Relating the Drag Coefficient and the Roughness Length over the Sea to the Wavelength of the Peak Waves

The standard 10-m reference height for computing the drag coefficient over the sea is admittedly arbitrary. The literature contains occasional suggestions that a scaling length based on the wavelength of the peak waves lambda(p) is a more natural reference height. Attempts to confirm this hypothesis must be done carefully, however, because of the potential for fictitious correlation between nondimensional dependent and independent variables. With the DMAJ dataset as an example, this study reviews the issue of fictitious correlation in analyses that use lambda(p)/2 as the reference height for evaluating the drag coefficient and that use k(p) (=2 pi/lambda(p)) as a scale for the roughness length z(0). (The DMAJ dataset is a compilation of four individual datasets; D, M, A, and J, respectively, identify the lead authors of the four studies: Donelan, Merzi, Anctil, and Janssen.) This dataset has been used in several previous studies to evaluate the dependence of k(p)z(0) and the drag coefficient evaluated at lambda(p)/2 on the nondimensional wave parameter omega(*) = omega(p)u(*)/g. Here omega(p) is the radian frequency of the peak in the wind-wave spectrum, u(*) is the friction velocity, and g is the acceleration of gravity. Because the DMAJ dataset does not, however, include independent measurements of lambda(p) and omega(p), lambda(p) had to be inferred from measurements of omega(p) through the wave dispersion relation. The presence of omega(p) in both the dependent and independent variables, therefore, exacerbates the fictitious correlation. One conclusion, thus, is that using lambda(p) to formulate the drag coefficient and the nondimensional roughness length as functions of a nondimensional variable that includes omega(p) requires a dataset with independent measurements of lambda(p) and omega(p).