376 The Representation of Oceanic Movements and Kinematics 



Multiplying equation (XII. 6) by dz and integrating from the surface to the bottom it 

 follows that 



At the sea bottom w^ equals and further if the vertical elevation of the sea surface 

 above the equilibrium level (positive upwards) is denoted by i then Wq = —{dlidt) 

 and from (XII. 9) follows 



8t 1 , 



-f= divAf (XII.IO) 



dt po 



The divergence of the current amount is thus always associated with vertical displace- 

 ments of the sea surface and these can be readily calculated from (XII. 10) if the current 

 amount is known. For a stationary state of the sea surface (C — const.) it follows 

 necessarily 



divA/ = 0, (XII. 11) 



that is, at stationary sea surfaces the total current amount must be divergence free. 

 This need not be the case in every layer but in the entire water column an excess in- 

 flow in some of the individual layers must be balanced by a deficit in the other layers, 

 if no effect on the sea-level should appear. 



Under stationary conditions in the sea there must be in any volume element a con- 

 stant amount of all the dissolved substances in the water besides the constancy in 

 density (see Defant, 1941^/). If the salinity for example is denoted by s and exchange 

 processes are for the moment disregarded, tliis requires 



ds ds 8s 8s ^s ^ ^ , ,^. 



^. = ^ + " ^ + ^ ^ + '^' TT = 0- (XII.12 



dt dt 8x 8y 8z 



Multiplying this equation by p and then adding the continuity equation (X. 31) 

 multiplied by s, it follows that 



8 OS 8ups 8vps dw'ps 



For stationary conditions the first term on the left-hand side is zero and the condition 

 of a constant salinity will be given by the remaining equation integrated over the total 

 volume under consideration. Introducing a space vector S with horizontal components 

 Sx and Sy which is given by the equation 



•/I 

 S= pscdz (XII. 14) 



J 



allows the equation (XII. 13) for stationary conditions to be rewritten in the form 



div5 = (XII. 14a) 



S can be termed the salinity amount and the equation states that under stationary con- 

 ditions the vector indicating the amount of salt flow must also be divergence-free. 



The constancy of the water mass in a given space and the constancy of the characteristic water 

 properties existing under stationary conditions has often been used in the derivation of the current 



