SECT. 2] SMALL-SCALE INTERACTIONS 47 



Here s, the amount oiS per unit mass of air, becomes u, the mean wind speed, ^ 

 when the vertical transfer of horizontal momentum is under consideration and 

 q, the specific humidity, for water-vapour transfer. For heat transfer the 

 corresponding quantity is the product of specific heat, Cp, and temperature, 

 but the appropriate vertical gradient must take account of the effect of the 

 variation of pressure with height and the adiabatic heating or cooling ex- 

 perienced by a volume of air on changing its level. So the heat flux, H, is 

 related to vertical gradient of temperature, T, as follows : 



H = pCpKh{dTI8z + r), (3) 



where F is the dry adiabatic lapse rate of 0.98°C per 100 m. In the atmosphere 

 near sea -level, it is a close approximation to take the term in brackets to be the 

 vertical gradient of potential temperature, 6. 



The vertical gradient of potential temperature has a marked influence on 

 the intensity of turbulence and three main states may be distinguished. 



(i) dd/dz positive; stable conditions of stratification: a volume of air dis- 

 placed upwards becomes colder than its environment and tends to sink back 

 to its original level ; turbulence is diminished. 



(m) dd/dz zero; neutral or adiabatic conditions.^ 



{Hi) dd/dz negative; unstable stratification with consequent increased 

 turbulence. 



The coefficient Kg in equation (2) is not a constant as in molecular transfer ; 

 it depends on the turbulent motions in the air and these vary markedly with 

 height because, at the surface itself, the eddy motion must be greatly reduced. 

 It also varies with the stabihty of the stratification and with the roughness of 

 the surface, both factors influencing the intensity of the turbulence. 



As will be seen in due course, knowledge of Km, the transfer coefficient for 

 momentum, can be obtained from certain aerodynamic relationships, but there 

 is no corresponding way in which Kh and Ke, the transfer coefficients for heat 

 and water vapour, can be derived. It has therefore been customary to assume, 

 as the simplest hypothesis, that the K values for the various entities are equal. 

 However, momentum, unlike other properties of the air flow, is affected by 

 pressure forces so there is no a priori reason why Km should exactly equal the 

 other transfer coefficients. Also heat transfer is affected by buoyancy forces 

 because the turbulent temperature fluctuations are associated with density 

 fluctuations in an expansible fluid such as air. This leads to the expectation, as 

 shown by Priestley and Swinbank (1947), that Kh should differ from the 

 transfer coefficient of a passive entity, i.e. one having no effect on the structure 

 of turbulence. 



1 From this point onward the bar denoting a time mean will be omitted except in those 

 cases where confusion might result. 



2 Taking account also of the buoyancy effects associated with water vapour stratifica- 

 tion, the condition for neutrality becomes that the lapse-rate of virtual temperature shall 

 be equal to F. This makes a significant difference when the Bowen ratio (see p. 83) is 

 small. 



