On the convective nature of bar instability

Bar instability is recognized as the fundamental mechanism underlying the formation of large-scale forms of rivers. We show that the nature of such instability is convective rather than absolute. Such a result is obtained by revisiting the linear stability analysis of open-channel uniform flow over a cohesionless channel of Colombini et al. (1987) and using the Briggs (1964) criterion to distinguish between the convectively and absolutely unstable temporally asymptotic response to an initial boundary-value perturbation of bed topography. Examining the branch-point singularities of the dispersion relation, which can be determined in closed form, we show that all the existing branch-point singularities characterized by positive bar growth rate omega(i), involve spatial branches of the dispersion relation which, for large positive values of omega(i), lie in the same half lambda-plane, lambda denoting the complex bar wavenumber. Hence, the nature of instability is convective and remains so for any value of the aspect ratio, the controlling parameter of the basic instability, as well as for any lateral mode investigated. The latter analytical findings are confirmed by numerical solutions of the fully nonlinear problem. In fact, starting from either a randomly distributed or a localized spatial perturbation of bed topography, groups of bars are found to grow and migrate downstream leaving the source area undisturbed. The actual bars observed in laboratory experiments arise from the spatial-temporal growth of some persistent initial perturbation. The nonlinear development of such perturbations is shown to lead to a periodic pattern with amplitude independent of the amplitude of the initial perturbation. Bars are also found to lengthen and slow down as they grow from the linear into the nonlinear regime, in agreement with experimental observations. The distance from the initial cross-section at which equilibrium is achieved depends on the initial amplitude of the perturbation, a finding which calls for a revisitation of classical laboratory observations reported in the literature.