Journal of Geophysical Research: Earth Surface

Modeling the evolution of channel shape: Balancing computational efficiency with hydraulic fidelity

Authors


Abstract

[1] The cross-sectional shape of a natural river channel controls the capacity of the system to carry water off a landscape, to convey sediment derived from hillslopes, and to erode its bed and banks. Numerical models that describe the response of a landscape to changes in climate or tectonics therefore require formulations that can accommodate evolution of channel cross-sectional geometry. However, fully two-dimensional (2-D) flow models are too computationally expensive to implement in large-scale landscape evolution models, while available simple empirical relationships between width and discharge do not adequately capture the dynamics of channel adjustment. We have developed a simplified 2-D numerical model of channel evolution in a cohesive, detachment-limited substrate subject to steady, unidirectional flow. Erosion is assumed to be proportional to boundary shear stress, which is calculated using an approximation of the flow field in which log-velocity profiles are assumed to apply along vectors that are perpendicular to the local channel bed. Model predictions of the velocity structure, peak boundary shear stress, and equilibrium channel shape compare well with predictions of a more sophisticated but more computationally demanding ray-isovel model. For example, the mean velocities computed by the two models are consistent to within ∼3%, and the predicted peak shear stress is consistent to within ∼7%. Furthermore, the shear stress distributions predicted by our model compare favorably with available laboratory measurements for prescribed channel shapes. A modification to our simplified code in which the flow includes a high-velocity core allows the model to be extended to estimate shear stress distributions in channels with large width-to-depth ratios. Our model is efficient enough to incorporate into large-scale landscape evolution codes and can be used to examine how channels adjust both cross-sectional shape and slope in response to tectonic and climatic forcing.

1. Introduction

[2] Numerical models of landscape evolution have evolved to the point where they can now be used to understand how river systems transmit changes in base level through drainage networks [e.g., Bishop et al., 2005; Crosby and Whipple, 2006; Loget et al., 2006; Berlin and Anderson, 2007]; how landscapes respond to changes in climate [e.g., Tucker and Slingerland, 1997; Whipple et al., 1999]; and how erosion and tectonics might interact at the orogen scale [Beaumont et al., 1992; Whipple and Meade, 2004; Simpson, 2006]. While these modeling efforts have clearly bolstered our understanding of how landscapes function, in nearly all of them the assumption is made that bedrock channel width is a simple power law function of upstream drainage area. This is a particular problem for exploring the dynamics of landscapes, since recent work has shown that these power law relationships break down in exactly the settings we find most interesting from a tectonic perspective [e.g., Harbor, 1998; Lave and Avouac, 2001; Montgomery and Gran, 2001; Finnegan et al., 2005; Amos and Burbank, 2007; Whittaker et al., 2007]. Since the cross-sectional form of a bedrock river controls its ability to carry water off a landscape, to convey sediment derived from hillslopes, and to erode its bed and banks, unaccounted changes in channel shape through time could have important implications for landscape dynamics.

[3] In addition to providing better parameterizations of erosion that can feed into landscape evolution models, understanding the controls on channel geometry is an important research target in its own right. For example, dozens of geomorphic studies have relied on radiometric dates obtained from strath terraces (river terraces carved directly into bedrock) to reconstruct rates of fluvial erosion through time [e.g., Merritts et al., 1994; Hancock et al., 1999; Lave and Avouac, 2000; Pazzaglia and Brandon, 2001]. These strath terraces preserve information about the hydraulic conditions of a river system at some point in the past; however, these conditions are not always fully understood [e.g., Montgomery, 2004]. Physically based models describing how bedrock channels acquire their cross-sectional shape would improve our ability to interpret radiometric dates obtained from strath terraces.

[4] Ideally, models of fluvial erosion and drainage basin dynamics should therefore include a mechanism for rivers to adjust their cross-sectional form in response to natural or anthropogenic perturbations. This requires that we have a meaningful way of modeling the cross-sectional distribution of a physical quantity that can be used as a proxy for erosion rate, such as shear stress. However, models that explicitly describe the hydraulics of rivers generally require iterative solution schemes that are too computationally expensive to model shear stress distributions and evolve cross-sectional forms over long timescales [Houjou et al., 1990; Pizzuto, 1991; Naot et al., 1993; Vigilar and Diplas, 1997; Lane et al., 1999; Ma et al., 2002; Kean and Smith, 2004]. On the opposite end of the spectrum, the simplest possible formulation in which the shear stress is approximated by the local depth-slope product is both a misrepresentation of fluid physics and will inevitably lead to continually deepening and narrowing channel cross sections. The challenge, then, is to develop models that remain as faithful as possible to the physics of fluid flow without requiring iterative solutions to the equations for turbulence, momentum, and velocity structures.

[5] Toward this end, Wobus et al. [2006b] developed a simple model for bedrock channel evolution that used a “shortcut” to reconstructing the flow field by assuming that radial logarithmic velocity profiles connect the bed and banks with the channel centerline. While this model (hereafter Wobus, Tucker, and Anderson Model (WTA)) has been shown to reproduce many of the scaling relationships observed in natural channels, it has not yet been tested against more complete parameterizations of river hydraulics to ensure that there are no systematic biases introduced by the simplifications used. In this contribution we take a step back, and test the model against independently derived calculations of the velocity structure and shear stress distributions in natural channels. Where available, we also test the model against experimental data from laboratory flumes. These comparisons allow us to evaluate the sources of errors that might be introduced by the WTA model, and motivate some simple refinements that can be implemented to improve its approximation of the flow field in symmetrical channels with a wide range of aspect ratios.

2. Model Descriptions

2.1. Wobus, Tucker, and Anderson Model

[6] The goal of the model presented by Wobus et al. [2006b] is to allow channels to adjust both their cross-sectional shape and their longitudinal profile form in response to tectonic or climatic perturbations. Such perturbations are expected to occur over timescales on the order of 104−106 years. As such, the formulation requires tradeoffs to be made between computational efficiency and faithful implementation of fluid physics in open channel flow. The details of the model formulation are given by Wobus et al. [2006b]; a brief summary is provided here.

