• Banner, M. L., , and W. L. Peirson, 1998: Tangential stress beneath wind-driven air–water interfaces. J. Fluid Mech, 364 , 115145.

  • Belcher, S. E., 1999: Wave growth by non-separated sheltering. Eur. J. Mech. B/Fluids, 18 , 447462.

  • Belcher, S. E., , and J. C. R. Hunt, 1993: Turbulent shear flow over slowly moving waves. J. Fluid Mech, 251 , 109148.

  • Chalikov, D., , and V. Makin, 1991: Models of the wave boundary layer. Bound.-Layer Meteor, 56 , 8399.

  • Charnock, H., 1955: Wind stress on a water surface. Quart. J. Roy. Meteor. Soc, 81 , 639640.

  • Donelan, M. A., , F. W. Dobson, , S. D. Smith, , and R. J. Anderson, 1993: On the dependence of sea surface roughness on wave development. J. Phys. Oceanogr, 23 , 21432149.

    • Search Google Scholar
    • Export Citation
  • Donelan, M. A., , W. M. Drennan, , and K. Katsaros, 1997: The air–sea momentum flux in conditions of wind sea and swell. J. Phys. Oceanogr, 27 , 20872099.

    • Search Google Scholar
    • Export Citation
  • Drennan, W. M., , H. C. Graber, , D. Hauser, , and C. Quentin, 2003: On the wave age dependence of wind stress over pure wind seas. J. Geophys. Res.,108, 8062, doi:10.1029/2000JC000715.

    • Search Google Scholar
    • Export Citation
  • Ebuchi, N., , and Y. Toba, 1991: Sea-surface roughness length fluctuating in concert with wind and waves. J. Oceanogr. Soc. Japan, 47 , 6379.

    • Search Google Scholar
    • Export Citation
  • Hara, T., , and S. E. Belcher, 2002: Wind forcing in the equilibrium range of wind-wave spectra. J. Fluid Mech, 470 , 223245.

  • Janssen, P. A. E. M., 1989: Wave-induced stress and the drag of air flow over sea waves. J. Phys. Oceanogr, 19 , 745754.

  • Johnson, H. K., , J. Hjstrup, , H. J. Vested, , and S. E. Larsen, 1998: On the dependence of sea surface roughness on wind waves. J. Phys. Oceanogr, 28 , 17021716.

    • Search Google Scholar
    • Export Citation
  • Jones, I. S. F., , and Y. Toba, 2001: Wind Stress over the Ocean. Cambridge University Press, 307 pp.

  • Kitaigorodskii, S. A., 1968: On the calculation of the aerodynamic roughness of the sea surface. Izv. Atmos. Oceanic Phys, 4 , 870878.

    • Search Google Scholar
    • Export Citation
  • Kudryavtsev, V. N., , and V. K. Makin, 2001: The impact of air-flow separation on the drag of the sea surface. Bound.-Layer Meteor, 98 , 155171.

    • Search Google Scholar
    • Export Citation
  • Kudryavtsev, V. N., , V. K. Makin, , and B. Chapron, 1999: Coupled sea surface–atmosphere model. 2. Spectrum of short wind waves. J. Geophys. Res, 104 , 76257639.

    • Search Google Scholar
    • Export Citation
  • Makin, V. K., , and C. Mastenbroek, 1996: Impact of waves on air– sea exchange of sensible heat and momentum. Bound.-Layer Meteor, 79 , 279300.

    • Search Google Scholar
    • Export Citation
  • Makin, V. K., , and V. N. Kudryavtsev, 1999: Coupled sea surface–atmosphere model. 1. Wind over waves coupling. J. Geophys. Res, 104 , 76137623.

    • Search Google Scholar
    • Export Citation
  • Makin, V. K., , and V. N. Kudryavtsev, 2002: Impact of dominant waves on sea drag. Bound.-Layer Meteor, 103 , 8399.

  • Makin, V. K., , V. N. Kudryavtsev, , and C. Mastenbroek, 1995: Drag of the sea surface. Bound.-Layer Meteor, 73 , 159182.

  • Mason, P. J., 1988: The formation of areally-averaged roughness lengths. Quart. J. Roy. Meteor. Soc, 114 , 399420.

  • Phillips, O. M., 1977: The Dynamics of the Upper Ocean. 2d ed. Cambridge University Press, 336 pp.

  • Phillips, O. M., 1985: Spectral and statistical properties of the equilibrium range in wind-generated gravity waves. J. Fluid Mech, 156 , 505531.

    • Search Google Scholar
    • Export Citation
  • Plant, W. J., 1982: A relationship between wind stress and wave slope. J. Geophys. Res, 87 , 19611967.

  • Taylor, P. K., , and M. J. Yelland, 2001: The dependence of sea surface roughness on the height and steepness of the waves. J. Phys. Oceanogr, 31 , 572590.

    • Search Google Scholar
    • Export Citation
  • Toba, Y., , and H. Kunishi, 1970: Breaking of wind waves and the sea surface wind stress. J. Oceanogr. Soc. Japan, 26 , 7180.

  • Yelland, M., , and P. K. Taylor, 1996: Wind stress measurements from the open ocean. J. Phys. Oceanogr, 26 , 541558.

  • View in gallery

    Upper and lower bounds of sheltering wavenumbers ks vs wind friction velocity u∗ for mature seas. Solid lines indicate estimates based on approach I with data of Banner and Peirson (1998). Dash–dot lines indicate estimates based on approach II with the data collated by Phillips (1985) with sheltering wave age cs/us = 0.46 and 2.38

  • View in gallery

    Upper and lower bounds of sheltering wave age cs/us vs wind friction velocity u∗ for mature seas. Solid lines indicate estimates based on approach I with data of Banner and Peirson (1998). Dash–dot lines indicate estimates based on approach II with the data collated by Phillips (1985) with sheltering wave age cs/us = 0.46 and 2.38

  • View in gallery

    Schematic of energy conservation inside wave boundary layer

  • View in gallery

    Upper and lower bounds of the Charnock coefficient vs friction velocity for mature seas. Lines are results calculated with ks estimated by approach II (dashed lines: k1 = 100 rad m−1, dash–dot lines: k1 = 400 rad m−1, and solid lines: asymptotic limit of u1/u∗ → 0), with sheltering wave age cs/us = 0.46 and 2.38. Diamonds are results calculated with ks estimated by approach I. Squares are empirical estimates by Banner and Peirson (1998)

  • View in gallery

    Mean wind profiles over mature seas. Friction velocity u∗ = 0.5 m s−1. Sheltering wave age cs/us = 0.46. Dashed line: k1 = 100 rad m−1, dash–dot line: k1 = 400 rad m−1, and solid line: asymptotic limit of u1/u∗ → 0. Diamonds (z = δ/k0) and squares (z = δ/k1) indicate top and bottom of the wave boundary layer, respectively. Dotted line shows wind profile over a smooth solid surface

  • View in gallery

    Fig. A1. Partition of the energy flux from the top of the wave boundary layer into the internal dissipation and the flux into surface waves. Results are shown vs sheltering wave age. Solid line: FT/ρau3, dashed line: (FTFD)/ρau3, and dash–dot line: FB/ρau3. (a) Results with the first eddy viscosity model (Janssen 1989; Chalikov and Makin 1991; Makin et al. 1995); (b) results with the second eddy viscosity model (Makin and Kudryavtsev 1999)

