318 



Forces and their Relationship to the Structure of the Ocean 



For this assumption concerning the inner friction, the effect of the solid, stationary 

 sea bed appears as a corresponding boundary condition. If n is the direction of the 

 normal to the sea bottom (z = 0) then 



(1) for completely frictionless movement of the water over the sea 

 bed (r = 0): dVldn = 0; 



(2) if the water is stationary at the bottom (z = 0): K = 0; 



(3) for part-time gliding at the sea bottom, that is for a discontinuity 

 of the velocity at z = 0: dVjdt = f(V), where /(K) is a certain 

 function of V, for example, kpV^. 



In a volume element 8x 8y 8z (see Fig. 134) in a current in which the velocity V 

 in a direction perpendicular to the vertical direction z is very much stronger, there 

 will be a shearing stress rSxSy on the lower surface 8x8y and a corresponding 



- (X. 11) 



Fig. 134. Computation of the frictional force from the shearing stresses. 



^T j^ {8Tl8z)8z}8x8y on the upper surface at a distance Sz from the lower. On the 

 entire volume element there acts thus a frictional force (8Tldz)8x8y8z so that accord- 

 ing to (X. 10) the frictional force per unit mass in the direction of the x-co-ordinate 



will be given by 



fi 8^V 



8z^ ' 



R:r. — 



(XJ2) 



Where fx can be regarded as a constant. 



From the general theory of friction in hquids it follows that for an incompressible 

 fluid (and thus also with sufficient accuracy for sea-water) the components of the fric- 

 tional force per unit mass in a viscous liquid are given by the three expressions : 



u ix M , 



;?^ = - An, Ry = ^ Av, i?, = - Aw, 

 PR P 



(X.13) 



Footnote continued from p. 317 



seldom change position. These occasional changes in position are facilitated by the action of a tan- 

 gential shearing stress especially in the direction of the stress itself and this alone permits the individual 

 layers to glide over each other. The more frequent the changes in position of the molecules, the lower 

 is the internal friction (viscosity) characteristic of the liquid. 



