1122 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 37 Wave–Current Interaction: A Comparison of Radiation-Stress and Vortex-Force Representations E. M. LANE Institute of Geophysics and Planetary Physics, University of California, Los Angeles, Los Angeles, California J. M. RESTREPO Department of Mathematics, and Department of Physics, The University of Arizona, Tucson, Arizona J. C. MCWILLIAMS Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, Los Angeles, California (Manuscript received 10 May 2006, in final form 8 August 2006) ABSTRACT The vortex-force representation of the wave-averaged effects on currents is compared to the radiation- stress representation in a scaling regime appropriate to coastal and shelf waters. Three-dimensional and vertically integrated expressions for the conservative current equations are obtained in both representa- tions. The vortex-force representation decomposes the main wave-averaged effects into two physically understandable concepts—a vortex force and a Bernoulli head. The vortex force is shown to be the dominant wave-averaged effect on currents. This effect can occur at higher order than the apparent leading order for the radiation-stress representation. Excluding nonconservative effects such as wave breaking, the lowest-order radiation or interaction stress can be completely characterized in terms of wave setup, forcing of long (infragravity) waves, and an Eulerian current whose divergence cancels that of the primary wave Stokes drift. The leading-order, wave-averaged dynamical effects incorporate the vortex force together with material advection by Stokes drift, modified pressure-continuity and kinematic surface boundary condi- tions, and parameterized representations of wave generation by the wind and breaking near the shoreline. 1. Introduction fects on currents using the divergence of either a radia- tion or an interaction stress, or a three-dimensional Surface gravity waves influence slowly evolving long analog, following the work of Longuet-Higgins and waves, such as infragravity waves, as well as currents Stewart (1960, 1961, 1962, 1964, hereinafter collectively and material distributions in the ocean. This paper is referred to as LHS) and Hasselmann (1971, hereinafter an examination of the relation between two alterna- referred to as H71). The vortex-force representation tive representations of these wave-averaged effects in arose to explain Langmuir circulations through wave coastal waters in the absence of dissipative or forcing vorticity generation by the currents and vortex stretch- mechanisms. We denote the two representations as “ra- ing by the wave’s Lagrangian mean flow, the Stokes diation stress” and “vortex force.” The concept of ra- drift (Craik and Leibovich 1976), but the representa- diation stress has helped to explain such phenomena as tion is more generally germane. wave setup, surf beats, generation of alongshore cur- Formally, the radiation-stress and vortex-force rep- rents in the surf zone, and nonlinear interactions within resentations are equivalent, related through two alter- the wave field. We use the term radiation-stress repre- native representations of the inertial acceleration (i.e., sentation to encompass equations that include wave ef- advection). The radiation-stress representation arises from the identity U͒, ͑1͒ · UU͒ ϩ U͑١͑ · ١ ١U ϭ · Corresponding author address: E. M. Lane, Institute of Geo- U physics and Planetary Physics, University of California, Los An- U ϭ 0, while the · ١ geles, Los Angeles, CA 90095-1567. together with incompressibility E-mail: [email protected] vortex-force representation comes from the identity DOI: 10.1175JPO3043.1 © 2007 American Meteorological Society Unauthenticated | Downloaded 09/28/21 12:48 AM UTC JPO3043 MAY 2007 LANE ET AL. 1123 |U| 2 long waves and variation of the wave quantities to the ϫ U͒ ϫ U, ͑2͒ ١͑ ϩ ١ ١U ϭ · U 2 time scale of the waves.) • The wave slope for the spectrum peak components is where is the Eulerian velocity vector. This leads to U small (again apart from breaking) (i.e., ␧ K 1, where two ways of expressing the wave effects in an asymp- ␧ϭAk is the wave slope). This implies that advection totic framework based on a small wave slope and rapid plays a secondary role compared to propagation and wave oscillation. The radiation-stress representation its nonlinear dynamical effects occur on a much views the wave-averaged effects of the waves on the longer time scale than the wave period (e.g., 7–15 s). current as the divergence of a stress tensor (hence the Ϫ • The current typical velocity (0.1–0.5 m s 1) and sea name). This is analogous to how Reynolds stress enters level (0.05–0.3 m) fluctuations are smaller than the time-averaged equations of turbulent fluid motion. The Ϫ wave typical phase speed (10–25 m s 1), orbital mo- vortex-force representation decomposes the effect of Ϫ tion (1–3ms 1), and sea level fluctuation (ϳ1–3m) the waves into two