  • View in gallery

    Fig. C1. Charnock coefficient vs the sheltering wave age cs/us for mature seas. Asymptotic results of u1/u∗ → 0. Dotted lines: eddy viscosity model with n = 1 (Janssen 1989; Chalikov and Makin 1991; Makin et al. 1995), dashed lines: eddy viscosity model with n = 2/3 (Makin and Kudryavtsev 1999), and solid lines: the present model

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 221 221 23
PDF Downloads 105 105 14

Wind Profile and Drag Coefficient over Mature Ocean Surface Wave Spectra

View More View Less
  • 1 Graduate School of Oceanography, University of Rhode Island, Narragansett, Rhode Island
  • | 2 Department of Meteorology, University of Reading, Reading, United Kingdom
© Get Permissions
Full access

Abstract

The mean wind profile and the Charnock coefficient, or drag coefficient, over mature seas are investigated. A model of the wave boundary layer, which consists of the lowest part of the atmospheric boundary layer that is influenced by surface waves, is developed based on the conservation of momentum and energy. Energy conservation is cast as a bulk constraint, integrated across the depth of the wave boundary layer, and the turbulence closure is achieved by parameterizing the dissipation rate of turbulent kinetic energy. Momentum conservation is accounted for by using the analytical model of the equilibrium surface wave spectra developed by Hara and Belcher. This approach allows analytical expressions of the Charnock coefficient to be obtained and the results to be examined in terms of key nondimensional parameters. In particular, simple expressions are obtained in the asymptotic limit at which effects of viscosity and surface tension are small and the majority of the stress is supported by wave drag. This analytical model allows us to identify the conditions necessary for the Charnock coefficient to be a true constant, an assumption routinely made in existing bulk parameterizations.

Corresponding author address: Dr. Tetsu Hara, Graduate School of Oceanography, University of Rhode Island, South Ferry Road, Narragansett, RI 02882. Email: thara@uri.edu

Abstract

The mean wind profile and the Charnock coefficient, or drag coefficient, over mature seas are investigated. A model of the wave boundary layer, which consists of the lowest part of the atmospheric boundary layer that is influenced by surface waves, is developed based on the conservation of momentum and energy. Energy conservation is cast as a bulk constraint, integrated across the depth of the wave boundary layer, and the turbulence closure is achieved by parameterizing the dissipation rate of turbulent kinetic energy. Momentum conservation is accounted for by using the analytical model of the equilibrium surface wave spectra developed by Hara and Belcher. This approach allows analytical expressions of the Charnock coefficient to be obtained and the results to be examined in terms of key nondimensional parameters. In particular, simple expressions are obtained in the asymptotic limit at which effects of viscosity and surface tension are small and the majority of the stress is supported by wave drag. This analytical model allows us to identify the conditions necessary for the Charnock coefficient to be a true constant, an assumption routinely made in existing bulk parameterizations.

Corresponding author address: Dr. Tetsu Hara, Graduate School of Oceanography, University of Rhode Island, South Ferry Road, Narragansett, RI 02882. Email: thara@uri.edu

1. Introduction

Present parameterizations of the wind stress, or equivalently the drag coefficient, over the ocean are far from satisfactory, as pointed out in the recent book by Jones and Toba (2001). Most operational atmospheric models use a simple bulk parameterization based on the equivalent surface roughness z0 being determined by
i1520-0485-34-11-2345-e1
(Charnock 1955), where g is gravitational acceleration and u∗ is the wind friction velocity. This constant is called the Charnock coefficient and is usually set to be about 0.010–0.015. It is still being debated whether the Charnock coefficient is a true constant or depends on the wind stress and other parameters (e.g., Yelland and Taylor 1996; Donelan et al. 1997; Taylor and Yelland 2001). One of the main uncertainties regarding the drag coefficient estimation is the effect of the ocean surface wave field. The bulk formulation is expected to be valid only over a fully developed wave field. The effect of growing or confused seas is still difficult to predict. Jones and Toba (2001) review previous studies of the effect of the inverse wave age (u∗/cp) on the Charnock coefficient (here cp is the phase speed of dominant waves) and conclude that the results are still far from conclusive.

There have been many attempts to predict the drag coefficient by explicitly calculating the stress supported by surface waves, the so-called wave-induced stress. Over the ocean surface, in stationary homogeneous conditions the total stress is independent of height in the lower part of the atmospheric boundary layer, the constant stress layer. Since the total stress is the sum of the turbulent stress and the wave-induced stress (except inside the viscous sublayer), the turbulent stress is reduced inside the wave boundary layer. Earlier studies employed an eddy viscosity model to relate the reduced turbulent stress to the mean wind profile (Janssen 1989; Chalikov and Makin 1991; Makin et al. 1995) and then predict the drag coefficient. Makin and Kudryavtsev (1999) proposed a modified expression of the eddy viscosity based on the turbulent kinetic energy budget inside the wave boundary layer. All of these studies used empirical parameterizations of the surface wave field. More recently, Kudryavtsev and Makin (2001) introduced a much simplified model of the wave boundary layer and included the effect of airflow separation due to surface breaking waves. They also used the model for the surface wave spectrum of Kudryavtsev et al. (1999) instead of using an empirical surface wave spectrum. Makin and Kudryavtsev (2002) further included the effect of dominant wave breaking to the model of Kudryavtsev and Makin (2001).

What these studies all show is that it is the short waves in the wave spectrum (say, less than 10 m in wavelength) that dominate the roughness of the sea surface and hence the drag. This part of the spectrum is often in a local equilibrium. Recently, Hara and Belcher (2002) developed a simple analytical model of this equilibrium range of surface gravity wave spectra. The model predicts that the equilibrium surface wave spectrum is determined by a single parameter ks, called a sheltering wavenumber. The sheltering wavenumber is determined by how the total wind stress is partitioned into stress supported by different parts of the wave spectrum as well as the surface viscous stress. Here we use this analytical model of the equilibrium wave spectrum to calculate the Charnock coefficient and mean wind profile over mature seas. The study draws from the approach of Makin and Mastenbroek (1996) and Makin and Kudryavtsev (1999), in that it is based on conservation of momentum and energy in the wave boundary layer, but there are important differences:

  1. Energy conservation is cast as a bulk constraint, integrated across the depth of the wave boundary layer, which demonstrates that it is natural and sufficient to close the turbulence by parameterizing the dissipation rate of turbulent kinetic energy.
  2. Momentum conservation is accounted for using the analytical model developed by Hara and Belcher (2002). This allows us to obtain analytical expressions of the Charnock coefficient, and to examine the results in terms of key nondimensional parameters.

As in Makin and Kudryavtsev (1999), we assume that surface waves are not breaking and the airflow remains attached to the water surface with a viscous sublayer established just above the water surface.

In section 2 we briefly review the model of the equilibrium wave spectrum by Hara and Belcher (2002) and estimate the sheltering wavenumber ks over mature seas in section 3. A new model of the wave boundary layer is introduced in section 4 and the mean wind profile and the Charnock coefficient over mature seas are calculated, followed by concluding remarks in section 5.

