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 (x, t, ) ϭ a(x, t) cos, where ϭ kx Ϫ Mellor 1987; Bleck and Boudra 1986). However, the t; k is the wavenumber, is the frequency, c ϵ /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 ϭ kacekz ocean circulation models do not recognize surface cos and wÄ ϭ kacekz sin. 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 2 ()ϭ ()d. (1) integrated from the bottom, z ϭϪh, to the wave surface, 2 ͵ z ϭ , and then are phase averaged. Furthermore, the 0 slow (wind-, tide-, and density-driven) horizontal ve- Whereas uÄ ϭ 0, vertical integrals such as locities are often stipulated to be independent of z a priori. The result is a mismatch between the wave mod- M ϵ uÄ dz ϭ uÄ dz (2) els and three-dimensional circulation ocean models, ͵͵ Ϫh 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 ϭ ac|ekz| cos ϭ (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 uÄץ uÄץ uÄץ uÄץ uL ϭ uÄ ϩ xÄ ϩ zÄ ϭ xÄ ϩ zÄ, (4) zץ xץ zץ xץ -Corresponding author address: George L. Mellor, Program in At mospheric and Oceanic Sciences, Princeton University, P.O. Box CN710, Sayre Hall, Princeton, NJ 08544-0710. where the particle displacement (xÄ, zÄ) ϭ # 0 (uÄ, wÄ ) dt ϭ E-mail: [email protected] aekz(Ϫsin, cos), so that the Stokes drift is
᭧ 2003 American Meteorological Society SEPTEMBER 2003 MELLOR 1979
22 2kz uL ϭ kace (5) and, in a similar way, wL ϭ 0. When (5) is integrated from Ϫh (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 ϩ zÄ(t), t] ,zץ/zÄץz; then multiply uÄ by 1 ϩץ/ uÄץϭ uÄ(x, z, t) ϩ zÄ 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 ϭ 0.02 m sϪ1 and zÄ wץ uÄץ zÄץ uÄ (x, z ϩ zÄ,t)1ϩϭuÄ ϩ zÄ 1 ϩ the Stokes drift velocity from integrals of the Pierson±Moskowitz Ϫ1 z spectrum for a wind, U10 ϭ 14.6 m s ; also shown are the rms waveץ zץ zץ orbital velocity (uÄ 22ϩ wÄ )1/2 divided by 10. The friction velocity and uÄ Ϫ1ץ zÄץ wind are related according to U10 /ua ϭ ln(10 m/z0), where Char- ϭ uÄ ϩ zÄ . (6) nock's wave roughness is taken to be z 0.012u2 /g, 0.40, and z 0 ϭ a ϭץ zץ 22 uura ϭ 860w . The wind stress was initially ramped up from rest to 2 The ®rst term on the right is due to the velocity±¯ow the valueuw 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 KϪ1. 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 Longuetץ u (x, z, t) ϭϭkace22 2kz, (7) .(z Higgins (1969ץ S In Fig. 1, I show velocity pro®les calculated from a a result identical to (5). The Stokes transport, M ϵ one-dimensional turbulence closure model (Mellor and 0 #Ϫ1 us dz, can be obtained directly from Yamada 1982) for a speci®c wind speed of 14.6 m sϪ1 2 Ϫ2 zÄuÄ ka2 c (and kinematic stress of 0.33 m s ) and the Stokesץ 0 M ϭ dz ϭ u(0)zÄ(0) ϭ . (8) drift velocity and wave orbital velocity corresponding z 2ץ ͵ Ϫh 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- -z. locities are an order of magnitude larger than the windץ/zÄץxuϭ Äץ/ uÄץthat xÄ 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
ץ The plan of this paper is to present the governing (uà ϩ uÄ ϩ u ) t iiiץ equations for the slowly changing variables, the wave variables, and the turbulence variables in section 2. The ץ transformation to sigma coordinates is done in section ϩ (uà jiuà ϩ uà jiuÄ ϩ uà jiu ϩ uà ijuÄ ϩ uà iju ϩ uÄ jiuÄ xjץ The well-known linear wave solutions are given in .3 section 4. All terms appropriate to the continuity and ϩ uÄ u ϩ u uÄ ϩ u u ) momentum equations are obtained in section 5, and the ji ji ii ץ -wave energy equation is derived in section 6. The con ϩ ⑀ f (uà ϩ uÄ ϩ u ) ϩ (pà ϩ pÄ ϩ pÂ) xץtinuity and momentum equations are vertically depen- ijkikkk dent, whereas, by virtue of assuming the linear wave i solutions to be valid to lowest order, one needs only the à  ϭϪ ϩ g␦ . (12) vertically integrated wave energy equation. Section 7 iz establishes the correspondence between the present re- oo sults and the conventional momentum and energy equa- It is possible to write equations for the three com- tions. In section 8, additional terms that result from wind ponents, and I have done so. However, as will be seen pressure processes are derived. The results are sum- in appendix A, the slow and phase-averaged wave com- marized in section 9 together with the equations for ponents interact in a somewhat complicated way. I there- scalars and the turbulent kinetic energy. Appendix A fore write the equations for ui ϵ uà i ϩ uÄ i and p ϵ pà ϩ considers interactions between the momentum and wave pÄ such that uץ -energy equations and appendix B de®nes the nomen clature of the velocities and scalars used in this paper. i ϭ 0 and (13) xiץ
ץ p Ãץ uuijץ uiץ 2. The basic equations ϩϩ⑀ijkfu j kϩϭϪg␦ izϪ͗u iu j͘. xץ x ץxjiojץ tץ The equations of motion are (14)
Uj The Reynolds stresses are u u , and so it is assumedץ ϭ 0 and (9) ͗ i j͘ xj that u i can be extracted from ui, a process simpler inץ laboratory experiments than in ®eld experiments for P g which the waves are cyclic (Jensen et al. 1989; Cheungץ UUjiץ Uiץ ϩϩ⑀ijkf jU kϩϭϪ␦ iz, (10) Ϫ1 N (x and Street 1988) and ui ϭ͗Ui(t)͘ϵN ⌺jϭ1 Ui(t ϩ jpץxiioץ tץ is a phase-conditioned average, where p is the wave where Ui ϭ (U, V, W ), xi ϭ (x, y, z), and Ui ϭ Ui(xi, period and N is a large number; other dependent vari- t). Here, P ϭ P(xi, t) is the kinematic pressureÐthat is, ables are similarly processed. Field observations present the dynamic pressure divided by a reference density a greater challenge in separating wave and turbulence 0Ðand f j is the Coriolis parameter. The coordinate z properties. and velocity w are vertically upward. We have omitted The equations governing turbulence are viscous terms, but they can be reclaimed where they uÂץ may be important locally such as next to a smooth bot- k ϭ 0 and (15) xkץ tom surface. The tensor ⑀ijk is nil if any of the indices are repeated, is equal to 1 when i, j, and k are any triplet ץץ in the sequence xyzxy, and is Ϫ1 in any other sequence. u ϩ (uu ϩ uu ϩ u u Ϫ͗u u ͘) x jiijjijiץt iץ We now decompose Ui into three velocity compo- j nents: a ``slow'' component uà , whose time- and space p Âץ i 2 (scales are L and T, respectively; a wave component uÄ i, ϩϩg␦izϭ ٌ u i. (16 xioץ ,whose smaller time- and space scales are kϪ1 and Ϫ1 respectively; and a random turbulence component, u , i Notice that ͗u i͘ϭ0 and that the averages of all terms such that in (16) are nil. I have added the viscous term ( is kinematic viscosity) because, when (16) is converted to U ϭ uà ϩ uÄ ϩ u and (11a) iiii an energy equation, the term becomes the important
P ϭ pà iiiϩ pÄ ϩ p . (11b) turbulence kinetic energy dissipation. The density is split into a slow and ¯uctuating turbu- lence component 3. A transformation ϭ à ϩ Â, (11c) There is need for separate nomenclature for the hor- izontal and vertical coordinates; thus, x␣ ϵ (x, y), xi ϵ therefore ®ltering out baroclinic interaction with surface (x␣, z), and ui ϵ (u␣, w). Next transform the dependent waves. Using the above de®nitions, (10) may be written variables in (13) and (14) according to SEPTEMBER 2003 MELLOR 1981
(x␣␣, z, t) ϭ *(x *, ,t*) (17) and the independent variables such that
