Turbulent Transport in the Atmospheric Surface Layer - Transport in the Atmosphere-Vegetation-Soil Continuum - page 111

Again, the expressions for the aerodynamic resistances follow directly from the com-
bination of Eqs. ( 3.24 ) and ( 3.29 ) and are a mere mathematical consequence.

3.5.6 Similarity Theory: Final Remarks
Other Scalars and Other Turbulence Statistics
In the previous sections we introduced a framework that enables the description of
the link between vertical gradients (or vertical differences) and luxes. The equations
given were only for momentum and heat, but given the fact that the lux-gradient
relationships presented in Section 3.5.3 are identical for all scalars (heat, humidity
and an arbitrary scalar x ), the results for heat can also be applied to other scalars.
Furthermore, MOST is applied not only to gradients of mean quantities. Other tur-
bulence statistics used in MOST are, for example, the standard deviation of scalars
and velocity components, the structure parameters of temperature and humidity (see,
e.g., DeBruin et al., 1993 ; Li et al., 2012 ) and the dissipation rate of TKE (e.g., Har-
togensis and DeBruin, 2005 ).

Stability Parameters
In Section 3.3.5 the (lux-) Richardson number was introduced as an indicator of the
stability regime in the surface layer and in 3.5.2 it was noted that Ri f and z/L contain
equivalent information, but are not necessarily identical.
Now that we have expressions that link luxes to gradients, we can investigate the
relationship between Ri f and z/L more precisely. Using the deinitions of the lux-gra-
dient relationships, and replacing local luxes by surface luxes, Ri f can be written as:

g g

θ θ

u κ θ θ z

** v v *

z
L

1

Ri =

v

=

v

=

(3.32) f

u

z
L

z
L

z



  



  



  

2 2

u

κ φ

*

u φ φ

     

* m * m m

z L

/ ( ) is close to 1) the lux Richardson number and z/L
are identical, but for more stable or unstable conditions this identity does not hold.
Because in some applications (e.g., in atmospheric models), luxes are not readily
available, two other variants of the Richardson number are also often used, viz. the
gradient Richardson number Ri g (where the luxes are assumed to be proportional to
the gradient of the transported quantity) and the bulk Richardson number Ri b , where
the gradients in Ri g have been replaced by differences:

Thus, for small z/L (where φ m

zL

∂
∂
∂
∂

θ

v

g g ∆∆

∆

θ z

z
u
z

v

(3.33)

Ri ≡ , Ri ≡

g b

2

()

2

θ θ

u


 


 

v v