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  • View in gallery

    Nondimensional F5(kPz) as used in (17), applicable to deep water. The wave number at the peak frequency is given by kP. The solid line is the spectral average of , whereas the dashed line is simply . The max difference is about 25%.

  • View in gallery

    Time series of the first 31 days of a yearlong simulation of weather station Papa for 1961. (top) Wind speed (m s−1) and direction (°) from Martin (1985) are shown. (bottom) Calculated significant wave height HS (m) and mean wave propagation direction (°) are shown.

  • View in gallery

    CD sampled at 3-h intervals for the first 31 days. The solid line is from (22) and the dashed line is the linear relation from Garrett (1977).

  • View in gallery

    Sample velocity (m s−1) profiles at day 30 of the yearlong computation. The solid lines are the currents and the dashed lines are Stokes drifts. (top) The eastward and (bottom) northward components are shown.

  • View in gallery

    Sample stress profiles at day 30 of the yearlong computation. The solid lines are the pressure transfer of momentum into the water column according to (17); the dashed lines are the turbulence transfers according to (18). (top) The eastward and (bottom) northward components are shown.

  • View in gallery

    The yearlong temperatures (°C; contour interval is 1°C) at station Papa. (top) The measured and (bottom) calculated values are shown.

  • View in gallery

    The yearlong surface temperatures (°C) averaged monthly. The circles are measured and the solid lines are calculated values.

  • View in gallery

    The relation between vs inverse wave age to be compared with a similar plot in Terray et al. (1996).

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Pressure–Slope Momentum Transfer in Ocean Surface Boundary Layers Coupled with Gravity Waves

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  • 1 Program in Atmospheric and Oceanic Sciences, Princeton University, Princeton, New Jersey
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Abstract

The paper focuses on the consequences of including surface and subsurface, wind-forced pressure–slope momentum transfer into the oceanic water column, a transfer process that competes with now-conventional turbulence transfer based on mixing coefficients. Horizontal homogeneity is stipulated as is customary when introducing a new surface boundary layer model or significantly new vertical momentum transfer physics to an existing model. An introduction to pressure–slope momentum transfer is first provided by a phase-resolved, vertically dependent analytical model that excludes turbulence transfer. There follows a discussion of phase averaging; an appendix is an important adjunct to the discussion. Finally, a coupled wave–circulation model, which includes pressure–slope and turbulence momentum transfer, is presented and numerically executed. The calculated temperatures compare well with measurements from ocean weather station Papa.

Corresponding author address: George Mellor, Program in Atmospheric and Oceanic Sciences, Sayre Hall, Forrestal Campus, Princeton University, Princeton, NJ 08540. E-mail: glmellor@princeton.edu

Abstract

The paper focuses on the consequences of including surface and subsurface, wind-forced pressure–slope momentum transfer into the oceanic water column, a transfer process that competes with now-conventional turbulence transfer based on mixing coefficients. Horizontal homogeneity is stipulated as is customary when introducing a new surface boundary layer model or significantly new vertical momentum transfer physics to an existing model. An introduction to pressure–slope momentum transfer is first provided by a phase-resolved, vertically dependent analytical model that excludes turbulence transfer. There follows a discussion of phase averaging; an appendix is an important adjunct to the discussion. Finally, a coupled wave–circulation model, which includes pressure–slope and turbulence momentum transfer, is presented and numerically executed. The calculated temperatures compare well with measurements from ocean weather station Papa.

Corresponding author address: George Mellor, Program in Atmospheric and Oceanic Sciences, Sayre Hall, Forrestal Campus, Princeton University, Princeton, NJ 08540. E-mail: glmellor@princeton.edu

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 ftp://aden.princeton.edu/pub/glm/corrected2003] that waves create a contribution as a result of the correlation of wind pressure and wave slope, also called form drag, which is projected into the water column and competes with turbulence-supported stress. It is noted that form drag also penetrates into the atmosphere, but by a small distance relative to atmospheric boundary layer heights; it has played a role in drag coefficient parameterizations (e.g., Hwang 2006).

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 is obtained.

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.