x␣␣ϭ x*, (18a) t ϭ t*, and (18b)
z ϭ s(x␣*, , t*), (18c) where and s are general but will be constrained shortly; see (23a,b). From (17), one has ץ *ץ ץ ץ *ץ *ץ ץ ϭϩ , ϭ , zץ ץ zץ␣ xץ ץ *xץ␣␣xץ ץ *ץ *ץ ץ ϭϩ , tץ ץ *tץ tץ
(␣xץ/ץ)(ץ/sץ) x␣) ϩץ/ xץ)(**␣␣xץ/sץ) x␣ ϭץ/zץ and from (z FIG. 2. An instantaneous sketch of wave particle position in (topץ/ץ ,x␣ ϭϪs␣ /s; in a similar wayץ/ץ ϭ 0, I obtain x␣*, x, z space and (bottom) as transformed to x, space according toץ/sץt ϭϪst /s, where I de®ne s␣ ϵץ/ץ ϭ 1/s and .. Putting these relations to- (23) for à ϭ 0ץ/sץt*, and s ϵץ/sץst ϵ gether yields ,* s The transformation thus far is general. Now, howeverץ *ץ ץ ϭϪ ␣ , (19a) - s (anticipating the linear wave solutions in the next secץ *xץ xץ ␣␣ tion) de®ne *1ץ ץ ϭ , and (19b) s(x, y, , t) ϵ à ϩ D ϩ sÄ and (23a) sץ zץ sinhkD(1 ϩ ) (* s sÄ ϵ a cos. (23bץ *ץ ץ ϭϪ t . (19c) sinhkD sץ *tץ tץ The mean elevation isà (x, y); h(x, y) is the bottom ;u ␣ / depth, and D ϵϩà h is the mean water column depthץ Using (19a,b), I now transform (13) (written as z ϭ 0) and, at the same time, drop the à , a, k, and D are slow functions of x, y, and t that varyץ/ wץx ␣ ϩץ asterisks so that on length scales and timescales of L and T, respectively, -w 1 and ϵ k␣x␣ Ϫ t, where k␣ and are large in comץ usץ uץ ␣␣␣Ϫϩϭ0. parison with LϪ1 and T Ϫ1. The ``sigma'' variable rang- x s s es from Ϫ1 where s ϭϪh to 0 where s ϭ ϭϩÃץ ץ␣ ץ Next de®ne Ä . Figure 2 illustrates the transformation process for cases in which à ϭ 0 and there is zero atmospheric w ϭ Ê ϩ us ϩ s (20) ␣␣ t pressure. so that, after some rearrangement, The vertical derivatives of (23a,b) are sץ s ץ Êץ␣suץ ϩϩϭ0. (21) sϭϭD ϩ sÄ and (24a) ץ tץ ץ ␣xץ Equation (21) could have been derived from a control coshkD(1 ϩ ) sÄ ϭ kDa cos, (24b) volume formulation (visualized with help from the top sinhkD panel of Fig. 2); thenÊ is the component of (the nearly vertical) velocity normal to surfaces of constant . and, in consequence of (20) at ϭ 0ors ϭ , Ê (0) t and, at ϭ 0ץ/ץx␣ ϩץ/ץ(Equation (14), for the horizontal velocity compo- ϭ 0 so that w(0) ϭ u␣(0 /hץ(nents, can be similarly manipulated and transforms to or s ϭϪh, Ê (Ϫ1) ϭ 0 so that w(Ϫh) ϭϪu␣(Ϫh . xץ ␣ ץ ץץ (su) ϩ (suu) ϩ (Êu ) Ϫ ⑀ fsu  ␣ ␣zzץ␣x ץ␣t ץ  4. The linear wave relations The well-known linear relations are obtained from ץץ ϩ (sp␣) Ϫ (sp) x␣ ϩץ/ uÄ iץ␣ t ϩ uÃץ/ uÄ iץ xj ϭ 0 andץ/ uÄ jץ solutions ofץ ␣xץ xi (p ϩ gz) ϭ 0 wherein only the fast terms of orderץ/ץ kL or T are retained in (13) and (14). The turbulence ץץץ ϭϪ (s␣͗u u ͘) ϩ (s ␣͗u u ͘) Ϫ͗w u ␣͘. (22) - terms are assumed to be small. Solutions whose zץ ץ xץ 1982 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33
x Ϫץ/ ␣kץ x␣ ϭ 0 andץ/ץt ϩץ/ ␣kץ dependent arguments are k(h ϩ z) transform to kD(1 which we obtain -x␣ ϭ 0 so that the wave number vector is irrotaץ/ kץ .(ϩ ϩ sÄ /D) in accordance with (17), (18c) and (23a (␣xץ/kץ)(kץ/ץ) x␣ ϭץ/ץ The mean sea surface is at z ϭ à . Thus, the wave tional. These equations and x␣)k yield the well-known relationץ/ץ) velocities are ϩ coshkD(1 ϩ ϩ sÄ/D) uà Aץ ץ kץ␣␣kץ uÄ␣␣(x , z, t, ) ϭ kac ␣ cos (25a) ϩ (c ϩ uà ) ϭϪ Ϫk , sinhkD g A ␣ xץ x kץx␣ץ tץ and k)(k /k). If the right side were nilץ/ץ) where cg ϭ sinhkD(1 ϩ ϩ sÄ/D) (depth and Doppler velocity are constant), then the wave wÄ (x , z, t, ) ϭ kac sin. (25b) ␣ sinhkD number vector is invariant along a trajectory prescribed by the combined group velocity and Doppler velocity. Notice that wÄ ϭ 0 where ϭϪ1. The phase is de®ned as 5. The transformed continuity and momentum ϵ kx␣␣Ϫ t. (26) equations The dispersion relation and intrinsic frequency are The next step is to phase average the terms in (21) and (22) and to de®ne a mean velocity U such that ϭ ϩ ku␣ à A␣ and (27a) ␣
