Modification of Meander Migration by Bank Failures

Home , Bank erosion, Floodplain, Meander (mathematics), Meander cutoff

JournalofGeophysicalResearch: EarthSurface

RESEARCH ARTICLE Modiﬁcation of meander migration by bank failures 10.1002/2013JF002952 D. Motta1, E. J. Langendoen2,J.D.Abad3, and M. H. García1

Key Points: 1Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA, • Cantilever failure impacts migration 2National Sedimentation Laboratory, Agricultural Research Service, U.S. Department of Agriculture, Oxford, Mississippi, through horizontal/vertical floodplain 3 material heterogeneity USA, Department of Civil and Environmental Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania, USA • Planar failure in low-cohesion floodplain materials can affect meander evolution Abstract Meander migration and planform evolution depend on the resistance to erosion of the • Stratigraphy of the floodplain floodplain materials. To date, research to quantify meandering river adjustment has largely focused on materials can significantly affect meander evolution resistance to erosion properties that vary horizontally. This paper evaluates the combined effect of horizontal and vertical floodplain material heterogeneity on meander migration by simulating fluvial

Correspondence to: erosion and cantilever and planar bank mass failure processes responsible for bank retreat. The impact of D. Motta, stream bank failures on meander migration is conceptualized in our RVR Meander model through a bank [email protected] armoring factor associated with the dynamics of slump blocks produced by cantilever and planar failures. Simulation periods smaller than the time to cutoff are considered, such that all planform complexity is Citation: caused by bank erosion processes and floodplain heterogeneity and not by cutoff dynamics. Cantilever Motta, D., E. J. Langendoen, J. D. Abad, failure continuously affects meander migration, because it is primarily controlled by the fluvial erosion at and M. H. García (2014), Modification the bank toe. Hence, it impacts migration rates and meander shapes through the horizontal and vertical of meander migration by bank failures, J. Geophys. Res. Earth Surf., 119, distribution of erodibility of floodplain materials. Planar failures are more episodic. However, in floodplain 1026–1042, doi:10.1002/2013JF002952. areas characterized by less cohesive materials, they can affect meander evolution in a sustained way and produce preferential migration patterns. Model results show that besides the hydrodynamics, bed Received 16 AUG 2013 morphology and horizontal floodplain heterogeneity, floodplain stratigraphy can significantly affect Accepted 31 MAR 2014 meander evolution, both in terms of migration rates and planform shapes. Specifically, downstream Accepted article online 1 APR 2014 meander migration can either increase or decrease with respect to the case of a homogeneous floodplain; Published online 9 MAY 2014 lateral migration generally decreases as result of bank protection due to slump blocks; and the effect on bend skewness depends on the location and volumes of failed bank material caused by cantilever and planar failures along the bends, with possible achievement of downstream bend skewness under certain conditions.

1. Introduction Slump blocks produced by mass failure of river banks may modify bank erosion rates by buffering or but- tressing of the bank face and toe [Wood, 2001] and by reducing the shear stress acting on the bank because of a shifting of the locus of high streamwise velocity away from the bank [Kean and Smith, 2006a, 2006b]. Thorne [1982] investigated the role of slump blocks on the bank retreat cycle and introduced the concept of “basal endpoint control” (BEC) comprising three removal scenarios: (i) impeded removal, where bank mass failures supply material to the base of the bank at a higher rate than it is removed; (ii) excess basal capacity, where the rate of fluvial erosion at the bank toe exceeds the supply rate of failed material; and (iii) unim- peded removal, where the supply and removal processes are in balance. The BEC concept reflects the role of slump blocks in shear stress partitioning at river banks and bank buffering, with consequent reduction of lateral erosion [Wood, 2001]. The impact of this process on meander migration rates and shapes has not been investigated so far. Feedback between mass failure processes and meander migration is complex, because (i) slump blocks are not uniformly distributed along the river but concentrate where the hydrodynamic conditions and the topographic and geotechnical properties of the banks induce mass failure processes; (ii) bank-material stratification affects slump block volumes, which may impact bank protection and therefore migration rates and patterns; and (iii) differing rates of slump block supply through mass failure and their consequent weathering, erosion, and removal make the impact of bank protection by slump blocks time dependent. Different methods have been used to incorporate bank protection through slump blocks in models of bank evolution, channel evolution, and meander migration. Langendoen and Simon [2008] and Motta et al. [2012a, 2012b] increased critical shear stresses for hydraulic erosion to indirectly account for bank protection and

match observed migration. Darby et al. [2007] and Rinaldi et al. [2008] may also have indirectly accounted for slump block protection by calibrating the erodibility parameter to match calculated and measured eroded volumes. Xu et al. [2011] and Parker et al. [2011] described the phenomenon more explicitly by introducing an armor factor as a function of the bank height [Xu et al., 2011] or the cantilever failure volume caused by fluvial erosion of the lower cohesionless layer [Parker et al., 2011]. The research presented here uses the physically and process-based method for bank erosion developed by Langendoen and Simon [2008] and used in meander migration calculations by Motta et al. [2012a, 2012b]. This method explicitly considers the impact of both horizontal and vertical heterogeneity of floodplain materials on cantilever and planar failure mechanisms. Using this methodology in combination with a slump block dynamics model, we analyze how the horizontal and vertical heterogeneity of floodplain materials affects meander migration and enhances planform complexity. This paper therefore represents the next step in the progression of our research on the factors affecting meander migration patterns: we show that, besides the functional form used for bank erosion [Motta et al., 2012a] and the horizontal structure of the floodplain [Motta et al., 2012b], combined horizontal and vertical structure of the floodplain affects meander migration rates and shapes through explicit inclusion of the mechanism of slump block protection. Slump block impact on meander migration has been much less investigated than riparian vegetation, which can affect bank erosion through several mechanisms and depending on the type of vegetation [Micheli and Kirchner, 2002]. Herbaceous vegetation increases bank strength due to the reinforcement of bank soils by roots [Micheli and Kirchner, 2002]; riparian trees increase the roughness of the channel boundary to flow, reducing turbulence intensity and Reynolds stresses [e.g., Lopez and Garcia, 2001], and produce large woody debris affecting primary and secondary flow [e.g., Daniels and Rhoads, 2004]. Research has shown that meander migration coefficients are reduced by riparian vegetation [Johannesson and Parker, 1985; Odgaard, 1987; Pizzuto and Meckelnburg, 1989; Micheli and Kirchner, 2002; Micheli et al., 2004] and meander shapes are influenced by the spatial distribution of biomass density [Perucca et al., 2007]. In this paper we investigate the effects of slump blocks in both reducing erosion rates and modifying patterns of meander migration. We also discuss how riparian trees can interact with slump block dynamics and how that can be incorporated in our modeling framework. As in Motta et al. [2012b], the studied migration scenarios do not consider meander cutoff and its associated planform complexity. Note that current computational algorithms for meander cutoff are quite simplified and only model neck cutoff [e.g., Sun et al., 1996; Xu et al., 2011]. Any inclusion of such algorithms may therefore distort our results.