For small wave slope, the linear equations of motion are
e1
e2a
e2b
where 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.
From (2a) and (2b), one deduces irrotationality and, therefore, . Also from (2)
e3
We seek solutions whereby the surface elevation is , and where are wave amplitude, frequency, and wavenumber, respectively. At the surface,
e4
In (4), small elevation is assumed (or ka ≪ 1) so that . At the bottom,
e5
After insertion of into (1), Laplace's equation is obtained for which a solution satisfying (4) and (5) is
e6a
e6b
According to (3),
e7
It is assumed that is small and that a possible component of the wind surface pressure is similarly small;1 thus, from (7) for and to lowest order in , one obtains the dispersion relation
e8
Let , so that (7) may be written
e9
which uses (8) and the definition . For , the flow is dominated by the standard, linear problem for zero surface pressure [for the first two terms on the right-hand side of (9) cancel]; the linear dispersion relation prevails and the (small) pressure component of the wind is correlated with the surface elevation slope .

3. Phase-averaged equations

The term in (9) is the subsurface projection of the surface wind pressure as in Mellor (2003). The material surface and subsurface departure from the rest is (obtained from ), where .

Next, multiply (9) by the material slope . After phase averaging, one obtains
e10
for the wave pressure–slope stress throughout the water column; at the surface, and thus, (10) conforms to the well-known surface form drag . The term, form drag, is actually a drag on the atmospheric side of the air–sea interface; on the water side it is a positive stress. In any event, form drag results from a greater integrated pressure on the backward face of a wave than on the forward face (e.g., Buckles et al. 1984); this is related to separated flow (the sheltering effect) or to aerodynamic boundary layer behavior for accelerating and decelerating velocities in attached flow.
For the present, horizontally homogeneous problem and following Mellor (2003), the phase-averaged momentum equation obtained from (2a), as derived in appendix A, is
e11
This may seem intuitive but, in fact, its derivation is not too simple.
It can be shown and it is well known [in the introduction to Mellor (2003) there is a derivation that can serve as an introduction to the basic methods of that paper and appendix A] that the Stokes drift is
e12
where is the wave energy, and c is the phase speed. After inserting (10) and (12) into (11), the vertically dependent parts of both sides of (11) cancel identically so that
e13
Alternately, 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, = 0 and (13) yields
e14
Thus, as might be anticipated, the wave energy is linearly proportional to the surface form drag and inversely proportional to the Rayleigh drag coefficient.

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.

At this point, it is useful to note that measurements taken at a fixed level instead of following a material surface should yield
e15
(Henceforth, wave properties will be denoted by the superimposed symbol ~.) To arrive at (15), first note that the average momentum flux across any surface is given by , where ds is an elemental surface area whose normal unit vector is . The vector velocity on the surface is whereas is velocity relative to a moving surface. On a wave surface, and . If L is a wavelength, the x component of the momentum flux is as in (15). On the other hand, for a fixed surface, , , and as in (15).

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 will compete with the vertical turbulence stress gradient; the latter had been assumed to be the total stress in previous ocean surface, boundary layer models. As stated in the introduction, horizontal homogeneity is stipulated. For a horizontally variable application of the model see Mellor (2013).

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.

The momentum equation is
e16
where, as derived in Mellor (2003), ; is the current; is the Stokes drift; f is the Coriolis parameter; and is the permutation tensor ( αβz = xyz, = −1 if αβz = yxz, or = 0 if αβz = xxz or yyz). The subscript or denotes horizontal coordinates x or y, whereas z is the vertical coordinate pointing upward from the sea surface. In (16) the turbulence stress term is , where primes denote turbulence velocities. Note that turbulence is defined such that turbulent fluctuations are uncorrelated with wave motions; for example, . The Rayleigh-damping term is necessary in one-dimensional calculations (Pollard and Millard 1970) lest the ocean velocities grow inexorably; it is a surrogate for missing three-dimensional processes. This artificial growth was demonstrated analytically and numerically in Mellor (2001) and where the value was justified.
According to section 2, Mellor (2003), and Mellor et al. (2008), the wind-driven stress is
e17
implemented here for the first time when modeling surface boundary layers. In the following, we specialize to deep water (kh ≫ 1), wherein is the spectral average of plotted in Fig. 1 and is one of several averaged functions defined in Mellor et al. (2008). As in sections 2 and 3, is surface wind pressure correlated with the elevation slope .
Fig. 1.
Fig. 1.

