SECT. 3] DYI^^AMICS OF OCEAN CURRENTS 351 



solutions for the homogeneous ocean in detail because they serve as an introduc- 

 tion to the more interesting theory of inertial baroclinic flows in a two-layer 

 ocean. Moreover, both the exact and boundary-layer solutions can be obtained 

 for the homogeneous ocean giving us an opportunity to evaluate the approxi- 

 mate boundary-layer procedure of deriving solutions. 



If we assume that the ocean bottom is level and the density constant, we 

 can write equations (47) to (50) as 



duvh 8v^h . ^ , 8r) 



-^ + ^+M=-^^^ (86) 



Pb = pgiv-^s) = pgh (87) 



8uh 8vh 



where frictional terms and surface stresses are assumed to be zero. 

 The momentum equations (85) and (86) can be transformed to 



-Ca« = -I (89) 



Uu = -^. (90) 



where Ca is the absolute vorticity, f+ ^, t, the relative vorticity equal to 8vl8x — 

 8ul8y and Q is introduced for gr] -\- \{u^ + v"^). 



If we assume that changes of depth are small compared with the depth, we 

 can reduce (88) to 



8u 8v ^ ,^,, 



which can be satisfied by introducing a stream-function «/r such that 



'' = | "=-!• <«^' 



Hence, (89) and (90) become 



8^ _ 8Q 8^ _ 8Q 



^"Yx- ~Tx ^''Yy- ~~8^ ^^^^ 



and we can derive the relations 



8_Q8iP_8_Q8l^^ 8Jadl_8Udl^^^ 



8x 8y 8y 8x ' 8x 8y 8y 8x 