2. Theory of equilibrium wave spectrum

In this section, we briefly review the theoretical model of the equilibrium range of surface wave spectra developed by Hara and Belcher (2002). The model starts with the conservation of momentum inside the wave boundary layer. The mean wind and the wave field are assumed to be aligned. The total air–sea momentum flux τtot is expressed as a sum of the wave-induced stress τw(z) and the turbulent stress τt(z),
τtotρau2τwzτtz
where ρa is air density, u∗ is the friction velocity, and z is the height above the instantaneous water surface [see Makin et al. (1995) and Makin and Kudryavtsev (1999) for discussion of the wave-following coordinate]. The wave-induced stress is expressed as
i1520-0485-34-11-2345-e3
(The contribution to the wave-induced stress from waves propagating against the wind, i.e., |θ| > π/2, is negligible because the energy in these components is so small. Therefore, the integration in θ spans −π/2 to π/2 only.) Here βg is the wave growth rate, ρw is water density, k is the wavenumber, σ is the wave angular frequency, θ is the wave propagation direction relative to the mean wind direction, and B is the degree of saturation, which is related to the wave height spectrum ψ as B = k4ψ. Following Makin et al. (1995), the decay function F(k, z) is approximated by a step function, namely,
i1520-0485-34-11-2345-e4
Following Belcher and Hunt (1993) the wave-induced stress penetrates a distance L(k) into the airflow and so we set kL(k) = δ = const. The wave-induced stress then becomes
i1520-0485-34-11-2345-e5
that is, it is equal to the total momentum flux into surface waves in the wavenumber range of 0 < k < δ/z. If we interpret height L(k) as a blending height, then δ is a constant O(0.05) (Mason 1988).
Belcher (1999) and Makin and Kudryavtsev (1999) show that the growth rate βg(k, θ) of a particular wave scale is determined by the turbulent stress τt evaluated at the height comparable to the depth of the inner region L(k); that is,
i1520-0485-34-11-2345-e6
This is called a local turbulent stress and is denoted by τlt(k) = ρa[ul(k)]2. It is seen from (6) that the local turbulent stress, which forces waves of a wavenumber k, is equal to the total wind stress minus the stress supported by all the longer waves. Then, the growth rate is described by
i1520-0485-34-11-2345-e7
where cβ is an empirical coefficient and h(θ) is the directionality of the wave growth rate. The coefficient α2, which defines the smallest wavenumbers that are forced by the wind, is set to 0.07 after Plant (1982).
The evolution of the surface wave spectrum is described in terms of the wave action spectral density N(k),
i1520-0485-34-11-2345-e8
where k is the wavenumber vector, T(k) is the flux of the wave action by nonlinear wave interactions, ∇k is the gradient operator in k, Sw is the wind input, and D is the dissipation. The action density is related to the wave height spectrum ψ(k) and the degree of saturation B(k) as
i1520-0485-34-11-2345-e9
for surface gravity waves, where g is the gravitational acceleration. It is well known that ocean surface wave spectra at frequencies much higher than the peak frequency attain an equilibrium state (called an “equilibrium range”; e.g., Phillips 1977). In the equilibrium range, the three input terms to the wave action conservation equation balance one another to achieve a quasi steady state (Phillips 1985); that is, the right-hand side of (8) is equal to zero.
As in Phillips (1985), Hara and Belcher (2002) assume that both the divergence of the wave action flux ∇k · T(k) and the dissipation D in the wave action equation are proportional to the cube of the local wavenumber spectrum. Since the sum of these two terms is balanced by the input term in the equilibrium range, we may set
kTkDSwαgk−4B3k
where α is a nondimensional proportionality constant. The wind input term is expressed as
SwβgkNkβgkg1/2k−9/2Bk
where βg(k)is the wave growth rate described in (7). Introducing (11) into (10), the degree of saturation is expressed in terms of the growth rate as
Bkα−1βgkg−1/2k−1/21/2
On differentiating (6) by k and introducing (12) and (7), we obtain an integral equation of ul, which can be solved analytically provided proper boundary conditions are specified. Let the equilibrium range be established for a wavenumber range of k0 < k < k1 and let us specify the local friction velocities at k0 and k1, denoted by u∗0 and u∗1, respectively. Then, the solution for the local friction velocity is written
i1520-0485-34-11-2345-e13
where
i1520-0485-34-11-2345-e14
is called a sheltering wavenumber and
i1520-0485-34-11-2345-e15
is called a sheltering friction velocity. The sheltering wavenumber ks represents the wavenumber at which the local friction velocity begins to be affected by sheltering by the longer wavelength waves. At low wavenumbers (kks), ul becomes constant and equal to 2us, while ul decreases like k−1/2 at high wavenumbers (kks). If the solution (13) is introduced back into the differential equation for ul, the coefficient α is found to be
i1520-0485-34-11-2345-e16
where cs = (g/ks)1/2 is called a sheltering wave phase speed, cs/us is called a sheltering wave age, and
i1520-0485-34-11-2345-e17
Introducing (11), (13), and (16) into (12), the degree of saturation B is found to be
i1520-0485-34-11-2345-e18
If integrated over all angles, the solution becomes
i1520-0485-34-11-2345-e19
with
i1520-0485-34-11-2345-e20
At low wavenumbers (kks), where the sheltering effect is weak, B(k) is proportional to k1/2 and is consistent with the prediction by Phillips (1985). At high wavenumbers (kks), B(k) is strongly influenced by the sheltering effect and becomes independent of wavenumber. Last, using the linear dispersion relation, the frequency spectrum is obtained from (19), namely,
i1520-0485-34-11-2345-e21
where σs = (gks)1/2. Therefore, the frequency spectrum is proportional to σ−4 at low frequencies (σσs) and is proportional to σ−5 at high frequencies (σσs).

In summary, the equilibrium spectrum is determined by two empirical coefficients, cβ and cθ, and a single dynamical variable, ks, called a sheltering wavenumber. In the next section we estimate the value of ks for mature seas.

3. Calculation of the sheltering wavenumber for mature seas

In this section we present two different approaches to determine the sheltering wavenumber ks over mature seas. The first approach is based on measured values of the stress and is the same as the method used by Hara and Belcher (2002); the second is based on comparison with the measured wave spectra.

a. Approach I

Hara and Belcher (2002) calculate the sheltering wavenumber ks for mature seas using the empirical observations of the total wind stress (ρau2) and the surface viscous stress (ρau2ν) provided by Banner and Peirson (1998). They start with two assumptions:

  1. The lower bound of the equilibrium range is set:
    i1520-0485-34-11-2345-e22
    with c0 = (g/k0)1/2, and α2 = 0.07, because wind forcing is negligible below this wavenumber according to Plant (1982). Then the local stress, ρau20, evaluated at k0 is equal to the total stress, ρau2.
  2. The upper bound of the equilibrium range is k1 = 100 rad m−1 and the local stress, ρau21, evaluated at k1 is equal to the viscous stress, ρau2ν, at the surface.
They then use Banner and Peirson's (1998) estimates of u∗ and uν to calculate the upper and lower bounds of ks. The results are reproduced in Fig. 1. The estimated value of ks monotonically decreases as u∗ increases.