Nondimensional F5(kPz) as used in (17), applicable to deep water. The wave number at the peak frequency is given by kP. The solid line is the spectral average of , whereas the dashed line is simply . The max difference is about 25%.

Citation: Journal of Physical Oceanography 43, 10; 10.1175/JPO-D-13-068.1

The turbulence contribution to the momentum equation is
e18
where is a mixing coefficient defined below. Note that it is the current that is used in (18) rather than . This follows from the fact that attenuation of waves and therefore Stokes drift is governed by the wave energy equation [see (29) and (35)] rather than the momentum equation. Also, in the case of swell, waves and therefore Stokes drift attenuate very little (Snodgrass et al. 1966).
The vertical boundary conditions for (16) are
e19
e20
Here, is the difference between the 10-m wind vector and the ocean surface velocity; and is the ratio of air to water density. We define a form drag coefficient dependent on significant wave height —where is total wave energy—and inverse wave age, ; is frequency at the peak of the wave spectrum, is the 10-m wind speed; and is the von Kármán constant. Thus,
e21a
e21b
(Donelan et al. 1985, 1992; Hwang and Wang 2004). For turbulent flow over a smooth surface, the friction drag coefficient is
e22a
e22b
(Schlicting 1979) and is appropriate to turbulent flow with no waves and therefore involves the kinematic viscosity . Of the two determinations, and , the maximum as calculated in (21) and (22) prevails; the other is set to zero. In practice, the turbulence formula, (21b), is only invoked for low wind speeds and has very little influence on calculated results.

The switch from to or vice versa might seem abrupt but in Schlicting (1979, Fig. 20.21) it is seen that, for rough walls, an abrupt change is a plausible approximation. Given the current state of knowledge, the approximation for wavy walls is appropriately simple and is approximately justified by most drag coefficients plotted as a function of wind speed (see Fig. 3, described in greater detail below). However, the correct partition between form drag and friction drag is an outstanding research question (e.g., Janssen 1989; Donelan et al. 2012).

b. The temperature equation

The equation for potential temperature is
e23
where R is the solar penetrative radiation flux. The turbulence-based heat flux (pressure is absent in scalar equations and so is pressure–slope transfer) is
e24
where KH is a mixing coefficient for heat flux defined below.

c. The turbulence kinetic energy equation

The turbulence kinetic energy equation is
e25
is twice the turbulence energy, and is its mixing coefficient. The second, third, and fourth terms on the right-hand side are shear production, buoyancy production, and dissipation, respectively; and is a length scale discussed below. It is noted that the shear source term is derived by consideration of the mean kinetic energy equation obtained by multiplying (16) by (Mellor 2005); the term appears as a sink term that requires balance of a source term in the turbulence kinetic energy equation and thus includes as well as the usual .
The surface boundary condition for the turbulence equation is as originally suggested by Craig and Banner (1994), whereby a source of turbulence due to wave breaking is injected at the surface as a surface boundary condition as in
e26
where is obtained from the wave model described in the next section; see Mellor and Blumberg (2004) for numerical details. In those papers, a wave model was not available and breaking wave energy was simply parameterized by , where is the friction velocity and . Until research provides a means of vertically distributing into the subsurface layers of the water column, (26) is regarded as an approximation.
Considering the length scale, an important aspect here is that, near the surface, we set
e27
where is the conventional length scale for which there are many prescriptions in the literature, reflecting a high degree of empiricism, but, generally, . Here, is obtained from a differential equation (Mellor and Yamada 1982). An algebraic equation would undoubtedly work well if it is tuned to the ocean surface layer case. It has been assumed by Terray et al. (1999) that where the proportionality constant is a tuning constant for which solutions are quite sensitive (Mellor and Blumberg 2004). To best fit the data in Fig. 4 (described in greater detail below), we have set ; no significance is attached to the proportionality constant of unity. In three-dimensional simulations where internal waves, surface cyclonic divergences, or anticyclonic convergences are in play, mixing is enhanced and the proportionality coefficient may have to be decreased.

d. The mixing coefficients

The model is completed by
e28
where the stability factors are functions of as originally derived by Mellor and Yamada (1982) and modified by Galperin et al. (1988). We have in the past vacillated in specifying , either making it proportional to or setting it equal to a constant. The resulting differences are small but here we choose .