2 ϭ gk tanhkD. (27b) DU␣␣ϵ su ϭ (D ϩ sÄ ␣␣)(uà ϩ uÄ )
The ``Doppler velocity'' uà A␣ is, according to the anal- ϭ Duà ␣ ϩ DuS␣, (30a) ysis of Kirby and Chen (1989; and see other references where u (D sÄ )uÄ /D is the Stokes drift velocity. in their paper), a weighted average of the vertically s␣ ϵ ϩ ␣ Referring to (25a) and uÄ uÄ (x, y, 1 sÄ/D) nonuniform current: ␣ ϭ ␣ ϩ ϩ ϭ ()/D, we obtain (see section 1ץ/␣ uÄץ)uÄ ␣(x, y, 1 ϩ ) ϩ sÄ 0 kD cosh2kD(1 ϩ ) sÄuÄץ uà A␣ ϭ 2 uà ␣ d. (27c) 1 ͵ sinh2kD u ϭ (1 ϩ sÄ /D)uÄ ϭ ␣ ץ Ϫ1 S␣␣D The intrinsic phase speed and group speed are k (ka)2 cosh2kD(1 ϩ ) ϭ ␣ c g k 2 sinh2kD c ϵϭ tanhkD and (28a) kkΊ 2kEcosh2kD(1 ϩ ) ϭ ␣ . (30b) d c 2kD c sinh2kD cg ϵϭ1 ϩ . (28b) 2 dk 2 sinh2kD Note that us␣ /c is of order (ka) . Here and throughout The surface elevation and pressure from waves are the paper, I use the fact that phase averages of odd powers of cos and products of cos and sin are nil. Ä (x␣␣, t, ) ϭ a(x , t) cos and (29a) The expression on the right of (30b) uses (27b) and the de®nition, E ϵ ga2 /2; E will later be shown to be the coshkD(1 ϩ ) pÄ ϩ gsÄ ϭ kac2 cos sum of the kinetic and potential wave energies. sinhkD The product, su␣u in (22), when averaged, is al- coshkD(1 ϩ ) gebraically complicated. Thus, ϭ ga cos. (29b) coshkD suu␣␣␣ϭ s (uà ϩ uÄ )(uà ϩ uÄ ) The last term on the right is obtained using (27b). ϭ Duà ␣uà ϩ uà ␣(D ϩ sÄ )uÄ  Note that, whereas the wave velocities in (25) have been ``corrected'' to include the small term sÄ/D, the ϩ uà ␣␣(D ϩ sÄ )uÄ ϩ DuÄ uÄ wave pressure in (29b) does not include this term be- DU U DuÄ uÄ , (31) cause its omission cancels an equal and opposite small ϭ ␣ϩ ␣ error in the linear solution. Notice that (29b) satis®es Ϫ1 where the terms ϪD (D ϩ sÄ␣)uÄ (D ϩ sÄ )uÄ ϭ the condition that pÄ ϭ 0at ϭ 0. I will use the condition 4 Dus␣us are of order (ka) and are neglected. pÄ(0) ϭ pÃ(0) ϭ 0 to simplify the following analyses as Next de®ne Ê ϭ⍀(1 ϩ sÄ /D) so that did Phillips (1977) but will, in section 8, separately consider the consequences of nonzero atmospheric pres- Ê ϭ⍀ and (32a) sure. Êu␣␣ϭ⍀U , (32b) Although equations for k␣ will not be needed in this paper, they are included here for completeness. Thus, where ⍀(Ϫ1) ϭ⍀(0) ϭ 0. Using (30), (31), and (32), t, from (21) and (22), after phase averaging, may be writtenץ/ץx␣ and ϵϪץ/ץfrom (26), one has k␣ ϵ SEPTEMBER 2003 MELLOR 1983
à method developed above. Thus, multiply (14) by ui toץ ⍀ץ ␣DUץ ϩϩϭ0 and (33) form a kinetic energy equation: tץ ץ ␣xץ
22 ץ ץץ u ץuii ץ (DU␣␣␣␣) ϩ (DU U ) ϩ (⍀U ) ϩ ⑀ zzfDU ϩ gz ϩ u ϩ p ϩ gz ץ xץ tץ x 2 ץ t2ץ
D 2ץץץ Ãץ ץ ui ץ gD (DpÃ) pà ϩ ϩ Ϫ ϩ w ϩ p ϩ gz ϭϪu ͗u u ͘. (35) x ␣ ijiץץ xץ␣␣xץ xjץ z 2ץ ץ sÄ pÄץ Sץ ␣ ␣ The buoyancy term is excluded from (14) because I ϭϪ ϩ Ϫ ͗w u ␣͘, (34a) assume that buoyancy does not affect the near surfaceץ ץ xץ wherein we invoke the boundary layer approximation wave motion. Now transform (35) according to (17), and neglect horizontal gradients of the Reynolds stress- (19), and (20) and phase average so that
ϭϪDgà /o,weץ/ pÃץ es. From the hydrostatic relation 22 u ץuii ץ obtain sϩ gs ϩ su ϩ p ϩ gs 0 x 2 ץ t 2ץ à Ϫ pà ϭϪgD Ϫ o d. (34b) ͵ o ץץ u2 ץ ϩ Ê i ϩ p ϩ gs ϩ spϭϪu ͗w u ͘. ץ tiiץ 2ץ The ®rst term on the right of (34a) is (36a) S␣ϵ DuÄ ␣uÄ ϩ ␦ ␣sÄ pÄ , (34c) where Because it is known that the Reynolds stress tensor com- ponents are of the same order, we have used the bound- kk 222 2 ␣kac cosh kD(1 ϩ ) ary layer approximation in the last term in (36a). Next, DuÄ␣uÄ ϭ D k222 sinh kD expand
