320 



CAMERON AND PRITCHARD 



[CHAP. 15 



terms. Hence, the original omission of these latter terms in equations (1) and 

 (2) is justified. 



Pritchard (1956) argued that of the three eddy -stress terms in each equation, 

 only d((u' z u' z))/dz would be significant in equation (6) and only d((u' z u' y )>)ldz 

 should be retained in (7). On the basis of observations from the James River 

 estuary, he showed that there was approximately a lateral geostrophic balance ; 

 that is, the mean lateral pressure force —(a. dp/dy} was very nearly balanced 

 by the term fu x . The eddy-stress term in (7) was of secondary importance. 

 Stewart (1957), on the other hand, raised the question whether the eddy-stress 

 term could be retained in equation (7) in view of the fact that the mean lateral 

 velocity was assumed to be negligible, and hence there would be no velocity 

 shear corresponding to this stress term. He showed that the geostrophic 

 imbalance could be explained as a result of the curvature of the boundaries of 

 the James River estuary. 



The mean longitudinal equation of motion retains two terms involving the 

 tidal components of the velocity. This bears out the statements made in the 

 earlier descriptive paragraphs that a portion of the mean pressure force is 

 associated with the non-linear terms involving the tidal velocities, even though 

 these velocity components may be purely harmonic in character and average 

 to zero over the tidal period. If the tidal wave in the estuary is a progressive 

 wave, then U x and U z will be 90° out of phase, and the second of the two 

 terms in equation (6) involving tidal velocity components will disappear. The 

 remaining term becomes, using equation (4), 



dx 



\U x u x y = u i 



dUp, 

 dx 



(8) 



If the tidal wave is a standing wave, then U x and U z will be 180° out of phase. 

 In this case the last two terms on the left side of (6) become, after employing 

 the equation of continuity. 



^-<u x u x )+^(u x u z y = \ 



ox dz 2 



U t 



dx 



_ jj 



dU, 



dz 



(9) 



In the case of a progressive tidal wave in the estuary, a part of the mean 

 pressure gradient is then balanced by a term involving the longitudinal varia- 

 tion in tidal-current amplitude. In the case of a standing tidal wave, both the 

 longitudinal and vertical variations in the tidal-current amplitude are involved. 



There are some interesting implications which result from the presence of 

 these non-linear terms in the mean longitudinal equation of motion. For 

 example, consider the simple case of a coastal embay ment with no net fresh- 

 water inflow and uniform density distribution. The mean pressure force 

 — (a dp/dxy would then be given simply by <gr drj/dx}, where g is the accelera- 

 tion due to gravity and 77 the elevation of the water surface. In the absence of 

 any density gradient, we would expect the mean velocity components to 



