1. Introduction

Mellor (2003, hereafter M03) produced an analysis providing depth-dependent wave-current interaction equations which, when vertically integrated, were in agreement with the depth-integrated equations of Longuet-Higgins and Stewart (1962, 1964, hereafter L-HS), Phillips (1977), and others. Nevertheless, a commentary by Ardhuin et al. (2008a) pointed to a discrepancy in M03 for shallow water (kD ≅ 1) compared with a case of unforced waves traversing a bottom with variable topography. This led to a further discovery that, with the M03 formulation, unforced waves with bottom variations produced mean currents even for deep water (say, kD ≅ 10), a physically unacceptable finding.

The present paper, although containing elements of M03, abandons the a priori use of sigma coordinates; characterization of waves derived for a flat bottom can be misinterpreted in the sigma domain. Specific differences between M03 and the present paper are postponed to the summary in section 7.

A recent paper by Smith (2006), starting from the vertically integrated equations of motion, explores the interaction between wave momentum and current momentum; this has significant instructional value. McWilliams et al. (2004) and Ardhuin et al. (2008b) develop equations for the current û_α; their analyses are complicated, and it is hard to see correspondence to the results of the present paper. Here, we obtain depth-dependent equations corresponding to the vertically integrated equations of L-HS and Phillips (1977). There is emphasis on developing equations that are easily incorporated into three-dimensional circulation models. It is shown that these models as now coded require only the addition of depth-dependent stress radiation terms to the momentum equation. Of course, a wave model is required to supply wave energy and wavenumber. The wave model can also provide variables for a wave-sensitive surface wind stress parameterization (Donelan 1990). A finding in M03 and here is that transport of the surface stress into the water column is supported by pressure and turbulence, not turbulence alone as, for example, in Mellor and Yamada (1982), Large et al. (1994), and many other papers.

Section 2 contains the derivation of the continuity and momentum equations that includes waves. Use is made of an elemental control volume bounded by material surfaces vertically and surfaces normal to the Cartesian coordinates horizontally. Current plus wave velocities—set equal to the standard linear solutions—are used to evaluate the continuity and momentum balances for the elemental control volume and the results are phase averaged. Special care is required to evaluate the balance of pressure forces following closely the reasoning of L-HS. Section 3 deals with the vertical transport of surface wind stress. The wave energy equation is presented in section 4. The transformation of the Cartesian equations to a sigma coordinate version is in section 5. In section 6, the aforementioned wave-current interaction case of Ardhuin et al. (2008a) is discussed; unlike M03, there is agreement with their results and the results of this paper.

2. Derivation in Cartesian coordinates

To construct equations that include wave motions, the waves are conventionally represented by the linear irrotational solutions for elevation η̃, velocity (ũ_α, w̃), and kinematic pressure P̃ (dynamic pressure divided by a reference water density, ρ₀), as follows:

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The Cartesian coordinates are (x_β, z), where Greek subscripts α or β denote horizontal coordinates; the vertical coordinate z is positive upward from the sea surface. In (1), ψ = k_βx_β − ω t; k_β and ω are directional wavenumber and frequency, such that ω = σ + k_βÛ_β and k = |k_β|; σ is the intrinsic frequency and Û_α is the Doppler velocity; a is wave amplitude; c = σ/k is the phase speed; h is the bottom depth; and η̂ is the mean surface elevation. The wave elevation is defined by (1a) and η η̂ + η̃. The water column mean depth is D ≡ η̂ + h. The change from the first to the second form in (1d) uses the dispersion relation σ² = gk tanhkD and is an example of similar manipulations below. Note that P̃(η̂) = gη̃.

The vertical locations of material surfaces are

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Equation (2b) is obtained from ∂S̃/∂t = w̃. The vertical derivatives are

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and the horizontal derivatives are

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Note that, at z = η̂, (2b) yields S̃ = a cosψ = η̃, whereas at z = −h, S̃ = S̃_α = 0.