2 kk␣cosh kD(1 ϩ ) 22 2 ϭ DkE (34d) su i /2 ϭ s␣(uà /2 ϩ uÄ ␣␣uà ϩ uÄ i /2) k2 sinhkD coshkD 22 ϭ D(uà ␣/2 ϩ uus␣␣à ϩ uÄ i /2) using (25a) and (27b). From (24b) and (29b), we obtain
and use uà ␣ ϭ U␣ Ϫ uÄ s␣ to obtain cosh2kD(1 ϩ ) 2222 sÄ pÄ ϭ kDE su/2 ϭ D(U /2 Ϫ u /2 ϩ uÄ /2) [ sinhkD coshkD i ␣ s␣ i ϭ D(U 22/2 ϩ uÄ /2). (36b) sinhkD(1 ϩ ) coshkD(1 ϩ ) ␣ i Ϫ . (34e) sinh2kD ] The deleted term is of order (ka) 4. In a similar way, 222 In a similar way from (23b) and (29b), suui /2 ϭ DU␣ U /2 ϩ DU  uÄ i /2 ϩ DU␣␣ uÄ uÄ (36c) coshkD(1 ϩ ) sinhkD(1 ϩ ) and sÄ pÄ ϭ E 1/2 Ϫ ␣ coshkD sinhkD 222 []Êuiii/2 ϭ⍀(U /2 ϩ uÄ /2). (36d) sinhkD(1 ϩ ) Neglecting buoyancy terms, one has p ϭ pà ϩ pÄ ϭ ץ ϫ E 1/2 . (34f) x sinhkD ϪgD ϩ pÄ and, of course, s ϭϩà D ϩ sÄ so that pץ ␣ [] ϩ gs ϭ pÄ ϩ gsÄ ϩ gà . Completing the operations in- Equations (34d) and (34e) after insertion into (34c) are volving these terms in (36a) yields the so-called stress radiation terms derived in their ver- tically integrated form by Longuet-Higgins and Stewart gs s ϭ gD(à ϩ D) ϩ gsÄ sÄ, (36e) , using (34f), is an additionalץ/ sÄ pÄץ The term .(1961) ␣ su(p gs) DU g à UsÄ (pÄ gsÄ) radiation term that vertically integrates to zero. ϩ ϭ  ϩ  ϩ
ϩ DuÄ(pÄ ϩ gsÄ), and (36f) 6. The wave energy equation Ê (p ϩ gs) ϭ⍀[gà ϩ sÄ(pÄ ϩ gsÄ)/D]. (36g) To close the momentum equation derived in section Inserting (36b±g) into (36a) yields 5, one needs to know E(x␣, t) which is the vertically integrated or two-dimensional wave energy. Obtaining 22 DU␣ uÄ i ץ an equation for E(x␣, t) involves complicated algebra, ϩϩgD(à ϩ D) ϩ gsÄ sÄ t[]22 ץ and I have followed Phillips (1977) but also utilized the 1984 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33
22 DU U U DuÄ 0 22 ץ hץ Ã gץ g ץ ϩϩ␣DU uÄ uÄ ϩ  i .x 22␣␣ gD (Ã ϩ D) d ϪϭϪϭ0ץ tץ t 2ץ t ͵ 2ץ ]  []Ϫ1
ϩ uà sÄ (pÄ ϩ gsÄ) ϩ Ugà ϩ DuÄ (pÄ ϩ gsÄ)  ␣  ] 7. Correspondence with conventional equations 22 To compare with formulas in Phillips (1977) and oth- ץ (Uu␣ Ä i sÄ(pÄ ϩ gsÄ ץ ϩ⍀ϩϩgà ϩϩspt ers, neglect baroclinicity, so that pà ϭϪgD, and tur- 22 D bulence terms; then integrate (33) and (34a) from ϭץ[]·Άץ Ϫ1to ϭ 0 and obtain ץ ϭϪu ͗w u ͘. (37) Mץ Ãץ ץii ϩϭ 0 and (40) xץ tץ 2 Now from (37), subtract U␣ times (34a) and addU ␣ /2  Ϫ gà times (33). After complicated but straightforward Mץץ algebra, one obtains (M␣) ϩ M ␣ϩ S ␣ϩ ⑀ ␣zzfM xD ץ tץ uÄ 22gà ץ Ãץ i ϩ gD(à ϩ D) ϩ gsÄ sÄ Ϫ (t[]22 ϩ gD ϭ 0, (41ץ ␣xץ Uץ DU uÄ 2 ץ ϩϩ i UsÄ sÄ ϩ DuÄ (pÄ ϩ gsÄ) ϩ S ␣ where the transport is xץ␣  x 2 ץ [] 0 2 sÄ (pÄ gsÄ) M ϵ DUd␣ and (42) ͵ ץ uÄ i ϩ ץ ϩ⍀ϩ ϩsp Ϫ1 tץ Ά·[]2 Dץ ckk c 1 S ϵ E g ␣ϩ ␦ E g Ϫ . (43) ␣ 2␣ ץ ץץ ϭϪuà sÄ pà Ϫ u ͗w u ͘ϩuà ͗w u ͘. ck c 2 ץ␣␣ ץ iiץ␣␣ The conventional simpli®cation of (38) requires that 4 , on the right is of order (ka) and uà ␣ be independent of , in which case (38) reduces toץ/ sÄ␣pÄץ␣A term, Ϫus has been discarded. ␣Uץ ץץ Next integrate from ϭϪ1 to 0 and obtain E ϩ [(cg␣␣ϩ uà )E] ϭϪS ␣. (44) xץ␣xץ tץ 0 2 DuÄ i ץ Eץ ϩ cEg ϩ uà ϩ gsÄ sÄ d Phillips had uà ␣ in place of U␣, but the difference is a x ͵ 2 4ץ tץ []Ϫ1 term of order (ak) . Except for the fact that I include the Coriolis terms ץ Uץ00 ␣ 0 ϭϪ Sd␣ Ϫ uà ␣␣sÄ pÄ d Ϫ |spt |Ϫ1 in (41), I recover the depth-independent results obtained ץ ͵ xץ ͵ Ϫ1 Ϫ1 by Phillips (1977) and others. ץץ00 ϩ uS␣␣͗w u ͘ d Ϫ uÄ ii͗w u ͘ d, (38) 8. Wind pressure forcingץ ͵ ץ ͵ Ϫ1 Ϫ1 where the wave energy is de®ned according to Heretofore, to simplify and to check some of the der- 0 ivations, I have assumed that the atmospheric pressure E ϵ [D(uÄ 2/2) ϩ gsÄ sÄ ] d is nil; that is, pÃ(0) ϭ pÄ(0) ϭ 0. Waves can be driven ͵ i Ϫ1 by wind pressure ¯uctuations acting on the sloping sea
