General Theory of Ocean Currents in a Homogeneous Sea 421 



According to recent hydrodynamic theory (see for instance, Prandtl, 1942, 

 p. 108) the investigations of flow over smooth and rough surfaces have shown that 

 the shearing stress of the wind follows the relations: 



w zp 



for a smooth surface: ,, , ,, = 5-5 + 5-75 log — ^J{r\p') (XIII.46) 



and 



z -\- Zq 

 for a rough surface: w = 5-75 Vi'^lp') log — :: — • (XIII.47) 



To decide whether a water surface is considered "smooth" or "rough" for different 

 wind conditions it is necessary to investigate the vertical wind distribution over it. 

 This has been done by WiJST (1920) and by Rossby and Montgomery (1935), who 

 have discussed the results and have concluded that for winds of more than 

 6-8 m/sec (measured 1 5 m above the surface, Beaufort 4) the water surface must be 

 considered as "rough". As a result it was ascertained that for moderate and strong 

 winds the roughness length z^ was independent of the wind strength and had a constant 

 value of 0-6 cm. The formula (XIII.47) then gives 



r=2-9 X \0-^ p'n'l, (XIII.48) 



where n\o is the wind speed at 10 m above the surface. This formula, however, no 

 longer applies when vvjo < Beaufort 4 or 6-8 m/sec and the surface has to be con- 

 sidered as "smooth". In this case the formula (XIII.46) will be valid. The values of T 

 calculated in this way are about a third less than those computed from (XIII.48). 

 As a reasonable first approximation they satisfy the relation 



r = 0-9 X 10-3 p'wlo. (XIII.49) 



This shows that there is a laminar boundary layer of small vertical extent in wind 

 profiles above the water surface, which reduces friction considerably (Rossby, 1936 b). 

 Further analyses of measurements of the tangential wind stress and the rouglmess of 

 the sea surface have been made by Neumann (1948) who showed that the frictional 

 factor at the surface decreases with increasing wind speed and that in general at the 

 surface of the sea 



r=0-9 X 10-3p'i;3/2. 



Neumann attempted to explain this striking behaviour of the hydrodynamic roughness 

 at the sea surface by changes in the nature of the sea-way dependent on the wind 

 strength. The waves move with the wind and the surface of the sea will very likely 

 tend towards a profile, offering the least possible resistance to the wind over it 

 (Model, 1942); see Munk (1955) for a more detailed discussion. Further measure- 

 ments of the wind stress over water have been made by van Dorn (1952). 



Another method for the determination of the wind stress on the water given by 

 Shepard and Omi (1952) making use of the geostrophic deflection of the wind at the 

 sea surface and upwards to a height of some hundred metres above it. The geostrophic 

 wind can be calculated with sufficient accuracy and the deviation of the observed 

 wind from this depends only on the friction. This method gives resistance coefficients 

 about 1 X 10-3. 