2. RVR Meander Model of Meander Migration 2.1. Hydrodynamics and Bed Topography The model for hydrodynamics and bed topography implemented in RVR Meander is analytical and obtained from linearization of the two-dimensional depth-averaged Saint Venant equations of motion. It follows the approach first developed by Ikeda et al. [1981] and adopts the secondary flow correction derived by Johannesson and Parker [1989a], who introduced an “effective centerline curvature”—the secondary current cell strength—which lags behind the local channel curvature and determines the bed transverse slope through a coefficient of proportionality named scour factor. Johannesson and Parker [1989a] and Camporeale et al. [2007] provide details of the analytical solution. Important model assumptions are spatially and temporally constant channel width, bed topography is only a function of channel planform (no free response of sediment), and spatially constant friction coefficient. The assumption of constant channel width during meander migration, while generally being supported by empirical observations [Ikeda et al., 1981] and adopted by many authors [e.g., Odgaard, 1987; Johannesson and Parker, 1989b; Howard, 1992; Zolezzi and Seminara, 2001], is a mathematical and physical simplification to obtain the analytical solution for the two-dimensional hydrodynamics and is not a result of modeling conservation of sediment mass. We justified the assumption of constant channel width because the focus of this paper is on time scales in the order of decades (during which the assumption of overall balance between sediment removed by bank erosion and amount stored within point bars is reasonable [e.g., Lauer and Parker, 2008]), and this is the first study that investigates the effect of explicit inclusion of slump block dynamics on patterns of meander migration. Mechanistical model of width adjustment that explicitly accounts for the morphological responses of both banks [e.g., Mosselman, 1998; Darbyetal., 2002; Chen and Duan, 2006; Parker et al., 2011] represents an incremental investigation step which is beyond the scope of this study.

The number of cross sections is set at the beginning of the simulation. Floodplain elevation and properties of the different floodplain layers are assigned on a rectangular, equidistant grid, and an inverse-distance weighting interpolation is used to compute floodplain properties at any given time and spatial location [Motta et al., 2012b]. No model for floodplain aggradation and stratigraphy formation is present, as we are interested in the response of meander migration to given floodplain stratigraphy at time scales smaller than the time to cutoff. Time is measured as flood time, when bank erosion and meander migration occur. The hydrologic regime of the river is crudely simplified as follows: floods are modeled with a single formative discharge, characterized by a certain intermittency i, i.e., the fraction of time during which that discharge is flowing in the river [Paola et al., 1992]. Hence, we “lump” the effect of a variable regime and spectrum of flows that are able to erode and cause meander migration [e.g., Hooke, 1980] into a single discharge [Blondeaux and Seminara, 1985]. The remaining time is characterized by low flow. For instance, if the formative discharge has intermittency of 0.05, 1 year in flood time corresponds to 20 years of real time. We assume here, for simplicity, that a single flood event occurs every year. Thus, for 1 year and intermittency i, the duration of the flood event . Tflood is equal to i years, followed by a low-flow period Tlow flow equal to (1 − i) years. For instance, if i = 0 05, . . Tflood = 0 05 years and Tlow flow = 0 95 years. Using a one-dimensional channel evolution model and a formative discharge, Langendoen [2013] showed that this temporal scaling method yielded predicted long-term morphodynamic changes very similar to those observed for a meandering stream in Northern California.

2.2. Bank Erosion and Meander Migration In the physically based meander-migration approach developed by Motta et al. [2012a] and implemented in RVR Meander, simulated bank retreat is controlled by the resistance to hydraulic erosion and the occurrence of cantilever and planar failures. Erosion of stream banks by sapping or piping, that is, by exfiltrating seepage [Hagerty, 1991a, 1991b], is not considered here. Hydraulic erosion requires that the local boundary shear stress exceeds the critical value to detach crumbs or peds rather than that related to the primary sediment particles [Pizzuto et al., 2008] and is modeled with an excess shear stress relation. An average erosion distance is computed for each layer comprising the composite bank material. The shear stress distribution on banks along bends is influenced by factors such as secondary flow strength, bank slope, width-to-depth ratio, difference in roughness between bed and bank, and bed form progression [Abad et al., 2013]. In the case of a straight channel, integration of the streamwise, depth-integrated momen- tum equation over a portion of wetted perimeter of interest allows computing the averaged stress over that portion. By neglecting time-averaged convective term and Reynolds stresses, the method of Lundgren and Jonsson [1964] is obtained [Parker, 1978]. Following this method, the shear stress acting on each of the 𝜏∗ 𝜏∗ 𝜏∗ bank-material layers is obtained by scaling the shear stress at the toe b = s (n =±1) (where s (n =±1) is the computed bed shear stress at the bank with the linear hydrodynamic model) using the hydraulic radius of the flow area impinging on the layer. For example, in the case of a three-layer bank-material system (num- bered from 1 to 3 from top( to bottom),) the shear stress exerted by the flow on the ith bank-material layer 𝜏∗ 𝜏∗ is computed as i = b AiP3∕A3Pi , where Ai is the flow area impinging on layer i and Pi is the wetted perimeter of layer i. A simplification of this method adopts the local depth instead of the hydraulic radius for scaling. In spite of the shortcomings associated to these methods and their strict validity for straight channels, they are adopted for their simplicity and hence efficiency to perform medium- to long-term simulations of channel evolution. Cantilever failure is the collapse of an overhanging slab of bank material formed by preferential retreat of more erodible underlying layers or simply by the erosion of the bank below the water level with respect to its upper, unsaturated portion. The occurrence of cantilever failure, for the case of shear collapse mechanism [Thorne and Tovey, 1981] considered here, is simply determined from geometrical considerations, once an undercut threshold is exceeded. The undercut threshold is defined as the ratio of bank-material cohesion to unit weight. In cohesive materials, mass failures of whole blocks may occur along a planar or curved failure surface. For high banks with mild slopes (slope lower than 60◦), the failure block typically slides along a curved slip surface, whereas steep banks tend to develop planar-failure surfaces that are often truncated by tension cracks [Pizzuto et al., 2008]. We consider only the latter case here, since eroding banks are often steep

at the outer margins of meander bends. In the RVR Meander model, the planar failure is analyzed using a limit equilibrium method in combination with a search algorithm to find the failure block configuration with the smallest factor of safety [Langendoen and Simon, 2008]. Factor of safety is the ratio of available shear strength to mobilized shear strength, and when smaller than one, the bank is unstable. The method accounts for the effects of pore-water pressure on bank-material shear strength, confining hydrostatic pressure provided by the water in the channel, and can automatically insert tension cracks if the upper portion of the failure block is under tension. Data and parameters that affect planar failure are bank geometry profile, groundwater elevation, bank-material stratigraphy, and, for each bank-material layer, unit weight, cohesion, friction angle, and matric suction angle.

