Wave–Current Interaction: a Comparison of Radiation-Stress and Vortex-Force Representations

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 representations. 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 conditions, 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 radiation 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

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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 fluctuation. 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. The transverse ϭ presentation of the fundamental equations (section 2), coordinates are x (x1, x2). Greek subscripts run from we relate and compare the vortex-force representation 1 to 2. The standard summation convention on re- with the widely used radiation-stress representation. peated indices is used. We compare the time-averaged equations derived We may redefine the pressure by subtracting the hy- through the two representations in section 3, and in drostatic pressure from the kinematic pressure, section 4 we consider the vertically integrated, time- p ϭ P Ϫ P ϩ ␳ gz averaged equations. We also make an estimate of the a 0 , energetics of the quasi-static response, long waves, and whereby (3) becomes current. Section 5 considers two examples that highlight ѨU 1 ͒ ͑ Ϫ ϩ ϫ ϭ ١ ϩ ١ the importance of properly handling wave–current in- ϩ Ѩ U · U ␳ p B z fz U 0. 9 teractions, both with regard to derivational issues as t 0 well as scaling. Further commentary on these issues is Here the buoyancy is defined as made in section 6, which also touches upon some the B difficulties with the radiation-stress representation in ␳ Ϫ ␳ ϭ 0 this scaling, and considers alternative scalings and other B g ␳ . 0 averaging frameworks. In section 7 we summarize our view of the important elements in wave–current inter- The pressure surface boundary condition (7) becomes action. ϭ ␳ ϭ p 0gE at z E.

2. Preliminaries We can also calculate the energy E of the system integrated over a horizontal domain x ∈ G and from the a. Basic equations seafloor to the sea surface, In the absence of body forces and dissipation, the E |U|2 g equations of motion are E ϭ ͵͵ͫ͵ ͩ Ϫ Bzͪ dz ϩ E2ͬ dx. ϪH 2 2 ѨU 1 ␳gz G ͒ ͑ ϩ ϩ ϫ ϭ ١ ϩ ١ ϩ Ѩ U · U ␳ P ␳ fz U 0 and 3 t 0 0 Under the assumption that there is no net flux of en- U ϭ 0, ͑4͒ ergy across the boundary of G, the equation for the · ١ evolution of the energy is with tracer equation ѨE 1 ѨE 1 ١ ͵͵ Ѩ ϭϪ ͵͵ ϭϪ C Ѩ ␳ Pa Ѩ dx ␳ T · xPa dx, ١C ϭ 0. ͑5͒ t 0 t 0 · ϩ U Ѩt G G The surface boundary conditions are ͑10͒ ѨE where ͒ ͑ ١ ϭ ϩ W Ѩ Q · xE and 6 t E ϭ ϭ ͑ ͒ T ϭ ͵ Q dz P Pa at z E, 7 ϪH and the bottom boundary condition is is the transport. If Pa is constant over the domain G, ͒ ͑ ϭϪ ١ ϭϪ W Q · xH at z H. 8 then energy is conserved. Tracer content is also conserved, namely, The physics and well-posedness considerations deter- mine the lateral boundary conditions. The Eulerian ve- Ѩ E ϭ ͵͵͵ C dz dx ϭ 0. ͑11͒ locity vector is U (Q, W). The density is described by Ѩt ␳ ␳ ϪH and 0 is a reference value of the density; the Coriolis G

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Upon vertically integrating (9) and (5) using (4) and temporal scales. Long waves evolve over the temporal (6)–(8), we obtain scale ␶ ϭ ␤t, as does the wave envelope. Thus, their ␶ E E 2 scales are (X, z, ). The current evolves over a longer ѨT␣ Ѩ 1 Ѩ g ѨE ϩ ϩ Ϫ time scale, T ϭ ␥t, where ␥ K ␤ K 1, giving it the Ѩ Ѩ ͵ Q␣Q␤ dz ␳ Ѩ ͵ pdz Ѩ t x␤ ϪH 0 x␣ ϪH 2 x␣ scaling (X, z, T). We also assume that the mean hori- ␦ 1 ѨH zontal velocities are O( ) relative to the wave quanti- Ϫ p͑ϪH͒ ϩ ͑fz ϫ T͒␣ ϭ 0, ͑12͒ ties, and long-wave velocities are O(␭) relative to the ␳ Ѩx␣ 0 waves where ␦, ␭ K 1. Furthermore, incompressibility Ѩ E Ѩ E requirements demand that, in the vertical direction, the ͵ ϩ ͵ ϩ Ѩ Wdz Ѩ WQ␣ dz gE ratio of mean to wave velocity is ␤␦. We use the mu- t ϪH x␣ ϪH tually consistent assumptions made in MRL04 that ␤ ϭ p͑ϪH͒ E ␧2 ␦ ϭ␧ ␭ ϭ␧ ␥ ϭ␧4 ␤ ϭ Ϫ ϭ ͵ ͑ ͒ , , , and . If, on the other hand, ␳ B dz, 13 ␧ 0 ϪH , dispersion enters the wave amplitude equation at the same order as the nonlinearity (Mei 1989, chapter 12). and While this affects the evolution of the waves and the

ѨE ѨT␣ nonlinear terms in the current equations, it does not ϩ ϭ 0. ͑14͒ Ѩt Ѩx␣ alter the nature of the wave forcing of the mean equations. The essential results of this paper hold as long as Equations for the vertically averaged horizontal veloc- ␤ K 1, as required for leading-order wave (e.g., Went- ity T/(H ϩ E) may be obtained from (12). Similarly, zel–Kramers–Brillouin-like) solutions. Ѩ E Ѩ E The scaling for tracers is different than the velocity ϩ ϭ ͑ ͒ Ѩ ͵ C dz Ѩ ͵ CQ␣ dz 0. 15 scaling. It is discussed in MRL04 and McWilliams and t ϪH x␣ ϪH Restrepo (1999). The largest component of the tracer b. Nondimensionalizaton, asymptotic scaling, and concentration is due to the long-term currents. The pri- flow decomposition mary waves and long waves cause smaller, faster-scale variations in the tracer concentration. The ratio of the As in MRL04, we nondimensionalize according to wave concentration to the current concentration is ␯ ͑ ͒ → Ϫ1͑ ͒ and that of the long-wave concentration to the current x, z k0 x, z , concentration is ␭␯, where ␯ K 1 and is ␧ in MRL04. t → ␴Ϫ1 0 t, We decompose the flow into three components: cur- → ϭ ␧ Ϫ1 rent, long wave, and primary wave. The three-way de- E a0E k0 E, composition for velocity is denoted as ϭ ͑ ͒ → ␴ ͑ ͒ ϭ ␧ Ϫ1␴ ͑ ͒ U Q, W a0 0 Q, W k0 0 Q, W , c c lw lw w w ⍀ → ␧␴ ⍀ U ϭ ͑Q, W͒ ϭ ͑q , w ͒ ϩ ͑q , w ͒ ϩ ͑q , w ͒ 0 , c lw w → ␧␳ ␴2 Ϫ2 ϭ u ϩ u ϩ u , P 0 0k0 P, and B → ␮B, with ͑ ͒ ϭ ͑ ͒ ϩ ͑ w w͒ where the quantities on the right are dimensionless: k0 Q, W q, w q , w , c c c lw lw lw is the characteristic wavenumber, H0 is the character- where u ϭ (q , w ) is the mean current, u ϭ (q , w ) ␮ ϭ w w w istic depth, k0H0 is a nondimensional parameter is the long wave, and u ϭ (q , w ) is the wave com- ␴ ϭ that specifies the relative depth of the water; 0 ponent. Waves are asymptotically expanded in small ͌ ϭ gk0L is the characteristic wave frequency with L wave slope ␧, namely, tanh␮; a is the typical wave height and ␧ϭa k the 0 0 0 ͑ w w͒ ϭ ͑ w w͒ ϩ ␧͑ w w͒ ϩ ␧2͑ w w͒ ϩ nd ϭ ␴ q , w q0 , w0 q1 , w1 q2 , w2 ···; wave slope; f0 is the Coriolis parameter with f f0/ 0 ⍀ w ϭ w w a Rossby number based on wave time scales; is the the notation ui (qi , wi ) is also used. Sea surface vorticity. elevation and pressure are split into their mean, long- We adopt the following scalings for the primary wave, and wave components according to E ϭ ␨c ϩ ␨ lw waves, long waves, and currents: The nondimensional ϩ ␨w and p ϭ͗p͘ϩp ϩ pw. Likewise, tracers and wave scale coordinates are (x, z, t). The bottom topog- buoyancy are decomposed and expressed as C ϭ cc ϩ raphy varies over a longer horizontal spatial scale, X ϭ c lw ϩ cw and B ϭ bc ϩ b lw ϩ bw. At times we suppress ␤x. This is also the horizontal spatial scale for the wave the distinction between long waves and current by just envelope, long waves, and the currents. The long waves defining waves and mean current. In this case we drop and the currents are distinguished by their different the superscript from the mean component.

