This yields, finally, the inflow-outflow balance for no vorticity accumulation 



/ 



p\/c/?( ~l pvdx =0. (23) 



'outflow "inf/ouv 



The model is applied by specifying an inflow vAiich resembled the Gulf 

 Stream on a section extending southeastward from Cape Hatteras. Equation (23) 

 states that the outflow from the channel should not be selected arbitrarily. 

 In the tests of the model that have been conducted, the outflow stream-func- 

 tion profile has been constructed to be linear. This, unfortiinately, has not 

 generally satisfied the inflow- outflow balance. In tests viiich differ essen- 

 tially in the sign of the inflow-outflow imbalance, the phase of the instabil- 

 ity near the inflow is reversed. Although no reduction of instability was in- 

 dicated, the striking difference in phase, apparently due to differing inflow- 

 outflow imbalance, points to some sort of interaction between the imbalance 

 and instability. 



FREE-SURFACE MODEL 



The second modification which may help control instability in the model is 

 to relax the rigidity of the top of the fluid. The fluid then possesses not 

 only a rigid, sloping bottom, but also a free top surface. To outline the 

 derivation of the model equation for this case, we return to the kinematic 

 boundary conditions, equation (10), vhich will now be: 



^5=^, ^H^%'Vh, (2A) 



where s(x, y, t) is the free-surface top of the fluid. When the geostrophic 

 assumption is introduced to relate the free surface to the stream function, 

 the model equation similar to equation (13) becomes: 



v'§ ^rv(A' v'f^f-i Ugh) -^M^o. (25) 



The difference between equations (13) and (25) is the term — ^^ Qt 



(the Helmholtz term). A perturbation analysis of this model demonstrates that 



the Helmholtz term serves to reduce the speed of long waves. 



Testing of the free-surface model awaits the completion of the investiga- 

 tion &iow under way) into the instability which has heen encoxintered. Inclu- 

 sion of the Helmholtz term in the model would seem to be beneficial since it 

 can be expected to counteract, to some extent, the unstable amplification 

 which has been encountered near the inflow. 



ref^:rence 



Warren, Bruce A. I963 Topographic influences on the path of the Gulf Stream, 

 Tellus, V. 15, no. 2, pp. 167-183. 



144 



