Numerical simulation of bank erosion and channel migration in meandering rivers

Authors


Abstract

[1] A numerical model of river morphology for meander bends with erodible cohesive banks is herein developed and tested. The new model has three key features. First, it couples a two-dimensional depth-averaged model of flow and bed topography with a mechanistic model of bank erosion. Second, it simulates the deposition of failed bank material debris at and its subsequent removal from the toe of the bank. Finally, the governing conservation equations are implemented in a moving boundary fitted coordinate system that can be both curvilinear and nonorthogonal. This simplifies grid generation in curved channels that experience bank deformation, allowing complex planform shapes associated with irregular natural channels to be simulated. Model performance is assessed using data from two flume experiments and a natural river channel. Results are encouraging, but the model underpredicts the scour depth in pools adjacent to the outer bank and, consequently, underpredicts bank migration rates.

1. Introduction

[2] Perhaps the most common river planform is the single-thread meandering river. Such rivers have been the subject of intensive research due to their aesthetic appeal and the intriguing question as to the mechanism of formation and evolution of the sinuous planform. Examples of previous studies relating to the issue of river meandering include the study of bend flow processes [Thorne and Hey, 1979; Dietrich and Smith, 1983; Odgaard and Bergs, 1988; Hodskinson and Ferguson, 1998], sediment transport and grain-size sorting in bends [Allen, 1970; Parker and Andrews, 1985; Bridge, 1992; Sun et al., 2001a, 2001b], the morphology and development of individual meanders [Carlston, 1965; Brice, 1974; Thompson, 1986], and the study of the geological processes that operate at the scale of entire meander trains [Schumm, 1968; Furbish, 1991; Sun et al., 2001c].

[3] The approaches used to predict the morphology and evolution of river meanders have also varied substantially [Sun et al., 1996]. Each of these approaches have provided insight into the nature of river meandering, but a number of scientific issues still remain. For example, most empirical [e.g., Carlston, 1965; Brice, 1974; Thompson, 1986; Nanson and Hickin, 1986] and analytical [Allen, 1970; Parker and Andrews, 1985; Bridge, 1992] models of meander morphology are limited to steady state conditions. On the other hand, process-based bend erosion models [Ikeda et al., 1981; Parker et al., 1982; Odgaard, 1989; Crosato, 1990; Howard, 1992; Sun et al., 1996, 2001a, 2001b, 2001c; Seminara et al., 2001] have reproduced the long-term development of meandering forms. However, these approaches utilize idealized and nonmechanistic relationships that link bank erosion rate and near-bank flow velocity, via an erodibility coefficient that is determined by calibration, rather than in relation to the physical characteristics of the sedimentary environment. Another limitation is that some theories are restricted to artificial morphologies tied to idealized representations of the river planform, such as the most probable path or sine-generated curve [Langbein and Leopold, 1966; Thakur and Scheidegger, 1970]. Such models are limited in scope because natural river meanders often have irregular shapes. Finally, the validity of some models is limited because they do not account for all the degrees of freedom involved in channel adjustment. Specifically, many meander models [Nelson and Smith, 1989; Odgaard, 1989; Bridge, 1992; Alabyan, 1996; Sun et al., 1996, 2001a, 2001b, 2001c] neglect the adjustment of channel width. This biases predictions of flow, sediment transport and bed-level change in rivers with erodible banks [Thorne and Osman, 1988; Darby et al., 1996].

[4] Recent studies have attempted to address some of these limitations. A numerical model of channel evolution which accounts for bank erosion, as well as adjustments of bed topography, gradient and bed material sorting, has been developed and tested [Darby and Thorne, 1996a; Darby et al., 1996; Simon and Darby, 1997a]. This work involved combining geotechnical bank stability analyses [Darby and Thorne, 1996b] with flow and sediment routing models to account for widening in incised rivers where the banks are undermined and collapse as a result of bed-degradation and bank-toe scour. However, the Darby-Thorne model is limited to straight rivers.

[5] Meanwhile, researchers in the Netherlands at Delft University of Technology and Delft Hydraulics have been developing and refining numerical models of flow and bed topography for meandering channels [Kalkwijk and De Vriend, 1980; De Vriend, 1981; Struiksma et al., 1985; Olesen, 1987; Crosato, 1990; Talmon, 1992]. This work has now led to the development of a two-dimensional (2D) depth-averaged numerical model (RIPA) of flow and bed topography for single-thread rivers with irregular planform [Mosselman, 1991, 1992, 1998]. RIPA differs from the Delft flow and sediment transport models in that it includes a method to predict bank migration, with bank erosion rate modeled as a function of excess bank height. This recognizes implicitly the significance of mass wasting in the bank erosion process, but its implementation still needs an erodibility coefficient whose value is determined by calibration.

[6] In this paper the RIPA model is developed further by replacing its existing bank erosion submodel with a more mechanistic approach [Osman and Thorne, 1988]. This approach simulates changes in bank stability conditions resulting from deformation of the bank profile caused by direct fluvial shear erosion of the bank materials and near-bank bed degradation. Additional modifications relating to the description of the deposition of bank material debris on the bed of the channel after mass failure, and its subsequent removal by the flow, are also described herein. The main advantages of the new model are as follows. First, the inclusion of a more realistic model of bank erosion, collapse and subsequent deposition of failed debris allows further insight into the influence of sedimentary characteristics on meander evolution to be obtained [Thorne and Osman, 1988; Huang and Nanson, 1998; Millar, 2000]. Second, the new model is a dynamic model and is not, therefore, restricted to steady state conditions. Finally, the coordinate system employed by RIPA is well suited for simulating the (irregular) planforms associated with natural single-thread meandering rivers. A similar modeling approach has recently been proposed by Nagata et al. [2000], though their model is more appropriate for noncohesive bank failures and does not have such a detailed description of the bank erosion process.

2. Model Description

2.1. Coordinate System

[7] One of the key criteria for selecting the RIPA model for use in the present research is its use of a channel-fitted coordinate system that can be both curvilinear and nonorthogonal. In this system the s direction is taken to follow the streamwise direction of the river, whereas the n direction is the transverse direction, and the local skewness (ε) is defined as the cosine of the angle between the base vectors of the coordinate system (Figure 1). According to Mosselman [1992], the use of a curvilinear nonorthogonal grid is important for two main reasons. First, a curvilinear coordinate system is well suited for modeling natural (irregular) river bend morphologies. Second, in using a channel-fitted coordinate system, all points of the computational grid correspond to locations in the river, so boundary conditions can be imposed conveniently on grid lines. The significance of this point is as follows. At the start of a simulation, the physical characteristics of the bank materials can easily be defined by assigning values to individual bank points. However, after a certain amount of bank migration, a new grid, adapted to the new planform, must be generated. To retain the correct bank properties at the proper locations when generating a new grid, bank points must not shift along the banks, but they must instead move perpendicularly to the local bank lines [Mosselman, 1992]. This implies that it is not possible to generate a purely orthogonal grid, so transforming the governing equations into nonorthogonal counterparts makes grid generation easier.

Figure 1.

The channel-fitted curvilinear, nonorthogonal, coordinate system.

