Although it is widely acknowledged that spatial heterogeneity in floodplain erosional resistance should affect the planform evolution of meandering rivers, past studies have not systematically explored the importance of this effect. Using a physically-based model of river meandering and stochastically-generated heterogeneous landscapes, we analyze in detail how spatial variability in erosional resistance influences meander evolution and the emergence of planform complexity. The simulated planforms are remarkably similar to natural freely-meandering rivers, both visually and in their spectral signatures. We quantify the dependence of meander morphology on both the scale of erosional heterogeneity and the magnitude of erosional variability in external forcing. We also show how the sensitivity of autogenic meandering processes to stochastic variability in the environment leads to different patterns of planform evolution even in landscapes with the same mean spatial heterogeneity and magnitude of variability. The results demonstrate that heterogeneity in erosional resistance has a major influence on meander evolution and should be incorporated into morphodynamic models to enhance our understanding of meandering dynamics at the landscape scale.
 Meandering rivers are highly-dynamic earth-surface systems. The migration of meandering rivers over their floodplains has been of societal and scientific interest because of its importance in hazards associated with bank erosion [Piégay et al., 2005], in landscape evolution [Van De Wiel et al., 2007], in terrestrial sediment fluxes [Odgaard, 1987], in floodplain development [Peakall et al., 2007], and in riparian-ecosystem dynamics [Ward et al., 2002]. Freely-meandering natural rivers uninhibited by valley walls or bedrock substrate typically evolve into complex planforms characterized by elongated bends with multiple absolute maxima of curvature – features known as compound loops or multilobes [Brice, 1974; Frothingham and Rhoads, 2003] or into irregular patterns (i.e., sequences of bends that differ in shape and size) [Ferguson, 1975]. Despite general recognition that external influences, such as spatial variation in floodplain erodibility, may influence meander evolution, the emergence of complex meanders has been viewed mainly as the autogenic product of feedbacks between fluid-dynamic and morphodynamic processes of meandering [Parker et al., 1983; Seminara, 2006]. The main approach to characterizing these feedbacks has been through mathematical models of meander migration [Ikeda et al., 1981; Johannesson and Parker, 1989; Zolezzi and Seminara, 2001], in which migration rate is assumed to be a linear function of excess near-bank velocity.
 The quest to accurately simulate complex, irregular meander forms has mainly pursued increasingly sophisticated representations of process mechanics in predictive models under the assumption that complex meander forms are the result of the internal (i.e., autogenic) dynamics of meandering. Advanced models are capable of simulating the emergence of compound loops [Zolezzi and Seminara, 2001], but cannot reproduce the full range of meander complexity in pre-cutoff conditions. Regardless of their degree of sophistication, virtually all models are deterministic and assume homogeneous erosional resistance of the floodplain landscape over which the river migrates.
 Discrepancies between meander patterns simulated by extant models and the complex, irregular meanders observed in natural rivers may not solely be due to incomplete characterization of process mechanics, but could partly reflect the influence of external (i.e., allogenic) forcing, particularly spatial variability in floodplain erosional resistance [Hudson and Kesel, 2000]. Spatial variability may result from a variety of environmental factors, including soil mechanical properties (e.g., soil type, sediment-deposition structure, and geology), land use and land cover (e.g., vegetation patterns and density), and floodplain topography. Moreover, even in relatively homogenous landscapes, such as floodplains, spatial diversity in soil characteristics can develop through nonlinear dynamics of intrinsic soil-forming processes [Phillips, 2001]. An alternative view recognizes that planform complexity may reflect interactions between fluvial processes and landscape variability across spatial and temporal scales [Wynn and Mostaghimi, 2006; Constantine et al., 2009].
 A few pioneering numerical-modeling studies have included results that show how spatial variability in vegetation cover [Perucca et al., 2007] and sedimentary and geological constraints (e.g., easily erodible point-bar deposits, resistant valley walls and clay plugs) [Howard, 1996; Sun et al., 1996] can influence the pattern of meandering rivers, but have not systematically explored the effect of spatial variability in erosional resistance on planform evolution. A preliminary numerical investigation by Sun et al.  suggested that a spatially-uncorrelated random distribution of erodibilities can introduce irregularities into evolving planforms. These findings point to the need for an extensive examination of the influence of spatial heterogeneity and stochasticity in floodplain erosional resistance on the complex morphological evolution of meandering rivers [Sun et al., 1996].