2.3. Parameterization of Slump Block Protection Analogously to Parker et al. [2011] and Xu et al. [2011], we parameterize the impact of mass failure (cantilever and planar failure) on bank retreat and river migration rates in RVR Meander through a “reduction factor” ∗ Karmor applied to the ﬂuvial erosion rate E . The reduction factor is a function of the slump block volume: the larger the slump block volume, the higher the measure of bank protection, and the smaller the reduction factor. Speciﬁcally ( ) Karmor = exp −cAblock,total (1)

( ) 𝜏∗ − 𝜏∗ E∗ = K E∗ = K M∗ c (2) armor unarmored armor 𝜏∗ c

𝜏∗ ∗ 𝜏∗ where is the acting shear stress, M is the erosion-rate coefficient, c is the critical shear stress, Ablock,total is the volume per unit streamwise distance of the slump blocks generated by cantilever and planar failure processes, and c is a coefficient that amplifies the effect of slump block bank protection if greater than 1.0 and dampens the effect if lower than 1.0. Analogously to Xu et al. [2011], we use a simple exponential form

to describe the relation Ablock,total versus Karmor in equation (1), because a physically based description of bank armoring due to slump blocks has not been developed at this time. We expect c to be site speciﬁc and likely related to the size of the river: the larger the size of the river (and therefore implicitly discharge), the smaller c. That is, bank protection per unit volume of slump block produced is proportionally larger for smaller rivers and discharges. The parameterization in equations (1) and (2) lumps the eﬀects of direct bank

armoring and reduction of the shear stress acting on the toe due to shear stress partitioning. For Ablock,total = → → 0, Karmor = 1 (no armoring); for Ablock,total ∞, Karmor 0 (no bank erosion). Our approach is different from that of Xu et al. [2011] because we use a physically based approach for bank erosion instead of Ikeda et al. [1981] method, which is based on the use of a dimensionless migration coefficient. Our approach is also different from Parker et al. [2011] approach since (i) the fluvial erosion formula is used for the bank toe (instead of assuming purely cohesionless bottom bank layer), (ii) contributions from planar mass failures are considered and the bank failure volumes are directly computed using the algorithms for cantilever and planar failure, (iii) bank profile geometry is arbitrary and its time evolution simulated, (iv) actual, measurable bank-material erodibility properties are used, and (v) the parameterization of the reduction factor is simply a function of the slump block volume and does not make any assumption on block disaggregation and rearrangement at the toe of the bank, because the distributions of block sizes and locations cannot be described yet. Wood [2001] observed that the block size distribution shows either a unimodal (skewed toward smaller elements) or bimodal pattern according to the governing fluvial and subaerial processes, which also impact the spatial arrangement of blocks. Here, this complexity is implicitly represented by the factor c.

Migration is computed using discrete time steps which are either equal to Tflood (i.e., one time step per flood event) or a fraction thereof (several time steps per flood event). At the beginning of each time step

ΔT of erosion, the volume per unit streamwise distance Ablock,total of slump blocks at the toe of the bank is obtained by summing the bank-material volumes (if any) generated by cantilever failure and planar failure

during the previous flood time step (Ablock,cantilever and Ablock,planar, respectively) and the volume remaining from previous time steps. This volume is then adjusted for block decay, which occurs during the low-flow periods between floods when slump blocks undergo degradation due to weathering, particle-by-particle

erosion, general disintegration, or removal by ﬂow [Wood et al., 2001]. The decay relation of Parker et al. [2011] is adopted here:

dAblock,total Ablock,total =− (3) dt Tblock

which gives, at the end of the low-ﬂow period, ( ) Tlow ﬂow Ablock,total = Ablock,total,initial exp − (4) Tblock

where Ablock,total,initial is the volume of slump blocks after the last flood, and Tblock is a characteristic period of existence for slump blocks representing the dynamics of slump block decay. Weathering and surface erosion via fluvial processes may cause reduction in block size, which increases the probability of their entrainment by the flow [Wood, 2001]. Conversely, vegetation growth can enhance their resistance to erosion

[Murgatroyd and Ternan, 1983]. These processes can be represented by reducing or increasing Tblock, respectively. Finally, the reduction factor Karmor is calculated using equation (1). The two alternative time-stepping methods mentioned above impact meander migration diﬀerently. If time

step ΔT = Tflood, slump block protection is due to bank material deposited at the toe of the bank during previous floods. The slump blocks produced during a flood affect the next flood because the flow of the next flood will have to remove them before eroding the bank toe, resulting in an overall reduction of erosion

rate parameterized through Karmor. If single floods are modeled by more than one time step, slump blocks generated during a flood will affect erosion during the flood itself. While simplified hypotheses have been formed on the general dimensions of the material deposited at the toe [Wood, 2001; Darby et al., 2002]), we chose not to consider the impact of slump blocks on bank geometry. In absence of further research in the field, including a model for block rearrangement may introduce either a certain degree of arbitrarity or a bias in our results, whereas our goal is to identify general patterns of planform response to horizontal and vertical floodplain heterogeneity. Furthermore, we assumed that slump blocks are attached to the bank, because detached blocks may actually increase the fluvial impinge- ment on the bank toe, enhancing bank retreat instead of reducing it. This effect is beyond the scope of the present paper.

3. Impact of Horizontal and Vertical Floodplain Heterogeneity on Patterns of Meander Migration Idealized meander migration simulations are presented for the case of an initial sine-generated alignment with arc-wavelength of 1000 m and crossover angle of 70◦ with respect to the valley direction, which corresponds to a sinuosity of 1.51. The length of the initial channel of 15 bends is 7494 m. Channel width, flow discharge, valley slope, and Manning’s roughness coefficient are 60 m, 120 m3/s, 0.0005, and 0.030 sm−1∕3, respectively. For the initial centerline, channel slope S = 0.00033, reach-averaged friction coefficient 2 . Cf,ch = 0.0070, reach-averaged squared Froude number Fch = 0 048, and channel half-width to depth ratio 𝛽 = 14.65. Selected ratios between channel width and water depth and between meander wavelength and channel width are common to natural rivers [Williams, 1986]. For representative width-to-depth ratios, it is reasonable to assume that the reach-averaged friction coefficient is not significantly affected by the presence of slump blocks at the banks. We selected a typical value of 5.0 for the scour factor [Zimmerman and Kennedy, 1978]. The selected values of scour factor, friction coefficient and width to depth ratio, cause bends to rapidly expand laterally (i.e., in the cross-valley direction). Since our goal is to evaluate the impact of slump block protection on both downstream and lateral migration, as well as on planform shape, flow parameters were selected to obtain significant initial downstream migration of bends in absence of slump block protection. The value of channel width was selected toward the lower end of the range for lowland rivers, since we expect, from observations and anecdotal information, that in large rivers (width of hundreds of meters and bank height of several meters) slump blocks have little influence on migration as they are easily carried away by the flow or only protect a small portion of the bank; on the other hand, in very small rivers, slump blocks may profoundly limit migration.