2.2. Flow Submodel

[8] RIPA's flow submodel is based on the 2D depth-averaged steady-flow equations which in conventional form for curvilinear orthogonal systems read:

equation image
equation image
equation image

where u and v are streamwise and transverse velocities (in the s and n directions, respectively), p is pressure, h is flow depth, g is gravitational acceleration, C is the Chezy coefficient for hydraulic roughness, and R is the radius of curvature. Derivation of equations (1) to (3) from the full three-dimensional flow equations by means of depth integration is based on a similarity hypothesis, in which the vertical profiles of the primary and secondary flows are self-similar [De Vriend, 1981]. This is valid for shallow, mildly curved channels, where most of the flow is not influenced by the banks. For a curvilinear, nonorthogonal system, equations (1) to (3) are:

equation image
equation image
equation image

[9] It should be noted that in equations (1'), (2'), and (3'), the terms Rs and Rn take a slightly different meaning from their counterparts in equations (1), (2), and (3) for orthogonal systems. In the nonorthogonal system equations (1’), (2’), and (3’), 1/Rs and 1/Rn denote the divergence of the transverse and streamwise coordinate lines, respectively (note the inverse subscript notation). In orthogonal systems this corresponds to the curvature of the streamwise and transverse grid lines, but in nonorthogonal systems Rs and Rn have no direct geometrical interpretation, other than being the reciprocal of the divergences [Mosselman, 1992]. If the transverse flow velocity (v) is not equal to zero then the streamwise coordinate lines and streamlines of the flow do not match exactly. The streamline curvature 1/Rf is not equal to the curvature of the streamwise coordinate lines 1/Rr and can be expressed [Olesen, 1982, 1987] as:

equation image

[10] Curved streamlines induce secondary currents. In river bends these are broadly of a spiral pattern with maxima located at the surface and near-bed zone. The development of the secondary flow intensity (is) can then be described following Olesen [1987] and using the notation of Struiksma et al. [1985] and Mosselman [1992]:

equation image

where λr is the adaptation length of the secondary flow. The length scale λr is small compared to other length scales in the model and it can, therefore, be omitted from the flow computations so that:

equation image

It should be noted that the adaptation length must still be retained in the calculation of the direction of the sediment transport vector (see next section), because its omission there would cause numerical stability problems [Olesen, 1987].

[11] The convective influence of the secondary flow in turn deforms the horizontal distribution of the primary flow. Kalkwijk and De Vriend [1980] show that this effect can be represented in-depth-averaged models via an additional acceleration or friction term. Hence the streamwise friction term in equation (1’) can be replaced by:

equation image

where ksn is the secondary flow convection factor and n′ denotes the local orthogonal direction (i.e., normal to the streamwise direction). It is now evident that the purpose of the preceding development of the secondary flow intensity (equations (4), (5), and (5a)) is that the term is in the momentum equation can be replaced by terms which are either known or can be calculated. Specifically, the term equation image is replaced in equation (6) by the term equation image. Formally, equation (6) thus holds along characteristics that do not coincide with the streamlines but instead they gradually shift toward the concave bank. However, the error involved in applying the equation along streamwise coordinate lines is negligible in mildly curved channels [Mosselman, 1992].

2.3. Sediment Transport and Bed Evolution Submodel

[12] The evolution of bed topography is governed by spatial gradients of fluvial sediment flux and material supplied from eroding riverbanks. For the case of a curvilinear orthogonal coordinate system and flows for which there is no input of sediment from bank erosion, this concept is formalized by writing the mass continuity equation as follows Olesen [1987]:

equation image

where zb is the bed elevation, t is the time coordinate, and qs and qn are the fluvial sediment fluxes in the s and n directions, respectively. According to Mosselman [1991, 1992] this continuity equation is transformed as follows for the case of a curvilinear nonorthogonal coordinate system with a supply of sediment from eroding banks:

equation image

The right hand side of (7a) contains source terms reflecting the input of erosion products from the right (subscript R) and left (subscript L) banks, with V the volumetric inflow rate of eroded sediment. The function δ describes how the eroded material is distributed over the cross section.

[13] Mosselman [1998] assumes that the magnitude of the sediment transport vector, |q|, is:

equation image

where e is a coefficient representing the influence of the streamwise bed slope on sediment transport and fs denotes the sediment transport rate. The latter is predicted by any standard transport formula in which the sediment transport rate is a function of the flow velocity vector, |u|, the flow depth, the Chezy coefficient and the sediment grain size, D. In fact there are numerous published sediment transport equations that can be used in equation (8), and the advantages and limitations of these different equations are quite well known [e.g., ASCE Task Committee, 1982; Gomez and Church, 1989; Yang and Wan, 1991]. In the present version of the model users are able to select from one of three sediment transport formulae, namely those by Meyer-Peter and Müller [1948], Engelund and Hansen [1967], or a simple power law of the form:

equation image

where ks is a user-specified coefficient and m is a user-specified exponent. No further comment on these formulae is required other than to say that equation (8) implies that the sediment transport rate is determined by the local flow field only, which is appropriate for bed material load. Hence, the model presented here is limited in that it does not account for wash load and is based on the use of single representative grain size, so grain size sorting effects are neglected.

[14] Having estimated the magnitude of the sediment transport vector, the direction, ψ, between the vector and the s direction is required. This is known to be determined by the depth-averaged flow direction, the deviation of the near-bed flow direction from the depth-averaged flow direction due to secondary flow, and the effect of gravity on sediment grains moving in contact with a transverse sloping bed [e.g., Engelund, 1974; Parker and Andrews, 1985; Bridge, 1992]. Struiksma et al. [1985] model the deviation angle using:

equation image

where kψ is a calibration coefficient weighting the influence of gravity on sediment grains on the transverse sloping bed, A is a weighting coefficient for the influence of the secondary flow on the near-bed shear stress, and θc is the Shields parameter at the channel centerline.

[15] According to Mosselman [1991, 1992] for a curvilinear nonorthogonal coordinate system equation (10) is transformed as follows:

equation image

and the s and n components of the sediment transport vector are then calculated using [Mosselman, 1992]:

equation image
equation image

where γ is given by:

equation image

Although the governing equations for both the flow and sediment transport submodels are designed for nonorthogonal computational grids, it is still important to minimize grid skewness in order to prevent, under certain circumstances, the introduction of significant truncation errors.

2.4. Bank Erosion Submodel

[16] In general, bank erosion is a complex process involving the combined action of weathering, fluvial erosion and geotechnical instability [Thorne, 1982; Lawler, 1992]. Meandering rivers commonly have bank materials composed of fine-grained cohesive sediments [e.g., Schumm, 1968; Knighton and Nanson, 1993; Millar, 2000], and consequently bank erosion is characterized by combinations of fluvial shear erosion and gravitational mass failure processes (weathering is excluded from consideration here). Nonetheless, a wide variety of specific erosion mechanisms are still involved [Darby, 1998]. For example, depending on the shape of the bank profile and the physical properties of the bank materials, any one of the following mass failure mechanisms might be observed: planar [Lohnes and Handy, 1968; Osman and Thorne, 1988; Darby and Thorne, 1996b; Simon et al., 1999; Rinaldi and Casagli, 1999]; rotational [Bishop, 1955]; cantilever [Thorne and Tovey, 1981]; or piping or sapping type failures [Hagerty, 1991]. The diversity of these bank erosion mechanisms makes it difficult to develop a universal model. Instead, the approach adopted here is to develop a model for specific types of meandering rivers. Since steep banks associated with the outer banks of meander bends are prone to planar failures, and because this type of failure is the simplest to analyze, the model developed here is restricted to planar failures only.

