404 General Theory of Ocean Currents in a Homogeneous Sea 



49-1° and the current dies away at the finite depth of \-25D. It should be pointed out 

 that the relationship between T, D and Vq are somewhat different. The total transport 

 for the quadratic frictional law is, however, also given by (XIII.27) and is thus inde- 

 pendent of the frictional assumption. This can also be shown by strict mathematical 

 treatment. For a variable -q the expression 



d\u, v) 



in equation (XIII.23) is replaced by 



d I d(u, v) 

 8z [ ^ -d^- 



see p. 319. Integrating this equation from z = to z = oo or respectively down to a 

 depth at which the drift current can no longer be detected, and considering that the 

 shearing stress is present only at the sea surface, then with the help of equation (XIII. 13) 

 relationships are obtained which are identical with (XIII.27). These, however, were 

 derived for a constant rj. It could possibly be expected that during the transfer of the 

 turbulent wind momentum to the water masses at the sea surface the two horizontal 

 components of the shearing stress (in the direction of the wind and at right angles to 

 it) would be governed by different turbulent coefficients. An extension of the Ekman 

 theory along such lines has been given by Ertel (1937). It leads to deflection angles 

 different from 45° while the vertical current structure becomes a deformed spiral. 



Another principle applicable both to the wind stress at the sea surface and to the 

 friction at the bottom has been developed by Jeffreys (1923), In conformity with 

 turbulence theory he assumed that at both the sea surface and at the bottom, "gliding" 

 of the water masses occurs in which the friction is assumed proportional to the square 

 of the velocity differences. The boundary condition at the sea bottom is taken as 



and at the sea surface as 



- 7] —^ = Kp{u\ v^) 



^("'^) V '2 '2^ 



where p' is the density of the air and u' and v' are the velocity components of the wind 

 relative to the water movement at the sea surface (see p. 317, equation (X.9).) 



The more recent results of research in turbulence also show that in the vicinity of 

 boundary surfaces the assumption of a constant frictional coefficient leads to current 

 distributions which do not accord with the observed facts. This makes it necessary to 

 introduce turbulent coefficients, wliich vary with the distance from the solid boundary. 

 That such a method leads to results satisfactorily explaining the observed features has 

 been shown by an investigation of Fjelstad (1929) using observations made by Sver- 

 drup on a drift current over the North Siberian Shelf, where there was a strong increase 

 of the frictional coefficient from the bottom to the surface. He succeeded in deriving 

 a functional relationship for these coefficients of the form 



fZ+ €\8/* 



1 =^ Vo ' 



