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A Comparison of Radiation-Stress and Vortex-Force Representations Under consideration for publication in J. Fluid Mech. 1 Wave-Current Interaction: A Comparison of Radiation-Stress and Vortex-Force Representations By E M I L Y M. L A N E1 , J U A N M. R E S T R E P 02 A N D J A M E S C. Mc W I L L I A M S1;3 1Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA 90095-1567 U.S.A. 2Department of Mathematics and Department of Physics, University of Arizona, Tucson, AZ 85721 U.S.A. 3Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, CA 90095-1565 U.S.A. (Received 21 April 2005) The effects of wind-generated surface gravity waves on more slowly evolving long waves, currents and material distributions in stratified coastal waters are investigated using the wave-averaged, asymptotic equations developed in McWilliams et al. (2004), based on small wave slope and on scale separations in both time and horizontal space. Excluding non-conservative effects such as wave breaking, the lowest order radiation stress, intro- duced by Longuet-Higgins & Stewart (1960) and Hasselmann (1971), can be completely characterized in terms of wave set-up, forcing of long (infra-gravity) waves and an Eule- rian current whose divergence cancels that of the primary waves' Stokes drift. The vortex force of Craik & Leibovich (1976) and its generalization for inhomogeneous waves and Earth's rotation are shown to be the dominant wave-averaged effects on currents, and these effects can occur at higher order than the apparent leading order for the radiation stress. The leading-order, wave-averaged dynamical effects are completed with material advection by Stokes drift, modified pressure-continuity and kinematic surface bound- ary conditions, and parameterized representations of wave generation by the wind and breaking near the shoreline. 1. Introduction Surface gravity waves influence slowly evolving long waves, such as infra-gravity waves, as well as currents and material distributions in the ocean. This paper is an examination of the relation between two alternative representations of these wave-averaged effects in coastal waters in the absence of dissipative or forcing mechanisms. We denote the two representations as `radiation-stress' and `vortex-force.' The concept of radiation stress has helped explain such phenomena as wave set-up, surf beats, generation of longshore currents in the surf zone, and spectral transfers within the wave field. The vortex-force representation arose to explain Langmuir circulations through wave vorticity generation by the currents and vortex stretching by the waves' Lagrangian mean flow, the Stokes drift, but the representation is more generally germane. Formally the radiation-stress and vortex-force representations are equivalent, related through two alternative representations of the inertial acceleration (i.e., advection). The 2 E. M. Lane, J. M. Restrepo and J. C. McWilliams radiation-stress representation arises from the identity, U U = (UU) + U( U); (1.1) · r r · r · together with incompressibility, U = 0, while the vortex-force representation comes from the identity, r · U 2 U U = j j + ( U) U; (1.2) · r r 2 r × × where U is the Eulerian velocity vector. This leads to two ways of expressing the wave effects in an asymptotic framework based on a small wave slope and rapid wave oscillation. The radiation-stress representation views the wave-averaged effects of the waves on the current as the divergence of a stress tensor (hence the name) in some nominal way similar to how Reynolds stress enters time-averaged equations of turbulent fluid motion. The vortex-force representation decomposes the effect of the waves into two components, the gradient of a Bernoulli head and a vortex force. The Bernoulli head is essentially an adjustment to the pressure in accommodating incompressibility. After wave averaging, the vortex force is shown to represent an interaction between the vorticity of the flow and the Stokes drift. In McWilliams et al. (2004) (hereafter referred to as MRL04), we derived coupled equa- tions for wave-current interaction. These equations encompass all previous derivations within the vortex-force representation (Craik & Leibovich 1976; Leibovich 1977a,b, 1980; Huang 1979; Holm 1996; McWilliams & Restrepo 1999) when the asymptotic scaling is taken into account. The adoption of a specific asymptotic scaling, appropriate to coastal and open-ocean waters, is an important aspect of the derivation presented in MRL04, as is the decomposition of the complete dynamics into primary gravity waves, long waves with longer horizontal and temporal scales and the even more slowly evolving currents. This scaling is justified on the following premises: our intended setting is beyond the shoreline wave-breaking zone (i.e., the littoral or surf zone), in water where the typical depth times the primary wavenumber is not small. In the horizontal plane we assume that the dominant current and long-wave scales are kilometres or more and the time scales are hundreds of seconds or more (n.b., shelf currents have typical time