0 surface, however (Miles 1954; Phillips 1954; Donelan ϭ D(uÄ 22/2) d ϩ gÄ /2. (39) 1999; etc.). ͵ i I will continue to represent the free wave pressure Ϫ1 ®eld by pÄ(x, ,t) to which I now add the pressure ®eld Using (25) and (29), both terms on the right of (39) driven by atmospheric pressure forcing. This pressure may be shown to be equal to ga2 /4 and their sum, E ϭ ®eld may be written 2 0 ga /2. The term | spt | Ϫ1 ϭ 0, but it is left as a place holder, useful in section 8. To obtain (38), I have also coshkD(1 ϩ ) p ϭ pà ϩ pÄ ; pÄ ϭ a sin, used w atm wwwcoshkD
0 2 where pà atm is the slowly varying atmospheric imposed ץ gaץץ DuÄ (pÄ gsÄ) d c (cE),  ϩ ϭ g ϭ g pressure and pÄ w is the solution for imposed surface ¯uc- xץ x 2ץ ͵xץ Ϫ .␣xץ/ Äץ tuation for the Fourier constituent in phase with where cg ϭ kcg /k is the group velocity and Therefore, SEPTEMBER 2003 MELLOR 1985
Ä coshkD(1 ϩ ) sinhkD(1 ϩ ) The functions in (48) are plotted in Fig. 3. (The productץ sÄ␣ pÄ wwϭ pÄ , x coshkD sinhkD kDFCS is plotted because it is ®nite in the shallow waterץ ␣ → . is plotted in Figץ/ FSSFCCץ limit, kD 0.) The factor where pÄ w ϭ aw sin is the wind pressure at the surface. 4. Differentiation yields In summary, I ®rst repeat (33):
(Ä 2kD cosh2kD(1 ϩ ץ sÄ␣ pÄ wץ Ãץ ⍀ץ ␣DUץ ϭ pÄ w , (45) ϩϩϭ0. (50) x␣ sinh2kDץ ץ tץ ץ ␣xץ which in the next section is a term that will be added Equation (34a), after addition of (45) may be written .( in (34aץ/ sÄ␣pÄץ to ץ ץץ ,In a similar way (DU␣␣␣␣) ϩ (DU U ) ϩ (⍀U ) ϩ ⑀ zzfDU 0 ץ xץ tץ sÄ pÄץ ␣ w 0 uà ␣ d ϩ |sptw|Ϫ1 0ץ ͵ bץ Dץ bץץ Ϫ1 2 0 ϩ D (gà ϩ pà ) ϩ D Ϫ d ץ xץ xץ x atmץ Äץ (Ä 2kD cosh2kD(1 ϩ ץ ϭ pÄ uà d ϩ pÄ ␣␣␣͵ tץ x ͵ ␣ sinh2kD wץ w Ϫ1 ץ FFSS CCץ Äץ sÄ␣ pÄץ ␣Sץ ,Ä ϭϪ ϩ ϩpÄ w␣Ϫ͗w u ͘ץ Äץ ץ ץ xץ ץ␣xץ (ϭ pÄ uà ϩ pÄ . (46 w x A␣ w t (51a) ץ ץ x and usingץ/ Äץ tpϭ (k /k 2)Äץ/ Äץ Last, because pÄ w ␣ w␣ and, from (34c,d,e), (27a)
0 kk␣ Äץ ␣sÄ␣ pÄ w kץ 0 S␣ ϵ kDE FCS F CC ϩ ␦␣(FFCS CCϪ FF SS CS). uà ␣ d ϩ |sptw|Ϫ1 ϭϪ cpÄ w , (47) k2 [] ␣xץ kץ ͵ Ϫ1 which is a term that will be added to the corresponding (51b) free wave pressure terms in (38). From (34f), In potential ¯ow over a solid wavy wall, Donelan ץ (1999) reminds us that the pressure and wall slope are 1/2 1/2 sÄ␣ pÄ ϭ (FCCϪ F SS)E (EF SS). (51c) ␣xץ in antiphase so that, at the wave surface, the existence of a correlation between the two quantities depends on the existence of a nonzero relative phase shift. He ad- In (51a) I have used (34b) and de®ned the buoyancy vances speci®c laboratory-derived relations for à Ϫ o . x as a function of the ``wave age'' de®ned as b ϵϪgץ/ Äץ␣pÄ w the ratio of wind speed to dominant phase speed, which, o when considering only monochromatic waves, is equal An important result is that the third term on the right to c ϭ /k. of (51a), the wind pressure forcing term, has the same vertical structure as the Stokes drift velocity in (49). 