378 FOFONOFF [sect. 3 



assume, further, that W and & vary approximately Hnearly in the upper layer 

 and, as 



Tf = >r41-e-2(«-A)/L], (202) 



^ = ^^e-2(c-/.)/L (203) 



in the lower layer, where d-h is the temperature along the surface of zero vertical 

 flow iC^h). These functions are consistent with the averages chosen in each 

 layer. We replace W and d&ldr) in (188) by their approximate averages and 

 integrate the linearized equation separately over each layer to obtain 



qh*-qo* = We{&o-&h)+yTiWe (204) 



for the upper layer and 



q^* = -Woo&h + lyTiWoo (205) 



for the lower layer, where 



are the fluxes of heat at ^ = and l, = li respectively. Similarly, integration of 

 (189) yields 



bW 



for the upper layer and 

 dVJ_ 



___ + __ = (^„_^,)_^__ (208) 



^^ = ^'i^-^^ >^ ^^^^^ 



for the lower layer. Equation (208) reduces to the relation, Wefh — —2WoolL, 

 given by Robinson and Stommel, if we neglect the heat flux term. However, 

 this relation implies d'h=^Q and is not consistent with our other approximations. 

 The six equations, (204) to (209), are sufficient to determine the six un- 

 knowns d'o, &h, qh*, h, L and Ti if go*, We, Woo and ky are assumed to be known. 

 The equations cannot be solved explicitly for the unknowns. However, we can 

 reduce them to two algebraic equations for h and L that can be solved numeri- 

 cally. The two equations, in non-dimensional form, are 



A2(A-Ao) = Xo{h' + hi){h'^-h2^) > (210) 



A = {l-h'^)lh', (211) 



where 



h' = hlho ~0.80 



A = LolL -0.59 

 ho = {2kvWelqo*y^'^ -0.58 



