### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Study Area
- 3. Stream Power Analytic Model
- 4. Transient Channel Incision Into a Tilted Surface
- 5. Competitive Fluvial Incision
- 6. Discussion
- 7. Conclusions
- Acknowledgments
- References
- Supporting Information

[1] The Kyrgyz Range, located on the northern margin of the western Tian Shan, illustrates long-term (10^{6}–10^{7} year) transient landscape evolution in response to an active basement-cored rock uplift. Eastward propagation of range uplift has progressively exposed resistant bedrock capped by a tilted, (formerly) planar, pre-Cenozoic unconformity. We develop an approximate, stream power–based analytic model of transient river profile incision into progressively exposed and resistant bedrock to explore the patterns of channel development into the unconformity surface. This analysis shows that the unconformity can be preserved as a geomorphic marker defined by upland headwaters and interfluvial areas. Though channels are not at equilibrium with rock uplift, prominent knickpoints are not predicted to develop on main stem channels. However, knickpoints are predicted to develop on tributaries upstream of their junctions with the trunk stream because of differential erosion rates. Initial channel slope and transient channel form are both sensitive to the *n* value of the stream power model and could prove useful for calibration of *n* from field data. Accumulation of catchment area (a proxy for discharge) into larger catchments develops a positive feedback where larger drainages with higher stream power at a given slope undermine and capture adjacent drainage area. A simple model of competitive fluvial incision illustrates the role of tributary junction position in maximizing stream power expended upon the trunk stream. Examples from the Kyrgyz Range illustrate the effects of tributary junction position on fluvial relief, and we propose that adjustments to the tributary network through stream capture are ongoing within this landscape even after several kilometers of exhumation.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Study Area
- 3. Stream Power Analytic Model
- 4. Transient Channel Incision Into a Tilted Surface
- 5. Competitive Fluvial Incision
- 6. Discussion
- 7. Conclusions
- Acknowledgments
- References
- Supporting Information

[2] Fluvial relief, i.e., the relief conveyed by bedrock channels, is the primary control on the overall topographic relief in mountainous landscapes [*Whipple et al.*, 1999]. Under equilibrium conditions, where fluvial bedrock incision balances rock uplift, bedrock channel profiles develop a predictable concave longitudinal profile [*Whipple and Tucker*, 1999]. However, channel networks that erode bedrock and convey sediment away from a growing uplift are not born into an equilibrium condition. Nascent uplifts disrupt preexisting drainage networks, build elevation, and establish new topographic divides [*Humphrey and Konrad*, 2000; *Sobel et al.*, 2003]. Where exhumation of low-erosivity basement promotes development of steep channel gradients, formation of high topography may inhibit fault slip and drive tectonic shortening elsewhere before equilibrium conditions can be established [*Molnar and Lyon-Caen*, 1988; *Hilley et al.*, 2005]. Thus, in order to predict the topographic evolution of a growing orogen, the evolution of bedrock river channel profiles must be examined under disequilibrium, or transient conditions. In addition, the arrangement of entire channel networks and sizes of catchments also evolve within a growing uplift [*Densmore et al.*, 2004, 2005]. These processes, which affect the downstream accumulation of discharge in a landscape, can have a significant impact on fluvial relief because of the tradeoff of channel slope and discharge as channel profiles approach equilibrium grade [*Howard*, 1994].

