power bedrock incision model is used [Howard, 1994; Howard and Kerby, 1983; Whipple and Tucker, 1999]. The stream-power model provides a convenient and stable model for exploring different styles of knickpoint propagation in detachment-limited settings [e.g., Weissel and Seidl, 1998]. A common criticism of the stream-power model is its lack of an explicit treatment of coarse sediment transport and deposition [e.g., Johnson et al., 2009; Lague, 2010; Sklar and Dietrich, 2001; Turowski et al., 2007; Yanites and Tucker, 2010]. Although this is a potential limitation of the approach here, it is worth noting that an incision model that directly accounts for coarse sediment abrasion, transport and deposition [Sklar and Dietrich, 2004] has instabilities that can, under some circumstances, prevent altogether the occurrence of upstream propagating incision waves [Crosby et al., 2007; Wobus et al., 2006]. Nevertheless, in many settings knickpoint propagation is observed, and at the watershed scale knickpoint propagation speed is correlated with drainage area [Berlin and Anderson, 2007; Bishop et al., 2005; Castillo et al., 2013; Crosby and Whipple, 2006], suggesting that a stream-power model can be used to predict knickpoint propagation into a catchment [Berlin and Anderson, 2007; Crosby and Whipple, 2006]. That said, factors such as bedrock structure [Berlin and Anderson, 2009], the existence of thresholds for river incision [Crosby and Whipple, 2006], and spatial accelerations in river flow velocity [e.g., Haviv et al., 2006] prevent the stream-power approach from reproducing knickpoint position and form in detail.

[11] The stream-power model states that the rate of vertical incision into bedrock is given by:

display math(2)

[12] In equation (1), z is channel elevation, A is drainage area, S is channel slope, and k determines erosional efficiency and depends primarily on climate, sediment supply, rock erodibility, and channel width [Whipple and Tucker, 1999]. m and n in equation (2) dictate channel concavity and are argued to reflect the process of river incision [Whipple and Tucker, 1999] as well as discharge variability [DiBiase and Whipple, 2011; Lague et al., 2005] and channel geometry [Finnegan et al., 2005; Lague et al., 2005]. Knickpoint propagation speed, in turn, can be derived from geometry [e.g., Lavé and Avouac, 2001]. Specifically, the headward retreat rate of an eroding channel segment is given by the incision rate divided by the local channel slope. Substitution of the stream power model for vertical incision rate yields the solution for knickpoint propagation rate for a detachment-limited bedrock channel [Rosenbloom and Anderson, 1994]:

display math(3)

[13] In equation (3), x is channel distance upstream. Otherwise, the terms in equation (3) are identical to those in equation (1). Substitution of equations (2) and (3) for I and C in equation (1) implies that for the case of a single knickpoint propagating through a detachment-limited channel, the slope of the terrace tread (St) is given by:

display math(4)

which after rearranging yields:

display math(5)

where Sk is knickpoint slope and Sr is river channel slope upstream of the knickpoint (Figure 2). Equation (5) thus represents a straightforward expression for predicting the slope of a strath terrace tread created from an upstream propagating incision wave in a detachment limited channel. From equation (5) it is apparent that the exponent on channel slope in the stream-power model, n, dictates the slope of the terrace tread formed in response to a propagating knickpoint in the absence of a spatial rock uplift gradient. Specifically, the slope of a terrace resulting from a propagating knickpoint will be zero if the process of river incision is linear in slope, n =1. In other words, the terrace will be horizontal. For n > 1, terraces will lose elevation downstream. As n approaches infinity, the terrace slope approaches the slope of the active channel. Thus, terraces in the model can never slope downstream at a steeper gradient than the channel slope upstream of the knickpoint, provided there are no spatial gradients in rock uplift rate. If n < 1, terrace treads will climb in elevation downstream. It is worth noting that analytical solutions to the stream power equation (L. Royden and J. T. Perron, Solutions of the stream power equation and application to the evolution of river longitudinal profiles, submitted to Journal of Geophysical Research, 2013) can be rearranged to yield the same basic result as equation (5).

