Crescentic bedforms in the nearshore region

A wave of small amplitude is considered which approaches a straight beach normally and which is partially reflected at the coastline. By assuming that the local depth is much smaller than the length of the incoming wave, the shallow water equations are used to determine the water motion. The surf zone width is assumed to be small compared to the length of the incoming wave and hence the effect of wave breaking is included only parametrically. The time development of the cohesionless bottom is described by the Exner continuity equation and by an empirical sediment transport rate formula which relates the sediment flux to the steady currents and wave stirring. It is shown that the basic-state solution, which does not depend on the longshore coordinate, may be unstable with respect to longshore bedform perturbations, so that rhythmic topographies form. The instability process is due to a positive feedback mechanism involving the incoming wave, synchronous edge waves and the bedforms. The growth of the bottom perturbations is related to the presence of steady currents caused by the interaction of the incoming wave with synchronous edge waves which in turn are excited by the incoming wave moving over the wavy bed. For natural beaches the model predicts two maxima in the amplification rate: one is related to incoming waves of low frequency, the other to wind waves. Thus two bedforms of different wavelengths can coexist in the nearshore region with longshore spacings of a few hundred and a few tens of metres, respectively. To illustrate the potential validity of the model, its results are compared with field data. The overall agreement is fairly satisfactory.