[3] In this paper we examine the transient development of river longitudinal profiles into a basement block undergoing surface uplift. This contribution is motivated by observations of the Kyrgyz Range, a field example of an actively growing basement-cored range from the northwestern margin of the Tian Shan orogen (Figure 1). Rather than develop a comprehensive model of all surface processes that affect a growing basement massif, we instead isolate two first-order processes that modulate the evolution of transient river channel longitudinal profiles and explore these analytically. The advantage of this approach is that we gain insight into the specific relationship between parameters controlling erosion and development of transient channel longitudinal profiles. We begin by introducing the Kyrgyz Range study area. Next we review the stream power–based erosion rule [*Howard and Kerby*, 1983] and develop a kinematic wave–based solution for an equilibrium channel profile for the case where channel incision rate is linear with channel slope. We then extend this solution to approximate transient cases where incision rate is nonlinear with slope and apply this solution to model transient river profiles developed on a tilted, progressively exposed resistant bedrock surface. These surfaces develop during exposure of the unconformity that separates easily erodible sedimentary rocks from resistant basement and are common features of basement-involved orogens, including much of the Tian Shan [*Davis*, 1904; *Burbank et al.*, 1999; *Abdrakhmatov et al.*, 2001], the Sierra Pampeanas [*Jordan and Allmendinger*, 1986], the Rocky Mountains [*Gregory and Chase*, 1994], and southern California [*Spotila et al.*, 1998]. Our analytic model lends insight into the conditions necessary to preserve a relict “surface” in the landscape and reveals a pattern of knickpoints developed at tributary junctions that should be a characteristic feature of these landscapes. We also find that the pattern of channel incision is strongly affected by the nonlinearity of the dependence of incision rate on slope. This dependence, which can be directly related to the incision process [*Hancock et al.*, 1998; *Whipple et al.*, 2000; *Tucker and Whipple*, 2002], is difficult to uniquely extract from equilibrium channel gradients [*Whipple and Tucker*, 1999]. Next we explore the amalgamation of catchment area into larger drainage basins with progressive incision into a basement-cored uplift [*Densmore et al.*, 2004, 2005]. We model this process via competitive fluvial incision of adjacent tributaries and develop the role of positive feedback between catchment area and channel incision to drive stream capture. Stream capture may occur through undermining of adjacent catchments and rerouting of tributaries, or possibly through gradual adjustment of the positions of tributary junctions within the landscape [*Howard*, 1971]. Steepened “knickzones” that form on the trunk stream ahead of tributary junctions may be evidence of recent or ongoing stream capture processes.

[4] This contribution does not develop a complete model for landscape evolution in basement-cored uplifts, but rather focuses on first-order processes expressed at two different stages of range development. A comprehensive landscape evolution model would require further consideration of the interactions among these processes as well as additional phenomena not considered here. For example, development of threshold hillslopes [*Burbank et al.*, 1996] occurs in concert with relief production. The role of sediment production from these hillslopes and its potential effects on bedrock channel incision [*Sklar and Dietrich*, 1998, 2004] are not considered. Also, surface uplift of erosionally resistant basement rocks results in landscapes that cross disparate climate zones and leads to both positive and negative reinforcement of precipitation and its effects on the landscape. For example, the Kyrgyz Range encompasses nearly 4 km of topographic relief (Figure 1) from the Kazakh platform (800 m) to its highest peaks (4,700 m). Annual precipitation generally increases with elevation, but winter cyclonic storm systems dominate at low elevations while summer convective storms become more important sources of rain and snow at high elevation [*Aizen et al.*, 1995]. These orographic effects are not considered in the transient channel profile evolution models developed here.

### 2. Study Area

- Top of page
- Abstract
- 1. Introduction
- 2. Study Area
- 3. Stream Power Analytic Model
- 4. Transient Channel Incision Into a Tilted Surface
- 5. Competitive Fluvial Incision
- 6. Discussion
- 7. Conclusions
- Acknowledgments
- References
- Supporting Information

[5] The Kyrgyz Range, the northernmost range of the Kyrgyzstan Tian Shan (Figure 1), provides a natural laboratory to examine the erosional processes that develop relief and balance rock uplift in basement-cored orogens. The Kyrgyzstan Tian Shan absorb up to 13 mm/yr of shortening distributed across a series of basement-cored ranges and intrabasinal faults [*Abdrakhmatov et al.*, 1996; *Thompson et al.*, 2002]. Approximately 1.5 mm/yr of this shortening occurs at the foot of the Kyrgyz Range, which defines the accretionary boundary between the high topography of the Tian Shan and the stable, low-elevation Kazakh platform [*Bullen et al.*, 2001, 2003; *Thompson et al.*, 2002].