[17] In recent years considerable advances have been made in modeling the planar-type bank failure mechanism. Early stability analyses [Culmann, 1866; Lohnes and Handy, 1968] were restricted in that the bank profile used to represent the failure wedge was linear. More recent studies [Osman and Thorne, 1988; Darby and Thorne, 1996b; Darby et al., 2000] have introduced realistic bank profiles associated with banks that are deformed through lateral toe-erosion and near-bank bed degradation (Figure 2). Bank stability is modeled by defining a factor of safety (FS) as the ratio of resisting (FR) to driving (FD) forces acting on the incipient failure block:

equation image

According to equation (13), banks become unstable when FS < 1. The resisting and driving force terms in equation (13) are given by:

equation image
equation image

where cb is the bank material cohesion, L is the length of the failure plane, Wt is the weight of the incipient failure block, βf is the failure plane angle, and ϕ is the friction angle of the bank material. From the planar failure geometry shown in Figure 2, Osman and Thorne [1988] show that the weight of the failure block and length of the failure plane are given by

equation image
equation image

where γb is the unit weight of the bank material, H is the bank height, H′ is the uneroded bank height (see Figure 2), K is the depth of the tension crack, Kr is the depth of any relic tension crack from a previous bank failure (these form a vertical face on the bank profile) and θ is the uneroded bank angle. K is estimated using standard equations [Taylor, 1948; Selby, 1993] in which the depth of tension cracking is a function of the geotechnical properties of the bank sediment. The initial value of Kr is determined from field observations but is updated after bank failure in accordance with the value of K at that time.

Figure 2.

Definition diagram illustrating the type of bank profile characterized by the Darby-Thorne bank stability analysis. The bank profile is deformed by combinations of near-bed degradation (ΔZ) and lateral toe-erosion (ΔWb) that may ultimately cause a condition of limiting stability.

[18] To apply equations (13) to (17) it is necessary to develop expressions for the failure plane angle, and to model the values of H and H′ in relation to the changing shape of the bank profile caused by near-bank bed degradation and fluvial shear erosion (Figure 2). Osman and Thorne [1988] give the failure plane angle as:

equation image

[19] Osman and Thorne [1988] describe how the height (H) of the eroded bank profile is varied according to the magnitude of near-bank bed degradation using:

equation image

where ΔZ is the near-bank degradation computed at the sidewall nodes and Ho is the initial bank height. The uneroded bank height (H′) decreases if there is direct lateral toe-erosion by the flow:

equation image

where ΔWb is the magnitude of lateral erosion (Figure 2). Unfortunately, accurate prediction of the fluvial shear erosion of cohesive bank materials is notoriously difficult [ASCE Task Committee, 1998]. Here a simple excess shear stress model is used:

equation image

where τ is the boundary shear stress exerted at the toe of the bank (i.e. the shear stress value simulated at the sidewall node), τc is the critical shear stress required to initiate fluvial bank erosion, Δt is the length of the computational time step, and χ and x are user-specified calibration parameters. Appropriate values of τc can be estimated using empirical models based on the use of relevant bank material properties [Ariathurai and Arulanandan, 1978; Arulanandan et al., 1980] or via measurement in situ, for example through the use of recently developed jet-test devices [Hanson, 2001; Hanson and Simon, 2001; Dapporto, 2001].

[20] Bank material can, therefore, enter the channel sedimentary system via two routes. First, if the banks are unstable with respect to mass failure (FS < 1), sediment slides down the failure plane toward the channel bed at the toe of the bank. Second, during lateral fluvial erosion, bank sediment is directly entrained by the flow. In either case the eroded material must be partitioned between three possible “destination” components representing wash load, bed material and bed material load, respectively [Simon et al., 1991]. The relative distribution of the eroded bank material between these sinks has important implications for the morphological evolution of the near-bank zone, so partitioning is a key step in the simulation model. Our partitioning model treats the eroded bank material separately according to whether the mode of erosion is mass failure or fluvial shear erosion. This is in contrast to previous approaches [Simon et al., 1991; Darby and Thorne, 1996a; Mosselman, 1998] which lump the eroded material into a single unit.

[21] Consider first the bank material debris generated by mass failure. There is very little published work on this aspect of the bank erosion process, but those few empirical data that are available [Simon and Darby, 1997b] suggest that the amount of failed bank material transferred to each sink is controlled by the particle size distribution of the failed bank material. The fraction of failed bank material finer than 0.062 mm (η1) is, therefore, assumed to be transferred (rapidly according to field observations, but instantaneously in our model) to washload and plays no further role in morphological computations. In contrast, field observations show that the coarsest fraction of the failed bank material debris (η3) can remain on the bed of the channel for long periods of time, effectively as bed material, until it is weathered into smaller particles or entrained by large floods [Simon and Darby, 1997b]. Our model simulates this in that the coarse (≥10 mm) fraction (η3) of failed bank material is transferred instantaneously to a deposit of bed material in accordance with the slump deposition model shown in Figure 3. We recognize that our model is a simplification because once it is deposited, this fraction of the failed bank material immediately takes on the physical characteristics (grain size, D) of the bed material specified in the model. Hence, we do not simulate the breakdown and removal from the toe of the bank of blocks of failed bank material debris over a series of postfailure flood events [Simon and Darby, 1997b]. Rather, the key feature of this part of our partitioning model is the realistic depiction of the near-bank bed topography resulting from the slump deposit immediately after mass failure. Figure 3 shows that this slump deposit is spread across cells located within one bank height from the unstable bank, by adjusting the inclination of the surface of the deposit to conserve volume. This is broadly consistent with personal observations from several field sites that indicate coarse failed bank material is deposited at the angle of repose within a narrow zone close to the bank. It is also broadly similar to the slumping models of Pizzuto [1990] and Nagata et al. [2000] though these approaches fix the gradient of the surface of the deposit while allowing its extent to spread toward the center of the channel. The remaining fraction of the failed bank material (η2) is transferred to bed material load during the computational time step. This is consistent with observations that the intermediate particle sizes of the failed bank material are readily transported by the flow [Simon and Darby, 1997b].

Figure 3.

Illustration of (a) the bank material partitioning model and (b) lateral distribution model used to transfer bank material supplied by mass-wasting processes to the channel (not to scale).

