## 1. INTRODUCTION

[2] River meanders are one of the most ubiquitous patterns in fluvial morphology [e.g., *Chitale*, 1970; *Allen*, 1984; *Howard*, 1992]. For many years the beauty and applicative importance of these nearly regular loops in river *planimetry* have attracted the interest of several researchers in fluid mechanics and morphodynamics [*Ikeda and Parker*, 1989; *Seminara*, 1998, 2006], geomorphology [*Allen*, 1984], river engineering [*Jansen et al.*, 1979; *Elliott*, 1984], riparian ecology [*Salo et al.*, 1986], and petroleum engineering [*Swanson*, 1993]. (Italicized terms are defined in the glossary, after the main text.)

[3] From a physical point of view, meandering rivers form a dynamical system far from equilibrium, which, in its continuous evolution, exhibits some kind of statistical stationarity [*Cross and Hohenberg*, 1993; *Liverpool and Edwards*, 1995; *Stølum*, 1996; *Camporeale et al.*, 2005]. The river evolution is driven by fluid dynamic and morphodynamic processes, which cause lateral bank erosion and the continuous migration of meanders, as well as by sporadic *cutoffs* that prevent self-intersections of the river and produce sudden reductions in river length and sinuosity (see Figure 1). These internal dynamics are usually forced by external deterministic or stochastic factors, with different temporal and spatial scales, due to hydrological and riparian processes as well as to pedological, geological, and anthropic constraints. In the present review, attention is focused on the mathematical modeling of the fluid dynamic and morphodynamic processes that are responsible for the short-term evolution of rivers. We will also show how such mathematical models can be coupled with the cutoff dynamics and different types of external forcing to investigate the long-term evolution of meandering rivers.

[4] Historically speaking, the study of meandering rivers has followed two interrelated paths: a geomorphologic approach and a fluid dynamic approach. The geomorphologic approach, through fundamental field studies [e.g., *Leopold and Wolman*, 1960; *Kinoshita*, 1961; *Allen*, 1965; *Chitale*, 1970; *Nanson and Hickin*, 1983; *Carson and Lapointe*, 1983; *Thorne and Furbish*, 1995] and laboratory experiments [e.g., *Friedkin*, 1945; *Rozowskij*, 1957; *Zimmerman and Kennedy*, 1966; *Kinoshita and Miwa*, 1974; *Whiting and Dietrich*, 1993a, 1993b], has described the main characteristics of meanders and offered valuable empirical relationships on the planimetric features of meanders and river bed forms. The fluid mechanic approach has focused on the mathematical modeling of the physical mechanisms governing the meandering dynamics. The pioneering works of *Van Bendegom* [1947] and *Engelund* [1974] on the flow field and bed topography in a bend were followed by several important contributions that elucidated some key fluid dynamic aspects of river meandering. In particular, *Ikeda et al.* [1981] proposed the first model of the evolution of single reach of river bends by linking the flow field and the erosion rate; *Parker et al.* [1982, 1983] described the downstream migration of meanders and the occurrence of third-order harmonics in Kinoshita's curve; *Blondeaux and Seminara* [1985] clarified the link between the bend and alternate *bar* dynamics and pointed out their possible resonance; *Kalkwijk and De Vriend* [1980], *Kitanidis and Kennedy* [1984], and *Johannesson and Parker* [1989b] investigated the role of secondary currents; *Struiksma et al.* [1985] observed and modeled the *overdeepening phenomena*; *Tubino and Seminara* [1990] investigated the nonlinear interaction between bars and bends; and *Zolezzi and Seminara* [2001] pointed out the upstream propagating influence.

[5] After 3 decades of conspicuous efforts the scientific community has produced a number of different models of increasing detail and complexity. On one hand, the linear [e.g., *Ikeda et al.*, 1981; *Blondeaux and Seminara*, 1985; *Struiksma et al.*, 1985; *Odgaard*, 1986; *Crosato*, 1987; *Johannesson and Parker*, 1989a; *Zolezzi and Seminara*, 2001] and weakly nonlinear [*Seminara and Tubino*, 1992] models are strictly valid only for low curvatures of the river axis and slowly varying bed topography [*Seminara and Solari*, 1998] far from resonant conditions. However, because of their analytical solutions and their good agreement with observed river evolution [*Imran et al.*, 1999], they have been extensively used for both theoretical and numerical investigations of river morphodynamics [e.g., *Howard*, 1984; *Stølum*, 1996; *Sun et al.*, 1996, 2001a; *Seminara et al.*, 2001; *Edwards and Smith*, 2002; *Camporeale et al.*, 2005; *Camporeale and Ridolfi*, 2006; *Lanzoni et al.*, 2006]. On the other hand, fully nonlinear models [*Smith and McLean*, 1984; *Olsen*, 1987; *Nelson and Smith*, 1989b; *Shimizu et al.*, 1992; *Mosselman*, 1991, 1998; *Imran et al.*, 1999; *Duan et al.*, 2001; *Darby et al.*, 2002; *Blanckaert and De Vriend*, 2003] have less geometric restrictions and provide a better quantitative resolution of the flow, but they require a more demanding computational effort.

