M = \ (m + iv) dz = 



J 00 



General Theory of Ocean Currents in a Homogeneous Sea 403 



on a horizontal plane, give a representation of the direction and strength of the current 

 at the surface and at equidistant levels O-ID, 0-2i), etc. The arrovi^ at the peak of the 

 vertical Hne represents the direction of the wind. The arrow-heads he on a doubly 

 curved spiral and the end-points of the vectors on the horizontal plane lie on a logar- 

 ithmic spiral (Ekman spiral). Referring the current components to the direction of the 

 current at the surface and at right angles to it the diagram pictured in Fig. 169 is 

 obtained, which allows one immediately to judge whether the observed vertical dis- 

 tribution of the current carries the character of a drift current. 



Equation (XIII.26) shows further that the sea surface velocity increases in propor- 

 tion to the shearing stress T but in inverse proportion to the frictional depth D. 

 This is reasonable since, for equal Tthe more water that is set in motion, the smaller 

 must the velocity of the drift current be, i.e. the greater the depth D. The total drift 

 current transport per unit area of the sea surface is given by 



T 



7 



that is 



M^ = (Tjf) and My = 0. 



The total water transport due to a drift current occurs perpendicular cum sole to the 

 direction of the shearing stress of the wind producing it and since rj is not involved 

 it is independent of the assumption concerning the effects of eddy viscosity. For an 

 arbitrarily chosen co-ordinate system with shearing stresses T^ and Ty in the x- and 

 >Mlirections, the water transports in these directions will be 



M^ = ^ and My= - j. (XIII.27) 



Finite water depth. When the depth of the water is about of the same order as D it 

 has a noticeable effect on the drift current. For a depth d the e-functions in the solu- 

 tion will be replaced by hyperbolic functions. At the sea bottom (z = d) u = and 

 V — are assumed as boundary conditions indicating "adhering" ("Haften") of the 

 water on the underlaying surface. It is apparent from this solution and follows also 

 from Fig. 1 68 that as long as the depth of water is greater than the frictional depth D 

 the vertical distribution of the drift current will be unaffected, since the water layers 

 below the frictional depth have an insignificant share in the drift current. When, how- 

 ever, the water depth d becomes smaller than D, the effect of the bottom will be of 

 more influence the shallower the sea. Figure 170 shows the vertical current structure 

 for depths d = 1-25D, 0-50D, 0-25 D and 0-lD. The thin dotted curve near the origin 

 of the co-ordinate system for the curve ^ = 1-25 D shows the deviation towards the 

 curve for an infinitely large depth; thus in practice there is no significant difference 

 between them. The angle of deflection decreases rapidly with the depth of the water 

 and at very small depths, approximately from about d <0-\D, the movement shows 

 almost no effect of the Earth rotation. 



Other frictional assumptions. In addition, Ekman has given a solution for the case 

 where the frictional coefficient is proportional, not to the difference in velocity be- 

 tween two adjacent layers, but rather to its square. This gives essentially the same 

 results as for a constant -q ; the angle of deflection of the sea surface current is now 



