A mathematical model for meandering rivers with varying width

The mathematical modeling of the long-term evolution of meandering rivers needs an efficient computation of the flow field. Indeed, the estimate of the near bank velocity, needed to determine the rate at which the outer bank migrates, cannot rely on the full numerical solution of the governing equations when considering the river evolution on geological time scales. The aim of the present contribution is twofold: determining the complete linear response of a meandering river to spatially varying channel axis curvature and width, exploiting the ability of the model to describe the morphological tendencies of alluvial rivers; and developing a computationally efficient tool that can be easily incorporated in long-term planform evolution models. The centrifugally induced secondary flow associated with channel axis curvature and longitudinal convection is accounted for by a suitable parametrization based on the structure of the three-dimensional flow field. Cross section width variations are accounted for through a suitable stretching of the transverse coordinate. The relevant momentum and mass conservation equations are then linearized by taking advantage of the fact that alluvial rivers often exhibit mild and long meander bends, as well as evident but relatively small width variations. The input data needed by the analytical solution are the spatial distribution of channel axis curvature and width variations, the mean slope of the investigated river reach, the characteristic grain size of the sediment bed and the flow discharge. The performances of the model, as well as its intrinsic limitations are discussed with reference to the comparison with the bed topography surveyed in a 21 km long reach of the Po River. The results indicate that, in the presence of wide, mildly curved and long bend and weak width variations, the river topography is described with a good accuracy, thus supporting the use of the model to investigate how a river could react to changes in planform geometry or external forcing. Moreover, the analytical character of the model implies a limited computational effort, facilitating a straightforward integration within the models used to simulate the planimetric evolution of alluvial rivers on geological time scales.