components: the gradient of a Ber- (i.e., ␦ K 1 and ␥ K 1, where ␦ is the ratio of the mean noulli head and a vortex force. The Bernoulli head is current to the wave orbital velocity and ␥ is the long essentially an adjustment to the pressure in accommo- time scale associated with the mean current). This dating incompressibility. After wave averaging, the vor- implies that there is a clear scale separation between tex force is shown to represent an interaction between the evolution rate for currents in association with in- the vorticity of the flow and the Stokes drift. Garrett ternal gravity waves, tides, inertial motions, and ad- (1976) first showed that the vortex-force representation vection and the primary wave frequency, allowing for could be derived from the radiation-stress representa- a meaningful average over the primary wave fluctua- tion. tions. In McWilliams et al. (2004, hereinafter referred to as • The intended setting is beyond the shoreline wave- MRL04), we derived coupled equations for wave–cur- breaking zone (i.e., the littoral or surf zone), in water rent interaction. These equations encompass all previ- where the primary wavenumber times the typical ous derivations within the vortex-force representation depth is not small (i.e., ␮ ϭ kH is not small and, (Craik and Leibovich 1976; Leibovich 1977a,b, 1980; indeed, could be large). For a 50-m peak wavelength, Huang 1979; McWilliams and Restrepo 1999) when the this condition is met once the depth is greater than a asymptotic scaling is taken into account. The adoption few meters. of a specific asymptotic scaling appropriate to coastal and open-ocean waters is an important aspect of the Therefore, in many situations and over most coastal derivation presented in MRL04, as is the decomposi- and open-ocean areas—with very strong tidal flows and tion of the dynamics into primary gravity waves, long the surf zone among the few exceptions—there is a waves with longer horizontal and temporal scales, and clear separation between waves and currents in ampli- the even more slowly evolving currents. A full list of tude and horizontal space and time scales. This pro- assumptions is given in MRL04; the choice of scaling is vides a basis for deriving the conservative wave–current justified as follows: interaction equations by avoiding closure assumptions. • The important part of the surface wave field for con- We adopt a particular set of the relation among the servative dynamical influences (i.e., apart from scaling parameters (section 2b) that is chosen to breaking) on the infragravity waves and currents is achieve the greatest generality for the resulting wave- the spectrum peak scale associated with wind genera- averaged dynamical balances, even though in particular tion, whether in local equilibrium or as previously situations the actual parameter values will differ and and remotely generated wave swell. The relevant the dynamical balance will involve a subset of dominant measures of dynamical importance are Stokes drift, influences. Some alternative parameter relations are Bernoulli head, etc., which are dominated by the also analyzed in section 6b. Of course, any complete peak-scale waves. This peak wavelength is typically representation of wave–current interactions must also 75–300 m, which is usually much shorter than the include their wind generation, spectrum distribution, horizontal scale of the principal currents and infra- and dissipative mechanisms, such as wave breaking. gravity waves, and is even comparable to the hori- Offshore, the wave breaking is a primary mechanism zontal scale of energetic turbulent eddies in the sur- for conveying wind stress to currents, and near shore it face boundary layer. (Thus, ␤ Յ 1 for currents, and is a principal cause of littoral currents. However, fol- ␤ K 1 for infragravity waves, where ␤ is the ratio of lowing MRL04, we focus here on conservative, un- the topographic and mean horizontal scale to that of forced wave–current interactions. the waves. It is also the ratio of the time scale of the In this paper we compare the outcomes of the adop- Unauthenticated | Downloaded 09/28/21 12:48 AM UTC 1124 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 37 tion of radiation-stress and vortex-force representa- parameter is f and the gravitational constant is g; the tions in the derivation of mean Eulerian wave–current pressure is described by P and Pa is the atmospheric ץ ١ ϭ ١ interaction equations in a scaling appropriate for pressure; C is the tracer concentration; ( x, z)by coastal and shelf waters. In doing so, we make plain the convention and t is time; z ϭ E(x, t) is the surface discrepancies and similarities of these two representa- displacement, z ϭϪH(x) describes the bottom topog- tions under the asymptotic assumptions that we have raphy, z ϭ 0 is understood to be the quiescent level of made, as well as more generally. After a preliminary the ocean, and z is positive upward.