b. Approach II

An alternative approach to estimate ks is to use the observed equilibrium wave spectra. Phillips (1985) shows that previous observational data of the equilibrium frequency spectrum agree with his predicted form,
σαpguσ−4
for frequencies not too far from the peak and that the empirical constant αp is between 0.02 and 0.11 (and between 0.06 and 0.11 for most of the field observations). Our model, on the other hand, yields
i1520-0485-34-11-2345-e24
for σσs, which is a good approximation not far from the spectral peak. Equating (23) and (24), we may write the sheltering wavenumber in terms of αp:
i1520-0485-34-11-2345-e25
Therefore, ks is proportional to u−2 and is determined by three empirical coefficients, cβ, cθ, and αp. This estimate of ks is also shown in Fig. 1 for αp = 0.02 and 0.11. Here, we have set cβ = 40, following Plant (1982), and cθ = (3/16)π corresponding to h(θ) = cos2θ. For a fixed u∗, this estimate yields a relatively wide range of ks, which includes the range of ks estimated from approach I.
It is noteworthy that, if we set k = k0 and ul(k) = u∗ in (13) and introduce (22), we obtain
i1520-0485-34-11-2345-e26
Therefore, we can write the sheltering wave age as
i1520-0485-34-11-2345-e27
Now, if the Phillips (1985) constant, αp, is truly constant and independent of u∗, then (25) shows that cs/u∗ is also constant, and so according to (27) the sheltering wave age cs/us is also independent of u∗. Since the sheltering wave age is related to α through (16), the proportionality constant α, originally defined in (10), is then also a constant. The lower and upper bounds of the sheltering wave age, corresponding to αp = 0.02 and 0.11, are cs/us = 0.46 and 2.38. In contrast, the sheltering wavenumber estimated from approach I yields the sheltering wave age that depends weakly on u∗. Figure 2 shows the variation of sheltering wave age calculated using these two methods.

4. Effect of mature seas on the atmospheric wave boundary layer

In this section, we first develop a model of the wave boundary layer based on the conservation of momentum (described in section 2) and energy (section 4a). Next, the equilibrium spectral model of Hara and Belcher (2002), described in section 2, and the estimates of the sheltering wave age cs/us, described in section 3, are introduced into the model to calculate the Charnock coefficient (sections 4b–4d) and the mean wind profile (section 4e) over mature seas. Throughout this study the wave field is assumed to be aligned with the mean wind direction.

a. Conservation of energy in the wave boundary layer

Previously, the effect of surface waves on the mean wind profile was estimated using eddy viscosity models (Janssen 1989; Chalikov and Makin 1991; Makin et al. 1995; Makin and Kudryavtsev 1999). It can be shown that these models do not conserve energy within the wave boundary layer (see appendix A). Makin and Mastenbroek (1996) introduced a wave boundary layer model that does satisfy both momentum conservation and energy conservation. Here, we develop a simple model of the wave boundary layer based on bulk conservation of energy, which demonstrates that it is natural and sufficient to effect turbulence closure by parameterizing the dissipation rate of the turbulent kinetic energy (TKE).

Let us introduce a coordinate system in a fixed frame of reference (in horizontal) and define the positive x direction as the mean wind direction. The z coordinate is the height above the instantaneous water surface as defined earlier. We then introduce the following decomposition of the wind velocity ui(i = 1, 2, 3) and pressure p,
uiuiũiuippp
where the overbar denotes time average, the tilde denotes the wave-correlated part of the signal, and the prime denotes the turbulent fluctuation. The total kinetic energy is also decomposed into
i1520-0485-34-11-2345-e29
and the mean kinetic energy e itself consists of three components,
i1520-0485-34-11-2345-e30
The turbulent and wave-induced stresses are written as
i1520-0485-34-11-2345-e31
Following Makin and Mastenbroek (1996), the energy budget for the airflow over surface waves can be expressed as the budget for each of the three components of the mean kinetic energy. The budget of the mean kinetic energy of mean motions, ½u2, is
i1520-0485-34-11-2345-e32
the budget of the mean kinetic energy of wave-induced motions, ½, is
i1520-0485-34-11-2345-e33
and the budget for the mean TKE, ½, is
i1520-0485-34-11-2345-e34
Here, Pt = τtdu/dz is production of the TKE from the mean velocity shear;
i1520-0485-34-11-2345-e35
is production of the TKE from work done by wave-induced turbulent stress against wave-induced shear; Dw = τwdu/dz is transfer of energy from the mean to the wave-induced motions;
i1520-0485-34-11-2345-e36
is the vertical transport of the kinetic energy of the wave-induced motions;
i1520-0485-34-11-2345-e37
is the vertical transport of the TKE; and ε is the viscous dissipation of the TKE. The total energy budget is obtained by adding (32) to (34),
i1520-0485-34-11-2345-e38
which states that the difference between the shear production and the viscous dissipation is balanced by the divergence of the total energy flux (Π + Π′). Therefore, inside the wave boundary layer the shear production is not equal to the viscous dissipation. Above the top of the wave boundary layer Π and Π′ are both negligibly small and the shear production balances the viscous dissipation.
At the ocean surface the largest contribution to the vertical transport of the wave-induced energy is from the pressure transport,
i1520-0485-34-11-2345-e39
because at the ocean surface − is equal to the energy flux into surface waves. (The contribution to Π from the surface shear stress is negligible for surface gravity waves.) Hence, we may write
i1520-0485-34-11-2345-e40
with
i1520-0485-34-11-2345-e41
The lower boundary condition on the mean wind speed is given in terms of the equivalent roughness scale zν of the viscous sublayer,
i1520-0485-34-11-2345-e42
where νa is the air viscosity and ρau2ν is the surface viscous stress. If we integrate (38) from z = zν to the top of the wave boundary layer (z = zT), we obtain the equation for bulk conservation of energy:
i1520-0485-34-11-2345-e43
since Π(zT) = 0 and Π(zν) = Π(0).
As schematically shown in Fig. 3, the bulk energy conservation requires that the total energy flux from the top must be equal to the sum of the energy flux into surface waves and the total viscous dissipation of the TKE inside the wave boundary layer. Above the wave boundary layer, the mean velocity profile becomes a standard logarithmic layer, with a roughness length that characterizes the sea surface. Thus, from the condition of bulk conservation of energy, once the total stress, the total energy flux into waves, and the total TKE viscous dissipation in the wave boundary layer are known, we may determine u(zT) and hence the equivalent roughness length and the drag coefficient. This demonstrates that it is natural and sufficient to close the turbulence by parameterizing the dissipation rate of turbulent kinetic energy. Therefore, we do not introduce an eddy viscosity and prognostic equations for turbulence closure as in Makin and Mastenbroek (1996). Instead, following the approach used in one-equation models of turbulence, the viscous dissipation of the TKE, ε, is simply related to the local turbulent stress, ul, at each height (rather than the total stress) so that
i1520-0485-34-11-2345-e44
where κ is the von Kármán constant.

b. Analytical expression of the equivalent roughness over mature seas

Consider now a fully grown sea with the equilibrium spectral form presented in section 2. As before, the wavenumber k0 is set to the lower bound of the equilibrium range such that u∗/c0 = α2 = 0.07. We also set ul(k0) = u∗; that is, the wave growth rate is zero for k < k0. In addition, the equilibrium spectrum obtained from section 3 is valid up to the large wavenumber cutoff k1, which thus neglects effects of surface tension and viscosity on waves up to k1. Last, assume that there are no waves for k > k1 so that u∗1 = uν. The flux into surface waves W(k) is evaluated using the definition of the growth rate βg in (7), as well as the equilibrium form of B(k), given in (18), and ul, given in (13). This procedure yields
i1520-0485-34-11-2345-e45
with
i1520-0485-34-11-2345-e46
Introducing (44) and (45) into (43), the mean wind speed at the top of the wave boundary layer (z = zT = δ/k0) is found to be
i1520-0485-34-11-2345-e47
which can be integrated (see appendix B for derivation) to yield
i1520-0485-34-11-2345-e49
with
i1520-0485-34-11-2345-e50
The equivalent roughness z0 is then found to be
i1520-0485-34-11-2345-e52
and the Charnock coefficient becomes
i1520-0485-34-11-2345-e53
This analytical expression allows us to examine how the Charnock coefficient depends on key nondimensional parameters, as described in the next subsection.

