SECT. 3] DYNAMICS OF OCKAN CURRENTS 327 



We can now obtain the steady-state equations by averaging (3) and (5) 

 with respect to time. Using the definition 



^ = I™ (2^ JI '' *) 



to denote the average of any property 93, we obtain 



-^^ + 2eo-.^,,^. =. _ _ _ ,^ S3, + ^ __ (6) 



and 



5.T; 



= 0. (7) 



The time-dependent equations are obtained by subtracting (6) from (5) and 

 (7) from (3).i Thus, the time-dependent equations are 



dpuj dipUjUj-pUjUj) — 



-7T- -^ 7— 1- ^€ij]cUj{pUk — pUk) 



= 8^, (p-/>)yg3.-f/x g^^g^. (8) 



and 



^+ ^''"'r^' = 0. (9) 



We can expand the averages of products in the equations by sphtting the 

 dependent variables — velocity components, pressure and density — into steady 

 and time-dependent parts so that 



ui = Ui + u'i 



V = P+P' (10) 



P = P + P'y 



where u'i, p' and p' have zero averages. We can then separate the averages of 

 products into terms expressible in terms of the averages of the individual 

 factors and correlations between fluctuations of the factors. For example, 

 the average pUiUj becomes 



pUiU} + pu'iu'j + p'u'j Ui + p'u'i U} + p'u'iu'j. 



Variations of density in the ocean are of the order of 0.1 % of the mean density, 

 whereas velocity fluctuations are much larger and can be of the same magnitude 

 as their mean. Consequently, the terms containing correlations between 

 fluctuations of density and velocity will be siuall compared to terms containing 

 correlations between velocity com])onents. We assume, therefore, that the 



1 The time-dependent motion is deHnecl to have a zero mean. This definition is imphed 

 in most studies of time-dependent How, l>ut it is not always exphcitly stated. 