[6] Concordant, low-relief surfaces underlain by erosionally resistant Paleozoic metamorphic and plutonic basement rocks characterize the crests and slopes of many ranges within the western Tian Shan [*Davis*, 1904; *Chediya*, 1986; *Burbank et al.*, 1999; *Abdrakhmatov et al.*, 2001], including the easternmost 50 km of the south facing range slope of the Kyrgyz Range [*Oskin and Burbank*, 2005] (Figure 2). These basement surfaces are coincident with the unconformable basal contact of unconsolidated Cenozoic nonmarine sedimentary rocks and are interpreted as exposed, folded remnants of this contact surface [*Burbank et al.*, 1999]. Because Late Cenozoic syntectonic strata are removed, the elevation of this “unconformity surface” with respect to the stable Kazakh platform defines, to first order, the net rock uplift and shortening of the western Tian Shan [*Abdrakhmatov et al.*, 2001].

[7] The Kyrgyz Range preserves the transition from a presteady state condition where surface uplift dominates and erosion is minimal, to a condition that at least approaches a flux steady state [*Willett and Brandon*, 2002] where the erosion rate balances the rock uplift rate [*Sobel et al.*, 2006]. Over 5 km of erosion of the central, highest-relief portion of the Kyrgyz Range has occurred, sufficient to completely remove the unconformity surface and exhume apatite with reset fission track ages [*Bullen et al.*, 2001]. Additional apatite fission track data from the eastern half of the range indicate that total exhumation increases from east to west along the range crest. These data further support that this gradient in exhumation has evolved over time, with over 110 km of lateral propagation of the range tip over the last 8 to 11 Ma [*Sobel et al.*, 2006]. Trends in rock uplift and exhumation in the eastern part of the range are correlated to the growth and stabilization of range elevation, deepening of relief of incised canyons, and rising average hillslope angles (Figure 1). These metrics gradually approach spatially average values at the center of the range, suggesting that further topographic adjustments are not required to balance rock uplift. These data support our contention that the evolution of range-scale steady state topography of the Kyrgyz Range may be examined in the context of a space-for-time substitution, where systematic spatial topographic patterns inform our understanding of the temporal development of geomorphology at a site within the range.

[8] We closely examine two aspects of the presteady state topographic evolution of channel profiles in the Kyrgyz Range. We first consider observations at the foot of the Kyrgyz Range that highlight disparate regimes of bedrock channel incision upon exposure of the unconformity (Figure 2). New channels incise interfluvial regions to develop a dense channel network characterized by low interfluvial relief. These areas form relict surfaces in the landscape that are separated by deeply incised gorges connected to larger drainage areas higher up in the range. Second, we consider the continued evolution of channel networks in more deeply incised portions of the landscape. Here we develop a mechanism that relies on competitive fluvial incision to promote the observed gradual consolidation of drainage area into fewer, but larger catchments (Figure 1).

### 3. Stream Power Analytic Model

- Top of page
- Abstract
- 1. Introduction
- 2. Study Area
- 3. Stream Power Analytic Model
- 4. Transient Channel Incision Into a Tilted Surface
- 5. Competitive Fluvial Incision
- 6. Discussion
- 7. Conclusions
- Acknowledgments
- References
- Supporting Information

[9] To explore the form of channel profiles we relate channel erosion rate to channel slope and upstream catchment area through a general form of the stream power erosion rule [*Howard and Kerby*, 1983], where the rate of change in elevation of the surface,

