Coupled simulations of fluvial erosion and mass wasting for cohesive river banks



[1] The erosion of sediment from riverbanks affects a range of physical and ecological issues. Bank retreat often involves combinations of fluvial erosion and mass wasting, and in recent years, bank retreat models have been developed that combine hydraulic erosion and limit equilibrium stability models. In related work, finite element seepage analyses have also been used to account for the influence of pore water pressure in controlling the onset of mass wasting. This paper builds on these previous studies by developing a simulation modeling approach in which the hydraulic erosion, finite element seepage, and limit equilibrium stability models are, for the first time, fully coupled. Application of the model is demonstrated by undertaking simulations of a single flow event at a single study site for scenarios where (1) there is no fluvial erosion and the bank geometry profile remains constant throughout, (2) there is no fluvial erosion but the bank profile is deformed by simulated mass wasting, and (3) the bank profile is allowed to freely deform in response to both simulated fluvial erosion and mass wasting. The results are limited in scope to the specific conditions encountered at the study site, but they nevertheless demonstrate the significant role that fluvial erosion plays in steepening the bank profile or creating overhangs, thereby triggering mass wasting. However, feedbacks between the various processes also lead to unexpected outcomes. Specifically, fluvial erosion also affects bank stability indirectly, as deformation of the bank profile alters the hydraulic gradients driving infiltration into the bank, thereby modulating the evolution of the pore water pressure field. Consequently, the frequency, magnitude, and mode of bank erosion events in the fully coupled scenario differ from the two scenarios in which not all the relevant bank process interactions are included.

1. Introduction

[2] The erosion of sediment from riverbanks is a key factor affecting a range of physical and ecological issues in the fluvial environment. These include the establishment of river and floodplain morphology and their associated habitats [Thorne and Lewin, 1979; Darby and Thorne, 1996; Millar, 2000; Goodson et al., 2002; Eaton et al., 2004], turbidity problems [Bull, 1997; Green et al., 1999; U.S. Environmental Protection Agency, 2000], as well as nutrient and contaminant dynamics [Marron, 1992; Reneau et al., 2004]. Indeed, the contribution of bank-derived materials to fluvial sediment budgets may be higher than previously thought. Bank-derived sediments typically contribute between about 37% and 80% of the total suspended sediment yield emanating from catchments, with the lower end of this range associated with low-energy UK rivers [Walling et al., 1998, 1999] and the upper end corresponding to incised channel systems [Wasson et al., 1998; Simon and Darby, 2002; Simon and Rinaldi, 2006]. With such a significant fraction of bank-derived material within the alluvial sedimentary system, knowledge of the rates, patterns and controls on bank erosion events is necessary for a complete understanding of the fluvial sediment transport regime.

[3] Given the importance of bank-derived sediment within the fluvial sediment transfer system, it is not surprising that bank erosion has been the subject of much research. A particular focus of recent work has been the application of slope-stability modeling to quantify the role and relative influence of the factors that control mass wasting. Such studies have focused on the respective roles of bank shape [Osman and Thorne, 1988; Darby et al., 2000], riparian vegetation [Abernethy and Rutherfurd, 2000; Simon and Collison, 2002], as well as bank and channel hydrology [Rinaldi and Casagli, 1999; Casagli et al., 1999; Simon et al., 1999; Rinaldi et al., 2004]. Although fewer studies have been concerned with the process of fluvial erosion (i.e., the removal of bank sediments by the direct action of the flow), notable recent contributions have started to address the issue of quantifying entrainment thresholds and process rates [Lawler et al., 1997a; Simon et al., 2000; Dapporto, 2001; Lawler, 2004, 2005].

[4] Nevertheless, predictions of bank erosion rates remain poor [Darby et al., 1998; Mosselman, 1998], with two key issues standing out as likely limiting factors. First, existing studies utilize data collected at relatively coarse timescales, understandably so given the problems of working during floods, but preevent versus postevent studies cannot resolve process thresholds, timing and rates [Lawler, 2004, 2005]. Second, previous research tends to view individual processes in isolation, but bank retreat is the net result of interacting processes [Wolman, 1959; Thorne, 1982; Lawler et al., 1997b]. Although the adoption of approaches that break systems down into manageable components is a logical first step, interactions and feedbacks between processes in nonlinear systems can lead to outcomes that are not always predictable a priori. In short, viewing bank processes in isolation is unrealistic, introducing the possibility that the conclusions derived from such studies may be biased. This limitation is starting to be addressed by modeling, with a particular focus on coupling fluvial erosion and mass wasting models [Darby and Thorne, 1996; Darby et al., 1998; Langendoen, 2000; Simon et al., 2006]. However, as yet no study has coupled fluvial erosion, seepage and stability submodels in a fully integrated analysis.

[5] To address this gap, we herein develop a new simulation approach that seeks to couple hydraulic and geotechnical simulations in a way that facilitates analysis of the influence of dynamically interacting processes on the erosion of bank sediments. This is achieved by coupling a hydraulic erosion model with a finite element seepage analysis and limit equilibrium stability methods to address transient mass wasting triggered by bank profile deformation and/or variations in bank pore water pressures [Dapporto, 2003; Dapporto and Rinaldi, 2003]. The new method requires the finite element mesh used in the seepage analysis to be adapted to the deforming bank profile, thereby allowing for the possibility of feedbacks to occur between simulated bank stability, fluvial erosion and pore water pressure fields. Having outlined the simulations, we go on to analyze the results to explore the consequences of some of these key feedbacks.

2. Methods

[6] In this paper we seek to develop insight into the dynamics of bank erosion by developing simulations that couple a fluvial erosion model with a finite element seepage analysis and limit equilibrium methods for evaluating bank stability. The approach involves applying these three submodels at each of a series of discrete time steps throughout a flow event hydrograph (Figure 1). This study therefore builds on previous models [Darby and Thorne, 1996; Langendoen, 2000; Simon et al., 2006] which consider the combined effects of fluvial erosion and mass wasting, but differs in that a finite element seepage analysis is also included to account for the influence of pore water pressure in controlling the onset of mass wasting. Some previous studies [Rinaldi and Casagli, 1999; Casagli et al., 1999; Simon et al., 1999; Rinaldi et al., 2004] have analyzed this latter effect, but only in the context of river banks that are not deformed by fluvial erosion. To our knowledge, ours is therefore the first study to fully couple all three relevant process submodels, though one study [Simon et al., 2006] has combined measured pore water pressure data with fluvial erosion and bank stability models. This section details the submodels and their parameterization, focusing on how the finite element seepage and bank stability analyses are applied in a form that can accommodate the effects of fluvial erosion.

