## 1 Introduction

[2] Meandering rivers are landscape features which have interested scientists, civil engineers, geomorphologists, and decision makers for centuries. Although great progress has been made in the study of meandering rivers over the last few decades, their behavior remains to be fully understood.

[3] Meander models generally consider three interconnected processes: the hydrodynamics, bed morphodynamics, and bank morphodynamics (cf. Figure 1 and *Camporeale et al*. [2007, Figure 2] for a graphic overview of these interconnected processes). The flowing water induces shear stresses on the river bed which move the bed sediments, ultimately leading to a pattern of stable and migrating morphological features. The flow of water also induces shear stresses on the banks which cause them to erode, whereas banks in zones of low shear stress may accrete, thus changing the river's course. As the banks are generally made of more cohesive material or bound by riparian vegetation, the time scales of the bank adaptation are generally longer than those of the bed adaptation. This implies that the bed and bank morphodynamics can be considered separately. The morphological changes of the bed and the banks in their turn influence the hydrodynamics, thus giving rise to further morphological changes, and so on. An alluvial river is thus a complex adaptive dynamic system. As a step toward unraveling this system, this article focuses on the bed morphodynamics.

[4] At present, it is possible to model meandering channels in great detail. *Rüther and Olsen* [2007] simulated the 3 day meander experiment by *Friedkin* [1945] with a three-dimensional meander model, solving the hydrodynamics by means of a Reynolds-averaged Navier Stokes model with a linear *k*−*ε*turbulence closure. Typical length and time scales for meander evolution are much larger than the 72 h and the 40 m of the *Friedkin* [1945] experiment. The simulation of longer time and length scales motivates the use of reduced-order models (obtained by depth-averaging (and width-averaging) operations), which are generally an order of magnitude faster than their three-dimensional counterparts. Moreover, reduced-order models require less input, which is useful when data are scarce or uncertain.

[5] Present reduced-order meander models are either based on the linearity or gradual variation assumption. The linearity assumption implies that the variables are assumed to be small with reference to a basic state, a condition which is often justified in mild-curvature bends. The gradual variation assumption allows large variations of the variables as long as they vary gradually in space, a condition which is generally justified in mild-curvature bends [cf. discussion in *Bolla Pittaluga et al*., 2009].

[6] Models which are based on the linearity assumption include *Struiksma* [1983a], *de Vriend and Struiksma* [1984], *Odgaard* [1989], *Lancaster and Bras* [2002], *Abad and Garcia* [2006], and *Crosato* [2008], whereas models based on the gradual variation assumption include *Zolezzi and Seminara* [2001] and *Bolla Pittaluga et al.* [2009].

[7] In strongly curved channels, the linearity and gradual variation assumptions no longer hold. Furthermore, in strongly curved bends, the frequently used linear parameterization of secondary flow [*Rozovskii*, 1957; *Engelund*, 1974; *de Vriend*, 1977], which is proportional to the depth to radius of curvature ratio, no longer holds. This is due to a nonlinear feedback mechanism which causes the reduction of the secondary flow strength compared to its linear equivalent [*Blanckaert and de Vriend*, 2003, 2010; *Ottevanger et al*., 2012]. These issues motivate the development of nonlinear reduced-order meander models which also allow sudden variations in curvature and include a nonlinear treatment of the secondary flow (in short, nonlinear without curvature restrictions). *Blanckaert and de Vriend* [2003, 2010] extended the hydrodynamic modeling to strongly curved bends. In this paper, the morphodynamics will be extended in a similar manner.

[8] Morphological development is related to the sediment transport field. In the cases considered herein, this transport field can be described by a sediment transport formula relating the transport rate and direction to local flow properties. Most sediment transport formulae are derived from straight flume measurements and are related to the drag force imposed by the fluid on the sediment particles. In curved channels, a transverse bed slope *∂**z*_{b}/*∂**n* (the gradient of the bed level *z*_{b}in transverse direction *n*) exists with a bed that usually deepens in outward direction (cf. Figure 1). Gravity exerts a downhill force on sediment particles positioned on the tranversally inclined bed. The combination of these forces allows us to determine the sediment transport direction on sloping beds as found in curved channels.

[9] *Fargue* [1868] [cf. *Hager*, 2003] was the first to report that the magnitude of the transverse bed slope in a river bend correlates well with the inverse radius of curvature 1/*R* of the bend. Later research showed the correlation of the transverse bed slope in a bend to the water depth *H* as well [*van Bendegom*, 1947]. By inserting a constant of proportionality *A* (also known as the scour factor), the transverse bed level gradient can be expressed in the following manner:

*Allen* [1978] reported that *van Bendegom* [1947] derived a first estimate of the scour factor *A*=10 from the force balance on a stationary sediment particle. Later, *Rozovskii* [1957] independently found *A*=11, which he also justified by comparison to field and laboratory data. *Engelund* [1974, 1975] obtained *A*=21 for an annular flume with a moving lid and *A*=7 for open-channel field conditions. *Ikeda et al.* [1981] and *Odgaard* [1981] reported values of the scour factor *A* between 2.5 and 6 for alluvial rivers. *Zimmerman and Kennedy* [1978] found an expression for *A* related to the friction factor (based on an empirical relation found by *Nunner* [1956]), a ratio relating the projected surface area of a nonspherical particle to its volume and the particle densimetric Froude number (), where *U* is the bulk velocity, *g* is the gravitational constant, Δ(≈1.65) is the relative density of sediment, and *D* is a characteristic sediment diameter. The above mentioned findings are all based on the assumption of mildly sloping streamwise bathymetries. Recently, *Seminara et al.* [2002] and *Parker et al.* [2003] developed an implicit theoretical model for bed load transport at low shield stresses, based on the force balance on a particle in motion along aribitrarily sloping beds. Their results can, however, not easily be expressed in terms of equation (1).

[10] The range of the scour factor in strongly curved bends remains to be investigated, but measurements in the field by *Nanson* [2010] and *Schnauder and Sukhodolov* [2012] and measurements in laboratory flumes by *Blanckaert* [2010] and *Abad and Garcia* [2009] reveal scour factors of *A*=0.1, 0.35, 2, and 2, respectively, which suggest that in sharp bends significantly lower scour factors apply than those typically found in more mildly curved channels.

[11] The objectives of the present paper are (i) to investigate the mechanisms responsible for the generation of the bed morphology in sharp river bends; (ii) to develop a nonlinear model for the bed morphology without curvature restrictions, encompassing existing linear models; (iii) to improve the model of the gravitational pull on the sediment particles and to analyze the sensitivity of results to this model; and (iv) to analyze the importance of nonlinear effects in the prediction of the bed morphology.