Turbulent Transport in the Atmospheric Surface Layer - Transport in the Atmosphere-Vegetation-Soil Continuum - page 126

accurate, especially in view of the accuracy of the empirical lux-gradient relation-
ships on which they are based. The solution for unstable conditions is:

2

u

H
c

κθ ∆∆

3
4

( )

≈− 116 − Ri

b *

ρ

2

 


 

z
z


 

(3.47)

p

ln

2

 

 

1

κ ∆ u

z
z

1
4

( )

u ≈ 116 − Ri

* b *


 


 


 


 

(3.48)

2

ln

1

where ∆ θθ θ , Ri b* is an 'effective' bulk-Richardson number, in this case

= () z − () z

2 1

given by:

 




z
z




g
T

∆
∆

θ

(3.49)

Ri = zz ln

2

   

 

b* 12

2

() u

 

1

For stable conditions, the solution is even exact, due to the particular form (simply
linear) of the lux-gradient relationships. Starting from Eq. ( 3.29 ) and using the rela-
tionship between z/L and the bulk Richardson number (Eq. 3.37 ) , the following for-
mulations for the luxes can be derived (Launiainen, 1995 ; Basu et al., 2008 ):

2

u

H
c

κθ ∆∆

− ( )

2

=− 15 Ri if 0 < Ri < 0 . 2

b b

2

ρ

 


 


 

(3.50)

z
z

p

ln

2

 

 

1

κ ∆ u

z
z

− ( ) u = 15 Ri if 0 < Ri < 02 .

* b b

 


 


 

(3.51)

ln

2

 

 

1

where

zz g
T

θ
()

∆
∆

b = ( )

(3.52)

Ri

2 1

u

2

which is a standard formulation for the bulk Richardson number. It should be noted
that these results for the stable case have a singularity at Ri b = 0.2 (the critical Rich-
ardson number). At that stability the luxes vanish because all turbulence has been
suppressed by buoyancy. For values of Ri b > 0.2 the expressions in Eqs. ( 3.50 ) and
( 3.52 ) are no longer valid: the luxes are simply equal to zero.