SECT. 3] DYNAMICS OF OCEAN CUBRENTS 385 



A non-trivial solution for r]^ and rji^ is possible only if the determinant of the 

 coefficients in (238) and (239) is zero. This condition yields, for oj 7^ 



cx)^ + gHajZ + g'gHiH2Z'^ = 0. (241) 



As g' <^g, the two roots of (241) are closely approximated by 



oji = -gHZ = -cb^-Z (242) 



W2 = -{g'HiH2lH)Z = -c,/Z, (243) 



where Cb and Cg can be interpreted as speeds of the barotropic and baroclinic 

 waves respectively.! xhe characteristic equations, relating angular velocities 

 and wave numbers, are of the form 



-P = c- 



a; \./--aj"/ J ^ - co 



(244) 



where c is equal to cb for barotropic waves and to Cg for baroclinic waves. 



We have assumed implicitly in deriving (244) that k and / are independent of 

 the co-ordinates. As the Coriolis parameter enters into the equation, this 

 assumption cannot be strictly true. Consequently, waves travelling from one 

 latitude to another must change their wave-length and amplitude. However, 

 we will not take the change of wave-length into account and will consider/ to be 

 constant in (244). At the same time, we will consider j8 to be different from zero. 

 This simplification, introduced by Rossby (1939), is reasonable if the angular 

 velocity is either large or small compared with /. However, for inertial oscilla- 

 tions, CO ~/, the approximation is not adequate and leads to anomalous results. 



For wave modes, oj is real but both k and I may be complex. Substitution of 

 X + ifjL for k and v + ie for I in (244) yields 



(a;2-/2)/c2 = {X-Xoy'-fJi'- + v^ + [€o-^-Xo'- + {e+eo)^] (245) 



for the real part, and 



/x(A-Ao) + Ke + eo) = (246) 



for the imaginary part, where 



The approximate forms of Ao and €q are valid only for tu <^/. At high frequencies, 

 tL»>/, both Ao and eo approach zero. 



1 Strictly speaking, cb and Cg represent wave speecis in the sense of phase velocities 

 only at high frequencies, w>>f. At low frequencies, a» </, the phase velocities of the baro- 

 tropic and baroclinic waves are much less than cb and Cg respectively. 