 In this study, we systematically examine how the scale, magnitude, and stochasticity of floodplain erosional resistance influence the planform evolution of meandering rivers. The results show that both the scale of heterogeneity and the magnitude of variability in erosional resistance of the floodplain landscape substantially influence the emergence of complex meanders and irregular planform patterns of meandering rivers. The findings also indicate that planform evolution is highly sensitive to specific spatial configurations of floodplain erosional resistance.
 To simulate planform evolution of meanders, we employ a simple physically-based mathematical model, which is represented by a first-order ordinary-differential equation characterizing the relation between near-bank excess velocity and planform curvature [Ikeda et al., 1981]. The model has been widely used in river-meandering research [Howard, 1996; Stølum, 1996; Sun et al., 1996]. It incorporates basic fluid-dynamic and morphodynamic processes of meandering and captures suitably the fundamental planform characteristics of freely-meandering natural rivers, such as fattening and upstream skewing of bends, yet fails to reproduce compound loops and multilobes and irregular meander patterns [Seminara et al., 2001; Camporeale et al., 2007; Güneralp and Rhoads, 2009] (Figure 1a). Thus, in our case, the model provides a control; any observed complexity and irregularities in the simulated planforms must reflect the influence of spatial variability in erosional resistance. The solution of the hydrodynamic component of the model (i.e., ub(s), which is the excess near-bank velocity along the streamwise axis (s)), as well as the other simulation details, is given in the auxiliary material (Text S1). The migration of each point along the streamwise axis ζ(s) is computed based on an empirical relationship, ζ(s) = Eub(s), where E is a coefficient that represents the erodibility characteristics of the channel bank material.
 Spatial heterogeneity (i.e., patchiness) in floodplain erosional resistance is represented by a grid surface. The scale of heterogeneity is varied as a function of grid-cell (i.e., patch) size in the down-valley direction (Ldv) and cross-valley direction (Lcv). Patch size ranges from Ldv = 2λ to λ/16 and Lcv = 4A to A/4 where λ and A are, respectively, the linear wavelength and amplitude of initial meander bends used in the simulations. Initial conditions correspond to a Kinoshita curve [Kinoshita, 1961], representing a typical river meander in its expansion phase [Parker et al., 1983].
 To examine the influence of the spatial variability in bank erodibility of a floodplain landscape, we define the bank erodibility coefficient E = E0 + Er where E0 is the erodibility coefficient characterizing homogenous landscape (HL) conditions and Er is a stochastic erodibility coefficient that represents a spatially-varying component of bank erodibility. To characterize the variability in bank erodibility in a spatially heterogeneous landscape (SHL), random numbers are generated for each grid-cell using a probability density function, p(Er). Since extensive landscape-level data on bank resistance are not available, as a first approximation, we assume a Gaussian distribution (μ, σ) where Er has a mean of zero (μ) and a standard deviation σ defined as a function of E0. Two values of σ, σH = E0 and σL = E0/2, are used in separate simulations to explore the sensitivity of the results to variation in σ. Each heterogeneous surface of random values of Er is smoothed using a polynomial filter so that cell values (Er) do not exhibit abrupt discontinuities along cell edges. As the river migrates across the floodplains in the simulations, points along the channel move along smoothly-varying erodibility gradients, encountering distinct patches of differing erodibility.
 The heterogeneous mosaics of differential erosional resistance with different scales of patchiness are meant to represent spatial arrangements of factors that influence migration and that vary with scale, such as patches of floodplain vegetation (e.g., forested versus nonforested areas), sedimentological complexity (e.g., clay plugs versus sandy alluvium), and human activities (e.g., cropland versus managed forest). For each heterogeneity configuration, ten realizations are generated to evaluate the effects of stochasticity in bank erodibility on the spatial characteristics of planform evolution. In addition, simulations are conducted with the two different values of σ (i.e., σH and σL) to evaluate the influence of the magnitude of variability in bank erodibility on the morphodynamic evolution of meanders in heterogeneous landscapes. Simulations are performed until the point of cutoff; the analysis does not consider post-cutoff meander dynamics.