[7] The flow structure in the Wobus, Tucker, and Anderson model is calculated by the simultaneous solution of four equations describing the mean velocity, hydraulic roughness, and momentum balance in the model channels. First, the requirement that the channel must convey the prescribed discharge is combined with a modified Chézy formulation to determine the cross-sectional area and mean velocity of the flow for a given cross section (equations (1) and (2))

equation image
equation image
equation image

where Q is the water discharge, equation image is the mean flow velocity, C is the Chézy smoothness coefficient, A is the cross-sectional area of the flow, R is the hydraulic radius (cross-sectional area divided by wetted perimeter), and S is the channel gradient. The Chézy coefficient C is a function of both the bed roughness and the hydraulic radius, and is calculated in equation (2b) using an empirical relationship described by Julien [1998]. Here l0 represents an effective roughness length from the bed at which the velocity goes to zero, and is measured perpendicular to the channel boundary; g is the acceleration due to gravity at the Earth's surface.

[8] Conservation of momentum requires that the integrated shear stress on the bed is balanced by the downslope component of the weight of water [e.g., Parker, 1979; Stark, 2006]

equation image

where P is the total distance along the wetted perimeter of the channel, τb is the local boundary shear stress, dpb is an increment of distance along the channel's wetted perimeter, and ρ is the density of water. Once the flow depth and velocity are determined, the task that remains is to determine how the shear stress is distributed along the wetted perimeter of the channel in a manner that satisfies equations (1)−(3). For simplicity, the formulation follows Prandtl's mixing length hypothesis, which states that the bed shear stress at each point along the wetted perimeter of the channel is proportional to the square of the near-bed velocity gradient [e.g., Furbish, 1997]

equation image

where L is the mixing length, r is a coordinate measured between the wall of the channel and the channel midpoint, and l0 is the effective roughness length scale as above.

[9] Here a series of assumptions is required in order to efficiently reproduce the flow structure and shear stress distribution. First, the model assumes that the maximum velocity occurs in the center of the channel at the top of the flow (Figure 1a). This assumption is broadly consistent with measurements from natural and experimental channels, in which the maximum velocity is typically found near the center of the channel and slightly below the surface [e.g., Knight and Sterling, 2000; Ma et al., 2002]. Placing the maximum velocity at the water surface simplifies the model without substantial departure from this observation. Second, the velocity gradient near the bed is assumed to scale with the mean velocity gradient between wall and centerline by employing the law of the wall along “rays,” r, drawn from the channel centerline to the bed

equation image

where r is a radial coordinate measuring distance from the channel wall to the centerline. Third, the formulation resolves the “total” velocity gradient onto a bed-normal vector in order to determine the shear stress. And finally, the mixing length scale L is assumed to be invariant along the wetted perimeter of the channel. The magnitude of L is found by combining equations (3) and (4) [e.g., Wobus et al., 2006b].

Figure 1.

Wobus, Tucker and Anderson model summary. (a) Schematic showing channel cross section and coordinate system. Coordinate r is measured along radii from the bed to the channel midpoint; total distance along each radius is R. Boundary shear stress at each point on the wetted perimeter is calculated from the mean velocity gradient, Umax/R. Velocity is assumed to be zero near the margin of the channel at a roughness length l0 along these radii. (b) Calculation of mean velocity from boundary shear stress. (left) Isovels constructed by employing the law of the wall along bed-normal rays drawn from the channel bed to the water surface. (right) Cartesian regridding of the resulting two-dimensional velocities to calculate a cross-sectionally averaged mean velocity. (c) Comparison of cross-sectionally averaged mean velocity resulting from Figure 1b with the mean velocity required to satisfy conservation of mass. Agreement in all model runs is within 1–3%.

[10] It is important to note that the approach outlined above does not require calculation of the full velocity structure of the flow. However, once the distribution of bed shear stresses has been estimated, it can be used to reconstruct the velocity field by calculating the shear velocity at each point (u* = equation image), and applying the law of the wall along bed-normal vectors. Note that because Prandtl's mixing theory is built upon momentum exchange and turbulent stresses across velocity gradients, the orientation of these vectors with respect to gravity does not enter into the formulation. Local flow depth is therefore only important as it relates to the length scale that determines the velocity gradient in equation (4). As a test of the degree to which the resulting flow structure is reasonable, the cross-sectionally averaged velocity can then be recalculated by projecting the velocities from these bed-normal vectors onto a Cartesian grid and finding the mean value of the gridded velocity structure (Figure 1b). This exercise typically yields a mean flow velocity that is within 1–3% of that required to satisfy equation (1) (Figure 1c), suggesting that the approximation of the flow structure is reasonable to first order, and obeys this integral constraint.

[11] The result of the assumptions made by the WTA model is an efficient approximation of the flow structure and the distribution of shear stress along the bed and banks. By assuming that the erosion rate depends on the local shear stress, the formulation can then be used to evolve channel cross sections quickly enough that both cross-sectional forms and longitudinal profiles can be modeled over geologically relevant timescales [e.g., Wobus et al., 2006b]. The original WTA model makes the assumption that erosion rate scales linearly with shear stress [e.g., Howard and Kerby, 1983]; however, any erosion rule that depends on the local shear stress could also be used.

2.2. Ray-Isovel Model

[12] Kean and Smith [2004] developed a method for determining the distribution of velocity and boundary shear stress across the entire cross section of a straight channel. Their method, which is a generalization of the approach of Houjou et al. [1990], numerically solves the momentum equation for steady, uniform flow using a ray-isovel turbulence closure. The approach provides a foundation for making sediment transport and geomorphic adjustment calculations, because it accurately resolves the relative distribution of stress between the bed and the banks of channels of arbitrary cross section. A brief overview of the method is given below. The reader is referred to Kean and Smith [2004] for a complete description of the approach, as well as descriptions of the additional steps required to incorporate the effects of form drag on roughness elements such as the stems of rigid vegetation, which will be neglected within the analysis of this paper.

