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Keywords:

  • bank erosion;
  • widening;
  • narrowing

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Summary of the Experimental Investigation
  5. 3. Theoretical Formulation
  6. 4. Numerical Implementation
  7. 5. Numerical Simulations
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Integral Form of the Exner Equation of Sediment Continuity Over the Bed and Sidewall Regions
  11. Acknowledgments
  12. References

[1] This paper is devoted to a morphodynamic model of incision into a reservoir deposit driven by partial or total removal of the associated dam. The model considers the erosional processes upstream of the position of the former dam, rather then the deposition that occurs downstream. A theory is developed to predict the evolution of both the width and depth of the incisional channel that develops as erosion progresses. The theory is implemented in a numerical model, which is tested against and verified with flume experiments on sudden, complete removal of a dam. In these experiments a channel of a given initial width is allowed to freely incise into a noncohesive reservoir deposit after sudden dam removal. These experiments show a phenomenon that we refer to as “erosional narrowing.” That is, as a channel of a given initial width rapidly incises into the deposit, it can become narrower. As the rate of incision slows, this short period of rapid narrowing is followed by a longer period of widening. In the model the incisional channel is abstracted to a trapezoidal channel with well-defined bed and bank regions, both of which are allowed to erode. Balance between bed and bank erosion plays a key role in the morphodynamics of the channel. More specifically, rapid erosion of the bed can cause the channel to narrow even as bank erosion progresses. As the rate of bed erosion slows, bank erosion causes channel widening. This observed pattern is explained in the context of a theoretical model tested against the experiments.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Summary of the Experimental Investigation
  5. 3. Theoretical Formulation
  6. 4. Numerical Implementation
  7. 5. Numerical Simulations
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Integral Form of the Exner Equation of Sediment Continuity Over the Bed and Sidewall Regions
  11. Acknowledgments
  12. References

[2] Erosional and depositional processes in rivers induced by sea/lake elevation changes provide some of the most challenging problems in the prediction of fluvial morphology. One of the main consequences of water level in a reservoir or of the sea (base level drop) is fluvial channel incision upstream. An extended analysis of incised channels is given by Schumm et al. [1984]. Many of these same incisional processes are also observed after sudden (“blow and go”) dam removal, which represents an extreme case of base level drop. In the case of sudden dam removal, these processes occur at a relatively short timescale.

[3] Artificial reservoirs filled with sediment are commonly found worldwide. In industrialized countries, some of these dams have been removed in recent years, and many more are under study for removal. The impetus for dam removal is often environmental in nature, yet removal itself creates its own environmental problems. Moreover, the problem of dam removal constitutes not only a challenging problem from the point of view of engineering and geomorphology, but also raises serious issues from the social and economic points of view [Graf, 2002].

[4] In particular, in the past few years significant efforts have been devoted to investigate the transient response created by dam removal, and ultimate new equilibrium subsequently achieved by the affected river. Field monitoring of this response has allowed for qualitative description of some of the morphodynamic processes involved [Doyle et al., 2003]. Prediction of the effects and dynamic consequences of the removal of a dam is nevertheless not possible in the absence of more research from the field, experimental, theoretical, and numerical points of view. The recommendations given by the ASCE Task Committee on Hydraulics, Bank Mechanics, and Modeling of River Width Adjustment [1998a, 1998b] represent a clear message to the research community in regard to further basic research on bank erosion and channel evolution associated with dam removal. More specifically, research is needed on the erosion of the deltaic deposit upstream of the dam, the morphodynamic evolution of the resultant incised channel, and the consequent rate of delivery of sediment downstream.

[5] Matilija Dam, shown in Figures 1a and 1b, helps illustrate the magnitude of the problem. The dam is located on the Ventura River, approximatey 100 km northwest of Los Angeles, California. Here the deltaic deposit of the reservoir has reached the dam structure (Figure 1a), so that the dam no longer plays a role in flood mitigation (Figure 1b).

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Figure 1. Images of Matilija Dam (California) illustrating the following: (a) The deltaic deposit has reached the dam structure, and (b) the dam no longer mitigates floods. Photos courtesy of Paul Jenkin, Matilija Coalition.

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[6] A numerical model describing channel evolution after dam removal can play a useful role in the design of removal schemes. Sudden dam removal presents significant problems from the numerical point of view because (1) the steep streamwise bed slope created by dam removal induces high flow velocities and sediment transport rates, and (2) the width of the channel incising into the deposit evolves over time. The dam removal express assessment model (DREAM) suite of models [Cui et al., 2006a, 2006b], which address these issues, represent a major advance in the numerical prediction of the consequences of dam removal. These models treat not only the erosional processes as a channel incises into the reservoir deposit, but also the deposition occurring downstream. They are supported by experimental evaluation and have been applied to field cases.

[7] The DREAM models include a conceptualization of a feature that has not been included in most morphodynamic models of channel evolution, i.e., the input of sediment from the sidewalls of the channel as it incises. In order to implement these models it is necessary to prescribe a sidewall slope Ss and a minimum bottom width Bbm of the incisional channel. If the bottom width Bb of the incising channel is larger than this minimum value, it is allowed to narrow as it degrades without eroding its sidewalls. Once the minimum width Bm is attained, the sidewalls are allowed to erode as the channel degrades at constant width. The prescription of this minimum width is subjective.

[8] Recent experiments performed at St. Anthony Falls, however, add a new element to the problem. These experiments show that an incising channel can both erode its sidewalls and change its width at all stages in the process of incision. The channel first undergoes rapid incisional narrowing and then undergoes slow incisional widening [Cantelli et al., 2004].

[9] In the present paper a theoretical framework is developed to describe both incisional narrowing and widening. The framework divides the cross section into bed and bank regions. The Exner equation of sediment continuity is integrated across each region to determine equations of evolution for each region. The two regions are linked by the means of sediment delivery from the bank regions to the bed regions. These equations are closed using appropriate simplified assumptions for flow hydraulics and sediment transport. The relations so derived are then used to develop a numerical model of the morphodynamics of channel incision subsequent to dam removal.

