SECT. 3] DYNAMICS OF OCEAN CURRENTS 375 



where ^^zF{x). They assumed that the temi)erature did not vary with x at 

 the surface and changed Hnearly with y as To+ IQ-^Tiy. By substituting (185) 

 and (186) into (182) and (184), they found that solutions having the desired 

 characteristics had to be of the form 



T = m, ^ = x-y^oj{i) (187) 



and I = zx-^K They considered solutions for | > only, i.e. z < 0, a; < 0. However, 

 this restriction excludes poleward flows and will therefore not be imposed. ^ 

 We shall consider both positive and negative values of |. 



The basic equations were transformed to remove some of the dependence on 

 y and to introduce units to make the terms of the equations of unit magnitude. 

 The final equations are 



d^ 8& SWd& 



where W = {xle)y^w, ^ = {ejx)y^z, r? = 10-8?/, y=10-»fl^, e^ga^/Sp. 



Boundary conditions, imposed at the bottom of the Ekman frictional layer 

 (^ = 0), are 



^0 = To+10-^Tiy = To+Tirj (190) 



Wo= We= {xl€)y^We, (191) 



where Wg is the vertical velocity given by the divergence of the Ekman trans- 

 port components. The conditions for |^| ^ go are d' ^ and W -^ Woo. We 

 shall also consider the implications of the boundary condition 



kv 



= g* - \xl€\y^Q* (192) 



which expresses the "apparent" heat flux across the ocean surface. 



Robinson and Stommel linearized (188) by replacing W and dd-ldr] by their 

 averages over a depth interval L, where L is defined as the interval in which 

 both functions differ by more than l/e2(^l/7) from their asymptotic values. 

 Then, by choosing simple exponential expressions for W and & that satisfy the 

 boundary conditions and yield the correct averages, they integrated (188) and 

 (189) over ^ to obtain algebraic equations relating the averages. We shall use 

 the same approximate procedure. 



The meridional temperature gradient, d&ldiq, varies between Ti at the surface 

 and zero for |^| -^ oo. Hence, its average will be of the order of Tij'l. Similarly, 

 the velocity, W, varies between We and Wx and has the approximate average 



1 Robinson and Stommel introduced this restriction so that an eastern boundary could 

 be placed at a; = 0. However, the solutions are singular at a: = so that little is gained by 

 this procediire. It is preferable to assume that the solutions apply strictly to interior 

 regions and can be connected to the boundaries by higher-order boundary solutions. 