Figure 1.

Logic diagram showing the computational sequence used to couple the finite element seepage, bank stability, and fluvial erosion analyses.

2.1. Study Site Description

[7] The study site investigated herein, the Sieve River, is located within the mountain belt of the Northern Apennines (Figure 2a), and has been described in previous papers [e.g., Rinaldi et al., 2004]. The Sieve River (840 km2) is the main tributary of the Arno (Tuscany, Central Italy) and at the study site presents a single-thread, sinuous pattern, with bed sediments predominantly composed of gravel, and a channel gradient of 0.0030. Mean daily discharge at the study reach, which is located close to the basin outlet (830 km2), is 15 m3 s−1, while the 2-year return period peak flow (Q2) is 410 m3 s−1. The eroding banks have an average height ranging between 5.00 and 5.50 m and are layered, with the bank material stratigraphy (defined by grain size analyses combined with the results of one static and six dynamic penetration tests) arranged as shown in Figure 2b. Thus the bank in question comprises a cohesive upper part overlying a gravel toe as is commonly the case in upland and piedmont zones in Europe and elsewhere, but it is distinct from the fine-grained bank settings that are more usually associated with lowland environments and which have been the subject of related research [e.g., Simon et al., 1999, 2000; Simon and Collison, 2002].

Figure 2.

Sieve River study reach showing (a) location of the reach within the Sieve basin (area 1 is lacustrine, fluvio-lacustrine, and alluvial deposits of the Upper Pliocene and Quaternary; area 2 is arenaceous-marly and calcaleous flysch formations of the Cretaceous and Miocene) and (b) arrangement of the bank materials observed within the study reach.

[8] Of numerous cross sections surveyed along the reach, a representative section was selected for further analysis by choosing the profile with the rate of bank retreat that was closest to the mean rate of bank retreat observed along the reach as a whole. On the basis of the Q2 discharge values referred to above, the reach has a rather high value of stream power per unit bed area (350 W/m2). Mean rates of bank erosion for most flows are nevertheless only moderate (0.40 m/yr during February 1996 to February 2000), since the study site is located in an almost straight reach, with fluvial erosion, and shear and cantilever-type failures all contributing to the observed retreat. The availability of data, the presence of a range of interacting bank processes, and its location within a straight reach (which simplifies the required hydraulic computations) makes the Sieve study site ideal for this research.

2.2. Fluvial Erosion Model

[9] Fluvial bank erosion rates can be quantified using an excess shear stress formula such as [Partheniades, 1965; Arulanandan et al., 1980]

equation image

where ɛ (m/s) is the fluvial bank erosion rate per unit time and unit bank area, τb (Pa) is the boundary shear stress applied by the flow, kd (m3/Ns) and τc (Pa) are erodibility parameters (erodibility coefficient, kd, and critical shear stress, τc) and a (dimensionless) is an empirically derived exponent, generally assumed to equal 1.0. Although equation (1) appears simple, in practice it is necessary to define the erodibility parameters and boundary shear stress. These are all highly variable, helping to explain why observed rates of fluvial erosion range over several orders of magnitude [Hooke, 1980]. Consequently fluvial erosion is predictable only to the extent that the parameter values can be estimated accurately.

[10] For granular (noncohesive) sediments, critical shear stress can be estimated by applying the same methods that are used to predict the entrainment of bed material, albeit with modifications to take into account the effect of bank angle on the downslope component of the particle weight [Lane, 1955] and the influence of packing and cementing [e.g., Millar and Quick, 1993; Millar, 2000]. However, there are no theoretical or empirical methods to determine kd for granular sediments. Theoretical determination of critical shear stress for cohesive materials is even more complex, given that it is widely recognized that it depends on several factors, including (amongst others) clay and organic content, and the composition of interstitial fluids [Arulanandan et al., 1980; Grissinger, 1982]. However, recent studies have deployed in situ jet-testing devices [e.g., Hanson, 1990; Hanson and Simon, 2001] to obtain direct measurements of both bank erodibility parameters [e.g., Dapporto, 2001]. Bearing these difficulties in mind, we estimated the values of the erodibility parameters (kd and τc) in equation (1) as follows:

[11] 1. Loose gravel. At the Sieve River study site a wedge of loose gravel is located at the bank toe prior to a flow event. We estimated the critical shear stress (τc = 27.9 Pa, see Table 1) for this material using Lane [1955]. We assumed that after this critical shear stress was exceeded, the material was removed from the toe exposing the basal packed gravel (layer 2) to the direct action of the flow.

Table 1. Geotechnical and Hydraulic Characteristics of the Bank Materialsa
ParameterSediment LayersNotes
  • a

    The positions and thicknesses of the sediment layers 1, 2, and 3 are illustrated in Figure 3; n/a means not applicable.

Effective cohesion, c′, kPan/an/a2.0Data based on a single triaxial test
Effective friction angle, ϕ′, degn/an/a35Data based on a single triaxial test
Matric suction angle, ϕb, degn/an/a15–35See text for explanation
Unit weight, γ, kN/m3n/an/a15.1–19.3Data based on a single soil sample removed from layer 3 during dry (low value) and saturated (high value) conditions, respectively
Porosity, n, % based on single samples removed from each layer
Saturated hydraulic conductivity, ksat, m/s6.0 × 10−41.0 × 10−41.4 × 10−6 ± 7.8 × 10−7Values for layer 3 are the mean and standard deviation (quoted range) of 9 Amoozemeter tests and 3 open single ring infiltrometer tests. See text for details concerning layers 1 and 2
Critical shear stress, τc, Pa27.932.51.8 ± 0.7Values for layer 3 are the mean and standard deviation (quoted range) of 4 jet tests
Erodibility coefficient, kd, m3/Nsn/a1.3 × 10−65.4 × 10−6 ± 2.6 × 10−6Values for layer 3 are the mean and standard deviation (quoted range) of 4 jet tests

[12] 2. Packed gravel. For this material, it is not possible to apply the Lane [1955] equation because it is restricted to conditions (not met for this layer) in which the bank angle is less than the friction angle of the material. We therefore estimated the critical shear stress (τc = 32.5 Pa, see Table 1) using [Millar, 2000]

equation image

where D50bank (0.0187 m) is the median grain size of the bank material, ϕ′ (82°) is an equivalent friction angle of material, estimated from the steepest angle that the bank forms at the bankfull waterline [Millar, 2000], θ (70°) is the bank angle, γ is the unit weight of water, assumed here to be 9810 N/m3, and s is the specific gravity of the sediment, assumed here to be 2.65. The erodibility coefficient (kd = 1.3 × 10−6, see Table 1) was then determined via model calibration (i.e., by forcing best agreement between calculated and measured bank toe retreat).

