## 1. Introduction

On the ocean surface, the correlation of wind pressure fluctuations and wave slope forces wave generation but also is a forcing term in the phase-averaged momentum equation. Phase averages of wave motion yield residual particle displacements (Stokes drift) identical to current displacements. The difference between the instantaneous wave velocity minus its phase average leads to the so-called radiation stress, first identified by Longuet-Higgins and Stewart (1962, 1964, hereinafter L-HS), which modifies the momentum equation much like turbulence Reynolds stress but, in addition, includes a component due to pressure.

L-HS, Phillips (1977, hereinafter Phillips), and Smith (2006) derived vertically integrated, wave radiation stress terms as an addition to the fluid dynamic equations of motion rendering them suitable for two-dimensional oceanic flows with a wind-driven wave surface. These derivations start from the basic vertically integrated, fluid dynamic equations after which the total velocity is partitioned into current and wave velocities resulting in phase-averaged equations applicable to vertically independent ocean models. A characteristic of the aforementioned papers is that prognostic equations are obtained for the combined (Eulerian) current and Stokes drift.

Following L-HS, this paper presents a concise review of the vertically integrated wave circulation equations but then derives the corresponding, vertically dependent equations directly from their integrated counterparts. The purpose is to demonstrate the intimate connection between the two equations sets and, at the same time, provide a corroborative and somewhat simpler derivation of the vertically dependent wave–current interaction equations in Mellor (2003). An error in the L-HS formulation that, however, disappeared in their final equations led to a similar error in the formulation by Mellor (2011) but is corrected here and in Mellor (2013a).

The basic equations of fluid dynamics are cited in section 2, wherein velocities and pressure are partitioned into their mean and wave components. Section 3 and appendix A are devoted to an expression for pressure, an important element of the ensuing derivations, which distinguishes this paper from papers that present an alternate expression for pressure. The vertically integrated equations are cited in section 4, from which the phase-averaged integral equations of L-HS are derived in section 5. The same vertically integrated equations are the basis of the derivation of vertically dependent, three-dimensional equations derived in section 6. Some relationships between the derivations of sections 5 and 6 are discussed in section 7, wherein a subtle but important error in the L-HS derivation is revealed. For convenient reference, the complete phase-averaged, three-dimensional equations are summarized in section 8, and small and neglected errors related to surface wave slope and bottom topographical slope are discussed in section 9. In section 10 and appendix B, there is discussion of theories by McWilliams and Restrepo (1999), Newberger and Allen (2007a,b), and Ardhuin et al. (2008) that apply to vertically dependent ocean models but which diverge from the findings of the present paper and, when vertically integrated, diverge from the findings of L-HS and Phillips.

Incorrect rendering of the pressure term in Mellor (2003) was later corrected in a PDF file (available online at http://shoni2.princeton.edu/ftp/glm/Corrected2003.pdf) and in Mellor (2013a). As mentioned above, a similar error occurs in L-HS; nevertheless, their final radiation stress result is correct, as is the final result in Mellor (2003).

## 2. The basic equations

*z*is the vertical coordinate;

*t*is time; the three-dimensional velocity is

*p*is the kinematic pressure; and

*g*is the gravity constant. Repeated subscripts denote summation, for example,

*a*is the amplitude;

*h*is the depth. Also,

We denote phase averages by an overbar; thus,

## 3. Pressure

*ka*)

^{4}will be neglected and represent an error of the same order. However, the phase average of (6) is exactly

*ka*, the free-surface pressure

Equation (6) differs from that in Mellor (2003), which was corrected in a PDF file (available online at http://shoni2.princeton.edu/ftp/glm/Corrected2003.pdf) and in Mellor (2013a). It also differs, for example, from the treatment of pressure in McWilliams and Restrepo (1999) and Ardhuin et al. (2008).

## 4. The vertically integrated equations of motion

## 5. The derivations as in L-HS and Phillips

The derivation of the vertically integrated equations is reviewed here. L-HS, in their pursuit of the then novel idea of radiation stress terms, generally neglect the nonlinear advective terms as in (8). The book by Phillips is rich in its omnibus description of surface (and internal) waves. However, his derivation of phase-averaged momentum equation is spread around different pages and a derivation of an important element of the stress radiation term [the last term in (16)] is missing. This term and its vertically distributed counterpart has been a source of controversy in the literature, as discussed in sections 7 and 9.

### a. The continuity equation

### b. The momentum equation

^{1}

*O*(

*ka*)

^{4}, which can be neglected. [Phillips initially includes the term within

## 6. The vertically dependent equations

*x*,

*y*, and

*z*coordinates to

*x*,

*y*, and

*σ*for frequency). For

*ka*,

*ka*. Similarly,

*D*signifies that (20a), (20b), and (20c) are confined to the phase-averaged vertical space

### a. The continuity equation

^{2}

### b. The momentum equation

*∂*

*x*

_{α}, but that term can be neglected since

*gD*(see footnote 1). Now using (18) and (19c), we have

*ka*)

^{4}and is neglected. The term under the phase-averaged bar in (25) can be evaluated as

*ka*)

^{4}and can be neglected. Thus, we arrive at

^{3}to conform to established fluid dynamic theory.

## 7. Relationships between the integral and differential equations

One advantage of sigma coordinate equations is that vertical boundary conditions are “built in,” and they are easy to vertically integrate. Thus, reintegration of (24) and (30) readily yields (11) and (15).