[22] At this point it is appropriate to consider the partitioning pathways of bank material eroded by fluvial shear. In this case the eroded bank material is divided between two, not three, potential sinks. The first is the fine-grained (<0.062 mm) fraction (η4) of the intact bank material which, once entrained, is transferred directly to washload. The remainder of the intact bank material can, during fluvial shear erosion, be considered to become bed material load. The bank material fractions derived from mass failure and fluvial shear erosion and transferred to bed material load can, therefore, be aggregated into a single term, V(s), that is treated as a source term in the mass balance equation (equation 7a):

equation image

where VBs is the volume of failed bank material (gravitational failures) emanating from an unstable bank (equation image) and the second and third terms represent the volume of bank material derived from fluvial shear erosion. In utilizing equation (22) it is important to note that the parameters η1, η2 and η3 refer to specific size fractions of the failed bank material debris (η1 + η2 + η3 = 1), while η4 refers to the fine fraction (<0.062 mm) of the intact bank material.

[23] As equation (7a) makes clear, the supply of bank material (from mass failure and fluvial shear erosion) that becomes bed material load is spread over the channel cross section using a distribution function, δi(s, n). Our distribution function mirrors the slump model shown in Figure 3 in that the supply of material from the eroding banks is confined to n near-bank cells within a distance H of the toe of the bank. There is a linear decrease in the value of δi(s, n) with distance toward the centerline, until δi(s, n) = 0 at i = n. The values of the distribution function are calculated by varying the value of δi(s, n) at i = 1 (the sidewall) until the condition equation image is satisfied.

2.5. Numerical Implementation

[24] The computational procedure used to solve the above submodels broadly follows that of the original RIPA code (see Olesen [1987] and Mosselman [1992] for details). The complexity of this procedure is minimized by making two key assumptions. First, it is assumed that the computation of river planimetry can be decoupled from the bed topography computations. This is reasonable for cohesive bank materials because the rate of bank retreat relative to the horizontal scale of the channel is, in general, much slower than the rate of bed level adjustment. Second, the flow is assumed to be quasi-steady, which means that bed topography computations can be decoupled from the flow computations. Unsteady flows must, therefore, be simulated using a stepped hydrograph.

[25] The resulting three step computational procedure initially involves solution of the flow field while keeping the bed and banks fixed. From the computed flow field, the sediment transport fluxes and fluvial shear bank erosion rates are computed. In the second step, bed level changes are obtained from the sediment transport gradients and the input of bank erosion products. In the final step, bank line changes are calculated from the bank migration rates. The position of the bank toe is adjusted according to the amount of lateral erosion predicted by equation (21), while bank top retreat accords to the amount of floodplain loss (BW) shown in Figure 2. Calculations then loop back to the first step, generating a new grid if the magnitude of widening is sufficiently large. This iterative cycle is repeated until the desired length of the simulation is reached.

[26] The numerical flow submodel involves discretizing the governing differential equations using a central difference scheme with second order accuracy [Mosselman, 1991]. The depth-integrated longitudinal and transverse momentum equations (1’) and (2’) are represented as:

equation image
equation image

where U = u2 and a, b, c and d are given by:

equation image
equation image
equation image
equation image

in which:

equation image
equation image

where n′ has the same meaning as in equation (6). Equations (23) and (24) can then be presented as central difference equations:

equation image
equation image

where the subscripts j and i correspond to grid point identifiers in the streamwise and transverse directions, respectively. The pressure P can be eliminated from these difference equations by subtracting equation (31) from the corresponding one for i + 1, adding equation (32) and subtracting the corresponding one for j + 1. The resulting difference equation is linear in U:

equation image

This is solved by initially guessing a value of u at the starting location (see below) and then “sweeping” across the cross section to solve (33) over the cross section. The relevant boundary condition is u = 0 at the sidewall nodes. At each cross section the integral condition of continuity is used as an auxiliary condition:

equation image

The flow submodel therefore contains an iterative procedure in which guesses of the starting value of u are improved until the discharge, Q, calculated from equation (34) matches the actual (specified) discharge of the river. Equation (34) is determined by means of numerical integration based on Simpson's rule, which is valid only if the grid points are equidistant. However, for small variations of the transverse grid point spacing, the method is still fairly accurate [Mosselman, 1992]. To make the iteration process more efficient, the updated value of u at the starting node is estimated on the basis of the calculated discharge:

equation image
equation image

In equations (35) and (36) the subscripts refer to the different steps of the iteration. The transverse flow velocity is then calculated from the continuity equation (3’), in which the spatial derivatives of the fluxes (huΔn and hvΔs) are discretized as follows:

equation image
equation image

The flow continuity equation is solved using the boundary condition of v = 0 at the sidewalls. Next, the streamline curvature is calculated from:

equation image

in which Ω is a relaxation coefficient. To ensure computational stability, Olesen [1987] states that the value of the relaxation coefficient must match the following criterion:

equation image

where B is the channel width. Then the secondary flow intensity is can be determined from (5a) and substituted into (29). Subsequently, a new estimate of a, b, c and d is calculated, and u is recomputed from equation (33).

[27] The implementation of the flow submodel described above was developed by Kalkwijk and De Vriend [1980] and later enhanced by Olesen [1987] and Mosselman [1991, 1992]. The main limitation of this scheme is that the difference calculations deal with U which, by definition, should not take negative values. However, for certain combinations of geometrical and hydraulic parameters, the computational scheme sometimes converges to give negative U values, most often in the near bank zone. Furthermore, the iterative correction procedure is not effective for u values close to zero. We have attempted to rectify this by allowing users to select the direction and starting location of the numerical “sweeping” scheme used in the difference equations. Hence the initial value of u can now be calculated from the left or right banks, or from the channel centerline. In those few instances where negative values of U are still computed in the near-bank cells, we assume that u = v = 0 and the streamline curvature is equated to the local grid curvature.

[28] Discretization of the sediment balance equation is based on a central difference scheme for the space derivatives, with the time derivative approximated using an explicit first-order difference in the numerical integration procedure [Olesen, 1987]. It is not possible to use centered-differences at the boundaries so first order (noncentral) finite difference schemes are applied at the channel sidewalls and outflow boundary. In addition, the depth is assumed to be constant at the inflow boundary. The finite difference approximation of the sediment continuity equation is obtained by formal integration over a box around the point (i, j) in the computational grid. The result is:

equation image

Note that the terms in equation (7a) not containing derivatives are omitted here for reasons of simplicity. In order to solve (41), it can also be seen that the values of the sediment fluxes in the grid must be interpolated. This is achieved following the method suggested by Olesen [1987]:

equation image
equation image

In equation (43) the direction of the sediment transport vector (ψ) is given by:

equation image

where tan δs represents the angle between the streamline and the bed shear stress direction and is equated to the term is/(u + εv) from equation (5).

[29] The accuracy of the numerical scheme depends strongly on the truncation error of the discretization. The truncation error resulting from the use of a curvilinear grid is in part dependent on the directions of the coordinate lines with respect to Cartesian space. In fact, Mosselman [1991] has shown that the truncation error for nonorthogonal solutions is determined by the grid point spacing, grid skewness, gradients of spacing and skewness, and the solution gradients. Hence users must ensure a smooth distribution of points, so care is required in specifying the initial positions of the grid points at the banks. Still, after bank migration the grid will become increasingly deformed and it might become necessary to eventually restart the computations with a new distribution of bank points [Mosselman, 1992].

