All -101 



Q5 



av _ifz 

 at 



-h 



2 3x 



PI 



cz 



-h 



2 ay 





+ 2Q sin(^) u o''2 



R 



1 



- g Q 9lL k + Mv e^^^ 



-h 



Approxiae.to the exponentials at their lii-iits tay 

 e ^ ~ 1, e"^* 2 0. Then (3) and ('-:■) become 



9u . 1 - au . 1 " 8u 



at 



4 4 u 1^ + A V gii . 2Q sina) v = 



ax 2 ay 



R' 



_ '^ D |lik + AAu + T^k (5) 



ox ^'■ 



av , 1 - av . 1 ~ av 

 at 2 ax 2 



^ + 2Q sin^) u = 



ay R 



- g D Ink + AAv + T, 1- (6) 

 ay y 



Linearize the Goriolis naraneter by 2Q 3in(^) ~ p 

 where p = :^ , Taking the derivative of (6) with respect to 

 X and the derivative of (5) with respect to y and subtracting, 

 we have 



4 (av _ ^) + 1 

 at ax ay 2 



a.u av ^ ^ a^ + av av ^ -a^v _ au au _ -a u 

 ax ax " ~ ' " - ~-- - - - • 



''I 



.2 



9x2 ax ay 



axay ax ay 



ax ay 



av au - a u 

 9y 9y " ay2 



+ Pv( 



y'ax ay 



au . av) + pv = AA (fi - 1^) 

 ^ ax ay 



+ k(^ - |:^2) 



.ax, 

 ■ax 



ay 



(7) 



Choose T = , T^ = - (V/' + r' sin ut) cos ny. 



X 



We shall non-dimensionalize the velocities so that they 

 are of order unity in the interior of the ocean. It is con- 

 venient* to choose 



*The choice of the non-dimensional quantities is motivated in 

 section h. 