[13] 3. Cohesive portion of the bank. For these materials, both erodibility parameters were estimated by averaging the results from a series of four jet tests (Table 1). As is commonly the case [e.g., Hanson and Simon, 2001; Dapporto, 2001] error estimates (Table 1) indicate a substantial degree (±39% for τc and ±48% for kd) of natural variability. However, since the primary purpose of this paper is to develop and demonstrate the fully coupled approach, for reasons of simplicity we ignore this source of uncertainty and conduct a single set of simulations using only the mean parameter values.

[14] To characterize the near-bank shear stresses exerted on the river banks during the simulated event we adopted the following approach. It is important to note that shear stress and therefore lateral erosion was computed for each time step of the discretized hydrographs, following the procedure shown in Figures 1 and 3a. Boundary shear stress was initially estimated using τmean = γRS, where τmean = mean boundary shear stress (Pa), γ = unit weight of the water (N/m3), R = hydraulic radius (m), and S = energy slope (m/m). To transform τmean to the near-bank shear stress, we selected a function describing the distribution of boundary shear stress around the wetted perimeter. Numerous experimental studies have attempted to derive such a function, but most have been conducted in trapezoidal flumes with gentle bank slopes. In this study we therefore used a function (Figure 3a) obtained from laboratory flume experiments [Leutheusser, 1963] within straight, rectangular, channels with vertical banks, which provides a closer analogy to the near-vertical bank encountered at the Sieve River. The Leutheusser distribution function (Figure 3a) was applied at each of 32 computational nodes spaced apart up the bank profile at a uniform vertical distance of 0.15 m. This procedure is clearly an idealization of the actual near-bank shear stress distribution, but was used simply to demonstrate the methodological approach required to combine the fluvial erosion and mass-wasting process models. Having obtained the near-bank shear stress, the bank profile was deformed by fluvial shear erosion estimated using equation (1).

Figure 3.

Overview of the coupled bank erosion analysis for the Sieve River study site: (a) method of estimating fluvial erosion; (b) geometry of the finite element seepage analysis, indicating the different types of assigned boundary conditions (bank profile and bank top boundaries are represented by solid and open circles, respectively; all the other boundary nodes are zero flux boundaries) and sediment horizons; and (c) slide- and cantilever-failure mass wasting analyses applied to the upper cohesive part of the river bank (sediment layer 3). Terms are defined as follows: C*, effective cohesion; CT, total cohesion; ϕ′, effective friction angle; ϕb, friction angle in terms of matric suction; γ, bulk unit weight; L, length of the failure surface for the cantilever failure block; A, area of the layer i of the cantilever failure block; τc, critical shear stress to entrain the bank material; kd, erodibility coefficient; LE, lateral erosion increment simulated in the time step; ɛ, bank erosion rate; Δt, time step; τb, shear stress exerted on the river bank; τmean, mean boundary shear stress.

2.3. Modeling Saturated and Unsaturated Flow

[15] Changes in pore water content and pressures are recognized as one of the most important factors controlling the onset and timing of bank collapse [Thorne, 1982; Springer et al., 1985]. Pore water has at least four main effects: (1) reducing shear strength (under conditions of positive pore water pressure), (2) increasing the unit weight of the bank material, (3) providing an additional destabilizing force due to the presence of water in tension cracks, and (4) providing additional (stabilizing or destabilizing) seepage forces. A crucial point when accounting for pore water pressures is their transient character, driven as they are by dynamic hydrological variables (rainfall, as well as the varying level of the water in the river). The actual mechanisms and timing of bank failures induced by pore water pressure effects are therefore difficult to predict if their temporal changes are not accounted for [Rinaldi and Casagli, 1999; Casagli et al., 1999; Simon et al., 2000, 2006]. For this reason, bank stability response during flow events requires knowledge of the dynamics of saturated and unsaturated seepage flows. Herein we follow previous studies [e.g., Dapporto et al., 2001; Dapporto, 2003; Rinaldi et al., 2001, 2004] that have modeled pore water pressure distributions in riverbanks using the software SEEP/W [Geo-Slope International, 2001a]. The software performs a two-dimensional, finite element seepage analysis using the governing equations of motion (Darcy's law) and mass conservation, the latter expressed here in a form extended to unsaturated conditions [Richards, 1931; Fredlund and Rahardjo, 1993] as

equation image

where H = total head (m), kx = hydraulic conductivity in the x direction (m/s), ky = hydraulic conductivity in the y direction (m/s), Q = unit flux passing in or out of an elementary cube (in this case an elementary square, given that the equation is in two dimensions) (m2/m2s), θ = volumetric water content (m3/m3), and t = time (s).Critical to the success of the modeling is accurate parameterization of the hydraulic and physical properties of the bank sediments. It is also necessary to appropriately set the initial and boundary conditions, and to discretize (Figure 3b) equation (3). Each of these aspects is now considered.

[16] Parameterization of the hydraulic and physical properties of the bank sediments (Figure 4 and Table 1) primarily involves defining the relations between hydraulic conductivity (k) and pore water pressure (u), and between volumetric moisture content (θ) and pore water pressure for each layer of sediment. Accurate parameterization of these relations, referred to here as the conductivity (k-u) and soil characteristic (θ-u) curves, respectively, is important because simulated pore pressures are highly sensitive to these hydraulic functions [Rinaldi et al., 2004]. The soil characteristic (θ-u) curves (Figure 4b) were constructed via direct measurements of pore water pressure (by deployment of tensiometer arrays) and water content values (by removal of soil samples for standard laboratory analysis) over a seasonal cycle. The k-u curves (Figure 4c) were estimated following procedures described in Rinaldi et al. [2004]. In summary, a range of empirical relationships [Green and Corey, 1971; Van Genuchten, 1980; Fredlund et al., 1994] was used to model possible forms of the k-u relations, based on measured grain size distributions as determined from bulk samples extracted from each horizon (Figure 4a). For the cohesive layer 3 that is central to the stability analyses undertaken herein, the resulting k-u relations were then constrained (by displacing the curves vertically) to match measured values of saturated hydraulic conductivity (Table 1) obtained in situ using a series of 9 Amoozemeter [Amoozegar, 1989] and 3 open single ring infiltrometer [Daniel, 1987] tests. Consistent with the rationale adopted in section 2.2, we undertook a single set of simulations based on the median k-u relations (Figure 4c) from the resulting range. As noted above, model results are sensitive to the k-u relation, but we have confidence in the parameterization as the range of modeled k-u curves was found to be small and sensitivity tests indicate that the results reported below are not materially affected.

