SECT. 3] DYNAMICS OF OCEAN CURRENTS 381 



solution of the time-dependent equations and much of the theory has been 

 developed for an unbounded ocean only.^ 



The theory of time-dependent motion is presented here in a form similar to 

 that developed by Veronis and Stommel (1956). However, their procedure of 

 solving the characteristic equation for the frequencies is not followed in favour 

 of the simpler procedure of solving for the wave numbers. Also, the effect of a 

 constant steady velocity on the time-dependent motion is considered in order 

 to interpret more fully the nature of the boundary currents already considered. 



A. Simplified Time- Dependent Equations 



We have seen from the analysis of the magnitude of the terms in the time- 

 dependent equations that the interactions Avith steady flow depend on the 

 magnitude of the Rossby number. We shall restrict our attention to flows with 

 Rossby numbers sufficiently small so that we can neglect the interaction terms. 

 Similarly, we assumed that the gradients of the correlations, represented by 

 r'ij in (17), are small. For simplicity, we can consider these simplifying assump- 

 tions as being equivalent to the assumption that the steady flow is entirely 

 absent. However, such a strong assumption is not necessary and can be modified 

 later in the light of the solutions obtained. Using the simplifying assumptions 

 given above, we can write the time-dependent equations (13) and (14) in the 

 form 



^1-^=-^ (-) 



^ + ^ + %^ = 0, (215) 



ex cy cz 



where pa is the steady-state pressure. The variation of density with time, 

 present in (14), does not have to be considered in the equations for the two- 

 layer ocean provided we assume thjat the vertical motion of the interface is 

 small compared with the thickness of the layers so that boundary conditions 

 can be applied at the mean position of the interface. 



We can proceed to analyse the system of time-dependent equations using 

 the concept of barotropic and baroclinic velocity components introduced for 

 the steady state. We denote the barotropic velocity components of the time- 

 dependent motion by w^, Vq and the baroclinic components by Ug, Vg. The 



1 Some idea of the difficulties encountered in solving the time-dependent equations for 

 a bounded ocean can be obtained by examining the solution for constant Coriolis para- 

 meter given by Taylor (1922). He was able to separate the motion into symmetrical and 

 antisynMnetrical modes. These modes are coupled together if ^ is different from zero. 



