## 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

*η̃*, velocity (

*ũ*

_{α},

*w̃*), and kinematic pressure

*P̃*(dynamic pressure divided by a reference water density,

*ρ*

_{0}), as follows:

*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*

_{β}

*x*

_{β}−

*ω t*;

*k*

_{β}and

*ω*are directional wavenumber and frequency, such that

*ω*=

*σ*+

*k*

_{β}

*Û*

_{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

*σ*

^{2}=

*gk*tanh

*kD*and is an example of similar manipulations below. Note that

*P̃*(

*η̂*) =

*gη̃*.

*S̃*/∂

*t*=

*w̃*. The vertical derivatives are

*z*=

*η̂*, (2b) yields

*S̃*=

*a*cos

*ψ*=

*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*)/sinh*kh* 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*)/sinh*kh*]^{2} is small; this could be a problem for small *kh*; however, see section 6.

*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.

*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

*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

Overbars will represent phase averaging: *π*)^{−1}∫^{2π}_{0}( )*dψ* (so that, e.g.,

*x*,

*x*+ Δ

*x*;

*y*,

*y*+ Δ

*y*and

*s*,

*s*+

*s*

_{z}Δ

*z*. On the

*s*surfaces, the unit vectors are (

*n*

_{β},

*n*

_{z}) = (

*s*

_{β},

*s*

^{2}

_{β}

*s*

_{β}, 1 −

*s*

^{2}

_{β}/2). Applying (5) to this control volume and phase averaging, one obtains

*x*Δ

*y*Δ

*z*has been factored out of the equation. Notice that (∂

*f*/∂

*s*)Δ

*s*= (∂

*f*/∂

*z*)(∂

*z*/∂

*s*)

*s*

_{z}Δ

*z*= (∂

*f*/∂

*z*)Δ

*z*.

*û*

_{α}+

*ũ*

_{α}+ (∂

*ũ*

_{α}/∂

*z*)

_{z}

*S̃*, and the flow area, (1 +

*S̃*

_{z})Δ

*z*Δ

*y*(if

*α*=

*x*). This is illustrated in Fig. 1. Further, using (1b) and (2b),

*σ*

^{2}=

*kg*tanh

*kD*and the definition of wave energy

*E*=

*g*

*ga*

^{2}/2 [see (A.8)]. For deep water,

*u*

_{Sα}= (2

*k*

_{α}

*E*/

*c*) exp[2

*k*(

*z*−

*η̂*)].

*x*Δ

*y*Δ

*z*.

*U*

_{α}as determined in (10a) and (11a). Now consider the second bracketed term,

*ka*)

^{4}relative to terms of order (

*ka*)

^{2}has been neglected. For the third bracketed term in (13),

### b. The pressure terms

*h*<

*z*< η̃, the mean hydrostatic equation 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.]

*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*) −

*g*η̃ = 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

*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).

*p̂*and

*P̃*are functions of

*z*(and not

*s*), the phase-averaged contribution of

*h*<

*z*<

*η̂*is nil. However, in the region −|

*η̃*| <

*z*−

*η̂*< |

*η̃*|, a phase-averaged integral of (19) exists and is

*E*, the top vertical layer of incremental size,

_{D}*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

*x*Δ

*y*Δ

*z*—yields

*kD*≫ 1),

*F*

_{SS}=

*F*

_{CS}=

*F*

_{SC}=

*F*

_{CC}= exp[

*k*(

*z*−

*η̂*)].} Then, substituting (1) into (24a) yields

In (23), the buoyancy term, where *b* ≡ *gρ̂*/*ρ*_{0}, 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}/*σ*^{2} ≪ 1 in regions occupied by waves; *N* ^{2} ≡ −∂*b*/∂*z* is the Brunt–Väisälä frequency.

Note that ∫^{η̂}_{−h} *S*_{αβ} *dz* = *E*[(*k*_{α}*k*_{β}/*k*^{2})(*c*_{g}/*c*) + *δ*_{αβ}(*c*_{g}/*c* − 1/2)] as in Phillips (1977).

## 3. Vertical wind stress transport

*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,

*ρ*

_{w}

*τ*

_{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.

*η̃*/∂

*x*

_{α}in (25c) is

*P̃*

_{wη̃}=

*p*

_{w0}sin

*ψ*, and its subsurface continuation is

*P̃*

_{w}(

*z*) =

*p*

_{w0}

*F*

_{CC}sin

*ψ*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

*z*=

*η̂*. The surface stress

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

*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 exp2

*k*(

*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

*ς*instead of

*σ*) coordinate is

*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.

*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)

^{2}

*k*(

*z*+

*h*) − sinh

^{2}

*k*(

*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*in (24b), is excluded because ∂

_{D}*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 *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 *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} = *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*_{β}*T*)/∂*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.

## Acknowledgments

I thank Ardhuin, Jenkins, and Belibassakis for their commentary that led to this paper. The funding support of NSF Grant OCE05-26508 is appreciated. Comments by reviewers and J. A. Smith were helpful.

## REFERENCES

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## APPENDIX A

### Derivation of the Energy Equation

*u*

_{k}/∂

*x*

_{k}= 0, the product of

*u*

_{i}and ∂

*u*

_{i}/∂

*t*+

*u*

_{k}∂

*u*

_{i}/∂

*x*

_{k}+ ∂(

*p*+

*gz*)/∂

*x*

_{i}= ∂

*τ*

_{β}/∂

*z*is

*z*= −

*h*to

*z*=

*η*and use

*w*(

*η*) =

*u*

_{β}(

*η*)∂

*η*/∂

*x*

_{β}+ ∂

*η*/∂

*t*and

*w*(−

*h*) = −

*u*

_{β}(−

*h*)∂

*h*/∂

*x*

_{β}. Assuming that ∂

*η̃*/∂

*t*≪ ∂

*η̃*/∂

*t*, we have after phase averaging

*p*

_{wη}as in section 3, and the last term on the right is the rate of work done by wind pressure.

