Analytical and numerical modeling of sandbanks dynamics
Résumé
Linear and nonlinear behavior of large-scale underwater bedform patterns like sandbanks are studied using linear stability analysis and numerical modeling. The model is based on depth-integrated hydrodynamics equations with a quadratic bottom friction law and a bed load sediment transport model including a bottom slope effect. First, the linear stability analysis of a flat erodible bottom subject to a steady current is performed. The direction of the most amplified bedform is controlled by the current, whereas its wavelength is selected by the gravity driven sediment flow. The instability proved to be due to the velocity component parallel to the mean flow, whereas the transverse velocity and the bottom slope effect are damping processes. The growth rate is then related to the spatial phase-lag between flow velocity and bathymetry. In order to validate the numerical model in this linear regime, the growth rate and the phase celerity of an infinitesimal bottom perturbation are computed as a function of its wavevector. A good agreement with linear stability results is found. Second, using the property that the growth rates for block-function and steady current are the same, the nonlinear behavior of the instability is investigated for a simple tidal current. The saturation height of the theoretically most amplified mode is estimated using the numerical model. The analysis of the sediment fluxes gradients shows that the saturation is mostly due to the hydrodynamic processes, although the saturation height slightly depends on the value of the bottom slope effect coefficient. Third, a Landau equation, whose coefficients are computed from the previous results, is used to predict the temporal evolution of the bedform amplitude from the initial infinitesimal perturbation to the saturation. A comparison with the characteristics of continental shelf sandbanks shows that the model gives a reasonable estimation of the temporal dynamics of these large-scale bedforms. However, the saturation height seems to be slightly overestimated. The limitations of the method are also discussed.