Initial bank height, when measured in straight reaches, equals 2.5 m, enough to contain the formative, quasi-bankfull discharge. Note that, as sinuosity increases, water depths increase as a consequence of the decreasing channel slope; therefore, the water surface elevation gets progressively closer to the bank top over time. Groundwater level is constant in time and equal to the top elevation of the bank. Flood- plain materials are described on a 20 m x 20 m grid. Patches of material characterized by different properties are distributed ad hoc and a priori across the floodplain in order to identify effects on migration patterns. Regarding floodplain stratigraphy, we focused on the sim- plest and common case characterized by the presence of two material layers, which is typical of alluvial rivers, where the bottom bank layer mainly comprises coarse-sediment bar deposits and is overlain by finer-grained material deposited during large or overbank flow events. The floodplain ground level, as well as the interface between the two layers, are assumed to be inclined at a slope equal to the longitudinal valley slope. In order to quantify the impact of stream bank failure and horizontal/vertical floodplain heterogeneity on meander migration and identify patterns of response, we computed statistics for one of the lobes of the migrated center- Figure 1. Definition sketch of bend geometry characteristics lines where slump blocks are produced. A lobe used for statistical analysis. is defined as the portion of the migrated centerline between crossings with the valley axis (Figure 1). The evaluated statistics are bend skewness, averaged lateral migration (computed as illustrated in Motta et al. [2012b]), and downstream bend length as indicator of downstream meander migration (Figure 1b). Note that bend length is herein computed as the linear distance between points A and A’ in Figure 1b, not as arc-length of the bend. Values of statistics were then normalized by their value in absence of slump block protection.

3.1. Impact of Cantilever Failure Within our model, cantilever failure occurs when an overhang threshold is exceeded. The supply of slump blocks is therefore determined by the rate at which the overhang forms, which is controlled by the erosion resistance of the bank-material layers, the distribution of shear stress on the bank, and the thickness of the top layer or the thickness of bank above the water surface level. We performed simulations to investigate the effect of vertical variation of bank-material erodibility, varying parameters such as the thickness of the 𝛿 less erodible top layer top layer, the slump block residence time Tblock, and the parameter c in equation (1). We applied slump block protection either along the entire channel (section 3.1.1) or only within a small area of the floodplain (section 3.1.2). 3.1.1. Horizontally Homogeneous and Vertically Heterogeneous Floodplain The first set of simulations is focused on the effects of slump block protection caused by cantilever failure in vertically heterogeneous, but horizontally homogeneous, floodplain. Specifically, in Figure 2 we examine 𝛿 the effects of the parameters top layer, Tblock, and c on the evolution of meander planform. As the bottom layer is mainly formed by bar material deposited at the inner bank during meander migration, 𝛿 the thickness top layer generally scales with the difference between bank-top elevation and top elevation

Figure 2. Examples of cantilever failure impact on meander migration for horizontally homogeneous and vertically heterogeneous floodplain. Flow is from left to right, and the initial channel centerline is dashed, while the final, migrated centerlines are continuous. Red migrated centerlines correspond to meander migration in absence of slump block pro- 𝛿 tection. Black migrated centerlines refer to the following cases, all including slump block protection: (a) top layer = 1.0 m, 𝛿 . . 𝛿 . Tblock = 200 days, and c = 1.5; (b) top layer = 0 5 m, Tblock = 200 days, and c = 1 5;(c) top layer = 1 0 m, Tblock = 200 days, . 𝛿 . . and c = 0 75;and(d) top layer = 1 0 m, Tblock = 400 days, and c = 1 5. All migrated centerlines in Figures 2a–2d are for . . . T = 4 years. Other parameters were kept constant among the four simulations: ΔT = 0 05 years, i = 0 05 (Tflood = 0 05 years), 𝜏∗ = 3.5 Pa, M∗ = 2.0E-07 m/s, M∗ = 8.0E-07 m/s, and the vertical variation of shear stress on bank c top layer bottom layer was not considered. (e) A closer view for the same simulations of Figure 2d but the two migrated centerlines correspond to different simulation times.

𝛿 ∗ 𝜎 ∗ ∗ of the point bar. For our model, in bankfull conditions, top layer = Dch (1 − A B ), where Dch is the depth at channel centerline, A is the scour factor, B∗ is the half width of the channel, and 𝜎 is the secondary flow 𝛿 strength (computed by convolution of channel curvature). Hence, for an alluvial floodplain, top layer is expected to vary spatially because it depends on the channel curvature at the time of point bar formation. 𝛿 For the simulations shown in Figure 2 we used a simplified configuration where top layer is constant over the 𝛿 floodplain, with the goal of isolating the effect of varying top layer on meander migration.

We used data from Wood [2001] to determine the order of magnitude of block residence time Tblock used in the simulations presented in this paper. Fitting equation (4) to the time series of the intermediate axes of the

largest blocks (used as proxy of slump block volume) in Goodwin Creek (Mississippi, USA), yields Tblock ≈ 250 to 400 days. Note that ﬂow, channel form, and boundary materials for Goodwin Creek diﬀer from our test case; in Goodwin Creek, bank height is about 4.5 m, large discharges are about 30 to 40 m3/s, top width is 35 m, and cohesion is about 6–7 kPa. However, the data available for Goodwin Creek provide an indication

of the order of magnitude of Tblock, given the general lack of data on slump block dynamics. 𝜏∗ . The critical shear stress c = 3 5 Pa for the lower layer, which is larger than a typical critical shear stress value of purely cohesionless material. A certain degree of cohesion is always present in the lower layer due 𝜏∗ to presence of silt and clay, with consequent increase of the value of c of up to one order of magnitude [Mitchener and Torfs, 1996; Panagiotopoulos et al., 1997; Kothyari and Jain, 2008]. Presence of gravel may provide armoring of the lower layer too. Corresponding values of erosion-rate coeﬃcient M∗ were selected using the relation developed by Hanson and Simon [2001]. Figure 2 compares simulated migrated centerlines with and without inclusion of slump block protection. The overall eﬀect of slump block protection is a reduction of meander migration rates. The slump block

Figure 3. Examples of cantilever failure impact on meander migration for different sizes and locations of floodplain patches for Scenario 1 (see Table 1). The patch characterized by the presence of a bank top layer which produces slump blocks is colored blue. The bend considered for the computation of planform statistics is highlighted in green. In all panels, flow is from left to right, and the initial channel centerline is dashed, while the final, migrated centerline is continuous.

