Drag Coefficient, Surface Roughness, and Reference Wind Speed

P.A. Hwang
Oceanography Division

Introduction: Drag coefficient (CD) and surface roughness (z0) are important parameters for quantifying air-sea momentum and energy exchanges. In almost all earlier analyses of CD and z0, the reference wind speed at 10-m elevation (U10) is used. Although the adoption of U10 in the analyses provides a consistent reference level of wind measurements (compared to earlier reports using "mast height" or "anemometer height"), the dynamical significance of the 10-m elevation in the marine boundary layer is not clear. From a heuristic point of view, surface waves are the ocean surface roughness, and the air-sea interaction processes are influenced by wave conditions. Because the influence of surface waves decays exponentially, with wavelength serving as the vertical length scale, the dynamically meaningful reference elevation should be λ, the characteristic wavelength of the peak component of surface wave spectrum.

Wavelength Scaling: For a logarithmic wind profile,

Equation Image

where U is the wind speed, u* the wind friction velocity, κ the von Kármán constant (0.4), z the vertical elevation, and z0 the dynamic roughness, the drag coefficient referenced to wind speed at the elevation equal to one-half of the surface wavelength, Cλ/2= u*2 / U2λ/2 is

Equation Image

Equation (2) suggests that a natural expression of the dimensionless surface roughness is kpz0,

Equation Image

Analysis of several field experiments with wind-sea dominated wave conditions1 yields

Equation Image

with Ac = 1.22 × 10-2, ac = 0.704, where ω** = u*ωp / g is the dimensionless frequency of the air-sea coupled system, g the gravitational acceleration, and ωp the frequency of peak spectral component (Fig. 1(a)). Although the data scatter is large, the result is a substantial improvement over C10 (Fig. 1(b-d)). Figure 1 illustrates convincingly that Cλ/2 is indeed superior to C10 for accounting for the surface wave effects on the wind stress over the ocean surface.


Figure 1 Image
Drag coefficient based on (a) wavelength scaling,1 Cλ/2**) and 10-m scaling: (b) C10(U10), (c) C10**), and (d) C10(U10/Cp), where Cp is the phase speed of the peak component of wave spectrum. The data scatter in C10 expressions is considerably worse than that of the Cλ/2 expression. The solid curve is Eq. (4).


Figure 2 Image
Dimensionless roughness in terms if (a) kpz0**), and (b) z0* = z0g/u*2; the curve represents Eq. (4) applied to Eqs. (3) and (5). (c) Same as (b) but includes additional field and laboratory data in the comparison. Empirical relations from several earlier studies2 are also shown in this figure.


Figure 2(a) shows the dimensionless roughness kpz0. The measurements from different sources collapse very nicely following the wavelength scaling function (Eq. (3)). The surface roughness is also frequently expressed in terms of the Charnock parameter, z0* = z0 g / u*2 which can be obtained from applying the dispersion relation of surface waves to Eq. (3),1

Equation Image

The Charnock parameter increases with ω** in the frequency range of the field data (Fig. 2(b)). The parameterization function (Eq. (5)) predicts that the dependence of z0***) reverses to a decreasing trend at ω** -0.25 and z0* approaches ω**2 asymptotically at very large ω**. The inverse relationship of z0***) at high ω** range has been observed in laboratory measurements2 (Fig. 2(c)). It is noted that the apparently different behavior of z0* with ω** shown in Fig. 2(c) had generated considerable controversies in the air-sea research community, as is evident from the large number of empirical (and contradicting) relations of z0***) that have been proposed in the literature.2 The present analysis predicts correctly the increasing trend of z0***) in the low-frequency region and the decreasing trend in the high-frequency region.

Conclusions: Quantification of the drag coefficient and surface roughness is critically influenced by the choice of the reference wind speed. It is shown that when the scaling wind speed is referenced to the surface waves, the experimental data of drag coefficient and surface roughness can be collapsed into simple clusters. The analysis yields strong evidence that surface waves are the roughness element of the ocean surface and that they exert significant influences on air-sea momentum exchanges. The results also have important implications on ocean remote sensing applications.

[Sponsored by ONR]

1P.A. Hwang, "Influence of Wavelength on the Parameterization of Drag Coefficient and Surface Roughness," J. Oceanogr. (in press).
2I.S.F. Jones and Y. Toba (eds.), Wind Stress Over the Ocean (Cambridge University Press, Cambridge, UK, 2001).