Figure 4.

Bank material characteristics at the Sieve River study site showing (a) grain size distributions of each layer (1, 2, and 3, as illustrated in Figure 3) of bank sediment, based on bulk samples collected from each layer; (b) soil water characteristic curves; and (c) hydraulic conductivity functions for each of the bank sediments. Terms are defined as follows: d, grain size; ua, pore air pressure; u, pore water pressure; θ, volumetric water content; k, hydraulic conductivity.

[17] Having defined the bank sediment properties, finite element seepage analyses were performed by discretizing the river banks into a series of finite elements, and assigning k-u and θ-u functions to each defined layer of bank material. For this study we used a mesh comprising 1600 quadrilateral and triangular finite elements (Figure 3b). This mesh is considerably finer in resolution than the 517 element mesh used at the Sieve River study site in an earlier study [Rinaldi et al., 2004] in which a constant bank profile was assumed, but the increased computational requirements are not problematic with modern desktop computers. We return to the significance of the mesh resolution in the next section.

[18] Regarding the boundary conditions employed along the borders of the finite element grid, for bank profile nodes (indicated as closed circles on Figure 3b) a total head versus time function was used to represent the flow hydrograph of the simulated event, the latter being defined using 15-min resolution time series data obtained from pressure transducers installed in the channel adjacent to the bank. To simulate the effects of infiltrating rainfall, a rainfall intensity versus time function was used along the bank top nodes (indicated as open circles on Figure 3b), using 15-min resolution time series data obtained from an automated rain gauge installed at the study site. Zero flux boundary conditions were assigned along the remaining vertical and horizontal boundaries of the finite element grid. This approach follows Rinaldi et al. [2004], who argued that the zero flux assumption has little affect on simulated pore pressures in the area of interest, namely the cohesive upper part of the bank. Initial conditions were defined using the groundwater level measured before the start of the flow event using piezometers installed in the banks (Figure 3b).

[19] The seepage analysis is performed by discretizing a specific flow event hydrograph into a series of time steps. For this study we selected a single peak flow event (Q = 790 m3/s), that occurred during 18–20 November 1999. This event represents a large magnitude (16-year RI) flood, providing a case study wherein high rates of fluvial erosion induce mass failures. The flow hydrograph and rainfall inputs for this period were discretized into 25 time steps, with shorter time steps during phases of rapidly varying flow (Figure 5).

Figure 5.

Flow hydrograph and rainfall time series for the investigated flow event (18–20 November 1999) on the Sieve River.

2.4. Mesh Adaptation

[20] The seepage analysis described above is identical to previous studies [Dapporto et al., 2001; Dapporto, 2003; Rinaldi et al., 2001, 2004] that maintain a constant bank profile. However, in this study bank profiles are deformed to accord with fluvial erosion and/or mass wasting (see below) simulated at the end of each discrete time step. Accordingly, special attention must be paid to the manner in which the finite element mesh is adapted to the new bank geometry. Finite element mesh adaptation was achieved in this study using two manual procedures (Figure 6):

Figure 6.

Illustration of the two possible schemes used to adapt the finite element mesh to an increment (FE) of simulated fluvial erosion: (a) unmodified near-bank cells prior to fluvial erosion; (b) scheme 1, in which the bank profile nodes (indicated by the large solid circles) are displaced inward by an amount equal to the simulated fluvial erosion; and (c) scheme 2, in which the geometry of the original mesh remains unchanged but the physical properties of the “eroded” cells are updated and boundary conditions are assigned to the newly exposed bank profile nodes (indicated by the large solid circles).

[21] 1. If the magnitude of the simulated fluvial erosion (FE) is less than the width of the boundary cell (CW), the boundary node is shifted horizontally inward by an amount equal to the simulated fluvial erosion (Figure 6b). This scheme has the advantage that the shifted node retains its status as a bank profile, so that the associated boundary condition does not change, but it necessitates manual remeshing at the end of each time step. Furthermore, as finite elements in the seepage model cannot be destroyed, this scheme cannot be applied if the increment of simulated fluvial erosion is greater than the thickness of the boundary cell.

[22] 2. For the second case (FECW), the change in grid geometry is simulated artificially by adjusting (again, manually) the hydraulic conductivity and volumetric water content of eroded cell(s) to replicate the conductivity and saturated state of subaqueous in-channel cells (Figure 6c). This was achieved by assigning k-u and θ-u functions appropriate for a (hypothetical) coarse sediment, so that the behavior of the cells is that of a very permeable material (in practical terms, the water level within this material is equal to the river stage). Elements exposed by fluvial erosion are then updated with new bank profile (total head versus time) boundary conditions. This scheme contrasts with the first in that the status and associated boundary conditions of the adjusted nodes must be updated manually, albeit without the need to remesh at the end of each time step.

[23] That both schemes involve manually updating the mesh and/or boundary conditions is a clear disadvantage in that the procedures require significant investments of operator time. However, manual remeshing is a necessary consequence of employing SEEP/W, as users cannot access the source code and implement customized, automated, mesh generation algorithms. Which of the two meshing schemes is adopted depends on the relative scales of the boundary cells and simulated fluvial erosion in a discrete time step. Thus coarse (CWFE) elements on the bank profile cause the first scheme (Figure 6b) to be adopted, whereas the second scheme (Figure 6c) is associated with fine resolution elements and/or high fluvial erosion rates (CW < FE). The second scheme is advantageous in the sense that manually updating the element properties and boundary conditions is much simpler than manually remeshing, but it requires the use of much higher resolution finite element grids than those used in past studies. This constraint can, to some extent, be relaxed in the first scheme, but it is more labor intensive than the second in that manual updating of the mesh is required at the end of each time step during which fluvial erosion and/or mass wasting is predicted to occur. Note that in the fully coupled simulations reported below (scenario 3 in section 3, below) the bank profile is deformed after 15 of the 25 time steps. For this reason we found it significantly easier to utilize the high-resolution mesh as this enabled us to manually update the boundary conditions (scheme 2) rather than the mesh itself, making the task manageable.

