## 1. Introduction

[2] River meandering is a major topic in the field of morphodynamics. It has been the subject of extensive investigations in the recent past. The review paper of *Seminara* [2006], to which the reader is referred for a broad overview of the subject, has outlined the main steps whereby progress has been made in this field.

[3] Let us briefly recall them. The attention was initially focused on understanding the mechanism of meander formation starting from a straight channel configuration. Linear stability analyses (so-called “bend” theories) were employed, hence linear models were developed to determine flow and bed topography in weakly curved channels. The physical implications of the linearity assumption can be appreciated recalling the main ingredients of the process whereby the pattern of bed topography develops in a sinuous channel. The first feature is the establishment of a centrifugal secondary flow directed outwards close to the free surface and inward close to the bed: it arises because the lateral pressure gradient driven by the lateral slope of the free surface established in a bend is unable to provide the centripetal force required for fluid particles to move along purely longitudinal trajectories. If the bed is nonerodible, a “free vortex” effect prevails initially, longitudinal trajectories in the inner part of the bend being shorter than in the outer part. As a result, flow at the inner bend accelerates relative to the outer bend. This is a purely metric effect which is accounted for also in linear models. Proceeding downstream, a net transfer of momentum toward the outer bend is driven by the secondary flow (the outward transfer occurring in the upper layer prevailing on the inward transfer occurring in the lower layer), hence the thread of high velocity progressively moves from the inner to the outer bend. In the context of perturbation approaches where the basic state is laterally uniform, this is a second-order effect hence it is not accounted for in the context of linear models. The picture changes considerably when the bed is erodible. Under the latter conditions, secondary flow also affects the motion of grain particles: they deviate from the longitudinal direction, hence sediments are transported toward the inner bends where a sequence of so-called point bars are built up while pools develop at the outer bends. The bar-pool pattern then drives a topographical component of the secondary flow and an additional contribution to sediment transport which further modifies the bed topography. Linear theories are indeed able to generate this topographical effect, though the important role of the lateral transfer of momentum driven by the topographical secondary flow is again neglected, formally appearing still at second order. The need to relax the linear constraint was recognized in the engineering literature, where a large effort was made to construct a rational framework, amenable to numerical treatment, in order to predict flow and bed topography in meandering channels with finite curvature and arbitrary width variations. These models serve the interests of river engineering, being fairly successful when applied to relatively short reaches of alluvial rivers and fairly short events.

[4] However, a more general interest toward the construction of sound analytical nonlinear models arises in the context of the fundamental research on the subject. In fact, the availability of such a model is potentially suitable to investigate a number of important processes observed in meander evolution, which still await to be understood. Let us list some of these processes, which did motivate our present work.

[5] An inspection of the patterns of meandering rivers (e.g., Figure 1) reveals that the river width, defined as the width of the stream free surface, undergoes typically spatial oscillations which display a distinct correlation with channel curvature. The river width may peak at bend apexes, reaching a minimum at inflection points (Figure 1a) or viceversa (Figure 1b). Note that this issue bears both a conceptual and a practical relevance. In fact, we know from the seminal contributions of *Parker* [1978a, 1978b] that the average width of straight channels in equilibrium is ultimately controlled by requirements of bank stability. We also know that meandering does not alter such equilibrium in the mean. Why and how? An attempt at clarifying the latter points has been recently proposed by *Parker and Shimizu* [2008]. However, provided the river is free to erode and deposit, i.e., it is able to choose its own width, then curvature makes the stream unable to keep a constant width. Why? And to what extent channel widening at the bend apexes modifies the scour pattern typically observed at the outer banks, thus affecting the lateral migration of meanders?

[6] A further reason of interest is related to a second observation: it is not uncommon to detect the formation of an island at bends of meandering channels (Figure 2). The presence of the island then forces the stream to bifurcate into an outer and an inner branch. In natural settings this is not a stable configuration: sooner or later, the stream will cut through and abandon the outer branch. The latter well known process is described as chute cutoff and occurs typically in wide bends with fairly large curvatures, high discharges, poorly cohesive unvegetated banks and high slope [*Howard and Knutson*, 1984]. Though some recent numerical investigations [*Jager*, 2003, and references therein] have attempted to model the latter process, it is not unfair to state that the occurrence of chute cutoff is a problem yet awaiting to be understood. The availability of an analytical nonlinear model of river meanders would allow to approach the latter problem. In fact, the process of widening is known [*Repetto et al.*, 2002] to promote the formation of steady central bars in straight channels: it is then natural to wonder whether the formation of bend islands is similarly related to a bottom instability driven by widening of a curved channel. The next step would then consist of modeling the tendency of the central bar to force the stream to bifurcate into an outer and an inner branch leading to the occurrence of chute cutoff.

