Forces and their Relationship to the Structure of the Ocean 



321 



conservative force is gravity acting in the positive direction of the z-axis (downward), 

 the pressure gradient forces will be given by 



\ dp I dp I dp 



p dx* P dy* p dz ' 



respectively, and introducing the CorioHs force according to (X. 5) and the frictional 

 forces according to (X. 13), then the basic hydrodynamic equations of motion will 

 take the complete form 



(X.16) 



Aw 



The third equation in the 2-direction can be considerably simpUfied, which shall be 

 done at once. Since the movements of the water in the ocean occur very largely in a 

 horizontal plane and w, dw/dt and the frictional term in this direction can always be 

 assumed to be small, and further, since the vertical component of the Coriolis force 

 can be neglected, the third equation in (X. 1 6) reduces to 



1 dp 



(X.17) 



0=^ 



dz 



which corresponds to the basic hydrostatic equation (see p. 337). 



For problems involving the whole or an extended part of the rotating Earth it is 

 convenient to use polar co-ordinates. The reference surface selected is the free sea sur- 

 face in a state of equilibrium (usually it is sufficiently accurate to take a spherical 

 surface with the mean radius R of the Earth) and as co-ordinates can be taken the pole 

 distance {^ = 90 — (j), the longitude A and the distance z from this surface (along the 

 radius of sphere R, positive outwards). The velocities relative to the Earth along the 

 three axes are then 



u = (R + z) 



d& 

 dt 



V = Rs'm >& 



dX 

 It 



and 



w 



dz 

 Jt 



(X. 18) 



If the external forces have a potential Q-f and if the frictional terms are omitted, the 

 equations of motion take the following form 



du ^ - ^ d /p \ 



dt 



2ojv cos §■ — — 



R + z dd'\p 



dv 

 dt 



dw 

 df 



+ 2<ou cos 'd' + 2cjL}W sin d' = 



1 



R sin 



d 



1 8X 



M' 



— 2cov sin '&■ = 



8 



dz 



+ Q 



(X.I9) 



t The forces X, Y, Z have a potential Q when they can be represented by 



ei? _dQ _8Q 



~dx ~ ~dy ~ ~dz 



X = 



