On the variability of the Charnock constant and the functional dependence of the drag coefficient on wind speed

Abstract

A model for the air–sea interface, based on the coupled pair of similarity relations for “aerodynamically” rough flow in both fluids, is presented, which is applied to fetch-limited and high wind speed conditions which occur, for example, in hurricanes. It is shown that the specification of the maximum 10-m drag coefficient and the 10-m wind speed and the peak wave speed at which it occurs are sufficient to uniquely determine the drag law, which asymptotes at low wind speeds to a Charnock constant similar to that for the fully developed wind wave sea and is almost independent of the peak wave speed at the maximum in drag coefficient. A feature of the drag law is that it is of Charnock form, almost independent of the wave age, consistent with the transfer of momentum to the wave spectrum being due to the smaller rather than the dominant wavelengths. The analysis is also applied to a variable sea state in which either the surface wind or the surface Stokes drift vary, but the peak wave speed is kept constant. The corresponding variability in the Charnock constant is in general accord with observations.

Appendix

1.1 A The dependence of c ₀ on R ₀ (or alternatively a or B)

We seek first to express a and B in terms of R ₀. On eliminating $\hat c_0$ between Eqs. 22 and 15 applied at $\hat K_{10}$ and using Eq. 17, we obtain

$$ a(R_0) = \left( \hat u_\star \left( 1+\frac{R_0\kappa}{2\sqrt{K_I}} \right) \right)^{-1} $$

(44)

and on eliminating $\hat c_0$ between Eqs. 22 and 29, we have

$$ B(R_0) = \frac{1}{2} a(R_0)\kappa \sqrt{2gz_{10}} \exp \left( \frac{1}{a(R_0)\hat u_\star} -\frac{\kappa}{2\sqrt{\hat K_{10}}} \right) $$

(45)

Hence, from Eq. 15,

$$ c_0(u_\star,R_0)=u_\star B(R_0)\frac{R(u_\star,R_0)}{\sqrt{K_I}} $$

(46)

and on substituting Eq. 46 in Eq. 12, K ₁₀(u _⋆, R ₀).

In summary for a prescribed R ₀, the peak wave speed c ₀ and also K ₁₀ are both only functions of u _⋆ (or u ₁₀). We note also that Eq. 44 reduces approximately to $a \approx 2/(\hat u_\star \kappa R_0)$, and since all the estimates of $\hat u_{10}$ are similar, a is approximately inversely proportional to R ₀ (Table 1). The range of R ₀, however, is not large, and hence, the dependencies of α on u _⋆ are similar from Eq. 23 for a prescribed α ₀, and consequently, the drag laws are also similar.