SECT. 3] DYNAMICS OF OCEAN CURRENTS 365 



conditions alone. At least one of the variables must be completely specified 

 before an explicit solution can be obtained. Usually the vertical component of 

 velocity is assumed to be known although it is possible to assume, for example, 

 that the density is given (Stommel, 1956a) or the transport in the western- 

 boundary current (Ichiye, 1960). The applicability of the results obtained by 

 this approach depends on whether or not a realistic choice of one of the variables 

 can be made. This approach has been applied with considerable boldness by 

 Stommel and Arons (I960, 1960a) in their development of a theory of deep- 

 water circulation. 



The second, and more difficult, approach is to simulate the convection and 

 diffusion of heat and salt by introducing a simplified conservation equation for 

 temperature (heat) or density. If temperature is used, it is also necessary to 

 introduce an equation of state relating density to temperature, i The augmented 

 set of equations is closed in the sense that the number of equations equals the 

 number of unknown variables so that, in principle, the solutions can be deter- 

 mined in terms of the boundary conditions. However, the additional conserva- 

 tion equation is non-linear so that only those cases in which perturbation 

 methods can be applied, or in which the independent variables can be separated, 

 have been studied. This approach was introduced by Lineikin (1955) and 

 extended by him and others subsequently.^ 



We shall examine first the dynamical effects of a vertical component of 

 velocity in the interior of the ocean. We assume that in some region of an 

 ocean with a level bottom the vertical velocity, wq, is given as a function of 

 latitude along a level surface, zq. We assume further that the horizontal flow is 

 geostrophic and slow with possible exceptions near meridional boundaries 

 where strong currents may develop. The flow must satisfy mass continuity 

 (28) and, where geostrophic, the relations given in (34). 



If we integrate (28) with respect to z from the ocean bottom, zb, to the level 

 surface, zq, and with respect to x from the west to the east coast, we obtain 



dT 



— = -poWoL, (145) 



where T is the total meridional mass transport across the zonal section of 

 length L. The distance between the west and east coasts, L, is assumed to be a 

 known function of y. 



Integrating (145) with respect to y, we obtain 



T-To = - r poWoLdy. (146) 



The difference in transports at the two latitudes y and yo is equal to the mass 



1 As a linear dependence of density on temperature is invariably assumed to simplify 

 the analysis, the choice of a density or temperature equation is a matter of taste. Both 

 yield identical results. 



2 Lineikin (1957), Stommel and Veronis (1957), Robinson and Stommel (1959), Welander 

 (1959) and Ichiye (1958). 



