SECT. 6] TURBULENCE 807 



relative mean motion from turbulent motion for the particular scale of eddies 

 considered. From similarity considerations it is found that 



N oc i4/3. (20) 



Thus the effective eddy viscosity increases with the scale of motion considered. 

 The result (20) is similar to the relation for the neighbour diffusivity (16) found 

 by Richardson. The smallest eddies lose all their energy by viscous dissipation. 



More recently, much attention has been given to the properties of the 

 "large eddies", i.e. those of dimensions comparable with the scale of the mean 

 flow and largely responsible for abstracting energy from it. These are markedly 

 anisotropic and would appear to be a more ordered type of motion than the 

 remainder of the turbulence. Some progress has been made by representing 

 them as arrays of vortices of defined properties (e.g. Grant, 1958). 



A comprehensive account of recent work on turbulent shear flow was written 

 by Townsend (1956). Frenkiel (1953) and Batchelor and Townsend (1956) have 

 written reviews of work on turbulent diffusion and a general account of the 

 problem of atmospheric diffusion has been given by Deacon (1959). A new 

 approach to a theory of turbulent shear flow has been made by Malkus (1956). 



F. Influence of Stability 



The energy of turbulent motion is continuously being dissipated by viscosity 

 and, in order to maintain a steady state, energy must be derived from the mean 

 motion by the action of the Reynolds stresses. In the case of a horizontal shear 

 flow, in which the mean current U is in the OX direction and varies in the OZ 

 direction only, the rate of generation of turbulent energy per unit volume is 



r-r ^^ 



dz 



(21) 

 —8U „ (dUY 



The rate of dissipation by viscosity is 



D = 110, 

 where 



)■ 



(22) 



(h - ^(^'^V '?(^^Y 9(^^Y (^^ ^^Y (^^ ^^Y l^^ ^^\ 



W ^'^\d^) ^^l"^/ "^lai^a^j ^\d^'^~dl) ^yd^^W 



If there is a vertical gradient of mean density in the fluid, the density may be 

 regarded as a property capable of turbulent transport. Let the instantaneous 

 value of the density at a point be p + p', so that p' represents a density fluctua- 

 tion. Then vertical turbulent mixing gives rise to an increase in potential 

 energy at a rate per unit volume of 



P = g7^= -gKz^' (23) 