𝛿 volume scales with the thickness of the top layer top layer and the diﬀerence in erosion between top and bottom layer. The larger the thickness of the top layer and the larger the diﬀerence in erodibility between

bottom and top layer, the larger the slump block volume per unit streamwise distance Ablock,total, and con- sequently, the larger the bank armoring (equation (1)) and the smaller the future erosion rate of the bottom 𝛿 layer (equation (2)). This process is highlighted by comparing migrated centerlines in Figure 2a ( top layer = 1 𝛿 . m) and Figure 2b ( top layer = 0 5 m). Comparing Figure 2a with Figures 2c and 2d show that smaller parame-

ter c and slump block residence time Tblock (which reduces Ablock,total in the exponent in equation (1) through equation (4)) reduce the impact of slump block protection, which we will quantify in the next section. Fur- thermore, as cantilever failures preferentially occur at locations where the channel migrates at a faster rate, differences in meander bend shapes arise compared to the case without inclusion of slump block protection (Figure 2e). This implies that slump block protection can potentially affect meander patterns even without invoking horizontal heterogeneity of the floodplain materials. 3.1.2. Horizontally Heterogeneous and Vertically Heterogeneous Floodplain We examined the effects of varying materials across the floodplain by only considering slump block protection for a floodplain patch. Simulated centerline migration outside the patch is solely determined by the

Table 1. Parameters for Cantilever Failure Simulations in Horizontally and Vertically Heterogeneous Floodplainsa

T ΔTc 𝜏∗ M∗ M∗ 𝛿 T Vertical Stress c top layer bottom layer top layer block Scenario Parameter Changed (years) (years) (m−2) i (Pa) (m/s) (m/s) (m/s) (days) Distribution on Bank 1 - 4.0 0.05 1.5 0.05 3.5 2.0E-07 8.0E-07 1.0 200.0 No 𝛿 2 top layer 4.0 0.05 1.5 0.05 3.5 2.0E-07 8.0E-07 0.5 200.0 No 3 Tblock 4.0 0.05 1.5 0.05 3.5 2.0E-07 8.0E-07 1.0 400.0 No 4 M∗ 4.0 0.05 1.5 0.05 3.5 4.0E-07 8.0E-07 1.0 200.0 No top layer 5 c 4.0 0.05 0.75 0.05 3.5 2.0E-07 8.0E-07 1.0 200.0 No 6 Vertical stress distribution on bank 4.0 0.05 1.5 0.05 3.5 2.0E-07 8.0E-07 1.0 200.0 Yes 7 ΔT 4.0 0.025 1.5 0.05 3.5 2.0E-07 8.0E-07 1.0 200.0 No aT is the simulation duration (in flood time), ΔT is the simulation time step, c is the parameter in equation (1), i is the intermittency of the formative discharge, 𝜏∗ is the critical shear stress of bank material, M∗ is the erosion-rate coefficient of the bank top layer, M∗ is the erosion-rate coefficient of c top layer bottom layer 𝛿 the bank bottom layer, top layer is the thickness of the bank top layer, and Tblock is the slump block residence time.

erodibility characteristics of the lower layer (Karmor = 1). Size and location of the patch reflect the extent of 𝛿 the portion of the bend impacted by slump blocks and therefore bank protection. In the patch, top layer is constant and produces slump blocks as the lower layer is eroded at a higher rate. We evaluated nine combinations of patch size and location. Figure 3 shows that patch size was varied as 1, 0.5, and 0.25𝜆, where 𝜆 is the initial, linear meander wavelength. Patch locations were centered over the bend apex or over upstream or downstream crossovers. Table 1 reports the parameters used for seven simulation scenarios. For each scenario, the nine combinations of patch size/location were simulated. Simulated migrated centerlines show that both size and location of the floodplain patch, where slump blocks are produced, affect patterns of meander migration (Figure 3). Figure 4 shows the normalized values of bend length, lateral migration, and bend skewness for varying size and location of the patch and for the different scenarios in Table 1, with reference to the lobe indicated in Figure 3.

Figure 4. Plots of normalized statistics of (a1 to e1) bend length, (a2 to e2) lateral migration, and (a3 to e3) bend skewness as a function of patch size and location. Each column corresponds to a diﬀerent scenario (see Table 1). Statistics refer to the lobe indicated in Figure 3 and are normalized with respect to their value in absence of slump block protection. Contour plots were generated by interpolation from the computed values for the nine combinations given by patch size (1, 0.5, and 0.25𝜆) and patch location (centered over the bend apex or over upstream (US) or downstream (DS) crossovers).