At this point, it is appropriate to point out that the derivation of L-HS introduced a subtle error in their determination of

## 8. Summary of the full, vertically dependent equations

*f*is the Coriolis parameter. The last term on the left side is the baroclinic term—familiar to users of sigma coordinate equations—where

The vertical integral of (33) is (17). For deep water, as in *kD* > 3), all of the *F* terms in (34) limit to

The pressure–slope transfer term in (32),

After wave parameters have been calculated, Stokes drift can be obtained from (22) and then the current

*T*is any scalar variable, and

*q*is the vertical flux of that variable. If

*T*is taken to be temperature, then

*q*is vertical heat flux including penetrative solar radiation. Equation (35) is derived in a similar way as the continuity equation in section 6.

In the preceding developments, sigma coordinates have been a convenience. However, the end results (31), (32), and (35) can be transformed to Cartesian coordinates for which reference is made to appendix A of Mellor (2005; wherein the term

*r*is a weighting function so that

When dealing with wave spectra, *F* terms in (34) are needed to feed into (32) and (33). This does create a burden on computational resources in addition to the need to deal with five independent variables and the need to deal with frequency shifts due to wave–wave interaction. An approximation whereby the spectral shape is parameterized has been proposed in Mellor et al. (2008) to greatly reduce the computational burden.

## 9. Errors of order (*ka*)^{4}

In all developments of phase-averaged equations, it is assumed that temporal and spatial scales of amplitudes are small, that is, *k* and

Throughout this paper’s development, wave slope *ka* has been stipulated to be small. The wave terms contained in (29) or (33) and the addition of the Stokes drift term to *ka*)^{2}. Terms of order (*ka*)^{3} are identically zero since they are coupled with triplet products of sines or cosines whose phase averages are nil. Terms of order (*ka*)^{4} have been neglected throughout the paper. Relative to the retained terms, the neglected terms represent an error of order (*ka*)^{2} = *O*(10^{−2}). Note that, ab initio, waves are represented by only the first term in the Stokes series, as in (5); including the second-order term would again introduce terms of neglected order.

An error is embodied in the linear wave solutions [(5a)–(5d)], which are based on the boundary condition *ka*)^{4}, one should require that *kD* > 3.

## 10. Prognostic equations for the Eulerian velocity

The developments in this paper and those of L-HS and Phillips predominantly deal with prognostic equations for

A difference is the treatment of pressure. The McWilliams and Restrepo (1999) derivation begins with the curl of the momentum equation wherein the pressure gradient term disappears; then, after considerable manipulation, the resulting phase-averaged vorticity equation is “uncurled” and a gradient pressure term can be reintroduced, but their choice of a new, phase-averaged pressure differs from that derived in appendix A and cited in (6). In McWilliams and Restrepo (1999) and Ardhuin et al. (2008), the Stokes vertical component is nonzero so that, instead of (31), the divergences of

Ardhuin et al. (2008) include terms proportional to

The theoretical development in Newberger and Allen (2007a) is quite complicated, but the final equation set in appendix B of Newberger and Allen (2007b) is not. These equations are meant to apply to shallow water (*kD* ≪ 1). The momentum equation for Eulerian current is vertically dependent but is forced by vertically integrated terms like (16) but where a vortex force term replaces the first term on the right side of (16). Also, gradients of component Stokes drift are concentrated at the surface and are a surface boundary condition for the underlying continuity equation. Additional modeling for breaking wave rollers is added to the wave energy equation.

A relatively recent paper by Aiki and Greatbatch (2013) seems to have bearing on the approach of this paper and the 2003 paper, but the paper is quite complicated and I could not extract equations such as (31) and (32) for comparison. However, they correctly complained about my earlier characterization (born out of the now discovered error in L-HS) of a portion of

The pressure–slope term

Bennis et al. (2011) have created an inviscid barotropic, shallow-water

## Acknowledgments

The paper was significantly improved because of reviewer comments.

## APPENDIX A

### Derivation of (6)

*z*to

*O*(

*ka*)

^{2}, we have retained the

*ka*)

^{4}have been neglected. Note that whereas the phase resolved (A4) is needed, the phase average of (A4) could have been obtained directly and exactly from (A1) since the phase averages of

## APPENDIX B

### Derivation of a Prognostic Equation for

This is an effort to derive a prognostic equation for

*E*to

*r*is a weighting function so that

*i*and

*j*instead of

*J*.

It is now recognized that (B6) coincides with the equations of Newberger and Allen (2007b) if the left side of (B6) remains vertically dependent but velocities on the right side are assumed be vertically constant. Thus, the first term and the third term (for

The left side of (B6) also corresponds to the equations of McWilliams and Restrepo (1999; however, advective terms are excluded), McWilliams et al. (2004), or Ardhuin et al. (2008). [In the latter case, refer to Bennis et al. (2011), which offers simplifications to the 2008 paper.] The right side agrees with all of the above authors’ equations only if one approximates

Conversely, after vertical integration of the equation of McWilliams and Restrepo (1999), Ardhuin et al. (2008), or Bennis et al. (2011) and use of (B3), there seems to be no way to bring them into agreement with those of Phillips or Smith (2006).

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^{1}

Note that partial differentiation is taken outside of the integral because

^{2}

As in Mellor (2003), the sigma coordinate (nearly) vertical velocity is *W* and

^{3}

Omitting