SECT. 3] DYNAMICS OF OCEAN CUKBENTS 353 



The solution for negative uq, omitting primes, is obtained by combining the 

 particular solution 4^p= —y with the homogeneous solution ifjh = A cosh koy + 

 B sinh koy to satisfy the boundary conditions at y = Q,TT. We obtain 



0P + 0;i = —y + TT{sinh.koylamh.ko7T) (102) 



The function (102) is symmetrical with respect to x. Hence, we can satisfy the 

 boundary conditions at x= ±a by adding to (102) homogeneous solutions of 

 the form An cosh knX sin ny, where kn^ = ko^ + n^. These solutions are sym- 

 metrical with respect to x and satisfy the boundary conditions at y = 0,7T. The 

 coefficients An are found by expanding (102) as a Fourier series in y and 

 equating coefficients at the boundary x = a. Because of symmetry, the condi- 

 tions at a:= — a are automatically satisfied. The complete solution is therefore 



7T sinh koy ^ {-l)n+-^ko^ cosh A; n^: . 



lA = —V-\ T-rr-, — - + 2 > ,, „ 7—, — sm ny. (103) 



^ ^ sinhifcoTT ri=i'^i^o^ + ^) coshA:„a -^ ^ ' 



By following the same procedure for positive uo, we obtain 



TT sin koy -^ {-ly^+^ko^ cos knX . 

 ^ smkoTT „=i w(«^o^-w^) cos A;„a 



where kn^ = ko^ — n^. This solution can also be obtained from (103) by replacing 

 ko,kn by iko,ikn and changing sign. The solution is valid provided A;o is not an 

 integer and kna not an odd integral multiple of 7r/2. 



As ko is large compared with unity for slow interior velocities, we can derive 

 much simpler approximate expressions for «/» from the exact solutions. For 

 L = 2500 km, jS = 2 X 10~i3 cm-i sec-i and wo= —5 cm sec~i, ko has the value 

 15.9 approximately. Thus, for the first few terms of the series in (103), kn will 

 not be much greater than ko, and we can factor the function of x out of the 

 series, neglecting all terms in which n is not small compared with ko. The 

 series then consists of the first few terms of the Fourier expansion of (102). 

 Hence, replacing the series by (102) and simplifying the resultant function by 

 neglecting terms of order e-^o" and e~^<'^, we obtain the approximation 



~ _[2/_7re-fco(''-l/)][l-e-fco(a-a;)_e-*o(a+x)]^ (105) 



which is equivalent to the boundary-layer solution given by FofonofiF (1954). 

 We obtain the characteristic width of the boundary region, I//7rA;o = (|wo|/iS)'/^ 

 ~ 50 km, and maximum velocity in the eastward jet, L{P\uo\y'^^ ^ 250 cm sec-i. 



Solutions of (98) for Co^/o contain a jet at both zonal boundaries, except in 

 the special case Co =fo + ^L in which the eastward jet is present at the southern 

 boundary only. For /o < Co </o + jSL, both jets are eastward and t/j is zero at an 

 interior latitude. For Co</o or Co>fo + ^L, one of the jets is westward. In all 

 cases, the interior flow is westward. 



For positive values of wo, the flow appears to consist of a complex series of 

 eddies covering the entire ocean. No eastward drift is evident in the solution. 

 As this solution can be interpreted most readily in terms of time-dependent 



