Addendum to `Wind Stress Over Waves: Effects of Sea Roughness and Atmospheric Stability’

Abstract

Myrhaug and Slaattelid 1 (hereafter referred to as MS) considered the effects of sea roughness and atmospheric stability on the sea surface wind stress over waves, which are in local equilibrium with the wind, by using the logarithmic boundary layer profile including a stability function, as well as adopting some commonly used sea surface roughness formulations. The engineering relevance of the results was also discussed. Since no consistent sea surface roughness formulations existed in the literature at that time, MS chose to demonstrate how the results varied by using two roughness formulas having significantly different behavior. MS used the Toba et al. 2 formula where the roughness increases as the wave age increases, and the Donelan et al. 3 formula where the roughness decreases as the wave age increases (see Table 2, MS). The wave age independent Charnock 4 formula was used as a reference. Since then an expression for the sea surface roughness has been provided by Volkov 5. The purpose of this note is to use the Volkov 5 roughness formula and to present similar results as in MS. Volkov concludes that simple power law formulas like that proposed by e.g. Toba et al. and Donelan et al. for the dependency of roughness with wave age are not adequate. At present state of knowledge he suggests the model given by z0*=0.03x exp(−0.14x) for 0.35<x<35Display Formulaz0*=0.008 for 35≤xwhere z0*=gz0/u*2 is the dimensionless roughness, and x=cp/u* is the wave age. Moreover, g is the acceleration of gravity, z0 is the sea surface roughness length, u* is the friction velocity equal to the square root of the vertical flux of horizontal momentum at the surface, and cp is the phase speed associated with wind waves with peak frequency σp. Equation (1) is obtained as a reasonable fit to existing data (see Fig. 10.6, Volkov 5). The nondimensional roughness has a maximum value at cp/u* around 10 (see Fig. 1.15, Jones et al. 6), and for cp/u*>35, corresponding to light wind over swell, the sea surface becomes smooth with a nondimensional roughness near 0.01. So, essentially Eq. (1) behaves as the Toba et al. roughness for very young waves and as the Donelan et al. roughness for fully developed waves. Volkov’s expression seems to be a reasonable compromise between simplicity and accuracy based on the present state of knowledge. Further background and details are given in Volkov 5 and Jones et al. 6.