*h*to

*η̂*and

*η̂*to

*η*(Phillips 1977); evaluating the velocity terms proved complicated. As a reasonable approximation, replace the velocity integrals in (A.2) so that

^{η}

_{−h}

*dz*= ∫

^{η̂}

_{−h}

*s*

_{z}

*dz*=

*η̂*+

*η̃*+

*h*, so that the role of

*s*

_{z}is to span the entire vertical range even though the upper limit has been changed. Because the pressure terms differ analytically in the regions −

*h*<

*z*<

*η̂*and −|

*η̃*| <

*z*−

*η̂*< |

*η̃*| as given by (19), the pressure integrals are similarly divided.

*û*

_{α}=

*U*

_{α}−

*u*

_{Sα},

*ka*)

^{4}has been expunged. Similarly,

*û*

_{β}+

*U*

_{β}, so that using (18),

*U*

_{β}

*E*

_{D}for

*û*

_{β}

*E*

_{D}introduces an error of order (

*ka*)

^{4}.

*U*

_{α}and (23) yields a mean flow (current plus Stokes) energy equation:

*E*=

*ga*

^{2}/2.

*c*

_{g}= (

*c*/2)(1 + 2

*kD*/sinh2

*kD*) is the group speed and

*c*

_{gβ}=

*k*

_{β}

*c*

_{g}/

*k*. An energy advective velocity is defined such that

*U*

_{β}as contributions to

*u*

_{Aβ}.

*S*

_{W}is the wind energy source defined below and

*S*

_{Diss}is dissipation.

#### Wind energy source

*û*

_{β}∂

*z*, to which we add the term

*U*

_{β}=

*û*

_{β}+

*u*

_{Sβ}, we have

*η̃*/∂

*t*= −(

*ωk*

_{β}/

*k*

^{2})∂

*η̃*/∂

*x*

_{β}and

*ω*=

*σ*+

*k*

_{α}

*Û*

_{α}. The last term in (A.11) is order (

*ka*)

^{2}smaller than the phase speed, and the wave energy forcing can be represented by [

*c*

_{β}+ (

*k*

_{α}

*k*

_{β}/

*k*

^{2})

*Û*

_{α}]

*Û*

_{β}| ≪ |

*c*

_{β}|; otherwise,

*Û*

_{β}=

*u*

_{Aβ}should be a good approximation.

## APPENDIX B

### Derivation of the Wave Action Equation

#### Wave kinematics

*ω*, intrinsic frequency

*σ*, wavenumber vector

*k*

_{β}= (

*k*

_{x},

*k*

_{y}), and the Doppler velocity

*Û*

_{α}, is

*ψ*=

*k*

_{α}

*x*

_{α}−

*ωt*,

*σ*=

*σ*(

*k*,

*D*), the relations ∂

*k*/∂

*x*

_{α}= (

*k*

_{β}/

*k*)∂

*k*

_{β}/∂

*x*

_{α},

*c*

_{gβ}= (

*k*

_{β}/

*k*)(∂

*σ*/∂

*k*), and (B.3), one obtains a wavenumber equation:

*σ*/∂

*t*+

*c*

_{gβ}∂

*σ*/∂

*x*

_{β}, using

*σ*=

*σ*(

*k*,

*D*) again and (B.4b) and canceling two equal but opposite terms containing ∂

*D*/∂

*x*

_{β}, yields the intrinsic frequency equation

#### The wave energy and action equations

*Û*

_{β}=

*u*

_{Aβ}in the left side of (B8); otherwise, the final result in (B.11) will contain additional terms proportional to

*Û*

_{β}−

*u*

_{Aβ}.

If *U*_{β}(*z*) is vertically constant and equal to _{β} and *Û*_{β}, then the horizontal current gradient in the first term on the right of (B.11) can be taken outside of the integral, so that all of the terms on the right of (13) cancel (Mei 1983) and one obtains the conventional wave action equation. But, generally, in a three-dimensional ocean, *U*_{β}(*z*) ≠ *Û*_{β} ≠ _{β}.

The pressure field. Below *z* = 0 (here *η̂* is set to zero), solid lines are contours of constant pressure (solid lines are positive *P̃*, dashed lines are negative) according to (1d), which, at *z* = 0, supports hydrostatic pressure in the shaded regions; i.e., *P̃*(*z* = 0) = *gη̃*.

Citation: Journal of Physical Oceanography 38, 11; 10.1175/2008JPO3971.1

The pressure field. Below *z* = 0 (here *η̂* is set to zero), solid lines are contours of constant pressure (solid lines are positive *P̃*, dashed lines are negative) according to (1d), which, at *z* = 0, supports hydrostatic pressure in the shaded regions; i.e., *P̃*(*z* = 0) = *gη̃*.

Citation: Journal of Physical Oceanography 38, 11; 10.1175/2008JPO3971.1

The pressure field. Below *z* = 0 (here *η̂* is set to zero), solid lines are contours of constant pressure (solid lines are positive *P̃*, dashed lines are negative) according to (1d), which, at *z* = 0, supports hydrostatic pressure in the shaded regions; i.e., *P̃*(*z* = 0) = *gη̃*.

Citation: Journal of Physical Oceanography 38, 11; 10.1175/2008JPO3971.1