Assuming the flow to be nondivergent, the stream function CF) is introduced 

 by definition as: 



u = - H, V = II 

 3y 9x 



the vorticity equation (15) is then rewritten using the stream function 



^29^ ^ ^ 3(V^y ) 9f 9(72y ) av 3f ^^^ 



3t 3y 9x 3x 3y ~ 3^ 37 



The equation is transformed into centered spaced finite differences and is 

 solved for ay/at at each grid point by numerical relaxation on the 20-kin 

 grid^at 6-hour time intervals. In addition, the stream functions alon^ the 

 boundaries were adjusted so that the net transport of vorticitv across 

 the boundaries was balanced. A further simplification was made in this 

 preliminary test by assuming a uniformly rotating ocean, the Coriolis para- 

 meter is constant, and 3f/3y = in equation (16). 



The prediction was accomplished as follows. The observed northern 

 edge of the Gulf Stream was drawn on a 20-kin grid. This line was given 

 a stream function value of zero. A series of stream function lines were 

 drawn south of this line to correspond to a Gulf Stream with a mean hori- 

 zontal velocity of 50 cm/sec and a width of 100 km. To the north of the 

 Stream, the grid was filled with zeroes; to the south, the grid was filled 

 with the maximum value of the stream function which occurred along the 

 southern edge of the Gulf Stream. The resulting flow field is unrealistic 

 since it is discontinuous, i.e., no flow occurs north or south of the 

 stream with 50 cm/sec in the stream. This discontinuity creates both 

 physical and mathematical problems. 



The prediction of the new stream function values at each grid 

 point was accomplished using the f irst-forward-then-centered time differ- 

 encing method in the following forms: 



forward time differences: 

 V = V + IM.\ At 



T+1 T btjx 



centered time differences: 



Vi- Vi^Mfr),'^^ 



where 1'_.i is predicted value, 4"^ and (34'/3t) are the present or 

 calculated values, and "F-j-i is the value from the previous time step. The 

 forward time difference is used only for the first time interval, there- 

 after the centered time difference is used. In addition, the computational 



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