Unauthenticated | Downloaded 09/28/21 12:48 AM UTC 1126 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 37 w ١ As in MRL04, (·) signifies averaging over the wave ϭ w X · T and time scale t and ͗···͘ signifies averaging over the longer ␨ Ѩ͗ ͘ Ѩ w ␶ Ϫ p Ϫ p P0 scale. Fluctuations are distinguished by ␧͗p͘ Ϫ ␧ 2 ϩ ␨ ϭϪ␧ 2ͳ␨ w ʹ Ϫ ͑20͒ L Ѩz Ѩz L ͑·͒Ј ϭ ͑·͒ Ϫ ͗ · ͘ and ͑·͒† ϭ ͑·͒ Ϫ ͗ · ͘. at z ϭ 0 and The long waves and the currents may be distinguished ͒ ͑ ϭ ١ ϩ by averaging over the two different time scales so that w q · XH 0 21 ͗Q͘ϭqc, Q ϭ qc ϩ qlw and Q† ϭ qlw. at z ϭϪH. Below the wave troughs the Eulerian average of a 3. Time-averaged equations of motion quantity is unambiguously defined. Above this level the averaging process is less clear as each point only lies a. MRL04’s current and long-wave equations underwater part of the time. The averaging process that The mean Eulerian current equations in the vortex- we use, and refer to as the mean Eulerian, involves force representation are analytically continuing the quantity and then averaging. This allows clear separation of wave quantities from Ѩ Ѩ q Ϫ current quantities. Note that, as in MRL04, the surface p͗͘ ϩ w ͪq ϩ fz ϫ q ϩ ␧ 2١ ١ · ϩ ͩq ѨT X Ѩz X ocean reference level is set to be z ϭ 0. Given our Ϫ asymptotic scaling, the difference between the mean ␧ 2K ϩ K ͒ ϩ J, ͑16͒͑ ϭϪ١ X 1 2 sea level and z ϭ 0isO(␧2) and thus inconsequential to Ѩ͗p͘ b Ѩ leading order. ␧Ϫ2 ϭ Ϫ ͑␧Ϫ2 ϩ ͒ ϩ ͑ ͒ K 1 K 2 K, 17 Ѩz L Ѩz Wave components K 1 and K 2 are the Bernoulli head. These represent adjustments to the mean pressure due Ѩw -ϩ ϭ ͑ ͒ to the presence of waves. Here J and K are the hori ١ X · q 0, and 18 Ѩz zontal and vertical components of the vortex force; St ϭ St St Ѩc u (v , w ) is the three-dimensional velocity com- ١c. ͑19͒ posed of the horizontal Stokes velocity and the vertical · ͒ ϭϪ͑u ϩ uSt ѨT Stokes pseudovelocity; Tw is the mass transfer due to

For full details, as well as the equations that capture the waves; and P0 contains wave and mixed terms from the waves, the reader is referred to MRL04. The boundary series expansion of the pressure about the quiescent conditions are level z ϭ 0. Specifically, the wave-averaged terms are

| w|2 ϭ ͳ u0 ʹ K 1 , 2 |uw|2 ϭ ͳ 1 ϩ w wʹ K 2 u · u , 2 0 2 ͑ ͒ ϭ ͓ ͑ w ϫ ␻w͒ ͑ w ϫ ␻w͒ ͔ J␣, K ͗ u ␣͘, ͗ u 3͘ , t z ,vSt dzЈͬ ͵ · ١ͪuwʹ, Ϫ١ · vSt, wStͪ ϭ ͫͳͩ͵ uw dtͩ 0 0 X ϪH

␧E ␧2␨ Tw ϭ ͳ͵ Q dzʹ ϭ ͵ vSt dz, and ␧2␨ ϪH

Ѩ w ␨ w2 Ѩ2 ␧Ϫ1 Ѩ w P0 p ͗ ͘ ͗p͘ 2 p ϭ ͩ␨ͳ␨ w ʹ ϩ ϩ ͳ␨ w ʹͪ, ͑22͒ L Ѩz 2 Ѩz2 2 Ѩz

ϫ uw is the wave vorticity, which to the wave stresses and the mean surface elevation and ١ where ␻w ϭ lowest order is caused by the material advection of the mean pressure. This is the phenomenon known as wave mean vorticity field. The lowest-order momentum setup. Thus, we may separate the mean pressure and equations, (16)–(17), and the pressure boundary con- sea level into quasi-static and dynamic components as dition (20) represent a quasi-static balance between follows:

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͗p͘ ϭ pˆ ϩ ␧2͑pˆ c ϩ p˜ c͒ and To lowest order, the boundary conditions are w ١ c c ͑ ͒ ϭ 2 ␨ ϭ ␨ˆ ϩ ␧ ͑␨ˆ ϩ ␨˜ ͒. ͑23͒ w 0 X · T , ␨˜c ˜ Components with carets represent quasi-static balances P0 p˜ c͑0͒ Ϫ ϭϪ , and while components with tildes are dynamic. The lowest- L L ϭ ١ ͒ order quasi-static balance is ͑Ϫ ͒ ϩ ͑Ϫ w H q H · XH 0. 2 w |A| k cosh͓2k͑z ϩ H͔͒ As derived in MRL04, K 2, J, K, T , and P0 may be pˆ ϭϪK Ϫ P ϭϪͳ ʹ Ϫ P 1 a 2L sinh͑2kH͒ a computed from the wave quantities: 1 z Ѩ2V sinh͓2k͑z Ϫ zЈ͔͒ and ˜ ϭ ͳ␴|A|2͵ dzЈʹ K 2 2 2 4 ϪH ѨzЈ k sinh ͑kH͒ Ѩpw |A|2k ␨ˆ ϭ ͩ ϩ ͳ␨ w ͯʹͪ ϭϪͳ ʹ Ϫ L pˆ LPa, Ѩz 2 sinh͑kH͒ 1 lw 2 0 ϩ ͗͑q ͒ ͘, 2 ͑24͒ ͑J, K͒ ϭ ͓Ϫz ϫ ͗vSt ͑͘␹c ϩ f ͒ Ϫ ͗wSt ͘␰c, ͗vSt ͘ · ␰c͔,and which comes in at O(␧3) in the horizontal momentum equations and O(␧) in the vertical momentum equa- |A|2␴k Tw ϭ ͳ ʹ, ͑30͒ tion. The higher-order balance is 2k tanh͑kH͒ where (␰c, ␹c) is the mean Eulerian vorticity, which, to 9 |A|4␴2k2 cosh͓4k͑z ϩ H͔͒ pˆ c ϭϪKˆ ϭ ͳ ʹ, lowest order, is 2 64 sinh8͑kH͒ Ѩq Ѩ␷ Ѩu ͩ , Ϫ ͪ. and Ѩz Ѩx Ѩy Ѩpˆ Ѩpˆ lw 1 The Stokes velocity is ␨ˆ c ϭ Lͫpˆ c ϩ ␨ˆ ͑0͒ ϩ ͳ␨ˆ lw ͑0͒ʹ ϩ Pˆ ͬ. Ѩz Ѩz L 0 |A|2␴ cosh͓2k͑z ϩ H͔͒ vSt ϭ k, ͑25͒ 2 sinh2͑kH͒

St The pressure pˆ differs from the historical wave-induced w is as defined in (22), P0 is given in MRL04’s (9.11), pressure pw ϭϪ͗(ww)2͘ by ͗g|〈|2k/[2sinh(2kH)]͘, which and V ϭ k · q. Note that the quantities in (30) differ is independent of z. The advantage of using pˆ rather slightly from those given previously in (22). We have than pw is that pˆ drops completely out of the wave- used the gauge invariance to ensure that the definition averaged momentum equations, even when the wave of the vortex force is consistent with previous defini- quantities vary over time. tions (MRL04, their section 9.6). Once the quasi-static components are removed the The equations for the long waves are three-dimensional current equations in (16)–(19) are Ѩqlw 1 ,†uw|2͒|͑ plw ϭϪ١ ١ ϩ Ѩq Ѩ Ѩ␶ X X 2 0 ,K˜ ϩ J p˜c ϭϪ١ ١ ϩ w ͪq ϩ fz ϫ q ϩ ١ · ϩ ͩq ѨT X Ѩz X X 2 Ѩplw Ѩ 1 ϭϪ ͑|uw|2͒†, ͑26͒ Ѩz Ѩz 2 0

Ѩp˜ c b Ѩ Ѩwlw ,qlw ϩ ϭ 0 · ١ ϭ Ϫ K˜ ϩ K, ͑27͒ Ѩz L Ѩz 2 X Ѩz

H͒ ϭ͑ ١ Ѩw wlw͑ϪH͒ ϩ lw͑ϪH͒ ,ϩ ϭ ͑ ͒ q · X 0 ١ X · q Ѩ 0, and 28 z Ѩ␨lw ,†␨wq ͑0͔͓͒ · ١ wlw͑0͒ Ϫ ϭ Ѩc Ѩ␶ X 0 0 ͒ ͑ ١ ϭϪ͑ ϩ St͒ Ѩ u u · c. 29 T Ѩ w † 1 p0 plw͑0͒ Ϫ ␨lw ϭϪͫ␨w ͑0͒ͬ , and The vertical momentum equation shows that in this L 0 Ѩz scaling the pressure is in a quasi-hydrostatic balance Ѩclw 1 Ѩ Ѩ Ѩc ١c ϭ ͫͩ e2ͪ ͬ, ͑31͒ · ͔†with the Bernoulli head, buoyancy, and vertical vortex ϩ ͓ulw ϩ ͑uSt͒ force. Ѩ␶ 2 Ѩz Ѩ␶ Ѩz