c. Discussion

First, note that the Charnock coefficient is proportional to
i1520-0485-34-11-2345-e54
where Re∗1 = u∗1/νk1 is the friction Reynolds number of the flow over the smallest waves. Now, (u∗1/u∗)3 varies depending on the degree of sheltering across the wave spectrum. When there is little sheltering, u∗1u∗ and a large fraction of the total stress is supported by the viscous stress on the sea surface. Hence in this case, the Charnock coefficient varies inversely with Re∗1, as would be expected. But, when there is appreciable sheltering across the wave spectrum, u∗1u∗ and
δ∗1−(u*1/u*)3
In this case the drag of the sea surface is dominated by the aerodynamic drag supported by the waves, and so there is no dependence on Reynolds number. In between these two asymptotic limits, the Charnock coefficient depends on Reynolds number with varying powers between 0 and −1, depending on how much of the stress is supported by the waves. The estimates from Banner and Peirson (1998) show that (u∗1/u∗)3 decreases roughly from 0.54 to 0.11 as the wind speed increases from 6 to 14 m s−1 over mature seas. Therefore, we expect the Reynolds number dependence of the Charnock coefficient to reduce and become smaller at higher wind speeds.

Kitaigorodskii (1968) and others have argued that drag of the sea surface depends on the Reynolds number through processes similar to those acting in flow over a solid rough wall. Thus the flow is aerodynamically rough when Reτ = uhs/ν ≫ 1 so that the roughness elements, of height hs, are taller than the thickness of the depth of the viscous sublayer, of depth ν/u∗. Toba and Kunishi (1970) suggest that hs should be taken to be the characteristic wave height, whereas Kitaigorodskii (1968) suggests that, since the roughness elements are the short waves, hs ∼ 1 cm. The implicit assumption in this picture is that the roughness elements act as bluff bodies inducing separated flow. The present model shows that this transition occurs differently when the surface is streamlined and the flow remains attached. It shows that large waves extract momentum from the wind (through pressure forces) so that shorter wavelength waves are “sheltered” and are exposed to a reduced stress. When sheltering is strong, the longer waves support the majority of the stress, leaving very little to be supported by surface viscous stress and hence little Reynolds number dependence.

Second, the Charnock coefficient also depends upon X0 and X1, which are defined by
i1520-0485-34-11-2345-e56
So X20 and X21 measure the extent of the equilibrium range, and the strength of sheltering in the equilibrium range that is, k0 and k1 and their relation to ks. It is noteworthy that the Charnock coefficient depends on both.
When the sheltering is extremely strong, k1/ks → ∞ so that X21 = u∗1/2us → 0. In this limit G(X1) → 0. This limit is almost equivalent to the limit of u∗1/u∗ → 0 since 2us/u∗ is close to 1 (between 1.02 and 1.09 corresponding to the sheltering wavenumber cs/us between 0.46 and 2.38). Therefore, in this limit the Reynolds number dependence is also removed as discussed earlier. Then, the Charnock coefficient simplifies to
i1520-0485-34-11-2345-e57
with
i1520-0485-34-11-2345-e58
The result is now a function of X0 and cs/us. Since X0 is uniquely related to the sheltering wave age, cs/us, through (26) and (27), the Charnock coefficient is a function of the sheltering wave age only. So, if the sheltering wave age is a true constant (i.e., independent of u∗), then the Charnock coefficient is also a true constant.

When, in practice, do we expect this simplified form (57) to be a good approximation? The limit u∗1/u∗ → 0 is valid when the majority of the stress is supported by form drag on the waves rather than by the surface viscous stress. In addition, this form has been obtained with an assumption that the analytical form of the equilibrium spectrum by Hara and Belcher (2002) is valid up to the cutoff wavenumber k1; that is, that the effects of surface tension and viscous dissipation are small. Both conditions are expected to become increasingly valid as the wind speed increases.

In the atmospheric modeling community, the Charnock coefficient is commonly assumed to be a constant of about 0.010–0.015. However, the present model indicates that this assumption is strictly valid only if the following three conditions are satisfied:

  1. The surface wave field is fully developed.
  2. Effects of surface tension and viscosity are small, and the majority of the stress is supported by waves (u1/u∗ → 0).
  3. The sheltering wave age, cs/us, is a true constant, which, according to section 3, means that the equilibrium frequency/wavenumber spectrum is proportional to u∗ not too far from the spectral peak.
Although the first two conditions are naturally expected, the validity of the third condition needs to be carefully examined in the future, by inspecting fully developed wave spectra under a wide range of wind forcing.

d. Calculation of the equivalent roughness over mature seas

The values of ks and cs/us were determined for mature seas in the previous section. So, if δ = 0.05 and when k1 is specified, the Charnock coefficient can be calculated using (53). (In approach I, k1 was set 100 rad m−1; in approach II k1 is specified below.) Figure 4 shows the upper and lower bounds of the Charnock coefficient versus the friction velocity using the values of ks presented in Fig. 1.

Let us first focus on the results calculated using ks values estimated by approach II, shown by different lines. The upper and lower bounds correspond to the lower and upper bounds of ks in Fig. 1 and the upper and lower bounds of cs/us in Fig. 2, respectively. For fixed u∗ and k1, the Charnock coefficient varies by a factor of 5–10 depending on whether the upper bound or lower bound of cs/us is used. That is, the Charnock coefficient may significantly very depending on the level of the equilibrium spectrum. The effect of choosing different k1 (for a fixed cs/us) is also noteworthy. If k1 is chosen to be above 400 rad m−1, the calculated Charnock coefficient is not too far from the asymptotic limit (say, within a factor of 1.5).

We do not pursue here how the results are modified if spectral forms of gravity–capillary and capillary waves are introduced. However, the observation of the sensitivity of the results to different choices of k1 suggests that knowledge of the spectral form at very large k is not critically important in determining the equivalent roughness at higher winds, particularly with larger values of the sheltering wave age cs/us.

So far, we have chosen a particular directionality of the growth rate h(θ) = cos2θ. If we choose a different form of h(θ), the result of the Charnock coefficient is modified through the change of cθ [by modifying the estimate of cs/us in (25)] and through the change of cθ [by modifying the coefficient in (50)]. Hara and Belcher (2002) show that cθ changes very little (from 3π/16 ≅ 0.5890 to 189π/1024 ≅ 0.5798) if h(θ) changes from cos2θ to cos6θ. It is found that the coefficient cθ varies little also [from 16/(9π) ≅ 0.5659 to 32 768/(19 845π) ≅ 0.5256]. Therefore, our results of the Charnock coefficient are not sensitive to different choices of h(θ).

