## Abstract

It is proposed that the sea surface roughness *z*_{o} can be predicted from the height and steepness of the waves, *z*_{o}/*H*_{s} = *A*(*H*_{s}/*L*_{p})^{B}, where *H*_{s} and *L*_{p} are the significant wave height and peak wavelength for the combined sea and swell spectrum; best estimates for the coefficients are *A* = 1200, *B* = 4.5. The proposed formula is shown to predict well the magnitude and behavior of the drag coefficient as observed in wave tanks, lakes, and the open ocean, thus reconciling observations that previously had appeared disparate. Indeed, the formula suggests that changes in roughness due to limited duration or fetch are of order 10% or less. Thus all deep water, pure windseas, regardless of fetch or duration, extract momentum from the air at a rate similar to that predicted for a fully developed sea. This is confirmed using published field data for a wide range of conditions over lakes and coastal seas. Only for field data corresponding to extremely young waves (*U*_{10}/*c*_{p} > 3) were there appreciable differences between the predicted and observed roughness values, the latter being larger on average. Significant changes in roughness may be caused by shoaling or by swell. A large increase in roughness is predicted for shoaling waves if the depth is less than about 0.2*L*_{p}. The presence of swell in the open ocean acts, on average, to significantly decrease the effective wave steepness and hence the mean roughness compared to that for a pure windsea. Thus the predicted open ocean roughness is, at most wind speeds, significantly less than is observed for pure wind waves on lakes. Only at high wind speeds, such that the windsea dominates the swell, do the mean open ocean values reach those for a fully developed sea.

*Corresponding author address:* Peter K. Taylor, Southampton Oceanography Centre, Express Dock, Southampton SO14 3ZH, United Kingdom.