The shape of alluvial rivers is controlled by the feedback between the water flow and sediment transport. The flow, depends on the shape of the banks. It then entrains sediments and modifies the shape of the bed. Sediments are eroded when the flow-induced stress exceeds a threshold stress. In the first part of this manuscript, we discuss rivers without any sediment transport. In this case, the tangential stress exerted on the banks equals the threshold stress. This condition selects the river morphology and predicts the width and slope of the river as a function of the water discharge. These so-called regime equations correctly predict the shape of laminar laboratory rivers. The application to field data is less satisfactory: the exponent of the scaling laws are well predicted but the aspect ratio of natural rivers is wider than predicted from the regime equations. In the second part of this thesis, we show that the shape of a river cross-section also depends on the sediment discharge. In addition to their downstream motion, sediments entrained by the flow show significant fluctuations of their cross-stream velocity, likely due to the roughness of the underlying sediment bed. This causes particles to disperse laterally. We show that this lateral spreading is the manifestation of a random walk, which causes transverse Fickian diffusion. The shape of a river cross-section is selected by the balance between the transverse flux of sediments induced by gravity and this diffusive process. This balance leads to new regime equations relating the river width and slope to the sediment and water discharges. In particular, increasing the sediment discharge increases the aspect ratio and the slope of the river. Above a threshold aspect ratio, the river destabilizes and forms a braided river. This behavior is observed qualitatively in both laminar flow experiments and in Nature.