Figure 2.

Derivation sketch for calculation of knickpoint trajectory via the stream-power model.

[14] It is also apparent from equation (3) that when n is unequal to one, the knickpoint propagation rate also depends on slope. As noted by Weissel and Seidl [1998], this means that knickpoint evolution is highly sensitive to initial conditions. Specifically, when n > 1, steeper sections of knickpoints will propagate faster than less steep sections. For a sigmoidal knickpoint, this results in steepening at the top of the knickpoint as steeper sections overtake the rounded lip of the knickpoint. Additionally, for n > 1, relaxation of the slope at the bottom of the knickpoint occurs because the steeper middle section of the knickpoint propagates faster than the toe [Weissel and Seidl, 1998]. Alternatively, if n < 1, less steep sections of a knickpoint will overtake steeper sections. Thus steepening occurs at the base of the knickpoint and slope relaxation occurs at the top of the knickpoint, again assuming a sigmoidal initial condition [Weissel and Seidl, 1998].

3 Numerical Derivation of Terrace Tread Slope

[15] The analytical results above are only valid for a knickpoint with constant slope. However, as discussed above, knickpoints commonly evolve to have nonuniform slopes [Berlin and Anderson, 2009; Haviv et al., 2006; Weissel and Seidl, 1998], suggesting that a model in which knickpoint slopes can evolve in time would allow for a more complete treatment of the problem of strath terrace riser slope. Thus, the governing equation for river elevation for a detachment-limited channel was numerically solved under a range of assumptions about the dependency of incision rate on channel slope for transients corresponding to step-function changes in the rate of rock uplift and in base-level elevation, both of which should be expected to generate transient incision pulses that propagate upstream through a catchment [Whipple and Tucker, 1999]. The governing equation for river elevation for a detachment-limited channel states:

display math(6)

[16] The second term in the right side of equation (6) is the stream-power model. The first term, U, is rock uplift rate or, in the absence of tectonically driven downcutting, base level lowering rate. In the modeling presented below, drainage area is prescribed according to Hack's Law and in all simulations the lowest node in the channel has a fixed elevation. A first-order explicit finite difference scheme was used to solve equation (6) and then elevation and distance were nondimensionalized by the steady-state relief for the initial channel and the channel length, respectively. For all simulations, combinations of m and n were chosen so that the ratio of m/n was always 1/2. This choice reflects the common observation that bedrock channels thought to be in a condition of topographic steady-state typically exhibit concavities of close to 1/2 [e.g., Hilley and Arrowsmith, 2008; Snyder et al., 2000]. Situations with both spatially uniform and spatially nonuniform rock uplift are considered in the simulations below. For the experiments simulating base-level drop, following Weissel and Seidl [1998], a gently inclined error function (as opposed to a single linear knickpoint) was used to simulate the initial conditions of the knickpoint. This initial condition ensures that the knickpoint slope will evolve over time for the cases where n is unequal to one.