In the derivation of the above equations, ka, ∂h/∂x_α, ∂a/∂x_α, and ∂k_β/∂x_α are assumed to be small. In the following nonlinear analyses, the same quantities are also assumed to be small (properly nondimensionalized on a representative k and σ). In particular, we note that terms additional to (1) and therefore (2), (3), and (4) that account for bottom slope are proportional to ka(∂h/∂x_α). To obtain this scaling, start with the linear irrotational wave equations; then expand the potential function using the small parameter, ε = ∂h/∂x. The lowest-order solutions are (1a)–(1c) and the next order that satisfies a nonzero but small bottom slope yields the aforementioned scaling. Further analysis, or indeed intuition, reveals that a more specific parameter is ka(∂h/∂x)/sinhkh because for deep water, the bottom slope should not be a factor in the description of surface gravity waves. Toward the final nonlinear equations derived below, terms of order (ka)⁴ are neglected relative to retained terms of order (ka)². For variable topography, it is assumed that (ka)²[(∂h/∂x)/sinhkh]² is small; this could be a problem for small kh; however, see section 6.

The integral control volume equation for mass conservation (density is constant) is

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where n_k = (n_α, n_z) is the unit vector normal to elemental surfaces dA of the control volume. The velocity u̇_k is relative to moving boundaries of the control volume.

The horizontal components of the integral momentum equation is

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The elemental volume is dV, and ∮τ_α dA represents momentum transfer by pressure and turbulence as described in section 3. (The form ∫∂p/∂x_α dV will be more convenient than the equivalent ∮pn_α dA.)

a. The velocity terms

Velocity components are divided such that

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where (û_α, ŵ) are defined to be “current” velocities that vary on spatial and temporal scales that are large compared to k⁻¹ and σ⁻¹. The wave velocities are extrapolated from z to z + S̃ such that

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Equations (7) together with (8) are not novel; it is accepted that the absolute velocity on the crest of a wave exceeds the absolute velocity in the trough—a fact that is intrinsic to Stokes drift. The wave velocities [Eqs. (8a) and (8b)] and material surfaces [Eqs. (2a) and (2b)] are schematically shown in Fig. 1.

Overbars will represent phase averaging: = (2π)⁻¹∫^2π₀( )dψ (so that, e.g., = = 0, = = 1/2, = 0, etc.).

Consider a control volume bounded by the surfaces, x, x + Δx; y, y + Δy and s, s + s_zΔz. On the s surfaces, the unit vectors are (n_β, n_z) = (s_β, 1 − s²_β) ≅ (s_β, 1 − s²_β/2). Applying (5) to this control volume and phase averaging, one obtains

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after ΔxΔyΔz has been factored out of the equation. Notice that (∂f /∂s)Δs = (∂f /∂z)(∂z/∂s)s_zΔz = (∂f /∂z)Δz.

Now = = û_α + , so that

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where

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is the Stokes drift. Thus, the Stokes drift is the product of the phase-averaged horizontal component of the extrapolated velocity, ũ_α + (∂ũ_α/∂z)_zS̃, and the flow area, (1 + S̃_z)ΔzΔy (if α = x). This is illustrated in Fig. 1. Further, using (1b) and (2b),

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The latter form uses the dispersion relation σ² = kg tanhkD and the definition of wave energy E = g = ga²/2 [see (A.8)]. For deep water, u_Sα = (2k_αE/c) exp[2k(z − η̂)].

For the second term in (9),

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because, using (7a), (8a), (3), and (4), = 0. Now define

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(the last substitution for cosmetic uniformity). Therefore, the continuity Eq. (9), is simply

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The momentum Eq. (6) applied to the same control volume as above and for the horizontal advective and Coriolis portions of (6) are

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after phase averaging and factoring out ΔxΔyΔz.

The velocity in the first (tendency) term and last (Coriolis) term of (13) is U_α as determined in (10a) and (11a). Now consider the second bracketed term,

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or

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In arriving at (14), a term of order (ka)⁴ relative to terms of order (ka)² has been neglected. For the third bracketed term in (13),

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Using (14) and (15), (13) may be written as

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b. The pressure terms

Dealing with the pressure term in (6) is complicated. From the vertical component of momentum, we have

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For the region −h < z < η̃, the mean hydrostatic equation is

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After integration, (17) yields

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Equations (17) and (18) were derived by L-HS and Phillips (1977). The kinematic atmospheric pressure (divided by the reference water density) is p_atm. [As a check, p̂(η̂) + = p_atm is obtained by applying the integral momentum equation to a thin control volume that includes the air–sea interface.]

