## 1. Introduction

Previous to this paper, surface boundary layer models (e.g., Mellor and Yamada 1982; Large et al. 1994) generally assumed that momentum transfer from surface wind stress into the water column is due to turbulence Reynolds stress. Here, we examine the finding of Mellor [(2003), a corrected version can be found in

To focus on vertical transfer processes, most surface boundary layer models are initiated assuming horizontal homogeneity wherein the model algorithm and the relevant empiricism are introduced and tested against available data. Because this paper examines a new element of surface wave boundary layer physics, horizontal homogeneity will also be assumed.

In section 2, a phase-resolved analytical solution of a stationary, wind-forced wave field in the absence of turbulence is developed. Surface wind pressure penetrates the water column. Then, in section 3, after phase averaging the information of section 2, it is seen that momentum is transferred into the water column entirely because of wave pressure acting on sloping material surfaces resulting in Stokes drift. When interpreted in terms of surfaces of constant *z* rather than material surfaces, a nonzero wave correlation

In section 4, a model, again specialized for horizontal homogeneity but including waves, currents, and turbulence, is presented and executed numerically. The interplay between pressure–slope- and turbulence-supported momentum transfer is demonstrated. For the turbulence portion, an augmented Mellor and Yamada (1982) model will be used. For the wave portion of the complete model, we will use the wave model developed by Mellor et al. (2008, hereafter MDO). Because horizontal homogeneity is prescribed, elements such as wave radiation stress terms are absent.

Comparisons with temperature data from station Papa (Martin 1985) are in section 5. There is uncertainty associated with the assumption of horizontal homogeneity because possible effects of advection are excluded. Of course, there are also uncertainties associated with the empirical content of all models.

## 2. A simple phase-resolved wave model

In this section, an analytical solution is obtained for a phase-resolved problem relevant to near-stationary (fully developed) waves wherein surface forcing is balanced by a simple momentum sink. The focus is on determining the subsurface pressure distribution and the effect of wind on wave properties.

*t*is time; (

*x*,

*z*) is the horizontal and vertically upward coordinates respectively; (

*u*,

*w*) are the velocity components in the (

*x*,

*z*) directions;

*p*is kinematic pressure (dynamic pressure divided by density);

*g*is the gravity constant; and

*r*is a simple, constant coefficient for Rayleigh drag that, it will be seen, counterbalances form drag. To simplify nomenclature and because mean properties will be horizontally homogeneous, we let the mean water level be nil.

*ka*≪ 1) so that

^{1}thus, from (7) for

## 3. Phase-averaged equations

The term

*c*is the phase speed. After inserting (10) and (12) into (11), the vertically dependent parts of both sides of (11) cancel identically so thatAlternately, if one multiplies (2a) by

*u*, (2b) by

*w*, adds the two equations, integrates from

*z*= −

*h*to 0, phase averages, and makes further manipulation, one obtains (13) again. For the present problem of section 2,

Whereas the Rayleigh terms, the second terms on the right of (11) and (12), are a convenient invention and differ from reality as seen in section 4, the first terms are derived from (1) and (2a) and (2b) exclusive of the Rayleigh terms and are considered realistic. Note also that here the momentum and energy equations are not independent.

*ds*is an elemental surface area whose normal unit vector is

*L*is a wavelength, the

*x*component of the momentum flux is

## 4. A phase-averaged coupled wave–current model

The foregoing theoretical analysis introduced the idea of momentum transfer into the water column through pressure acting on material surfaces in the absence of turbulence transfer. In the following, the gradient

### a. The mean or total momentum equation

In this and the following subsections, the turbulence closure model of Mellor and Yamada (1982) is modified to include pressure–slope forcing and the inclusion of wave properties in surface boundary conditions. Another paper is relevant: in Mellor (2001), an extensive summary of the model is presented and a litany of problems is included, other than the surface boundary layer problem, for which the model has been applied.

*f*is the Coriolis parameter; and

*αβz*=

*xyz*, = −1 if

*αβz*=

*yxz*, or = 0 if

*αβz*=

*xxz*or

*yyz*). The subscript

*x*or

*y*, whereas

*z*is the vertical coordinate pointing upward from the sea surface. In (16) the turbulence stress term is

*kh*≫ 1), wherein

The switch from

### b. The temperature equation

*R*is the solar penetrative radiation flux. The turbulence-based heat flux (pressure is absent in scalar equations and so is pressure–slope transfer) iswhere

*K*is a mixing coefficient for heat flux defined below.

_{H}### c. The turbulence kinetic energy equation

### d. The mixing coefficients

From the above, it will be seen that wave information is required. Specifically, the significant wave height and wave age is needed in (21) and the vertical dependence of the momentum pressure transfer in (17) requires the peak wavenumber.