[13] The ray-isovel model calculates fluid stress along curves (rays) that run perpendicular to lines of constant velocity (isovels) (Figure 2). The rays begin perpendicular to the channel boundary and end at the surface. Leighly [1932] first used this approach empirically to determine the stress distribution in natural channels from measured velocity fields. Simultaneous determination of both the velocity and boundary shear stress fields, however, requires numerical integration. The procedure involves alternately solving for the velocity field, defined in a Cartesian coordinate system, and for the fluid stress and eddy viscosity fields, which are defined more easily in the orthogonal ray-isovel coordinate system. The momentum equation for steady, stream-wise uniform flow in a channel is given by

equation image
Figure 2.

Ray-isovel model setup. Rays begin orthogonal to the channel bed and are everywhere perpendicular to the lines of constant velocity (isovels). Here δpb is the perimeter distance along channel bed and δpl is the distance between rays at a distance l from the bed as measured along a ray. See text for model description.

[14] In this equation, K is the kinematic eddy viscosity and u is the velocity in the downstream (x) direction. The variables y and z represent the cross-stream and vertical directions, respectively. The boundary conditions for equation (6) are that ∂u/∂z= 0 at the water surface and u = 0 at a roughness length perpendicular to the boundary, l = l0.

[15] The two terms in parentheses in equation (6) represent the deviatoric stress components τyx and τzx. Stress throughout the channel also can be specified by the rays, which define the direction of shear along which mixing occurs. The boundary shear stress (τb) between any two rays is simply equal to the downstream component of the weight of water between the rays divided by the length of wetted perimeter that separates them. Similarly, the interior shear stress along a ray a distance l from the boundary (τlx) is equal to the downstream component of the weight of water between two rays from l to the surface, divided by length along the isovel between the two rays. The ray-isovel coordinate system also is used to define a scalar eddy viscosity that relates τlx to the velocity gradient along a ray

equation image

[16] The functional form of K near the boundary used by Kean and Smith [2004] is

equation image

where κ is von Karman's constant, and u* is the shear velocity. The eddy viscosity is assumed to increase linearly along each ray until it reaches the channel-scale eddy viscosity, Ko, defined as

equation image

where Hc is the flow depth, R is the hydraulic radius, and β is a constant equal to 6.24. In an infinitely wide channel, the value of β = 6.24 is that required to match the eddy viscosity profile in the linearly increasing portion of the profile (i.e., z/H < 0.2) with the channel-scale eddy viscosity defined in equation (9) (see Kean and Smith [2004] for a complete discussion). This algebraic form for the eddy viscosity has previously been shown to give good agreement with experimental results [e.g., Shimizu, 1989].

[17] Having defined K for the entire cross section it is possible to solve equation (6) iteratively for u. The procedure starts with an initial guess of the boundary shear stress distribution and the velocity in the interior. Then the computation alternately solves the momentum equation (6) for u and equations for τb, τlx, and K until the flow solution converges. To reduce the number of grid points required in the calculation, the velocity within a short distance of the boundary is computed using the law of the wall

equation image

[18] To date, the ray-isovel method described above has been applied only to laboratory flumes and natural channels with fixed cross-sectional geometries [Griffin et al., 2005; Kean and Smith, 2005]. In this paper, a simple erosion component is added to the modeling framework in order to permit the channel to change shape with time on the basis of the flow and sediment transport conditions. The erosion model used here is identical to the one used by Wobus et al. [2006b] and assumes that the bed-normal erosion rate scales linearly with the local boundary shear stress. Equilibrium channel geometries using this erosion model are computed for a constant discharge, channel slope, and channel roughness. The calculation begins with an initial cross section, the choice of which is arbitrary. Next, the stage corresponding to the specified discharge is determined iteratively. This phase of the computation typically requires three trials and is necessary because the discharge is not specified a priori in the ray-isovel calculation (i.e., it is an output of the calculation). After the flow and boundary shear stress fields for the specified discharge are determined, the y and z components of the erosion rate are determined from the bed slope and local boundary shear stress, and a new cross section is computed. The calculation is repeated in this manner until the channel geometry reaches an equilibrium shape. Despite being computationally less intensive than fully three-dimensional (3-D) numerical flow and geomorphic adjustment models, the ray-isovel approach requires approximately 2 orders of magnitude more time to compute equilibrium geometries relative to the semianalytical WTA method described in the previous section.

2.3. Depth-Slope Product

[19] An assumption that is commonly employed in modeling the long-term evolution of bedrock channels is that the cross-sectionally averaged shear stress is proportional to the depth-slope product

equation image

where equation image is the average shear stress and H is the average depth of the flow. This expression can be derived from equation (3) for channels with a large width-to-depth ratio, where the width is approximately equal to the wetted perimeter and the wall stresses can be neglected. By extension, one might expect that a momentum balance could be applied locally, such that the local shear stress can be approximated by the local depth-slope product [e.g., Li et al., 1976; Parker, 1979]:

equation image

where τ(y) and H(y) are the shear stress and depth evaluated at position y along the cross section, and αn is a constant coefficient that ensures the momentum balance in (3) is satisfied.

[20] While it is appealing in its simplicity, there are at least two problems with this approximation of the shear stress distribution that make it unsuitable for modeling the dynamics of bedrock rivers. First, this formulation is inappropriate for channels with small width-to-depth ratios, since wall stresses represent an important component of energy dissipation and cannot be neglected to arrive at equation (11). Second, even in a wide channel, the depth-slope product gives a poor approximation to local stress close to the banks. For these reasons, such a model cannot be used to evolve cross sections through time, since the formulation would inevitably lead to a runaway feedback in which depth and local shear stress grow indefinitely in the deepest portions of the channel. The inadequacy of the depth-slope product for expressing local stress along a complete channel perimeter may at first seem startling. To demonstrate how and where the depth-slope product deviates from boundary stress even under unidirectional flow, several of the illustrations below compare boundary shear stress distribution with the stresses one would estimate using τ(y) = αnρgH(y)S.

