Weakly nonlinear theory of regular meanders

Abstract

Flow and bed topography in a regular sequence of meanders is shown to be strongly influenced by nonlinear effects within a fairly wide range of aspect ratios of the channel and meander wavenumbers. This finding is associated with the behaviour of meanders as nonlinear resonators in a neighbourhood of the resonance conditions discovered by Blondeaux & Seminara (1985). A weakly nonlinear approach valid for relatively small measures of channel curvature and within a neighbourhood of the resonant conditions displays all the typical features of nonlinear resonators, including non-uniqueness of the channel response. The nonlinear structure of forced bars close to resonance is also shown to be related to that of nonlinear free steady bars spatially developing in a straight channel from a non-uniform initial condition. Finally we show how to reconcile the intrinsic nonlinearity of the near-resonant channel response with traditional bend stability theories. Some comparison with a systematic set of experimental observations of Colombini, Tubino & Whiting (1990) provides qualitative support for the present theory but also suggests that strongly nonlinear effects may play a non-negligible role for fairly small values of channel curvature. The main implication of this work is the clear need to revisit the literature on the modelling of flow and bed topography in river meanders, which is mostly based on linear theories.