328 FOFONOFF [sect. 3 



density fluctuations are not significant in the acceleration terms and that the 

 correlations between them and velocity fluctuations may be neglected. ^ Using 

 this assumption, we can write (6) and (7) in the simplified form 



^ = 0, (12) 



where — pu'ifi'j, the Reynolds stress tensor, is denoted by Rij. Similarly, the 

 time-dependent equation becomes 



_ du'i _^^ du'i _ , dUi dR'ij ^ ^ _ , dp' 



dxjdxj 

 ct dxj 



where R'a is introduced for — p{u'iu'j — u'iu'j). 



From (11) we see that the influence of the time-dependent motion on the 

 steady flow is represented by the Reynolds stress tensor, Ra, only. The inter- 

 actions in the time-dependent equations are more complex. The interaction 

 terms sej3arate into convection of fluctuating momentum by the steady flow, 

 convection of steady-state momentum by the fluctuating flow and the diver- 

 gence of momentum-flux variations of the purely time-dependent flow. 



The time-dependent equations (13) and (14) include fluctuations of all 

 frequencies. If we are interested in variations of ocean currents with time- 

 scales of a few days or longer, we can carry out a second averaging process 

 using an averaging time that is short compared with the periods of the motion 

 being considered. In this way we can distinguish between the slower variations 

 being considered and high-frequency turbulence and wave components. If we 

 carry out the flnite average in equations (13) and (14), we find that the only 

 term that is altered in character is R'ij. The remaining terms enter linearly 

 into the equations and may be replaced directly by their averages. The finite 

 average of R'a is equal to the diflFerence between the finite and steady-state 

 average of pu'tu'j. Fluctuations with periods shorter than the finite averaging 

 time will contribute approximately equally to both averages and will cancel 

 when the difference is taken. If we interpret the Reynolds stresses in the 

 steady-state equations as representing a dissipative mechanism, we have to 

 consider the divergence of the finite average of R'a as a source of momentum 

 to which only the longer-period motions can contribute significantly. In other 

 words, the time-dependent motion of a given time-scale can draw momentum 



1 However, the correlation between density and velocity fluctuations may not be 

 negligible in the other equations where velocity correlations do not appear. In particular, 

 the correlation may not be negligible in the equation for mass continuity. 



