Sand ripples under sea waves. Part 4. Tile ripple formation

We investigate the formation of small-scale three-dimensional bedforms due to interactions of an erodible bed with a sea wave that obliquely approaches the coast, being partially reflected at the beach. In this case the trajectories of fluid particles at the top of the bottom boundary layer are ellipses in the horizontal plane, the axes of which depend on the angle of wave incidence and the distance from the shore. A weakly nonlinear stability analysis of an initially flat, cohesionless, sandy bottom is performed. We focus on the resonant interaction of three perturbation components. The results show that these elliptical forcing conditions are responsible for the formation of both brick-pattern ripples and tile ripples. In particular tile ripples are associated with a flow at the top of the bottom boundary layer which is near-circular (ellipticity close to one), whereas brick-pattern ripples are related to a unidirectional oscillatory flow (zero ellipticity).