The Charnock coefficient is also calculated using ks values estimated by approach I. The results, shown in Fig. 4 by diamonds, are within a factor of 2 or so of the empirical results by Banner and Peirson (1998), which are shown by squares. Since it is the drag that is usually required, which varies logarithmically with the Charnock coefficient, agreement within a factor of 2 is satisfying. The agreement found here is not obtained trivially: recall that approach I used only the empirical estimate of the relationship between the total stress and the surface viscous stress obtained by Banner and Peirson (1998); no information about the wind speed was used. Therefore, the combination of the present models for the equilibrium spectrum and the wave boundary layer produces the Charnock coefficient (or drag coefficient) that is consistent with the observations, provided that the relationship between the total stress and the surface viscous stress is given and reasonable values are set for the two parameters, α2 and δ.

How do the present theoretical predictions compare to observations? The calculations have been completed here for mature wind seas, when the peak of the spectrum is beyond the region of wind forcing; that is, u∗/ cp < α2 = 0.07 (cp is the phase speed at the spectral peak). It is therefore necessary to compare the results with observations that also satisfy this condition. Ebuchi and Toba (1991) report the Charnock coefficient of mature seas, with u∗/cp < 0.07, spanning a wide range from 0.005 to 3. Later, Donelan et al. (1993) report values from 0.002 to 0.07, and mostly between 0.005 and 0.03. Johnson et al. (1998) present values between 0.002 and 0.2, and mostly between 0.005 and 0.04. The more recent study of Drennan et al. (2003), which carefully selected only pure wind-sea conditions, show values between 0.005 and 0.1, and mostly between 0.005 and 0.05. They also show that most of previously proposed empirical parameterizations fall roughly between 0.007 and 0.05. The value from the present model ranges from 0.02 to 0.1, which has a strong overlap with values observed by Drennan et al. (2003).

The asymptotic analysis shows that the Charnock coefficient becomes a constant at very high winds only. However, Fig. 3 shows that in practice the Charnock coefficient varies by at most a factor of 2 through all wind speeds provided the sheltering wave age is constant and k1 is chosen to be above 400 rad m−1. Therefore, it seems to be reasonable, in practice, to use a constant Charnock coefficient over mature seas at all wind speeds. This finding, however, does not mean that the original dimensional argument by Charnock (1955), that the only g and u∗ are relevant external parameters to determine the roughness length, is valid at lower winds. The Charnock coefficient does depend on the Reynolds number significantly at lower winds. (For example, if we change the value of air viscosity significantly, the Charnock coefficient will not remain constant!)

The results of the Charnock coefficient obtained using the same wave spectrum but different wave boundary layer models based on various forms of the eddy viscosity (Janssen 1989; Chalikov and Makin 1991; Makin et al. 1995; Makin and Kudryavtsev 1999) are discussed in appendix C.

e. Determination of the wind profile in the wave boundary layer

In section 4d, it was shown that the mean wind speed at the top of the wave boundary layer (and hence the Charnock coefficient) is uniquely determined from the conservation of momentum and energy inside the wave boundary layer, provided the viscous dissipation of the TKE is related to the local, reduced turbulent stress at each height. In order to determine the vertical wind profile inside the wave boundary layer, two further conditions need to be introduced.

First, linear analyses of airflow over sinusoidal waves (Belcher and Hunt 1993; Belcher 1999) show that the vertical decay of the pressure transport associated with a particular wavenumber k decays vertically over the same length scale as the wave-induced stress. Hence, the vertical variation of is represented here by the same function as for the wave-induced stress, namely F(k, z) defined in (4). Then, we may write
i1520-0485-34-11-2345-e59
so that the vertical gradient of Π is
i1520-0485-34-11-2345-e60
Second, we suppose that within the wave boundary layer the divergence of the turbulent transport dΠ′/dz is smaller than the other terms in the energy balance, as it is in a homogeneous rough-wall boundary layer.
Introducing (44) and (60) into (38) and setting dΠ′/ dz = 0, the net energy budget then becomes
i1520-0485-34-11-2345-e61
This energy conservation equation, evaluated at each height, may be used to determine the wind profile inside the wave boundary layer. [Compare this with the integrated energy budget (43) that was used in section 4d.]
Introducing (45) into (61), the vertical gradient of wind speed is expressed as
i1520-0485-34-11-2345-e62
On integrating (62)–(64) in z, we may write the mean wind speed analytically as (see appendix B for the derivation)
i1520-0485-34-11-2345-e65
with
i1520-0485-34-11-2345-e68
In the asymptotic limit of u∗1/u∗ → 0, the wind profile becomes
i1520-0485-34-11-2345-e70

In Fig. 5 the mean wind profiles are shown with the friction velocity u∗ = 0.5 m s−1, the sheltering wave age cs/us = 0.46, and different choices of k1. Just below the top of the wave boundary layer (z = δ/k0, indicated by diamonds), the wind profile becomes slightly steeper because the loss of the kinetic energy of the mean flow is enhanced by the energy flux into surface waves. However, the slope of the wind profile rapidly decreases toward the bottom of the wave boundary layer (z = δ/k1, indicated by squares) because the viscous dissipation of the TKE is reduced corresponding to the reduction of the turbulent stress τt. The asymptotic result (solid line) is a reasonable approximation for k1 ≥ 400 rad m−1.

The wave boundary layer model presented here yields the expression for the eddy viscosity K inside the wave boundary layer. (In contrast, previous models needed to specify the eddy viscosity.) Define the eddy viscosity such that
i1520-0485-34-11-2345-e72
Then, from (63), we obtain
i1520-0485-34-11-2345-e73
The eddy viscosity is affected by surface waves in two different ways. The first bracket on the right decreases the eddy viscosity by a fixed factor throughout the wave boundary layer. The second bracket on the right illustrates the effect of sheltering on the eddy viscosity, which is small near the top of the wave boundary layer but increases toward the bottom of the wave boundary layer where the turbulent stress is reduced.

5. Concluding remarks

We have developed a model of the wave boundary layer based on the conservation of momentum and energy. Energy conservation was cast as a bulk constraint, integrated across the depth of the wave boundary layer. The turbulence closure was then achieved by parameterizing the viscous dissipation rate of the TKE, ε, in terms of the local turbulent stress τt, following what is done in conventional one-equation models of turbulence. Momentum conservation was accounted for using the analytical model of the equilibrium surface wave spectra developed by Hara and Belcher (2002). This allowed us to obtain analytical expressions for the Charnock coefficient and to examine the results in terms of key nondimensional parameters. The results are generally consistent with previous observations and existing empirical parameterizations of the Charnock coefficient.

The strength of the analytical model developed here is that we have been able to identify the conditions for the Charnock coefficient to be a true constant. They are (i) the surface wave field is fully developed, (ii) the effects of surface tension and viscosity are small and most of the stress is supported by the wave drag, and (iii) the sheltering wave age is independent of wind stress; that is, the frequency/wavenumber spectrum not too far from the peak is proportional to the wind friction velocity.

One fundamental question that arises from this study is whether the sheltering wave age (cs/us) is truly independent of wind stress. The sheltering wave age should be constant if the frequency/wavenumber spectrum not too far from the peak is proportional to the friction velocity. A constant sheltering wave age is also a necessary requirement for the Charnock coefficient to be a true constant over mature seas. Field observations of equilibrium wave spectra under wide ranges of wind forcing are needed to answer this question.

When the wind sea is growing, the wave spectrum near the peak is clearly outside the equilibrium range, but contributes to the wave-induced stress. Hence the present model based on the equilibrium spectral model is not complete. If, however, the wave spectrum near the peak is known (either through observations or through numerical simulations) and the tail part of the spectrum is specified using the equilibrium spectral model, the wind profile and the drag coefficient may be calculated using the present wave boundary layer model. This may be a reasonable next step to examine the drag coefficient over growing and confused seas.

