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On the nonlinear dynamics of free bars in straight channels. (English) Zbl 0789.76040

The authors studied the nonlinear behavior of free bars generated by a unidirectional flow in an infinitely long straight channel with an erodible bottom and non-erodible banks. Using a simple model, which only contains the basic mechanisms responsible for bedform instabilities, the main aim of the paper was to derive the Ginzburg-Landau equation governing the evolution of weakly nonlinear bar amplitudes and to demonstrate its potential importance for understanding the dynamics of free bars in rivers and laboratory tanks. Linear theory establishes an alternate bar structure propagating downstream for near critical conditions. While the solutions of the Landau equation characterize the nonlinear behavior of a simple unstable wave, the evolution of the envelope amplitude of a packet of marginally unstable waves is described by the Ginzburg-Landau equation. The authors demonstrate that, contrary to earlier Landau equation results, the periodic bar pattern can become unstable through this effect, as long as the bed is dune covered. The subsequent dynamical behavior was investigated by using a spectral model of the Ginzburg-Landau equation leading to the emergence of quasi- periodic bar patterns. However, no chaotic solutions were observed.

MSC:

76E30 Nonlinear effects in hydrodynamic stability
76E20 Stability and instability of geophysical and astrophysical flows
86A05 Hydrology, hydrography, oceanography
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