2.5. Riverbank Stability Analyses

[24] The pore water pressure distributions obtained by the seepage analysis in each time step are used as input data for the bank stability analyses, thus providing one of the two key factors that force a transient response in bank stability through the flow event. The second forcing factor is, of course, the deformation of the bank profile by fluvial erosion, details of which were provided in subsection 2.2.

[25] Two specific mechanisms of bank failure were modeled in this study (Figure 3c). First, the software SLOPE/W [Geo-Slope International, 2001b] was used for the application of limit equilibrium methods to determine stability with respect to slide-type failures. The factor of safety was, in all cases reported herein, computed by the Morgenstern-Price method, with the Mohr-Coloumb criterion in terms of effective stress used for portions of the bank with positive pore water pressures, while the Fredlund et al. [1978] criterion was used for parts of the bank with negative pore water pressures, the latter being expressed as

equation image

where τ = shear strength (kPa), c′ = effective cohesion (kPa), σ = normal stress (kPa), ua = pore air pressure (kPa), ϕ′ = effective friction angle (equation image), u = pore water pressure (kPa) and ϕb = angle (equation image) expressing the rate of increase in strength relative to the matric suction (uau).

[26] Second, the analysis of Thorne and Tovey [1981] was used to determine bank stability with respect to cantilever-type failures. We analyzed only one of the three possible cantilever failure mechanisms defined by Thorne and Tovey [1981], namely the shear-type failure, primarily for reasons of simplicity (the other two mechanisms, toppling and tensile failures, respectively, are more complex to model for the unsaturated conditions of interest here). This is not a major limitation as shear-type failures appear to be the most common observed at the study site, though it is admittedly sometimes difficult to discriminate between shear versus toppling failures in the field. What is clear is that tensile failures are rarely observed. For shear-type cantilever failures (Figure 3c) the factor of safety can be written as

equation image

where Li is the vertical length (layer i) of the cantilever block (m), CTi is the total cohesion (layer i) of the cantilever block (kPa), γi is the unit weight (layer i) of the cantilever block (kN/m3), and Ai is the cross-sectional area (layer i) of the cantilever block (m2). Note that the total cohesion is the sum of the effective and apparent cohesion, the latter being calculated from the third term on the right hand side of equation (4) according to the pore water pressure distribution in the cantilever block.

[27] Both stability analyses were performed for the cohesive part of the bank that is subject to mass failure (Figure 3c). A single set of sediment samples were removed from this horizon to estimate the soil density and shear strength (Table 1) used in the stability analyses. The latter was defined using a triaxial shear test to estimate the effective cohesion and effective friction angle components of shear strength. Unlike the finite element seepage analysis, the bank stability model requires no computational grid and in each time step computations were simply performed with a bank profile shape updated in accordance with the pattern of fluvial erosion simulated in that time step, together with the specified geotechnical properties of the soils and the pore water pressure field imported from the SEEP/W output in the corresponding time step. It should be noted that the bank stability analysis also includes the stabilizing hydrostatic confining pressures exerted by the water in the river channel. However, the variation of unit weight with changing soil moisture content during an event, and the variability of the parameter ϕb, also require consideration. For each time step and sediment layer of the simulation, the unit weight was updated using

equation image

where γ is the unit weight of the soil during the time step (kN/m3), γd is the unit weight of soil under completely dry conditions (kN/m3), and θ is the volumetric water content (m3/m3) during the time step, which is estimated from the soil water characteristic curve using the simulated pore water pressure in that time step. In contrast, a function relating ϕb to the matric suction was defined on the basis of the soil water characteristic curves, which has the effect of varying ϕb from a minimum value of 15° to a maximum value equal to ϕ′ (35°) when the soil approaches saturation. This was achieved by dividing the failure envelope into a series of linear segments with varying ϕb angles and different intercepts (c*), each segment corresponding to a range of matric suction [Fredlund and Rahardjo, 1993].

[28] Both stability analyses were performed at the end of each time step, with the most likely failure mechanism being discriminated by the method that provides the lowest simulated factor of safety. For time steps in which the simulated minimum factor of safety was less than the critical value of 1, the bank is unstable and the bank profile was updated in accordance with the geometry of the simulated failure (Figure 1). In doing so it was assumed that failed material is completely and instantaneously removed by the flow. This is reasonable for the Sieve study site because field observations indicate that failed debris stored at the bank toe has very low effective cohesion and a high degree of saturation, so the material is quickly disaggregated and removed by the flow.

3. Results

[29] In this section we present simulations of bank response during and after the flow event of 18–20 November 1999. Simulations are undertaken for three scenarios:

[30] 1. There is no fluvial erosion and the bank profile remains constant throughout. This scenario emulates the approach adopted in many previous studies.

[31] 2. There is no fluvial erosion but the bank profile is deformed by simulated mass wasting.

[32] 3. The bank profile is allowed to freely deform in response to both simulated fluvial erosion and mass wasting.

[33] Comparing simulation results from these scenarios enables the effects of feedbacks between fluvial erosion, mass wasting and the evolving pore water pressure field to be isolated.

3.1. Simulated Fluvial Erosion

[34] Simulated fluvial erosion (Figure 7), which is common to scenarios 2 and 3, commences at time step 5 (t = 7.5 hours), when the boundary shear stress exceeds the critical conditions for incipient motion of the loose gravel and the wedge of loose sediment at the bank toe is therefore removed. Subsequently, the packed gravel at the toe is directly exposed to fluvial action and, by time step 7 (t = 8.5 hours), boundary shear stress exceeds the critical value for incipient motion of this material. At the same time, the river stage just exceeds the contact between the gravel and the upper cohesive part of the bank, and the boundary shear stress is sufficient to trigger erosion of the cohesive sediment. Consequently, the fluvial erosion rate of both layers rapidly increases, in phase with increasing stage (Figure 7). The rate of fluvial erosion of the basal packed gravel is higher than for the cohesive layer, promoting the generation of an overhanging bank profile, andreaches a maximum during time step 12 (t = 12 hours), the peak flow stage. Most (about 60%) of the simulated fluvial erosion occurs during the peak flow phase, between time steps 12 and 14 (t = 12 to 15 hours). As stage recedes, fluvial erosion of the upper cohesive part is terminated by t = 17.5 hours, with fluvial erosion of the packed gravel at the toe continuing until t = 20.5 hours. Overall, the flow event induces a mean retreat along the bank profile of 0.90 m.