From Figures 3 and 4 several trends can be observed for bend length, bend skewness, and lateral migration: 1. Bend length increases for the upstream-located patch and decreases for the downstream-located patch (Figures 4a1–e1). As mentioned above, we focused here on the effect of the patch on a single bend (highlighted in green in Figure 3). Obviously, in a series of bends, the bends located upstream or downstream of the studied bend are also affected. However, the scope here is to compare the effect of different combinations of patch size and location on the same bend. The increase of bend length for the upstream-located patch is explained by the reduced rate of downstream migration of the portion of the bend upstream of the bend apex. Analogously, the decrease of bend length for the downstream-located patch is due to the reduced rate of downstream migration of the portion of the bend downstream of the bend apex. The 𝛿 increase (or decrease) of bend length is larger for larger thickness top layer of the top layer in the patch (compare Figures 4a1 and 4b1), given the larger slump block volume Ablock,total and the smaller factor Karmor. Increasing the slump block residence time Tblock produces a similar effect (compare Figures 4a1 and 4c1). 2. The produced slump block volumes generally decrease lateral migration (Figures 4a2–e2). The decrease is larger for larger downstream-located patches, because the direction of meander migration is down- 𝛿 stream. The decrease of lateral migration is also more pronounced for larger top layer or Tblock (compare Figure 4a2 to Figures 4b2 and 4c2). 3. The effects of cantilever failure and horizontal/vertical floodplain heterogeneity on bend skewness are more complex (Figures 4a3–e3). The simulated bend skewness deviates from the typical upstream-skewed pattern to which meander bends evolve according to “classic” meander migration models that are based on the use of a dimensionless migration coefficient approach in homogeneous floodplain. Bend skewness generally increases for the small downstream-located patch and reduces for the small upstream-located patch. The decrease of bend skewness indicates that the migrated centerline is less upstream-skewed than that obtained in absence of the patch. The decrease of bend skewness for the upstream-located patch is caused by the reduced rate of downstream migration of the portion of the bend upstream of the bend apex. Slump block protection can even lead to a negative value of bend skewness, that is a downstream-skewed bend (e.g., Figures 4a3, 4c3, and 4e3). As for the case of bend 𝛿 length and lateral migration, slump block residence time Tblock and top layer thickness top layer affect the magnitude of skewness modification but not the general planform pattern (compare Figure 4a3 with Figures 4b3 and 4c3). For a patch size of the order of the meander wavelength, skewness is generally reduced as slump blocks delay the process of upstream skewness development. Further, reducing the parameter c in equation (1) (i.e., reducing the degree of bank protection per unit volume of slump block) reduces the magnitude of the response to horizontal and vertical heterogeneity of floodplain materials but not its patterns (Figure 4, Scenario 5). Modeling flood events with multiple time < steps (ΔT Tflood), to account for bank protection by blocks generated during the flood itself, enhances the effect of slump block dynamics on bend statistics (compare Figures 4a and 4e columns) but again patterns of response stay the same. Further, considering the vertical shear stress distribution on the bank instead of a single, constant shear stress or increasing the erosion-rate coefficient of the top layer (respectively Scenarios 6 and 4 in Table 1 but not shown in Figure 4) does not change the results. This implies that the thickness of 𝛿 the top layer top layer and the slump block residence time Tblock are the primary parameters that control the effect of bank toe protection by blocks. < For the time-stepping method in which ΔT Tflood, it is of interest to investigate the effect of the simulation time step ΔT as well as the overhang threshold for cantilever failure. Note that the time step ΔT does not affect the simulated migration using Ikeda et al.’s [1981] method, which linearly relates migration to the excess velocity at the outer bank. Also, it does not significantly affect the predicted centerline migration using the method of Motta et al. [2012a, 2012b], which is based on the use of fluvial erosion formula

(equation (2)) with no explicit account for slump block protection as Karmor = 1 (since the method adopts a threshold for bank erosion, the degree of impact of ΔT on predicted migration depends on river planform and erodibility parameters). Here, however, as cantilever failure contributes to slump block volume every time step, the magnitude of ΔT could aﬀect the simulated migrated centerlines. Figure 5 shows an example of the temporal evolution of failed bank-material volume per streamwise distance per time step

due to cantilever failure Ablock,cantilever (Figure 5a), slump block volume per streamwise distance Ablock,total (Figure 5b), and armor factor Karmor (Figure 5c) at the outer bank for a location downstream of the bend apex and for two different values of time step and overhang threshold. In Figure 5, different flood events

(a) (b) (c) 2.5 2.5 1.2

) 2 2 2 ) 2

(m 0.8

1.5 (m 1.5 (−) 0.6

1 1 armor K

block,total 0.4 A block,cantilever

A 0.5 0.5 0.2

0 0 0 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 Flood time (years) Flood time (years) Flood time (years)

Figure 5. Examples of time evolution of (a) failed volume per streamwise distance per time step due to cantilever failure Ablock,cantilever, (b) slump block volume per streamwise distance Ablock,total, and (c) armor factor Karmor. Ablock,cantilever is the volume generated at a particular time step, whereas Ablock,total is the accu- mulation of cantilever failures over time experiencing block decay. In the legends, OT is the abbreviation for overhang threshold. For these simulations, i = 0.05; . therefore, Tﬂood = 0 05 years.

are separated by minima of Ablock,total or, equivalently, maxima of Karmor (recall that slump blocks decay during low-flow periods between floods). During each flood, Ablock,total increases and Karmor decreases as erosion takes place. Increasing the overhang threshold (that is, more cohesive material) reduces the frequency of cantilever failure events (Figure 5a). However, the cumulative effect produced by different time steps or

overhang thresholds on erosion distances and planform shape was found to be not significant, as Ablock,total, and therefore, Karmor were not impacted much. Note that all simulations presented use a single value of scour factor. This is because our scope is to investigate the impact of slump block protection of river banks on meander migration patterns under different combinations of horizontal and vertical heterogeneity of floodplain materials keeping all the other parameters unchanged. In natural settings, the scour factor, which describes bar development within our modeling approach, varies both between rivers and within reaches. Figure 6 shows a comparison of migrated centerlines for three values of scour factor for the same floodplain patch size and location. Different responses are obtained for the same configuration of floodplain heterogeneity, because the scour factor value affects the locus of high velocities along bends, the magnitude of shear stress at outer banks, and the bank height and the number of bank-material layers exposed to the flow (therefore affecting slump block volumes). For the set of simulations in Figure 6, lowering the scour factor (Figure 6a) reduces migration in general and, in the patch of interest, causes the development of slight downstream skewness; increasing the scour factor (Figure 6c) accentuates the upstream skewness. Figure 6 is not meant to provide a thorough evaluation of the effect of the scour factor on meander migration in horizontally and vertically heterogenous floodplains, which is beyond our scope here; it suggests, however, that future investigation on the interaction and combined effect of bed and bank bathymetry on bank protection and meander migration is needed.

500 500 500

0 0 0 y [m] y [m] y [m]

-500 -500 -500

2000 2500 3000 2000 2500 3000 2000 2500 3000 (a) x [m] (f) (b) x [m] (c) x [m]

Figure 6. Examples of scour factor impact on meander migration for patch size of 0.5𝜆, patch location centered over the downstream crossover of the initial centerline, and simulation duration of three flood years (other parameters correspond to Scenario 1 in Table 1). (a–c) The scour factor value in each panel equals to 4, 5, and 6, respectively. The patch characterized by the presence of a bank top layer which produces slump blocks is colored blue. In all panels, flow is from left to right, and the initial channel centerline is dashed, while the final, migrated centerline is continuous.

Table 2. Parameters for Planar Failure Simulationsa

Scenario Number of Layers Cohesion C Shear Stress Distribution on Bank Set Tension Crack Depth (m) 1 5 All layers: C = 1 kPa 1 0.5 2 5 All layers: C = 1 kPa 2 0.5 35Toptwolayers:C = 15 kPa, bottom three: C = 1 kPa 2 0.5 45Toptwolayers:C = 5 kPa, bottom three: C = 1 kPa 2 0.5 55Toptwolayers:C = 15 kPa, bottom three: C = 1 kPa 2 1.5 65Toptwolayers:C = 1 kPa, bottom three: C = 15 kPa 2 0.5 aRegarding shear distribution on bank: 1 = shear stress constant over the diﬀerent bank layers; 2 = shear stress scaled using ratio area/wetted perimeter on each layer.