3. Intermodel Comparisons

[21] Although the details of the two hydraulic models are different, the essence of the WTA and ray-isovel model (RIM) models is quite similar. Both models reconstruct the flow field by assembling a series of rays through the flow, and calculating a 1-D velocity structure along each ray. These 1-D velocity structures are then used to construct the full two-dimensional structure of the flow. The main differences between the two formulations lie in the orientation of these rays within the flow, and in the parameterization of the kinematic eddy viscosity. The WTA model uses a simplification in which the rays are straight line segments drawn normal to the bed, while the RIM model ensures that the rays are constructed so that they are everywhere normal to lines of constant velocity. In addition, the RIM model uses a two-part eddy viscosity formulation, while the WTA model does not.

[22] To facilitate intermodel comparisons, the RIM and WTA models were each run to steady state under a series of identical discharge, slope, and roughness conditions. In all cases, model cross sections were started from a simple “V-shaped” geometry, and were allowed to evolve until both the channel geometry and the flow field stabilized. Below we compare the models' representations of channel cross-sectional shape, velocity structure, and boundary shear stress distributions at steady state, along with the shear stress predictions of the depth-slope product rule for our equilibrium geometry (section 3.1). We then briefly explore the scaling relationships among width, depth, slope, discharge, and roughness for the WTA and RIM models (section 3.2). In all cases, we find that the WTA and RIM models behave similarly, suggesting that the simplified WTA model captures the essence of the more computationally expensive RIM hydraulic formulation.

[23] We next compare the WTA model to the predictions from other available models and laboratory measurements for channels with both straight and gently curving cross-sectional profiles (section 3.3). Finally, we compare the shear stress distributions predicted by the RIM and WTA models for a set of prescribed cross-sectional shapes with varying aspect ratios. This inspires improvements to the WTA model that allow better representation of the flow structure in wide channels (section 3.4).

3.1. Flow Structure

[24] The channel form in both models must evolve to a steady state geometry in which (1) the cross-sectional shape and velocity structure are adjusted to convey the prescribed discharge; (2) the momentum balance in equation (3) is satisfied; and (3) the shear stress distribution at steady state (and therefore the erosion rate) is adjusted such that the channel will maintain its cross-sectional shape as it incises. The model comparisons described below are based on a prescribed discharge of 20 m3/s, a gradient of 0.0015, and a roughness length (l0) of ∼0.002 m. Note that intermodel comparisons using other combinations of discharge, gradient, and roughness length scales yield similar results.

[25] The steady state cross-sectional form and isovel structures predicted by the two models are shown in Figures 3a–3c. Both cross sections are quasi parabolic, with width-to-depth ratios of ∼3.3 for the WTA model and ∼3.9 for the RIM model. The steady state cross-sectional areas for the two models are ∼10.9 and ∼10.6 m2, respectively. The RIM has a slightly higher-mean velocity (∼3%), as required to compensate for its smaller cross-sectional area.

Figure 3.

RIM and WTA model comparisons. In all cases Q = 20 m3/s, S = 1.5e – 3 and l0 = 0.002. (a) Steady state cross-sectional geometry. (b and c) Isovel structure for steady state RIM and WTA models, respectively. (d) Vertical profiles of velocity along channel midline.

[26] The most important difference in the flow structure between the two models is the velocity structure in the upper portion of the flow (Figure 3d). This difference is primarily a result of the differences in the way the two models treat the eddy viscosity: to boost computational efficiency, the WTA model extends the law of the wall all the way to the free surface, which builds in an assumption that Prandtl's eddy length scale (L in equation (4)) grows with the flow depth throughout the water column. In contrast, the RIM model assumes a two-part eddy viscosity, in which the eddy length scale reaches the constant channel-scale value at about 20% of the distance along a ray from the boundary to the surface. As a result, the modeled flow velocities are higher for the RIM in the upper portion of the flow.

[27] Also as a result of these differences in the treatment of the flow hydraulics, the near-bed velocity gradient is higher for the WTA model. This leads to a peak shear stress at the channel midpoint that is approximately 7% higher for the WTA model than it is for the RIM model (Figure 4). To satisfy the integral constraint imposed by conservation of momentum (i.e., equation (3)), the boundary shear stress in the WTA model is commensurately lower along the margins of the flow. As noted by Wobus et al. [2006b] (equations (7) and (8)), the shape of the shear stress distribution can be related to the vertical lowering rate and the shape of the channel at steady state. The higher-peak shear stress implies that the rate of vertical translation will be slightly faster for the WTA model than for the RIM model for a given erosion law.

Figure 4.

Steady state bed shear stress distributions for RIM and WTA models for steady state geometries shown in Figure 3. Shear stress distribution using local depth-slope product rule, normalized to ensure momentum conservation, is shown for comparison.

[28] For comparison, we also include in Figure 4 the shear stress distribution predicted by the depth-slope product rule for the steady state geometry shown in Figure 3c. Note that this formulation substantially overpredicts the shear stress at the channel midpoint, and substantially underpredicts the stress along the channel margins, relative to both the WTA and RIM models. This shortcoming arises because wall stresses are neglected by this simplification of the flow hydraulics, so that the shear stress distribution is artificially weighted toward the channel midpoint. This also underscores the inherent problem with a depth-slope product rule in capturing the dynamics of bedrock channel adjustment: since boundary shear stress in such a formulation is always concentrated in the deepest portions of the channel, the implementation of any erosion rule that depends on shear stress will generate runaway deepening. As a result, wall stresses must be incorporated into our models if we hope to capture the dynamics of bedrock channel adjustment.

