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 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 Briggs (1964) criterion to distinguish between 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 ω _i, involve spatial branches of the dispersion relation which, for large positive values of ω _i, lie in the same half λ -plane, λ denoting the complex bar wavenumber. Hence, the nature of instability is convective and keeps such 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 non linear 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. Briggs, R.J. 1964. Electron-Stream Interaction With Plasmas. Cambridge, Mass: MIT Press. Colombini, M., Seminara, G. and Tubino, M. 1987. Finite-amplitude alternate bars. J. Fluid Mech. 181, 213-232.