Veronis and Stommel (1956) took a^ = a^ = 1 for a constant depth 

 basin. However, the presence of coupling terms B and C in (11) and 

 (12) due to varying depth in the present study precludes an arbitrary 

 choice of a. Notice that both a and /3 are functions of x and y. 



The procedure used to determine the individual values of a and ^ 

 for both modes is based on energy considerations. In essence the 

 energy equations derived from the primitive equations and the normal 

 mode equations must be consistent. 



The two energy equations formed from the primitive equations and 

 the normal mode equations, respectively, are 



where 



S(Ej^+Ep)/at + V-J = S, (20) 



^k = 29(^1^1^ + 2p-Lh-|^h2 + P2^i^^ i 



Ep = ^[(Mj^VpiHi) + (MI/P2H2), 



-* -* ■* 



3 = g[(h3_+h2)M3^ + l/p2(Pihj;^-p2^2^'^2] ' 



S = Mi-F-l/piH^ + M2-F2/P2H2. 



(21) 



and 



3(Ej^+Ep)/at + V-J - T = S . (22) 



Equation (22) contains not only the additional term T, but the 

 expressions for the kinetic energy per unit area, Ej^, the potential 

 energy per unit area, Ep, the energy flux per unit width, J, and the 

 net energy supply per unit width per unit time, S, are also different 

 from those given in (21). The added term, T, which defines the 



11 