3.2. Impact of Planar Failure Planar failure of a stream bank is controlled by the bank geometry profile and the geotechnical properties of the bank materials. Using a constant modeling discharge, occurrence of planar failure is specifically caused by the following mechanisms: (i) spatial distribution of the geotechnical properties of the floodplain materials; (ii) temporal changes in bank height caused by variations in toe elevation following the evolution of the planform-induced bed topography of the river. Bank height changes due to floodplain formation have a similar effect for geologic time scales but are not considered here; and (iii) steepening at the bank toe, because of basal erosion due to larger boundary shear stresses or more erodible basal material. Note that we here investigated banks that can fail as large planar failure blocks, therefore with a significant degree of cohesion. We did not consider mainly cohesionless sediments, where surface landsliding [Daerr and Douady, 1999] acts to mitigate the steepening; and (iv) changes in flow depth as sinuosity changes. Xu et al. [2011] considered the latter as an indirect indicator for slump block protection but did not explicitly solve for the planar failure process. Here we examined how spatial changes in geotechnical properties, which affect bank stability [Parker et al., 2008], affect meander migration in a sustained way. Among the different geotechnical parameters (sat- urated unit weight, cohesion, friction angle, and matric suction angle), we considered only the impact of cohesion, since it most strongly controls bank stability when all other factors are kept constant [Parker et al., 2008; Langendoen and Simon, 2008; Samadi et al., 2009]. Note that Langendoen and Simon [2008] showed that similar variations in factor of safety, which expresses the degree of bank stability, can be obtained when varying each of the other parameters listed above, though by different amounts. In our simulations, the cohesion of the floodplain material is 15 kPa, except for a floodplain patch equaling one meander wavelength and centered over the apex of a bend. The soil cohesion in the patch is reduced and has a value as low as 1 kPa. The larger value of 15 kPa is one standard deviation above the mean of the Simon [1989] data set on West Tennessee rivers, whereas the lowest cohesion value of 1 kPa corresponds to one standard deviation below that mean. Five bank-material layers are considered to capture the vertical variation of boundary shear stress acting on the bank, which enables the simulation of bank steepening through enhanced basal erosion. Table 2 summarizes the relevant parameters adopted for the different simulation scenarios. In Scenario 1 no vertical variation of cohesion or shear stress acting on the bank is considered. In Scenario 2 the vertical variation of simulated boundary shear stress is taken into account and all bank-material layers are characterized by the same cohesion. In Scenario 3 the top two layers have larger cohesion than the lower three. In Scenario 4 the top two layers are less cohesive than in Scenario 3 (5 kPa instead of 15 kPa), but they are still more cohesive than the lower three layers. Scenario 5 is the same as Scenario 3 except that tension crack depth is increased. In Scenario 6, the top two layers are less cohesive than the lower three layers. Other parameters were set as follows: critical shear stress = 3.5 Pa, erosion-rate coefficient = 8.0 ⋅ 10−7 m/s, unit weight = 18 kN∕m3, friction angle = 25◦, simulation duration in flood time = 4 years, time step = 0.05 . years, intermittency = 0.05 (Tflood = 0 05 years), coefficient c (equation (1)) = 1.5, and slump block residence time = 200 days. Stream banks were initially vertical causing planar failures at the first time step, which initialize slump block volume and armor factor. Other planar failure events may occur at later stages; otherwise, the bank expe- riences near-parallel retreat as result of fluvial erosion and cantilever failure. In order to isolate the effect of

(a) (b) (c) 12 12 12 11 11 11 10 10 10 9 9 9 8 8 8 7 7 7 6 6 6 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1 0 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Flood time (years) Flood time (years) Flood time (years)

Figure 7. Examples of time evolution of failed volume per streamwise distance per time step due to planar failure Ablock,planar, slump block volume per streamwise distance Ablock,total, and armor factor Karmor for(a)Scenario2,(b)Sce- nario 3, and (c) Scenario 4 listed in Table 2. Ablock,planar is the volume generated at a particular time step. Ablock,total is the accumulated planar failure volume experiencing block decay during the dry period. For clarity, only the first two flood . . years are shown at intervals of one time step. For these simulations, i = 0 05; therefore, Tflood = 0 05 years.

planar failure on meander migration, cantilever failures, although active, did not contribute to the slump

block volume Ablock,total. Figures 7–9 show how the vertical variation of both bank-material cohesion and acting stress, as well as bank geometry and height, aﬀect the occurrence of planar failure events and, therefore, bank protection and migration patterns.

Whereas Ablock,cantilever showed a fairly regular temporal pattern (Figure 5a), the time evolution of the failed volume per streamwise distance per time step due to planar failure, Ablock,planar, is more episodic (Figure 7). Since planar failure events occur when the ﬂuvial bank erosion destabilizes the bank for given

geotechnical properties, planar failure events are associated with sudden increases in Ablock,total, followed by periods of block decay. As a consequence, Karmor greatly decreases after a planar failure event, to then recover gradually. Figure 8 shows a typical evolution of the bank profile for the different scenarios listed in Table 2. In Sce- nario 1, planar failure occurs only at the beginning of the simulation followed by parallel retreat of the bank. Because no vertical variation of shear stress or cohesion is considered, the slope of the bank is preserved impeding new planar failure events. In Scenario 2, the vertical variation of simulated boundary shear stress steepens the bank toe, which leads to multiple planar failure events (Figures 7a and 8). In Scenario 3, for which the top two layers have larger cohesion, fewer failure events occur (Figures 7b and 8), increasing the rate of bank retreat as fluvial erosion is not reduced as much. In Scenario 4 the top two layers are less cohesive than in Scenario 3 (5 kPa instead of 15 kPa). This increases the number of large-volume planar failure events (Figures 7c and 8). In Scenario 5 a deeper tension crack also increases the number of planar failure events compared to Scenario 3. Because of the greater number of failures bank retreat in Scenarios 4 and 5 is reduced compared to that in Scenario 3. In Scenario 6, characterized by less cohesive material in the top two layers, only small-volume, top-bank planar failure events occur. The smaller failure volumes do not affect bank retreat as much. Therefore, if a large-enough portion of the bank material is characterized by lower cohesion (especially the lower layers) and bank steepening is simulated, the frequency of planar failure events is increased and could impact bend shape and migration rates. Note that the evolution of channel planform affects the occurrence of planar failure. The location of the pools and riffles move as meander bends grow and migrate, thereby changing bank height and conse- quently bank stability. Figure 8 shows that for scenarios 3, 4, and 5, planar mass failures of the entire bank occur when bank height exceeds a certain threshold. For example, a planar failure occurred at Station −46 m for Scenario 3 when bank height exceeded this threshold. Therefore, unstable outer banks along some por- tions of a bend may become stable as the bend migrates, or conversely, stable banks may become unstable, potentially leading to preferential migration patterns. Figure 9 shows the impact of bank failures on meander migration. Lateral migration is reduced because planar failures are more likely to occur close to the bend apex where outer banks are steeper and taller. The

