1978 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33 The Three-Dimensional Current and Surface Wave Equations GEORGE MELLOR Princeton University, Princeton, New Jersey (Manuscript received 26 April 2002, in ®nal form 4 March 2003) ABSTRACT Surface wave equations appropriate to three-dimensional ocean models apparently have not been presented in the literature. It is the intent of this paper to correct that de®ciency. Thus, expressions for vertically dependent radiation stresses and a de®nition of the Doppler velocity for a vertically dependent current ®eld are obtained. Other quantities such as vertically dependent surface pressure forcing are derived for inclusion in the momentum and wave energy equations. The equations include terms that represent the production of turbulence energy by currents and waves. These results are a necessary precursor for three-dimensional ocean models that handle surface waves together with wind- and buoyancy-driven currents. Although the third dimension has been added here, the analysis is based on the assumption that the depth dependence of wave motions is provided by linear theory, an assumption that is the basis of much of the wave literature. 1. Introduction pressure so that their results differ from mine and, after vertical integration, differ from the corresponding terms In the last several decades there has been signi®cant in Phillips (1977). They did not address three-dimen- progress in the understanding and numerical modeling sional effects on the wave energy equation. of ocean surface waves (e.g., Longue-Higgins 1953; To contrast the developments in this paper with con- Phillips 1977; WAMDI Group 1988; Komen et al. 1994) ventional logic and to simplify discussion, I will tem- and similar progress has been made in ocean circulation porarily address deep water (kh k 1) propagating waves modeling (e.g., Bryan and Cox 1968; Blumberg and such that h(x, t, c) 5 a(x, t) cosc, where c 5 kx 2 Mellor 1987; Bleck and Boudra 1986). However, the st; k is the wavenumber, s is the frequency, c [ s/k two streams are separate. More often than not, wave is the phase speed, and a is wave amplitude. The as- models do not recognize vertical current structure and sociated horizontal and vertical velocities are uÄ 5 kacekz ocean circulation models do not recognize surface cosc and wÄ 5 kacekz sinc. Reviewing conventional log- waves as having any in¯uence on the ocean. ic, I de®ne the phase-averaging operator In the wave modeling literature, before derivation of the surface wave equations, the equations are generally 1 2p ()5 ()dc. (1) integrated from the bottom, z 52h, to the wave surface, 2p E z 5 h, and then are phase averaged. Furthermore, the 0 slow (wind-, tide-, and density-driven) horizontal ve- Whereas uÄ 5 0, vertical integrals such as locities are often stipulated to be independent of z a hh priori. The result is a mismatch between the wave mod- M [ uÄ dz 5 uÄ dz (2) els and three-dimensional circulation ocean models, EE 2h 0 which necessarily involve z-dependent horizontal ve- locities and other properties. For example, a recent paper are nontrivial; thus, approximately, by Xie et al. (2001) suggests including the conventional ka2 c M 5 ac|ekz|h cosc 5 (3) vertically integrated stress radiation terms from waves 0 2 as forcing terms in the vertically dependent momentum equations, a strategy that is obviously incorrect. Dolata is the Stokes transport. This is said to be an Eulerian result. and Rosenthal (1984) did attempt to derive three-di- One can alternately obtain the lowest-order Lagrang- mensional radiation stress terms but left out effects from ian velocity according to ]uÄ ]uÄ ]uÄ ]uÄ uL 5 uÄ 1 xÄ 1 zÄ 5 xÄ 1 zÄ, (4) Corresponding author address: George L. Mellor, Program in At- ]x ]z ]x ]z mospheric and Oceanic Sciences, Princeton University, P.O. Box CN710, Sayre Hall, Princeton, NJ 08544-0710. where the particle displacement (xÄ, zÄ) 5 # 0 (uÄ, wÄ ) dt 5 E-mail: [email protected] aekz(2sinc, cosc), so that the Stokes drift is q 2003 American Meteorological Society SEPTEMBER 2003 MELLOR 1979 22 2kz uL 5 kace (5) and, in a similar way, wL 5 0. When (5) is integrated from 2h (where kh k 1) to 0, the result is identical to (3). a. An alternate procedure However, in this paper, I do not wish to ®rst integrate vertically, because I seek z-dependent equations to com- plement the z-dependent circulation equations of three- dimensional ocean models. Furthermore, Lagrangian formulations are awkward when they involve equations