2. Summary of the Experimental Investigation

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Summary of the Experimental Investigation
  5. 3. Theoretical Formulation
  6. 4. Numerical Implementation
  7. 5. Numerical Simulations
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Integral Form of the Exner Equation of Sediment Continuity Over the Bed and Sidewall Regions
  11. Acknowledgments
  12. References

[10] Experiments on sedimentation and erosion processes in reservoirs performed at St. Anthony Falls Laboratory [Cantelli et al., 2004] are focused on the coevolution of both channel width and longitudinal profile. The experimental program consisted of a first phase in which sedimentation was induced in the reservoir due to the progradation of a delta front, followed by a second phase in which erosion of the deltaic deposit was driven by dam removal. Two different methods of dam removal were used in second phase: instantaneous removal (“blow and go”) and staged removal. Ten experiments on dam removal were performed. Inflow water discharge was varied from run to run. The inflow sediment discharge was estimated based on the sediment transport relation of Meyer-Peter and Muller [1948] and a number of trial experiments. Two different types of sediments were used. The first type had a specific gravity of 2.67, a median size D50 of 0.80 mm, and a geometric standard deviation σg of 1.71. The second type had a more homogenous distribution, with a specific gravity of 2.65, a value of D50 equal to 0.33 mm, and a value of σg of 1.47.

[11] Target parameters for study during the erosional experiments were the longitudinal bed profile, the water surface, the solid discharge flowing beyond the position of the (former) dam, and the rate of bank erosion. During the erosion process the evolution of the width of the water surface of the channel incising into the deposit at different sections was observed using video cameras. An analysis of these experiments allowed the ascertainment of the influence of water discharge on both incision and sidewall erosion.

[12] An interesting phenomenon, termed “erosional narrowing” by the authors, was observed in five of the eight runs pertaining to sudden dam removal of them. More specifically, in the early stages of the erosion process the channel width at each section near the dam rapidly decreased to a minimum value, and subsequently slowly increased. The erosional behavior during the two runs with stepped removal was similar but not as distinct. The narrowing phenomenon is easily explainable in terms of the distribution of boundary shear stress. In each section, the boundary shear stress has a maximum value in the middle zone of the channel and drops to zero at the edges [e.g., Parker, 1978a, 1978b]. Thus, during channel incision, the erosion tends to be concentrated in the central zone. This differential erosion narrows and deepens the channel. Erosional narrowing also induces a streamwise convergence of the streamlines toward the channel center that increases net erosion, resulting in a positive feedback. The narrowing process was observed to continue until lateral sediment transport from the sidewalls sufficiently slows channel degradation and stops channel narrowing. Beyond this time the channel slowly widens as it incises. Figure 2 shows the time evolution of the width of the water surface for run 6; each line corresponds to a different section (i.e., to a different distance downstream of the inlet) in the vicinity of the dam. The successive appearance of a width minimum at increasing upstream distance illustrates the upstream migration of a wave of channel narrowing and incision. All these data converge to an asymptotic trend that shows a progressive erosion of the banks toward an equilibrium section. Figure 3 shows the evolution of the bed profile along the channel center during the erosion of the reservoir deposit for run 6. The erosion upstream of the delta front and the deposition downstream are clearly visible. The dotted line represents the initial bed profile before dam installation. At the end of the experiment the bed slope is approximately equal to the initial slope.

image

Figure 2. Evolution of channel water surface width upstream of the former dam after sudden removal. The origin of the streamwise coordinate in the legend is the sediment feed point; the position of the former dam was 9.0 m.

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image

Figure 3. Longitudinal profile measured in run 6 by Cantelli et al. [2004]. The origin of the horizontal axis is the upstream sediment feed point.

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[13] This observation is contrary to the general belief that the incisional channel widens from the very beginning [Schumm et al., 1984]. The delineation of a unified framework that describes both erosional narrowing and widening should lead to an improved understanding of the process of incision, and should provide the basis for a more accurate and versatile numerical model of the morphodynamics of dam removal. This framework is developed below.

3. Theoretical Formulation

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Summary of the Experimental Investigation
  5. 3. Theoretical Formulation
  6. 4. Numerical Implementation
  7. 5. Numerical Simulations
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Integral Form of the Exner Equation of Sediment Continuity Over the Bed and Sidewall Regions
  11. Acknowledgments
  12. References

3.1. Model Definition and Assumptions

[14] Figure 4 shows a schematization of the process of incision into a reservoir deposit observed in the experiments of Cantelli et al. [2004]. In these experiments a shallow channel of specified width that was always less than the deposit width was preexcavated before dam removal. Erosional narrowing is hypothesized to occur as a result of the lateral variation of the boundary shear stress, from a higher value at the channel center to a lower value along the bank regions. Under conditions of sufficiently rapid degradation, the contribution of sediment from the sidewalls cannot keep up with bed degradation, and as a result the channel narrows. As the degradation rate drops, narrowing gives way to widening.

image

Figure 4. Sketch representing the evolution of a cross section during the process of incision into a reservoir deposit.

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[15] The present theory uses several approximations to allow for a tractable description of the evolution of an incising channel. The flow is described in terms of a one-dimensional (1-D) formulation, but with channel width as a freely varying parameter. The normal flow approximation (i.e., uniform and steady flow) is used to describe momentum balance, so circumventing the need for a description of transcritical processes (in the sense of the Froude number) as a steep channel evolves into one with a milder slope. Although not shown here, a formal scale analysis does indeed justify this assumption for the experimental configuration used in the present work. The channel is assumed to be straight with a trapezoidal cross section. Only noncohesive bed sediment is considered here. The bed sediment is approximated as uniform in size and is assumed to move only as bed load. Future improvements to the model can allow relaxation of some of these approximations.

[16] Figures 5a and 5b illustrate the conceptual model. Bank erosion and channel bed degradation play key roles in determining the total sediment balance controlling the evolution of channel geometry, and in particular of channel width. The channel is abstracted to a laterally horizontal bed region, bounded on either side by a sidewall (bank) region with constant side slope Ss. Both the streamwise sediment transport imbalance due to incision and the depositional contribution from the channel bank erosion are considered in this formulation of the problem. In Figure 5b the streamwise volume bed load transport rates per unit width in the bed and sidewall regions are denoted as qbx and qsx, respectively. The volume transverse bed load rates per unit distance downstream in the bed and sidewall regions are denoted as qby and qsy, respectively.

image

Figure 5. Sketchs of the conceptual model for erosional narrowing: (a) trapezoidal cross section presenting al the relative notations and (b) 3-D representation including sediment transport rates. The cross section at time t is denoted with a solid line, and the cross section at time t + Δt is denoted with a dashed line.

