General Theory of Ocean Currents in a Homogeneous Sea 393 



At all depths -q has about 10 times the magnitude of the exchange coefficient A 

 determined from salinity measurements made at the same time. The quotient -qjA is 

 almost always larger than the Ri-number and therefore according to the above con- 

 dition is not compatible with turbulence. The Ri-numbers, which vary between 2-6 

 and 125, are so high that also according to this criterion a turbulent flow can hardly be 

 present. However, the measurements indicated still a small, though very weak, 

 turbulence with a frictional coefficient between 1-9 and 3-8 g cm~^ sec~^. According 

 to these investigations, other factors seem also to be involved in the appearance and 

 maintenance of turbulence (close distance to a solid boundary or the presence of an 

 intermediate layer between the otherwise almost homogeneous water masses above and 

 below). 



{d) Turbulence and Mixing in the Sea; Statistical Theory of Turbulence 



The modern hydrodynamic approach to ocean currents has led increasingly to the 

 view that the turbulence of the ocean currents, which finds its visible expression in the 

 oceanic mixing processes, is the basic cause of a number of oceanic phenomena. 

 Oceanography has mostly been concerned solely with the effects of turbulence and 

 mixing on oceanic phenomena; only recently has interest been directed also towards 

 the nature of oceanic turbulence and one has asked the important question : of what 

 kind is this nature ? In laminar flow the velocity can be represented by a simple function 

 of position and time. In turbulent flow the mean velocity, which again can be repre- 

 sented by a simple function of this sort, is superimposed on an additional, irregularly 

 varying turbulent velocity component that changes with both time and space. The 

 sharp distinction between the two types of flow is shown by experimental investigations 

 which indicate that a discontinuous transition from laminar to turbulent flow occurs 

 when a dimensionless quantity, the Reynolds number, exceeds a critical value, the 

 magnitude of which is about 1000. The form of the Reynolds number indicates the 

 cause of this basically different behaviour of the two types of flow. The Reynolds 

 number is given by R = plJL\r], where p is the density, U and L are values for the 

 velocity and the hnear dimension which are characteristic for the structure of the 

 particular current under consideration; r] is the eddy viscosity coefficient (frictional 

 coefficient). It is clear that the current will be turbulent when the momentum (impulse) 

 of the flow pU or the distance L passed through are large; it will be laminar if the 

 viscosity is large. The viscosity is a force carrying neighbouring elements of the 

 medium along the same path. Therefore, it is obvious that large viscosities will have a 

 tendency to smooth the course of the flow. The empirical fact that the current tends to 

 change to turbulent flow even with very small disturbances — i.e. that the laminar 

 flow is unstable — shows that the turbulent flow has in a certain sense to be regarded 

 as the natural form of motion of media with low viscosity. The Helmholtz vortex-laws 

 of classical hydrodynamics show that a vorticity-free current cannot develop vortices 

 spontaneously. Thus no turbulence can occur in it by itself. It can only be produced 

 inside the fluid by friction at solid surfaces, or by similar processes through the forma- 

 tion of vortices at the boundary of the liquid. Once formed it will spread out in the 

 fluid. This is, however, not the case which we meet in the open sea remote from the sea 

 bottom and from the coasts. The ocean currents here usually have a considerable vortex- 

 intensity from the beginning, i.e. from their formation; it is their further distribution 