3.2. Scaling Relationships

[29] Even with the simplifications to the hydraulics such as those used by the WTA model, large-scale landscape evolution models typically cannot afford the computational expense of reconstructing the flow field at every individual channel cross section. In order to “scale up” the fluvial hydraulics to make predictions about landscape evolution, it is therefore useful to distill the predictions of our models into simple scaling relationships. To do this, we ran the WTA and RIM models to steady state under a variety of discharge, slope, and bed roughness conditions to evaluate their predictions for these scaling coefficients.

[30] Figure 5a shows the scaling of equilibrium width as a function of discharge. For the range of roughness values used in both model formulations, power law regressions on the width-discharge relationships yield exponents between 0.36 and 0.38. Figure 5b shows the scaling of equilibrium width as a function of slope. Regressions on the width-slope relationship yield exponents between −0.18 and −0.19. Finally, Figure 5c shows the relationship between width and depth for bed roughness values (l0) ranging from 0.2–20 mm. In all cases, the width-to-depth ratio for smoother channels is slightly larger than that for rougher channels for a given discharge and slope. However, both models predict nearly constant width-to-depth ratios at steady state, ranging from ∼3.2 for the roughest WTA channels to ∼4.0 for the smoothest RIM channels. Note that the dependence of W/D on roughness is slightly weaker than that reported by Wobus et al. [2006b], reflecting an improved discharge matching scheme that imposes a tighter constraint (1%) on mass balance.

Figure 5.

Scaling relationships for steady state channel width as a function of (a) discharge and (b) gradient for WTA and RIM models. Both models predict scaling relationships close to WQ0.4 and WS−0.2. (c) Width versus depth for WTA and RIM models. Both models predict an approximately constant width-to-depth ratio for a given roughness length scale.

[31] The result of either the WTA or the RIM formulation is therefore a model that predicts channels with (1) a nearly constant width-to-depth ratio at steady state; (2) a power law relationship between width and discharge with an exponent near 2/5 (holding slope constant); and (3) a power law relationship between width and slope with an exponent near −1/5. For all parameters examined, the agreement between the two models is within ∼10%. To the extent that the RIM model is an accurate representation of the hydraulics of natural channels [e.g., Kean and Smith, 2005], this implies that the WTA model should be able to capture the dynamics of channel adjustment to a similar level of accuracy.

3.3. Comparisons With Other Data Sets

[32] In addition to the intermodel comparisons outlined above, it is instructive to compare the predictions of the WTA model with experimental data from laboratory flumes, and with the results of other numerical schemes for approximating shear stress distributions. For brevity, we limit our comparisons to two of the simplest end-member geometries: a half-pipe geometry such as that used by Knight and Sterling [2000] in their experiments; and a V-shaped geometry for comparison with the work of Lane [1953] as described by Khodashenas and Paquier [1999] in their evaluation of the geometrical “Merged Perpendicular Method.” To facilitate comparisons, we normalize all shear stress values by the depth-slope product at the channel midpoint.

[33] Figure 6a shows the results of our comparison with the experimental data of Knight and Sterling [2000]. The WTA model predicts a constant shear stress across the channel, with a magnitude equal to half the midpoint depth-slope product. (The factor 1/2 arises because the hydraulic radius R is equal to Hc/2 for a half pipe; the shear stress is everywhere equal to ρgRS). The prediction of the WTA model agrees favorably with the experimental results, except that the latter indicate a slightly higher shear stress at the channel midpoint (1.1ρgHS) and a slightly reduced shear stress along the channel margins (0.8ρgHS). As noted by Knight and Sterling [2000], this modest redistribution of shear stress toward the channel midline is most likely a result of secondary flows generated near the margins of the flow. Note that explicit modeling of these secondary currents requires a substantial increase in computation time, and will not be considered further here [Naot et al., 1993; Ma et al., 2002; Knight et al., 2007].

Figure 6.

Comparisons between WTA and RIM models, depth-slope product, and independently derived experimental data for half-pipe and V-shaped channel geometries. (a) Comparisons with Knight and Sterling's [2000] experimental Preston tube measurements of shear stress for a half pipe. (b) Comparisons with Lane's [1953] membrane-analogy experiments for a V-shaped geometry. Results of Khodashenas and Paquier [1999] Merged Perpendicular Method (MPM) shown for comparison.

[34] The second data set we use for comparison, shown in Figure 6b, is the shear stress distribution in a V-shaped geometry as described by Lane [1953]. Note that these data were derived using a “membrane analogy,” in which an elastic membrane is stretched over the channel form of interest and the local curvature is used as a proxy for shear stress [e.g., Olsen and Florey, 1952]. The measurements therefore differ from the direct Preston tube measurements used by Knight and Sterling [2000]. Nonetheless, this data set provides another experimentally derived estimate of the shear stress distribution for a well-constrained geometry. In this case, both the shear stress distribution described by Lane [1953] and the distribution predicted by the WTA model have a pronounced trough near the channel midpoint. In the WTA model, this trough arises because the center of the channel is actually further from the point of maximum velocity than the sloping channel sidewalls. Since the mean velocity gradient between the channel midline and the bed dictates the shear stress distribution in the WTA model, this results in a redistribution of maximum shear stress toward the portions of the channel with the shortest distance to the high-velocity midpoint. The same pattern is seen in the RIM model (Figure 6b). Note that both the RIM and WTA models are an improvement over the Merged Perpendicular Method described by Khodashenas and Paquier [1999], which is a purely geometrical method for distributing shear stress across irregular channels.

[35] Notably, the depth-slope product rule does poorly in estimating the shear stress in both of the cases outlined here: in general, using a depth-slope product for a narrow channel will substantially overpredict the shear stress in the deepest portions of the channel, and substantially underpredict the shear stress along the channel margins where the depth goes to zero. Again, we stress that this shortcoming arises because wall shear stress cannot be neglected in modeling narrow bedrock channels.