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[17] Other parameters in Figure 5a are defined considering the cross section as follows: x is the streamwise coordinate, y is the transverse coordinate (with origin at the channel center and taken to be positive toward the right bank), and z is the vertical coordinate. The interface between the bed and the right-hand sidewall region is located at y = yb; the corresponding locations of the water surface and the top of the channel sidewall are y = yw and y = yt. In addition, H is the water depth (taken to be constant on the bed region); Bb is the half-channel width of the bed region; Bw is the half-channel width of the water surface; Bs is the width of one sidewall region (including both submerged and emergent banks); ηb is the bed elevation on the bed region; ηw is the elevation at the top of the active channel, i.e., that part of the channel that is immersed in water; ηt is the elevation of the top of the channel sidewall (which is always taken to be above the water surface); and L is the transverse arc length of one sidewall region. Note that parameters such as ηb are taken to be constant in the transverse direction y. The formulation is 1-D within the bed region and 1-D within the sidewall region, but quasi 2-D in that the boundary between these regions is allowed to move freely.

[18] The variables defined above are related by the following geometrical relationships, where the subscripts denote the locations indicated in Figure 5:

  • equation image
  • equation image
  • equation image
  • equation image
  • equation image
  • equation image

The channel is assumed to be symmetric in y, so that a morphodynamic description of the right-hand half-channel of Figure 5 is sufficient to describe the problem.

3.2. Flow Hydraulics

[19] The streamwise boundary shear stresses on the bed and bank regions are denoted τbx and τsx, respectively. As noted above, streamwise momentum balance is described in terms of the normal (steady, uniform) flow approximation.

[20] The equation of conservation of momentum in the streamwise direction for the entire half-channel (bed and sidewall region) takes the following form:

  • equation image

where ρ is the density of water, g is gravitational acceleration, and S0 is the bed slope in the streamwise direction. The streamwise boundary shear stress on the sidewall region of the active channel τsx is related to the corresponding boundary shear stress on the bed region τbx according to the relation

  • equation image

where ϕ is an order-1 dimensionless parameter which typically varies between 0.40 and 0.80 [e.g., Lane, 1955]. Here this parameter is treated as a prescribed constant; the authors recognize, however, that it is a function of, e.g., side slope Ss and width-to-depth ratio Bb/H.

[21] The experiments of Cantelli et al. [2004] demonstrate that the mode of bank erosion is somewhat different depending upon whether channel evolution is in the narrowing or widening phase. Figure 6 shows two frame-grabs from video at two different times of the erosional process. Both are top views of the channel near the location of the dam removed. Figure 6a shows the incising channel at a time when the downstream part is undergoing erosional narrowing but the upstream part remains undisturbed by the flow. Figure 6b shows the incising channel at a time when downstream is undergoing erosional widening, while the upstream reach is undergoing erosional narrowing. It is seen therein that the banks are steeper in the narrowing zones than in the widening zones. This difference may be due to capillary effects caused by rapid drawdown of the water table during erosional incision. This problem can be addressed by simply using different values of Ss and ϕ depending on whether the cross section is narrowing or widening. The Visual Basic (VBA) code used to implement the numerical model allows for this; a detailed description is provided in section 3.3. Approximate values of Ss and ϕ were extrapolated from the experiments.

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Figure 6. Top view of the channel (a) early on, when the region proximal to the dam (right) is undergoing erosional narrowing and the distal region (left) is unaffected, and (b) later, when the proximal region is undergoing erosional widening and the more distal regions is undergoing erosional narrowing. The flow is from left to right.

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[22] Between (3) in (2), it is found that

  • equation image

[23] The streamwise boundary shear stress on the bed region may be related to flow velocity as

  • equation image

where U is the cross-sectionally averaged streamwise flow velocity and Cf is a dimensionless friction coefficient, here evaluated using a Manning-Strickler formulation due to Parker [1991]:

  • equation image
  • equation image

where Rb denotes the hydraulic radius of the bed region of the half-channel, αr = 8.10, nr = 1/6, ks is a roughness height related to the bed surface size Ds90 such that ks = nkDs90, and nk is a dimensionless coefficient ranging from 1.5 to 4.

[24] Flow velocity U is approximated as constant across the cross section. Denoting by Qw the water discharge of the entire channel (left plus right halves), the equation of water conservation can be expressed as

  • equation image

Solving for H between (4), (5a), (5b), and (6) results in the relations

  • equation image

where

  • equation image
  • equation image

Thus c1 represents the depth correction due to sidewall effects. It is important to note that as H/Bb approaches zero, c1 approaches unity and H approaches the asymptotic value H0, i.e., the depth prevailing in a channel sufficiently wide to allow neglect for sidewall effects. The influence of the sidewalls becomes significant as H/Bb increases. The flow depth H can be solved from (7a)(7c) once Qw, ks, So, Ss, Bb, and ϕ are specified. The solution must be implemented iteratively.

[25] The Shields number for the bed region is then given as

  • equation image

where r denotes the submerged specific gravity of the sediment ( = (ρsρ)/ρ, where ρs denotes sediment density) and D denotes the median grain size diameter of the sediment in the bed surface and c2 is given as

  • equation image

Note that c2 [RIGHTWARDS ARROW] 1 as H/Bb [RIGHTWARDS ARROW] 0. Using (3), the Shields number on the sidewall region is then

  • equation image

For given values of Qw, ks, So, Ss, Bb, and ϕ then, it is possible to compute H from (7a), τb* from (8a), and τs* from (9).

[26] Equations (7c) and (8b) combined with equations (7a)(7b) and (8a) show that in sufficiently narrow channels the parameters c1 and c2 play a significant role in determining the depth of flow and the Shield stress on the bed and banks, and thus in regard to the description of erosional narrowing as well. They are less essential for the case of erosional widening, particularly as the channel is already sufficiently wide. The parameters c1 and c2 both decrease with increasing depth-width ratio H/Bb. These trends alone dictate a flow depth and bed shear stress that decrease as the channel becomes narrower. The right-hand side of (7b), however, depends on half width Bb to the −3/5 power. This latter dependence dominates, so that both depth H and bed shear stress τb increase with decreasing half width Bb. That is, erosional narrowing tends to increase the bed shear stress as well as the bank shear stress according to (3). This in turn increases the rate of bank erosion.

