50 DEACON AND WEBB [CHAP. 3 



Equation (9) applies to fiow over both rough and polished surfaces provided 

 z is sufficiently large for the flow at that level to be fully turbulent and un- 

 affected by the more complex conditions existing very close to the surface. 

 Integration of (9) gives for the layer of constant stress the logarithmic law 



u = (w^/A;) ln(2/c) (10) 



in which c is the integration constant. An aerodynamically smooth surface is 

 one for which the roughness elements are small compared with the thickness 

 of the laminar layer so the only drag is that due to skin friction. In the laminar 

 layer of molecular transfer the velocity profile is linear as far as about z = 5vju^, 

 while between 5 and 30 multiples of vju^ conditions are transitional between 

 laminar and fully turbulent flow. In the fully turbulent regime the value of c is 

 such that (10) becomes 



u — {u^jk) \n{Au^zlv) (11) 



and with A; = 0.41 the appropriate value of -4 is 7.5 according to Clauser (1956). 



For rough surfaces the viscous drag at the surface is negligible in comparison 



with the form drag of the roughness elements and (10) is then found to become 



u = {u^lk) ln(2/2o), (12) 



in which zq, the roughness parameter, is a constant for a given uniformly 

 roughened surface under neutral stability conditions. Typical values of zq for 

 land surfaces range mainly between 0.1 cm for fairly smooth snow to 100 cm 

 for scrub -covered country. 



B. The Non-Neutral Profile 



When there exists a temperature contrast between surface and air, the 

 effects of thermal stratification and buoyancy have also to be considered, so 

 that in relating the shear stress to the wind shear the counterpart of (9) must 

 contain at least one more basic parameter. This involves consideration of the 

 effects of a heat flux, H, on the production of turbulent kinetic energy. Produc- 

 tion by the shear stress is at the rate t dujdz per unit volume. The working of 

 the buoyancy forces associated with a heat flux introduces an additional term 

 {gH)ICpT, where g is the acceleration of gravity and T air temperature : this 

 augments the energy supply to the turbulence under unstable conditions but 

 decreases it with stable stratification. The dimensionless ratio of these terms is 

 the flux form of the Richardson number, viz. 



Bf = -gHlicpTrduldz) (13) 



in which the minus sign is a matter of convenience to make the signs of Bf 

 and of the vertical gradient of potential temperature the same under given 

 conditions. Introducing proportionality of fluxes to gradients in Bf leads to 

 the more familiar gradient form of the Richardson number 



_ g dd/dz 

 ^' ~ Tiduldz)^ ^ ^ 



in which 6 is potential temperature. 