3. Results and Discussion
 Simulations with homogeneous floodplain resistance generate bend sequences that are highly regular in shape and size (Figure 1a). The cumulative effect of upstream curvature on local migration dynamics results in upstream skewness of bends, an asymmetrical pattern that gives rise to a bimodal spectrum of bend wave number (Figure 2a). The spectrum has a major low-frequency peak, approximately equal to 1/λc km−1, where λc corresponds to the curvilinear (i.e., meander) wavelength of the simulated meanders. It also has a minor high-frequency peak (∼ 3/λc km−1) that defines a secondary maximum in the curvature series resulting from upstream skewness of the meanders (Figures 1a and 2a).
 Meander sequences generated with SHLs evolve into much more complex planform morphologies than the ones generated with HL (Figure 1). In the SHL cases, the spectra are more diffuse with one or more major low-frequency peaks and numerous minor peaks across a range of high frequencies (Figure 2). Visual comparisons between meander sequences generated with SHLs and those for natural meandering rivers reveal remarkable similarities (Figures 1 and 2). The simulated meanders, like their real-world counterparts, develop complex morphologies characterized by irregular elongated asymmetrical bends that skew upstream or downstream and that, in some cases, evolve into compound loops with multiple lobes. Moreover, the curvature spectra for natural rivers exhibit similar distributions to those for the simulated meanders with distinct low-frequency major peaks and multiple minor peaks across a range of high frequencies (Figure 2).
 The spatial scale of erosional heterogeneity (patchiness), defined by Ldv and Lcv, strongly influences the spatial characteristics of planform evolution (Figures 3 and S1). Bends that develop on landscapes with the largest patch sizes (i.e., Ldv = 2λ − λ, Lcv = 4A) resemble distorted forms of Kinoshita curves. Many of these bends are highly elongated and upstream-skewed, but others are compressed, leading to an irregular meandering pattern with high variability in bend amplitudes (Figures 3 and S1b). Overall, this set of bends has the least variable power spectra (Figures 2b and S1d) and the highest cross-valley extent of the meander belt (Figures 3 and S1a). Bends remain elongated as the patch size in the down-valley direction decreases (Ldv = λ/2 − λ/16) for large patch size in cross-valley direction (Lcv = 4A), but bend complexity increases as elongated compound loops develop and bend symmetry is altered (Figure 3). As cross-valley heterogeneity increases (Lcv = A − A/4), the maximum cross-valley extent of the meander belt decreases (Figures 3 and S1a). Bends become fattened and have a tendency to form compound loops or multi-lobes (Figure 3). This increased planform complexity is defined in the spectra by multiple major low-frequency peaks as well as by many minor peaks across a range of high frequencies (Figures 2c and 2f). Thus, spectra for bends with high cross-valley variability and low to moderate down-valley variability exhibit both the highest mean frequency of curvature and the highest variability of curvature frequencies (Figures 3, S1c, and S1d). The finest heterogeneity structure (i.e., the smallest patch size, Ldv = λ/16, Lcv = A/4) results in the smallest variability in bend amplitudes (Figures 3 and S1b) and relatively low variability in curvature frequencies (Figures 2j and S1d). The simulated planform patterns for this configuration, however, are still much more complex than those for HL (Figures 1a and 2a), confirming the strong influence of small, high-frequency variations in planform curvature on morphodynamic evolution of meanders [Camporeale et al., 2007; Güneralp and Rhoads, 2009, 2010].
 The emergence of complex bend forms and irregular bend sequences induced by floodplain erosional heterogeneity decreases the time to cutoff. In particular, simulated planforms in the domain [Ldv = 2λ − λ/4, Lcv = A − A/4], where bend complexity is greatest as indicated by high variability in curvature spectra (Figure S1d), have the shortest times to cutoff (Figure 3). Another notable feature is downstream-skewed bends, the development of which is common for patch sizes of Ldv < λ and 4A < Lcv < A/2. Theoretical work has related downstream skewness to super-resonance conditions where migration at a particular location on the channel is also influenced by curvature downstream of that location [Zolezzi and Seminara, 2001]. This effect, combined with subresonance conditions where migration is influenced by upstream curvature, leads to the development of compound loops and multilobed bends. The results here clearly demonstrate that the development of complex meanders is not necessarily an inherent deterministic mechanism of the meandering process under homogenous erosional resistance. Such complexities may in fact arise from the interactions between deterministic morphodynamic processes of meandering and spatial variability in erosional resistance.