Unauthenticated | Downloaded 09/28/21 12:48 AM UTC 1128 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 37 where Ѩq␣ Ѩw ␧3 ϩ ␧3 ϭ 0. ͑36͒ ѨX␣ Ѩz 1 |A|2 sinh2͓k͑z ϩ H͔͒ † e2 ϭ ͭ ͮ . ͑32͒ 2 sinh2͑kH͒ Similarly, the mean Eulerian tracer equation derived from (5) becomes We have included the long-wave material tracer equation (analogous to the long-wave buoyancy equation if Ѩc Ѩc Ѩc buoyancy is taken to be a passive tracer), although it is ␧4 ϩ ␧4q ϩ ␧4w ѨT ␣ ѨX Ѩz not asymptotically coupled to the long-wave or current ␣ Ѩ Ѩ equations. It contains a vertical wave diffusion term ϭϪ␧4 ͗ w w͘ Ϫ ␧2 ͗ w w͘ ͑ ͒ Ѩ c q␣ Ѩ c w . 37 that vanishes after long time averaging and thus does X␣ z not appear in the corresponding current equation (19). 3 w w Once the quasi-static component is removed (as was The lowest order in (34) is actually O(␧ )as͗q␣w ͘ϳ done to the current equations above), it is apparent that O(␧2). Likewise, ͗cwww͘ϳO(␧2), and thus (37) occurs the long waves are barotropic to lowest order, obeying at O(␧4). We keep this notation, however, to indicate how many wave terms need to be calculated in order to Ѩqlw 1 evaluate these quantities. ␨˜lw ϭ 0 and ١ ϩ Ѩ␶ L X Following MRL04, we consider the horizontal mo- ␧5 Ѩ␨˜ lw mentum equations up to O( ), the vertical momentum -lw͒ ϭϪ lw ͑ ͒ equation up to O(␧3), and the mass and tracer equa ͑ ١ ϩ Ѩ␶ X · Hq LF , 33 tions to their lowest respective orders. The boundary where conditions are the same as given for the vortex-force representation (20)–(21). lw Equating the wave forcing in the radiation-stress rep- 1 Ѩ␨ˆ -Tw͒† ϩ ͬ resentation (34)–(37) with that in the vortex-force rep͑ · ١ͫ F lw ϭ L X Ѩ␶ resentation (16)–(19), we can see that 1 |A|2␴k † Ѩ |A|2k † ͮ ͬ ͫ Ϫ ͬ ͫ ١ͭ ϭ X · . Ѩ 2L k tanh͑kH͒ Ѩ␶ sinh͑2kH͒ Ϫ Ϫ qwqw͘ ϩ ␧ 2 ͗qwwwͪ͗͘ · Ϫ␧ 2ͩ١ X Ѩz Accordingly, to lowest order, the only effects of the Ϫ2 ␧ K ϩ K ͒ ϩ J. ͑38͒͑ ϭϪ١ waves on the long waves are the wave setup and the X 1 2 influence of the waves on the dynamic surface eleva- Likewise, comparing the two representations of the tion; the O(1) vertically integrated momentum equa- vertical momentum equations leads to tion is only affected by the rise and fall of the sea level due to the long waves themselves. Ѩ Ѩ w w͘ ϩ ␧Ϫ2 ͗ w2ͪ͘ ϭϪ ͑␧Ϫ2 ϩ ͒ ϩ ͗ Ϫͩ١ X · w q Ѩ w Ѩ K 1 K 2 K. b. Comparison of the equations in the different z z representations ͑39͒ Given the wave decomposition, current scalings, and nondimensionalization in section 2b, the radiation- Thus, the lowest-order radiation-stress divergence stress representation mean Eulerian equations derived terms balance the quasi-static pressure gradient. We from (9) and (4) are also see that for the tracers

Ѩq Ѩq Ѩq Ѩ͗p͘ w w St ١ ͘ ϭ ͗ ١ 5 3 ␣ 5 ␣ 5 ␣ 5 ␧ ϩ ␧ q␤ ϩ ␧ w ϩ ␧ ϩ ␧ ͑fz ϫ q͒␣ · c u u · c; ѨT ѨX␤ Ѩz ѨX␣ Ѩ Ѩ ϭϪ␧3 ͗ w w͘ Ϫ ␧ ͗ w w͘ ͑ ͒ that is, divergence of the wave tracer flux is equal to Ѩ q␣q␤ Ѩ q␣w , 34 X␤ z advection of the mean tracer by the three-dimensional Stokes velocity. Ѩw Ѩw Ѩw Ѩ͗p͘ ␧3b ␧7 ϩ ␧7 ϩ ␧7 ϩ ␧ Ϫ If, instead of averaging over time scales characteristic Ѩ q␤ Ѩ w Ѩ Ѩ T X␤ z z L of the currents, we average over long-wave scales, we Ѩ Ѩ 3 w w w2 obtain the following equation in the radiation-stress ϭϪ␧ ͗w q␤͘ Ϫ ␧ ͗w ͘, and ͑35͒ ѨX␤ Ѩz representation:

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Ѩqlw Ѩplw Ѩ where ͗⌻͘ is the average of the total velocity over the ␣ ϩ ϭϪ ͑ w w͒† q␣q␤ total sea depth and, as such, contains a wave compo- Ѩ␶ ѨX␣ ѨX␤ nent. It can likewise be interpreted as the Lagrangian Ѩ Ϫ2 w w † mean velocity integrated over the mean sea depth. In Ϫ ␧ ͑q␣w ͒ , Ѩz the same manner, ⌻w can be interpreted as a purely surface phenomenon occurring between the mean and Ѩplw Ѩ ϭϪ ͓͑ w͒2͔† fluctuating sea levels, or it can be interpreted as the Ѩ Ѩ w , and z z Stokes drift velocity integrated over the mean sea Ѩclw Ѩc Ѩc Ѩ depth. ϩ qlw ϩ wlw ϭϪ ͑cwqw͒† The asymptotic treatment in MRL04 naturally leads Ѩ␶ ␣ ѨX Ѩz ѨX ␣ ␣ ␣ to the flow being decomposed into primary waves, long Ѩ waves, and current. When using this decomposition, we Ϫ ␧Ϫ2 ͑cwww͒†, ͑40͒ Ѩz denote the total vertically integrated, time-averaged momentum as T. Thus, the decomposition given by T is with the mass equation and boundary conditions still ϭ c ϩ lw ϩ w ͑ ͒ w w † T T T T , 42 being given by (31). Similarly to the currents, (q␣w ) , c (cwww)† ϳ O(␧2), so these terms come in at order one. where ⌻ is as defined above and The radiation-stress representation, (40), is not clear ␨cϩ␨lw ␨cϩ␨lw as to the effect of the waves. In contrast, it is easy to Tlw ϭ ͵ qlw dz ϩ ͵ qc dz, c identify the quasi-static pressure and wave setup in ϪH ␨ (31). With the quasi-static components removed the E vortex-force representation (33) makes it readily appar- Tw ϭ ͵ Q dz. ␨cϩ␨lw ent that the long waves are barotropic. Furthermore, the wave effects on the tracer equation are identified as For LHS, the radiation stress S␣␤ is the excess mo- Stokes advection and vertical wave diffusion. mentum flux in the presence of waves. A slightly different definition of radiation stress is used in H71, rad wherein it is denoted by ␶ ␣␤ . The main difference is the 4. Vertically integrated, time-averaged equations sign, but there are several other subtle differences. These are enumerated in appendix A. Radiation stress It is in the context of depth-integrated circulation can be further decomposed into an interaction stress int equations that the concept of radiation stress was con- ␶ ␣␤, which acts on the bulk of the fluid, and a surface sl ceived and studied by LHS. Here we derive the trans- layer stress ␶ ␣␤. These three stresses are related by port equations using the vortex-force and radiation- ␶ rad ϭ ␶ int ϩ ␶ sl stress representations. We use the approach adopted in ␣␤ ␣␤ ␣␤, H71 of separating the total transport ͗⌻͘ into ⌻c and where ⌻w , the corresponding current and wave terms. We ␨c compare our results with H71, given our scaling as- ␶ int ϭϪ͵ ͑͗ w w͘ ϩ ␦ w͒ ␣␤ q␣q␤ ␣␤ p˜ dz and sumptions. ϪH E ␶ sl ϭϪ ͵ ͑ ϩ ␦ ͒ ␣␤ ͳ Q␣Q␤ ␣␤ p dzʹ. a. Transport ␨c In H71 the transport is decomposed into three quan- The wave effects on the evolution of ͗⌻͘ are obtained ͗⌻͘ tities—the total transport (this is M in H71), the by taking the divergence of the radiation stress. In H71 ⌻c vertically integrated momentum of the mean flow it is shown that the divergence of the interaction stress m (M in H71), and the vertically integrated momentum accounts for the wave effects on the evolution of ⌻c. ⌻w w of the surface layer (M in H71). The three quan- Note that we use the term radiation-stress representa- tities are related by tion to refer to equations of either type. The mean current stress is defined in H71 to be E ␨c ͗T͘ ϭ ͳ͵ Q dzʹ ϭ ͵ q dz ␨c ␶ m ϭϪ͵ ͑ ϩ ␦ m͒ ϪH ϪH ␣␤ q␣q␤ ␣␤ p˜ dz. ϪH E ϩ ͳ͵ Q dzʹ ϭ Tc ϩ ͗Tw͘, ͑41͒ H71’s radiation stress and interaction stress can also be ␨c related via

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X related to numerical or field data and have a clear or- ␶ ϭϪ ͑ ϩ ␦ ͒ Ϫ ␶ m ␣␤ ͳ͵ Q␣Q␤ ␣␤ p dzʹ ␣␤, dering within our scaling regime. ϪH where X ϭ E for the radiation stress and X ϭ ␨c for the b. Vortex-force representation interaction stress. Because our vortex-force representation gives equa- The dynamic mean pressure is decomposed into a tions for mean Eulerian quantities, it makes sense to wave contribution p˜ w and a current contribution p˜ m in compare the evolution equation of ⌻c in each of the H71. This is done by requiring that p˜ m be the dynamic frameworks; ⌻c represents the mean Eulerian velocity pressure in the absence of waves. We differ from H71 in integrated over the mean ocean depth. As such, the this regard. We decompose the mean pressure into vortex force and Bernoulli head are most immediately quasi-static (owing to the waves) pˆ ϩ␧2p˜ c, and dynamic comparable to the divergence of H71’s interaction mean pressure p˜ c, as defined in section 3. Although stress. these two decompositions are defined differently, in the We obtain the time evolution equation for ⌻c in vor- limit of deep water and low Rossby number they are tex-force representation by vertically integrating (26) similar. Also, as discussed in appendix A, the quasi- and making use of the quasi-static pressure relationship static and dynamic mean pressure used here are easily (27):

c 0 c 0 0 ѨT␣ Ѩ Ѩp˜ ͑0͒ ѨK˜ ͑0͒ Ѩ ϭϪ ͵ Ϫ Ϫ ͑ ϫ c͒ ϩ ͫ ͑ ͒͗ St͑ ͒͘ Ϫ 2 ϩ ͵ ͩ ϩ ͵ Јͪ ͬ Ѩ Ѩ q␣q␤ dz H Ѩ f z T ␣ q␣ 0 w 0 H Ѩ J␣ Ѩ Kdz dz , T X␤ ϪH X␣ X␣ ϪH X␣ z ͑43͒ where the wave terms are given by (22) or (30). The c. Comparison of the different representations effect of the waves can be broken into three parts. The first term in the square brackets is a momentum trans- We find the evolution equation for ⌻c in radiation- fer owing to mass influx from the surface layer, the stress representation by vertically integrating the phase- second term is a Bernoulli head evaluated at the mean averaged (34) over the mean vertical depth [ϪH, 0]. surface, and the third term is a vertically integrated Assuming the scaling given in section 2b, the evolu- vortex force. tion equation for ⌻c reads

