r^ - DP + cH3^H2 = , (15) 



where D is the total depth at rest (Hj^+H2), and e is the relative 

 density difference, (P2~Pi)/P2* 



The two roots of (15), which correspond to the equivalent depths 

 for external and internal modes, are 



Fg = D[l-e(HiH2/D2) + 0(e^)] , and 



r^ = €(HiH2/D)[l+e(Hj^H2/D2) + 0(e^)]. 

 Since H2^H2/D^ < 1 and e « 1 for the general ocean basin, then 



r„ = D , and 



q = e(HiH2/D) 



(16) 



(17) 



are the equivalent depths for external and internal modes as obtained 

 by Veronis and Stommel (1956). In general these may depend on x,y. 

 The ratio of a and (3 can be determined from (10a,b) as 



p/a = r - H3^/H2 = (Pi/P2)Hi/r - H2. (18) 



Substituting (17) in (18), the ratio (a//3) for each mode is 

 (/3/a)g = 1 - e(H3^/D) - e2(HfH2/D3) , and 



(19) 



(/3/a)i = -H1/H2 + £(Hi/D) + e2(H2H2/D^). 



Note that Cg and /3g are of like sign, but a^ and ^^ are of opposite 

 sign, ttg, /3g, a^ are chosen to be positive and ^^ to be negative. 



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