 Despite varying complexities of the meander sequences, the major, low-frequency peak in the curvature spectra corresponding to λc varies little across different heterogeneity configurations. The persistence of this peak demonstrates that 3the initial λc is preserved to some extent for all SHLs. On the other hand, spatial heterogeneity in bank erodibility greatly modifies minor peaks corresponding to high-frequency variations in planform curvature (Figure 2). The extent and character of these variations (e.g., diffuse, multiple peaks) represent the degree of complexity in planform geometry and reflect the influence of the scale of erosional heterogeneity on spectral variability of curvature series (Figure S1d).
 The degree of complexity in planform morphology reflects not only the scale of heterogeneity, but also the magnitude of variability in the bank erosional resistance. Increased magnitude of variability in bank erosional resistance enhances the influence of the spatial heterogeneity and promotes the development of more irregular and complex planform patterns (e.g., Figure 1l vs. Figure 1m and Figure 1v vs. Figure 1w). This effect is evident in increased spectral complexity of the planform patterns generated over landscapes with high magnitude of variability (e.g., Figure 2f vs. Figure 2g).
 Marked differences in simulated planform geometries across ten realizations of each patchiness configuration demonstrate the pronounced sensitivity of planform evolution to stochastic variability in erosional resistance (Figure 1c). The difference between simulations with the same patchiness highlights that, even in floodplain landscapes with the same scale of heterogeneity and the same magnitude of variability, the sensitivity of autogenic processes to details of the spatial structure of erosional variability leads to different evolutionary patterns of planform development. This sensitivity to boundary conditions shows that allogenic effects can induce nonlinear-dynamical behavior and that, in detail, no two meandering rivers are likely to have identical planforms.
 Our systematic analysis indicates that allogenic forcing associated with spatial heterogeneity in floodplain erosional resistance is not just a trivial or minor effect that slightly modifies patterns of river planform evolution generated by purely autogenic mechanisms. Instead, this forcing has a strong influence on planform evolution, leading to the emergence of substantial bend complexity and planform irregularity in meandering rivers. The complex planform morphologies generated by simulations with spatially-heterogeneous landscapes are remarkably similar to those of natural freely-meandering rivers, both visually and in their spectra of planform curvature. The character of planform complexity depends both on the spatial scale of erosional heterogeneity and the magnitude of stochastic variability in bank erodibility.
 Landscapes with patch sizes larger than the initial meander size promote the evolution of highly elongated, upstream-skewed meanders with high variability in amplitudes. As heterogeneity increases and patches become smaller than the initial meanders, bend irregularities and planform complexity increase, resulting in asymmetrical meanders, downstream-skewed bends, and compound loops or multilobes. Fine-grained heterogeneity results in meanders most similar to those for a homogenous floodplain. Increases in the magnitude of stochastic variability in erodibility enhance the influence of spatial heterogeneity. Moreover, planform evolution is sensitive to details of the spatial structure of erosional resistance in that the morphometric evolution of planforms differs for two floodplains with the same magnitude of erosional variability and spatial scale of heterogeneity.
 The findings are significant because they demonstrate the strong influence of both the spatial scale of heterogeneity and the magnitude of variability in erosional resistance on planform dynamics and show that the development of complex bend forms and irregular patterns of meandering are not attributable solely to autogenic processes in natural rivers with heterogeneous floodplains. The explicit integration of environmental heterogeneity and stochasticity characterizing landscape patterns and processes into morphodynamic models of river meandering can significantly enhance our process-based understanding of meandering dynamics at the landscape scale. Such an in-depth understanding is critical for improving our ability to forecast, manage, and restore integrated meandering river–floodplain systems, particularly in the presence of global environmental change.
 The authors thank the National Science Foundation for funding through grant BCS-0425209.
 The Editor thanks the anonymous reviewer for assistance in evaluating this paper.