482 Ocean Currents in a Non-homogeneous Ocean 



scale 1 : 500,000, then the vertical exaggeration n is 200 and one obtains for 

 isopycnals 



and for isosteres 



1-885 ^ , , , 

 v^-vi= ^^ 2:y (cm/sec). 



4. Ekman's Theory of Density Currents Including Friction 



Consideration of frictional effects in a stratified ocean is more difficult than in a 

 homogeneous sea for two reasons. 



First, the mathematical difficulties increase considerably, and secondly, the depen- 

 dence of the frictional coefficients on the stratification is very incompletely known. 

 In a stratified ocean friction should be less than in a homogeneous sea and the intro- 

 duction of a constant frictional coefficient, which must be made, does not fit so well 

 under these conditions as in the case of homogeneous water. 



Nevertheless, the results obtained on this basis afford some insight into the effect 

 of friction on the formation of density currents. Ekman (1905, 1906) has also dealt 

 with this in his theory of ocean currents and has made important contributions to 

 clarify this problem. A general solution, however, cannot be given. By means of some 

 typical cases only can conclusions be reached, from which the effects of friction can 

 be deduced by comparison with the frictionless cases. 



A simple case is that where the specific volume decreases uniformly with depth and 

 the isobaric surfaces are thus inclined planes. If, as a consequence of this assumption, 

 there is no pressure gradient at a particular depth d (horizontal isobaric surface), 

 then taking 



- ~ / = -fV and - - / = + fV 

 p dx ■' p dy ■' 



(U, V are the components of the geostrophic current) the equations of motion 

 (XII 1.28) give 



Z)2 d^u Z)2 8^v 



o^ ^1 + ^ = ^ ^^^ o^ ITS 



+ y=F and ^z tt^ - « = - ^. (XV.IO) 



Therein D is the frictional depth (equation XIII.26). For a co-ordinate system with the 

 X-axis parallel to the isobaric surfaces (F = 0) and taking as before U = b (d — z) a. 

 solution can be given for (XV.IO). The velocity profile can be calculated for different 

 values oi djD (Fig. 218) from the very complicated equation obtained. The velocity is 

 given in the diagram in units of f//5; they can also be considered as given in cm/sec if 

 the total layer from the sea surface down to the layer of no motion d, of the dynamic 

 section oriented in the direction of the gradient, contains in each 1 km layer a total of 

 10^cusin(/> solenoids (for 45° there are 51-6 solenoids). The difference from the 

 velocity profiles presented in Figs. 173 and 174 for a homogeneous mass structure is 

 considerable. Wherever the depth of no motion d may be, the motion there occurs 

 nearly in a plane. The friction affects principally the direction of this plane. Table 1 36 

 gives the largest (amax) and the smallest (auxm) angle of deflection from the gradient 