U(x) is the rock uplift rate relative to a datum, which may either be local base level or a global reference frame, such as the geoid [*England and Molnar*, 1990]. Equation (1) may be cast to represent mechanisms and transient behavior of a range of possible bedrock channel incision models [*Howard*, 1994; *Whipple and Tucker*, 1999, 2002; *Lague et al.*, 2005] and may be readily manipulated analytically [*Whipple and Tucker*, 1999; *Humphrey and Konrad*, 2000] to produce theoretical channel profiles [*Whipple et al.*, 1999]. Erosive power is spatially controlled by downstream slope of the channel bed, ∣∣ and the upstream drainage basin area, *A*, as a proxy for the effective discharge. The intrinsic concavity, θ, expresses the geometry of the longitudinal bedrock channel and the scaling of effective discharge with drainage basin size, including possible orographic increase in precipitation with elevation [*Roe et al.*, 2003]. The exponent *n* relates to the erosion mechanism [*Hancock et al.*, 1998; *Whipple et al.*, 2000] and the effect of a distribution of discharge events [*Snyder et al.*, 2003; *Lague et al.*, 2005]. At steady state and with a spatially uniform erosion rate, θ may be empirically estimated from the concavity of channel profiles [*Whipple and Tucker*, 1999] and is approximately 0.5 for bedrock channels [*Howard*, 1994]. Note that the product, *n*θ, is commonly combined into the exponent *m* in other studies [*Howard*, 1994; *Whipple and Tucker*, 1999; *Kirby and Whipple*, 2001; *Snyder et al.*, 2003]. The coefficient, *K*_{d}, collects physical constants of gravity and the density of water as well as dimensional constants from derivation of either unit stream power or shear stress at the river bed in terms of upstream drainage area. We cast the units of the term inside the brackets as unit stream power in *J*/*m*^{2}/*yr*. *K*_{e} represents the resistance of the channel substrate to erosion and its units convert those of unit stream power to the exponent, *n*, into an erosion rate in *m*/*yr*. The actual units of *K*_{d} and *K*_{e} depend upon the exponents θ and *n* (Table 1).

Table 1. Explanation of Symbols Used in TextSymbol | Name | Units |
---|

*A*(*x*) | catchment area | m^{2} |

*C* | capture point | m |

*E* | channel erosion rate | m yr^{−1} |

ε | finite erosion | m |

*h* | Hack exponent | |

θ | concavity | |

*K* = *K*_{e}*K*_{d}^{n}*K*_{a}^{nθ} | combined constant | m^{1−p} yr^{−1} |

*K*_{a} | Hack constant | m^{2−h} |

*K*_{d} | dimensional constant | J m^{−2(θ+1)} yr^{−1} |

*K*_{e} | erodibility | J^{−n} m^{2n+1} yr^{n−1} |

*L* | catchment length | m |

*m* = *n*θ | area exponent | |

*n* | slope exponent | |

*p* = *hn*θ | distance exponent | |

*S*_{0} | initial channel slope | |

*S*_{c} | present channel slope | |

*t* | time | years |

*t*_{i}(*z*) | time of initial channel position | years |

Δ*t* | finite time | years |

*U*(*x*), *U* | rock uplift rate | m yr^{−1} |

*U*_{0}, *U*_{1} | *U*(*x*) coefficients | m yr^{−1}, yr^{−1} |

*V* | vertical rate of exposure | m yr^{−1} |

*W* | catchment width | m |

*x* | distance downstream | m |

*x*_{i}(*z*) | initial channel position | m |

*x*_{c} | channel head position | m |

Δ*x* | finite distance | m |

*z* | elevation | m |

*z*_{o} | elevation of divide | m |

Δ*z* | finite elevation | m |

[10] The distribution of erosive power along the length of a channel is strongly related to the downstream increase in basin size, *A*. *Montgomery and Dietrich* [1992], expanding on work by *Hack* [1957], documented that globally, drainage basin size scales with the square of distance downstream. Discharge increases downstream in these “box-like” basins primarily via accumulation of tributary streams. Many of the channels draining the southeastern Kyrgyz Range do not display this scaling relationship. Rather, these basins are intermediate between “box-like” and “pipe-like” in form (Figure 3); they lack significant tributaries, and drainage basin size scales less strongly with downstream distance. The relationship,

which is the inverse of Hack's Law [*Hack*, 1957], captures the growth of basin size, *A*, with channel-wise distance downstream, *x*, and scaling with *h* ranging typically from 1 (pipe-like) to 2 (box-like).