3.3. Sediment Transport Rates

[27] Bed load rates can be computed once the relevant hydraulic parameters are known. The model considers the total volume bed load transport rate per unit width in the streamwise direction qt, and the corresponding one in the transverse (normal) flow direction qy. In particular, as presented in Figure 3, qx on the bed region is denoted as qbx and the corresponding transport rate the sidewall region is denoted qsx; likewise, the volume bed load transport rate per unit stream length in the transverse direction qy is denoted as qby on the bed region and qsy on the sidewall region. In principle, all four of these parameters are free to vary in x, y, and t.

[28] Parker's [1979] approximation of the Einstein [1950] bed load transport is used to calculate the streamwise bed load transport rates. Using (9), then,

  • equation image
  • equation image

where αs ≈ 11.20 and τc* is the Shields number for incipient particle motion.

[29] Introducing the dimensionless number Nqr, qsx and qbx can be related as

  • equation image
  • equation image

As noted above, the Shields stress τb* increases as channel half width Bb decreases. Thus, according to (10a), a narrower channel results in an increased streamwise bed load transport rate. It is shown below that this effect moderates erosional narrowing.

[30] In principle, the critical Shields number τb* on the sidewall region should vary from a value near the bed value at the base of the bank to a lower value higher up on the submerged part of the bank [e.g., Ikeda et al., 1981; Kovacs and Parker, 1994; Seminara et al., 2003]. The present analysis assumes a specified sidewall slope Ss, which, in general, better characterizes the part of the sidewall that is not covered in water rather than the submerged part. As a result, the Shields number is taken to be everywhere equal to ϕτb*, where ϕ is equal to a specified constant between 0 and 1. In a more advanced model, the bank shear stress might vary linearly from 0 at the water's edge to τb* at the junction between the bed region and the bank region.

[31] The relationship of Parker and Andrews [1985] is used to compute the transverse volume bed load transport rate per streamwise width qby. This relation takes the following respective forms on the bed and sidewall regions:

  • equation image
  • equation image

where η denotes bed elevation, τby and τsy denote the transverse boundary shear stress on the bed and sidewall regions, respectively, and αn is approximated as 2.65 after Johannesson and Parker [1989].

[32] As seen in Figure 5, the bed region is approximated here as horizontal in the transverse direction, so that ∂η/∂y is taken to vanish there. Straight channel secondary currents are also neglected in the present analysis, so that τby and τsy are taken to vanish. As a result, (11a) predicts vanishing transverse bed load transport on the bed region. On the sidewall region, however, ∂η/∂y = Ss, so that qsy, may be estimated as

  • equation image

Note that in the present simplified analysis qsy does not vary in the transverse direction. According to (12a), qsy is always directed in the −y direction, i.e., from the right sidewall region of Figure 5 to the bed region. It is thus useful to define a positive quantity equation imagesy such that

  • equation image

[33] As explained in section 3.2, both the water depth and the Shields parameter of the bed and the sidewall regions increase with the narrow of the channel. This in turn implies increases in the magnitudes of both the streamwise and transverse bed load transport rates, and thus an increased capacity to erode the channel banks (sidewalls).

3.4. Sediment Continuity in the Bed and Sidewall Regions

[34] The 2-D form of the Exner equation of sediment continuity equation is

  • equation image

where t is time and λp is the porosity of the sediment in the bed deposit, here taken to be a specified constant. The problem is reduced to two 1-D problems by integrating (13) in the transverse (y) direction over (1) the bed region and (2) the sidewall region of the right half of the channel of Figure 5. The details of the integration are somewhat tedious and are thus given in Appendix A.

[35] Equation (13) is integrated over the right half of the bed region of Figure 5 (i.e., from y = 0 to y = yb), and is reduced with Leibnitz' rule in Appendix A to obtain (A9), i.e., the following result:

  • equation image

In the above relation, ηb is the bed elevation on the bed region and qbx is the streamwise volume bed load transport rate on the bed region; both of these parameters may vary in x and t but are assumed to be constant in y. In addition, equation imagesyj is a positive quantity, denoting the transverse sediment flux exchanged at the boundary between the bank region and the bed, as shown in detail in Appendix A. Here equation imagesyj is estimated using equation (A8b), which is in turn obtained from (12a) and (12b):

  • equation image

The first term on the right-hand side of (14) is negative for a degrading (incising) channel, because degradation causes the transport rate to increase in the streamwise direction. The second term, on the other hand, drives channel aggradation as sediment is supplied to the bed region from bank erosion.

[36] In Appendix A, equation (13) is also integrated over the entire sidewall region of the right-hand side of Figure 5 including both the submerged and emergent regions, i.e., from y = yb to y = yw, and reduced with Leibnitz' rule to obtain (A19), i.e.,

  • equation image

In the above relation, qsx denotes the characteristic streamwise volume bed load transport rate per unit width on the sidewall region, here taken to be independent of y. The combination of equations (14) and (16) gives rise to the governing equation for the time evolution of the channel width as follows:

  • equation image

[37] Using (14) to reduce the first term on the right-hand side of (17), the following equivalent form governing the evolution of channel bottom width is obtained:

  • equation image

Equations (16), (17), and (18) all contain a term proportional to ∂H/∂x. The following approximation, which is valid as long as the channel is not too narrow, is employed below:

  • equation image

This assumption is not necessary for the analysis, in that ∂H/∂x can be calculated from (7a)(7c) as a function of ∂Bb/∂x and ∂So/∂x and incorporated into the analysis. In the case under consideration here, however, in which flow conditions are not far from the threshold of motion, (10b) specifies a very steep relation between qsx and τb*, so indeed rendering the first term in the left-hand side of (18b) considerably larger than the second term in question.

[38] Equations (14) and (17) can be better understood by considering a simpler case derived from them. In this simpler case the channel is allowed to degrade (∂ηb/∂t < 0) and erode its banks (equation imagesyj > 0) without changing width (∂Bb/∂t = ∂Bb/∂x = 0); in addition, bed load transport on the sidewall region is neglected (qsx = 0). For this case (17) reduces to

  • equation image

[39] Thus as the bed degrades it pulls down the adjacent sidewall at the same rate, so introducing sediment into the bed region. Between (20) and (14), the Exner equation of sediment continuity then becomes

  • equation image

This is the formulation used by Cui et al. [2006a] once the channel has reached some specified “minimum width” which corresponds to the variable Bb used here (see also Parker [2004] and http://cee.uiuc.edu/people/parkerg/morphodynamics_e-book.htm). According to (20), sidewall erosion slows degradation via the term (ηtηb)/Ss on the right-hand side, but does not stop it.

