SECT. 3] ESTUABIES 319 



ponents. In most estuaries the tide is of the reversing type, and the longitudinal 

 velocity component U x dominates. The lateral component U y is primarily 

 related to changes in width of the estuary, while U z results directly from the 

 tidal rise and fall of the water surface. 



We neglect the turbulent fluctuations in density in comparison to the corre- 

 sponding fluctuation in velocity, and assume water to be incompressible. The 

 equations of continuity for the instantaneous and for the mean velocities are 

 then 



8u x 8u y 8u z _ 



dx 8y 8z 



Equations (3), (4) and (5) are used in (1) and (2) in obtaining the time mean 

 equations of motion. In this operation cross-products of the type (u x u' y y 

 (where ( ) indicates a time mean) occur which are taken to be equal to zero, 

 since by definition (u x u' y y = u x (u' y y and (u' y y = 0. Likewise, terms of the type 

 (JJ x u' y y are set equal to zero since there is no reason to suspect a correlation 

 between the oscillatory tidal motion and the turbulent velocity fluctuations, 

 even though there may be a relationship between the root mean square of 

 the turbulent fluctuations and the magnitude of the tidal current. Cross- 

 products involving mean velocities and the tidal components are also zero. 



Available observational evidence also indicates that certain terms which 

 occur in the mean equations are quite small compared to other, dominant 

 terms. The mean vertical velocity, u z , is of the order of 10 -5 m/sec, as compared 

 to the horizontal velocity component of the order of 1 m/sec. The Coriolis 

 terms involving the vertical velocity component are then negligible compared 

 to the Coriolis terms involving the horizontal velocity. In the geometrically 

 simple, elongated estuary here considered, the terms involving the lateral 

 component of the velocity are also negligible. 



The mean longitudinal equation of motion then is given by 



8u x 8u x 8u x 8 8 



_+„,_+„, _ + _ < E7^ >+ _ <tW > 



/ 8p\ 8 8 , 8 , 



" -V'S/-^"*" •>-«*<• *"«>-5 <"'"■> (8) 



and the corresponding mean lateral equation is given by 



0= -(a-^-\-fu x -—(u' x u' y y-—{u' y u' y y-—(u' z u' y y, (7) 



where / replaces 2Q Z . 



The eddy-stress terms in these equations, involving products of the type 

 (u'tu'jy, are many orders of magnitude larger than the molecular viscous-stress 