scales of 1 days or weeks). We also assume that the currents' velocity ( 0:1 m s− ) and sea-level ∼ 1 fluctuation ( 0:1 m) are smaller than the waves' orbital motion ( 1 m s− ) and height ( 1 m). Thus,∼ in many situations and over most coastal and open-o∼ cean areas | strong tidal∼ flows and the surf zone being two of the few exceptions | there are clear sepa- rations between waves and currents in amplitude and horizontal space and time scales. This provides a rigorous basis for deriving the conservative wave-current interaction equa- tions without resorting to closure assumptions. Of course, any complete representation of wave-current interaction must also include their wind-generation as well as dissipative mechanisms such as wave breaking. Offshore, the wave breaking is a primary mechanism for conveying wind stress to currents, and nearshore it is a principal cause of littoral cur- rents. However, following MRL04, we focus here on conservative, unforced wave-current dynamics. The original work on radiation stress by Longuet-Higgins & Stewart (1960, 1961, 1962, 1964) and Hasselmann (1971) was done by averaging in an Eulerian reference frame, and it has an extensive subsequent literature (e.g., Bowen & Guza 1978; Longuet-Higgins 1970; Basco 1983; Mei 1989; Svendsen & Putrevu 1995). In this paper this representation will be compared with the vortex-force representation. Wave-current interactions have also been analysed using the Generalized Lagrangian Mean (GLM) (Andrews & McIntyre 1978a,b; Gjaja & Holm 1996) and in terms of other Wave-Current Interaction 3 averaging variants in a Lagrangian reference frame (Weber 1983; Jenkins 1987; Mellor 2003, 2004; Jenkins & Ardhuin 2004). In MRL04, we chose the Eulerian frame because of the conceptual simplicity of the Eulerian mean and because Eulerian means are now, and in the foreseeable future, more practical in relating to oceanic measurements and large-scale oceanic numerical models. If desired, the GLM velocity may be recovered from the Eulerian velocity to lowest order by the addition of Stokes drift. Because these are asymptotic quantities they will not exactly satisfy constraints such as conservation of total circulation, however this is of little consequence dynamically. Furthermore, regard- less of the choice of averaging, the question of how wave effects are represented is still pertinent. In the GLM frame, though, familiar vortex force terms may be hidden within the Lagrangian-averaged quantities. These two different reference frames (Eulerian and Lagrangian) lead to differences in the averages obtained. The Lagrangian mean velocity is equivalent to the Eulerian mean velocity plus a residual velocity, often identified as the Stokes velocity. Likewise, other Lagrangian averaged quantities differ from their Eulerian averaged counterparts, e.g., the pressure. Due to this difference, the evolution equation for the Lagrangian mean velocity encompasses the evolution equation for the Eulerian mean velocity but includes terms driving the evolution of the Stokes velocity in addition to wave forcing of the current. As shown in Section 4 this is analogous to the difference between the equations for the total transport T and the current transport Tc. When AndrewsD &E McIntyre (1978a) first developed the GLM formalism, they repre- sented wave-current effects in terms of the wave pseudomomentum p. GLM equations have also been derived where the wave effects are represented by the divergence of a radiation stress (Andrews & McIntyre 1978a; Groeneweg & Klopman 1998). For periodic waves, assuming time-averaging and a weak Coriolis force (all true in the MRL04 equa- tions), the wave pseudomomentum can be identified with the Stokes drift VSt (Leibovich 1980); (Andrews & McIntyre 1978a, Equation 3.8) may be converted to an equation for the Eulerian mean velocity. Leibovich (1980) showed that the conservative effects of this equation can be expressed as a vortex force and a pressure adjustment. In comparison, GLM equations that include stress divergence terms tend to be used for the Lagrangian- averaged velocity. When these equations are converted to Eulerian velocity, no stress divergence term is needed. This concurs with Section 5.2 that shows (at least in the vertically-integrated case) the divergence of the radiation stress is mainly the evolution of the Stokes drift. Mellor (2003) uses a wave-following coordinate system to calculate the three-dimensional equations for wave-current interactions. The average obtained this way is essentially a Generalized Lagrangian Mean. These equations are translated back into Cartesian coor- dinates in Mellor (2004). Mellor (2004) compares his equations with those in McWilliams & Restrepo (1999) (and thus by implication MRL04). There are obvious differences, such as a turbulent Reynolds stress closure and wind and surface pressure forcing, that simply are not included in the MRL04 analysis. The difference between the conservative wave effects in Mellor's equations compared to those in MRL04 is due to the different averaging techniques used.
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