96 OCEAN CURRENTS RELATED TO THE DISTRIBUTION OF MASS 



From equations (VI, 5) and (VI, 4) it also follows that the components 

 of the horizontal pressure gradient are identical with the components of 

 gravity acting along the isobaric surfaces : 



"(SL ="'"■'= 



a 



\dy/p.x 



^^dx dp 



= — 10a ^-; 



dp ax 



dz 



dp 



(VI, 7) 



The inclination is positive in the direction in which the surface slopes 

 downward because the positive z axis is directed downward. 



To the above equations must be added the equation of continuity 

 and the kinematic and dynamic boundary conditions. The equation of 

 continuity can be written in the form: 



_ldp Ida^dv, dv, dv,^^^ (VI, 8) 



p dt a dt dx dy dz 



and states that the rate of expansion per unit volume of a moving element 

 equals the divergence of the velocity. The vector symbol v is used for 

 the velocity. Water is nearly incompressible, and in the sea the vertical 

 component of velocity is always small. When applied to the oceans the 

 equation of continuity is often reduced to (dVx/dx -\- dVy/dy) = 0. 



The kinematic boundary condition states that a particle on a boundary 

 surface must move in a direction normal to the surface with the velocity 

 of that surface. If the boundary surface is rigid, the water in contact 

 with the surface can have no velocity component normal to the surface. 



A dynamic boundary condition states that at any boundary surface 

 the pressure must be the same on both sides of the surface. This also 

 applies to internal boundaries that separate water of different density, in 

 which case the condition states only that the pressure must vary con- 

 tinuously. The densities and velocities may, however, vary abruptly 

 when passing from one side of the boundary surface to the other. Calling 

 the densities on both sides p and p', and the velocities v and y', and omit- 

 ting the frictional terms and the accelerations, the dynamic boundary 

 condition takes the form 



d{p — p') = \(pVy — pVy)dx — \{pVx — p'vj)dy 



^ g(p - p')dz = Q, (VI, 9) 



because along the boundary surface p = p' , where p represents the pres- 

 sure exerted against the boundary surface from one side, and p' represents 

 the pressure exerted from the other side. Equation (VI, 9) must be 