In M03, the wave pressure field was treated similarly to the velocity as in (7) and (8) and as illustrated in Fig. 1, where the velocity field fills the entire region, η > z > −h, but a similar interpretation is not possible for wave pressure given by (1a) and (1d); see Fig. 2. Note that P̃(η̂) − gη̃ = 0, so that wave pressure is nil at the surface. In the shaded region of Fig. 2, denoted by −|η̃| < z − η̂ < |η̃|, the pressure is evidently hydrostatic, as noted by L-HS, and the entire field can be described by

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where p̂(z) is given by (18) and P̃(z) by (1d). In a trough, overlapping regions as given by (19) are conceptually unattractive but are nevertheless dictated by (1a) and (1d).

Because p̂ and P̃ are functions of z (and not s), the phase-averaged contribution of for −h < z < η̂ is nil. However, in the region −|η̃| < z − η̂ < |η̃|, a phase-averaged integral of (19) exists and is

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so that using (18),

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where a modified Dirac delta function is defined such that

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[In a finite difference rendering of E_D, the top vertical layer of incremental size, δ z—and only the top layer—would be occupied by ∂E_D/∂x_β = (δ z)⁻¹∂(E/2)/∂x_β.]

c. The phase-averaged momentum equation

Inserting (16) and (21) into (6)—after factoring out ΔxΔyΔz—yields

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where

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which is implicit in the L-HS derivation after vertical integration.

As in M03, it is convenient to define the following terms:

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{For deep water (kD ≫ 1), F_SS = F_CS = F_SC = F_CC = exp[k(z − η̂)].} Then, substituting (1) into (24a) yields

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In (23), the buoyancy term, where b ≡ gρ̂/ρ₀, has been added; it could have been included in (13), but was omitted to simplify the subsequent discussion. It is assumed that the waves are not affected by buoyancy, or more precisely, that N ²/σ² ≪ 1 in regions occupied by waves; N ² ≡ −∂b/∂z is the Brunt–Väisälä frequency.

Note that ∫^η̂_−h S_αβ dz = E[(k_αk_β/k²)(c_g/c) + δ_αβ(c_g/c − 1/2)] as in Phillips (1977).

3. Vertical wind stress transport

Thus far, the surface wind pressure has not been considered. However, the horizontal (kinematic) surface wind stress τ_α(η̂) can be divided into a turbulence-viscous part or skin friction τ_Tα(η̂) and a pressure part or form drag τ_Pα(η̂); that is,

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where is an empirical momentum mixing coefficient. The dynamic stress ρ_wτ_α, where ρ_w is seawater density, is continuous across the air–sea interface. The form of (25b), although conventional, is problematic and is subject to further research.

The component of the wind pressure fluctuation that correlates with ∂η̃/∂x_α in (25c) is P̃_wη̃ = p_w0 sinψ, and its subsurface continuation is P̃_w(z) = p_w0F_CC sinψ in (1d) [which for horizontally homogeneous, deep water is implicit in a formula in Weber (1983), albeit expressed in Lagrangian coordinates]. The subsurface continuation of ∂η̃/∂x_α is (4), so that

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or

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and it conforms to (25c) at z = η̂. The surface stress is empirical; however, the transport of this pressure stress into the water column is now a known function, unlike turbulence transport.

On sufficiently rough stationary surfaces, form drag dominates over skin friction (Schlichtng 1979), and this is assumed to prevail over wave surfaces by Smith (2006), Donelan (1999), and others for wind speeds greater than some threshold value (3 to 5 m s⁻¹). On the other hand, Janssen (1989) indicates that form drag, or “wave-induced stress,” dominates only for young waves (c_p/u_* ≅ 5, where c_p is the spectral peak phase speed and u_* is the friction velocity), while skin friction dominates for old waves (c_p/u_* ≅ 25).

4. The wave energy equation

The wave energy equation is derived in appendix A and is

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The terms on the right must be determined empirically; the first is a wind source term and the second term is dissipation.

The energy advective velocity, as defined in appendix A, is

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and r(z) is a weight factor biasing the evaluation of u_Aα toward the surface velocity and k∫^η̂_−h r(z) dz = 1. [In deep water, r = 2 exp2k(z − η̂).]