[11] For the steady state case where erosion rate everywhere equals rock uplift rate, elevation drop along an equilibrium channel is derived by substituting (2) into (1), setting to zero, and integrating downstream from the channel head at *x*_{c} [*Whipple and Tucker*, 1999],

For the case where *U*(*x*) is a constant, *U*, and *h*θ ≠ 1,

Under equilibrium conditions, the only dependence on *n* comes from the ratio of uplift rate, *U*(*x*), to erodibility, *K*_{e}. *Kirby and Whipple* [2001] exploited this dependence to derive *n* = 1 from channel profiles across portions of the Siwalik Hills with varying *U*(*x*). However, it is generally difficult to extract *n* values from equilibrium channel profiles because other conditions, such as the erodibiliy, *K*_{e}, may also vary between sites with different *U*(*x*) [*Snyder et al.*, 2000].

[12] Our analysis of disequilibrium, or transient channel profiles builds upon an alternative, kinematic wave solution to (1). We combine (1) and (2) and substitute *p* = *hn*θ and *K* = *K*_{e}*K*_{d}^{n}*K*_{a}^{nθ} to yield a formula for elevation change with respect to channel-wise distance,

where *x* is positive downstream. For the time being we drop the *U*(*x*) term and only consider elevation change due to erosion. Rock uplift rate, or another description of differential rock uplift, is incorporated later as an initial condition to the solutions to (5) presented here. By multiplying each side by , we rearrange (5) into a kinematic wave equation in the *x* direction,

[13] *Whipple and Tucker* [1999] derived (6) to describe the kinematic wave speed, or celerity, of knickpoint propagation in the upstream, negative *x* direction. We note that (6) more generally describes the upstream rate of backwearing of all positions, *x*(*z*), on a channel. This rate may be integrated for the *n* = 1 case to solve for the channel position at any time, *x*(*z*, *t*). For the *n* ≠ 1 case, we substitute the absolute value of initial channel slope, *S*_{0}, for to explore the initial stages of transient channel profile evolution where channel slope does not change significantly. The resulting differential equation is an initial value problem solved by separation of variables,

Integration of (7) yields the general solution,

where *x*_{i}(*z*) prescribes a point through which the solution must pass at time *t*_{i}(*z*). Because both of these initial conditions may vary as a function of elevation, *z*, these can describe moving boundaries, such as uplift of rocks relative to base level, *U*(*x*). It is important to keep in mind that (8) and (9) are strictly solutions only for the *n* = 1 case. Where *n* ≠ 1, these equations only approximate the initial stages of channel profile evolution where channel slope is not much changed from *S*_{0}.

[14] To illustrate that (8) is indeed a solution for the *n* = 1 case, we use this solution to derive an equilibrium channel profile and compare this to the result from (4). The key to this comparison lies in the selection of the initial conditions. We choose to track particles of rock that pass through the channel head at (*x* = *x*_{c}, *z* = 0). Because at equilibrium, rock uplift is balanced by erosion, the time that a particle of rock at a depth, −*z*, reaches the channel head position is prescribed by the sum of the present time, *t*, and the depth divided by the uplift rate, −*z*/*U*. The set of initial conditions are thus completely described by

[15] Substituting these into (8),

[16] The dependence on *t* disappears so that (12) can be rearranged to solve for *z*(*x*),

which is identical to (4) for *n* = 1, *p* = *h*θ, and *K* = *K*_{e}*K*_{d}*K*_{a}^{θ}.

