90 Linear Theories — Viscous 



n^z^njh. This constant c^ may be absorbed into p^ and q^. When subscripts 

 are dropped, the stream function has the form 



^ = ^(^Vsin^be^- + ge«--l), (13) 



where 



1— e^*" 

 P^-J-r Wr and q=l-p. (14) 



The curves (?/r = const.) are the streamUnes of the ocean currents. 



In order to help visuahze the meaning of this solution, it is advisable to 

 compute some numerical examples that will show what role the various 

 parameters play. Three cases are discussed. All involve the same effects of 

 wind stress, bottom friction, and horizontal pressure gradients caused by 

 variations of surface height. The role of the Coriohs force is different in each 

 case. First it is assumed that the CorioUs parameter vanishes everywhere — 

 the case of the nonrotating ocean. Secondly, it is assumed that the Coriohs 

 parameter is constant everywhere — the case of the uniformly rotating 

 ocean. In the third case it is assumed that the Coriohs parameter is a linear 

 function of latitude. Of the three cases, the last one most nearly approxi- 

 mates the state of affairs in the real ocean. 



For convenience of the numerical computations the dimensions of the 

 ocean are taken as follows : 



r =109 cm. = 10,000 km., 

 b =2nx 108 cm. = 6,249 km. , 

 D = 2xlO'*cm. = 200m. 



The maximum wind stress F is taken to be 1 dyne/cm. 2, 

 The coefficient of friction R is the only quantity for which a value must 

 be devised. If a value of i? = 0-02 is assumed, the velocities in the resulting 

 systems approach those observed in nature. 



The case 0/ the nonrotating ocean. — In the nonrotating ocean the con- 

 stants p and q are fairly simple. Within 1 per cent, or as closely as graphs 

 may be drawn, p and q are given by 



p^^-nrlb^ q=l (15) 



The equation for the stream function is therefore 



^=^l^\\m^[e^''-'^"l^ + e-^^l^-l]. (16) 



Rb \7tJ 



b 



The east-west and north-south symmetry of the streamlines is im- 

 mediately evident from this equation. The actual streamUnes computed 

 from it are exhibited in fig. 55. 



