398 General Theory of Ocean Currents in a Homogeneous Sea 



Weiszacker took a discrete velocity spectrum as the basis of his theory, Heisenberg 

 chose a continuous velocity distribution and provided an elegant mathematical proof 

 (in this connection see also Ichve, 1951). 



The principal result of the theory, as far as it concerns the exchange coefficients of 

 turbulent motion, is in complete agreement with the 4/3 power law derived from 

 observed data. The more recent statistical theory of turbulence can give a better 

 description of actual conditions in nature than the classical Fickian theory. In par- 

 ticular, the theory gives an explanation for the large differences in size between the 

 turbulence coefficients for small- and large-scale motion, for which there was no ex- 

 planation in earlier time. For small-scale oceanic phenomena the values found 

 for the diiTusion coefficient t] are on the average about 50-100 cm^ sec~^. For large- 

 scale ocean currents, on the other hand, the values were between 10^ and 10^ cm^ sec"^. 

 The ratio between these is about 5 X 10^ to lO**. For small-scale processes L can be 

 taken as about 50 m and for large-scale currents as about 1000 km. The ratio of the 

 L-values is 2 x 10* and for the T^-values should be according to the theory about 

 5-4 X 10^. The agreement with the values derived from observations is rather good. 



The question could also be raised, how far the assumptions made by the theory are 

 justified in oceanic conditions. Stommel (1949) has closely examined this question. 

 Not all the sources for turbulence in the ocean are due to air currents, a part is cer- 

 tainly due to the thermo-haline structure of the ocean currents the dependence of 

 which, of course, on solar radiation and evaporation is known. The assumption of a 

 continuous series of vortex sizes with horizontal isotropy can hardly be valid for the 

 large oceanic vortices ; it can be postulated as a first approximation only when they are 

 of smaller dimensions, i.e. for the genuine turbulent vortices of oceanic currents. The 

 changes which should be introduced for oceanic conditions involve the dividing of the 

 vortex sizes into two parts : an anisotropic one, including all the kinematically dissimi- 

 lar, large-scale horizontal movements, and an isotropic part, including all the kine- 

 matically, similar-to-each-other, turbulent vortices. The latter part only appears after 

 a certain nth averaging process. The first part is thus essentially concerned with the 

 advection of different water types. The exchange is only involved in the second, and 

 the statistical theory of turbulence should be fully applicable here. However, in spite 

 of these changes in many of the assumptions the basic idea of the theory remains and 

 offers a solid basis for the study of dynamic conditions of the ocean currents. 



4. Steady Currents in a Homogeneous Ocean under the Action of External Forces 



(a) Introduction 



The first ideas about the effect of friction on the movement of water masses were 

 based on the assumption that it arose from the roughness of the bottom surface 

 (gliding friction). The frictional force was thus given, as already shown on p. 317, by 



R = -KpV. 



GuLDBERG and MoHN (1876) using this principle for atmospheric flow presented a 

 diagram of the forces necessary for a steady motion. It can also be applied to water 

 movements in shallow ocean currents for which the frictional effects of the bottom 

 act throughout the entire water column. In that case the resultant of Coriolis force 

 and frictional force must balance the gradient force. The direction of the current is 