3. Assessment of Model Performance

[30] A detailed assessment of the performance of the model is necessary to determine the limits of its capabilities and to establish the level of faith that potential users might place in it when undertaking simulations. Many studies have relied on some form of model validation as a means of quantifying this level of faith [e.g., Bridge, 1992; Darby et al., 1996; Hodskinson and Ferguson, 1998; Mosselman, 1998; Nagata et al., 2000]. This exercise usually involves comparing the modeled and measured properties of the system that is being modeled, but this is not unproblematic. The main source of difficulty is that any discrepancy between model predictions and independently acquired data can be attributed to any, or all of: a failure to adequately represent the governing processes; failure to adequately specify initial and boundary conditions; and inappropriate model parameterization [Lane and Bates, 1998]. The difficulty is to distinguish between these differing sources of error.

[31] To limit this problem, we have undertaken a series of simulations wherein we move progressively from initial software benchmarking, through to test cases based on controlled laboratory flume experiments. At the final, most complicated, stage of this process a case study is undertaken to simulate the behavior of a meandering reach of a natural river channel. This progressive approach is not perfect, but it does at least allow us to undertake a series of interlinked checks on model performance that culminate in the case study that provides a quantitative comparison of modeled and measured characteristics of the natural river. Each progressive step of the assessment provides independent lines of evidence that, when taken together, provide insight into the sources of error in the new model. These individual steps are now discussed.

3.1. Software Benchmarking

[32] Initially, simple simulations designed to “benchmark” the software were undertaken to verify that it was free of basic coding errors and numerical instabilities, and as a means of checking that model predictions were physically realistic. In the context of the latter we were initially concerned with verifying that the model conserved mass, as the most basic indicator of model performance [Darby, 1998]. For this purpose we selected published data representing the simplest possible test case: a semi-circular-shaped fixed-width flume experiment [Odgaard, 1984] (hereinafter referred to as O84). This experiment is also the basis for more detailed simulations reported below and a detailed description of the physical characteristics of this flume channel is reserved for that discussion.

[33] The key point here is that we simply used the (high-quality) data obtained under the controlled conditions associated with this flume experiment to benchmark output from the unmodified version of RIPA against output from our modified version. This was done as a means of identifying any problems with the changes to the code that we had introduced. Since this flume experiment involved a fixed-width channel, it should be noted that these checks relate only to changes made to the flow and bed topography submodels, as well as to changes to the overall structure of the code resulting from combining these submodels with the bank erosion submodel. In fact, this exercise highlighted the problem described in section 2.5 wherein the flow model was found to sometimes converge to an incorrect solution. In light of this, and as described in section 2.5, the numerical solution algorithm was modified to enable the user to select the direction and starting location of the numerical sweeping scheme used in the difference equations. Having made this change, the corrected version of the modified RIPA was found to be in perfect agreement with the unmodified RIPA. This confirmed that no new errors had been introduced into the flow and bed topography submodels as a result of the changes we made to the numerical solution algorithm, or in combining RIPA with its new bank erosion submodel. Furthermore, the modified RIPA was also found to conserve fluid and sediment mass in these benchmarking simulations.

3.2. Model Calibration: Comparison of Predictions with Laboratory Data

[34] The next group of simulations involved the use of data from two laboratory flume experiments. The data sets in question were the O84 test case mentioned previously, together with data from the laboratory experiments reported by Whiting and Dietrich [1993a] (hereinafter referred to as WD93). The flow, grain-size and bend characteristics of these test cases are quite distinct (see Table 1 and Figures 4 and 5 for specific details). These simulations were undertaken to address three specific objectives. First, the use of the two data sets offered the opportunity to calibrate the model. Model calibration provides the means to select values of the adjustable parameters used in the model equations, and the realism of these calibrated values can be assessed. Model calibration based on two distinctive test cases also helps to identify the sensitivity of key calibration parameters. This is useful in selecting, or at least constraining, appropriate parameter values for use in higher-order simulations designed to quantify the overall predictive ability of the model (section 3.3). Second, simulations were undertaken so that model predictions could be compared to high-quality data obtained under the carefully controlled conditions associated with laboratory work. This was done to assess whether or not the model predictions were realistic. However, because these simulations were also geared to model calibration, comparisons between predicted and measured channel characteristics must necessarily be qualitative. This is because the calibration process has the potential to force agreement between modeled and observed parameters arbitrarily. Nevertheless, such comparisons are still helpful in understanding the behavior of the model. Third, since both data sets used in the two test cases are based on fixed-width flume experiments, comparisons of model predictions with measured data afforded the opportunity to assess the performance of the flow and bed topography submodels in isolation from the rest of the model. This represents a supplement to a previous study that had examined the predictive ability of the bank erosion submodel [Darby and Thorne, 1996b]. This is necessary because if there is divergence between model predictions and observations, it is important to direct research attention to those parts of the model that contribute most to the error [Lane and Bates, 1998].

Figure 4.

Comparison of (a) observed and (b) simulated bed topography (flow depth) for the model calibrated using the O84 test case. (c) Differences between observed and simulated data are also shown. Flow is from left to right.

Figure 5.

Comparison of (a) observed and (b) simulated bed topography (flow depth) for the model calibrated using the WD93 test case. (c) Differences between observed and simulated data are also shown. Flow is from left to right.

Table 1. Input Data Used in Simulationsa
VariableDescription and UnitsO84WD93Goodwin Creek
  • a

    Note that in all cases, sediment inflow at the upstream boundary was selected such that the inlet cross section is in equilibrium.

  • b

    The flow discharge value used in the Goodwin Creek case study is a steady, representative discharge equivalent to the dominant discharge value estimated by R. L. Bingner (personal communication, 1997). Based on flow duration data from station 2 (see Figure 7), this means that the model simulation period of 360 days used here is equivalent to the real time elapsed during the November 1982 to May 1988 study period.

  • c

    These are adjustable calibration parameters.

Qflow discharge (m3/s)0.1470.0017.5b
Dbed material grain size (mm)0.30.621.2
CChezy flow resistance43.229.110.0
λrsecondary flow adaptation length coefficientc0.70.70.7
ksnsecondary flow convection factorc0.470.450.46
Asecondary flow intensityc10.29.92.93
kψcoefficient to weight influence of gravity on grains moving over transverse slopec1.050.670.67
kscoefficient in sediment transport equationc5 × 10−46.4 × 10−48.2 × 10−3
mexponent in sediment transport equationc4.05.05.0
ecoefficient to weight influence of streamwise slope on sediment transportc4.08.012.0
τccritical shear stress for fluvial entrainment of bank-toe material (Pa)c20.0
χcoefficient in fluvial bank erosion equationc1.0 × 10−4
xexponent in fluvial bank erosion equationc1.0
ηχγ1fraction of failed bank material debris finer than 0.062 mm0.87
η2fraction of failed bank material debris with particle sizes between 0.062 and 10 mm0.0
η3fraction of failed bank material debris coarser than 10 mm0.13
η4fraction of intact bank material finer than 0.062 mm0.87
γbbank material unit weight (kN/m3)18.9
cbbank material cohesion (kPa)12.0
ϕbank material friction angle (degrees)20.0

[35] Model calibration involved varying the various adjustable coefficients until discrepancies between simulated and observed flow depths were minimized. The parameter values obtained as a result of this process are listed in Table 1. In so far as these values are within the range considered by Olesen [1987] to be physically realistic, these values can be considered to be appropriate. It should also be recognized that selection of the calibration parameter values was not undertaken entirely arbitrarily. Some of the parameter values (A, kψ, e) were calculated using theoretical relations proposed by Olesen [1987], in which the parameter values are given as functions of the Chezy coefficient (which has a known physical value). The remaining parameters are freely adjustable, but sensitivity tests showed that in practice λr and ksn had little influence on the output data. This left the coefficient (ks) and exponent (m) used in the sediment transport model as the two key adjustable parameters in that their values are unknown a priori while they simultaneously have a significant influence on model predictions.