3.4. Prescribed Geometries

[36] On the basis of the comparisons outlined above, the WTA model appears to strike a reasonable balance between computational efficiency and hydraulic fidelity for a range of channel geometries with relatively low width-to-depth ratios. But what about channels that have evolved to a larger width-to-depth ratio, due to the bed becoming protected by a mantle of coarse sediment [e.g., Hancock and Anderson, 2002; Wobus et al., 2006a; Turowski et al., 2008a], or the banks becoming weakened by subaerial weathering [e.g., Montgomery, 2004]? In this case, the assumption that the velocity gradient at the bed scales with the distance from the channel midpoint may be problematic since wider channels appear to be characterized by a high-velocity “core” of finite width [e.g., Knight and Sterling, 2000; Kean and Smith, 2005].

[37] To improve model flexibility in such situations, we modified the original WTA formulation to better approximate the flow structure in wider channels. Our model adjustments included the following:

[38] 1. Rather than assuming the maximum velocity occurs at a single point, the flow is assumed to be characterized by a “high-velocity core” in which the surface velocity is approximately constant. The width of this high-velocity core is defined as the portion of the channel over which the bed-normal distance to the water surface remains approximately constant (e.g., Figure 7a).

Figure 7.

Model comparisons for varying width-to-depth ratios. (a–d) (right) Predictions for RIM model. (left) Predictions for modified WTA model (Figures 7a–7c) and predictions for depth-slope product (Figure 7d). Model-generated rays (Figure 7a) and isovels (Figure 7b) for WTA and RIM models for W/D of 15. Shear stress distributions for WTA (Figure 7c) and depth-slope product (Figure 7d) compared to RIM model for W/D values between 5 and 15. (e) Prescribed channel geometries for all models.

[39] 2. The surface velocity within the high-velocity core is first approximated by assuming the shear stress along the bed below it is equal to ρgHS (as must be the case in infinitely wide channels). The shear velocity u is then equation image and the surface velocity is calculated by employing the law of the wall

equation image

[40] 3. The distribution of surface velocities between the edge of the high-velocity core and the channel wall is assumed to be logarithmic, reaching zero at a roughness length l0 from the channel wall.

[41] 4. The distance equation image used to calculate the near-bed velocity gradient beyond the high-velocity core is assumed to be the length of a bed-normal vector from the channel bed to the surface, rather than the distance from the bed to the channel midpoint.

[42] 5. The near-bed velocity gradient is calculated using the above approximations for surface velocities and bed-normal distances, and equation (3) is used to ensure a momentum balance.

[43] This set of assumptions allows us to relax the assumption of a maximum velocity “point” at the channel midline, and to reconstruct the velocity field and the distribution of shear stresses for channels with larger width-to-depth ratios. The predictions of this modified WTA model and the depth-slope product rule are compared with the RIM model in Figure 7, for prescribed channels with width-to-depth ratios ranging from 5 to 15.

[44] This modified WTA model predicts a shear stress distribution that is very similar to the RIM model for large width-to-depth ratios. For wide channels, the shear stress distributions predicted by the RIM and modified WTA models diverge most substantially in the corners of the flow where the horizontal bed merges with the bank (Figure 7c). Note that these corners are also the locations where secondary currents are likely to be most important in modifying the flow structure of natural channels [e.g., Naot et al., 1993; Knight and Sterling, 2000; Knight et al., 2007]. Experimental work indicates that these secondary currents can influence the shear stress distribution in this region, particularly as these corners become more angular [Knight and Sterling, 2000; Knight et al., 2007], but modeling the effects of these secondary currents would take us yet another step further from the computational efficiency we seek. As a result, we suggest that the general agreement between the modified WTA and RIM models is reasonable for our purposes.

[45] Note that for channels with large width-to-depth ratios, the depth-slope product rule does a reasonable job of replicating the shear stress distributions predicted by the RIM model (Figure 7d). In wider channels, wall stresses become less important so that calculating the center shear stress via the depth-slope product is reasonable. However, we emphasize that there is no physical basis for applying a depth-slope product rule locally to calculate the wall stresses along the banks of a wide channel.

[46] Figure 7 indicates that the predictive capacity of both the modified WTA model and the depth-slope product rule become limited as width-to-depth ratios decrease from 10 to 5. Compared to the RIM model, both of these formulations accurately predict the peak shear stress for W/D = 10, but overpredict the peak shear stress by ∼20% for a width-to-depth ratio of 5 (Figures 7c and 7d). Notably, early use of the ray-isovel model by Shimizu [1989] is consistent with this observation: a summary of their numerical simulations finds that the depth-slope product becomes a good predictor of the shear stress in the center of the channel only as the width-to-depth ratio increases to ∼10. This is also consistent with a range of experimental work summarized by Khodashenas and Paquier [1999] all of which suggests that wall stresses rapidly become important in the center of the channel as width-to-depth ratios drop below ∼5. At higher width-to-depth ratios, wall stresses account for less than ∼10% of total stress, at which time the depth-slope product should predict the peak shear stress to within the same order. However, even in these wide channels, wall stresses remain locally important in affecting the near-bank bed stress, and must be considered if we hope to model lateral erosion of natural channels.

[47] The comparisons outlined above suggest that wall stresses, which are inherently incorporated in both the WTA and RIM models, need to be modeled for channels with low width-to-depth ratios. Although field studies that explicitly describe the geometry of bedrock channels are rare, available data suggest that width-to-depth ratios less than ∼5 are not uncommon in natural systems [e.g., Finnegan et al., 2005; Whittaker et al., 2007]. It follows that a model developed to understand the evolution of bedrock channels must include wall stresses. Further, it must be flexible enough to incorporate a high-velocity core to the flow as width-to-depth ratios increase. We suggest that the combination of the original WTA model with the modified formulation outlined above strikes this balance. Future dynamic models of channel evolution in which changes in bed state might drive extreme widening or narrowing (e.g., because of large sediment influxes from landsliding) will implement changes between these end-members, guided by empirical observations of shear stress distributions in natural and experimental channels [e.g., Shimizu, 1989; Knight et al., 1994].