Last, as suggested by Makin and Kudryavtsev (2002), breaking waves may significantly enhance the roughness length. How might the present model be affected by breaking waves? Bulk energy conservation, as schematically shown in Fig. 3, always needs to be satisfied. Breaking waves are likely to modify the total energy flux into surface waves and the viscous dissipation inside the wave boundary layer. At present it is difficult to quantify these effects since our knowledge of breaking wave processes, such as statistical distribution of breaking waves and energy and momentum flux into individual breaking waves, is poor with uncertainties of factor of 10 or larger.

Although we cannot present quantitative arguments that neglecting breaking wave effects does not change appreciably our model results, the model shows, for the first time, a simple analytical picture of how the Charnock coefficient depends on different parameters and it yields the necessary conditions for the Charnock coefficient to be a true constant. This model framework will be a reasonable starting point to investigate breaking wave effects once our quantitative estimates of breaking wave processes become sufficiently constrained.

Acknowledgments

We thank the U.S. Office of Naval Research (CBLAST program, Grants N00014-0110125 and N00014-0110133) for supporting this research. Author TH also thanks the U.S. National Science Foundation (Grant OCE0002314) for additional support.

REFERENCES

  • Banner, M. L., , and W. L. Peirson, 1998: Tangential stress beneath wind-driven air–water interfaces. J. Fluid Mech, 364 , 115145.

  • Belcher, S. E., 1999: Wave growth by non-separated sheltering. Eur. J. Mech. B/Fluids, 18 , 447462.

  • Belcher, S. E., , and J. C. R. Hunt, 1993: Turbulent shear flow over slowly moving waves. J. Fluid Mech, 251 , 109148.

  • Chalikov, D., , and V. Makin, 1991: Models of the wave boundary layer. Bound.-Layer Meteor, 56 , 8399.

  • Charnock, H., 1955: Wind stress on a water surface. Quart. J. Roy. Meteor. Soc, 81 , 639640.

  • Donelan, M. A., , F. W. Dobson, , S. D. Smith, , and R. J. Anderson, 1993: On the dependence of sea surface roughness on wave development. J. Phys. Oceanogr, 23 , 21432149.

    • Search Google Scholar
    • Export Citation
  • Donelan, M. A., , W. M. Drennan, , and K. Katsaros, 1997: The air–sea momentum flux in conditions of wind sea and swell. J. Phys. Oceanogr, 27 , 20872099.

    • Search Google Scholar
    • Export Citation
  • Drennan, W. M., , H. C. Graber, , D. Hauser, , and C. Quentin, 2003: On the wave age dependence of wind stress over pure wind seas. J. Geophys. Res.,108, 8062, doi:10.1029/2000JC000715.

    • Search Google Scholar
    • Export Citation
  • Ebuchi, N., , and Y. Toba, 1991: Sea-surface roughness length fluctuating in concert with wind and waves. J. Oceanogr. Soc. Japan, 47 , 6379.

    • Search Google Scholar
    • Export Citation
  • Hara, T., , and S. E. Belcher, 2002: Wind forcing in the equilibrium range of wind-wave spectra. J. Fluid Mech, 470 , 223245.

  • Janssen, P. A. E. M., 1989: Wave-induced stress and the drag of air flow over sea waves. J. Phys. Oceanogr, 19 , 745754.

  • Johnson, H. K., , J. Hjstrup, , H. J. Vested, , and S. E. Larsen, 1998: On the dependence of sea surface roughness on wind waves. J. Phys. Oceanogr, 28 , 17021716.

    • Search Google Scholar
    • Export Citation
  • Jones, I. S. F., , and Y. Toba, 2001: Wind Stress over the Ocean. Cambridge University Press, 307 pp.

  • Kitaigorodskii, S. A., 1968: On the calculation of the aerodynamic roughness of the sea surface. Izv. Atmos. Oceanic Phys, 4 , 870878.

    • Search Google Scholar
    • Export Citation
  • Kudryavtsev, V. N., , and V. K. Makin, 2001: The impact of air-flow separation on the drag of the sea surface. Bound.-Layer Meteor, 98 , 155171.

    • Search Google Scholar
    • Export Citation
  • Kudryavtsev, V. N., , V. K. Makin, , and B. Chapron, 1999: Coupled sea surface–atmosphere model. 2. Spectrum of short wind waves. J. Geophys. Res, 104 , 76257639.

    • Search Google Scholar
    • Export Citation
  • Makin, V. K., , and C. Mastenbroek, 1996: Impact of waves on air– sea exchange of sensible heat and momentum. Bound.-Layer Meteor, 79 , 279300.

    • Search Google Scholar
    • Export Citation
  • Makin, V. K., , and V. N. Kudryavtsev, 1999: Coupled sea surface–atmosphere model. 1. Wind over waves coupling. J. Geophys. Res, 104 , 76137623.

    • Search Google Scholar
    • Export Citation
  • Makin, V. K., , and V. N. Kudryavtsev, 2002: Impact of dominant waves on sea drag. Bound.-Layer Meteor, 103 , 8399.

  • Makin, V. K., , V. N. Kudryavtsev, , and C. Mastenbroek, 1995: Drag of the sea surface. Bound.-Layer Meteor, 73 , 159182.

  • Mason, P. J., 1988: The formation of areally-averaged roughness lengths. Quart. J. Roy. Meteor. Soc, 114 , 399420.

  • Phillips, O. M., 1977: The Dynamics of the Upper Ocean. 2d ed. Cambridge University Press, 336 pp.

  • Phillips, O. M., 1985: Spectral and statistical properties of the equilibrium range in wind-generated gravity waves. J. Fluid Mech, 156 , 505531.

    • Search Google Scholar
    • Export Citation
  • Plant, W. J., 1982: A relationship between wind stress and wave slope. J. Geophys. Res, 87 , 19611967.

  • Taylor, P. K., , and M. J. Yelland, 2001: The dependence of sea surface roughness on the height and steepness of the waves. J. Phys. Oceanogr, 31 , 572590.

    • Search Google Scholar
    • Export Citation
  • Toba, Y., , and H. Kunishi, 1970: Breaking of wind waves and the sea surface wind stress. J. Oceanogr. Soc. Japan, 26 , 7180.

  • Yelland, M., , and P. K. Taylor, 1996: Wind stress measurements from the open ocean. J. Phys. Oceanogr, 26 , 541558.

