388 FOFONOFF [sect. 3 



ko sin 6 for v in (249), where Icq is the wave number along the direction of travel 

 and 6 is the angle between the direction of travel and a latitude circle, we 

 obtain 



a2 = ^^o2 - 2Ao^o cos 6 + Ao2. (255) 



Substituting for Ao and a from (247) and (249), we obtain the approximate 

 relation 



ko ko^ + iplc^) ^ ''''' 



for the phase velocity. Thus, we can see that Rossby waves travelling at an 

 angle to a latitude circle will move more slowly than zonal waves of the same 

 wave-length. However, the westward progression of a wave crest along a given 

 latitude circle is equal to that of a zonal wave of the same wavelength, and is 

 independent of the direction of travel of the wave. Thus, for angles of travel 

 approaching ttJ'Z, the waves become almost stationary with the particle motion 

 reducing essentially to a system of zonal currents. Rossby waves cannot be 

 propagated meridionally. We can obtain the same result more directly by 

 substituting A = into (249). As Ao'^ > cr^, there are no real solutions for v and, 

 consequently, no meridional Rossby waves for co> 0. 



Rossby waves for + v and — v can be combined to yield a zero meridional 

 velocity along a single zonal boundary or a pair of zonal boundaries. Thus, 

 Rossby waves can be reflected from zonal boundaries. The meridional motion 

 of the reflected wave is opposite to that of the incident wave. As both waves at 

 a given frequency travel westwards, they cannot be reflected from meridional 

 boundaries. Nevertheless, the two waves can be combined to yield zero zonal 

 velocities at meridional boundaries as was shown by Arons and Stommel 

 (1956). The combination of two waves cannot be interpreted as reflection 

 because the zonal flux of energy is not zero. In order to interpret the solutions 

 given by Arons and Stommel, we will first consider solutions of (249) for 

 0-2 < 0. These are of the form 



k = XQ±iSv, 1= ±v — i€o, (257) 



where 8^= \^\a^ — v^. These solutions represent a wave-like motion that 

 progresses westwards but changes its amplitude along the direction of travel. 

 Consequently, the energy flux associated with the motion is divergent zonally. 

 As the total energy must be conserved, there is a flow of energy along the crests 

 of the waves. In order to examine this characteristic of the motion in greater 

 detail, we formulate the energy equation for a time-dependent motion in a 

 homogeneous ocean {g' = 0, H2 = H). Multiplying (226) and (227) by the velocity 

 components and adding, we obtain 



where Eic = ^{u^ + v^)H is the total kinetic energy of a column of water of unit 



