SECT. 2] SMALL-SCALE INTERACTIONS 51 



Both Rf and Ri increase numerically with height and roughly in direct 

 proportion. The effect of a heat flux on the flow is, therefore, small near the 

 surface but increases rapidly with height, and this has been confirmed observa- 

 tionally. This has the consequence that, to specify the overall stability condi- 

 tions on any given occasion, it is necessary to give the value of Rf or Ri at 

 some definite height. Difficulties can arise in comparison of results when 

 different reference heights are used and specification in terms of only one 

 quantity, independent of height, would be advantageous. With known values 

 of friction velocity and heat flux this can be done in terms of the stability 

 length, L, introduced by Obukhov (1946). Its definition is 



L = -pCpTu^^lkgH (15) 



and zjL is another form of flux Richardson number. For near neutral conditions, 

 for which Kh = Km is a good approximation, the three parameters Rf, Ri and 

 zjL are very nearly equal. i The choice of stability parameter in any given 

 problem is one of convenience; there is no real physical difference between 

 them. 



The generalization of equation (9) in terms of Ri is 



du U4. „ „ 



and/(i?^) must tend to unity as Ri—^0. In Fig. 2 observational evidence 

 as to the dependence on Ri of u^l{z dujdz), denoted by Km, is given for those 

 researches where the shearing stresses were measured by direct methods. The 

 eddy-correlation technique was used in one study (Swinbank, 1955; Deacon, 

 1959)2 and the measurement of the drag on suitably mounted samples of the 

 surface in the case of Rider's (1954) work. This shows that as yet the functional 

 form of the stability dependence in (16) is but poorly known and further work 

 of this sort is to be desired. 



Rossby and Montgomery (1935), by a generalization of Prandtl's mixing 

 length approach, arrived at 



I = 11 (!+.«)■/■ (17) 



for stable stratification, a being a constant to be determined by observation. 

 Various wind-profile studies have given values of o- ranging between 6 and 12 

 with an average of around 9. The dotted Curve in Fig. 2 shows that the Rossby- 

 Montgomery formula with cr = 9 has about the right slope to fit the near-neutral 



1 The relationship between them in the general case is 



Ri(KH/KM) = Rf ^ {KMlk){z/L), 

 where Km = u^I{z dujdz). 



2 The values of m* have been increased by 10% to correct for under -measurement caused 

 by high-frequency cut-off of the eddy spectrum. 



