Dynamical models for sand ripples beneath surface waves.

We introduce order parameter models for describing the dynamics of sand ripple patterns under oscillatory flow. A crucial ingredient of these models is the mass transport between adjacent ripples, which we obtain from detailed numerical simulations for a range of ripple sizes. Using this mass transport function, our models predict the existence of a stable band of wave numbers limited by secondary instabilities. Small ripples coarsen in our models and this process leads to a sharply selected final wave number, in agreement with experimental observations.