324 FOFONOFF [sect. 3 



The steady-state theory developed in terms of total transport provides no 

 insight into the interplay between the circulation and the processes of mixing 

 and convection which affect the density field in the ocean. Lineikin (1955) 

 carried out a simplified analysis of the interaction using a linearized perturba- 

 tion model in which an initial uniform vertical gradient of density was assumed. 

 He was able to show that currents induced by the wind weakened with depth 

 because of the stability of the density stratification. Attempts to construct 

 more realistic convective models are hampered by the non-linearity of both the 

 equation of state of sea-water and the conservation equations describing the 

 processes involved. 



Theoretical studies of the dynamical response of an ocean were initiated by 

 Rossby (1938). He showed that the ocean would respond barotropically to 

 variations of wind stress under a wide range of conditions. Rossby 's and most 

 subsequent studies have been confined to an ocean without boundaries and in 

 the absence of strong steady currents. The basic problem of the interaction 

 of the time-dependent modes of response and swift steady currents, such as the 

 Gulf Stream and Kuroshio, has yet to be solved. The lack of understanding of 

 the interaction severely limits the applicability of time-dependent theory to 

 the real ocean. 



A non-mathematical survey of modern theories of steady-state currents, 

 convective models and time-dependent variations of ocean circulation has been 

 given recently by Stommel (1957). He pointed out the close relationship 

 between modern theories and some earlier investigations of tidal dynamics. A 

 discussion of many aspects of recent theories of ocean currents is given also by 

 Stommel (1958) in his comprehensive treatment of the Gulf Stream system. 



In the present discussion of the dynamics of ocean currents, the material is 

 divided into the two major topics of steady-state and time-dependent theory. 

 Emphasis is placed on recent developments of the theory and an attempt is 

 made to present the concepts in generalized, but elementary, form. Some of the 

 more complex developments are not given in detail, but the more elementary 

 concepts are developed in full detail in order to provide a background for 

 assimilating recent literature. Many aspects of the dynamical theory of ocean 

 currents are not included in the present treatment, but it is hoped that sufficient 

 topics have been discussed to provide a basis for appreciating some of the 

 problems as well as achievements in the field of the dynamics of ocean currents. 



1. Conservation Equations for Momentum and Mass 



In order to retain maximum simplicity and uniformity in the mathematical 

 treatment of the dynamics of ocean currents, we shall apply the equations 

 expressing conservation of momentum and mass exclusively in terms of a 

 rectangular co-ordinate system. Such a system allows considerably more scope 

 in applying elementary analytical procedures and obtaining simple results than 

 does the more natural spherical co-ordinate system or one derived from the 