[40] In the present more general formulation of (14) and (18), the analysis is extended to the case of varying width Bb. Equation (18) in particular illustrates the essential terms that drive erosional widening and narrowing. Insofar as equation imagesyj is always positive for a degrading channel, the first term on the right-hand side of (18), which quantifies bank erosion, always drives erosional widening. In the steep zone just upstream of the former position of the dam, however, the streamwise bed load transport rate qbx can be expected to be strongly increasing downstream, so that ∂qbx/∂x > 0. As a result, the second term on the right-hand side of (18) drives erosional narrowing. This same term drives rapid erosion and a downward-concave longitudinal profile, as seen in Figure 3. The third term on the right-hand side of (18) abets erosional narrowing in zones of converging flow, i.e., where ∂Bb/∂x < 0. Such a zone is seen in Figure 6.

3.5. Reduction of the Governing Equations

[41] The various terms in equations (14) and (18) can be reduced to functions of the channel bottom width Bb and bed slope S0 with the aid of (7)(10). For example, the term ∂qbx/∂x in (18) can be written as

  • equation image

where the dimensionless parameters NqB, image and NτB are given as

  • equation image
  • equation image
  • equation image

[42] Equation (14) then becomes

  • equation image

In order to reduce (18) the following additional dimensionless parameters are defined with the aid of (7a)(10), (12) and (15):

  • equation image
  • equation image

where

  • equation image
  • equation image
  • equation image

where Nqr is given by (10d). The calculations that lead to the above forms are laborious but straightforward. The governing equation for the evolution of channel bottom width (18) then becomes

  • equation image

where Bs is given by (1e).

[43] The interpretation of (27) is similar to that of (18). When the bed is degrading, the first term on the right-hand side of (27) always drives erosional widening. The second term on the right-hand side drives erosional narrowing for the case ∂So/∂x > 0, i.e., a concave-downward bed profile, as can be expected to prevail just upstream of the former location of a dam shortly after its removal. The third term on the right-hand side drives erosional narrowing for a converging flow, i.e., ∂Bb/∂x < 0, and erosional widening for a diverging flow.

3.6. Critical Condition for Inception of Erosional Narrowing

[44] One observation from the experiments by Cantelli et al. [2004] is that erosional narrowing does not always occur after the sudden removal of a dam. If the channel is incising into noncohesive sediment, whether or not erosional narrowing occurs depends upon the degree of downward concavity established immediately after removing the dam. A critical condition for the inception of erosional narrowing should thus exist. A first attempt at defining this condition follows.

[45] The channel width decreases in time during erosional narrowing; that is, ∂Bb/∂t ≤ 0. If the channel reach where erosional narrowing is concentrated corresponds to a section of approximately constant width in the longitudinal direction (i.e., near the cross section of minimum width in Figure 6), the streamwise derivative of Bb can be approximated as vanishing (∂Bb/∂x = 0). In consequence, equation (27) can be rearranged to determine the downward concavity in the streamwise bed profile required for erosional narrowing to occur. The degree of downward concavity is expressed in terms of the parameter ∂S0/∂x, which must be positive. The constraint on downward concavity obtained from (27) is expressed in dimensionless form below:

  • equation image
  • equation image

4. Numerical Implementation

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Summary of the Experimental Investigation
  5. 3. Theoretical Formulation
  6. 4. Numerical Implementation
  7. 5. Numerical Simulations
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Integral Form of the Exner Equation of Sediment Continuity Over the Bed and Sidewall Regions
  11. Acknowledgments
  12. References

4.1. Initial and Boundary Conditions

[46] The transformed equations (24) and (27) represent the governing equations for the time evolution of the bed elevation in the bed region ηb and the channel bottom width Bb, respectively. Since both of these coupled partial differential equations are highly nonlinear, they are solved numerically. One initial condition and two boundary conditions are required for ηb in equation (24), while one initial condition and only one boundary condition are needed for Bb in equation (17).

[47] The experiments performed by Cantelli et al. [2004] indicate that as a first reasonable approximation the reach upstream of the original dam degrades around a “pivot” point located immediately upstream of the removed dam. The numerical model does not yet explicitly include the aggradational region located downstream of the (removed) dam, though the influence of the aggradational region appears implicitly via the pivot point. Initial condition for ηb corresponds to the final longitudinal profile of the progradational delta front resulting from the previous dam-induced sedimentation.

[48] Although the initial condition for bottom width Bb can in principle be set to any value, here it is set equal to the equilibrium width associated with the specified water discharge Qw, the slope of the delta topset ST before incision takes place, and grain size D. This equilibrium width is obtained by specifying the condition that the sediment is at the threshold of motion in the sidewall region, so that equation imagesyj vanishes. This condition in combination with (15) yields the constraint

  • equation image

[49] Equations (8a) and (29) applied to the initial configuration yield

  • equation image

[50] Equations (7a) and (7b) applied to the initial configuration likewise yield the following relation for the initial width BbI:

  • equation image

[51] In general this equation must be solved iteratively in combination with (7a)(7c) and (8b), but the iteration can be commenced by setting c1 and c2 equal to unity.

[52] The reach of interest is defined to have length L, where x = L denotes the pivot point (just upstream of the location of the former dam) discussed above. Taking the pivot point as the elevation datum, the boundary condition there applying to takes the form

  • equation image

[53] A second boundary condition for ηb is derived from the assumption of a specified constant total volume bed load supply rate at the upstream end of the channel. Denoting the total volume bed load transport rate as Qx, then,

  • equation image

where Qxu denotes the supply rate of bed load. In general, Qxu can be specified arbitrarily, but in the present work the value used is the one that is in equilibrium with the initial bed slope of the channel (such that neither aggradation nor degradation would occur if the channel were to have the initial slope in the absence of a dam).

[54] The total streamwise bed load transport rate is equal to the sum of the rates in the channel and bed regions,

  • equation image

but in light of the imposition of the condition that the sidewall region is at the threshold of motion, (34a) reduces to

  • equation image

[55] Applying (7a), (7b), (8a), (10a), (29), and (33) at the upstream end of the reach, the following boundary condition on bed slope S0 is obtained:

  • equation image

[56] The solution is again obtained iteratively in conjunction with (7a)(7c) and (8b), and again the iteration can be commenced by setting c1 = c2 = 1.