ѨT c Ѩ␶m ѨH Ѩ␶int Ѩ␨ ѨH ␣ ϭ ␣␤ ϩ c͑Ϫ ͒ Ϫ ͑ ϫ c͒ ϩ ͭ␧Ϫ2 ␣␤ ϩ ͓͗ w͑ ͒ w͑ ͒͘ ϩ ␦ ͑ ͔͒ ϩ ␧Ϫ2 ͑Ϫ ͒ p H f z T ␣ q␣ 0 q␤ 0 ␣␤ pˆ 0 pˆ H ѨT ѨX␤ ѨX␣ ѨX␤ ѨX␤ ѨX␣

Ϫ ␧Ϫ4͗ w͑ ͒ w͑ ͒͘ Ϫ ͗ lw͑ ͒ lw͑ ͒͘ ϩ ͑ ͒͗ St͑ ͒ͮ͘ ͑ ͒ q␣ 0 w 0 q␣ 0 w 0 q␣ 0 w 0 . 44 where Ѩ c Ѩ␶m Ѩ T ␣ ␣␤ c H c ϭ ϩ p ͑ϪH͒ Ϫ f͑z ϫ T ͒␣ 0 ѨT ѨX␤ ѨX␣ ␶ m ϭϪ͵ ͑q q ϩ ␦ pc͒ dz ␣␤ ␣ ␤ ␣␤ Ѩ 0 ϪH ϩ ͑ ͒͗ St͑ ͒͘ Ϫ ͵ ˜ Ј ͫq␣ 0 w 0 dz K 2 ѨX␣ and ϪH ѨH 0 0 ϩ ˜ ͑Ϫ ͒ ϩ ͵ Ј ͑ ͒ H J␣ dz ͬ, 45 int w w 2 lw lw K 2 ␶ ϭϪ͵ ͑͗ ͘ ϩ ␧ ͗ ͘ ϩ ␦ ͒ ѨX␣ ␣␤ q␣q␤ q␣ q␤ ␣␤ pˆ dz. ϪH ϪH

w w 2 Note that ͗q␣(0)w (0)͘ is O(␧ ). Thus, the lowest-order retaining the lowest-order terms. terms are O(␧Ϫ2). The wave effects appearing within the square brack- To facilitate comparison with (44) we rewrite (43) as ets on the right-hand sides of (44) and (45) are equal:

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Ѩ␶int Ѩ␨ Ѩ Ϫ2 ␣␤ w w Ϫ2 H Ϫ4 w w lw lw St ␧ ϩ ͓͗q␣͑0͒q␤͑0͒͘ ϩ ␦␣␤ pˆ͑0͔͒ ϩ ␧ pˆ ͑ϪH͒ Ϫ ␧ ͗q␣͑0͒w ͑0͒͘ Ϫ ͗q␣ ͑0͒w ͑0͒͘ ϩ q␣͑0͒͗w ͑0͒͘ ѨX␤ ѨX␤ ѨX␣ Ѩ 0 ѨH 0 ϭ ͑ ͒͗ St͑ ͒͘ Ϫ ͵ ˜ Ј ϩ ˜ ͑Ϫ ͒ ϩ ͵ Ј ͑ ͒ q␣ 0 w 0 Ѩ K 2 dz K 2 H Ѩ J␣ dz , 46 X␣ ϪH X␣ ϪH

w w 2 recalling ͗q␣(0)w (0)͘ϳO(␧ ). The left-hand side is force representation. We derive wave-averaged energy the radiation-stress representation and the right-hand conversion terms for the quasi-static response due to side is the vortex-force representation. The last term on pressure and sea level changes (i.e., wave setup), the the left-hand side and the first term on the right-hand long waves, and the currents. In each case we respect side of (46) cancel trivially. The three terms on the the asymptotic scaling by not including terms at higher left-hand side with order lower than one cancel, giving order than the magnitude of the energy changes for the an order-one residual. In the vortex-force representa- wave-averaged components. This is only a partial view tion, however, pˆ and K 1 cancel exactly, leaving no of the total energetics since the sum of the component lower order terms. energies is not the total energy, even if the energy of the We may also compare (45) with the equation derived primary waves is included (i.e., there are cross terms by Garrett (1976) for wave effects on currents in among the components in the total energy balance). deep water [his (3.9) and (3.11), noting the erratum that Only energy exchange through conservative mecha- ϫ ١ w ϩ w ϫ ١ Ϫ (3.11) should read q · T T ( q) nisms is considered. when given in our notation]. Garrett does not include We define the quasi-static response, pˆ(z) and ␨ˆ,asthe rotational effects, assumes the mean current is indepen- sum of the contributions from the long waves (MRL04, dent of depth, and, because of the deep water assump- section 6) and currents [(23)] that are quasi-statically tion, includes no topographic variation. Thus, (45) be- balanced with the wave-averaged terms that do not in- comes clude long-wave velocity, currents, or Stokes drift. Given this definition, the only contribution to a quasi- c m 0 s ѨT␣ Ѩ␶␣␤ Ѩ static energy E is through the sea level potential en- ϭ ϩ ͑ ͒͗ St͑ ͒͘ Ϫ ͵ ˜ Ј Ѩ Ѩ q␣ 0 w 0 Ѩ K 2 dz ergy, T X␤ X␣ ϪH

0 ϩ ͵ Ј g J dz s 2 ␣ E ϭ ͵͵ ␨ˆ dx. ϪH 2 G Ѩ␶m ͒ ϫ ١͑ w ϩ w ϫ ١ ϭ ␣␤ Ϫ Ѩ q␣ · T T q X␤ The associated energy balance equation has a forcing Ѩ 0 equal to that of the area-integrated time derivative of Ϫ ͵ K˜ dzЈ. ͑47͒ the diagnostic forcing terms for ␨ˆ . ѨX 2 2 ␣ ϪH The long-wave dynamics (MRL04, their section 6) imply the following energy balance: The only difference between this and Garrett’s equation is the pressure adjustment involving K˜ , but as 2 dE lw d H g noted earlier the pressure decomposition used by H71 ϭ ͵͵ͫ |qlw|2 ϩ ͑␨˜ lw͒2ͬ dx (and also by Garrett) is not the same one used here; dt dt 2 2 ˜ G furthermore, K 2 is higher order than what Garrett evaluated. Taking this difference into account the two ϭϪ͵͵ ␨˜ lw˜ lw dx, ͑48͒ equations concur. Thus, the equations derived in F MRL04 are a more general version of what is derived in G Garrett (1976). excluding the quasi-static component. We derive the energy balance for the current, noting d. Energetics that both the three-dimensional u and uSt (augmented In H71 an equation is derived for the evolution of the with the vertical Stokes pseudovelocity) are nondiver- current energy that includes the wave contribution. We gent. repeat this derivation within the context of the vortex- The energy equation associated with the current is

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dEc d 0 1 0 ϭ ͵͵͵ ͩ |q|2 Ϫ zbcͪ dz dx ϭ ͵͵͵ ͕2⍀ · ͑uc ϫ uSt͒ Ϫ u · ͓͑uc · ٌ͒uSt͔ Ϫ bcwSt͖ dz dx dt dt ϪH 2 ϪH G G 1 ϩ ͵͵ wSt͑0͒ͫ |q͑0͒|2 ϩ g␨˜c ϩ P˜ ϩ K˜ ͑0͒ͬ dx, ͑49͒ 2 0 G assuming appropriate lateral boundary conditions that here they involve the Stokes drift. The surface conver- preclude energy flux. The first term on the right-hand sion terms are unfamiliar but appear to be due to mass side is due to the Coriolis effect. The second and third influx from the waves. conversion terms are analogous to familiar shear pro- The energy equation associated with the currents can duction and potential energy conversion terms except also be written as

dEc 0 1 ͬ͒ ͑ ˜ ϫ ͒ ͑ ϫ St͒ Ϫ c St͔ ϩ ͵͵ St͑ ͒ ϫ ͫ | ͑ ͒|2 ϩ ␨˜c ϩ ˜ ϩ ١ ϭ ͵͵͵ ͓͑ ⍀ ϩ 2 u · u u b w dz dx w 0 q 0 g P0 K 0 dx. dt ϪH 2 G G ͑50͒

This equation shows that, if the current and the Stokes For comparison, and to illustrate the relationship be- drift are parallel or antiparallel, the first term vanishes tween our equations and those given in H71, we also and only the potential energy conversion term and the calculate the energy equation for the currents in the surface conversion term remain. radiation-stress representation; namely,

ѨE c |q|2 0 ١ ϩ ␧Ϫ2͗ w w͘ ١ ϭ ͵͵ Ϫ ͑ ͒ͫ ϩ ␧Ϫ2͑͗ ͘ ϩ ͗ w2͒ͬ͘ Ϫ ␧Ϫ2 ͑ ͒ ͗ w w| ͘ ϩ ͵ ͗ w w͘ ͭ w 0 p w q 0 · q w w q · w q␣q · q␣ Ѩ 0 x x t 2 ϪH G Ѩ Ѩ Ϫ w Ϫ q Ϫ ϩ ␧ 2͗ww2͘ ϩ ␧ 4͗qwww͘ · Ϫ ␧ 2͗wwbw͘ dzͮ dx. ͑51͒ Ѩz Ѩz

This is essentially (22) in H71 given the assumption of term is the potential energy conversion term, which is the scaling described in section 2b. The terms up to the the same as in (49). plus sign before the single integral are the work done by A comparison between (49) and (51) [noting wSt(0) the surface interaction stress. The next four terms make ϭϪw(0)] shows that up what H71 called the “dissipation” term. The final

0 0 ͵͵͵ St͔͖ Ϫ ͵͵ ͑ ͓͒ ␨˜c ϩ ˜ ϩ ˜ ͑ ͔͒ ϭ ١͒ ⍀ ͑ c ϫ St͒ Ϫ ͓͑ c ͕ ͵͵͵ 2 · u u u · u · u dz dx w 0 g P0 K 2 0 dx ϪH ϪH G G G Ѩw Ѩq ͪ ϩ ␧Ϫ2͗ w2͘ ϩ ␧Ϫ4͗ w w͘ ١ ϩ ␧Ϫ2͗ w w͘ ١ ϫ ͩ͗ w w͘ w q · w q␣q · q␣ w q w · dz dx x x Ѩz Ѩz

Ϫ ␧Ϫ2͵͵ ͑ ͒͑͗ ͘ ϩ ͗ w2͒͘ ϩ ͑ ͒ ͗ w w| ͘ ͑ ͒ w 0 p w zϭ0 q 0 · q w 0 dx. 52

The scaling shows that the lowest-order terms are re- the vortex-force representation, giving a clear picture of lated to the quasi-static pressure. These cancel out in the effect of the waves on the energy of the mean flow.