9. Summary and ®nal comments Also, the third and fourth terms on the right of (51a) have regularly been lumped together in mixed layer Here I summarize the ®nal equations for which it is modeling, which would seem to be a questionable strat- useful to de®ne egy in view of the above results. sinhkD(1 ϩ ) Upon insertion of (47), the surface wind pressure F ϭ , (48a) terms, the wave energy equation (38) becomes SS sinhkD ץ Eץ coshkD(1 ϩ ) ϩ [(c ϩ uà )E] ␣x g␣ A2ץ tץ (F ϭ , (48b CS sinhkD ␣ Uץ Uץ00 sinhkD(1 ϩ ) ϭϪ Sd␣␣ Ϫ sÄ pÄ d F , and (48c) ␣ ␣ ץ ͵ xץ ͵ SC ϭ coshkD Ϫ1 Ϫ1 Äץ k coshkD(1 ϩ ) ␣ ϩ cpÄ w Ϫ uÄ i͗w u i͘ϩ (uÄ S␣␣͗w u ͘) FCC ϭ , (48d) k x ␣ ץ coshkD uץ uÄץafter which (30b) may be written 00 ϩ͗w u ͘ iSd Ϫ͗w u ͘ ␣ d. (52a) ץ ␣ ͵ ץ i ͵ FF Ϫ1 Ϫ1ץkE u ϭ ␣ SS CC . (49) In (52a) we have de®nedץ S␣ kcD 1986 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33
FIG. 3. Plots of the functions (48a±d). The labels on each curve are kD values. As kD → 0, → FSS ϳ 1 ϩ ,FSC ϳ 0, kDFCS ϳ 1, and FCC ϳ 1, and, as kD ϱ, FSS, FSC, FCS, and FCC ϳ exp(kD).
ϭ kD(FCSFCC ϩ FSSFSC) in (49) and (53) andץ/ FSSFCCץFIG. 4. Plots of the functions F1 ϵ
F2 ϵ kD[(FCSFCC ϩ FSSFSC)/2 ϩ FCSFSS] in (52b). Note that uÄ s␣ ϰ F1. The labels on each curve → → are kD values. As kD 0, F1 ϳ 1 and F2 ϳ 2(1 ϩ ). As kD ϱ, F1 and F2 ϳ kD exp(2kD). SEPTEMBER 2003 MELLOR 1987
0 represents the ®nal conversion of wind work into ther- uà ϵ uà E Ϫ12(DuÄ /2 ϩ gsÄsÄ ) d A2␣ ͵ ␣ i mal (or internal) energy. Ϫ1 Left for the future are prescriptions for the Reynolds 0 ¯ux, wind pressure forcing, and turbulence energy pro- ϭ kD uà [(FF ϩ FF)/2 ϩ FF] d. duction. There is a large literature on empirical ways ͵ ␣ CS CC SS SC CS SS Ϫ1 of determining the Reynolds stresses and ¯uxes where (52b) surface waves are neglected. And, utilizing the turbu- lence kinetic equation, there are beginning attempts to The phase and group speeds are as in (28a,b) and the include the effects of wave breaking on current structure Doppler velocity may be written (e.g., Craig and Banner 1994; Terray et al. 1996). Now, 0 however, there is a need to scrutinize the existing em- uà ϭ kD uà (FF ϩ FF) d. (53) pirical knowledge base and to enter into new research A␣ ͵ ␣ CS CC SS SC Ϫ1 framed by the continuity, momentum, wave energy, and turbulence energy equations derived in this paper. The weighting factors in (52b) and (53) are similar. They both integrate to unity cum sole and are identical in the short-wave limit but differ in the long-wave limit as Acknowledgments. I made much use of the book by shown in Fig. 4. Nevertheless, in the interest of sim- Owen Phillips; if my equations, when vertically inte- grated, did not initially agree with with those in his plicity, the approximation uà Ӎ uà is suggested. Fur- A2␣ A␣ book, then I knew I was wrong and renewed effort to thermore, uà A2␣ Ӎ uà A␣ Ӎ U␣ is also an acceptable ap- 2 ®nd the mistake. I pro®ted from discussions with Mark proximation because the difference us␣ is of order (ka) relative to the group velocity in (52a). Donelan, Gene Terray, and Jurjen Battjes. The research In (52a), the third, fourth, and ®fth terms on the right was funded by the NOPP surf-zone project and by ONR are surface wind work terms, one from pressure and the Grant N00014-01-1-0170. others from turbulence. The sixth term is wave dissi- pation; any eddy viscosity model will yield a negative APPENDIX A value. Appendix A provides further discussion, useful for Some Applications understanding interaction of the momentum and wave Associated with a given wave energy E is a Stokes energy equations. drift velocity given by (49b). However, how does a giv- To model an ocean with interacting waves and cur- en velocity ®eld develop and do (51) and (52) produce rents, one needs (50), (51), and (52). To solve for any compatible results? scalar ⌰ (such as temperature and