### e. The wave model

We invoke the phase-averaged, nonlinear wave model of MDO. The model is based on a parameterization of the shape of the frequency spectrum according to Donelan et al. (1985) so that one deals with the wave energy

*a*= 0.925 was determined empirically in MDO by reference to fetch data and

Most of the wind input energy is dissipated in situ. Aside from its wave age dependence, (31a) resembles the dissipation suggested by Craig and Banner (1994) after integration over all wave angles.

## 5. Station Papa data

We now appeal to the data from Weather Station Papa (50°N, 145°W) analyzed by Martin (1985) for the year 1966. The wind stress is as described above but the heat flux at the surface is that calculated by Martin using climatological radiation and conventional bulk surface formulas. The vertical distribution of ocean velocities and attendant variables on the one-dimensional grid, −200 < *z* < 0, includes 40 evenly distributed grid points except for the topmost 7 points that are logarithmically distributed. The wave variables are on the grid

A time series for wind speed and direction are plotted in the top two panels in Fig. 2, and the calculated significant wave height and mean wave propagation direction are shown in the bottom panels for the month of January when the winds are quite strong.

Figure 3 is a conventional plot of drag coefficient versus wind speed sampled at 3-h intervals in January. The scatter is due to the dependency of *C _{D}* on significant wave height and wave age in (21); nevertheless, it is somewhat remarkable that a complicated path through the wave variables and (21) should closely adhere to the simple

Figure 4 presents sample profiles of current *z* = −110 m—akin to the nocturnal jet in the atmosphere—is the result of an upward progression of the interface from unstable forced to stable unforced portions of the water column.

In Fig. 5 are sample plots of

Figure 6 compares station Papa yearlong temperature data with calculations. The short-wave motion, presumably internal waves, is missing; otherwise the comparison is quite favorable. Finally, the station Papa monthly-averaged surface temperature is compared with data in Fig. 7 in the manner presented by Martin (1985).

## 6. The generation of Stokes drift and Eulerian current

Reinstating *f*, a third time step can accommodate changes due to the Coriolis term.

## 7. Summary

In this paper, momentum transfer into an oceanic water column via pressure acting on material wave surfaces is demonstrated by a phase-resolved analytical model simplified by excluding currents, turbulence, and horizontal variability. The results are phase averaged and partially account for terms in a more complete model.

There then followed a description of a phase-averaged model complete with waves coupled to an ocean surface boundary layer model wherein momentum and energy is transferred to and from waves and the underlying water column.

We then compared with the data of Martin (1985), a well-known and much-cited dataset. Other data are included in Mellor (2001) and experience indicates that a model that performs well with the Martin data also does well generally. It is unfortunate that concomitant surface wave data are not available to compare with the calculated data. But as noted above, the wave model has reproduced fetch- and duration-limited data and hurricane buoy data so it is presumed that the calculated wave properties are reasonable. However, the main focus of the paper is on coupling waves to surface boundary layer dynamics and on the combined but separate role of turbulence- and pressure–slope-supported stresses as demonstrated in Fig. 5.

It is noted that there are no adjustable constants in the Mellor–Yamada turbulence closure model, which covers many different flow problems. However, with the addition of surface gravity waves, an adjustable constant is introduced in the relation

Scientific progress does require improved physical description and understanding of oceanic processes whenever possible.

## Acknowledgments

Two reviewers made important suggestions that improved the paper.

## APPENDIX A

### Derivation of (11)

Despite the simplicity of (11), its derivation is rather complicated but simpler than that in Mellor (2003) in that currents are nil (

*u*or

*p*is represented by

*u*and

*p*transform according toso that (2a) may be transformed toAfter rearrangingThe equations for wave velocity,are exact irrotational solutions (Mellor 2011) to (1) and (2) in the region

*u*and

The key to the above derivation is that phase averaging *x* and the fixed

## APPENDIX B

### The Relation

*C*is determined using the Garrett relation cited in Fig. 3. The reader is referred to their Fig. 6 and it will be seen that Fig. B1 is a reasonable representation of their Fig. 6; the greatest discrepancy is noted for small values of

_{D}## REFERENCES

Booij, N., , R. C. Ris, , and L. H. Holthuijsen, 1999: A third-generation wave model for coastal regions: 1. Model description and validation.

,*J. Geophys. Res.***104**(C4), 7649–7666.Buckles, J., , T. J. Hanratty, , and R. Adrian, 1984: Turbulent flow over large-amplitude wavy surfaces.

,*J. Fluid Mech.***140**, 27–44.Craig, P. D., , and M. L. Banner, 1994: Modeling wave-enhanced turbulence in the ocean surface layer.

,*J. Phys. Oceanogr.***24**, 2546–2559.Donelan, M. A., 1999: Wind-induced growth and attenuation of laboratory waves.

