Forces and their Relationship to the Structure of the Ocean 323 



Some kinematic properties of the motion should perhaps be referred to here. The 

 path of a small water element is obtained from the three simultaneous equations : 



dx = udt, dy = vdt, dz = wdt. (X.21) 



The integration constants for f = are then the three initial co-ordinates a, b, c of 

 the water element under consideration. 



The instantaneous state of motion in a water mass is given by the stream lines 

 (see Chap. Xll) which everywhere indicate the direction of a current by the tangent at 

 the point under consideration. Their differential equations are 



dx dy dz 



— - — = —. (X. 22) 



U V w 



Since the state of motion in a steady current does not change with time it is under- 

 standable that the stream lines in this case coincide with the trajectories of the water 

 elements. Steady currents are not without accelerations since only the local part of 

 the acceleration disappears; the advective part, for example, u(duldx) + vidujdy) + 

 w{8ul8z) requires that the moving water element reaches any point with a velocity 

 equal to that prescribed for that point, 



3. The Continuity Equation and the Boundary-surface Conditions 



To the equations of motion must be added, as a special condition, the continuity 

 equation which is based on the law of the conservation of mass. This states that in any 

 volume element specified in the interior of a liquid the mass entering it at a given time 

 must be equal to that leaving it at the same time. Any excess in one or the other direc- 

 tion must appear as a corresponding change in the density if the liquid will permit such 

 a change. Taking a volume element SxSySz, investigation of the extent by which, as a 

 consequence of flow through the boundaries the amount of liquid enclosed in it 

 varies, shows that for a conservation of mass the continuity condition is given by the 

 equation 



dp dpu dpi) dpw 



Using the relationship equation (X. 1 5) this can be given the following form 



\ dp ] da 8u 8v 8w 



p dt a dt 8x cy 8z 



In an incompressible liquid dpldt = the continuity equation reduces to 



8u 8v 8h' 



^ + ^ + — = - (X.25) 



ex oy oz ^ -^ 



This does not assume that the liquid has the same density everywhere (homogeneous 

 medium). The expression cujdx + cvjcy + bwj8z indicates the volume increase in 

 unit time per unit volume of the element and is usually termed the three-dimensional 

 or total divergence of the vector (m, r, vv). The continuity equation for an incom- 

 pressible medium is then 



div (//, r, h) =-- 0. (X.26> 



