334 FOFONOFF [sect. 3 



indicate that the deep flow is also highly variable. Hence, we may conclude 

 that the steady circulation in the ocean has meaning only as an average with 

 respect to time or space. In addition, the ocean basins and shorelines are 

 complex in shape and their influence on the circulation is difficult to determine 

 in detail. Also, the exchange of momentum, energy and mass across the surface 

 of the ocean is conditioned by intricate processes both in the atmosphere and 

 ocean. In order to outline even the major dynamical processes that determine 

 the behaviour of an ocean system, we have to resort to drastic simplifications 

 of the boundary conditions and of the exchange processes that drive the 

 circulation. 



Various models of steady-state circulation have been constructed to elucidate 

 separately the effects of friction, non-linear accelerations and thermohaline 

 processes. Attempts to combine two or more of these mechanisms in a single 

 mathematical model have not been very satisfactory because of the analytical 

 difficulties that are encountered. Because of these difficulties, our knowledge of 

 the interactions between the various mechanisms is inadequate. 



In discussing steady flow, we shall first consider regions of the ocean that are 

 not directly affected by boundaries. Where boundary influences are considered, 

 the boundaries will be simplified to straight vertical coastlines and a level, or 

 slowly varying, bottom. By this simplification, we exclude a number of in- 

 teresting features found in coastal regions which depend on details of bottom 

 topography and coastline. However, even with these simplifications, we cannot 

 solve the steady-state equations in three dimensions. A further reduction to a 

 two-dimensional system is necessary to obtain solutions of interest. This 

 reduction is accomplished most readily and systematically by integrating the 

 momentum and continuity equations vertically from the bottom to the surface 

 of the ocean. 



We can write the general equations governing steady motion (12) and (16), 

 using the x-y-z notation, in the form : 



f = -P9 (27) 



^ + ^ + M' = 0, (28) 



dx oy cz 



where / is introduced for the Coriolis parameter, 2Qz, and p for mean density. 

 On the basis of the dimensional analysis carried out earlier, we have neglected 

 horizontal components of Coriolis force due to vertical motion in (25) and 

 vertical accelerations, Coriolis and frictional forces in (27). The terms that have 