4. Discussion

[48] The WTA model originally formulated by Wobus et al. [2006b] is based on a number of approximations that allow the model to be efficient enough to examine channel evolution over large spatial and temporal scales. On the basis of our comparisons with the ray-isovel formulation [e.g., Leighly, 1932; Shimizu, 1989; Houjou et al., 1990; Kean and Smith, 2004], and with other available data from the literature [Olsen and Florey, 1952; Lane, 1953; Khodashenas and Paquier, 1999; Knight and Sterling, 2000] these approximations appear to capture the essence of velocity and boundary shear stress distributions driven by stream-wise flow in natural channels. In particular, the shape of the channels at equilibrium, the isovel structure, the shear stress distributions, and the scaling of width with slope and discharge are all within reasonable agreement between the two formulations, and between the model predictions and experimental results. Merging our original WTA model with a formulation that explicitly considers a widening high-velocity core as W/D ratios increase provides us with a model that can efficiently describe boundary shear stress distributions across the spectrum of concave up channel shapes.

4.1. A Constant Width-to-Depth Ratio?

[49] One of the intriguing findings of both the WTA and RIM models is that the steady state width-to-depth ratio is approximately constant, regardless of slope or discharge (Figure 5c). This finding is consistent with the assumption built into Finnegan et al.'s [2005] analytical model for the scaling of bedrock channel width with slope and discharge. However, as acknowledged both by Finnegan et al. [2005] and by Turowski et al. [2007] there is very limited field data to support the finding that width-to-depth ratios should be constant in nature. In fact, models that explicitly consider the effects of sediment in bedrock channels suggest that W/D may be highly variable due to downstream variations in sediment cover [Wobus et al., 2006a; Turowski et al., 2007].

[50] Despite these findings for models that explicitly consider sediment cover, Whittaker et al.'s [2007] data from the Fiamignano gorge in Italy suggests that width-to-depth ratios might actually approach a constant value when sediment cover is limited. Width-to-depth ratios from the Fiamignano exhibit a strongly nonlinear inverse relationship with channel gradient, which at first glance appears inconsistent with our findings. However, the channels with the highest W/D ratios are also those reported to have significant sediment cover, whereas the W/D ratios of high-gradient, cover-free channels asymptote to a value of ∼3. These high-gradient channels with a nearly constant W/D ratio may in fact be the best natural analog for our model, since we do not explicitly consider the role of sediment in controlling channel incision. It is therefore instructive to consider the potential origin of a constant width-to-depth ratio for channels with limited sediment cover.

[51] As a way of gaining physical insight, we can manipulate the equations used in the WTA model to explore the origin of this constant width-to-depth ratio. For example, if we simplify equations (4) and (5) to express the rate of channel wall erosion as a function of the mean velocity gradient rather than the near-bed velocity gradient, we can write

equation image

where B is a coefficient that includes the mixing length scale L and the erodibility of the substrate. At the channel midpoint, the radial distance is equivalent to the flow depth Hc, and the incision is purely vertical

equation image

where z is the height of the bed relative to a datum in the underlying rock. Noting that at steady state the rate of vertical translation must everywhere be the same, and using the chain rule, equations (14) and (15) can be related as

equation image

which can be simplified to yield

equation image

Here we can define the integration limits as follows for a steady state geometry: at the channel midpoint, z = 0 and equation image = Hc; where the water surface touches the bank z = Hc and equation image = W/2. Evaluating the resulting expression, we find that W = equation imageHc = 3.17Hc at steady state. Note that this is only an approximation since the near-bed velocity gradient used in the model is also a weak function of the ratio equation image/lo (see equation (5)). However, this manipulation of the governing equations yields insight into why the width-to-depth ratio should be constant, and what that constant should be, given our model formulation.

[52] Similar arguments could be made to suggest that W/D ratios should tend toward a constant value in natural bedrock channels, as long as three conditions are met: (1) the erosion rate is related to the local shear stress; (2) the local shear stress is related to the mean velocity gradient; and (3) the highest velocities occur near the center of the flow. To the extent that incision in natural channels adheres to these three first-order criteria, we might expect a tendency for channels to seek a constant W/D ratio [e.g., Finnegan et al., 2005]. However, we again stress that this prediction is based on a simplified model that neglects sediment cover effects [Wobus et al., 2006a; Turowski et al., 2007, 2008a].

4.2. Origin of Scaling Relationships

[53] As shown by Finnegan et al. [2005], if the width-to-depth ratio is constant for a given channel, the power law scaling of width with discharge and slope can be derived analytically using a roughness equation for average velocity combined with mass balance. Using the Manning equation, this leads to

equation image

while using the Chézy formulation employed by the WTA model (equation (1)) this gives

equation image

Equation (19) is consistent with the scaling computed by the WTA model, as one would expect [Wobus et al., 2006b]. Note that scaling relationships such as these are not as easily derived for the RIM model, since that formulation does not include closed form expressions for velocity or discharge as a function of width or slope.

4.3. Limitations of the Model

[54] As described in section 4.1, sediment cover appears to exert a strong influence on the shape of bedrock channels. A variety of experimental studies also underscore the importance of sediment for bedrock incision and channel shape. For example, Shepherd [1972], Wohl and Ikeda [1997], Finnegan et al. [2007], and Johnson and Whipple [2007] each describe controlled laboratory experiments in which sediment and water were passed across cohesive, eroding substrates. In all cases, channel shape evolves as a complex function of water discharge, local gradient, and importantly, sediment flux. The latter creates a feedback mechanism by which sediment is preferentially transported through topographic lows, enhancing incision of the substrate when sediment flux is low. However, consistent with the predictions of Gilbert [1877] and Sklar and Dietrich [2004], both experiments also uncover a negative feedback wherein an alluvial cover is formed on the bed that ultimately inhibits further incision. Sediment is not explicitly considered in either the WTA or RIM model, an obvious shortcoming of the formulations used here. However, preliminary modeling which accounts for the effects of sediment corroborates its importance in determining bedrock channel shape [Wobus et al., 2006a].