[36] Comparisons between simulated depth values obtained using the fitted models (Table 1) and observed data are illustrated in the shaded relief models (Figures 4 and 5). In the case of the O84 test case, the difference model (Figure 4c) highlighting the comparison between simulated and observed data is based on depth values interpolated from a total of 855 grid cells. In quantitative terms, overall agreement is good. Some 64.9% of these cells have simulated depths within ±2 cm of the observed value, while 94.3% are within ±5 cm, compared to an observed bed relief of 24 cm. However, given that this is a fitted model this statistical agreement is not surprising. It must be remembered that these simulations were not intended to provide a quantitative assessment of model performance. Instead, these statistics confirm that the model is calibrated appropriately. A more instructive approach to assessing model performance involves analysis of the spatial distribution of errors highlighted by the difference model, as these provide insight into where, and under what circumstances, the model has poor process representation [Lane and Bates, 1998]. Comparison of Figures 4a and 4b indicates that the relief and locations of the dominant point bar and scour pool are reproduced successfully by the model. However, Figure 4c shows that errors are located primarily in regions close to the inlet (where the blue shading indicates the model over-predicts depth) and outlet of the channel (red shading indicates the model underpredicts depth), and adjacent to the outer bank (over-prediction of depth). These patterns reflect in part the difficulty in simulating channel conditions close to the boundaries of the model, especially when the scour depths in the flume experiment appear to be significantly influenced (reduced in this case) by the presence of the sidewalls. Most significantly, systematic errors are evident around the region close to the head of the point bar. To a lesser extent, this pattern is replicated in the region between the bar tail and the outlet of the channel. On the inner bank, close to the face of the bar itself, the model evidently over-predicts the flow depth, whereas flow depth is underpredicted as one moves outwards toward the opposite bank. In other words, the transverse bed slope predicted by the model is, in general, somewhat too shallow in relation to the experimental data.

[37] In the case of the WD93 test case, the difference model (Figure 5c) is based on data from 407 grid cells within the region of observed data availability (see Figure 5a). As with the O84 test case, the statistical fit between simulated and observed data is good, confirming the choice of calibration parameters. Some 63.6% of the grid cells have simulated depths within ±0.2 cm of the observed value, while 95.3% are within ±1 cm, compared to an observed bed relief of 6 cm. As in the previous test case, the broad details (macroscale relief) of the bed topography are reproduced successfully. For example, the point bar on the inner bank is reasonably well reproduced, though the location of the bar head simulated by the model is further upstream than that evident from the experimental data. Likewise, the model reproduces the broad trend of scour along the outer bank. However, systematic underprediction (areas of red shading on Figure 5c) of flow depth is evident within 3 discrete pools that are clearly observed near the outer bank of the study bend (Figure 5a). This underprediction is caused by the inability of the model to resolve these individual pools within the context of the overall pattern of scour near the outer bank.

[38] Replication of the macroscale morphological features (bars and pools) evident in the experimental data provides some evidence that the flow and bed topography submodels of the modified RIPA are physically realistic. Detailed comparisons of simulated and observed long profiles (Figure 6) confirm that the macroscale detail of bed features is reproduced correctly. However, the results shown in Figures 4, 5, and 6 also indicate that the modified RIPA is not capable of reproducing the detailed variation of bed topography evident in the two test cases. Specifically, while the modified RIPA correctly simulates the amplitude and wavelength of the bed morphology evident in the O84 test case (Figure 6a), it underpredicts the transverse bed slope. Likewise, the model does not simulate the presence of three separate deep pools observed along the outer bank in the WD93 test case (Figures 5c and 6b). In short, while the modified RIPA appears capable of reproducing the gross features of the bed topography, higher order detail is also missing.

Figure 6.

Comparison of observed and simulated downstream variation of bed relief along the inner and outer banks of (a) the O84 test case channel and (b) the WD93 test case channel.

[39] Underprediction of the transverse side slope implies that the model predicts an unrealistically high cross-stream sediment flux across the point bar near the inner bank that is directed toward the outer bank. This results in the bar extending too far across the channel. The cause of this evident inability to accurately estimate the sediment flux vector is not immediately clear. It might be either an inherent problem with the sediment transport model, or an inability to resolve the flow velocity field driving the sediment motion, or both. Consideration of the possible reasons why the model fails to predict multiple pools in the WD93 test case, together with analysis of the way in which the bed topography evolves over time, provides further clues as to the true explanation.

[40] Whiting and Dietrich [1993b] have reviewed the competing merits of different theories that attempt to explain the presence of multiple pools along the outer bank of certain high amplitude meander bends. They concluded that the depth oscillation is an alternate bar-like response to an impulsive forcing caused by the sudden change in curvature at the bend entrance, as elucidated by Parker and Johannesson [1989]. However, in addition to the primary effects of flow curvature, Whiting and Dietrich [1993b] showed that there is an inherent instability in the flow that plays an important role in the evolution of large-amplitude bends. This instability is closely related to the lateral transport of streamwise momentum by the secondary flow [Johannesson and Parker, 1989; Dietrich et al., 1979]. It seems likely that the secondary flow convection model (equation (6)) used in the modified RIPA is too simple to capture all the details of this process.

[41] Animations of the evolving bed topography provide a means of visualizing the evolution of the bar forms and determining the rates of bar migration simulated for each test case (see http://www.geog.soton.ac.uk/users/darbyse/epsrc). In fact, for both the O84 and WD93 test case simulations the modified RIPA predicts the formation of bars that do not migrate downstream. While we do not have any data to determine if the bars in the 084 test case were actually fixed or migrating, our results are consistent with the observation of fixed (‘forced’ in the terminology of Seminara and Tubino [1989]) bars in the WD93 test case. However, although the modified RIPA simulates accurately the relief of the bars in a low-sinuosity test case (not shown here) described by Whiting and Dietrich [1993b, experiment OM10W8.970], our simulations predict fixed bars when in reality they were observed to be freely migrating. This suggests that the modified RIPA may not always be able to distinguish between channels with free (alternate, migrating) and forced (point) bar forms. In the case of (low-sinuosity) channels with freely migrating bars, it seems likely that the modified RIPA predicts an unrealistically large stress divergence over the bar face, sufficient to evacuate the sediment delivered there and hence prevent bar migration [Kinoshita and Miwa, 1974]. Consistent with the failure to predict the oscillation of pool depth in the high-amplitude bend, this points to a limitation with the flow submodel, suggesting that the tendency of the model to underpredict the transverse slope is probably not caused by a fundamental limitation with the sediment transport model. Whatever the cause, since both the bed topography and the distinctive nature and interaction of free and forced bars is important in driving bank deformation and the evolution of meander forms, and in controlling the establishment of meander wavelength [Blondeaux and Seminara, 1985; Tubino and Seminara, 1990; Whiting and Dietrich, 1993b], this is a significant limitation of the model.