4 Numerical Model Results

[17] Figures 3-6 show the results of the base-level drop experiments for n = 1 (Figures 3a and b), n = 2/3 (Figures 4a and b), n = 3/2 (Figures 5a and b), and assuming a threshold slope in river incision that is only surpassed during knickpoint propagation (Figures 6a and b). In each figure, the top panels (Figures 3a, 4a, 5a, and 6a) show the long profile evolution in each experiment from a frame of reference fixed to base level. Thus, the top panels show only the change in the elevation of the channel during the experiment. Alternatively, the bottom panels (Figures 3b, 4b, 5b, and 6b) show former channel positions relative to the position of the incising channel, and thus represent a frame of reference attached to the rock that is being advected through the incising profile. In both panels, the black channel profile represents the channel at the end of the simulation. Because the channel is incising vertically at all times in the model, older profile positions (shown in grey in all figures) are separated vertically from the active profile at the rate of channel incision. For the experiments shown in Figures 3-5, the simulated channel begins at a topographic steady-state (rock uplift rate = incision rate). Therefore the older channels in these experiments are actually uplifted away from the active channel. Each profile in the bottom panel is thus an isochron and records cumulative rock uplift over time. However, because base level lowering, absent tectonically driven rock uplift, will also produce vertical incision into bedrock, the results of this study are not dependent on the presence of tectonically driven rock uplift. For the remainder of the paper only figures that correspond to the frame of reference of the bottom panels in Figures 3b, 4b, 5b, and 6b are shown. Strath terrace locations were automatically extracted from the figures showing former channel positions relative to the position of the incising channel (i.e., Figures 3b, 4b, 5b, and 6b). This was accomplished first by subtracting the channel gradient in an abandoned profile from the initial steady-state channel gradient at each position along the channel. The upstream limit of the transient channel adjustment (i.e., the knickpoint crest) in the abandoned channel was then identified from the upstream point of departure from steady-state. Finally, the strath terrace tread was defined by connecting the knickpoint crests in space between successive abandoned channel profiles (e.g., Figure 3b), in keeping with the geometric relationship defined in Figure 1b. Terraces are indicated in all figures by dashed black lines. Note that because the terraces cross-cut the abandoned and uplifted channels (which are isochrons), the terraces systematically young upstream, as expected for a time-transgressive system.

Figure 3.

(a) Channel profile evolution following sudden base level drop, m = 1/2, n =1. Frame of reference is attached to base level. (b) Former channel positions relative to the position of the incising channel, m = 1/2, n =1. Frame of reference is attached to the rock that is being advected through the incising profile. The heavy black line in each panel indicates the channel profile at the end of the simulation. Dashed lines indicate strath terrace levels. Elevations are normalized by the steady-state relief of the initial channel. Channel length is normalized by the modeled basin length.

Figure 4.

(a) Channel profile evolution following sudden base level drop, m = 1/3, n =2/3. Frame of reference is attached to base level. (b) Former channel positions relative to the position of the incising channel, m = 1/3, n = 2/3. Frame of reference is attached to the rock that is being advected through the incising profile. The heavy black line in each panel indicates the channel profile at the end of the simulation. Dashed lines indicate strath terrace levels. Elevations are normalized by the steady-state relief of the initial channel. Channel length is normalized by the modeled basin length.

Figure 5.

(a) Channel profile evolution following sudden base level drop, m = 3/2, n = 3/4. Frame of reference is attached to base level. (b) Former channel positions relative to the position of the incising channel, m = 3/2, n = 3/4. Frame of reference is attached to the rock that is being advected through the incising profile. The heavy black line in each panel indicates the channel profile at the end of the simulation. Dashed lines indicate strath terrace levels. Elevations are normalized by the steady-state relief of the initial channel. Channel length is normalized by the modeled basin length.

Figure 6.

(a) Channel profile evolution following sudden base level drop assuming a threshold slope for channel incision that is only exceeded during knickpoint propagation. Frame of reference is attached to base level. (b) Former channel positions relative to the position of the incising channel assuming a threshold slope for channel incision that is only exceeded during knickpoint propagation. Frame of reference is attached to the rock that is being advected through the incising profile. The heavy black line in each panel indicates the channel profile at the end of the simulation. Dashed lines indicate strath terraces. Elevations are normalized by the steady-state relief of the initial channel. Channel length is normalized by the modeled basin length.

[18] Figures 4a and 5a illustrate clearly the nonlinear effects of knickpoint propagation when n is unequal to 1. The modeling results are consistent with previous studies [Tucker and Whipple, 2002; Weissel and Seidl, 1998] that note the evolution of knickpoint shape with time as a function of n, so this aspect of the experiments is not discussed in detail. More relevant to the study here is that the evolving knickpoint shape and its influence on the knickpoint celerity does not fundamentally change the basic finding of the analytical model. Despite significant evolution of the knickpoint morphology in the experiments where n is unequal to 1, the elevation of the terrace for the case of n = 3/2 increases upstream, and the elevation of the terrace for the case of n = 2/3 increases downstream, as expected from the analytical model. Additionally, Figure 3b provides a check on equation (2) in that it confirms that terraces are horizontal when n = 1. Although not directly predictable with our analytical approach, we also include an experiment (Figure 6b) showing the terrace produced from base level fall in a channel where incision has a threshold slope that is only exceeded during knickpoint propagation. This case is essentially equivalent to a base level drop experiment with a very high value of n in equation (5). Accordingly, in Figure 6b, because no incision occurs upstream of the knickpoint, the terrace is exactly parallel to the channel below it. Only in this case does the terrace tread represent a paleo-river profile (Figure 6b). Thus, except in the case when the process of river incision requires a slope threshold that is only exceeded during knickpoint propagation, terrace treads produced following a base level fall event do not record paleo-channel gradients. Instead, strath terrace tread slope simply reflects the degree of nonlinearity of bedrock incision as a function of channel slope.

