Radiation stress and mass transport in gravity waves, with application to ‘surf beats’

Abstract

This paper studies the second-order currents and changes in mean surface level which are caused by gravity waves of non-uniform amplitude. The effects are interpreted in terms of the radiation stresses in the waves.The first example is of wave groups propagated in water of uniform mean depth. The problem is solved first by a perturbation analysis. In two special cases the second-order currents are found to be proportional simply to the square of the local wave amplitude: (a) when the lengths of the groups are large compared to the mean depth, and (b) when the groups are all of equal length. Then the surface is found to be depressed under a high group of waves and the mass-transport is relatively negative there. In case (a) the result can be simply related to the radiation stresses, which tend to expel fluid from beneath the higher waves.The second example considered is the propagation of waves of steady amplitude in water of gradually varying depth. Assuming no loss of energy by friction or reflexion, the wave amplitude must vary horizontally, to maintain the flux of energy constant; it is shown that this produces a negative tilt in the mean surface level as the depth diminishes. However, if the wave height is limited by breaking, the tilt is positive. The results are in agreement with some observations by Fairchild.Lastly, the propagation of groups of waves from deep to shallow water is studied. As the mean depth decreases, so the response of the fluid to the radiation stresses tends to increase. The long waves thus generated may be reflected as free waves, and so account for the 'surf beats’ observed by Munk and Tucker.Generalle speaking, the changes in mean sea level produced by ocean waves are comparable with those due to horizontal wind stress. It may be necessary to allow for the wave stresses in calculating wind stress coefficients.

The large-scale influence of the Great Barrier Reef matrix on wave attenuation

“…As waves break and attenuate, the mean water surface elevation increases (setup), driving currents (Longuet-Higgins and Stewart 1962) and reef circulation (Hamner and Wolanski 1988;Pickard et al 1990;Symonds et al 1995;Angwenyi and Rydberg 2005). Such currents have implications for the transport of sediments, pollutants, nutrients, plankton, and larvae (Lowe et al 2005).…”

Refraction and reflection of infragravity waves near submarine canyons

“…Contributions from forced, or group bound, waves that shoal proportional to a theoretical maximum h À5 ratio [Longuet-Higgins and Stewart, 1962] are small everywhere except the surfzone. Also similar to previous results, the relative contribution of seaward propagating leaky waves (with depth dependence h À1/2 ) is estimated to be a significant portion of the total infragravity energy [Sheremet et al, 2002].…”

“…[4] When the swell waves break and dissipate in the surfzone, the infragravity waves are released and propagate toward the shoreline as free waves [Longuet-Higgins and Stewart, 1962;Herbers et al, 1995b]. In the surfzone, infragravity waves transfer some of their energy back to motions with higher frequencies [Thomson et al, 2006;Henderson et al, 2006], before reflecting from the shoreline [Suhayda, 1974;Guza and Thornton, 1985;Nelson and Gonsalves, 1990;Elgar et al, 1994;Sheremet et al, 2002].…”

Near-shore sediment dynamics computation under the combined effects of waves and currents

“…Taking into account the radiation stresses (S xx , S xy , S yx and S yy ) as de®ned by Longuet-Higgins and Stewart [6], the governing equations for the current ®eld computation are as follows:…”