An analytical non linear model for river meandering

We develop a non linear asymptotic theory of flow and bed topography in meandering channels able to describe finite amplitude perturbations of bottom topography and account for arbitrary, yet slow, variations of channel curvature. This approach then allows us to formulate a non linear bend instability theory, which predicts several characteristic features of the actual meandering process and extends results obtained by classical linear bend theories. In particular, in agreement with previous weakly non linear findings and consistently with field observations, the bend growth rate is found to peak at some value of the meander wavenumber, reminiscent of the resonant value of linear stability theory. Moreover, a feature typical of non linear waves arises: the selected wavenumber depends on the amplitude of the initial perturbation (for given values of the relevant dimensionless parameters) and, in particular, larger wavelengths are associated with larger amplitudes. Meanders are found to migrate preferentially downstream, though upstream migration is found possible for relatively large values of the aspect ratio of the channel, a finding in agreement with the picture provided by linear theory. Meanders are found to slow down as their amplitude increases, again a feature typical of non linear waves, driven in the present case by flow rather than geometric nonlinearities. The model is substantiated by comparing predictions with field observations obtained for a test case, a reach of a river recently subjected to detailed monitoring of its plan form evolution.