General Theory of Ocean Currents in a Homogeneous Sea 395 



in the velocity such as the tidal currents and the annual changes in u' . If T is selected 

 with a value of about a month the tidal currents will also be included in the value of 

 u'{t). If ris chosen for 10 years or more, the seasonal changes will also be included in 

 11 and only the secular changes will remain in U. From this it can be understood that, 

 in nature, motions in water masses as they appear in the ocean will be much more 

 complicated than, for example, in an experimentally controlled wind tunnel or a 

 water channel. Every size and all different velocities of the turbulent vortices can be 

 expected to occur in oceanic turbulence, and it is not easy to distinguish between the 

 basic velocity and the additional turbulent velocity. These difficulties occurring with 

 turbulent phenomena of the ocean and atmosphere seem to be fundamentally connected 

 with the nature of turbulence. 

 In dealing with mixing processes in the ocean, the simple relationship 



ds d"s 



Jt ^ ^8z2 



has usually been used, where S{z, t) is the concentration of the diffusing substance and 

 K denotes the mixing coefficient (eddy diffusivity, eddy conductivity), [cm^ sec~^]. 

 This is termed the "Fickian diffusion equation" (see Pt. I, pp. 95 and 104). It is derived 

 by analogy with molecular processes for the larger-scale processes in turbulent currents 

 using simplifying assumptions on the internal nature of turbulence; it does not accord 

 fully with more recent data, and especially not with the fact that the larger the mixing 

 coefficient becomes, the larger the scale of the phenomena under consideration, i.e. 

 with the existence of a continuous spectrum of the diffusion coefficient. 



With molecular diffusion, as described by the Fickian equation, the movement of 

 each molecule is independent of that of a neighbouring one. In contrast to this, how- 

 ever, in a turbulent current, adjacent elements have increasingly similar turbulent 

 velocities, and in fact the more there are the smaller the distance from each other. The 

 reason for this is easily understood when the behaviour and the effect of the turbulent 

 vortices of all sizes are studied altogether in detail. The distance between two initially 

 adjacent elements is altered only by the smallest vortices ; the effects of the larger 

 vortices cause no significant change in distance, since they give rise only to a simple 

 transport of these elements. If, however, the distance between two elements becomes 

 larger, the effect of the larger vortices is added to that of the smaller ones so that as the 

 distance between them increases the diffusion effect due to the larger-size vortices 

 becomes more and more involved. 



The most important independent variable cannot be, as in molecular diffusion 

 processes, the position of an element, but the distance from its neighbouring element. 

 This requires that the concentration of a diffusing substance is only a function of the 

 mutual separation of the particles inside this substance and not a function of the posi- 

 tion only. 



Richardson first showed this difference as compared with molecular diffusion and 

 further investigations have then been carried out to account for this circumstance 

 (Witting, 1933; Sverdrup, 1946; Proudman, 1948). The theory that the concentra- 

 tion of a diffusing substance is not a function of the position of the element which it 

 occupies, but rather of its distance / from the adjacent element leads to the conclusion 



