SECT. 6] TURBULENCE 815 



problem which was necessary, the theoretical treatment yielded results on the 

 depth and characteristics of the thermocline w hich were in general accordance 

 with experience. 



The influence of stability on the vertical mixing processes due to turbulence 

 has also been treated by Mamayev (1958), who considered that the appropriate 

 forms for Nz and Kz were 



Nz = Aqe-^R^, Kz = Aoe~^^\ 



where n — m>0. From Jacobsen's data it was deduced that n = O.S, m — OA. 



As discussed in section 1 of this Chapter, the quantity Rf=KzRilNz, known 

 as the flux Richardson number, represents the fraction of the turbulent energy 

 being generated by the shearing stresses which is used to maintain turbulent 

 mixing against the density gradient. Ellison (1957) has deduced on dimensional 

 grounds that Rf should approach a critical value Rf cm., less than 1, as Ri 

 increases indefinitely. Data from atmospheric turbulence indicate that RfcT\t. = 

 0.15 approximately. It may be noted that the forms of Nz and Kz used by 

 Munk and Anderson correspond to Rf—^Rfcrit. as Ri-^cc, but the critical 

 value is i?/crit. =0.52. Mamayev's forms of iV^ and Kz, on the other hand, 

 correspond to Rf -^ Q as Ri ->cc, but Rf would reach a maximum value, 

 Rfmax., at some intermediate value of Ri. He envisages the possibility of Rfm&x. 

 exceeding 1, but this would no longer correspond to a steady state as the 

 intensity of turbulence would be decreasing with time. 



Methods of determining the numerical values of the coefficients of eddy 

 viscosity Nz and eddy diffusion Kz, from stationary distributions of current 

 and temperature or salinity or from periodically varying conditions, have been 

 given by Sverdrup et al. (1942) and by Proudman (1953). Among the more 

 recent determinations of Kz in deep water are those of Wiist (1955), based on a 

 reconsideration of the Meteor data. In the core of the Antarctic bottom current, 

 in the West Atlantic Trough, he derived values of Kz ranging from 7 to 50 cm-j 

 sec with an average of 29 cm^/sec over the latitude range 50°S to 10°N. Values 

 of the same order of magnitude were obtained for the North Atlantic deep 

 current. Koczy (1956) used the distribution of radium with depth to determine 

 Kz, assuming that the radium content of the water was maintained by the 

 production of radium from ionium in the bottom sediments and its transport 

 upwards by eddy diffusion. Data from four stations, one in the Atlantic, one 

 in the Indian Ocean and two in the Pacific, gave values of 4 to 30 cm^/sec near 

 the bottom, decreasing with height to quite small values at 2000 to 3000 m 

 above the bottom. 



B. Velocity Profile near the Sea-Bed 



One of the most firmly established results in the general study of turbulent 

 flow is the logarithmic law for the velocity profile near a solid boundary (7) 

 and its use as a means of determining the shearing stress. The first measurements 

 of the velocity profile immediately above the sea-bed appear to be those of 

 Revelle and Fleming at the entrance to San Diego Harbour (reported in 