salinity after which Consider a thought experiment in which, initially, uà ␣ an equation of state will yield b), one can write ϭ 0, turbulence and Coriolis terms are nil, and the wave eld is horizontally homogeneous (in®nite fetch). Let® ץץ ץץ (D⌰) ϩ (DU ⌰) ϩ (⌰) ϭ (Ϫw ´ ), (54) ␣ the wave ®eld be given by k␣ ϭ (kx, 0) and de®ne Px ץ ץ xץ tץ x so that (51) and (52) reduce toץ/ Äץϵ pÄ w which neglects interaction between the scalar properties FFSS CCץץ .and surface waves (DUxx) ϭ P and (A1a) ץ tץ To complete the set of equations initiated in section Eץ I write the turbulence kinetic energy equation ,2 ϭ cPx. (A1b) tץ q ץ q ץq222 ץ D ϩ DU␣ ϩ⍀ - 2 Notice that the Ux pro®le has the same vertical distriץx␣ 2 ץ t2ץ bution as the Stokes pro®le in (30b). Let P ϭ 0 for t u ͗Âw ͘ xץץ 2 i Ͻ 0 and P Ͼ 0 for t Ն 0. Now, de®ne ϭϪ (͗w u ii͘ϩ͗w pÂ͘) Ϫ͗w u ͘ϪDg Ϫ D⑀, x oץ ץ 0 (55) M ϵ DUd ϭ Mà ϩ M , x ͵ xxSx which is derived from (16) after multiplication of every Ϫ1 term by u i; then, the terms are operated on by ͗͘, trans- 0 and because, from (49), MSx ϵ #Ϫ1 uSx d ϭ E/c, the formed to sigma coordinates, and phase averaged. Thus, momentum and energy equations result in the same ver- 2 2 q ϵ ͗u i ͘ is 2 times the turbulence kinetic energy, and tically integrated Stokes transport such that MSx Ͼ 0 and the terms on the right are turbulence diffusion, shear à Mx ϭ 0. We conclude that the wind pressure forcing production, buoyancy production, and dissipation; the creates a pure Stokes velocity response. In particular, if ®rst and last terms need to be modeled. The turbulence P ϭ constant for t Ն 0, then energy equation is the common basis of many current x Ãà turbulence closure models. The turbulence dissipation MxySxxSyϭ 0, M ϭ 0, M ϭ Pt, and M ϭ 0. 1988 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33
The situation changes if one includes Coriolis terms. APPENDIX B The same wind forcing is invoked, so that Nomenclature ( M , M)ץ xyϩ f (ϪM , M ) ϭ [P (t), 0] and (A2a) t yx xץ uà i ϭ current velocity, slowly varying in space and
time, ϭ (uÃ, à , wà ) ϭ (uà ␣, wà ) (MSx, M Sy)ץ ϭ [Px(t), 0]. (A2b) uà ␣ ϭ (uÃ, à ) t pà ϭ pressure, slowly varyingץ à ϭ surface elevation, slowly varying For zero forcing and steady state, (Mx, My) ϭ (0, 0), à uÄ i ϭ wave velocity ϭ (uÄ, Ä , wÄ ) and, if a Stokes drift exists, the current transport Mx just cancels the Stokes drift; this was ®rst demonstrated by pÄ ϭ wave pressure Ursell (1950) using a vorticity/circulation argument. Ä ϭ surface wave elevation u uà uÄ This has been said to be a paradox (Huang 1970; Xu i ϭ iiϩ and Bowen 1994) because there seems to be a discon- p ϭ pà ϩ pÄ u turbulence velocity tinuity in solution form between f ϭ 0 and f Ͼ 0no i ϭ u Stokes drift velocity, derived from uÄ matter how small the value f. But let us now set up a S␣ ϭ i U uà uÄ speci®c example ¯ow that is steady and with a Stokes ␣ ϭ ␣ ϩ S␣ drift. Set M ϵ M ϩ iM ; then (A2a,b) may be written ϭ transformed vertical coordinate such that x y ϭϪ1 when z ϭϪh and ϭ 0 when M z ϭϩà Äץ Mץ ϩ ifM ϭ P and Sx ϭ P . t xxt z ϭϪh, the ocean bottom; D ϵ h ϩ à à 0 ץ ץ M␣ ϵ Dd#Ϫ1 uà ␣ 0 For P ϭ 0 when t Յ 0 and P ϭ constant for t Ͼ 0, MS␣ ϵ Dd#Ϫ1 uS␣ x x à the solutions for M and MSx are M␣ ϵ MM␣ ϩ S␣ uà ϭ weighted vertical average ofuà equal to t A␣ ␣ 00 iPx # r()uà d, where # r() d ϭ 1 M ϭ P (tЈ)edtif(tЈϪt) ЈϭϪ (1 Ϫ eϪift), (A3a) Ϫ1 ␣ Ϫ1 ͵ x f E ϵ gÄ 22ϭ ga /2, where g is the gravity constant 0 and a is wave elevation amplitude MSxϭ Pt, x and (A3b)
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