A note on hydrodynamic instabilities and horizontal circulation in the surf zone

Abstract

The setup/setdown of the mean water level induced by incident waves on a beach has been hypothesized to be unstable and, thereby, a source of horizontal circulation, rip currents, and rhythmic topography but a direct proof was lacking. This paper examines the dynamics of a small‐amplitude circulation coupled with the subsequent perturbation on the mean water level and the wave radiation stress for normal, regular wave incidence on a nonerodible and nonbarred beach. Two cases are considered. In the first one, the wave refraction by currents and depth variations is neglected, whereas in the second one, wave refraction is accounted for. In the first case, it is proved by analytical methods that there is no instability. It is shown that the coupling between flow, mean free surface, and wave radiation stress provides a restoring force that reinforces gravity in the surf zone, while in the shoaling zone, the coupling opposes gravity. This destabilizing effect in the shoaling zone is too small to overcome gravity force. Although this restoring/destabilizing force cannot produce instability, it causes a small correction on the classical long wave phase speed that becomes now anisotropic. In contrast, when wave refraction is included in the model, the motionless basic equilibrium becomes unstable and circulation cells tend to develop. It is argued that, even without wave refraction, if sediment transport and topographic evolution is accounted for, the situation can be reversed and the basic equilibrium can be unstable. Therefore rip currents in nonbarred beaches cannot be explained from an instability originated only by the setup/setdown unless wave refraction or topographic evolution is taken into account.