SECT. 3] DYNAMICS OF OCEAN CURBENTS 391 



wave number to angular velocity can be found by substituting —ifkja} for I in 

 (244). The zonal wave number is given by 



, ^ 



2(/2-a;2) 



iS< 



2(/2-a;2) 



a>2\ 

 + ■^1 • (274) 



Kelvin waves move westwards along a boundary on the poleward side of an 

 ocean and eastwards along the equatorial side. The amplitudes of the waves 

 decrease with distance from the boundaries. Meridional Kelvin waves do not 

 exist for frequencies in the Rossby wave range. 



It seems probable that a solution for time-dependent motion in an enclosed 

 ocean could be obtained by combining Rossby and Kelvin waves together with 

 periodic solutions of the type given in (257). Such a solution would be useful in 

 interpreting the balance of energy flow in the ocean. Unfortunately, an explicit 

 solution is not available and a satisfactory interpretation of the oscillation 

 modes cannot be made. In the absence of an explicit solution, we can speculate 

 that the energy flow in the interior of the ocean of rectangular shape is due to 

 Rossby waves. Near the meridional boundaries, the energy is transferred 

 meridionally by the wave-like motion given by (257). The energy is then 

 returned zonally in the vicinity of a zonal boundary by Kelvin waves. If this 

 speculation is justified, we could conclude that the time-dependent equations 

 yield a flow of energy by wave motion that is similar to the flow of mass given 

 by the steady-state equations. 



Periodic solutions of the time-dependent equations do not transport mass. 

 However, we can find solutions that are not periodic in time by substituting 

 — i<xi for o) and —ik for k in (244). These solutions are exponential in time and 

 zonal distance but trigonometrical in y. They differ from the periodic solutions 

 in that water is transported by the motion. The characteristic equation for 

 these solutions is 



c2 4cu2 



k^-^f'- 



7 W 



/^ + 



(275) 



Solving for k and I, we obtain = (A; -H Ao)^ — v"^, 



k=-Xo±S„ l=±v-i€o, (276) 



where Sy = \/{G^ + v^). As 8^>Ao, the phase velocity co/k is westward for the 

 solution corresponding to — Aq-hS^ and eastward to — Aq— §^. The aperiodic 

 solutions can be used to examine time -dependent flow from one part of an 

 ocean to another but they are introduced primarily because of their importance 

 in interpreting boundary -layer solutions of the steady-state equations. 



The time-dependent solutions that we have considered can be extended to 

 apply to a homogeneous ocean with a level bottom in which there is a steady 

 and uniform zonal current, Uq. In the linear approximation, the equations 

 governing the flow are similar to the six equations (226) to (231), with d/dt 

 replaced by dl8t+ Uq djdx. Hence, the only effect of the steady flow is to move 