Figure 7.

Simulated fluvial erosion for the investigated flow event (18–20 November 1999) on the Sieve River.

3.2. Pore Water Pressure

[35] Changes in riverbank pore water pressures are usually considered to be controlled by rainfall and river stage variations. However, as shown here bank profile deformation due to fluvial erosion and mass wasting may also play a role in the transient evolution of bank pore water pressures. For reasons of clarity, in this section we focus attention on the differences between the two “extreme” scenarios 1 (where the bank profile is constant throughout the simulation), and 3 (where the bank is deformed by both fluvial erosion and mass wasting).

[36] The pore water pressures simulated within the cohesive layer of the Sieve river bank are presented in Figure 8. In both scenarios, at the beginning of the simulation (from time step 0 to 6, t = 0 to 8 hours) changes in soil moisture content are caused by rainfall infiltration into the soil, leading to an initial reduction of matric suction. The wetting front is partially saturated, with a mean suction of about 2 kPa (Figure 8, time step 6).

Figure 8.

Simulated bank pore water pressure distributions for selected time steps of the Sieve River simulation for scenarios 1 (no bank deformation), 2 (bank profile deformed only by simulated mass wasting), and 3 (bank profile deformed by both simulated fluvial erosion and mass wasting). Locations and associated factor of safety values for the simulated slide and cantilever failure surfaces are also illustrated, as are the positions of the groundwater surface. For scenarios 2 and 3, the initial bank profile is also indicated.

[37] As the flow stage rises (from time steps 6 to 12, t = 8 to 12 hours) and eventually exceeds the contact between the packed gravel at the toe and the cohesive part of the bank, a steep wetting front develops at the contact between the river and the bank profile, inducing a rise in the groundwater level. From this moment the pore water pressure distribution is characterized by a minimum in the central part of the cohesive layer and by higher values in the upper right and lower regions, due to the infiltration of both rainfall and river flow induced wetting fronts. However, subsequently the two scenarios diverge (see Figure 8, time steps 17 and 20). In scenario 3, the river flow induced wetting front and bank retreat have comparable rates of motion: As a consequence, the bank profile retreats into relatively dry areas of the bank, inducing lower pore water pressures along the bank profile. In terms of the effect of this on bank stability, the destabilizing effect of bank pore water pressure is reduced in scenario 3; however, as described in the following section, bank steepening due to fluvial erosion outweighs this effect and still promotes a net decrease in bank stability.

[38] During the hydrograph's falling limb the groundwater level and pore water pressure distribution lower progressively, with notable differences between the two scenarios. In scenario 1, the groundwater level initially recedes at a slower rate than the hydrograph, reaching a new equilibrium with river stage only by time step 20 (t = 18.5 hours). In contrast, in scenario 3 the groundwater level remains in equilibrium with the river stage during the whole drawdown phase (Figure 8, time steps 17 and 20).

[39] To highlight differences in pore water pressures arising between the two scenarios we extracted time series of pore water pressure along the sliding surface (excluding the part occupied by the tension crack) and computed the integral of the simulated values (shaded areas in the sketch of Figure 9). As a proxy for the stress exerted by pore water along the sliding surface, this index (Pw) has significance in the assessment of bank stability conditions. Figure 9a shows the trend of integral values computed only on the saturated portion of the sliding surface (Pw(+), highlighted as the dark grey area in the sketch) for both scenarios, while time series of the integral value across the entire sliding surfaces (Pw, sum of the light and dark grey areas) are shown in Figure 9b.

Figure 9.

Evolution of average pore water pressure values integrated along the failure surface in the Sieve River simulation for scenarios 1 (no bank deformation) and 3 (bank profile deformed by fluvial erosion and mass wasting): (a) pore water pressure integral along the saturated portion of the failure surface and (b) pore water pressure integral along the entire failure surface (excluding tension cracks).

[40] In interpreting these time series it should be noted that in both scenarios the geometry of the sliding surface varies throughout the simulation. However, positive pore water pressures develop in a limited part of the bank close to the river, so changes in sliding surface geometry only have minor effects on the computation of the Pw(+) values. Values of Pw(+) in scenario 1 are always higher than the values computed in scenario 3 (Figure 9a). The importance of Pw(+) as a measure of pore water pressure effects on bank stability is apparent in the computation of the factor of safety, where the “weight” of positive pore water pressures is higher than that of negative pressures, the former being multiplied by tanϕ′ (0.70 in this study, see Table 1), the latter by tanϕb (0.47 to 0.70 in this study, see Table 1).

[41] Pore water pressure integrals computed along the whole failure surface (Pw) exhibit a less regular trend (Figure 9b), the curve for the deforming bank profile (scenario 3) alternating in position above and below the curve for the constant bank profile (scenario 1). During the initial drawdown phase, the former is characterized by lower Pw values; this is not surprising considering the differences in pore water pressure distributions already described. However, at time step 17 (t = 16.5 hours) Pw suddenly increases and approximates the value of the constant bank profile line. This apparently abnormal trend is due to a change in the geometry of the computed sliding surface, which has a shorter length and a deeper tension crack line than in the previous time steps.

3.3. Bank Stability and Sediment Entrainment

[42] Time series of factor of safety with respect to the slide and cantilever failure mechanisms, and associated volumes of sediment entrained by each process, are shown in Figure 10. Factor of safety values in scenario 1 are purely hypothetical because the two failure mechanisms are modeled independently (so there is no consequence if one of the factor of safety curves falls below the critical value of unity), but they present a reference for comparison with the other scenarios. In scenario 1 (Figure 10a) the trend of factor of safety with respect to slide failure follows the hydrograph shape, due to the stabilizing influence of the hydrostatic confining pressure exerted by water in the river [see also Simon et al., 1999, Figure 6.6b, p.137]. As flow stage increases, the factor of safety rises from 1.25 initially to a maximum of 5.31 at the event peak, prior to falling on the recession limb. The factor of safety (0.98) at the end of the event (t = 40 hours) is lower than at the start because of the decreased extent of negative, and increased extent of positive, pore water pressures, caused by the infiltration of rainfall- and river flow–induced wetting fronts into the bank. The onset of instability with respect to slide failure occurs on the falling limb of the hydrograph (t = 16.5 hours), similar to other studies [Casagli et al., 1999; Simon et al., 1999; Simon and Collison, 2002; Rinaldi et al., 2004] that have modeled bank stability variations without considering the effects of fluvial erosion.