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FIG. 1. A comparison of radiation stress and vortex force for x momentum equation for ␧ϭ0.04. All units are 10Ϫ7 meters per second squared; dashed lines are negative contours, filled lines are positive. (a) Gradient of radiation stress divided by depth, (b) gradient of wave setup, (c) vortex force excluding Coriolis force, and (d) vortex force including Coriolis force.

5. Illustrative examples The radiation stress clearly does not capture these effects to its lowest order. These examples show how the In MRL04 (their section 13) we show how a baro- mean wave effects (exemplified by the vortex force) are tropic vortical current and waves interact on a finite- tied to the vorticity of the currents. In the absence of depth water sloped basin for ␧ϭ0.04. While a great Coriolis force the wave effect is basically the advection many simplifications went into producing that example, of the vorticity by the Stokes drift. When planetary the vortex-force representation makes it very easy to rotation is included in the model, the vortex force in- identify which aspects of the interaction are consequen- cludes a sort of Stokes–Coriolis term (Huang 1979). In tial for the current evolution. In Fig. 1 we illustrate the this sense the vortex-force representation might be seen differences between the two representations by show- as concisely identifying the wave effects on the cur- ing the radiation-stress and vortex-force terms that rents. By comparison, the radiation-stress representa- would appear in the x-momentum equation. The gradi- tion may incorporate a variety of phenomena, including ent of the radiation stress divided by the depth is shown wave effects on waves as well as currents, and it does in Fig. 1a. If we compare this with Fig. 1b, the gradient not distinguish between the Bernoulli head and the vor- of the wave setup due to the lowest-order Bernoulli tex force and to lowest order cancels out with the quasi- head, we can see that these quantities cancel each other static wave setup due to the Bernoulli head. at lowest order in the steady-state case [cf. (53)]. By In what follows we consider two other examples that contrast, the vortex force is shown in Fig. 1c (excluding highlight important consequences related to differences Coriolis force) and Fig. 1d (including Coriolis force). between the radiation-stress and vortex-force represen-

Unauthenticated | Downloaded 09/28/21 12:48 AM UTC 1134 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 37 tations as well as issues related to the relative asymp- Like the radiation stress, the divergence of the interac- totic balances. Complementary commentary will be fol- tion stress is equal to the time evolution of the ͗Tw͘ plus lowed in the discussion section. boundary terms. The evolution equation for Tc (44), does not involve the time evolution of ͗Tw͘. Using wave a. Lowest-order radiation and interaction stress quantities we may also evaluate To illustrate the role played by the lowest-order radiation and interaction stresses in the vertically in- Ѩ͗ w͘ Ѩ͗| |2͘ Ϫ2 w w T ␣ 1 A 2 tegrated momentum equations we consider their di- Ϫ␧ ͗q␣͑0͒w ͑0͒͘ ϭϪ Ϫ ϩ O͑␧ ͒. ѨT 4L ѨX␣ vergences. The divergence of LHS’s radiation stress (A5) is ͑56͒

w ѨS␣␤ Ѩ͗T ␣͘ H Ѩ␨ˆ ϭϪ Ϫ ϩ O͑␧2͒, ͑53͒ Substituting (55) and (56) into (44) shows that the low- ѨX␤ ѨT L ѨX␣ est-order divergence of the interaction stress cancels ͗ w͘ where T ␣ is defined in section 4a and makes use of the out with other boundary terms, leaving only higher or- wave evolution equations in MRL04 [(5.22) and (5.37)]. der effects. A similar result is obtained when H71’s radiation stress If we account for the long waves separately, there is ␶ rad ␣␤ is used (appendix A). Thus, the dominant dynamic an interaction stress associated with them to lowest or- role of the radiation-stress divergence is in balancing der. Its only effect is the static pressure adjustment (i.e., the evolution of the wave momentum, while its static wave setup). Once the static component is removed, effect is balancing the wave setup, as seen in Fig. 1. there is no other contribution by the interaction stress To lowest order the interaction stress is to the long-wave dynamics. If we consider the com- 2 |A| k␣k␤ 2kH bined effect of the long waves and the primary gravity ␶ int ϭϪͳ ͫ ϩ ͬʹ ␣␤ 1 4k2L sinh͑2kH͒ waves on the current, the long waves do not contribute to either stress at lowest order. Their effects are only |A|2 seen in the stresses at higher orders. ϩ ␦ ͩͳ ʹ ϩ ͪ ͑ ͒ c ␣␤ P H . 54 -T may be deץ/ Tץ ,(4L a As first done by Garrett (1976 rived as the difference, The divergence of (54) is Ѩ␶ int Ѩ͗T w͘ Ѩ͗|A|2͘ Ѩ ␣␤ ␣ 1 H 2 Ѩ րѨ Ϫ Ѩ w րѨ ϭ ϩ Ϫ pˆ͑ϪH͒ ϩ O͑␧ ͒. ͗T͘ T ͗T ͘ T. ѨX␤ ѨT 4L ѨX␣ ѨX␣ ͑55͒ From (B5) and (53),

ѨTc 1 Ѩ 2͒ Ϫ ϫ w Ϫ ٌ ␨ˆ2 Ϫ ͓ ͑ ͒ w ϩ ͑ ͒ w͔͑ ١ Ϫ ͔ ١ ͒ ␶m Ϫ ϫ c ϩ c͑Ϫ ١͓ ϭ X · fzˆ T p H XH X · S fz T X q 0 T ␤ q␤ 0 T , ѨT 2L ѨX␤ ͑57͒

where S(2) is the O(␧2) radiation stress as calculated ments closely tied to the vortex force, we now analyze with our scalings in (A6) and (A7). As in (43), only the a problem with certain nonlinear effects suppressed. O(␧2) wave components are required for the evaluation Specifically, we step outside the asymptotic scaling (and of (57). However, the challenge of calculating higher- accordingly revert to dimensional equations) and form order radiation-stress terms persists. Furthermore, the a composite of the preceding long-wave and current result is simply that given by the vortex-force represen- dynamical balances without the quadratic terms related tation as seen by comparison of (57) with (45). to Eulerian and Stokes advection. This is a type of linearized dynamics about a rotating, stratified, resting b. Linear, rotating, stratified wave-averaged state with B(z) its buoyancy stratification. After remov- dynamics ing quasi-static sea level ␨ˆ and pressure pˆ fields (with ␳ ͐ To further expose what is forced by waves in the the latter augmented by o Bdz), the wave-averaged radiation-stress representation, apart from the ele- equations are

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Ѩq 1 ϩ ϫ ϩ ٌ ϭ Ϫ ϫ St Ѩ fz q ␳ p fz v , t o 1 Ѩp Ϫ ϭ 0, ␳ Ѩ b o z Ѩw ,q ϩ ϭ 0 · ١ Ѩz Ѩb ѨB ѨB 1 Ѩ Ѩ ѨB ϩ w ϭ ϪwSt ϩ ͫͩ e2ͪ ͬ, Ѩt Ѩz Ѩz 2 Ѩz Ѩt Ѩz

,١H ϭ 0 · w͑ϪH͒ ϩ q͑ϪH͒ Ѩ␨ Ѩ␨ˆ Tw ϩ , and · ١ w͑0͒ Ϫ ϭ Ѩt Ѩt p͑0͒ Ϫ g␨ ϭ 0. ͑58͒

The notation should be interpretable from the previous Strikingly, this system has no wave forcing. For con- sections; e2 is defined in (32). The wave-averaged ef- stant f its solutions are of the form fects appear on the right-hand side and thus serve as forcing terms for the long-wave and current responses. 1 0 Ј Ј͒ ϭ ϫ ٌ͑ ␨ Ϫ Ј͒ ͑ When the wave fields vary in time on an intermediate q , w ͫz ␳ g ͵ bdz ,0ͬ and f 0 z time scale (previously denoted by ␶), the principal re- .١H ϭ 0 · qЈ͑ϪH͒ sponse in (58) is forced propagating waves. The low- frequency wave behavior implicit in the left-hand side These are determined by the boundary conditions of are a combination of long, shallow-water surface grav- the problem. ity waves, internal waves on the stratification B(z), and Thus, the only steady, wave-forced current is the rotational waves due to f. anti-Stokes flow represented by the first components in To focus instead on the long-time response, we as- (59). Note that this local anti-Stokes flow is linked with ϭ ץ sume that the wave field is in steady state and set t finite rotation and stratification. In the special situation 0 in (58). We further decompose the steady velocity where f ϭ B ϭ 0, the equations are essentially those field by given by Bühler and McIntyre (2003). In this case q is z barotropic and, assuming irrotationality for both waves vSt͑zЈ͒ dzЈ ϩ wЈ. and currents, the steady state of (58) is found through ͵ · ١ q ϭϪvSt͑z͒ ϩ qЈ, w ϭ ϪH ,T w · Hq͒ ϭϪ١͑ · ١ 59͒͑ Ј Ј The resulting equations for (q , w ) are the following: which is a nonlocal form of anti-Stokes flow. This in not the generic case, however, since the existence of even 1 -ϭ 0, very small nonzero values of f or B breaks the irrota ١ ϫ Ј ϩ fz q ␳ p o tionality assumption, reverting to the local steady-state 1 Ѩp solution. Ϫ ϭ 0, ␳ Ѩ b o z In summary, the linearized dynamical response to the ѨwЈ conservative wave effects that are contained in the ra- qЈ ϩ ϭ 0, diation-stress representation, not including the vortex · ١ Ѩz force, is limited to the following classes of phenomena: Ѩ B wave setup, forced propagating (long) waves, and wЈ ϭ 0, Ѩz steady anti-Stokes flow. Only the last of these is a “cur- ١H ϭ 0, rent” in the sense of a flow with a slow evolutionary · wЈ͑ϪH͒ ϩ qЈ͑ϪH͒ time scale. Of course, in a more general situation nei- wЈ͑0͒ ϭ 0, and ther the waves nor the currents are steady in time and p͑0͒ Ϫ g␨ ϭ 0. ͑60͒ advective dynamics are not negligible, so the currents

