## 1 Introduction

[2] Alluvial rivers usually exhibit quite complex planforms characterized by a wide variety of alternating bends that have attracted the interest of a large number of researchers (see the review by *Seminara* [2006, and references therein]). Much less attention has been paid to another striking feature observed in alluvial rivers (see Figure 1), namely the relatively regular spatial variations attained by the channel width. Actively meandering channels (sinuous point bar rivers) generally undergo spatial oscillations systematically correlated with channel curvature, with cross sections wider at bend apex than at crossings [*Brice*, 1975; *Transportation Research Board of the National Academies*, 2004; *Hooke*, 2007]. Conversely, rivers flowing in highly vegetated flood plains, i.e., single-thread rivers, may exhibit an opposite behavior, owing to the combined effects of bank erodibility and floodplain depositional processes, which, in turn, are strictly linked to vegetation cover [*Allmendinger et al.*, 2005]. Some other streams (sinuous canaliform rivers) exhibit irregular width variations, without a clear correlation with channel curvature [*Brice*, 1975].

[3] Similarly to curvature forcing induced by bends, the presence of along channel width variations may have remarkable effects on the flow field and sediment dynamics and, thereby, on the equilibrium bed configuration [*Repetto et al.*, 2002; *Zolezzi et al.*, 2012]. In particular, the formation of a central bar at a channel widening tends to divert the flow toward the channel banks, thus favoring erosion and widening of the river. Such enlargement, in turn, promotes sedimentation producing a subsequent narrowing and, eventually, a cyclic narrowing/widening sequence [*Repetto et al.*, 2002; *Hooke*, 2007; *Luchi et al.*, 2010]. The formation of a central bar is also a crucial process for understanding the dynamics of both chute cutoffs in meandering rivers [*Seminara*, 2006] and bifurcations in braided rivers [*Federici and Paola*, 2003]. The analysis carried out by *Brice* [1983] on a number of meandering streams subject to engineering realignments and relocations suggests that bend cutoff usually determines a widening of the new channel and acceleration in the growth rate of adjacent bends.

[4] A reliable assessment of the flow field resulting from spatial distributions of channel axis curvature and cross section width, together with a physically based model simulating the outer bank erosion and the inner bank reconstruction [*Parker et al.*, 2011] are the key ingredients for developing robust mathematical models that describe the morphodynamic evolution of alluvial rivers. Various attempts have been put forward to this aim. Numerical [*Mosselman*, 1998; *Duan and Julien*, 2005; *Röuther and Olsen*, 2007; *Darby et al.*, 2002] and semianalytical models [*Crosato*, 2007; *Chen and Duan*, 2006; *Motta et al.*, 2012] have been tested versus laboratory experiments and field data. Some of these models [*Mosselman*, 1998; *Darby et al.*, 2002; *Chen and Duan*, 2006] include also mechanistic models of bank erosion that induce width adjustments when simulating the short-term (order of years/decades) migration of a channel. Theoretical three-dimensional nonlinear models have been applied to analyze the amplitude of width variations and their phase lag with respect to channel curvature fluctuations [*Solari and Seminara*, 2005]. More recently, *Luchi et al.* [2011] investigated the effects of periodic width oscillations on bend instability, accounting for the mutual nonlinear interactions between planimetric forcing induced by curvature and width variations. Spatial distribution of channel curvature typically determines the formation of a rhythmic bar-pool pattern strictly associated with the development of river meanders. Along channel width variations are characterized by a sequence of narrowing, yielding a central scour, alternated to the downstream development of a widening associated with the formation of a central bar. Finally, the 3-D fully nonlinear analytical model of flow and bed topography in meandering rivers developed by *Bolla Pittaluga et al.* [2009] to account for nonlinearity in sinuous mildly curved and long bends has been extended by *Luchi et al.* [2012] to treat spatial variations of channel width in a sequence of sine-generated meanders. The model suggests that, for a constant longitudinal free-surface slope, the equilibrium width oscillates with a frequency twice that of the channel curvature. These periodic oscillations are correlated with channel curvature, such that the maximum width is experienced close to inflection points while the minimum occurs close to bend apexes.

[5] In this contribution, we present a morphodynamic model that predicts, at a linear level, the spatial distribution of the flow field and the equilibrium bed configuration of an alluvial river characterized by arbitrary (in general not periodic) distributions of both the channel axis curvature and the channel width. Linear models of the steady flow in meandering channels have played a major role in disclosing a variety of features of the meandering phenomenon [*Seminara*, 2006], in exploring the long-term (order of centuries) evolution [*Howard*, 1992; *Sun et al.*, 1996; *Frascati and Lanzoni*, 2009], and in evaluating the possible existence of a statistically universal behavior of meandering rivers [*Frascati and Lanzoni*, 2010]. A detailed comparison of the performance of different linear models can be found in *Camporeale et al.* [2007] and *Frascati and Lanzoni* [2009]. In the following, we extend the model used by *Frascati and Lanzoni* [2009, 2010] to study the morphodynamic regime and the long-term evolution of alluvial rivers, by relaxing the assumption of constant channel width. The model, owing to its analytical character, provides a computationally sustainable and robust tool that can be easily incorporated in long-term models of river planform evolution. We also show, through a comparison with field data, that it can be used to rapidly assess the morphological tendencies of an alluvial river in response to variations in planform geometry or hydrodynamic forcing.

[6] The rest of the paper is organized as follows. In section 2, we derive a two-dimensional, depth-averaged model for flow and bed topography in alluvial meandering channels with both arbitrarily varying curvature and width. Section 3 is devoted to the linearized solution of the morphodynamic problem, to summarize the input data and to discuss the applicability conditions of the model. Some results, along with a direct application of the model to a test case (a reach of the Po River, Italy) are presented in section 4. Finally, section 5 concludes the paper drawing some conclusions.