Turbulent Transport in the Atmospheric Surface Layer - Transport in the Atmosphere-Vegetation-Soil Continuum - page 112

The relationship between Ri f and Ri g can be derived easily using the lux-gradient
relationships:



  

z
L
z
L

φ

 

K
K

m

Ri =

h

Ri = Ri

(3.34) f g g



  

φ

m

 

h

Thus the relationship between Ri f and Ri g does depend on stability (as does the rela-
tionship between Ri f and z/L ). If we combine Eqs. ( 3.32 ) and ( 3.34 ), we ind that

z
L
z
L

z
L
z
L



  



  

φ φ

   

h h

z
L

Ri = Ri =

(3.35) g f

2



  


 


 



  

φ

 

φ

 

m

m

Using the empirical expressions for the lux-gradient relationships (Eqs. ( 3.21 ) and
( 3.34 )), we ind that:

z
L

z
L

= Ri , for < 0

g

(3.36)

Ri

z
L

z
L

g

= − , for > 0

15 Ri

g

An important consequence of the stable side of Eq. ( 3.36 ) is that the expression for
z/L has a singularity at Ri g = 0.2. For that value z/L tends to ininity and the condi-
tions become so stable that all turbulence is suppressed. The value of Ri g where this
happens is called the critical Richardson number, Ri gc . The value of 0.2 is a direct
consequence of Eq. ( 3.36 ), but the exact value of Ri gc is still a subject of debate (see,
e.g., Zilitinkevich et al., 2007 ). For stable stratiication a relationship similar to Eq.
( 3.36 ) can be derived for the bulk Richardson number (DeBruin, 1982 ; Launiainen,
1995 ; Basu et al., 2008 ):

z − z

= 

z
z


  −

Ri

z
L

2 1

ln

2 b

, for > 0

(3.37)

 

L 15 Ri

1 b

Question 3.20: Compute the two production terms (shear and buoyancy production)
in the TKE equation (Eq. ( 3.10 )) as well as their ratio ( Ri f ) for the following condi-
tions (use the lux-gradient relationship to determine the wind speed gradient, and use
a height of 10 m).
a) H = 140 W m -2 , L v E = 250 W m -2 , u * = 0.3 m s -1 .
b) H = -50 W m -2 , L v E = 5 W m -2 , u * = 0.2 m s -1 .