[7] A third motivation to develop an analytical approach to nonlinear meanders is related to the fundamental interest of a nonlinear theory of bend instability. In fact, linear theories display the occurrence of a resonance mechanism which controls the selection of the preferred wavelength for bend instability [*Blondeaux and Seminara*, 1985]. Resonance is obviously damped by nonlinear effects, as shown by the weakly nonlinear theory of *Seminara and Tubino* [1992]. However, no fully nonlinear theory has been proposed so far, though the role of nonlinearity is known to affect the flow field, hence the selection mechanism, considerably. In the present work, we do develop a nonlinear theory of bend instability, based on the nonlinear asymptotic solution of flow and bed topography in meandering rivers developed herein.

[8] Finally an analytical approach to nonlinear meanders involving a sufficiently modest computational effort, will hopefully allow us to investigate the long-term morphodynamic evolution of meandering rivers, a topic which has attracted the attention of both geomorphologists [*Sun et al.*, 1996] and engineers [*Camporeale et al.*, 2007]. For such applications, numerical solutions of the full 3-D governing equations or their shallow water version [e.g., *Mosselman*, 1991; *Shimizu*, 2002] are not appropriate tools as the computational effort they require would be prohibitive. Researchers have then been forced to employ analytical linear models for flow and bed topography, allowing only for geometric nonlinearities arising from planform evolution. The present model removes the latter restriction.

[9] This idea is pursued by resorting to the use of perturbation methods. We set up an appropriate perturbation expansion for the solution of the problem of morphodynamics, valid in the general case of rivers with arbitrary distributions of channel curvature, the only constraint being that flow and bottom topography must be “slowly varying” in both longitudinal and lateral directions and channel curvature must be “sufficiently small”. The former assumption requires the channel to be “wide” with channel alignment varying on a longitudinal scale much larger than channel width, while the latter assumption is satisfied provided the radius of curvature of channel axis is large compared with channel width. Both conditions are typically met in actual rivers but, in spite of the popularity enjoyed by linear models, neither of them implies that perturbations of bottom topography are necessarily “small”. It may be useful at this stage to clarify the latter statement by pointing out the fundamental distinction between the notion of linearity and the notion of slow variation. The former is based on the assumption that the amplitude of perturbations of bed topography driven by channel curvature must be small. The latter is based on the assumption that perturbations of bed topography must vary slowly in space. In order to further clarify this concept, let us consider the simplest possible case: a meander with channel axis defined in Cartesian coordinates as follows:

with ε meander amplitude, *λ*(= 2*π*/*L*) meander wave number, *L* meander intrinsic length, *ω* angular frequency and *c* meander curvature. The above relationships show various obvious facts: (1) channel curvature *c* can be small with meander amplitude ε “large” as long as the meander is “long” enough (*λ* ≪ 1) (i.e., slowly varying), and (2) conversely, a meander bend can be “sharp” (large curvature) even if its amplitude ε is small, provided the meander is “short” enough (*λ* ≫ 1). In this paper we are only interested in the former case.

[10] Taking advantage of the slowly varying assumption, a suitable extension of the approach developed by *Seminara and Solari* [1998] to investigate bed deformations in constant curvature channels with constant width can be developed. The latter approach allows for slow, yet finite, perturbations of flow and bed topography relative to a basic state consisting of a gradually varying sequence of locally uniform flow, slowly varying in both the lateral and longitudinal directions. The only unknowns left for numerical computation are then flow depth, a slow function of longitudinal and lateral coordinates, and variation of the longitudinal free surface slope satisfying a strongly nonlinear differential equation subject to continuity constraints.

[11] The content of the paper is then organized as follows. In chapter 2 we formulate the 3D problem of flow in sinuous alluvial channels with a noncohesive bed. In the analysis, the direct effect of secondary flow on the transverse distribution of the main flow, leading to lateral transfer of longitudinal momentum is accounted for. This effect, which has been argued to be important by many authors [e.g., *Nelson and Smith*, 1989; *Imran and Parker*, 1999], appears at the first order of approximation in the present scheme. Section 3 is then devoted to ascertaining the role of flow nonlinearity on “bend instability theory” by coupling the morphodynamic model with a bank erosion law, expressing the dependence of erosion intensity on the flow field. We are then able to predict the wavelength selected by bend instability as well as the meander wave speed in a nonlinear context. The model predictions are then tested by a direct application to a test case (a reach of the Cecina River, Italy) for which accurate data obtained by recent detailed monitoring are available (section 4). Predictions of both the equilibrium configuration and of the wave number selected in the meandering process do support the soundness of the present nonlinear approach. Finally, some discussion of future developments concludes the paper.