Nonlinearity and unsteadiness in river meandering: a review of progress in theory and modelling

River meandering has been extensively investigated. Two fundamental features to be explored in order to make further progress are nonlinearity and unsteadiness. Linear steady models have played an important role in the development of the subject but suffer from a number of limits. Moreover, rivers are not steady systems; rather their states respond to hydrologic forcing subject to seasonal oscillations, punctuated by the occurrence of flood events. We first derive a classification of river bends based on a systematic assessment of the various physical mechanisms affecting their morphodynamic equilibrium and their evolution in response to variations of hydrodynamic forcing. Using the database by Lagasse et al. (2004) we also show that natural meanders are typically mildly curved and long, i.e. such that both the centrifugal and the topographic secondary flows are weak, but they are almost invariably nonlinear. We then review some recent developments which allow us to treat analytically the flow and bed topography of mildly curved and long nonlinear bends subject to steady forcing, taking advantage of the fact that flow and bed topography in mildly curved long bends are slowly varying. Results show that nonlinearity has a number of consequences: most notably damping of the morphodynamic response and upstream shifting of the location of the nonlinear peak of the flow speed. Next we extend the latter model to the case of unsteady forcing. Results are found to depend crucially on the ratio between the flood duration and a morphodynamic timescale. It turns out that, in a channel subject to a repeated sequence of floods, the system reaches a dynamic equilibrium. We conclude the paper discussing how the present assessment relates to the debate on meander modelling of the late 1980s and suggesting what we see as promising lines of future developments