[1] The time evolution of rip currents in the nearshore is studied by numerical experiments. The generation of rip currents is due to waves propagating and breaking over alongshore variable topography. Our main focus is to examine the significance of wave-current interaction as it affects the subsequent development of the currents, in particular when the currents are weak compared to the wave speed. We describe the dynamics of currents using the shallow water equations with linear bottom friction and wave forcing parameterized utilizing the radiation stress concept. The slow variations of the wave field, in terms of local wave number, frequency, and energy (wave amplitude), are described using the ray theory with the inclusion of energy dissipation due to breaking. The results show that the offshore directed rip currents interact with the incident waves to produce a negative feedback on the wave forcing, hence to reduce the strength and offshore extent of the currents. In particular, this feedback effect supersedes the bottom friction such that the circulation patterns become less sensitive to a change of the bottom friction parameterization. The two physical processes arising from refraction by currents, bending of wave rays and changes of wave energy, are both found to be important. The onset of instabilities of circulations occurs at the nearshore region where rips are ''fed,'' rather than offshore at rip heads as predicted with no wave-current interaction. The unsteady flows are characterized by vortex shedding, pairing, and offshore migration. Instabilities are sensitive to the angle of wave incidence and the spacing of rip channels.

A quantitative theory is described for the formation mechanism of sand bars under surface water waves. By assuming that the slopes of waves and bars are comparably gentle and sediment motion is dominated by the bedload, an approximate evolution equation for bar height is derived. The wave field and the boundary layer structure above the wavy bed are worked out to the accuracy needed for solving this evolution equation. It is shown that the evolution of sand bars is a process of forced diffusion. This is unlike that for sand ripples which is governed by an instability. The forcing is directly caused by the non-uniformity of the wave envelope, hence of the wave-induced bottom shear stress associated with wave reflection, while the effective diffusivity is the consequence of gravity and modified by the local bed stress. During the slow formation, bars and waves affect each other through the Bragg scattering mechanism, which consists of two concurrent processes: energy transfer between waves propagating in opposite directions and change of their wavelengths. Both effects are found to be controlled locally by the position of bar crests relative to wave nodes. Comparison with available laboratory experiments is discussed and theoretical examples are studied to help understand the coupled evolution of bars and waves in the field.

In most past theories on Bragg reflection of waves by a finite patch of rigid bars, only outgoing waves are allowed on the transmission side, simulating the effect of an idealized shoreline where all the incident wave energy is consumed by breaking. In these theories the amplitudes of both the incident and reflected waves are found to decrease monotonically over the bar patch in the shoreward direction. This result has motivated the idea of artificially constructing bars to protect a beach from incident waves. However, some numerical calculations have suggested that this tendency does not always hold when there is some reflection from the shore. We show here that with finite reflection by the shoreline the spatial distribution of wave energy over the patch can indeed be reversed, indicating that the mechanism can increase the hazards to the beach. The phase relation between the bars and the shoreline reflection is found to be the key to this qualitative change of wave response.

We examine the instability leading to depth-averaged circulations, which are related to rip currents in the surf zone, due to wave–current interactions. Our intention is to clarify some issues which are critical to the determination of instability properties, as yet unresolved from previous studies. Those issues are also of interest for hydrodynamic instability problems in general. Attention is restricted to normally incident waves and the region inside, and just offshore of, the surf zone where linearized shallow-water waves are applied. The coupling of waves and circulations is modelled using the concept of wave radiation stresses and the classical ray theory. The instability properties, in terms of the neutral modes, the most unstable mode and the corresponding maximum growth rate, are examined in the domain of parameters which represent the effects of offshore wave height and bottom friction dissipation. Comparisons with observations of natural rip currents are made, and qualitative agreements are achieved.