WIND CURRENTS AND WIND WAVES 1 1 9 



stress, To, is practically independent of the distance from the surface, 

 von Kd^rmdn showed that the mixing length, I, increases linearly with 

 increasing distance from the surface: I = koz, w^here ko = 0.4, and that 

 the relation between the velocity distribution in the turbulent region 

 and the stress can be written in the form 



-^ = 5.5 + 5.75 log ^ . K (VII, 4) 



/to M \ P 



where n is the dynamic viscosity. Applied to the sea surface, the theorem 

 shows that, if a laminar boundary layer exists, the stress in the lowest 

 layers, which must equal the stress against the sea surface, could be 

 derived from a single measurement of the wind velocity, W, at a short 

 distance, z, from the sea surface. Measurements at two or more levels 

 would have to render the same value of the stress, and the observed 

 velocities would have to be a linear function of the logarithm of the 

 distance. 



Over a rough surface, different conditions are encountered. Prandtl 

 assumes that then the turbulent motion extends to the very surface, 

 meaning that the mixing length has a definite value at the surface itself : 



I = ko(z + zo), (VII, 5) 



where zo is called the roughness length and is related to the average height 

 of the roughness elements. According to Prandtl the eddy viscosity, 

 fie, is expressed by (VII, 2). Therefore, 



^0 = Me ^ = pko'iz + Zo)' y-^J 



dW 

 " dz 



or 



dW 



dz 



ko{z + .o) V? ^^^ ^' = '^°^' + '°^ ^P' 



Through integration, assuming ko = 0.4 and TF = at 2; = 0, and intro- 

 ducing base-10 logarithms, one obtains 



and 



TT, = 5.75 Jp log ^-±^ (VII, 6) 



0-0302 ^ 2 A^TT 7^ 



To = P 7 ^"2 Wz^. (VII, 7) 



(-^) 



Measurements of the wind velocity of two distances are needed in 

 order to determine both tq and 20, and measurements at three or more 

 levels are necessary in order to test the validity of the equation. 



