General Theory of Ocean Currents in a Homogeneous Sea 391 



{b) Dissipation of Energy by Turbulence 



The turbulent process mixes neighbouring water quanta; part of the energy is 

 deviated from the direction of the mean basic current, the water masses are flattened 

 out by vortices into thin layers and part of the energy is used up in this, which would 

 otherwise remain in the basic current. The magnitude of the energy dissipation by 

 turbulence can be calculated from the size of the shearing stress (XIII. 13). This shear- 

 ing force acts horizontally ; the relative movement of two water sheets one above the 

 other is dujdz. From this the work done by the turbulence (energy consumption by the 

 apparent friction "Scheinreibung") will be i? = rj(8uldzy. This is that work which must 

 be done in unit volume and unit time to maintain the turbulence against the velocity 

 gradient. (Schmidt, 1919). 



In the example described above, in the Dardanelles, the velocity decreased from 27 m 

 down to 2 m above the bottom by 6-9 cm/sec. The mean velocity gradient was thus 

 {dujdz) = (1/362). The dissipation of energy per day amounted to 0-6677 ergs per cm^. 

 This appears rather small but over a longer period has an appreciable effect. If t^ = 

 100 cm~^ g sec~^ then the kinetic energy of a current of 20 cm/sec will be 200 erg/cm^ 

 and this would be entirely absorbed by the turbulence in about 3 days if not continu- 

 ously renewed by other forces. 



(c) Turbulence and Stratification 



That the turbulence is dependent on the stratification in the medium is apparent 

 from the following considerations (Ekman, 1906; Schmidt, 1917; Pettersson, 1930, 

 1935). In the presence of stable stratification the mixing process is affected by the double 

 work required to lift the lower heavier water masses against gravity and to lower the 

 upper lighter ones against buoyancy forces. This hinders mixing and if the density 

 differences become large enough the stability of the water stratification reaches so 

 high a value that turbulence cannot act against it and may cease entirely. In subtropical 

 oceanic regions cases occur in the tropospheric deeper currents in which a thin layer 

 of highly saline water embedded between two layers of low-saline water can spread 

 over thousands of miles without being absorbed in the layers above and below by 

 mixing. The strong stabihty of the vertical stratification of the water masses completely 

 prevents mixing. An example of this behaviour of the subtropical intrusions of highly 

 saline water has been given in Pt. I, p. 169, Fig. 73 and the reader is referred to the 

 discussion at that place. 



The conditions under which the work expended in the vertical displacement of 

 water elements by turbulence becomes so large that the turbulence is completely 

 suppressed can be found by comparison of the energy dissipation by turbulence and 

 the lifting work done against gravity by mixing. The buoyancy force per unit 

 time and unit mass for a density gradient dpjdz is given by g{/l p)(8pl8z).* The vertical 

 disturbance velocity u'' according to the previous discussion can also be put propor- 

 tional to I{dujdz). From (XIII. 16) and taking into account that for an equilization of 

 the density differences (temperature and salinity), iq must be replaced by the exchange 

 coefficients for the material properties of the water ^4^ (pt. I, p. 103), it follows that the 

 work done against gravity in unit volume and unit time is g(AJ p)(Spjdz). The work 



* The symbol 8 should indicate the necessary consideration of the changes in density due to adia- 

 batic temperature changes. 



