328 



Forces and their Relationship to the Structure of the Ocean 



it has to be balanced by the vectorial sum of all the surface pressures, that is, by 

 Fj/?! — F^Pz- This gives for the current tube the equilibrium equation 



Ki2 + 



a-i 



P2. 



K22+-IF2 



which corresponds to the BemouUi pressure equation. 



If the current tube is curved (Fig. 1 36^) the forces at both places 1 and 2 will have 

 different directions and the resultant R of the two forces (indicated at point A) shows 

 the effect of the pressure exerted by the curved flow on the adjacent water masses. 



(b) 



Fig. 1366 



By the introduction of the contmuity equation, the equations of motion can be put 

 in a form which expresses changes in impulse more clearly {impulse form of the equation 

 of motion). Multiplying the continuity equation (X. 23) by m, v, w and adding these 

 expressions respectively to the first, second and third of the equations of motion (with- 

 out Coriolis force and friction terms, X, Y, Z are the external forces), then 



dpu dpuu 8puv dpuw 



dt 



+ 



8x 



+ 



dy 



+ 



8z 



pX 



dp 



dx' 

 dp 



8pv 8pvu 8pvv 8pvw 



'8i '^ ~8x '^ "ajT "^ ~aF~ ^ ^ 8/ > 



(X.38) 



8pw 8pwu 8pwv 8pww 



"aT "^ "ax" "^ ~e^ "^ 8z 



pZ 



8p 



8z' 



These show that the changes in the momentum within a volume element can be re- 

 garded either as the result of forces acting on the mass contained within the volume 

 element, or as the result of the mass flux passing through the boundary surfaces carrying 

 its own momentum with it. 



The impulse-form of the equations of motion (X. 38) can be used with advantage 

 in considerations concerning the internal structure of turbulent currents (Reynolds, 

 1 894). At any point of a turbulent flow there will be more or less strong variations in 

 the flow velocity. These variations will, however, balance each other completely if 

 on the average the current is steady, and if a sufficiently long period is considered. The 

 velocity components at a given point can then be represented by 



M = W + «', V = V -\- V', W = W + H'', (X.40) 