Unauthenticated | Downloaded 09/28/21 12:48 AM UTC 1136 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 37 will exhibit a broader range of wave-influenced behav- b. An alternative scaling ior than anti-Stokes flow. Although we have concentrated on the specific scal- Though certain nonlinear effects and pressure adjust- ing used in MRL04 in this paper, the applicability of the ments have been suppressed in this example, the vor- vortex-force representation is wider. To see this we re- tex-force representation makes it clear that the crucial vert to a more general scaling, using ␦ and ␤ as defined aspects of wave effects on currents is complete: see (38) in section 2b. We assume that the waves are irrotational and (39). Both representations address the same phys- to lowest order, ␧ K 1, and that their phase-averaged ics; however, it is less obvious from the divergence of a properties only vary slowly; that is, ␤ K 1 [N.B., ␦ ϭ radiation stress what aspects of the wave/current inter- O(␧) and ␤ ϭ O(␧2), with MRL04 scaling]. action are important. Moreover, if the leading-order The wave vorticity comes from either the rotational radiation or interaction stress does not include the vor- currents or planetary rotation. If the currents are the tex force, it captures only a limited part of the wave- source of vorticity then, to lowest order, averaged dynamical effects. This could be the case if our scaling is applicable to the physical situation at t ␰c ϫ ͵ w Ј͒ ١͑ ␰w ϭ ␧␦ ϫ hand. z xz · q dt and

t ␰c͵ w Ј͒͑ ١␦␹w ϭϪ␧ x · w dt , 6. Discussion a. Asymptotic consistency where ␰ and ␹ are the horizontal vector and vertical vorticities, respectively. The vortex-force term is given Although both radiation-stress and vortex-force rep- by ␧͗(␰ w, ␹w) ϫ uw͘—the wave-averaged cross product resentations give valid expressions for the conservative of the wave vorticity and the wave velocity. Because the effects of waves on currents, the vortex force and the wave vorticity and velocity are in quadrature, the aver- Bernoulli head provide a simpler and more specific age of the horizontal component of the vortex force is physical interpretation than the divergence of the ra- demoted from its apparent order by a factor of ␩ [N.B., diation or interaction stress. Furthermore, comparisons in MRL04, ␩ ϭ O(␧2)]. This is also the nondimensional with the vortex-force representation in (38), (39), and order of the term ͗qw ww͘ that appears in the radiation (46) show that, in both the three-dimensional and ver- stress. If it is planetary rotation that provides the vor- tically integrated current equations, the lowest-order ticity, then stress divergence simply balances the quasi-static pressure gradient; it does not include vortex-force effects. t ϫ nd ϫ w Ј ␰w ␹w͒ ϭϪ١͑ The same is true in the current energy equation in (52). , ͩf z ͵ q dt ͪ, The vortex-force effects must be evaluated as the higher-order difference between stress divergence and with the vertical component demoted by ␩. boundary terms. In this way it can be said that the Consider the order of the vortex-force and radiation- radiation-stress representation is asymptotically incon- stress (and thus also interaction stress) terms in the sistent. horizontal and vertical directions. In the horizontal ve- To obtain meaningful results from the radiation- locity equation, the vortex force enters at O[max(␧2␦␩, stress representation in this scaling, the primary waves ␧f nd)], the Bernoulli head at O(␧␤), and the radiation need to be calculated up to O(␧4). In the vortex-force stress at O[max(␧␤, ␧␩)]. In the vertical direction the representation we circumvent this requirement by cal- orders are O[max(␧2␦, ␧f nd␩] for the vortex force, O(␧) culating the wave vorticity to O(␧2) and making use of for the Bernoulli head, and O[max(␧, ␧␤␩) ϭ␧] for the the irrotationality to O(␧2). Although there are other radiation stress. From this we can see that, if the current ways to circumvent this requirement [e.g., see (57)], the strength is similar to or weaker than the waves (i.e., ␦ Յ asymptotic analysis has already been done in MRL04. 1) and planetary vorticity is weak (i.e., f nd K ␧2), then In contrast, the long-wave equations do not include the radiation stress is asymptotically inconsistent with the vortex force; thus the radiation-stress representa- respect to the vortex force. As with the MRL04 scaling, tion does capture the leading-order dynamics [see (33); its lowest-order divergence is the wave setup and long- cf. (40)], but its usage is still potentially confusing. The wave forcing, but higher orders are needed to capture vortex-force representation clearly separates the quasi- the vortex force. Only if either the currents are stronger static and dynamic response components, making it ob- than the waves (e.g., ␦ ϭ 1/␧) or the Rossby number is vious that the long-wave response is barotropic. This is large (f nd Ն ␧2) do the radiation stress and vortex force not as evident using the radiation-stress representation. enter the wave-averaged equations at the same order.

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In the case f nd ϳ O(␧2), the currents obey (58). For ␦ ϭ significant advective and Coriolis forces. Although we 1/␧ the current enters into the wave dispersion relation give these results for a specific asymptotic regime, sec- at its leading order and the waves are no longer irrota- tion 6 suggests that unless the currents are strong com- tional at any order. Nevertheless, even in this regime, pared to the wave orbital velocities—the opposite of which is atypical for the ocean, the vortex-force repre- typical shelf conditions—the lowest-order radiation or sentation retains its advantage of distinguishing be- interaction stress divergence will not encompass the tween pressure adjustment and vortex force. vortex force. The divergence of the radiation or interaction stress and vortex force only come in at the same order in the case of strong currents (i.e., ␦ ϳ1/␧). 7. Conclusions Wave–current interactions have also been cast using This paper shows that the conservative effects of the generalized Lagrangian mean (GLM) formalism waves on currents can be characterized in terms of (Andrews and McIntyre 1978a,b; Gjaja and Holm 1996; Groeneweg and Klopman 1998), Lagrangian mean • quasi-static wave setup, (Weber 1983; Jenkins 1987), and other averaging vari- • long- (infragravity) wave forcing, ants of a Lagrangian flavor (Mellor 2003, 2005). (Mellor • mass and other material transport by Stokes drift, and only follows the vertical wave motions. Furthermore, • vortex force and other asymptotically comparable his work is not completely asymptotically consistent.) momentum effects. In MRL04, we chose the Eulerian frame because of the Both the radiation-stress and the vortex-force repre- conceptual simplicity of the Eulerian mean and because sentations encompass all of these effects. In the radia- Eulerian means are usually more practical in relating to tion-stress representation the vortex force may be hid- oceanic measurements and large-scale oceanic numeri- den by lower-order effects. The vortex-force represen- cal models (e.g., pressure). While a detailed comparison tation, however, cleanly decomposes the physics into a between frameworks is beyond the scope of this paper, Bernoulli head and a vortex force. The Bernoulli head an essential start is a clear understanding of how aver- is a pressure adjustment owing to the wave effects as- ages in the Eulerian and the Lagrangian frames are to sociated with the well-known wave setup effect. The be compared. The GLM velocity is equivalent to the combination of transient mass transport and wave setup Eulerian mean velocity plus a residual velocity, often forces long waves. The vortex force is an interaction identified as the Stokes velocity. Thus, the GLM veloc- between the wave velocity and the wave vorticity. In ity may be recovered from the mean Eulerian velocity the asymptotic regime, the wave vorticity arises from to lowest order if desired. When the GLM equations wave advection of the current vorticity, so the vortex are written in terms of the Eulerian mean velocity, the force is equal to the curl of the Stokes drift and the relationship to the vortex-force representation becomes current vorticity. Given an irrotational background cur- apparent. Some work is required to interpret terms rent, there would not be a vortex force (most ocean such as the GLM pressure in an Eulerian sense, how- currents have nonzero vorticity). Material properties in ever. In general, it is crucial to remember that 1) the the ocean, including the buoyancy, follow wave- choice of formalism has no bearing on the physics and averaged trajectories that move with the sum of the 2) a direct comparison of the results presented here to Eulerian and Stokes drift velocities (including the ver- those of a Lagrangian-framework-based derivation can tical Stokes pseudovelocity). By decomposing the wave only be made by the adoption of a common scaling. effects in this fashion, we are able to cleanly remove the These two points are obvious; nevertheless, they are at wave setup, revealing the underlying wave effects on the heart of the confusion with regard to the character- the long waves and currents that are obscured by the ization of stresses in wave–current interactions. In this radiation-stress representation. paper we adhere to these two principles. In doing so we The wave effects on the energy associated with the are able to make clear how the vortex-force and radia- current can be characterized by four terms: a Coriolis– tion-stress representations compare to each other. We Stokes drift term, Stokes drift terms analogous to the feel we have succeeded in this regard. well-known shear production and potential energy con- The vortex-force representation given here provides version terms, and a surface layer term. an interpretation of the effect of radiation or interac- In addition to lacking a meaningful physical decom- tion stress in conservative dynamics with nearly irrota- position, the radiation-stress representation suffers tional waves. Within this regime, the lowest order vor- from being asymptotically inconsistent in this scaling. ticity in the waves is provided by the currents and plan- The apparent order of the radiation stress is that of the etary vorticity. The effects of wave breaking would wave setup. The currents evolve at a higher order with need to be considered separately, as this is a different

