352 FOFONOFF [sect. 3 



which imply functional dependence of Q and l,a on </(, Thus, we obtain the 

 Bernoulli equation 



Q = W) = 9'^ + ! 



8x) \8y, 



(95) 



and, from (93), the equation for conservation of absolute vorticity, 



^a= -^ = U'/')=/-VV- (96) 



Both Q and the absolute vorticity Za are constant along a stream-line of the 

 flow. An infinity of solutions is possible because Q or Za can be an arbitrary 

 function of ip. For any given function Ca(i/»), we can obtain i/r as a function of x 

 and y from (96) and the surface elevation tj from (95). 



In the simple class of inertial flows considered by Fofonoff (1954), the 

 absolute vorticity was assumed to be a linear function of ib of the form 



Za ^ f-V^tJj = Co + Cii/j. (97) 



Assuming / to be approximated by fo + ^y, we obtain from (97) the linear 

 differential equation 



V2^ + Cii/r =fQ-Co+^y (98) 



for the stream-function i/j. 



We can obtain solutions of (98) for a rectangular ocean by placing the origin 

 of the co-ordinate system midway along the southern zonal boundary so that 

 the limits of the ocean are given hy x= ±W and y = 0,L. The boundary condi- 

 tion in the absence of friction is that the normal component of velocity vanishes 

 at the boundary. This condition is satisfied by requiring ijj to be zero along the 

 boundary. 



Wherever the relative vorticity is zero, the velocity will be zonal and equal to 



I = »° = I (»») 



so that we may rewrite (98) as 



{uoI^WSj + iJj = uoy, (100) 



where uq may be positive or negative and Co is taken equal to /o. 



Introducing the non-dimensional variables x' = TrxlL, y' = 7TylL, ip' = ifjl\uo\L 

 and the non-dimensional parameter kQ = {l/7T){^L^I\uo\y'^^^l, we obtain the 

 non-dimensional form of (100) 



(l/A;o2)V2f ±f = y', (101) 



where the plus sign is associated with positive (eastward) uq and the minus 

 sign with negative uq. The boundary conditions become ip' = at y' — 0,7t and 



x' — ±a— ± ttWIL. 