Figure 10.

Simulated (left) bank stability and (right) sediment entrainment responses for the 18–20 November 1999 flow event on the Sieve River: (a, b) scenario 1 with the bank profile constant throughout the simulation; (c, d) scenario 2 with the bank profile deformed by mass wasting; and (e, f) scenario 3 with the bank profile deformed by both fluvial erosion and mass wasting. The event hydrograph is also shown (solid lines). The dotted horizontal lines indicate the critical factor of safety value of unity (Fs < 1 implies bank collapse), with the arrows indicating the onset of simulated failure episodes. Note that factor of safety data for the cantilever failures are plotted only for those points in time when a cantilevered (overhanging) bank profile is actually present. Also note that since the bank geometry in scenario 1 is not updated, either in response to mass wasting or fluvial erosion, there is no corresponding sediment entrainment time series (see inset in Figure 10b).

[43] In contrast, factor of safety with respect to cantilever failure declines on the hydrograph's rising limb, falling from 2.23 initially to a value denoting instability (0.99) at the event peak (t = 12 hours). This is caused by a rapid rise in pore water pressure along the failure surface, due to the cantilever failure surface's location closer to the advancing wetting front. After the hydrograph peak the flow recedes, but pore pressures remain elevated. Hence there is a period in which the downward trend in stability continues until the factor of safety minimizes (0.51) during t = 13 to 15.5 hours. The factor of safety subsequently recovers, but stability is not regained until t = 29 hours, when the rate of flow recession slows sufficiently for the slowly equilibrating pore pressure field to restabilize the bank (see Figures 8 and 9).

[44] Scenario 1 is thus characterized by phases of bank instability at the peak of the event and on the falling limb of the hydrograph for cantilever and slide-type failures, respectively. Although the results for this scenario are biased in the sense that the bank profile shape is not updated in accordance with these predicted failures, they show how stability fluctuations for each failure mechanism are driven by the evolving pore water and hydrostatic confining pressure fields. In particular, the distinctive response of each failure type is conditioned by the positions of the respective failure surfaces relative to the advancing wetting fronts. The shallow-seated cantilever failure surface is located close to the advancing wetting fronts, so bank response is more sensitive to pore water pressure. In contrast the deep-seated slide failure surface is relatively distant from the wetting front, causing a lag in the failure response to pore water pressure changes induced by variations in river stage.

[45] Factor of safety time series in scenario 2 (bank profile deformed by mass wasting only) are identical to those simulated in scenario 1 until time step 12 (t = 12 hours), when a cantilever failure is triggered (Figure 10c). In this scenario the cantilever failure results in the collapse and removal of the overhanging upper, cohesive, part of the bank. The absence of fluvial erosion, however, prevents the overhang from reforming, so that (unlike scenario 1) the time series of factor of safety with respect to cantilever failure is terminated at this point. In this scenario the change in bank geometry also has an effect on the simulated factor of safety with respect to the slide failure mechanism, resulting in a substantial increase in stability (e.g., the factor of safety at the peak of the flow event is 9.99 in scenario 2, compared to a value of 5.31 in scenario 1). This increased stability with respect to the slide failure in scenario 2 versus 1 is maintained throughout the recession limb of the hydrograph. Indeed, unlike scenario 1 the factor of safety with respect to the slide failure mechanism does not fall below the critical value of unity, a minimum value of 1.25 instead being attained by t = 18.5 hours.

[46] In the fully coupled scenario 3 (Figure 10e), the factor of safety with respect to cantilever failure is initially (t ≤ 8 hours) identical to that in scenarios 1 and 2, but subsequently declines more rapidly, falling below the critical value at t = 10.5 hours, before the corresponding failure at the peak of the hydrograph (t = 12 hours) in scenarios 1 and 2 (Figures 10a and 10c). Since the simulated pore water pressures in scenario 3 are lower than in scenarios 1 (see Figure 9a) and 2, this variation must be caused by the rapid increase in bank shear stress and fluvial erosion that is predicted to accompany the rise in stage (see Figure 7). Subsequently, the bank remains unstable with respect to cantilever failures until t = 16.5 hours as multiple small-scale cantilevers form and fail. Recovery to a stable condition (with respect to cantilever failure) also occurs earlier than in scenario 1, a consequence of updating the bank profile shape. Thus in scenario 3, multiple cantilever failures are generated by the repeated collapse and reformation (by fluvial erosion) of overhanging bank profiles.

[47] It is not possible, however, to compare the cantilever factor of safety time series for the different scenarios throughout the entire recessional limb because the analysis is terminated at t = 16.5 hours by the onset of a slide failure that reshapes the bank profile, removing the potential for fluvial erosion to generate overhangs. Indeed, the onset of a major slide failure in this scenario is distinct from scenario 2 (where fluvial erosion is absent, Figure 10c). It seems that fluvial erosion, which is predicted to occur throughout the flow event in scenario 3, deforms the bank profile sufficiently to significantly reduce the factor of safety with respect to the slide failure mechanism (e.g., a value of 4.08 at the peak of the hydrograph, compared to 5.31 and 9.99 in scenarios 1 and 2, respectively). This reduction is “significant” in the sense that by t = 16.5 hours, only 4.5 hours after the peak of the event, the combination of reduced confining pressure, elevated pore pressures, and fluvial erosion of the bank profile is sufficient to initiate a major slide failure (Figure 10e), where no such failure occurred previously (Figures 10a and 10c). This is very similar to results published by Simon et al. [2006] for the Upper Truckee River, California, in which measured pore water pressure distributions were combined with simulated bank toe erosion and bank stability analyses.

[48] Differences in the unit volumes of bank material entrained in scenarios 2 and 3 are summarized in the annotated insets in the figures (Figures 10d and 10f). These show that the combined effects of fluvial erosion and mass wasting in scenario 3 provide a total eroded sediment volume (11.65 m3/m) some 33 times greater than the scenario 2 simulations in which fluvial erosion is absent. Clearly, most (9.18 m3/m) of this additional material is eroded directly by hydraulic action, but significant additional (by a factor of about 7) volumes are also contributed by mass wasting, of which about two thirds and one third are attributable to the slide and cantilever failure mechanisms, respectively. That high rates of fluvial erosion cause an increase in the magnitude of bank erosion, both directly and, via its influence on mass wasting, indirectly is not surprising. However, the interaction of pore water pressure, fluvial erosion and mass failure dynamics in the fully coupled analysis (scenario 3, Figure 10f) is seen to provide modified frequencies, timings and modes of erosion relative to scenario 2 (Figure 10d). The significance of this point is now discussed.