*Wind-over-Wave Couplings,*S. G. Sajjadi, N. H. Thomas, and J. C. R. Hunt, Eds., Clarendon Press, 183–194.Donelan, M. A., , J. Hamilton, , and W. H. Hui, 1985: Directional spectra of wind-generated waves.

,*Philos. Trans. Roy. Soc. London***A315**, 509–562.Donelan, M. A., , M. Skafel, , H. Graber, , P. Liu, , D. Schwab, , and S. Venkates, 1992: On the growth of wind-generated waves.

,*Atmos.–Ocean***30**, 457–478.Donelan, M. A., , M. Curcic, , S. S. Chen, , and A. K. Magnusson, 2012: Modeling waves and wind stress.

,*J. Geophys. Res.***117**, C00J23, doi:10.1029/2011JC007787.Galperin, B., , L. H. Kantha, , S. Hassid, , and A. Rosatiu, 1988: A quasi-equilibrium turbulent energy model for geophysical flows.

,*J. Atmos. Sci.***45**, 55–62.Garrett, J. R., 1977: Review of drag coefficients over oceans and continents.

,*Mon. Wea. Rev.***105**, 915–929.Hwang, P. A., 2006: Duration and fetch-limited growth functions and wind-generated waves parameterized with three different scaling wind velocities.

,*J. Geophys. Res.***111**, C02005, doi:10.1029/2005JC003180.Hwang, P. A., , and D. W. Wang, 2004: Field measurements of duration-limited growth of wind-generated ocean surface waves at young stage of development.

,*J. Phys. Oceanogr.***34**, 2316–2326.Janssen, P. A. E. M., 1989: Wave-induced stress and the drag of air flow over sea waves.

,*J. Phys. Oceanogr.***19**, 745–754.Large, W. G., , J. C. McWilliams, , and S. C. Doney, 1994: Oceanic vertical mixing: A review and a model with nonlocal boundary layer parameterization.

,*Rev. Geophys.***32**, 363–403.Martin, P. J., 1985: Simulation of the mixed layers at OWS November and Papa with several models.

,*J. Geophys. Res.***90**, 903–916.Mellor, G. L., 2001: One-dimensional, ocean surface layer modeling: A problem and a solution.

,*J. Phys. Oceanogr.***31**, 790–809.Mellor, G. L., 2003: The three-dimensional, current, and surface wave equations.

*J. Phys. Oceanogr.,***33,**1978–1989.Mellor, G. L., 2005: Some consequences of the three-dimensional current and surface wave equations.

,*J. Phys. Oceanogr.***35**, 2291–2298.Mellor, G. L., 2011: Wave radiation stress.

,*Ocean Dyn.***61**, 563–568.Mellor, G. L., 2013: Waves, circulation, and vertical dependence.

,*Ocean Dyn.***63**, 447–457.Mellor, G. L., , and T. Yamada, 1982: Development of a turbulence closure model for geophysical fluid problems.

,*Rev. Geophys. Space Phys.***20**, 851–875.Mellor, G. L., , and A. Blumberg, 2004: Wave breaking and ocean surface layer thermal response.

,*J. Phys. Oceanogr.***34**, 693–698.Mellor, G. L., , M. A. Donelan, , and L.-Y. Oey, 2008: A surface wave model for coupling with numerical ocean circulation models.

,*J. Atmos. Oceanic Technol.***35**, 1785–1807.Pollard, R. T., , and R. C. Millard, 1970: Comparisons between observed and simulated wind-generated inertial oscillations.

,*Deep-Sea Res.***17**, 813–821.Schlicting, H., 1979:

7th ed. McGraw-Hill, 817 pp.*Boundary Layer Theory.*Snodgrass, F. E., , G. W. Groves, , K. Hasselmann, , G. R. Miller, , W. H. Munk, , and W. H. Powers, 1966: Propagation of swell across the Pacific.

,*Philos. Trans. Roy. Soc. London***259**, 431–497.Terray, E. A., , M. A. Donelan, , Y. C. Agrawal, , W. M. Drennan, , K. K. Kahma, , A. J. Williams III, , P. A. Hwang, , and S. A. Kitaigorodskii, 1996: Estimates of kinetic energy dissipation under breaking waves.

,*J. Phys. Oceanogr.***26**, 792–807.Terray, E. A., , W. M. Drennan, , and M. A. Donelan, 1999: The vertical structure of shear and dissipation on the ocean surface layer.

*The Wind-Driven Air–Sea Interface,*School of Mathematics, University of New South Wales, 239–245.Young, I. R., 2006: Directional spectra of hurricane wind waves.

,*J. Geophys. Res.***111**, C08020, doi:10.1029/2006JC003540.