Unauthenticated | Downloaded 09/28/21 12:48 AM UTC 1138 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 37 source of vorticity for the waves. Conventional practice Here Pc is taken to be the pressure of the current with- is to parameterize the effects of wave breaking. Off- out any wave effects; P and Pc are assumed to include shore wave breaking is often represented as surface the hydrostatic pressure. wind stress, though this approach has only limited va- A comparison between (A1) and (A2) suggests that w ϭ͗ ͘Ϫ lidity on the scale of the surface boundary layer com- p˜ P Pc. Thus, pared to modeling the breaking as stochastically dis- z tributed impulses (Sullivan et al. 2004). In the surf zone ˜ m ϭ ϩ m P Pc , i.e., p˜ the shoreward decrease of wave amplitude though ␧L breaking is a primary cause of littoral currents is the dynamical part of P . If we assume that ␧2␨ is the (Longuet-Higgins 1970) and the parameterization of c sea elevation in the absence of waves, then surf-zone breaking is a necessary element of the wave- averaged dynamics; this is often expressed as a radia- ␧␨ ␧␨ p˜ m͑␧2␨͒ ϭ P ͑␧2␨͒ ϩ ϭ P ϩ , ͑A3͒ tion-stress divergence. The conservative vortex force c L a L might also play a role in this region. Under certain circumstances this could be included in an additive way, as required by H71. although the use of asymptotically derived vortex force In addition to the sign difference between the two in the surf zone would be semiempirical at best. formulations, the expression in H71 includes the term

␧E Acknowledgments. This research is supported by the ͳ͵ ʹ Q␣Q␤ dz , National Science Foundation through Grant ␧2␨ DMS0327642 and the Office of Naval Research ␶ sl through Grant N00014-04-1-0166. JMR thanks PIMS in the surface layer stress, ␣␤, whereas LHS only in- and Simon Fraser University for their hospitality and clude the term support in this project. EML also thanks NCAR, the ␧E University of Canterbury, and the National Institute of ͳ͵ w w ʹ q␣q␤ dz . Water and Atmospheric Research (NIWA, Taihoro ␧2␨ Nukurangi), New Zealand, for their hospitality and This means that the radiation stress in H71 includes the support in this project. additional terms

APPENDIX A 2 w 2 w q␣͑␧ ␨͒T ␤ ϩ q␤͑␧ ␨͒T ␣.

Definitions of Radiation Stress In H71 these two terms are neglected when calculating ␶ sl As mentioned in section 4, Hasselmann (H71) iden- ␣␤ in H71’s (17a), as noted by Garrett (1976). Further- tifies his radiation stress with that of LHS with the ex- more, LHS use the total pressure when calculating the ception of a sign difference. However, other differences radiation stress, whereas H71 uses the dynamical pres- arise from the two definitions of radiation stress. sure. This does not make any difference in the bulk of Given our scalings the radiation stress in H71 is the fluid because only the pressure difference is calculated. In the surface layer this differences adds a term rad int sl ␶␣␤ ϭ ␶ ␣␤ ϩ ␶ ␣␤ ͗␨w2͘ ϩ ␧2͗␨ lw2͘ ␧2␨ 1 Ϫ ϭϪ ͗ w w ϩ ␧2 lw lw͘ ϩ ␦ w , ͵ ͩ q␣q␤ q␣ q␤ ␧ ␣␤ p˜ ͪ dz 2L ϪH to the LHS version of radiation stress. ␧E 1 Ϫ ϩ ␦ ͑ ͒ Thus, H71’s definition of radiation stress may be re- ͳ͵ ͩQ␣Q␤ ␧ ␣␤ pͪ dzʹ; A1 ␧2␨ lated to that of LHS by whereas LHS’s definition is rad 2 w 2 w ␶␣␤ ϭϪS␣␤ Ϫ q␣͑␧ ␨͒T ␤ Ϫ q␤͑␧ ␨͒T ␣ ␧2␨ 1 ϭ ͵ ͫ͗ w w ϩ ␧2 lw lw͘ ϩ ␦ ͑͗ ͘ Ϫ ͒ͬ w2 2 lw2 S␣␤ q␣q␤ q␣ q␤ ␧ ␣␤ P Pc dz ͗␨ ͘ ϩ ␧ ͗␨ ͘ ϪH Ϫ . ͑A4͒ 2L ␧E 1 Ϫ ͵ w w ϩ ␧2 lw lw ϩ ␦ ͳ ͩq␣q␤ q␣ q␤ ␣␤P ͪ dzʹ. c There are still difficulties that arise in the interpre- 2 ␧ ␧ ␨ m tation of the pressure. Respectively Pc and p˜ are de- ͑A2͒ fined as the pressure and dynamic pressure of the cur-

Unauthenticated | Downloaded 09/28/21 12:48 AM UTC MAY 2007 LANE ET AL. 1139 rent in the absence of waves. In fact, when LHS first To alleviate these problems and provide a consistent, defined radiation stress, they assumed that the fluid was computable interpretation of the radiation stress, we at rest when wave effects were not present; thus Pc was identify the wave mean pressure with pˆ, the static mean just the hydrostatic pressure. When currents were con- pressure balance from MRL04. The remaining part of sidered, they were simple ones. Likewise, the sea eleva- the mean pressure is pc, which we identify with the tion ␨ is defined as the sea elevation of the currents current mean pressure. Unlike p˜ m in H71, the surface without waves. While this may be appropriate when boundary condition on pc contains wave forcing terms. considering a specific current and asking what effects As stated in H71 the surface boundary condition on p˜ m waves would have on this current, at other times this is chosen to make it look like the usual pressure bound- approach is less convenient. Furthermore, wave setup ary condition rather than being derived from the wave- may significantly change the mean sea level (notably in averaged equations. We note that in the limit of deep shallow or finite depth water; this effect becomes insig- water and small Rossby number the wave forcing terms nificant in deep water). If the mean sea level is altered in the surface mean pressure boundary condition be- by the addition of waves, then (A3) is no longer gen- come negligible. erally true. Using these definitions we write the radiation stress as

␧2␨ ␨ˆ ␧E 1 ␧E Ϫ z ϭ ͫ͗ w w͘ ϩ ␧2͗ lw lw͘ ϩ ␦ ͩ Ϫ ͪͬ ϩ ͳ ͫ w w ϩ ␧2 lw lw ϩ ␦ ͩ Ϫ ͪͬ ʹ S␣␤ ͵ q␣q␤ q␣ q␤ ␣␤ pˆ dz ͵ q␣q␤ q␣ q␤ ␧ ␣␤ p ␧ dz , ϪH L ␧2␨ L which, in terms of linear gravity waves, is

2 2 |A| k␣k␤ 2kH |A| 2kH ϭ ͳ ͫ ϩ ͬʹ ϩ ␦ ͳ ʹ ϩ ␧2 2a ϩ ␧2␦ 2b ͑ ͒ S␣␤ 1 ␣␤ S␣␤ ␣␤S . A5 4k2L sinh͑2kH͒ 4L sinh͑2kH͒

We can see that, to lowest order, this agrees with LHS’s radiation stress calculated for linear gravity waves. The higher-order terms in (A5) are given by

4 |A| k␣k␤ 9 cosh͑2kH͒ 4kH 60 36 2a ϭ ͳ ͭ ͫ ϩ ͬ ϩ ϩ ͮʹ S␣␤ 1 56 64L sinh6͑kH͒ sinh͑4kH͒ sinh2͑kH͒ sinh4͑kH͒

0 z sinh͓2k͑zЈ ϩ H͔͒ 1 0 Ϫ ͳ| |2␴ ͩ͵ ͵ Ј Ϫ ͵ ͑ ͕͒ ͓ ͑ ϩ ͔͒ ϩ ͑ ͖͒ A k␣k␤ 2 2 dz dz V z cosh 2k z H cosh 2kz dz ϪH ϪH k sinh ͑kH͒ 2 ϪH

0 o 2 1 2kH 1 1 |A| k␣k␤ ϩ ͑Ϫ ͒ ͫ ϩ ͬͪʹ ϩ ͵ ͗␷St ϩ ␷ St ͘ ϩ ͵ ͗ lw lw͘ Ϫ ͳ ʹ V H 1 ͑ ͒ ␣ q␤ q␣ ␤ dz q␣ q␤ ␧ Pa ͑ ͒ 4 sinh 2kH 2 ϪH ϪH 2k tanh kH ͑A6͒

and

2 1 1 |A|2k ͗␨lw ͘ |A|2␴ 0 S2b ϭ ͫ P ϩ ͳ ʹͬ͗|A|2k ͑kH͒͘ ϩ Ϫ ͳ ͵ ͑z͒ ͑ kz͒ dz ʹ ␧ a ͑ ͒ tanh ͑ ͒ V sinh 2 2 4L sinh 2kH 2L sinh kH ϪH

|A|2␴V ͑ϪH͒ |A| 4k2 9 9 8 Ϫ ͳ ʹ ϩ ͳ ͫ ϩ Ϫ Ϫ 8ͬʹ. ͑A7͒ 2k tanh͑kH͒ 64L sinh6͑kH͒ sinh4͑kH͒ sinh2͑kH͒

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APPENDIX B cording to section 2b. This gives the following horizontal equation: Total Vertically Integrated Momentum Here ͗T͘ may be calculated by decomposing (12) into mean and wave components and then averaging ac-

Ѩ Ѩ ␨ ␨ E ␨ ͗ ͘ ϭϪ ϩ ͗ w w͘ ϩ Ϫ ͑ ϫ ͗ ͒͘ ϩ ͗ w w͘ Ѩ T␣ Ѩ ͩ͵ q␣ q␤ dz ͵ q␣q␤ dz ͳ͵ Q␣Q␤ dzʹͪ f zˆ T ␣ ͵ q␣q␤ dz t x␤ ϪH ϪH ␨ ϪH