4. Concluding Discussion

[49] Our results (Figure 10) indicate that when bank response models include feedbacks between fluvial erosion, bank pore water pressure and mass wasting (scenario 3), predictions of the modes (cantilever versus slide failures), frequencies, magnitudes, and timings of bank erosion episodes are distinct from predictions made by neglecting such effects (scenarios 1 and 2). It must be emphasized that the scope of our simulations are, of course, limited to the sedimentary conditions encountered at the Sieve River bank study site, which comprises a cohesive upper layer overlying a gravel toe. This arrangement is common in upland and piedmont zones in Europe and elsewhere, but is distinct from the fine-grained bank settings that are more usually associated with lowland environments. The scope of the present work is further restricted by our focus on a single, exemplar, event hydrograph. This means that our key findings might best be presented as tentative hypotheses that require testing in a wider range of environments. Nevertheless, if our findings are more widely transferable then there are a number of implications with respect to understanding the mechanisms by which bank materials may be contributed to an alluvial stream, and in this section we discuss these issues.

[50] Notwithstanding the preceding caveats, it can be noted that some of our findings are consistent with previous research [Rinaldi and Casagli, 1999; Casagli et al., 1999; Simon et al., 1999; Simon and Collison, 2002; Rinaldi et al., 2004] in that hydrologically driven variations in bank pore water and hydrostatic confining pressures are key controls on the transient response of mass wasting. Our simulations also highlight the significant role that fluvial erosion can have as a triggering mechanism for mass failure [Darby and Thorne, 1996; Langendoen, 2000; Simon et al., 2006], with significant additional volumes of mass wasted sediment contributed as a result of a bank profile being destabilized by fluvial erosion (Figure 10). However, the simulation data also reveal a previously undocumented effect wherein the evolving pore water pressure field in the near-stream region of the bank is modulated by deformation of the bank profile. Specifically, (positive) pore water pressure values in the near-stream region are decreased below what they would be in the absence of bank deformation because bank failure and/or fluvial erosion expose interior regions with higher values of matric suction. The magnitude of this effect is, in these simulations, insufficient to cause a delay in the onset of mass failure.

[51] The limited number of simulations in this study makes it difficult to discriminate quantitatively the conditions under which this modulating effect may be a more significant control affecting the onset of bank instability. However, in circumstances where fluvial erosion (1) removes bank materials that are saturated or at least partially saturated and (2) proceeds at a rate greater than the advance of the wetting front induced by the presence of flow within the channel, the effect is to remove regions of the bank face with relatively high pore pressures, “replacing” them with drier regions of reduced (or negative) pore water pressure within the bank interior. In terms of bank environments where such conditions might be favored, flashy stream environments and/or banks composed of erodible materials of low hydraulic conductivity are important. In such cases the bank face would be prone to periods of wetting that are sufficiently short in duration to maintain a relatively dry bank interior. However, even in such cases if the effect is to be significant (in the sense of being sufficiently large to influence mass wasting; that is, the factor of safety is modified in such a way that its value crosses the critical value of unity) an additional necessary condition is still required, namely that antecedent stability conditions must be marginal (i.e., Fs ≈ 1). These constraints suggest that the circumstances under which fluvial erosion acts to delay the onset of mass wasting may be restricted to very specific bank settings or times.

[52] What is clear is that our fully coupled simulations predict transient mass wasting responses that differ from those studies that suggest that mass failure is essentially a quasi-catastrophic event that occurs on the falling limb of event hydrographs [e.g., Thorne, 1982; Huang, 1983; Abam, 1997; Darby et al., 1998]. In this study mass wasting instead occurs as a series of failure episodes, with progressive fluvial erosion undermining the bank, modulating the evolving pore water pressure field, and triggering failures prior to, at, and subsequent to event peaks. What is not clear from this preliminary investigation is whether this is general, or whether our findings are specific to the hydrologic and sedimentary setting of this study. Nevertheless, a focus on the timing of mass wasting events, and specifically their phasing in relation to the event hydrograph, is a generically important aspect of bank erosion dynamics as it has a direct bearing on the capacity of the flow to remove the coarse fraction of the failed debris, with the potential for removal significantly decreased if failure occurs late on the recession limb [Rinaldi and Darby, 2007].

[53] The broader significance of these findings in relation to the delivery of bank-derived material to the alluvial sedimentary system can be explained with reference to the concept of basal endpoint control [Carson and Kirkby, 1972; Thorne, 1982]. The basal endpoint concept emphasizes that the residence time of bank-derived material stored in the basal zone is a critical factor controlling long-term bank retreat rates. Neglecting the complicating effects of variations in the caliber of the failed bank material [Wood et al., 2001], basal storage of eroded sediment is favored for riverbanks prone to large-scale slide failure(s), that deliver large volumes of sediment, timed during the recession limb of the hydrograph (when the erosivity of the flow is declining). In contrast, bank environments favoring multiple small-scale cantilever failures occurring before or at the peak of the event would likely substantially reduce the residence time of eroded bank material stored at the toe of the bank.

[54] It can also be noted that fluvial erosion has the effect of increasing the volume of sediment derived from mass wasting (Figure 10), so it can be expected that the basal endpoint status of riverbanks analyzed in this way (Figures 10e and 10f as well as studies by Darby and Thorne [1996], Langendoen [2000], Simon et al. [2006], amongst others) would be distinct from riverbanks represented using an uncoupled approach (Figures 10a and 10c). However, whether the additional material delivered to the toe by fluvially triggered bank failures can be supplied to the in-channel sediment transfer system will again depend on the basal residence time of that material. Thus a fruitful avenue for future research would be to quantify these residence times. If this could be achieved it would provide the means to explain how some rivers are able to both erode and transmit bank sediments effectively enough to produce the high sediment yields described in the introduction to this paper. We suggest that the development of conceptual sediment delivery models, which focus on characterizing transient bank response (especially the phasing and mechanism of erosion episodes in relation to the capacity of the flow to remove eroded material), would represent an advance toward this goal.


[55] This research was supported by the Royal Society (Joint Project Grant 15077). We would also like to thank Rob Millar and Andrew Simon for their constructive reviews.