410 General Theory of Ocean Currents in a Homogeneous Sea 



T^=f\ pvdz and Ty=f\ p(U - u) dz. (XIII.33) 







Here h denotes the lower frictional depth at which the deviations U — u and v from 

 the geostrophic current vanish. It can further be assumed that T at the bottom has 

 the same direction as the velocity at the bottom, so that 





= tan a, (XIII.34) 



Z= 



where a is the angle between the direction of the resulting T and that of the uniform 

 deep current. These relationships form the basis of the vertical current structure of the 

 bottom current, but further extension of the calculation fails due to the still imperfect 

 knowledge of the laws of turbulent flow. However, by use of the above presented 

 basics for turbulent friction a rather good estimate of the vertical velocity profiles 

 to be expected can be obtained. 



As a first approximation it can be assumed that in the vicinity of the bottom u 

 varies with the «th root of z 



u = U 



(a' 



Further, near the bottom v=u tan a ; in order that v vanishes at a height z=/7i one has to 

 assume 



y = M 1 1 — r I tan a. 



Ill is smaller than h and must be chosen so that the current structure near the bottom is 

 in accordance with that shown by turbulence research (equation XIII. 19), The 

 equation (XIII.33) then gives 



,,„ = ^^^' and r. = („ ^ i);p„ ^ 1) PfhU. (Xin.35) 

 For an indifferent mass structure the equation (XIII. 19) gives 



for the velocity distribution above a rough surface. In Co=(^/7-35) the quantity 

 kisa measure of the roughness height of the bottom. Since for z=hi, u must be equal 

 to U the ratio TJp can be expressed in terms of U 



• ^^ = {5-75 log (/;i/co)F (XIII.37) 



and from (XIII.36) 



10g(z/Co) /YTTT'58^ 



" = ^ 1 7rT~\ • (XIII.38) 



log (hjco) 



This gives a second equation for u and both must give a curve of the same shape. 

 The most suitable assumption is that both give the same values for the transport 