[19] Sudden base-level fall, however, is not the only means of generating a pulse of headward propagating incision in a catchment. An increase in rock uplift rate [Whipple and Tucker, 1999] will also drive a wave of headward incision into a catchment. Figures 7a–c shows the evolution of abandoned channel profiles following a three-fold increase in rock uplift rate for n = 1 (Figure 7a), n = 2/3 (Figure 7b) and n = 3/2 (Figure 7c). As in the case of the sudden base-level drop, terraces slope in accordance with the analytical findings. Although the steady-state slope of a river changes during transition to a higher rock uplift rate (Figure 7a–c), the slope of a terrace tread formed during this transition again in no way reflects a paleo-channel gradient nor does it record any information about the nature of the rock uplift rate transition.

Figure 7.

Former channel positions relative to the position of the incising channel and resulting strath terrace tread (dashed black line) for 3-fold acceleration in rock uplift rate, (a) m = 1/2, n =1, (b) m = 1/3, n =2/3, (c) m = 3/4, n =3/2. Frame of reference is attached to the rock that is being advected through the incising profile. The heavy black line indicates the channel profile at the end of the simulation. Elevations are normalized by the steady-state relief of the initial channel. Channel length is normalized by the modeled basin length.

[20] It is worth emphasizing that the above results only hold for situations with spatially uniform rock uplift, or where incision is driven by base level lowering in the absence of tectonic uplift. In the presence of spatially nonuniform rock uplift, the initial slope of the terrace tread will be modified by subsequent tectonic tilting or warping. Consequently, below the paper considers terrace formation under a simple nonuniform pattern up rock uplift, in this case a monotonic increase in rock uplift toward a channel's headwaters. Figure 8a–c shows the evolution of abandoned channel profiles following a sudden base-level fall along a river with a five-fold linear rock uplift rate increase toward its headwaters for n = 1 (Figure 8a), n = 2/3 (Figure 8b) and n = 3/2 (Figure 8c). Note that because both transient incision wave experiments produced terraces with similar slope characteristics with respect to n, the choice of the exact source of the wave of incision is unimportant to the ultimate long profile morphology of the terrace produced. Therefore, in Figure 8 only one such mechanism, sudden base-level fall, is considered.

Figure 8.

Former channel positions relative to the position of the incising channel and resulting strath terrace tread (dashed black line) for sudden base level drop along a river with a five-fold linear increase in rock uplift rate toward the headwaters, (a) m = 1/2, n =1, (b) m = 1/3, n = 2/3, (c) m = 3/4, n = 3/2. Frame of reference is attached to the rock that is being advected through the incising profile. The heavy black line indicates the channel profile at the end of the simulation. Elevations are normalized by the steady-state relief of the initial channel. Channel length is normalized by the modeled basin length.