more complicated than (4). Instead, let uÄ[x, z 1 zÄ(t), t] 5 uÄ(x, z, t) 1 zÄ]uÄ /]z; then multiply uÄ by 1 1]zÄ/]z, which is the wave-distorted, normal ¯ow area relative to the undistorted ¯ow area. Thus, FIG. 1. Comparison of the wind-driven, mixed layer velocity com- ponents for a water side friction velocity of u 5 0.02 m s21 and ]zÄ ]uÄ ]zÄ tw uÄ (x, z 1 zÄ,t)115uÄ 1 zÄ 1 1 the Stokes drift velocity from integrals of the Pierson±Moskowitz 21 121212]z ]z ]z spectrum for a wind, U10 5 14.6 m s ; also shown are the rms wave orbital velocity (uÄ 221 wÄ )1/2 divided by 10. The friction velocity and ]zÄ ]uÄ 21 wind are related according to U10 /uta 5 k ln(10 m/z0), where Char- 5 uÄ 1 zÄ . (6) nock's wave roughness is taken to be z 0.012u2 /g, 0.40, and ]z ]z 0 5 ta k 5 22 uura 5 860tw . The wind stress was initially ramped up from rest to 2 The ®rst term on the right is due to the velocity±¯ow the valueutw over one inertial period and then was held constant for area correlation, and the second term is due to particle 5 days. The initial temperature pro®le was linear with a vertical gradient of 0.05 K21. wave motion±velocity gradient correlation and is the same as the second term in (4). Thus, the Stokes drift velocity is tinuously down into the water column according to (7). ]zÄuÄ Similar reasoning will be found in the paper by Longuet- u (x, z, t) 55kace22 2kz, (7) S ]z Higgins (1969). In Fig. 1, I show velocity pro®les calculated from a a result identical to (5). The Stokes transport, M [ one-dimensional turbulence closure model (Mellor and 0 #21 us dz, can be obtained directly from Yamada 1982) for a speci®c wind speed of 14.6 m s21 2 22 0 ]zÄuÄ ka2 c (and kinematic stress of 0.33 m s ) and the Stokes M 5 dz 5 u(0)zÄ(0) 5 . (8) drift velocity and wave orbital velocity corresponding E ]z 2 2h to the same wind speed using the equilibrium wave The correspondence between uL and uS is not accidental spectrum according to Pierson and Moskowitz (1964). because, using the continuity equation, it can be shown In the direction of the wind stress, the rms orbital ve- that xÄ]uÄ /]xu5 Ä]zÄ/]z. locities are an order of magnitude larger than the wind- A more formal procedure will be pursued below, but driven velocity and the Stokes velocity. Following es- the essence is the same. I will obtain the above results tablished convention, I refer to the wind-, baroclinic-, along with other relations that are useful when incor- and Coriolis-driven velocities as ``currents'' that are dis- porating wave effects into the three-dimensional, phase- tinct from the ``Stokes drift.'' Figure 1 illustrates the averaged equations of motion; this is done by trans- point that, in general, the currents are not vertically forming the basic equations to a sigma-coordinate sys- constantÐa seemingly trivial statement were it not for tem. The motivation is not to obtain ®nal sigma-coor- the fact that it is a requirement of present-day wave dinate equations. Rather it is a helpful ®rst step in the models. Although the relative magnitudes of the dif- process of deriving the three-dimensional wave inter- ferent velocity components are probably correct, the cal- action terms in the equations of motion. If one wishes culation of currents and Stokes drift, independent of ®nal equations in Cartesian coordinates, the reverse each other as in Fig. 1, is not valid, as will be dem- transformation on the ®nal equation set, derived later onstrated in this paper. in this paper, is straightforward. Note the fact that, in much of the literature, the mean b. This paper wave momentum is conceptually thought to be trapped at the surface in consequence of (2) and the third term In this paper, I specialize to monochromatic waves. in (8). However, it is my opinion that it is more useful Later extensions will undoubtedly generalize to a spec- to conceive of the wave momentum as distributed con- trum of waves for use in models. 1980 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33 The plan of this paper is to present the governing ] (uà 1 uÄ 1 u ) equations for the slowly changing variables, the wave ]t iii variables, and the turbulence variables in section 2. The transformation to sigma coordinates is done in section ] 1 (uà jiuà 1 uà jiuÄ 1 uà jiu 1 uà ijuÄ 1 uà iju 1 uÄ jiuÄ 3. The well-known linear wave solutions are given in ]xj section 4. All terms appropriate to the continuity and 1 uÄ u 1 u uÄ 1 u u ) momentum equations are obtained in section 5, and the ji ji ii wave energy equation is derived in section 6.