An asymptotic theory for the interaction of waves and currents in coastal waters

Abstract

A multi-scale asymptotic theory is derived for the evolution and interaction of currents and surface gravity waves in water of finite depth, under conditions typical of coastal shelf waters outside the surf zone. The theory provides a practical and useful model with which wave–current coupling may be explored without the necessity of resolving features of the flow on space and time scales of the primary gravity-wave oscillations. The essential nature of the dynamical interaction is currents modulating the slowly evolving phase of the wave field and waves providing both phase-averaged forcing of long infra-gravity waves and wave-averaged vortex and Bernoulli-head forces and hydrostatic static set-up for the low-frequency current and sea-level evolution equations. Analogous relations are derived for wave-averaged material tracers and density stratification that include advection by horizontal Stokes drift and by a vertical Stokes pseudo-velocity that is the incompressible companion to the horizontal Stokes velocity. Illustrative solutions are analysed for the special case of depth-independent currents and tracers associated with an incident surface wave field and a vortex with O(1) Rossby number above continental shelf topography.

Three-dimensional modelling of wave-induced current from the surf zone to the inner shelf

“…Recently, developed 3-D wave-current interaction theories (e.g. Mellor, 2003;McWilliams et al, 2004;Ardhuin et al, 2008c) may provide useful approaches for this problem.…”

“…Here we use an approximation of the exact equations from (Andrews and McIntyre, 1978) to second order in wave slope, including a transformation to cartesian coordinates (glm2-z approximation, Ardhuin et al, 2008c), in a simplified form that neglects the vertical current shear effect on the dynamic pressure . In the limit of weak vertical current shear, these equations are formally equivalent to the Eulerian-mean equations of McWilliams et al (2004) that are based on an analytic continuation across the air-sea interface, and which have been used by Uchiyama et al (2010).…”

An Oil Fate Model for Shallow-Waters

“…[21] and [19]). It has contributions from the depth-averaged current v c and the Stokes drift velocity u St as well as the residual velocity due to wave breaking (see [22]).…”

“…Specifically, with the wave/current interaction model proposed in [19]. A full accounting of the [19] wave/current interaction model and its asymptotic balances will not be repeated here.…”

“…The second term in Equation (19) accounts to increases in the viscosity due to evaporation where F evap is the fraction evaporated from the slick. F wc is given in Equation (18) and C 4 is a parameter that varies between 1 and 10, the smaller value associated with lighter hydrocarbons.…”