All. -101 



25 



where p^^ - o\_r\ - T(x,y,t) ], the density at the free surface. 



If these terms be substituted into (1) and (2), v;e have 



^ + u' ^iii + v' -S-^i^ - 2Qv- sin (Z) = 



9t 



9x 



ay 

 g 1^ 9p 



R 



y r) 



p 



l/' 



ex 



dz -^ ^- iin p^ +I(V • A.V)u' 



P GX 



5 



(12) 



.avl + u' ^.^ + V l3Li + 2L~.u'- sin (^) = 



at ex ay R 



6 '' 9o . 



_ I ^i. dz 

 P Jz ay 



S 9i]. p +i( V A.V)v'. 

 p 8y ^0 ^ 1 



(13) 



As stated previously, the problem v/ill be simplified by 

 integrating the equations over the vertical coordinate, Zo 



Let us first consider the problem defined by the equa- 

 tions (h) ^ (12), (13). V/e assume that there is a depth 



z = - h(x,y,t) belov/ which the velocities may be considered 

 negligible (in some suitably defined sense), and we integrate 

 from z = - h up to the free surface. The depth z = - h(x,y,t) 

 may, of course, vary from point to point in the ocean. Since 

 the velocities are negligibly small below z = - h, the horizontal 

 pressure gradients must also be negligibly small and we may 

 therefore write 



1 P2 



p ax 



= 0, 



z=-h 



1 IP 

 P ay 



0. 



(1^) 



z=-h 



V/e must nov/ specialize the general form of the density 

 distribution because an integration involving p will actually 



* This assumption is the fundamental difference between the 

 two problems considered. 



