SKCT. 3] DYNAMICS OF OCEAN OTTRRKNTS 329 



|)ninarily from motion of longer time-scale and, conversely, must pass momen- 

 tum primarily to shorter time-scales. The mechanism by which momentum 

 and energy are transferred to shorter time-scales has been studied more 

 thoroughly in connection with turbulence (Townsend, 1956). 



Turbulent motion constitutes an essential part of the dissipative mechanism 

 in the ocean. Its action is complex and cannot be readily taken into account in 

 the momentum equations. However, it is often qualitatively simulated in its 

 dissipative action by assuming the Reynolds stresses to be proportional to 

 the strain rate of the mean flow. By analogy to molecular friction, the propor- 

 tionality constant is referred to as the virtual or eddy viscosity coefficient. The 

 concept of eddy viscosity has proved useful in providing a simple dissipative 

 mechanism, but, at best, it is a crude approximation to the action of turbulence 

 and results obtained through its use must be applied with caution. 



Separate coefficients of eddy viscosity are used to estimate stresses from 

 horizontal and vertical gradients of the mean flow. The necessity for using at 

 least two coefficients is evident from considerations of the dimensions of the 

 ocean, as will be shown later. We can introduce the eddy viscosity coefficients 

 by relations of the form (Saint-Guily, 1955), 



T. -— ; — r ,-^^Ui ...dUj , , 



Rij = -pUiUj = A^0)-^+/^(O-^' (1^) 



where fx{i ), fx{j ) are equal to the lateral eddy viscosity /x// for i, jV 3 and to the 

 vertical eddy viscosity /X7 for i,j = 3. It is clear from (15) that the magnitudes of 

 jjiH, iJi-v will depend on the scale of the motion and can be estimated approxi- 

 mately by dimensional analysis. 



If we substitute (15) for the Reynolds stresses in (11) and neglect molecular 

 friction and density variations in the frictional terms, we obtain 



where V^ is introduced for 8^ldxi"+ d^jdxz^. 



3. Magnitudes of Forces 



The separation of the flow into steady and time-dependent modes is accom- 

 plished by methods closely analogous to those used in turbulence theory (cf. 

 Townsend, 1956, chap. II). The basic difference in application of the equations 

 is in the time- and size-scales of motion that are considered. In studying ocean 

 circulation, we shall conflne our attention primarily to the low-frequency, 

 large-scale current systems for which the non-linear interaction terms are 

 negligible or capable of simplification. 



The relative magnitudes of the various forces and accelerations present in 

 (11) and (13) can be compared among each other by dimensional analysis. 

 Ocean currents are slow enough that pressure can be approximated with high 

 accuracy by the hydrostatic equation. Consequently, pressure gradients are 



