Turbulent Transport in the Atmospheric Surface Layer - Transport in the Atmosphere-Vegetation-Soil Continuum - page 115

displacement thickness (based on mass conservation) as used in luid mechanics
(DeBruin and Moore, 1985 ).
A irst rough guess of the displacement height for a surface with roughness ele-
ments of height h c , is d = 2/3 h c . More details on the displacement height follow in
Section 3.6.2 , where it is treated together with the roughness length.

Question 3.21: Given observations of the mean wind speed at 2, 4 and 6 m height:
2.00, 2.64 and 2.97 m s -1 (assuming neutral conditions, see Eq. ( 3.23 )). The displace-
ment height for the surface under consideration is 0.5 m.
a) Compute the friction velocity from the wind speed observations at 2 and 4 meters,
and from those at 4 and 6 m height. Take into account the displacement height.
b) As under (a), but now ignore the displacement height (set it to zero).
c) For which of the height intervals (2-4 or 4-6 m) is the error in the friction velocity
as found under (b) largest? Explain your indings.

Limiting Cases of Stability
For neutral conditions the lux-gradient relationships for stable and unstable condi-
tions coincide: the dimensionless gradients are independent of buoyancy and equal
to 1. This is due to the fact that under neutral conditions the virtual heat lux vanishes
as a relevant scaling variable in the dimensional analysis. For extremely stable and
unstable conditions similar analyses can be made.
In the case of strongly unstable conditions (so-called free-convection) turbu-
lence is produced predominantly by buoyancy and surface shear no longer plays a
role. This implies that the friction velocity u * should vanish as a relevant variable
and different scaling variables need to be introduced, based on

g θ θ
v

′′ and z only:

13 /

f = 





g

u

θ θ
v

wz

′′

and, for example, Tw u
f

=′ θ / (compare to θ * ). The dimension-

  f

less gradients made dimensionless with these scales should become constant as there
is only one dimensionless group (see Appendix C ), that is ∂
∂

θκ
z

z
T f

. Then if we would

rewrite that again in terms of MOST variables (including u * ) one would obtain that

− 13 /

z
L


 


 

φ h ~ −

. Looking at Eq. ( 3.28 ) we see that the Businger-Dyer relationships

do not show free-convection scaling in the limit of large z/L . DeBruin ( 1999 ) shows
that for a limited stability range (down to z/L = -1) Eq. ( 3.28 ) could be replaced by
an expression that does exhibit free convection scaling. Whereas mean gradients do
not seem to follow free convection scaling, scalar standard deviations and structure
parameters do (see, e.g., DeBruin et al., 1993 ).