3.3. Field Evaluation: Case Study of Goodwin Creek, Mississippi

[42] Our final set of simulations focused on undertaking a comparison of model predictions and observed data from a real river, namely a reach of Goodwin Creek, Mississippi [Blackmarr, 1995; Kuhnle et al., 1996]. The Goodwin Creek case study is an important supplement to the fixed-width flume experiments because this river has undergone a sequence of channel evolution that includes bank erosion and planform change, providing an opportunity to evaluate these aspects of the model. Goodwin Creek was selected for use in this case study because it is dynamic (so channel changes are observable), has cohesive banks, and long-term studies have been undertaken in sufficient detail to document channel changes over the space and time scales relevant to river corridor engineering. It should also be noted that the choice is largely a default as it is one of the few sites where all the flow, sediment transport, channel morphology, and bed and bank material data necessary for model parameterization and testing are actually available [ASCE Task Committee, 1998]. The location of the study reach in Mississippi is illustrated in Figure 7, while the nature of the study reach and the monitoring studies undertaken there have been reported by Blackmarr [1995] and Kuhnle et al. [1996].

Figure 7.

Location of the field study reach within the Goodwin Creek Experimental Watershed.

[43] In comparing model output to observed channel changes, we were particularly concerned to select unbiased values of the model calibration parameters (Table 1), to prevent agreement between predicted and observed data being forced artificially. This was straightforward for the model calibration parameters A, kψ and e as these were calculated in relation to the known Chezy ‘C’ value for the study reach. Likewise, since model output is insensitive to λr and ksn, these were set in accordance with the values obtained during model calibration (Table 1). Selection of the sediment transport model coefficient (ks) and exponent (m) was, however, more problematic because these parameters have a significant effect on model output. The calibration simulations provide some guidance in that these parameters vary with the changing hydraulic characteristics of each test case, so we have a rational basis for choosing the direction in which to vary these parameter values. Since Goodwin Creek has high rates of sediment transport, we eventually selected relatively high ks and m values. In practice our choice was constrained because certain combinations of ks and m caused problems with numerical instability. Furthermore, since the Goodwin Creek case study involves bank erosion and planform change, some additional calibration parameter values (χ, x, and τc) are required. The bank materials in the Goodwin Creek study reach are known to be resistant to fluvial attack (the dominant erosion mechanism is mass-wasting), so it is justifiable to set τc so that there is no fluvial erosion, rendering χ and x irrelevant. All the other input data values (Table 1) were obtained from field data [Blackmarr, 1995].

[44] The channel characteristics predicted within the study reach during the November 1982 to May 1988 study period are compared with field data in Figures 8 and 9. In so far as the calibration parameters used in this simulation were selected according to the rationale outlined above and were not adjusted to force agreement between simulated and observed data arbitrarily, these comparisons can be considered an unbiased means of evaluating the overall performance of the model. Figure 8 illustrates the sequence of planform adjustment simulated by the model, while Figure 9 compares simulated and observed bed topography at the end of the simulation at five specific channel cross sections (see Figure 7 for their locations). Direct evaluation of model performance is restricted in scope because observed data are available only at the locations of the cross sections.

Figure 8.

Simulated sequence of planform adjustment (November 1982 to May 1988) within the Goodwin Creek study reach. By means of comparison, the lines and numbers drawn adjacent to banks at specific cross sections indicate the location and magnitude of observed planform shift during the specific time slice. Note that the lines are drawn to scale. Absence of lines indicates that there was no observed planform shift.

Figure 9.

Detailed comparison of observed and simulated bed topography in May 1988 at five specific cross sections within the Goodwin Creek study reach (for locations of cross sections, see Figure 7).

[45] The pattern of cumulative planform change (Figure 8e) simulated by the model is broadly consistent with the observed data. The model predicts correctly that most bank erosion is located along the left bank, with a maximum value of simulated cumulative bank retreat of 9.1 m located close to the bend apex between cross sections C43-2 and C42-3. This result is consistent with the observed bank retreat values on the left bank at cross sections C43-2 (17.67 m) and C42-3 (3.77 m). However, the model is inaccurate in that it incorrectly predicts that bank erosion does not commence until a point just downstream of C43-2. In quantitative terms, the model systematically underpredicts bank erosion (Table 2). Furthermore, it does not reproduce any of the bank erosion observed on the left bank at C50-1 (5.55 m), C45-1 (8.18 m) or C41-3 (0.7 m), or on the right bank at C46-1 (0.43 m). Failure to predict bank erosion at C50-1 is understandable given that this is the inlet boundary cross section, but the model also incorrectly simulates a maximum of about 0.5 m of cumulative bank retreat on the right bank in the vicinity of C43-2, even though no erosion is observed at this cross section. The simulated data (Figures 8a–8d and the animation on the web site) also indicate that most bank erosion and planform shift occurs within the first phase of the simulation (November 1982 to July 1983, Figure 8a), with negligible rates of bank retreat subsequently (Figures 8b–8d). However, the observed data from the monumented cross sections indicate that while rates of retreat are variable, they are sustained throughout the study period.

Table 2. Comparison of Simulated and Observed Cumulative (November 1982 to May 1988) Bank Erosion Amounts Within the Goodwin Creek Study Reacha
Cross SectionLeft Bank Erosion, mRight Bank Erosion, m
SimulatedObservedSimulatedObserved
  • a

    Note that the mean discrepancies between simulated and observed bank erosion are −3.8 m and 0.01 m for the left and right banks, respectively.

  • b

    Simulated value is from the bend apex just downstream of cross section C43-2.

C50-105.600
C47-200.100
C46-10000.43
C45-108.200
C43-29.1b17.70.50
C42-30.53.800
C41-300.700

[46] In terms of the bed topography (Figures 9 and 10), agreement between observed and simulated bed elevations at the end (May 1988) of the simulation is reasonable. The bed relief model (Figure 10) highlights the formation of a well-defined point bar between cross sections C43-2 and C42-3. This is consistent with reality in so far as this bar is the dominant feature visible on a 1990 aerial photograph of the study reach. However, detailed comparison of simulated and observed cross sections indicates that the simulated transverse bed slopes are too flat, especially at cross sections C46-1, C45-1, and C43-2 (Figure 9). This is consistent with results from the fixed-width laboratory flume simulations, and this lends credence to the hypothesis that model predictions of flow (and hence sediment flux) need to be improved. Whatever the cause, systematic underprediction of transverse side slope and pool depths undoubtedly contributes to underprediction of bank erosion. However, underprediction of bank erosion rates was also a feature of simulations of straight channels undertaken by Darby et al. [1996]. These simulations used a numerical model based on the same mass-wasting analysis employed in this research [Darby and Thorne, 1996b], which has a known tendency to over-estimate bank stability.

Figure 10.