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 dependent on wave propagation direction , the horizontal coordinates, and time. It is, therefore, a relatively simple model compared to third-generation models. However, it has been shown in MDO to reproduce fetch- and duration-limited data and to produce comparable performance to Simulated Waves Nearshore (SWAN; Booij and Holthuijsen 1999) in comparison with buoy data during Hurricane Katrina. It is well suited to present requirements in that it is computationally efficient (requiring two orders of magnitude less computational resource relative to third-generation models) and is comparable to surface boundary layer models in that details of both wave spectra and turbulence spectra are avoided.

The wave energy equation, greatly simplified by exclusion of horizontal gradients, is
e29
where is the energy of waves propagating in direction . Here, wave direction is the only independent variable; advective and refractive terms—otherwise included in MDO—disappear. Wave variables are distributed in the range . We next define a spreading function
e30
The wind direction is and . Thus, is at a maximum in the wind direction and diminishes on either side of the wind direction.
The wind source term is the usual wind input for waves propagating in a direction near the wind; as determined in MDO, it is
e31a
where is the water-side friction velocity. Recall that (= for deep water, and is the peak spectral phase speed) is the inverse wage age. A second sink term
e31b
accounts for waves propagating in directions opposite to the wind direction. The factor, 0.4, is according to Donelan (1999). Only positive is permitted; therefore, after is reduced to zero in the vicinity of , (31b) is ineffective.
The surface wave dissipation is given by
e32
where a = 0.925 was determined empirically in MDO by reference to fetch data and is determined by the limiting case . The first term in (32) represents the fact that the high-frequency part of the spectrum is dissipated very nearly in situ and the second part is dissipation of the middle- () to low-frequency part of the spectrum. This means, of course, that overall wave growth only responds to ; nevertheless, the full dissipation is needed as input to the turbulence kinetic energy equation as in (26). The model crudely approximates swell by using a reduced when .

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.

A -dependent frequency equation, simplified for horizontally homogeneous flow, is
e33a
e33b
In regions of that are wind driven (), the source term has the effect of nudging (a term used, for example, in data assimilation of various ocean properties) toward ; where waves are not wind driven, and is unchanged. The peak frequency is then parameterized by
e34
as determined by Donelan et al. (1985, 1992) and Hwang and Wang (2004). The constant is . Although based mostly on moderate winds, (34) has also been shown to conform to hurricane data by Young (2006). Here, is the integral over wave angle but limited to the wind-driven portion, where ; is the same integral but taken over all wave angles.
The Stokes drift for a monochromatic wave in deep water is where the wavenumber vector is and . Because wave properties are distributed in the range , an average is
e35

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 and are divided into 24 equally spaced increments. The time step is 5 min.

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.

Fig. 2.
Fig. 2.

Time series of the first 31 days of a yearlong simulation of weather station Papa for 1961. (top) Wind speed (m s−1) and direction (°) from Martin (1985) are shown. (bottom) Calculated significant wave height HS (m) and mean wave propagation direction (°) are shown.

Citation: Journal of Physical Oceanography 43, 10; 10.1175/JPO-D-13-068.1

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 CD 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 .

Fig. 3.
Fig. 3.

CD sampled at 3-h intervals for the first 31 days. The solid line is from (22) and the dashed line is the linear relation from Garrett (1977).

Citation: Journal of Physical Oceanography 43, 10; 10.1175/JPO-D-13-068.1

Figure 4 presents sample profiles of current and Stokes drift components. The total mean velocity components are the sum of the two. The narrow southward jet around 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.

Fig. 4.
Fig. 4.

Sample velocity (m s−1) profiles at day 30 of the yearlong computation. The solid lines are the currents and the dashed lines are Stokes drifts. (top) The eastward and (bottom) northward components are shown.

Citation: Journal of Physical Oceanography 43, 10; 10.1175/JPO-D-13-068.1

In Fig. 5 are sample plots of and at day 30; notice that is nil at the surface but comparable to below the surface.

Fig. 5.
Fig. 5.

Sample stress profiles at day 30 of the yearlong computation. The solid lines are the pressure transfer of momentum into the water column according to (17); the dashed lines are the turbulence transfers according to (18). (top) The eastward and (bottom) northward components are shown.

Citation: Journal of Physical Oceanography 43, 10; 10.1175/JPO-D-13-068.1

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).

Fig. 6.
Fig. 6.