[21] Figures 8a–c show that in the presence of a rock uplift gradient that increases linearly toward a channel's headwaters, strath terraces produced from headward incision have significantly different morphologies compared to terraces produced in the spatially uniform rock uplift case (e.g., Figures 3-7). Specifically, for the n = 1 case (Figure 8a), the slope of the terrace gradually increases downstream, forming a gently convex surface. Because the terrace youngs upstream, the downstream end of the terrace has experienced a longer duration of tilting than the upstream end, thereby resulting in a terrace that gently steepens downstream. Notably, because of the time-transgressive nature of the terrace, the apparent deformation pattern on the terrace does not relate in a simple way to the actual deformation field. This mismatch is even more dramatic for the case of n = 2/3 (Figure 8b). Here, again, in the presence of a simple rock uplift rate field that increases linearly toward the channel headwaters, the terrace produced from headward incision is apparently folded as though traversing an anticline. This pattern arises because the primary topographic gradient of the n = 2/3 terrace (elevation increasing downstream) is opposite of the rock uplift gradient (rock uplift increasing upstream). Hence, the older, downstream portions of the terrace have experienced the rock uplift gradient for a sufficient length of time to reverse the primary tilt on the terrace, whereas the upstream portions of the terrace still preserve the primary terrace tilt, resulting in a gentle apparent folding that bears no straightforward relationship to the actual deformation field. For the case of n = 3/2, because the rock uplift rate and the primary terrace elevation increase in the same direction (upstream), the resulting terrace is simply a steeper version of the terrace produced for n = 3/2 in a spatially uniform rock uplift field (Figure 8c). However, unlike in the case of spatially uniform rock uplift, in this example the terrace gradient can surpass the gradient of the active channel. It is worth noting that were the direction of the rock uplift gradient reversed (i.e., rock uplift rate increasing downstream), synclinal folding would be observed in the n = 3/2 case, whereas the n = 1 channel would be concave-up and steadily increasing in elevation downstream. The n = 2/3 case, in this scenario, would then have its slope amplified by the rock uplift field.

5 Discussion and Conclusions

[22] Many studies have attempted to constrain the exponent n in the stream-power erosion model both theoretically and empirically [e.g., DiBiase and Whipple, 2011; Howard and Kerby, 1983; Lague et al., 2005; Seidl et al., 1994; Stock and Montgomery, 1999; Weissel and Seidl, 1998; Whipple et al., 2000; Whipple and Tucker, 1999]. The results presented above suggest a novel alternative means of constraining n. Where tectonic uplift rate is spatially uniform or where incision is driven by base level lowering in the absence of tectonic uplift, the slope of a terrace formed from headward incision into a catchment, when compared to the slope of the active channel, can constrain the value of n to less than, equal to, or greater than one.

[23] In most settings, however, it is difficult to conclude a priori whether headward propagating incision is in fact responsible for strath terrace formation, and hence whether terraces can be used to constrain n. Moreover, because in many settings terraces are studied precisely because they can constrain spatial gradients in rock uplift [e.g., Molnar et al., 1994; Pazzaglia and Brandon, 2001; Wegmann and Pazzaglia, 2009], even if terraces are time-transgressive, it may be impossible to separate the original gradient of the terrace and the subsequent deformation. Lastly, terraces related to Pleistocene climate changes may overprint the morphologic effects of waves of headward incision [e.g., Cook et al., 2009; Crosby and Whipple, 2006], thereby further confounding interpretation of terrace morphologies.

[24] That said, several studies in simple tectonic settings (or over short distances where large rock uplift gradients are unlikely) report terrace elevation profiles inferred to have formed from headward retreating knickpoints [Howard et al., 1994; Zaprowski et al., 2001] or waterfalls [Jansen et al., 2011; Seidl and Dietrich, 1992]. In each of these studies, the terraces apparently created from headward propagating incision are parallel or close to parallel with the active channel (Figure 9). Additionally, in each study mapped terrace treads merge with main-stem channels at the location of knickpoint crests, and in each example there is no evident difference in the slope of the terrace tread and the channel upstream of the putative propagating knickpoint (Figure 9). Based on the modeling results, this geometry implies a process of incision with an exponent on slope larger than one (Figure 5), or alternatively, a process of incision with a slope threshold that is only exceeded during periods of knickpoint or waterfall propagation (Figure 6).

Figure 9.

Terrace tread (dashed lines) and river elevation (black line) long profiles for four locations where terrace formation is attributed to upstream propagating knickpoints (indicated with grey arrows). Data are after Zaprowski et al. [2001], Howard et al. [1994], Jansen et al. [2011], and Seidl and Dietrich [1992].

