SKCT. 3] DYNAMICS OF OCEAN CURRENTS 325 



spherical system. ^ It is clear that we must show that a result obtained in the 

 rectangular system has its analogue in the spherical system before we can 

 apply it without ambiguity. In many cases the translation to the spherical 

 system is simple. However, some results can be expected to be appreciably 

 modified by spherical geometry.'^ 



The change of Coriolis parameter with latitude is simulated in the rectangular 

 co-ordinate system by assuming the parameter to be a function of the horizontal 

 co-ordinates. To simplify analysis, the Coriolis parameter is often approximated 

 by a constant except when differentiated with respect to a co-ordinate repre- 

 senting latitude. Then, the derivative is assumed to have a constant non-zero 

 value 6. If this simplification is carried out, it is referred to as the /S-plane 

 approximation. The |S-plane can be considered to simulate mid-latitudes of 

 the earth. In equatorial regions the Coriolis parameter approaches zero so that 

 its variation with latitude is not small compared with its magnitude. In polar 

 regions the derivative of the Coriolis parameter ^ approaches zero and cannot 

 be approximated by a constant. In both cases the ^-plane approximation is 

 poor and other techniques have to be used. 



The |8-plane approximation was introduced by Rossby (1939) and has 

 proved to be a useful tool for developing new ideas on the dynamics of ocean 

 currents, particularly in time-dependent theory. However, its use must be 

 regarded as an intermediate step in the development of more complete and 

 satisfactory theories of ocean circulation. 



In setting up the conservation equations for momentum and mass, we shall 

 consider the ocean to be unbounded in the j8-plane, unless effects of boundaries 

 are being considered. We assume also that gravity, g, is everywhere constant 

 and acts in a parallel direction. To specify position in the ocean, we introduce 

 a right-handed rectangular co-ordinate system with the origin at the mean free 

 surface of the ocean. We shall use two systems of notation to denote the co- 

 ordinates in the conservation equations ; the cartesian tensor notation to 

 describe general properties of the equations and the x-y-z notation to express 

 more detailed results and solutions. 



The co-oi-dinates are oriented so that xi,x points eastwards, X2,y northwards 

 and X3,z upwards and parallel to the gravitational force. The velocity com- 

 ponents along the co-ordinate axes are denoted by ui,u for the eastward com- 

 ponent, U2,v for the northward component and us,w for the vertical component. 



Using tensor notation, 3 we can express the conservation of momentum as : 



1 The main handicap in using surface co-ordinates, such as those introduced by Morgan 

 (1956), is presented by the convergence of meridians towards the poles of the earth. The 

 conservation equations must contain coefficients dependent on latitude in such a system 

 and, hence, are more difficult to analyse. 



2 Stommel and Arons (1960) give an interesting example of the different results obtained 

 in examining simple flows in rectangular and spherical co-ordinates. 



3 An index appearing twice in a term implies summation over all three index values. 



