208 [TS]-relationship and Connection with Mixing Processes and Large Water Masses 

 salt increase due to the exchange of water quanta by mixing and is thus 



I+cc 

 ps X 10"^ qds. 



Putting with sufficient accuracy 



(ds\ 1 [d^s\ 



^dzj Q 2 ydz^/Q 

 and using the above expression for q gives for the point A (z = 0) 



Then k is expressed in terms of the exchange coefficient A. The [r^j-relationship for 

 the water column under consideration is presented in Fig. 96, At the reversal point 

 /i,z = Oandthedepthmarks +1, +2, +3, . . ,,and — 1, — 2, — 3, . . ., respectively, 

 correspond to the centre points of water cubes of 1 cm^; the cube at A is thus the zero 

 cube. The circle with a radius R (AO = R) approximates closely to the [r5']-curve 

 at the point A. For the part of the curve under consideration the depth marks are so 

 situated that the arc between each pair of depth marks always corresponds to the 

 same angle a in point O. It is necessary to find the co-ordinates (temperature and 

 salinity) after / sec of the zero cube initially at A . Two points M and A^ at vertical 

 distances z and z -\- dz cut out a volume element of dz cm^ of water. According to 

 the previous discussion a quantity of water gdz is transferred from this element to 

 the zero cube in t seconds. The same quantity of water gdz is also transferred from the 

 symmetrically situated volume element M'N' in the same time. These two quantities 

 of water mix, and according to the mixing rule the mixture 2qdz corresponds on the 

 [rS'j-diagram to the small interval BC which is situated on the radius of curvature AO. 

 It is determined by the distance 



AB ^ r = R - Rcos (za) = ^Rah^ 

 and 



BC = dr = la'^z'^dR. 



The water masses entering the zero cube during time / are not only transferred from 

 the two cubes MN and M'N', but also from all other cubes above and below, and it is 

 easily understood that the T and S values for all these water masses must lie on the 

 radius AO. Mixing of all these differential quantities gives the co-ordinates of the zero 

 cube after / sec. Its position on the [rS'j-diagram will be fixsd by the distance Z 

 along AO. According to the mixing rule this must be given by 



2q dr dz. 







One therefore obtains 



Z = - RaH. 