Simulated bed topography within the Goodwin Creek study reach in May 1988 (for locations of cross sections, see Figure 7). Flow is from left to right.

4. Conclusion

[47] In the preceding sections we have reported the development and evaluation of a new 2D depth-averaged model of flow, bed topography and bank erosion for single-thread meandering rivers. Unlike many previous studies, our model is based on combining the flow and bed topography submodels with a mechanistic bank erosion submodel. The use of a physically based method to predict bank collapse avoids the need for bank erodibility parameters whose values would otherwise have to be obtained via model calibration. Our bank erosion model also facilitates specific analysis of the deposition of failed bank material debris at the toe of the bank and its subsequent removal by the flow. Finally, in using a boundary-fitted coordinate system that can be both curvilinear and nonorthogonal, our modeling approach simplifies simulations of irregular planform shapes associated with natural river meanders.

[48] The performance of the model was assessed using a combination of field and experimental data to evaluate various components of the model. The results are encouraging in that both the experimental test cases and the Goodwin Creek case study suggest that the model can accurately predict the macroscale features of the bed topography observed in each example. However, the O84 and WD93 test cases reveal three specific limitations of the flow and bed topography submodels. First, transverse bed slopes predicted by the model are systematically less than those observed in the two experiments, as well as the Goodwin Creek case study. Second, the model is not able to resolve the presence of multiple pools observed along the outer bank in the large amplitude bend (WD93) test case. Finally, the model incorrectly predicts the formation of fixed point bars, instead of migrating (alternate) bars, in a low amplitude bend test case. Analysis of these limitations suggested that it is likely that they are caused by a common factor. Specifically, it seems probable that the flow submodel cannot simulate realistically the detailed variation of boundary shear stress within the channel, especially in regions where the bed topography varies rapidly, such as over and around bar faces.

[49] In the case of the Goodwin Creek case study, in which we simulated the deformation of a channel with erodible banks, the model predicts correctly the locations and patterns of bank migration evident from the observed data. However, the magnitude of simulated planform adjustment is incorrect, since predicted rates of bank retreat are too low in relation to the observed data. Systematic underprediction of bank erosion rates is almost certainly a consequence of the known tendency of the mass-failure model to over-estimate bank stability, compounded by an underprediction of transverse bed slope, which unrealistically limits scour depth adjacent to outer banks. Due to the relatively good quality of the experimental and field data used in this investigation, our level of confidence in this overall assessment is correspondingly high.

[50] It can be concluded that this research has provided a methodological framework for coupling a mechanistic bank erosion model with flow and bed topography models, the better to simulate all dimensions of adjustment in meandering channels. The 2D model developed in this research has some potential for use as a tool for providing qualitative, rather than quantitative, predictions of bed topography and planform adjustment. Analysis of model predictions in relation to high-resolution two-component flow and sediment transport data (of a type not available in this project) would help in identifying the precise limitations of the model. However, the evidence suggests that improved predictions of meander morphology and adjustment could now be obtained by combining bank erosion models of the type used in this research with a more advanced flow model. In this respect, recent advances in the application of high-resolution Computational Fluid Dynamics (CFD) modeling approaches to geomorphological problems offers the potential to improve predictions of boundary shear stress, bed topography and bank erosion rates in meandering rivers.

Notation
A

coefficient for the influence of the secondary flow on near-bed shear stress.

B

channel width [L].

C

Chezy hydraulic roughness coefficient [L0.5/T].

cb

bank material cohesion [M/LT2].

D

representative grain size of the bed material [L].

e

coefficient to account for the influence of the streamwise bed slope on the sediment transport rate.

FD

driving force acting on a unit length of the incipient failure block [M/T2].

FR

resisting force acting on a unit length incipient failure block [M/T2].

FS

factor of safety for mass failure of the river bank.

g

gravitational acceleration [L/T2].

H

bank height [L].

H

uneroded bank height [L].

h

flow depth [L].

is

secondary flow intensity [L/T].

K

tension crack depth [L].

Kr

relic tension crack depth [L].

ks

coefficient in sediment transport rate law [L].

ksn

secondary flow convection factor [T2/L2].

kψ

coefficient weighting the influence of gravity on sediment grains moving along a transverse sloping bed.

L

length of the incipient failure plane [L].

m

exponent in sediment transport rate law.

n

transverse coordinate [L].

p

pressure [M/LT2].

Q

flow discharge [L3/T].

q

volumetric sediment transport rate per unit width [L2/T].

R

radius of curvature [L].

Rf

streamline radius of curvature [L].

Rn

reciprocal of divergence of streamwise coordinate lines, in orthogonal coordinate systems equal to radius of curvature of transverse coordinate line [L].

Rr

radius of curvature of streamwise coordinate lines in the nonorthogonal system [L].

Rn

reciprocal of divergence of transverse coordinate lines, in orthogonal coordinate systems equal to radius of curvature of streamwise coordinate line [L].

s

streamwise coordinate [L].

t

time coordinate [T].

u

flow velocity in streamwise direction [L/T].

V

volumetric inflow rate of sediment eroded per unit length of river bank [L2/T].

VBs

volume of bank material emanating from mass failures per unit length of river bank [L2].

v

flow velocity in transverse direction [L/T].

Wt

weight of the incipient failure block per unit length of river bank [M/T2].

ΔWb

lateral bank erosion caused by fluvial shear [L].

x

exponent in lateral bank erosion equation.

zb

bed elevation [L].

ΔZ

near-bank bed degradation computed at the sidewall nodes in a time step [L].

βf

failure plane angle.

χ

coefficient in lateral bank erosion equation [L/T].

δ

function describing how eroded bank material is distributed laterally over the cross section.

ε

local grid skewness, defined as the cosine of the angle between the base vectors of the coordinate system.

ϕ

bank material friction angle.

γ

function relating the streamwise and transverse components of the sediment transport vector.

γb

bank material unit weight [M/L2T2].

η1

fine (<0.062 mm) fraction of the failed bank-material debris.

η2

intermediate (0.062 mm ≤ η2 < 10 mm) fraction of the failed bank-material debris.

η3

coarse (≥10 mm) fraction of the failed bank-material debris.

η4

fraction of fine-grained (<0.062 mm) undisturbed bank material.

λr

adaptation length of the secondary flow [L].

θ

uneroded bank angle.

θc

Shields parameter at the channel centerline.

ρ

fluid density [M/L3].

τ

boundary shear stress exerted at the sidewall node [M/LT2].

τc

critical shear stress required to initiate fluvial bank erosion [M/LT2].

Ω

relaxation coefficient in numerical solution scheme.

ψ

angle between the sediment transport vector and streamwise direction.

Subscripts

 

i

grid point identifier for transverse direction.

j

grid point identifier for streamwise direction.

k

iteration counter index.

L

left bank.

n

in the direction normal to the streamwise grid line.

o

initial condition.

R

right bank.

s

in the direction of the streamwise grid line.

Acknowledgments

[51] This research was supported by the Engineering and Physical Sciences Research Council (grant GR/M46532). We are very grateful to Erik Mosselman for providing the source code of RIPA and to Peter Whiting for supplying source data and notes from his flume experiments.

Ancillary