Figure 8. Simulated outer bank retreat at a section initially located immediately downstream of the bend apex where bank height is large, for the six scenarios listed in Table 2. Plotted bank profiles are separated by the same duration of low-flow period and following flood time step. For clarity, only the first floodyear is shown at intervals of one time step. Labels “L1” to “L5” indicate the five floodplain material layers. Vertical red arrows indicate bank profiles modified by the occurrence of planar failure.

simulated shape is also diﬀerent Scenario 3 when compared to that obtained in 200 the absence of planar failure. We used the metrics described in section 3, with reference to the lobe highlighted 0

y [m] in green in Figure 9, to quantify these eﬀects. Scenario 3 is characterized -200 by a 7% decrease of lateral migra-

Scenario 4 tion and a 29% decrease of bend

22002400 2600 2800 3000 3200 skewness, whereas Scenario 4 is char- x [m] acterized by a 31% decrease of lateral Figure 9. Example of planar failure impact on meander migration for migration and a 60% increase of Scenarios 3 and 4 listed in Table 2. The ﬂoodplain patch characterized by bend skewness. one or more layers with lower cohesion is colored blue. Flow is from left to The impact on meander migration is right, and the initial channel centerline is dashed, while the ﬁnal, migrated centerlines are solid. The bend considered for the computation of planform larger for higher frequency of planar statistics is highlighted in green. failure events. In order to characterize

quantitatively the frequency, two methods were used. The first method computes the number of planar failure events that occurred during the simulation, normalized by the simulation time. This method has the disadvantage of being dependent on the number of centerline nodes adopted to discretize the channel centerline. The second method evaluates the total planar failure volume produced during the simulation, normalized by both the stream length within the less cohesive floodplain patch and by the simulation time, to obtain a measure of volume per unit length per year. For Scenarios 3 and 4 plotted in Figure 9, a value of 0.03 and 0.18 m2/yr per bank is respectively obtained. Scenario 4, for which the impact of planar failure on bend statistics is more pronounced, is therefore characterized by a frequency 6 times larger than Scenario 3. This implies that field observations of high planar failure frequency may indicate a possible effect of planar failure on meander planform development.

4. Discussion and Conclusions We investigated how meander migration is modulated by bank erosion, and specifically by bank-failure-induced slump-block protection of the bank toe using the RVR Meander computer model. Simulations showed that the process of cantilever failure, which occurs rather continuously during migration because it is controlled by the fluvial erosion at the bank toe, has an important impact on migration rates and meander shapes. In particular, bend length can either increase or decrease, and bends can even become downstream skewed. The magnitude of such effects is primarily controlled by the thickness of the less erodible top layer and the characteristic slump block residence time. The process of planar failure, which is controlled by bank geometry and bank-material shear strength, displays a more episodic pattern than cantilever failure. However, it can still significantly affect meander migration by reducing migration rates and modifying bend shapes. Because planar failures concentrate where bank heights are larger, bank protection provided by slump blocks depends on the planform-induced topography of the river and can cause preferential directions of meander migration. Therefore, less cohesive floodplain patches can contribute to the complexity of river planform geometry. This work shows that, besides previously studied processes such as hydrodynamics, bed morphodynamics, and horizontal floodplain heterogeneity, the vertical structure of the floodplain affects meander planform evolution. The model results show that slump blocks can be a first-order control on meander migration, especially in presence of combined horizontal and vertical heterogeneity of the floodplain materials. This effect cannot simply be included in the modeling by modifying the migration coefficient (in the “classic” Ikeda et al. [1981] approach for meander migration) or the bank erodibility parameters (in Motta et al. [2012a] approach), as slump block generation depends on river planform configuration and direction of migration, as well as location of the river bends with respect to the floodplain material patches. In this regard, this paper extends the work of Constantine et al. [2009], who established a relation between Ikeda et al.’s [1981] migration coefficient and the erodibility coefficient for fluvial erosion. In rivers where slump blocks influence meander migration, such relation cannot be established and an explicit inclusion of slump block dynamics in the model of meander migration is necessary. When correlating the observed complexity of meander planform shapes to floodplain field surveys, the vertical distribution of floodplain materials should be considered. In particular, where the process of cantilever failure is dominant, attention should be paid to the location of stratigraphically different floodplain material patches with respect to the river centerline; the degree of cohesiveness of floodplain soils should also be investigated to assess if large slump blocks produced by bank planar failure have affected or will affect meander migration. Our model could be used to identify a priori the degree of impact of slump block bank protection on channel planform evolution by knowing the floodplain stratigraphy or, in absence of that information, by carrying out Monte Carlo simulations analogously to what presented by Motta et al. [2012b] for horizontal floodplain material heterogeneity. Additional research is needed to describe deterministically the following: 1. The slump block residence time, which is dependent on processes such as weathering, particle-by-particle erosion, general disintegration, and removal [Wood et al., 2001]. 2. The coefficient c (equation (1)), which depends on the dynamics of slump blocks and their effect on the local hydrodynamic field. This is needed to study the impact of mass failure mechanisms for different river scales.

3. The mechanism of rearrangement of slump blocks at the toe of the bank, for which there is currently a general lack of field studies. The process is rather complex, as it depends not only on the properties of the bank material and the mechanics of failure [Darby et al., 2002; Parker et al., 2011] but also on bank height and slope, as well as the planform-induced river bathymetry, which determines bank height and therefore the portion of bank surface shielded by the blocks. This means that inclusion of slump block bank protection in a one-dimensional model, which is unable to account for the effect of channel curvature on bank height, may not capture the impact of slump blocks on channel evolution. 4. The interaction and composite bank protection effect of slump blocks and downed trees which may accompany slump blocks along riparian forests. Within the approach presented in this paper, the effect of riparian vegetation may be parameterized through increased values of slump block residence time

Tblock, as well as through the coefficient c, because wood is an additional factor of shear stress partitioning [Manga and Kirchner, 2000]. Field investigation and a more refined theoretical framework are needed to identify the relative importance of vegetation and slump blocks in the mechanics of bank protection. 5. The impact of the dynamics of channel bar development on the process of slump block protection, as mentioned in section 3.1.2. Furthermore, the impact of stream bank erosion by sapping or piping at reach scale should also be investigated in the future by including a more refined model for saturation conditions in banks during and after flood events.

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