A particle model of rolling grain ripples under waves

Abstract

A simple model for the formation of rolling grain ripples on a flat sand bed by the oscillatory flow generated by a surface wave is presented. An equation of motion is derived for the individual ripples, seen as “particles,” on the otherwise flat bed. The model accounts for the initial appearance of the ripples, the subsequent coarsening of the ripples, and the final equilibrium state. The model is related to the physical parameters of the problem, and an analytical approximation for the equilibrium spacing of the ripples is developed. It is found that the spacing between the ripples scales with the square-root of the nondimensional shear stress (the Shields parameter) on a flat bed. The results of the model are compared with measurements, and reasonable agreement between the model and the measurements is demonstrated.