### 7. Conclusions

- Top of page
- Abstract
- 1. Introduction
- 2. Study Area
- 3. Stream Power Analytic Model
- 4. Transient Channel Incision Into a Tilted Surface
- 5. Competitive Fluvial Incision
- 6. Discussion
- 7. Conclusions
- Acknowledgments
- References
- Supporting Information

[52] Basement-cored rock uplifts, commonly developed on the periphery of orogens [*Rodgers*, 1987], represent an important natural laboratory for calibrating the delicate balance between tectonic processes and erosion [*Molnar and Lyon-Caen*, 1988; *Sobel et al.*, 2003; *Hilley et al.*, 2005]. Using stream power modeling, we quantitatively analyze the development of river channels and the effects of stream capture processes during transient landscape evolution of basement-cored rock uplifts. This modeling is guided by observations of the Kyrgyz Range, a natural example of an active basement-cored range within the Tian Shan orogen [*Sobel et al.*, 2006]. First, we examined landscape evolution on a progressively exhumed unconformity that separates resistant basement from more erodible cover strata. We find a pattern of channel incision that characterizes this landscape. Our analysis confirms that channel headwaters and interfluves will closely mimic the initial unconformity position, reinforcing the utility of such landscapes to reconstruct channel incision and deformation [*Burbank et al.*, 1999]. We also find that, although trunk-stream channels are constantly evolving and transient landforms, they take on concave profiles that may appear graded (i.e., at equilibrium) but gradually change over time. Conversely, disparate erosion rates of tributary and trunk streams leads to generation of knickpoints on tributaries upstream of their junction with the trunk stream. The existence of these features provides an important test of whether a landscape preserves evidence of a progressively exhumed relict surface. Second, we examined the role of stream capture processes in elaboration and deepening incision of channel networks. A simple model of competitive fluvial incision shows how stream capture, and possibly the migration of tributary junctions, increases expenditure of stream power on the trunk stream, promoting greater incision and a positive feedback that promotes additional stream capture. Field evidence for stream capture includes relict, abandoned downstream portions of tributary valleys and contrasting channel responses upstream and downstream of capture points. Channel gradients should relax downstream of a capture because of increased stream power, while a knickzone will gradually form and expand upstream of a capture.

[53] Balance between rock uplift and erosion rates in a basement-cored range appears to be difficult to achieve via progressive exposure of basement at the surface, as attested by preservation of unconformity surfaces up to the range crest in the Kyrgyz Range [*Oskin and Burbank*, 2005] and in other, similar settings [*Abdrakhmatov et al.*, 2001; *Burbank et al.*, 1999; *Jordan and Allmendinger*, 1986]. Successful balance between rock uplift rate and erosion rate in the central Kyrgyz Range [*Sobel et al.*, 2006] is achieved through development of large, deeply incised and elaborated (“box-like”) drainage networks on the north facing range slope, on the opposite flank to the exhumed unconformity surface. The development of these catchments may be limited by the rates of competitive fluvial incision and stream capture, warranting further investigation of these processes.

[54] The analysis developed here, though approximate for cases where river erosion rate is nonlinearly related to slope, lends new insight into the utility of transient landscapes for calibrating surface processes. By providing a test for preservation of relict surfaces, our analysis provides a firmer basis for using these surfaces to calibrate incision rates and landscape features developed from fluvial and glacial erosion. The predictability of knickpoint formation at tributary junctions on progressively exposed unconformity surfaces also provides a new laboratory for exploring knickpoint propagation and its implications for the stream power erosion rule [*Tucker and Whipple*, 2002; *Crosby and Whipple*, 2006]. Similarly, the systematic response of channel incision to an imposed, steadily exposed unconformity surface slope provides a new approach for extracting mechanically important parameters of erodibility, *K*_{e}, and the slope dependence, *n*, of the stream power bedrock erosion rule.