## 1. Introduction

[2] In the past, numerous river canalization works have been undertaken that confined the river to a prescribed planform, in order to improve navigation or to use the alluvial plane. Recently, there is a tendency to renaturalize rivers, e.g., by giving them more freedom to shape their course in the alluvial plane. Such renaturalized rivers are recognized as a potentially rich biotope and as an important buffer in the flood defense system. Moreover, natural rivers are preferred from a landscape point of view. This new way of dealing with rivers requires more subtle measures than before, aiming at influencing the river in its natural behavior, rather than shaping it according to man's needs. This requires a more thorough understanding of flow, sediment transport, morphology, and mixing processes than before, as well as better predictive capabilities.

[3] Mathematical modeling is one way of predicting a river's behavior. The basis of such models is often a flow model, coupled to a sediment transport model and a sediment balance equation, and/or a dispersion model for dissolved or suspended matter. Although fully three-dimensional flow models are available now, depth-integrated flow models will remain an important complementary tool in research and engineering practice for long to go, especially in morphological studies. They yield a less detailed image of the flow pattern but are computationally at least an order of magnitude less expensive, and thereby allow for covering more complex situations, larger river domains (river reach or basin scale), and longer time periods (historical and geological timescales), or for performing stochastic or probabilistic simulations.

[4] The information related to the vertical structure of the flow field is to a large extent lost by depth-integrating the flow equations. The remaining information has to be introduced via a closure submodel. *Johannesson and Parker* [1989b], *Finnie et al.* [1999], *Blanckaert* [2001], and *Blanckaert and Graf* [2003] have shown the importance of this closure, which mainly has to account for the secondary circulation. This is a characteristic and dominant feature of curved flow: it redistributes velocity, boundary shear stress, and sediment transport, shapes the bed topography, and mixes dissolved and suspended matter.

[5] It has been a known fact for quite some time now that so-called linear closure submodels that neglect feedback effects between the secondary circulation and the main flow over-predict the effect of the secondary circulation, and have validity limited to weak curvatures [*de Vriend*, 1981a; *Yeh and Kennedy*, 1993; *Blanckaert*, 2001; *Blanckaert*, 2002, chapter II.3] (see also section 4). To solve the problem, a number of nonlinear models have been proposed in the past [*Jin and Steffler*, 1993; *Yeh and Kennedy*, 1993]. However, those models are difficult and computationally time consuming to be implemented. The present paper proposes a nonlinear closure submodel that takes due account of feedback effects. It can be reduced to a semiheuristic form, which requires hardly more computational effort than the linear models. Therefore, from a practice point of view, the current model will be more likely to be adapted by the hydraulic engineering and geologic modeling communities and have wider impact than those previous nonlinear models. Moreover, the nonlinear model provides new insight into the mechanisms and effects of the secondary circulation.