[25] Several field studies lend support to the inference that channel incision into bedrock only occurs above a threshold slope. For example, Seidl et al. [1994] proposed that bedrock channels mantled with immobile boulders in several settings might only lower when knickpoints sweep upstream beneath the channel's boulder armor. Additionally, observations of coseismic knickpoints following the 1999 Chi Chi Earthquake in Taiwan show that stripping of alluvial cover above bedrock channel can be triggered by the passage of a knickpoint [Sklar et al., 2005; Yanites et al., 2010a]. The inference that a threshold marks the transition from alluvial to bedrock conditions is also supported by field [Johnson et al., 2009] and laboratory [Finnegan et al., 2007] observations showing abrupt spatial and temporal changes in bedrock channels from bare bedrock to completely alluvial conditions. Notably, each of the four studies cited in Figure 9 also report that knickpoints mark sharp spatial transitions from steeper bedrock channel segments downstream to less steep channels mantled in alluvium upstream. The geomorphology of terraces and channels in Figure 9 combined with the results summarized in Figure 6 thus suggest that a threshold slope may be required to accurately capture the process of bedrock channel lowering in these and other settings where channels transport coarse sediment.

[26] The dependency of bedrock river incision rate on channel slope can also be constrained in the absence of preserved terraces where rates of knickpoint retreat are quantified. Specifically, equations (4) and (5) show that the ratio of vertical incision rate above a knickpoint to the retreat rate of the knickpoint (second term on the right side of equations (4) and (5)) will always be less than the channel slope above the knickpoint if n > 1. Alternatively, this ratio will always be greater than the channel slope above the knickpoint if n < 1. Thus, if knickpoint retreat can be quantified along with vertical incision rate above the knickpoint, then n can be constrained from comparison of this ratio to the channel slope above the knickpoint. Figure 10 shows data from three field studies of knickpoint retreat [Berlin and Anderson, 2007; Cook et al., 2012; Crosby and Whipple, 2006] that can be used, via equations (4) and (5), to constrain n. In two of the three studies [Berlin and Anderson, 2007; Crosby and Whipple, 2006] numerous channels were examined, so a range of retreat rates and upstream channel slopes characterize these locations. Additionally, in these two studies constraints on channel incision rates prior to the arrival of the propagating knickpoint are reported. For the third study [Cook et al., 2012], upstream channel slope and retreat rate are well quantified. However, in contrast to the other two studies that describe catchments adjusting to a new rate of base level lowering, Cook et al. [2012] document impulsive incision associated with a single earthquake. Thus constraining the rate of incision upstream of the propagating knickpoint is more difficult. For this study, a range of reasonable “background” incision rates were therefore assigned based on a previous study along a nearby river [Yanites et al., 2010b].

Figure 10.

Ratio of measured knickpoint retreat rate to upstream incision rate plotted against channel slope upstream of retreating knickpoint from 3 studies [Berlin and Anderson, 2007; Cook et al., 2012; Crosby and Whipple, 2006]. Via equation (5), data plotting below the 1:1 line indicate n > 1, whereas data that plot above the 1:1 line indicate n < 1. For the examples in New Zealand [Crosby and Whipple, 2006] and Colorado [Berlin and Anderson, 2007], geologically constrained rates of river incision were used in combination with reported knickpoint retreat rates to calculate the ratio of knickpoint retreat rate to upstream incision rate. For the Taiwan example, upstream incision rate was constrained from measurements of long-term river incision nearby [Yanites et al., 2010b] and was used in combination with reported knickpoint retreat [Cook et al., 2012] to calculate the ratio of knickpoint retreat rate to upstream incision rate. Upstream channel slope was calculated from long profiles included in Crosby and Whipple [2006], Berlin and Anderson [2007], and Cook et al. [2012].

