On the variability of the Charnock constant and the functional dependence of the drag coefficient on wind speed: Part II-Observations

Abstract

An analytical expression for the 10 m drag law in terms of the 10 m wind speed at the maximum in the 10 m drag coefficient, and the Charnock constant is presented, which is based on the results obtained from a model of the air-sea interface derived in Bye et al. (2010). This drag law is almost independent of wave age and over the mid-range of wind speeds (5−17 ms⁻¹) is very similar to the drag law based on observed data presented in Foreman and Emeis (2010). The linear fit of the observed data which incorporates a constant into the traditional definition of the drag coefficient is shown to arise to first-order as a consequence of the momentum exchange across the air-sea boundary layer brought about by wave generation and spray production which are explicitly represented in the theoretical model.

Appendix

1.1 The derivation of the 10 m drag law

A convenient analytical procedure is to express the various relations in terms of the non-dimensional parameter X = 1/(a u _⋆m ). On substituting X in Eq. 3 using Eq. 6, we obtain the frictional parameter:

$$ R = \frac{2\sqrt{K_{I}}}{\kappa} \cdot \frac{X(X-1)}{X-s} $$

(13)

in which K _I is the inertial drag coefficient, κ is von Kármán’s constant and s = u _⋆/u _⋆m , and where from Eq. 5 evaluated at K _10m using Eq. 6:

$$ X = \frac{1}{2}\left[\ln\frac{ c_{0m}^{2}}{2gz_{10}} + \frac{\kappa}{\sqrt{K_{10m}}} \right] $$

(14)

in which g is the acceleration of gravity and z ₁₀ = 10 m. Similarly, using Eqs. 1, 2 and 13, we obtain:

$$ c_{0} = \frac{ (X - 1)s}{(X - s)} c_{0m} $$

(15)

and the wave generation parameter:

$$ B = \frac{1}{2} \frac{\kappa c_{0m}}{X u_{\star m}} $$

(16)

On substituting for R and B in Eq. 5 and using Eq. 14) to eliminate c _0m , we obtain the following relation for the inverse drag law, u ₁₀(u _⋆), in terms of X:

$$ u_{10} = \frac{2u_{\star}}{\kappa} \left\{ \frac{X (s - 1)}{(X - s)} - \ln \left[ \frac{(X -1) s}{(X - s)} \right] \right\} + s u_{10m} $$

(17)

from which the 10 m drag law (5) is determined. X can be evaluated from Eq. 14) provided that K _10m , and c _0m which depends on sea state, are known. Direct observations of c _0m (or T _m ), however, are not generally available. It is, therefore, necessary to use field data at u ₁₀ < u _10m to infer X, as is shown below.

On substituting Eqs. 15) in 14, we obtain:

$$ X = X^{\prime} + \ln \left[ \frac{X-s}{X-1} \right] $$

(18)

where $X^{\prime } = 1/2 [ \ln {c_{0}^{2}}/ 2gs^{2} z_{10} + \kappa /\sqrt {K_{10m}}]$ , and since X≫1, on using the binomial expansion (18) yields approximately:

$$ X = X^{\prime} + \frac{1-s}{X^{\prime}} $$

(19)

Hence, X may be evaluated in terms of s, c ₀, and K _10m by substituting c ₀ in X ^′.

The Charnock constant can also be evaluated in terms of X and compared with observations. From the defining relation for the sea surface roughness (z ₀), we have $\ln z_{10}/z_{0} = \kappa /\sqrt {K_{10}}$ and hence from Eq. 9,

$$ \alpha = \frac{g}{u_{\star}^{2}} z_{10} \exp \left[ \frac{-\kappa}{\sqrt{K_{10}}} \right] $$

(20)

which on eliminating K ₁₀ using Eq. 5 yields:

$$ \alpha = \frac{1}{2} W^{2} \exp \left[ \frac{-R\kappa}{\sqrt{K_{I}}} \right] $$

(21)

where W = c ₀/u _⋆ is the wave age (Bye et al. 2010). On substituting for R from (13), and for W from (15) which yields:

$$ W =W_{m} \frac{(X-1)}{(X-s)} $$

(22)

where W _m = c _{o m} /u _⋆m is the wave age at the drag coefficient maximum, we obtain,

$$ \alpha = \frac{1}{2} {W_{m}^{2}} \left[ \frac{X-1}{X-s} \right]^{2} \exp \left[ - 2 X\frac{(X - 1)}{(X - s)} \right] $$

(23)

in which, from Eq. 14:

$$ {W_{m}^{2}} = \frac{2 g z_{0}}{u_{\star}^{2}} \exp \left[ 2 X- \frac{\kappa}{\sqrt{K_{10m}}} \right]. $$

(24)

Finally, on substituting Eqs. 24) in 23, we have the expression for the Charnock constant:

$$ \alpha = \left[ \frac{X-1}{X-s} \right]^{2} \frac{g z_{10}}{u_{\star m}^{2}} \exp \left[ 2 \frac{X(1 - s)}{(X - s)} \right] \exp \left[ \frac{-\kappa}{\sqrt{K_{10m}}} \right] $$

(25)

We note that (25), which is independent of c _0m , indicates that the choice of K _10m , rather than u _⋆m or c _0m , is highly significant in the evaluation of α. In terms of X, the spray parameter:

$$ a = \frac{1}{X u_{\star m}} $$

(26)

These expressions will be evaluated using the observed estimates at the maximum drag coefficient of K _10m = 0.002 and u _10m = 40 ms⁻¹ (u _⋆m = 1.79 ms⁻¹), and also κ = 4, g = 9.8 ms⁻² and K _I = 0.0015, and at the onset of wave growth (s = 05) yield from Eq. 17 u ₁₀ = 3.0 ms⁻¹, and, α = 018 in agreement with observations for the fully developed sea (Section 3.2). Note that all these expressions, except for Eqs. 13 and 21 which involve R, are independent of K _I .