Aii-101 87 



Then for the equations of zoro-ordGr in 6 , we have* 



r fas? ar? - • • • - ^o ;^|1 . Vo . .a [-^_o . Lo ] 



- (1 + a sin T)sin nsy' (10) 



™^ + —2 = (11) 



ax' ay' ^ ^ 



The first order equations in 6 are: 



a .9^0 9"oN r^^o ^V. 9% 9Vq a^v a^v 

 d-z ^ax' ay'^ ^ Y L g^, g^, + g^, g^, + Uo ^^^2 "^ ^1 9^,2 



aH^ aVn au-i 



4... - y ^q 4 Vi =e A' (-4 - ~4) (12) 

 aT ^ ax' ay' 



aiL avn an^ 



ax' ay- ax ' ^ -^^ 



Munk, Groves, and Carrier [?] have shown that the effect 

 of the non-linear terms in [lO ] is quantitative and that these 

 non-linear terms can be neglected as compared to the Coriolis 

 term, Vq. The relationship of the non-linear terms to the 

 Coriol is term in equation (12) is essentially the same as that 

 in equation (10). This fact can be shov/n by considerations 

 based on orders of magnitude. We choose a typical non-linear 



au„ aVo au., av. 



term in each equation, y g™ "S'^ ^" ^^°^ ^^^ ^ Tx^ J^ ^^ 

 (12), and compare it to the Coriolis terms in that equation, 

 Vq in (10) and V^ in (12). 



In the solution it is shov/n that Uq, Vq, U-]_, Y-^ and all 

 their derivatives are of order unity in the interior of the 



*Equation (10) with a = is the same as that of Munk, Groves, 

 Carrier [7 1 . 