[6] Despite the advances produced by these various models, their formulations are difficult to compare (the formalisms are often different), and it is hard to evaluate what the effective role played by different modeling approximations is or whether the increasing modeling complexity is justified by the results. Apart from the work by *Parker and Johannesson* [1989], who performed a partial comparison between some models but focusing essentially on the resonance, overdeepening, and the dynamics of the secondary currents, no other comparative assessment has been published so far. For these reasons the main objective of the present work is to review the fundamental morphodynamic mechanisms that govern the meandering dynamics and to formulate a general framework from which the previously proposed linear models can be hierarchically derived and then critically compared according to their hypotheses and their level of detail in the description of the various physical processes. In this manner the main models are obtained in cascade through a series of subsequent simplifications. We also derive some extensions of the existing theories that are useful for model intercomparison and for understanding the role of some physical processes involved in meandering dynamics. Finally, a comparison with a real case of meander evolution allows the role of the different model hypotheses to be highlighted.

[7] We have focused on linear theories as they allow one to take into account all the key processes that govern meandering dynamics but, at the same time, to maintain analytical tractability. Many fundamental conceptual results are thus obtained without the need of numerical simulations. However, in order to verify the reliability of the linear models we also derive a nonlinear version for each level of morphodynamic simplification and compare them with the correspondent linear models. To this aim we have extended the nonlinear iterative procedure by *Imran et al.* [1999] to the equation of sediment mass continuity.

[8] Particular attention is devoted to the models of *Ikeda et al.* [1981], *Johannesson and Parker* [1989a], and *Zolezzi and Seminara* [2001], hereinafter referred to as IPS, JP, and ZS, respectively, as they represent key steps in the comprehension and modeling of meandering dynamics. These models form the skeleton of our work, and we will refer to them when discussing other existing linear models [e.g., *Howard*, 1984; *Struiksma et al.*, 1985; *Odgaard*, 1986; *Crosato*, 1987; *Bridge*, 1992]. The IPS and JP models have been widely used in numerical simulations of river evolution [e.g., *Howard*, 1992; *Stølum*, 1996; *Sun et al.*, 1996, 2001a], while the ZS model is more detailed and encompasses all the principal morphodynamic mechanisms (for this reason, ZS is used here as the reference model, and its notation is extended to other models). Other linear models that focus on bank erosion and on how soil properties and riparian vegetation influence bank geotechnical characteristics are not discussed here [e.g., *Lancaster and Bras*, 2002; *Richardson*, 2002]. Although these models are interesting, they are not completely physically based, and as such they are not capable of describing the complex interactions that exist among bed topography, flow field, and sediment transport. As far as the role of bank erosion is concerned, mention is made here only of the refined two-dimensional (2-D) nonlinear models by *Darby et al.* [2002] and *Duan and Julien* [2005], where both fluid dynamic and geotechnical aspects are modeled in detail.

[9] It should be noted that all the previously mentioned models share two important basic assumptions: (1) The river discharge is always assumed to be constant and usually equal to the mean annual or bankfull value, and (2) the shallow water approximation allows the flow field to be solved using a 2-D (or quasi-three-dimensional) depth-averaged scheme. Although the former assumptions can result in rather crude approximations, only in very few studies have they been at least in part relaxed. In particular, in the work of *Howard and Hemberger* [1991] the IPS model was forced with temporally varying discharges extracted from a lognormal distribution. The authors did not observe a relevant change in the statistical behavior of the river planimetry with respect to the case with a constant discharge equal to the mean value. However, the determination of the formative discharge for the meandering patterns still remains an open question, since it does not necessarily coincide with the dominant geomorphic discharge for the hydraulic geometry proposed by *Wolman and Miller* [1960].

[10] The problem of resolving the flow field using a three-dimensional (3-D) rather than a 2-D approach has received much more attention, especially from a numerical point of view [*Shimizu et al.*, 1990; *Ye and McCorquodale*, 1998; *Ferguson et al.*, 2003; *Olsen*, 2003; *Wilson et al.*, 2003; *Blanckaert and De Vriend*, 2004; *Rüther and Olsen*, 2005], although the linear analytical treatment by *Seminara and Tubino* [1989] should also be mentioned. The main result is that the 2-D scheme cannot give a correct description of the flow field when either the bend curvature is high or the aspect ratio is too low, which corresponds to the breakdown of the shallow water approximation. In these circumstances the numerical solution of the 3-D helicoidal motion becomes essential for modeling meandering dynamics.

[11] To date, the necessary high computational efforts limit the use of full 3-D models to simple geometries with sharp bends and hinder the simulation of the planimetric evolution of rivers, as testified by the comparisons with experimental data that are restricted to channels with fixed banks. However, the increasing advances in computer science suggest that the adoption of powerful computational fluid dynamics tools, such as direct numeric simulation (DNS), large eddy simulation (LES), or *k*-ɛ models, will produce important contributions in the context of the emerging discipline of numerical morphodynamics [e.g., *Keylock et al.*, 2005].

[12] The paper is organized as follows. Section 2 gives a detailed qualitative description of the morphodynamic processes involved in meandering dynamics, while section 3 is devoted to the general mathematical modeling of the meandering dynamics and to presenting a method for the nonlinear solution of the morphodynamic problem. In section 4, the different linear models are derived from the same general formalism and discussed according to their level of approximation. When deducing the linear models, some extensions of the existing theories are also derived that are useful for the comparison of the models and for understanding the role of some physical processes involved in meandering dynamics. The various linear models are compared in section 5 by analyzing their free and forced response separately, being the curvature the forcing of the system. The experimental verification of the longitudinal flow field is presented in section 6, while the behavior of each model is discussed in section 7 for some typical meander configurations and a field case in order to evaluate the quantitative importance of the degree of refinement of each model. The role of some external forcings in long-term dynamics is reviewed in section 8, and the conclusions follow in section 9. A brief glossary of the typical expressions herein adopted closes this review.