386 FOFONOFF [sect. 3 



We can satisfy (246) by choosing ix = and e= — eo. The real part can then 

 be reduced to 



ct2 = ^^^^' + ^ ( A - Ao)2 + v2. (249) 



By introducing S^ for ■\/{a^ — v^), we may write the solutions for the wave 

 numbers in the form 



k = X = Ao±S., I = ±v-i€o (250) 



for j/2 < ct2 



The expression for I in (250) contains the imaginary term — ieo. This term 

 represents an exponential factor in the solutions that is dependent on the 

 latitude and angular velocity but not on the direction of travel of the waves. 

 The factor is a consequence of our assumption that w is real, i.e. that wave 

 amplitudes at a given location do not change with time. Because of the varia- 

 tion of the Coriolis parameter with latitude, the wave amplitude changes as 

 the wave travels from one latitude to another. Hence, for a» to be real, we must 

 assume a distribution of amplitudes with latitude such that each wave would 

 have the same amplitude if propagated to a standard latitude. Otherwise, oj 

 must have an imaginary part representing an increase or decrease of amplitude 

 at a given location as successive waves arrive from different initial latitudes. 

 The factor is not required for zonal waves (i^ = 0) as (246) is satisfied with 

 €^ — eo. Waves with angular velocities exceeding / decrease in amplitude as 

 they move polewards, while those with angular velocities less than / increase in 

 amplitude polewards. 



In general, two waves are possible at each frequency. For the lower frequency 

 range < a> </, we note that Ao^ > a^^ S^,^. Hence, from (250), A; = Ao + S^ > 0. 

 Thus, both pairs of barotropic and baroclinic waves travel westwards. We shall 

 refer to these waves as Rossby waves. ^ We note from (249) that Rossby waves 

 occur if 



^^ < J[/^-(/^-i32c2)V2] ^ ^2c2/4y2 (251) 



provided/ 2 |>^c. Other wave solutions occur for 



CO' > H/2 + (/'*- i82c2)V2] ^/2. (252) 



Except near oj=f, these high-frequency waves are essentially equivalent to 

 the waves obtained assuming a constant Coriolis parameter. These have been 

 described thoroughly by Proudman (1953). An interesting study of the propaga- 

 tion of the high-frequency waves in the vicinity of partial boundaries has been 

 carried out by Crease (1956). 



We can examine minimum wave periods and the corresponding wave-lengths 



1 Rossby (1939) described the zonal barotropic wave in a study of time-dependent 

 motion in the atmosphere. Stommel (1957) pointed out that similar waves had been 

 studied earlier in connection with tidal theory. 