It has been suggested that instead of using the wave energy Eq. (27), the wave action equation—conventionally used in many models—be adopted, the presumption being that the third term on the left of (27) would neatly disappear. However, because U_α is not vertically constant, the wave action equation would be insensitive to vertical profiles of U_β, unlike (27). The wave action equation is derived in appendix B with vertical velocity gradients included.

Appendix B also contains Eqs. (B.4a) and (B.5), which can be solved along with (27) and (28) to provide the intrinsic frequency and wavenumber. Alternatively for steady flow, the “encounter frequency” ω is spatially constant according to (B.2), and the simpler Eqs. (B.1), (B.3), and the dispersion relation can be used.

5. The sigma equations

The relation between the vertical Cartesian and sigma (using ς instead of σ) coordinate is

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Transforming (12) to sigma coordinates, we have

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and for (23)

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The definition of S_αβ remains the same as in (24b), noting that (z + h) = D(1 + ς). The term Ω is a velocity normal to sigma surfaces and Ω(η̃) = Ω(−h) = 0.

6. The case posed by Ardhuin et al.

As mentioned previously, this paper was stimulated by Ardhuin et al. (2008a), who cited a solution from a multimode model by Belibassakis and Athanassoulis (2002) in which currents and waves were unidirectional and propagated into a straight entry channel of 6-m depth, which smoothly transitioned to a straight exit channel of 4-m depth. Although the algorithm was complicated, the solution was simple and deemed accurate. The wave frequency was selected so that kD varied from 1.10 to 0.85, a shallow-water case; the group velocity was nearly constant and so was the wave energy (see Fig. 3.4 in Phillips 1977). They pointed out that the radiation stress terms in M03 produced a vertical gradient of mean velocity greater than zero, contrary to that of the multimode solution.

From the discussion in section 2, some error is to be expected for finite ∂h/∂x. However, the results of this paper now agree with the multimode solution because, for the steady, unidirectional, irrotational case of Ardhuin et al., we have from (23)

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and from (24b)

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because cosh²k(z + h) − sinh²k(z + h) = 1. Thus, the vertical structure vanishes as in the multimode solution; the radiation term is balanced by the hydrostatic pressure gradient (in the 6- to 4-m transition section there must be some nonhydrostatic effects). The singular term, E_D in (24b), is excluded because ∂E/∂x ≅ 0 in this case.

Thus, there is a good possibility that the equations in this paper do apply to shallow water for kD ≈ 1 (where, realistically, viscous-turbulence effects should not be ignored).

7. Summary

The above results differ from M03 in several ways. Horizontal derivatives of bottom depth were retained in the M03 equivalent of (4), which is inconsistent with the derivation of (1) based on a flat bottom. The terms and E_D were missing in the M03 version of (24a) and a P̃ − S̃_α correlation term erroneously substituted. The derivation and definition of U_α to include currents and Stokes drift is unchanged. An important consequence here is that for large kD, the momentum equation is not sensitive to finite ∂h/∂x_α as was the M03 version. For kD ≈ 1, some discrepancy relative to the case posed by Ardhuin et al. (2008a) is expected, but in fact the discrepancy is nil.

The terms , , and E_D in (24a) are, in vertically integrated form, the same as those in the derivations of Longuet-Higgins and Stewart (1962, 1964) and Phillips (1977).

Stokes drift, given by (10c), is the same as the result from the Lagrangian determination, u_Sα = , where X̃ = ∫ũdt and z̃ = ∫w̃dt.

The basis for a coupled wave-circulation model are, in summary, provided by Eqs. (12), (23), (24), (26), (27), and (28). Empirical knowledge is needed for τ_Tα, S_W and S_Diss. The equation for mean temperature, ∂T/∂t + ∂(U_βT)/∂x_β = ∂[K_H(∂T/∂z)]/∂z—or any other scalar—appropriately uses U_β = û_β + u_Sβ as the advective velocity.

For a practical wave model, Eqs. (27) and (28) should be extended so that wave energy is dependent on wavenumber or frequency and wave propagation directions. Existing third-generation wave models (e.g., Tolman 1991) might be modified to conform to (27) and (28). Alternatively, a conforming, somewhat simplified wave model has been created (Mellor et al. 2008) and has since been coupled with the Princeton Ocean Model.