ON THE NONLINEAR DYNAMICS OF FREE BARS IN STRAIGHT CHANNELS

A simple morphological model is considered which describes the interac tion between a unidirectional flow and an erodible bed in a straight c hannel. For sufficiently large values of the width-depth ratio of the channel the basic state, i.e. a uniform current over a flat bottom, is unstable. At near-critical conditions growing perturbations are confi ned to a narrow spectrum and the bed profile has an alternate bar stru cture propagating in the downstream direction. The timescale associate d with the amplitude growth is large compared to the characteristic pe riod of the bars. Based on these observations a weakly nonlinear analy sis is presented which results in a Ginzburg-Landau equation. It descr ibes the nonlinear evolution of the envelope amplitude of the group of marginally unstable alternate bars. Asymptotic results of its coeffic ients are presented as perturbation series in the small drag coefficie nt of the channel. In contrast to the Landau equation, described by Co lombini et al. (1987), this amplitude equation also allows for spatial modulations due to the dispersive properties of the wave packet. It i s demonstrated rigorously that the periodic bar pattern can become uns table through this effect, provided the bed is dune covered, and for r ealistic values of the other physical parameters. Otherwise, it is fou nd that the periodic bar pattern found by Colombini et al. (1987) is s table. Assuming periodic behaviour of the envelope wave in a frame mov ing with the group velocity, simulations of the dynamics of the Ginzbu rg-Landau equation using spectral models are carried out, and it is sh own that quasi-periodic behaviour of the bar pattern appears.