The yearlong temperatures (°C; contour interval is 1°C) at station Papa. (top) The measured and (bottom) calculated values are shown.

Citation: Journal of Physical Oceanography 43, 10; 10.1175/JPO-D-13-068.1

Fig. 7.
Fig. 7.

The yearlong surface temperatures (°C) averaged monthly. The circles are measured and the solid lines are calculated values.

Citation: Journal of Physical Oceanography 43, 10; 10.1175/JPO-D-13-068.1

6. The generation of Stokes drift and Eulerian current

To simplify understanding of the detailed physics of the generation of Stokes drift and current , let and integrate (16) so that
e36
where . As in section 4, assume that form drag is the dominant surface wind stress. For monochromatic waves, it can be shown that whereas for a spectrum, following Terray et al. (1996), we let , where is a spectral average. Now, as in familiar numerical implementations, conceptually time split the forcing and dissipative processes. For the first time step, the forcing process, the total energy (29) with no dissipation, converts to . Thus, according to the energy equation, only Stokes drift is created and from (36) . For the second time step, the dissipative process is but . Thus, the Stokes drift decreases and is converted into current. Appendix B discusses the relation in more detail.

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 , where, as stated above, the constant is unity. The model has been run excluding pressure–slope transfer reverting to the more conventional turbulence transfer. Consequently, the summertime temperatures are reduced by about a degree. Now, if is adjusted so that , the comparison (not shown) between measured data and calculations is nearly the same as in Fig. 7. (Most of the year, velocity profiles are nearly the same but do differ significantly during summer months.) Therefore, the good agreement in Fig. 7 does not justify inclusion of pressure–slope per se. Justification derives from the simple fact that pressure–slope transfer into the water column is the logical continuation of form drag, which is nomenclature for pressure–slope transfer at the surface. Otherwise, form drag would be discontinuously continued into the water column by turbulence transfer.

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 (). Thus, it can serve the purpose of providing an introduction to the methodology of the 2003 paper; that paper of course contains more realism than does the idealized Rayleigh drag as shown in section 2.

Here, (2a) is transformed according to
ea1
where
eA2a
eA2b
and u or p is represented by . Surfaces of constant are material surfaces, whereas surfaces of constant are fixed, time-independent (rest) surfaces. From the expression for the vertical velocity below, one sees that derives from . At , whereas at . From (A2a) and (A2b), useful relations are
eA3a
eA3b
Now, following Mellor (2003), derivatives of u and p transform according to
eA4a
eA4b
so that (2a) may be transformed to
eq1
After rearranging
ea5
The equations for wave velocity,
eA6a
eA6b
are exact irrotational solutions (Mellor 2011) to (1) and (2) in the region . Thus, recalling (A2a), , one has
eA7a
eA7b
and also
eA7c
eA7d
plus terms higher order in . Similarly (9) becomes
ea8
or
eA9a
eA9b
The phase average of (A5) is
ea10
where, for horizontal homogeneity, the term ; also, u and are uncorrelated . Henceforth, the terms, and , defined in sections 2 and 3, will be used. Now from (A2), so that
eq2
because and whereas the other terms in (A9a) are not correlated. The other term in (A10) is
eq3
where (8) and are used. Therefore (A10) becomes
ea11
as in (11).

The key to the above derivation is that phase averaging is processed after the independent variables are transformed to x and the fixed .

APPENDIX B

The Relation

Using (31a), (19), (21a), and (21b), one obtains
eB1a
eB1b
where is the inverse wave age and is the air side friction velocity. Here, (B1a) and (B1b) are expressions reflecting different curve-fitted empirical expressions. Terray et al. (1996) used measured spectra and performed an integration over spectral frequency to determine . They plotted versus and the variables in (B1) and Fig. B1 are chosen to conform to their choice of variables. Here, CD 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 . The range of covers the range of their dataset and approximately brackets the scatter in their Fig. 6.
Fig. B1.
Fig. B1.

The relation between vs inverse wave age to be compared with a similar plot in Terray et al. (1996).

Citation: Journal of Physical Oceanography 43, 10; 10.1175/JPO-D-13-068.1

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1

It is small. From (10), (14), and calculations in section 5 (or from most any data source), one evaluates . Originally, the problem was expanded in the small parameter , but this was subsequently deemed to overly complicate the discussion.

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