390 FOFONOFF [sect. 3 



short wave corresponding to Ao + So and 17- the long wave corresponding to 

 Ao — So, we obtain the components 



a»Ao— eo/ 



V = —■ 



-^- — I sm 80X + — ^ cos Soa: 



Qi(wt+XoX)+eaX_ (270) 



U-n = u+in+-\-U-r)--\--— r^ g-n+^rj-^ COS 2 SoXe^^oV (265) 



— ^ _ grj+rj-JSo ^.^ ^ Soa;e2.o2/. (266) 



By choosing 



V = ^f2_^2)Uol{ojK-eof)g, V-' = {P-cv^)Uol{cok--eof)g, (267) 

 we obtain the wave solution 



7] = rj+ + rj- = -{Uoc^lgco^)[2aj So cos Sox + ijSsin 8oxy(-<+^o^)+eo2/ (268) 

 u = 2iUo sin Sore e^(<^t+x,z)+.oy (269) 



of 

 Taking the imaginary part of the solution, we obtain the flux components 



v^ = -{2Uo^c^^lgoj^)sm^SQX e^ov (271) 



mj = (2C7o2c2/ So/g'ojS) sin 2Soa; e^ov. (272) 



From (269), we can see that the solution will satisfy the boundary condition 

 w = at two boundaries, x = and x = a, if Soa is an integral multiple of n. 

 Hence, solving for to, we obtain the "eigenvalues" 



where n is an integer. For the first mode, n=l, the flow of energy is towards 

 the equator in the eastern half of the ocean and towards the poles in the western 

 half. There is a westward energy flow throughout the ocean. As no eastward 

 flow of energy is present, the solution is not closed. Energy enters a given 

 region in the eastern part of the ocean and leaves in the western part. For the 

 higher modes, n>l, there are alternate regions of positive and negative meri- 

 dional flows of energy. These solutions cannot be applied to a bounded ocean 

 because the particle motion is not zero along any zonal plane and energy cannot 

 be returned to the eastern part of the ocean. Rossby waves travelling obliquely 

 to latitude circles can be made to satisfy the boundary conditions along zonal 

 boundaries but not along meridional boundaries. We may conclude, therefore, 

 that Rossby waves do not constitute a complete mechanism for considering 

 periodic variations of flow in a bounded, or partially bounded, ocean. 



In the presence of zonal boundaries, the equations admit an additional 

 periodic motion in the form of zonal Kelvin waves. The meridional velocity 

 component for these waves is zero and the characteristic equation relating 