[27] Notably, the channels from Taiwan and New Zealand, both tectonically active settings that supply large volumes of sediment to the ocean for their respective land areas [Milliman and Syvitski, 1992], plot below the n = 1 line in Figure 10, consistent with an exponent on slope that is much larger than one. Additionally, studies of channels in these settings document stripping of alluvial cover in association with knickpoint retreat [Crosby and Whipple, 2006; Sklar et al., 2005]. Alternatively, the one study that plots on the n = 1 line [Berlin and Anderson, 2007] is from a region with low sediment supply, where channels upstream of knickpoints are free of sedimentary cover and flow over bare bedrock [Berlin and Anderson, 2007]. Thus Figure 10 also supports the inference that the process of bedrock incision in channels mantled with alluvial sediment apparently depends more nonlinearly on channel slope than in channels that are free of cover.

[28] Although there are clear reasons to expect that the mechanics of waterfall retreat should not be captured using the modeling approach used here [Lamb et al., 2007], in the discussion below waterfalls (vertical bedrock steps that propagate upstream) are not distinguished from knickpoints (subtler downstream increases in channel gradient that propagate upstream). This simplification reflects the fact that the stream-power model, and therefore the strath terrace slope prediction developed here, does not incorporate the physics governing waterfall evolution [Lamb et al., 2007]. That said, it is notable that there is no qualitative difference in the slope of terraces produced from retreating waterfalls as compared to retreating knickpoints in the four examples above (Figure 9). This suggests that for the relatively broad brush approach adopted here, knickpoints and waterfalls can be treated in a similar way. Hence, below the term knickpoint refers to all upstream migrating slope breaks in a river channel.

[29] That n appears to be greater than one in most settings with time-transgressive strath terraces or knickpoints is fortuitous from the perspective of tectonic geomorphology because it suggests that terrace tread slopes are commonly close to the slope of active channel reaches. Hence, in the case of n > > 1, a measurement of differential incision or deformation from a time-transgressive terrace need not encompass a model for the initial terrace slope that is different from the active channel slope, as would be the case for n = 1 or n < 1. Additionally, according to the numerical modeling results a preserved strath terrace that is steeper than the active channel below it can only occur with an increasing rate of rock uplift toward the channel's headwaters (e.g., Figure 8c). Thus this specific terrace geometry may be diagnostic of a headward increase in rock uplift rate for a time-transgressive terrace.

[30] Studies in weakly cohesive badlands [Howard and Kerby, 1983] suggest that n can be less than one in some settings. However, only one study [Frankel et al., 2007] documents the possibility of terraces that climb in elevation downstream, as would be predicted by the model for a n < 1. This observation may indicate that if n is less than one in some settings, these are not settings where time-transgressive terraces form. This makes some sense because the upper lip of a knickpoint is diffusive in the n < 1 case. Thus to the extent that the morphologic changes leading to terrace formation depend on a rapid change in vertical incision rate, terraces may become less recognizable as the crest of the knickpoint becomes more diffuse.

[31] The results presented above illustrate that measurements of deformation from terrace morphology are potentially very sensitive to whether the terrace is time-transgressive. Thus care is warranted in interpreting apparent deformation of terraces in a purely tectonic context. Apparent tilting of terraces relative to active channels can result in situations of spatially uniform rock uplift rate where terrace formation is time-transgressive (e.g., Figure 3b). Additionally, apparent folding of terraces can result from monotonic rock uplift rate gradients where terrace formation is time-transgressive (e.g., Figure 8b). Lastly, it is tempting to compare the slope of a terrace tread to the active profile to make inferences about changes in channels steepness with time [e.g., Merritts et al., 1994]. However, except in the case of bedrock river incision with a slope threshold that is only exceeded during knickpoint propagation, a terrace created from transient headward incision contains no information about the paleo-channel gradient. On this latter point, this study reinforces a point originally articulated by Culling [1957].

Acknowledgements

[32] The author is indebted to Kelin Whipple, both for a conversation that inspired this manuscript, and for helpful comments on an earlier draft of this manuscript. Additionally, this manuscript benefited substantially from reviews by Greg Tucker, George Hilley, Maureen Berlin, Associate Editor Dimitri Lague, and Editor Alex Densmore. This work was supported by NSF grant EAR-1049889.

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