[57] Equation (27) can be recast in the following form:

  • equation image

[58] The terms on the left-hand side of equation (36a) take the form of a kinematic wave equation, for which disturbances propagate upstream with wave speed cB, where

  • equation image

[59] The problem is, however, more complicated than this, because the term on the right-hand side of (36a) is coupled to (24). A linearized analysis of homogeneous versions of (24) and (36a) yield a second-order equation for complex celerity, suggesting the possibility of two wave speeds. The results of the numerical analysis, however, confirm the existence of an upstream propagating morphodynamic wave of the type indicated by the left-hand side of (36a).

[60] The form (25b) implies that NB is positive for sufficiently small values of H/Bs, and specific calculations indicate that for most cases of interest NB is thus positive. As a result, (36b) indicates a negative wave speed, so that the width disturbance propagates upstream. As a result, the appropriate boundary condition on (36a) should be applied at the downstream end of the reach.

[61] A reasonable assumption for this boundary condition is that the pivot point at x = L also constitutes a point of maximum bed load transport rate, so that upstream of this point the sediment load increases downstream (incising channel) and downstream of this point the sediment load decreases downstream (aggrading channel). The boundary condition on Bb can thus be obtained from the relation

  • equation image

Reducing (34a) with the aid of (10c) and (22), the following mixed boundary condition on Bb is found to hold at the downstream end:

  • equation image

where

  • equation image

[62] The term on the right-hand side of (25a) and (25b) becomes negligible for sufficiently large aspect ratio Bb/H. Insofar as this condition was found to prevail in the present analysis, (38a) has been approximated to the form

  • equation image

5. Numerical Simulations

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Summary of the Experimental Investigation
  5. 3. Theoretical Formulation
  6. 4. Numerical Implementation
  7. 5. Numerical Simulations
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Integral Form of the Exner Equation of Sediment Continuity Over the Bed and Sidewall Regions
  11. Acknowledgments
  12. References

[63] Experiment 7 of Cantelli et al. [2004] is employed here in order to test the numerical simulation. The following parameters were used: reach length L = 8.8 m, nk = 4.0, αr = 8.1, αs = 11.2, τc* = 0.024, αn = 2.65, r = 1.67, D = 0.80 mm, Qw = 0.3 L/s, λp = 0.40, ϕ = 0.70, and Ss = 0.33, respectively, in the narrowing phase and ϕ = 0.22 and Ss = 1.3 in the widening one; slope values were estimated from the experiments. In addition, D90 was estimated from the median size D, the geometric standard deviation of the sediment mixture σg of 1.71, and the assumption of a lognormal grain size distribution.

[64] While the differential equations of the numerical model govern the bed elevation ηb and bottom width Bb, the data from the experiment pertain to water surface elevation ηb + H and water surface half width Bb + H/Ss. This water surface elevation and width is easily calculated from ηb, Bb, (1b), (1f), and (7a)(7c). Figure 7 shows a comparison of the time evolution of observed and predicted channel water surface width at different cross sections located 8.20, 8.30, and 8.40 m downstream of the sediment feed point. For reference, the former location of the dam is x = 9.0 m and the pivot point is located at x = 8.8 m. Figure 8 shows a comparison of observed and predicted long profiles of water surface elevation.

image

Figure 7. Calculated (solid curve with solid symbols) and measured (solid curve with open symbols) time variation of channel surface width at 8.20, 8.30, and 8.40 m downstream of the sediment feed point.

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image

Figure 8. Assumed initial water surface long profile immediately after dam removal (dashed curve), and calculated (solid curve with solid symbols) and measured (solid line with open symbols) water surface long profiles 1200 s after removal.

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[65] The model clearly captures (1) the strongly downward-concave long profile evolving after dam removal, and (2) rapid erosional narrowing followed by slow erosional widening. The model also shows a damped, upstream migrating wave of erosional narrowing, in agreement with the experimental data. The model shows some disagreement with the data shortly after dam removal. The disagreement in the early stages of incision may be due to (1) the very high bed slope and resulting Froude-supercritical flow and (2) the very high, partially gravity driven sediment transport rate prevailing just upstream of the former dam right after removal. The present model can capture neither transcritical flow nor sediment transport as a slurry on very steep slopes. Subsequent to this early period, however, the model provides a reasonable description of both the evolution of channel width and longitudinal profile.

[66] Figure 9 shows the predicted long profiles of volume bed load transport per unit width qsx at various times. The model predicts extremely high bed load transport rates in the vicinity of the dam just after removal. This high transport rate declines in time at a streamwise point as incision proceeds, but the zone of high transport rate gradually migrates upstream.

image

Figure 9. Predicted long profiles of volume bed load transport rate per unit width qsx at various times.

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6. Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Summary of the Experimental Investigation
  5. 3. Theoretical Formulation
  6. 4. Numerical Implementation
  7. 5. Numerical Simulations
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Integral Form of the Exner Equation of Sediment Continuity Over the Bed and Sidewall Regions
  11. Acknowledgments
  12. References

[67] The present treatment abstracts a complex 2-D morphodynamic problem into two simpler 1-D morphodynamic problems based on the decomposition of the cross section into bed and bank regions. This simplification invariably results in some degradation of the predictive capabilities of the model. Figures 7 and 8, however, indicate that the model captures the essence of the incision process, including both erosional narrowing and erosional widening. The numerical model performs reasonably well except just upstream of the former dam shortly after removal. This limitation is to be expected, in that the model does not encompass sediment transport processes at the extremely steep initial bed slope near the dam.

[68] Only a modest amount of “tuning” was required to obtain the results of Figures 7 and 8. Of all the input parameters, the only ones that were “tuned” were the critical Shields number τc*, the side slope Ss, and the ratio of streamwise bank shear stress to streamwise bed shear stress ϕ. The values used for these parameters, however, fall well within the range that might be expected for the experiments of Cantelli et al. [2004].

[69] The model uses the assumption of normal flow everywhere. A more detailed analysis could be obtained by using a St. Venant formulation of momentum balance. Such an implementation is not straightforward, because the flow near the dam is Froude-supercritical right after dam removal, but becomes subcritical shortly afterward. The work of Zech et al. [2005] provides suggestions as to how the morphodynamic model might be modified to encompass transcritical flow.

