## 1. Introduction

[2] Ever since the seminal work of *Ikeda et al.* [1981], numerous two-dimensional (2-D) depth-averaged analytical models have been derived to study the hydrodynamics, bed morphodynamics, and migration of meandering rivers [*Blondeaux and Seminara*, 1985; *Crosato*, 1987, 1989; *Johannesson and Parker*, 1989; *Sun et al.*, 2001a; *Zolezzi and Seminara*, 2001] and the degree of planform shape complexity that can be obtained [*Seminara et al.*, 2001; *Frascati and Lanzoni*, 2009]. Planform complexity may also result from the heterogeneity of the floodplain soils. Figure 1 shows some examples where floodplain heterogeneity affects river planform shapes through erodibility and soil mechanics properties (Beaver River in Alberta, Canada and Wabash River in Illinois and Indiana), vegetation dynamics (Mackinaw River in Illinois and Luangwa River in Zambia), and land use (Beaver Creek in Ohio). Examples in Figure 1show that, while most of the “signal” associated with floodplain heterogeneity is erased, over geological timescales, through cutoffs that govern the statistical long-term behavior of meandering rivers [*Stolum*, 1996; *Camporeale et al.*, 2005], floodplain heterogeneity can affect short- and medium-term meander migration (i.e., migration over periods before cutoff occurrence), whose assessment is important for economic and social reasons (e.g., interaction with urban areas, prevention of damage to infrastructure, reduction of agricultural land losses, maintenance of biological diversity).

[3] While several studies have investigated the streamwise variation of bank erodibility and resulting bank erosion [e.g., *Lawler et al.*, 1999; *Hudson and Kesel*, 2004; *Wallick et al.*, 2006; *Wynn and Mostaghimi*, 2006; *Constantine et al.*, 2009], the impact on planform shape has only lately been addressed systematically: in particular, the very recent contribution by *Güneralp and Rhoads* [2011] has highlighted the importance of accounting for soil heterogeneity in river meandering models. Earlier work by *Howard* [1992, 1996] and *Sun et al.* [1996] concentrated on the development of different features in the floodplain caused by river migration and their influence on the future development of the channel. In particular, *Sun et al.* [1996] focused on the change in erodibility associated with features such as point bar, floodplain, and oxbow lake deposits. These researchers were able to reproduce the formation of meander belts over long time periods. Regarding planform shapes, *Sun et al.* [1996] stated that the typical meander wavelength is mainly determined by flow rate, channel dimensions, and valley slope, and is independent of the different erodibility properties of the various sedimentary deposits. *Perucca et al.* [2007] examined the role of vegetation on meandering river morphodynamics and the feedback between riparian vegetation dynamics and planform dynamics. Relating bank erodibility to the biomass density, they showed the possible occurrence of unusual meander shapes that do not show marked upstream skewness. *Posner and Duan* [2012] studied planform evolution using (1) a migration model with a constant migration coefficient and (2) a model where the instantaneous migration coefficient was treated as a stochastic variable satisfying either a uniform or normal distribution. They found that the stochastic approach yielded more realistic predictions of meander planform evolution. Recently, *Güneralp and Rhoads* [2011] examined how the scale, magnitude, and stochasticity of floodplain erosional resistance influence the planform evolution of meandering rivers. Using the power spectra of curvature series of migrated meander sequences, they showed that heterogeneity in erosional resistance has a major influence on meander evolution.

[4] All of the above mentioned studies assumed that the rate of bank erosion is linearly proportional to near-bank excess velocity through an empirical migration coefficient [*Howard*, 1992, 1996; *Sun et al.*, 1996, 2001b; *Güneralp and Rhoads*, 2011; *Posner and Duan*, 2012], following the approach first introduced by *Hasegawa* [1977] and independently by *Ikeda et al.* [1981] a few years later. *Motta et al.* [2012] recently proposed an alternative method to model meander migration at the reach scale, by explicitly formulating bank erosion in terms of the physical processes responsible for bank retreat. The approach is analogous to that adopted at the bend scale by *Darby et al.* [2007], *Rinaldi et al.* [2008], and *Darby et al.* [2010]. It simulates the processes of hydraulic (or fluvial) erosion, cantilever and planar failure [*Grissinger*, 1982; *Thorne*, 1982; *Lawler*, 1993; *Lawler et al.*, 1997; *ASCE Task Committee on Hydraulics, Bank Mechanics, and Modeling of River Adjustment*, 1998; *Couper*, 2004; *Rinaldi and Darby*, 2007]. *Motta et al.*'s [2012]model employs the physically and process-based algorithms from the CONCEPTS channel evolution model [*Langendoen and Alonso*, 2008; *Langendoen and Simon*, 2008; *Langendoen et al.*, 2009]. Hence, channel migration depends on measurable soil properties, natural bank geometry, and both vertical and horizontal heterogeneity of floodplain soils. *Motta et al.*'s [2012] model can be used either by (1) considering the presence of vertical layers in banks and parameterizing bank protection through slump blocks produced by mass failure of the upper portion of the bank [*Motta et al.*, 2011], similar to the approach proposed by *Parker et al.* [2011]; or (2) adopting a single “representative” soil layer, which will produce the same bank retreat as method (i). The latter is the approach followed herein, because the focus is on the horizontal heterogeneity of the floodplain. This approach is equivalent to assuming “basal endpoint control” [*Thorne*, 1982]: medium- to long-term bank retreat rates are controlled by the rate of sediment entrainment and removal from the toe, that is by fluvial erosion of blocks and aggregates from the bank toe instead of parent bank materials.