2 1 Ѩ͑␨2 ϩ ͗␨w ͒͘ ѨH Ѩ ␨ E ϩ ϩ ͗ ͑Ϫ ͒͘ Ϫ ͗ ͘ ϩ ͑ ͒ Ѩ p H Ѩ Ѩ ͩ͵ p dz ͳ͵ pdzʹͪ. B1 2L x␣ x␣ x␣ ϪH ␨

The other vertically integrated equations are

Ѩ E ␨ Ѩ ␨ ␨ E ϭϪ ϩ ͗ ͑Ϫ ͒͘ Ϫ ϩ ͗ w w͘ ϩ Ѩ ͳ͵ Wdzʹ p H Ѩ ͩ͵ wq␤ dz ͵ w q␤ dz ͳ͵ WQ␤ dzʹͪ, t ϪH L x␤ ϪH ϪH ␨ Ѩ␨ Ѩ͗T ͘ Ѩ E Ѩ ␨ ␨ E ϩ ␣ ϭ ϭϪ ϩ ͗ w w͘ ϩ ͑ ͒ Ѩ Ѩ 0, and Ѩ ͳ͵ C dzʹ Ѩ ͩ͵ cq␤ dz ͵ c q␤ dz ͳ͵ CQ␤ dzʹͪ. B2 t x␣ t ϪH x␤ ϪH ϪH ␨

If we split the mean pressure into a static and a dynamic component, we can rewrite the mean vertically integrated momentum as

m 2 rad w2 Ѩ Ѩ␶ ␣␤ 1 Ѩ␨ ѨH Ѩ␶␣␤ 1 Ѩ͗␨ ͘ ѨH ͗ ͘ ϭ ϩ ϩ c͑Ϫ ͒ Ϫ ͑ ϫ c͒ ϩ ͫ ϩ ϩ ͑Ϫ ͒ Ϫ ͑ ϫ w͒ ͬ T␣ p H f z T ␣ pˆ H f z T ␣ , Ѩt Ѩx␤ 2L Ѩx␣ Ѩx␣ Ѩx␤ 2L Ѩx␣ Ѩx␣ ͑B3͒

m rad where ␶ and ␶␣␤ are defined in section 4a. Introducing the scalings from section 2b, (B3) becomes

m rad w2 2 Ѩ͗T␣͘ Ѩ␶ ␣␤ ѨH Ѩ␶␣␤ 1 Ѩ͗␨ ͘ ␧ Ѩ ѨH ϭ ϩ c Ϫ ͒ Ϫ ϫ c͒ ϩ ␧Ϫ2ͫ ϩ ϩ ␨ˆ2 ϩ Ϫ ͒ Ϫ ␧2 ϫ ͒ ͬ p ͑ H f͑z T ␣ pˆ͑ H f͑z T ␣ . ѨT ѨX␤ ѨX␣ ѨX␤ 2L ѨX␣ 2L ѨX␣ ѨX␣ ͑B4͒

As is the case with the equation for Tc, the wave terms on the O(␧2) difference between two O(1) quantities. at O(␧Ϫ2) cancel, leaving an order-one residual. This Written using LHS’s version of radiation stress, (B4) may make practical evaluation difficult since it depends becomes

Ѩ͗ ͘ Ѩ␶ m Ѩ ␧2␨ˆ ϩ Ѩ␨ˆ Ѩ T␣ ␣␤ c H c H w Ϫ2 w w ϭ ͫ ϩ ͑Ϫ ͒ Ϫ ͑ ϫ ͒ ͬ Ϫ Ϫ ͑ ϫ ͒ Ϫ ͑␧ ϩ ϩ ͒ 2 p H f z T ␣ f zˆ T ␣ S␣␤ q␣T ␤ T ␣q␤ ϭ␧ ␨. Ѩ Ѩ Ѩ 2 Ѩ Ѩ z T X␤ X␣ ␧ L X␣ X␤ ͑B5͒

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Writing ͗T͘ in vortex-force representation gives

␧2␨ ␧2␨ Ѩ͗T␣͘ Ѩ q · q q · q ѨH ϩ ϩ c ϩ ͑ ϫ c͒ ϩ ͑␻c ϫ c͒ Ϫ ͑␧ ͒ c͑␧ ͒ Ϫ ϩ c Ѩ Ѩ ͵ p dz f z T ␣ ͵ u ␣ dz q␣ E w E ͩ p ͪ Ѩ ␣ϪH XסT X␣ ϪH 2 ϪH 2 z Ѩ ␧2␨ ѨH Ѩ ␧E u · u ␧2␵ ϭϪ ͵ ˜ ϩ ˜ ͑Ϫ ͒ Ϫ ␧Ϫ2 ͵ ϩ ␧Ϫ1 ϩ ͵ Ϫ ͑ ϫ w͒ dz H ͳ pdzʹ J␣ dz f z T ␣ Ѩ K 2 K 2 Ѩ Ѩ X␣ ϪH X␣ X␣ ␧2␨ 2 ϪH ␧E 1 Ѩ L 2 Ϫ ␧Ϫ4 ͵ ͑⍀ ϫ ͒ ϩ ␧Ϫ2͗␵w2͘ ϩ ͗␵lw2͘ ϩ ␵ ϩ ͳ u ␣ dzʹ ͫ ͩ ˆ P ͪ ͬ a ␧2␵ 2L ѨX␣ ␧

| |2 Ѩ Ѩ lw lw Ϫ4 w w Ϫ3 U E Ϫ3 E ϩ ͩ͗q␣ w ͘ ϩ ␧ ͗q␣w ͘ ϩ ␧ ͳ ʹ Ϫ ␧ ͳQ␣Q␤ ʹͪ . ͑B6͒ 2 Ѩx␣ Ѩx␤ zϭ␧E

Again, this can be shown to be all O(1), with the lower- Huang, H. E., 1979: On surface drift currents in the ocean. J. Fluid order terms canceling. The first two rows are the con- Mech., 91, 191–208. tributions of the currents and these can be shown to be Jenkins, A. D., 1987: Wind and wave induced currents in a rotating sea with depth-varying eddy viscosity. J. Phys. Oceanogr., ␧2 identical to the first row of (B4) to O( ). To evaluate 17, 938–951. (B6) directly to O(1), it is necessary to know the waves Leibovich, S., 1977a: On the evolution of the system of wind drift to O(␧4). This is two orders higher than is needed to currents and Langmuir circulations in the ocean. Part 1. evaluate (B4). Thus, the vortex-force representation is Theory and averaged current. J. Fluid Mech., 79, 715–743. not useful when evaluating the evolution equations of ——, 1977b: Convective instability of stably stratified water in the ͗ ͘ ocean. J. Fluid Mech., 82, 561–585. T , because the free surface terms are onerous to ——, 1980: On wave-current interaction theories of Langmuir cir- evaluate. The vortex-force value is in the interpretation culations. J. Fluid Mech., 99, 715–724. of the mean Eulerian momentum equations and, be- Longuet-Higgins, M. S., 1970: Longshore currents generated by cause ͗T͘ can be seen as the vertically integrated obliquely incident sea waves,1&2.J. Geophys. Res., 75, Lagrangian mean momentum, we do not gain anything 6778–6801. ——, and R. W. Stewart, 1960: Changes in form of short gravity by expressing the equations in this form. Nevertheless, waves on long tidal waves and tidal currents. J. Fluid Mech., if ͗T͘ is required, it can formally be written down in a 8, 565–583. more tidy form as the sum of the equations for Tc and ——, and ——, 1961: The changes in amplitude of short gravity ͗Tw͘. waves on steady non-uniform currents. J. Fluid Mech., 10, 529–549. ——, and ——, 1962: Radiation stress and mass transport in grav- REFERENCES ity waves, with application to “surf beats.” J. Fluid Mech., 13, 481–504. Andrews, D. G., and M. E. McIntyre, 1978a: An exact theory of ——, and ——, 1964: Radiation stresses in water waves: A physi- nonlinear waves on a Lagrangian-mean flow. J. Fluid Mech., cal discussion, with applications. Deep-Sea Res., 11, 529–562. 89, 609–646. McWilliams, J. C., and J. M. Restrepo, 1999: The wave-driven ——, and ——, 1978b: On wave-action and its relatives. J. Fluid ocean circulation. J. Phys. Oceanogr., 29, 2523–2540. Mech., 89, 647–664. ——, ——, and E. M. Lane, 2004: An asymptotic theory for the Bühler, O., and M. E. McIntyre, 2003: Remote recoil: A new interaction of waves and currents in coastal waters. J. Fluid wave-mean interaction effect. J. Fluid Mech., 492, 207–230. Mech., 511, 135–178. Craik, A. D. D., and S. Leibovich, 1976: A rational model for Mei, C. C., 1989: The Applied Dynamics of Ocean Surface Waves. Langmuir circulations. J. Fluid Mech., 73, 401–426. World Scientific, 740 pp. Garrett, C., 1976: Generation of Langmuir circulations by surface Mellor, G., 2003: The three-dimensional current and surface wave waves—A feedback mechanism. J. Mar. Res., 34, 116–130. equations. J. Phys. Oceanogr., 33, 1978–1989. Gjaja, I., and D. D. Holm, 1996: Self-consistent Hamiltonian dy- ——, 2005: Some consequences of the three-dimensional currents namics of wave mean-flow interaction for a rotating stratified and surface wave equations. J. Phys. Oceanogr., 35, 2291– incompressible fluid. Physica D, 98, 343–378. 2298. Groeneweg, J., and G. Klopman, 1998: Changes of the mean ve- Sullivan, P. P., J. C. McWilliams, and W. K. Melville, 2004: The locity profiles in the combined wave-current motion de- oceanic boundary layer driven by wave breaking with sto- scribed in a GLM formulation. J. Fluid Mech., 370, 271–296. chastic variability. I: Direct numerical simulation of neu- Hasselmann, K., 1971: On the mass and momentum transfer be- trally-stratified shear flow. J. Fluid Mech., 507, 143–174. tween short gravity waves and larger-scale motions. J. Fluid Weber, J. E., 1983: Steady wind- and wave-induced currents in the Mech., 50, 189–201. upper ocean. J. Phys. Oceanogr., 13, 524–530.

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