[70] At present the model is limited to uniform noncohesive sediment deposits. The constraint of uniform sediment does not appear difficult to relax. A treatment of incision into cohesive material, on the other hand, invariably requires an empirical relation for bed erosion, and must encompass the formation of an upstream migrating knickpoint. In sufficiently large reservoirs the cohesive sediment deposits as a bottomset, which is eventually buried by a prograding topset/foreset which often consists largely of noncohesive sediment [e.g., Smith et al., 1954]. In such cases the assumption of noncohesive sediment likely prevails until the bottomset is exhumed. In many smaller dams, such as those studied by the reservoir, deposit may be relatively cohesive throughout. The present model would require considerably more adaptation in order to be applicable to this case.

[71] Implementation of the present model requires the prescription of an initial “starter channel.” This “starter channel” then evolves according to the morphodynamics of erosional narrowing/widening. In nature, however, no starter channel exists. A better description of the very early evolution of the channel from an arbitrarily small disturbance (low point) toward the upstream end of the deltaic deposit would be of value.

7. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Summary of the Experimental Investigation
  5. 3. Theoretical Formulation
  6. 4. Numerical Implementation
  7. 5. Numerical Simulations
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Integral Form of the Exner Equation of Sediment Continuity Over the Bed and Sidewall Regions
  11. Acknowledgments
  12. References

[72] The experiments of Cantelli et al. [2004] revealed a new phenomenon pertaining to the incision of a channel into a reservoir deposit after sudden dam removal. More specifically, the experiments showed that a later stage of bed erosion accompanied by channel widening (erosional widening) can be preceded by an earlier state of bed erosion accompanied by channel narrowing (erosional narrowing).

[73] The 2-D problem of erosional narrowing/widening is here formulated analytically by abstracting the channel into bed and sidewall regions. Integration in the transverse direction and appropriate approximations allow a reduction to two 1-D problems, one describing the morphodynamics of the bed region and the other describing the morphodynamics of the sidewall regions. After some manipulation, the problem is found to be formulated in terms of one differential equation governing bed elevation and another differential equation governing channel width.

[74] A numerical solution of these equations proves to capture the essence of both erosional narrowing and widening. The model provides reasonable agreement with data from one of the experiments of Cantelli et al. [2004] everywhere and at all times except very near the former dam shortly after removal.

[75] Several suggestions are made for improving the model, including the incorporation of sediment mixtures and the treatment of cohesive sediment.

Appendix A:: Integral Form of the Exner Equation of Sediment Continuity Over the Bed and Sidewall Regions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Summary of the Experimental Investigation
  5. 3. Theoretical Formulation
  6. 4. Numerical Implementation
  7. 5. Numerical Simulations
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Integral Form of the Exner Equation of Sediment Continuity Over the Bed and Sidewall Regions
  11. Acknowledgments
  12. References

[76] The essentially 2-D (streamwise/transverse) problem of the coevolution of bed and bank erosion is here reduced to two linked 1-D problems by means of integration in the transverse direction and simplification of the result. The 2-D Exner equation for sediment continuity, given as (15) above, is here repeated for clarity:

  • equation image

[77] The results of integrating the above equation in the transverse direction over (1) the bed region and (2) the sidewall region are given here. The analysis is similar to that of Parker [2004] (see also http://cee.uiuc.edu/people/parkerg/morphodynamics_e-book.htm), but several errors in that analysis are corrected here.

[78] Equation (A1) is first integrated over the bed region, i.e., from y = 0 to y = yb of the right-hand half-channel of Figure 5 to yield

  • equation image

[79] The term on the left-hand side of (A2) is reduced with (1a) and Leibnitz' rule to yield

  • equation image

[80] On the bed region, however, η is approximated as horizontal in the transverse direction, so that the first term on the right-hand side of (A3) takes the form /∂t(Bbηb), and (A3) itself reduces to

  • equation image

[81] The first term on the right-hand side of (A2) is reduced with Leibnitz' rule to

  • equation image

[82] Approximating qx as everywhere equal to a value qbx that is independent of y on the bed region and again using (1a), the first term on the right-hand side of (A2) becomes

  • equation image

[83] The second term on the right-hand side of (A2) integrates to

  • equation image

[84] The parameter on the right-hand side of (A7) is the transverse sediment transport rate at the interface between the bed region and the sidewall region of the right half-channel of Figure 5. Since y is defined to be positive on the right half-channel, a negative value of qyyb denotes a net transfer of sediment from the sidewall region to the bed region due to bank erosion. It is thus useful to redefine this parameter into a form that can be expected to be positive for the case at hand, i.e., incision with bank erosion:

  • equation image

[85] Between (A8a) and (13), then, equation imagebsnj is estimated as

  • equation image

[86] A reduction of (A7) with (A8a) yields

  • equation image

[87] Between (A4), (A6), and (A8b), then, the integral equation for sediment conservation in the bed region (A2) becomes

  • equation image

[88] A similar integration of (A1) is now performed on the sidewall region of the right-hand side of the channel in Figure 5. The integration is performed so as to include both the submerged (active) part of the channel (ybyyw) and the subaerial (inactive) region up to the top of the sidewall in Figure 5 (ywyyt):

  • equation image

[89] Between yb and ye, the bed elevation η obeys the relation

  • equation image

[90] Given that the side slope Ss is constant here, differentiation of (A11) with respect to time yields

  • equation image

[91] Thus reducing the first term on the left-hand side of (A10), with (A3), (1a) to (1d), (1c) and (1f) yields

  • equation image

In the first term on the right-hand side of (A10) the upper limit of the integration yt may be replaced with yw because streamwise sediment transport occurs only in the submerged (active) channel. Thus the term integrates with Leibnitz' rule to

  • equation image

[92] Reducing (A14) with (1a), (1e), (1f) and the assumptions that (1) the streamwise bed load transport rate qx vanishes at the water's edge, where y = yt and (2) on the sidewall region itself qx can be characterized by a value qsx that does not vary in y, it is found that

  • equation image

[93] The second term on the right-hand side of (A10) integrates to

  • equation image

[94] It is assumed here that no sediment is supplied to the top of the emergent bank in Figure 5, where y = yt:

  • equation image

Substituting (1a), (1c), and (1f) and (A8c) into (A16) thus results in

  • equation image

[95] Using (A13), (A15), and (A18), the integral equation for sediment conservation (A15) in the sidewall region can be reduced to

  • equation image

[96] Reducing the term on the right-hand side of (A19) with (A9) and rearranging, it is found that

  • equation image
Notation
Bb

half-channel width of the bed region.