[55] An additional complexity that needs to be addressed is the influence of variable water and sediment discharges on channel evolution [e.g., Lague et al., 2005; Stark, 2006]. In natural channels, such variability in discharges will influence both the time-averaged thickness of alluvial cover and the velocity structure of the flow, such that single realizations of these parameters might not capture the hydraulics of the most important flow events. For example, alternating periods of high and low sediment flux might influence the relative importance of widening versus deepening in channel evolution [Hancock and Anderson, 2002; Hartshorn et al., 2002; Turowski et al., 2008a, 2008b], and wetting and drying of channel banks might lead to important changes in cross-channel erodibility that could strongly control the hydraulic geometry of natural channels [e.g., Montgomery, 2004].

[56] Finally, both the WTA and RIM models neglect the effects of secondary currents. As illustrated by our comparisons with experimental data, these currents can have the effect of redistributing boundary shear stresses, even for smooth channel forms (Figure 6). Furthermore, secondary currents are likely to become increasingly important as channel shapes become more angular and turbulence is enhanced. Models that explicitly account for the effects of secondary flows have been developed, but these models also increase model runtimes.

4.4. Numerical Approximations

[57] Both the WTA and RIM formulations make a series of numerical approximations in order to efficiently reproduce the flow structure in model channels. The effects of these approximations are likely to be of second order compared to the omission of sediment cover, secondary currents, and variable discharge as discussed above. Nonetheless, for completeness we briefly discuss here some of these numerical approximations.

[58] In finding the flow depth required to match the prescribed discharge (equations (1) and (2)), the WTA model approximates the cross-sectional area of the flow by summing the areas of individual rectangles. As the node spacing decreases, this rectilinear approximation more closely approaches the area enclosed by the curvilinear cross section. However, this formulation is likely to introduce small errors into the calculation of cross-sectional area. In addition, the discharge-matching routine adds nodes to the upper portion of the cross section until the prescribed and modeled discharges match to within 1%. A better match to the discharge could be forced; however, since this scheme is already computationally intensive, a 1% match was deemed appropriate. Finally, as shown in Figure 1c, the mean velocity calculated from the boundary shear stress distribution does not always perfectly match the mean velocity required to convey the prescribed discharge (equation (1)). Again, in light of the close fit between models and the need for efficiency, iterative schemes that would improve this fit were not deemed appropriate.

[59] The numerical implementation of the RIM model also requires approximations that may influence its predictions of the flow structure. The model solves equation (6) using a boundary-fitted rectangular grid and the solution scheme of Patankar [1980]. A separate curvilinear grid following rays is used to solve the equations for shear stress and eddy viscosity. These computational grids must be constructed such that they minimize any systematic numerical errors in the calculations of velocity and boundary shear stress, which, in turn, produce artificial effects on the predictions of channel geometry. The number and spacing of the nodes for the computational grids was optimized by comparing model-calculated patterns of velocity and boundary shear stress to analytic solutions available for (1) an infinitely wide channel, and (2) a semicircular channel (half pipe). The semicircular channel comparison provides the strongest test of the numerics because it has geometric characteristics similar to the narrow channels that are the focus of this paper. On the basis of these comparisons, a mean cross-stream and vertical node spacing of about H/50 was used for the rectangular grid. This spacing corresponds to approximately 3000 active computational nodes for a channel having the geometry shown in Figure 3b. About twice as many nodes are used to define the curvilinear grid along rays in order to better resolve the near-surface ray structure and to reduce errors in interpolating the eddy viscosity field from the curvilinear grid to the rectangular grid, which is required to solve (6). It is likely that the desired computational accuracy could be achieved with fewer nodes using a numerical discretization of equation (6) on the basis of nonrectangular elements (which are better suited for fitting curved boundaries). Such a scheme, however, has not yet been implemented.

5. Conclusions

[60] In order to understand the response of landscapes to changes in climatic or tectonic conditions, landscape evolution models must accommodate the simultaneous evolution of channel width and gradient to these environmental controls. While there have been a number of notable recent advances in understanding the width response of natural channels [Finnegan et al., 2005; Stark, 2006; Whittaker et al., 2007], fully dynamic models have remained elusive due in part to the computational expense of modeling the full flow field in these channels. The model proposed by Wobus et al. [2006b] requires a number of approximations to the flow field in order to boost computational efficiency and achieve the goal of fully flexible channels. Despite these approximations, however, we have shown here that the first-order predictions of the model match well those derived from more sophisticated parameterizations of fluvial hydraulics and with available experimental data from laboratory flumes. Our modeling framework will in the future be used to investigate how sediment cover influences the shape of natural channels, how strath terraces are formed, and how a dynamic width response modulates the tempo of landscape response.

Notation
A

cross-sectional area of flow [L2].

B

coefficient relating erosion rate to mean velocity gradient [LT].

C

Chezy smoothness coefficient [L0.5T−1].

K

kinematic eddy viscosity [L2T−1].

Hc

centerline flow depth [L].

r

coordinate measured along radii to channel midpoint [L].

R

hydraulic radius [L].

S

gradient [dimensionless].

equation image

mean velocity [LT−1].

U

velocity at a point [LT−1].

l0

roughness length scale [L].

L

Prandtl's mixing length scale [L].

dp

length of an element along channel perimeter [L].

P

total wetted perimeter length [L].

u*

shear velocity [LT−1].

κ

von Karman's constant [dimensionless].

β

constant needed to match upper and lower eddy viscosity profiles in RIM.

τb

boundary shear stress [ML−1T−2].

τlx

interior shear stress along rays [ML−1T−2].

αn

normalization factor used in depth-slope product rule.

equation image

total radial distance from channel margin to midpoint [L].

Acknowledgments

[61] We thank Jens Turowski, Dimitri Lague, Rob Ferguson, and an anonymous reviewer for their extremely thorough and constructive reviews. This work was funded by a CIRES postdoctoral fellowship to C.W.W., by a NSF grant EAR-062199 and ARO grant 47033-EV awarded to G.E.T., and by a NSF grant EAR-0724960 to Suzanne P. Anderson.