[5] The rate of lateral hydraulic erosion *E** (m/s) for any bank or floodplain soil is modeled using an excess shear stress relation typically used for fine-grained materials [*Partheniades*, 1965]:

where *M** is the erosion rate coefficient (m/s), *τ**_{c} is the critical shear stress (Pa), *k** is the erodibility or detachment coefficient (m^{3}/(N s)), and *τ** is the shear stress acting on the soil particles (Pa). Variables with an asterisk are dimensional, whereas variables without an asterisk are dimensionless.

[6] The parameters *M**, *τ**_{c}, and *k** depend on soil properties and are therefore site and soil specific. The critical erosional strength denotes the cohesive strength provided by interparticle forces of attraction or repulsion acting at the microscopic level, including electrostatic forces, van der Waals forces, hydration forces, and biological forces [*Papanicolaou et al.*, 2007]. Although the form of equation (1) is rather simple, complexity and uncertainty are introduced when assigning appropriate values for the erodibility parameters and the boundary shear stress *τ** exerted by a given flow. These parameters are highly variable, which explains why observed rates of fluvial erosion range over several orders of magnitude [*Darby et al.*, 2007]. Locally, values of the resistance to erosion parameters are time dependent, because of the effects of subaerial exposure, weathering, vegetation, or variations in soil water [*Wynn et al.*, 2008]. Difficulties reside with the data collection itself: the commonly used jet erosion test [*Hanson and Cook*, 2004] cannot be sufficiently repeated, primarily because of time limits on site access resulting in small data sets to estimate the parameters that are needed [*Constantine et al.*, 2009; *Darby et al.*, 2010].

[7] In equation (1), *τ** is the shear stress acting on the soil particles. Determining the bank shear stress is challenging. Several methodologies have been developed to calculate the shear stress in natural cross sections and implemented into one-dimensional (1-D) models [e.g.,*Lundgren and Jonsson*, 1964; *Khodashenas and Paquier*, 1999]. The effect of secondary currents on bank shear stress was studied for straight channels with trapezoidal cross sections [*Knight et al.*, 2007; *Rodriguez and Garcia*, 2008] and cohesive river banks [*Papanicolaou et al.*, 2007]. These findings still need validation for meandering rivers. Depending on planform shape and bed morphology, the distributions of bed and bank shear stresses near the bank toe could differ markedly from those in straight channels. Detailed measurements of shear stresses are available in the field. However, values mostly reflect specific cases and cannot be generalized. *Lane* [1955]proposed a methodology that relates bank shear stress to the bed shear stress through a proportionality factor. In bends, this parameter is influenced by secondary flow, bank slope, width-to-depth ratio, and difference in boundary roughness between bed and bank. Drag on small-scale topographic features can substantially alter the near-bank flow field and therefore the shear acting on the grains [*Kean and Smith*, 2006a], and depends on the magnitude of the shape parameters of bank topography [*Kean and Smith*, 2006b]. Bank form roughness may be a major component of the spatially averaged total shear stress [*Darby et al.*, 2010]. Large woody debris also may represent a factor of further partition of the shear stress [*Manga and Kirchner*, 2000]. Note that modification of macroscale flow around slump blocks, and stress partitioning near the banks due to bank forms or woody debris can be taken into account indirectly by increasing *τ**_{c} in our approach and may also be associated with significant spatial variability in *τ**_{c}.

[8] Notwithstanding all these problems to quantify the erosion resistance parameters and the boundary shear stress, equation (1) has performed well for assessing stream bank erosion in incised streams in the midcontinental United States [e.g., *Langendoen et al.*, 2000, 2009]. Resistance to erosion values were measured using a jet erodibility tester [*Hanson and Cook*, 2004] and boundary shear stress was computed using a one-dimensional flow model.

[9] The present study evaluates the impact of horizontal heterogeneity of floodplain soils on meander migration rates and patterns using a stochastic approach. The variability in meander planform development is determined as a function of the spatial variability in floodplain soil resistance to erosion and the spatial scale of this variability. The study extends the recent work by *Güneralp and Rhoads* [2011], while introducing a relation for bank erosion computations which explicitly accounts for different erodibility properties associated with soil heterogeneity. For homogeneous soil, *Motta et al.* [2012] showed that equation (1) produces greater planform shape complexity when compared to the classic approach in which the migration rate is based on excess velocity at the outer bank [*Hasegawa*, 1977; *Ikeda et al.*, 1981]. The comparison between the two methods is herein extended to the case of heterogeneous floodplain soils. Two different spatial distributions for floodplain heterogeneity are used. Simulations are performed using a Monte Carlo probabilistic method to evaluate how the variability in erodibility parameters affects meandering river migration at the reach scale, analogously to the approach followed, at the local scale, by *Parker et al.* [2008] and *Samadi et al.* [2011]for the computation of the factor of safety for planar and cantilever failure, respectively. This work identifies to what extent meander migration is affected by floodplain heterogeneity, the parameters that govern this effect, and how their variability affects migrated-centerline variability. Furthermore, study outcomes could be useful to determine feasible space and time scales for deterministic simulations of bend evolution when, for instance, there is interest in remeandering a stream for restoration and naturalization purposes.