Here t is the stress vector where subscripts s, i, and b stand for 

 surface, interface, and bottom, respectively. The atmospheric 

 pressure at the sea surface is P^. 



The normal mode form of the equations can be derived from these 

 primitive equations by a method similar to that employed by Veronis 

 and Stommel (1956). To transform (1) and (2) into normal mode form, 

 we multiply (1) and (2) by a and (3) and (4) by <3 and add the 

 corresponding momentum and mass conservation equations, respectively, 

 to obtain 



-» -+ -♦ 

 9M/9t + fkxM + g{V[aH3^(p3^h3_+P3^h2)+^H2(Pih3^+P2h2)] 



-[(Pj^h3_+Pih2)V(aHi) + (p3^hi+P2h2)V(/3H2)]} = G, (6) 



3(&/9t + aV-Mj^ + /3V-M2 = 0, (7) 



where 



M = oMj^ + /3M2, 



<t> = apj^h-L + i3p2h2f (8) 



G = aF^ + /JF2. 



Note that a and /3 are non-dimensional and, for the case of variable 

 depth, may depend upon x and y (this is the generalization of the 

 Veronis and Stommel analysis). 



The constraint imposed on a and /3 to make the elevation 

 anomalies in (6) and (7) proportional, is 



aHj^(P2^hjL+Pih2) + ^H2(Pih3^+P2h2) = T<t>, (9) 