APPENDIX A

Nonconservation of Energy in the Wave Boundary Layer

In past studies, two eddy viscosity models were proposed to determine the wind profile over surface waves. Here, we consider the energy budget inside the wave boundary layer using the two eddy viscosity models. With both models, the mean wind profile inside the wave boundary layer is related to the turbulent stress as
i1520-0485-34-11-2345-ea1
where K is the eddy viscosity and the viscous dissipation rate is scaled as
K3κz−4
The first eddy viscosity model, used by Janssen (1989), Chalikov and Makin (1991), and Makin et al. (1995), is simply related to the turbulent stress; that is,
Kκulkδzz.
The second eddy viscosity parameterization, proposed by Makin and Kudryavtsev (1999), was obtained from the TKE budget. It was assumed that the main balance was between the shear production τtotdu/dz and the viscous dissipation ρaε. Then, the eddy viscosity parameterization is expressed as
i1520-0485-34-11-2345-ea4
If we introduce (45), (A1), (A2), and the first eddy viscosity model (A3) into the energy equation
i1520-0485-34-11-2345-ea5
derived in section 4d, we obtain
i1520-0485-34-11-2345-ea6
Let us examine the overall energy budget inside the wave boundary layer over mature seas. We consider the asymptotic limit of u∗1/u∗ → 0 and integrate the three terms in (A6) from z = 0 to z = δ/k0. The results are written analytically (see appendix B for derivation) as
i1520-0485-34-11-2345-ea7
Here, FT, FB, and FD denote the integral of the first, second, and third terms, respectively. For a given sheltering wave age (i.e., for a given X0), we may compare the magnitude of these three quantities. In Fig. A1a we present how the total energy flux from the top of the wave boundary layer (FT) is partitioned into the internal dissipation (FD) and the flux into surface waves (FB). All the results are normalized by ρau3 and are presented versus the sheltering wave age cs/us between 0.46 and 2.38. Here, the solid line is the total flux from the top (FT/ρau3), the dashed line is the total excess energy, that is, the total flux from the top minus the internal dissipation [(FTFD)/ρau3], and the dash–dot line is the flux into surface waves at the bottom of the wave boundary layer (FB/ρau3). The energy is conserved if the dashed line and the dash–dot line overlap. Clearly, the eddy viscosity model (A3) underestimates the viscous dissipation and/or overestimates the shear production.
If we introduce (A1), (A2), and the second eddy viscosity model (A4) into the energy equation and integrate it from z = 0 to z = δ/k0, the results are written as
i1520-0485-34-11-2345-ea10
and FB/ρau3 is the same as (A8). We find that FT and FD exactly cancel out as expected. In Fig. A1b we present the total flux from the top (FT/ρau3) with the solid line and the flux into surface waves at the bottom of the wave boundary layer (FB/ρau3) with the dash–dot line. The total excess energy {the total flux from the top minus the internal dissipation [(FTFD)/ρau3]} is zero; that is, there is no energy left for surface waves. Therefore, the second eddy viscosity model (A4) overestimates the viscous dissipation and/or underestimates the shear production.

APPENDIX B

Evaluation of the Integral of the Local Friction Velocity

Let us evaluate the following integral of the local friction velocity,
i1520-0485-34-11-2345-eb1
which can be written
i1520-0485-34-11-2345-eb2
We introduce a variable X′ such that
i1520-0485-34-11-2345-eb3
Then, (B2) can be written
i1520-0485-34-11-2345-eb5
with
i1520-0485-34-11-2345-eb6
The integral on the right of (B5) may be solved analytically as
i1520-0485-34-11-2345-eb7
for n = 1,
i1520-0485-34-11-2345-eb8
for n = 3/2, and
i1520-0485-34-11-2345-eb9
for n = 3.

APPENDIX C

Charnock Coefficient with Different Eddy Viscosity Models

Let us calculate the Charnock coefficient over mature seas (idealized model) using the same equilibrium wave spectrum but two eddy viscosity models discussed in appendix A:
i1520-0485-34-11-2345-ec1
with n = 1 (Janssen 1989; Chalikov and Makin 1991; Makin et al. 1995) and n = 3/2 (Makin and Kudryavtsev 1999). The results can be expressed analytically as (see appendix B for derivation)
i1520-0485-34-11-2345-ec2
with
G1XX−20X2
for n = 1 and
i1520-0485-34-11-2345-ec4
for n = 3/2. The asymptotic results of u∗1/u∗ → 0 are
i1520-0485-34-11-2345-ec5
for n = 1 and
i1520-0485-34-11-2345-ec6
for n = 3/2. These asymptotic results are shown in Fig. C1 and compared with the results of this study. The eddy viscosity model with n = 1 yields lower values of the Charnock coefficient than the other two models. The difference between the eddy viscosity model with n = 3/2 and the present model is small at larger sheltering wave age but the former yields a lower value at lower sheltering wave age. Therefore, the present model results are less sensitive to the choice of the sheltering wave age values, that is, less sensitive to the level of the equilibrium spectrum.

Fig. 1.
Fig. 1.

Upper and lower bounds of sheltering wavenumbers ks vs wind friction velocity u∗ for mature seas. Solid lines indicate estimates based on approach I with data of Banner and Peirson (1998). Dash–dot lines indicate estimates based on approach II with the data collated by Phillips (1985) with sheltering wave age cs/us = 0.46 and 2.38

Citation: Journal of Physical Oceanography 34, 11; 10.1175/JPO2633.1

Fig. 2.
Fig. 2.

Upper and lower bounds of sheltering wave age cs/us vs wind friction velocity u∗ for mature seas. Solid lines indicate estimates based on approach I with data of Banner and Peirson (1998). Dash–dot lines indicate estimates based on approach II with the data collated by Phillips (1985) with sheltering wave age cs/us = 0.46 and 2.38

Citation: Journal of Physical Oceanography 34, 11; 10.1175/JPO2633.1

Fig. 3.
Fig. 3.

Schematic of energy conservation inside wave boundary layer

Citation: Journal of Physical Oceanography 34, 11; 10.1175/JPO2633.1

Fig. 4.
Fig. 4.

Upper and lower bounds of the Charnock coefficient vs friction velocity for mature seas. Lines are results calculated with ks estimated by approach II (dashed lines: k1 = 100 rad m−1, dash–dot lines: k1 = 400 rad m−1, and solid lines: asymptotic limit of u1/u∗ → 0), with sheltering wave age cs/us = 0.46 and 2.38. Diamonds are results calculated with ks estimated by approach I. Squares are empirical estimates by Banner and Peirson (1998)

Citation: Journal of Physical Oceanography 34, 11; 10.1175/JPO2633.1

Fig. 5.
Fig. 5.

Mean wind profiles over mature seas. Friction velocity u∗ = 0.5 m s−1. Sheltering wave age cs/us = 0.46. Dashed line: k1 = 100 rad m−1, dash–dot line: k1 = 400 rad m−1, and solid line: asymptotic limit of u1/u∗ → 0. Diamonds (z = δ/k0) and squares (z = δ/k1) indicate top and bottom of the wave boundary layer, respectively. Dotted line shows wind profile over a smooth solid surface

Citation: Journal of Physical Oceanography 34, 11; 10.1175/JPO2633.1

i1520-0485-34-11-2345-fA1

Fig. A1. Partition of the energy flux from the top of the wave boundary layer into the internal dissipation and the flux into surface waves. Results are shown vs sheltering wave age. Solid line: FT/ρau3, dashed line: (FTFD)/ρau3, and dash–dot line: FB/ρau3. (a) Results with the first eddy viscosity model (Janssen 1989; Chalikov and Makin 1991; Makin et al. 1995); (b) results with the second eddy viscosity model (Makin and Kudryavtsev 1999)

Citation: Journal of Physical Oceanography 34, 11; 10.1175/JPO2633.1

i1520-0485-34-11-2345-fC1

Fig. C1. Charnock coefficient vs the sheltering wave age cs/us for mature seas. Asymptotic results of u1/u∗ → 0. Dotted lines: eddy viscosity model with n = 1 (Janssen 1989; Chalikov and Makin 1991; Makin et al. 1995), dashed lines: eddy viscosity model with n = 2/3 (Makin and Kudryavtsev 1999), and solid lines: the present model

Citation: Journal of Physical Oceanography 34, 11; 10.1175/JPO2633.1

Save