(IV)
/ p defined under conditions of neutrality or thermal instability, as the
difference between the remaining terms.
(V)
/ M ¼ kðz À dÞ=u Ã
ð
Þ ð @u=@zÞ
ð 3:95Þ
(VI)
u e ¼ k z À d
ð
Þe=u
3
Ã
À
Á
ð3:96Þ
Figure 3.2 illustrates the variation of dimensionless factors with the ratio z/L,
when d is zero.
Each of the dimensionless terms of Eq. (3.91) corresponding to known dimensionless parameters, follow the Monin–Obukhov similarity theory, discussed later.
The corresponding empirical functions for dependence on thermal stability were
obtained from field data (experiments in Kansas, e.g., Businger et al. 1971).
These functions are as follows (Kaimal and Finnigan 1994):
/ M ¼ 1 þ 16 z À d
j
j=L
ð
Þ
ðÀ1=4Þ
n 0
1 þ 5ðz À dÞ=L
ð
Þ
n [ 0
ð3:97Þ
/ t ¼
Àn n 0
0
n [ 0
&
ð3:98Þ
/ e ¼ ð1 þ 0.5 n
j j
ð2=3Þ
Þ
ð3=2Þ
n 0
1 þ 0.5 n
j j
n [ 0
&
ð3:99Þ
In the surface boundary layer under conditions of instability the / t term is
positive, reflecting the upward transport of the turbulent kinetic energy production
Fig. 3.2 Similarity functions
in the surface layer (u H ;
dimensionless factor for the
sensible heat flux) (after
Kaimal and Finnigan 1994)
58
3 Characterization of Turbulent Flow in the Surface Boundary Layer