Sand banks of finite amplitude

The process which leads to the appearance of sand banks in shallow seas is investigated by studying the growth of small amplitude perturbations of the sea bottom, forced by oscillatory tidal currents. Since the analysis of field data carried out by Dyer & Huntley (1999) shows that sand banks are likely to occur where the tidal ellipse is circular or characterized by a low ellipticity, attention is focused on small values of sqrt{1-e^2}, where e is the ratio between minor and major axes of the tidal ellipse. The linear analysis, which considers perturbations of small (strictly infinitesimal) amplitude, shows the existence of a critical value r_C of the Keulegan-Carpenter number r of the tide (r=U*/(h* omega*), U* and omega* being the amplitude and angular frequency of the velocity oscillations induced by the tide propagation and h* the averaged water depth) such that for r smaller than r_C the flat bottom configuration is stable while for r larger than r_C sand banks start to appear. Close to the critical condition, the wavelength of the most unstable mode turns out to be finite. Then, a weakly nonlinear analysis is developed which allows to evaluate the equilibrium amplitude of the bottom forms when the parameter r is close to its critical value. The configuration of the sea bottom, when the bottom forms attain their equilibrium, is characterized by the presence of long ridges, almost parallel to the main axis of the tidal ellipse, with crest-to-crests distances similar to those observed during field surveys. The crests of the bottom forms turn out to be flat and the extensive shallow waters at the crests are compensated by deep troughs between the ridges.