Bw

half-channel width of the water surface.

Bs

width of one sidewall region (including both submerged and emergent banks).

Bss

width of one submerged sidewall region.

BbI

initial width.

c1

depth correction due to sidewall effects.

C2

Shields number correction due to sidewall effects.

Cf

dimensionless friction coefficient evaluated using a Manning-Strickler formulation due to Parker [1991].

D50

median grain size.

g

gravitational acceleration.

H

water depth.

ks

roughness height (ks = nkDs90).

L

transverse arc length of one sidewall region.

ρ

specific gravity.

nr

1/6.

nk

dimensionless coefficient ranging from 1.5 to 4.

σg

geometric standard deviation.

qx

total volume bed load transport rate per unit width in the streamwise direction.

qy

total volume bed load transport rate per unit width in the transverse (normal) flow direction.

qbx

streamwise volume bed load transport rates per unit width in the bed region.

qsx

streamwise volume bed load transport rates per unit width in the sidewall region.

qby

volume transverse bed load rates per unit distance downstream in the bed region.

qsy

volume transverse bed load rates per unit distance downstream in the sidewall region.

equation imagesyj

positive quantity which parameterizes the sediment supply to the bed region from the eroding bank.

Qw

denotes the water discharge of the entire channel (left plus right halves).

Qx

total volume bed load transport rate.

Rb

denotes the hydraulic radius of the bed region of the half-channel.

r

submerged specific gravity of the sediment.

S0

bed slope in the streamwise direction.

Ss

side slope.

t

time.

U

cross-sectionally averaged streamwise flow velocity.

x

streamwise coordinate.

y

transverse coordinate (with origin at the channel center and taken to be positive toward the right bank).

yb

interface between the bed and the right-hand sidewall region.

yw

corresponding location of the water surface.

yt

corresponding location of the water surface the top of the channel sidewall.

z

vertical coordinate.

αr

8.10.

αs

∼11.20.

ηb

bed elevation on the bed region.

ηw

elevation at the top of the active channel, i.e., that part of the channel that is immersed in water.

ηt

elevation of the top of the channel sidewall (which is always taken to be above the water surface).

τbx

streamwise boundary shear stresses on the bed.

τsx

streamwise boundary shear stresses on the bank regions.

ρ

density of water.

ϕ

parameter that relates the streamwise boundary shear stress on the sidewall region of the active channel τsx with the corresponding boundary shear stress on the bed region τbx.

τc*

Shields number for incipient particle motion.

λp

porosity of the sediment in the bed deposit.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Summary of the Experimental Investigation
  5. 3. Theoretical Formulation
  6. 4. Numerical Implementation
  7. 5. Numerical Simulations
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Integral Form of the Exner Equation of Sediment Continuity Over the Bed and Sidewall Regions
  11. Acknowledgments
  12. References

[97] This material is based upon work funded by the National Science Foundation under agreement number EAR-0207274, as well as the Science and Technology Center Program under agreement number EAR-0120914. This paper represented a contribution of the National Center for Earth-Surface Dynamics, a Science and Technology Center funded by the U.S. National Science Foundation. Partial support also came from Fondazione Cariverona (Progetto MODITE).

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Summary of the Experimental Investigation
  5. 3. Theoretical Formulation
  6. 4. Numerical Implementation
  7. 5. Numerical Simulations
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Integral Form of the Exner Equation of Sediment Continuity Over the Bed and Sidewall Regions
  11. Acknowledgments
  12. References
  • ASCE Task Committee on Hydraulics, Bank Mechanics, and Modeling of River Width Adjustment (1998a), River width adjustment: I. Processes and mechanisms, J. Hydraul. Eng., 124(9), 881902.
  • ASCE Task Committee on Hydraulics, Bank Mechanics, and Modeling of River Width Adjustment (1998b), River width adjustment: II. Modeling, J. Hydraul. Eng., 124(9), 903917.
  • Cantelli, A., C. Paola, and G. Parker (2004), Experiments on upstream-migrating erosional narrowing and widening of an incisional channel caused by dam removal, Water Resour. Res., 40, W03304, doi:10.1029/2003WR002940.
  • Cui, Y., G. Parker, C. Braudrick, W. E. Dietrich, and B. Cluer (2006a), Dam removal express assessment models (DREAM): part 1. Model development and validation, J. Hydraul. Res., 44(3), 291307.
  • Cui, Y., C. Braudrick, W. E. Dietrich, B. Cluer, and G. Parker (2006b), Dam removal express assessment models (DREAM): part 2. Sample runs/sensitivity tests, J. Hydraul. Res., 44(3), 307323.
  • Doyle, M. W., E. H. Stanley, and J. M. Harbor (2003), Channel adjustments following two dam removals in Wisconsin, Water Resour. Res., 39(1), 1011, doi:10.1029/2002WR001714.
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  • Graf, W. L., (Ed.) (2002), Dam Removal Research: Status and Prospects, 164 pp., H. John Heinz III Cent. for Sci., Econ., and the Environ., Washington, D. C.
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  • Kovacs, A., and G. Parker (1994), A new vectorial bedload formulation and its application to the time evolution of straight river channels, J. Fluid Mech., 267, 153183.
  • Lane, E. W. (1955), Design of stable channels, J. Hydraul. Eng., 120, 12341279.
  • Meyer-Peter, E., and R. Muller (1948), Formulas for bed load transport, paper presented at the Third Conference, Inst. Assoc. of Hydraul. Res., Stockholm.
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  • Parker, G. (1978b), Self-formed straight rivers with equilibrium banks and mobile bed: part 2. The gravel river, J. Fluid Mech., 89(1), 127146.
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  • Parker, G. (2004), The sediment digester, Internal Memo.117, 17 pp., St. Anthony Falls Lab., Univ. of Minn., Minneapolis.
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  • Smith, W. O., et al. (1954), Comprehensive survey of Lake Mead: 1948–1949, U.S. Geol. Surv. Prof. Pap., 295, 253 pp.
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