Turbulent Transport in the Atmospheric Surface Layer - Transport in the Atmosphere-Vegetation-Soil Continuum - page 117

quantity. This is an important result for practical applications, which is the subject of
the next section.

Question 3.22: In the case of free-convection scaling and z -less scaling MOST dimen-
sionless gradients have speciic dependencies on z/L .

− 13

implies that scaling is independent of u * .

/


 

z
L


 

a) Show that φ h ~ −

z
L implies that scaling is independent of z .

b) Show that φ h ~

3.6 Practical Applications of Similarity Relationships
The theoretical (and empirical) framework has been developed. Now it can be used
to determine luxes from observations (or model values). First, the situation is dealt
with where values are available at two levels in the surface layer. Second, the lower
level will be lowered to the surface, leaving only one observation level in the surface
layer.
One remark has to be made here. In the previous sections, the effect of buoyancy
has been expressed in terms of the virtual potential temperature (e.g., θ v * , in the def-
inition of z/L ). For reasons of simplicity, however, in most of the following sections,
except 3.6.1, it is assumed that the moisture contribution to the buoyancy can be
neglected. This implies that, with respect to buoyancy, θ v * is equivalent to θ * and to
characterize the production of turbulence only the surface luxes of momentum and
heat are needed.

3.6.1 Fluxes from Observations at Two Levels
The expressions in Eq. ( 3.29 ) can be used to derive u * and θ * from observations
if we would know z/L . However, z/L in turn depends on u * and θ v * (thus on θ * and
q * ). He nc e, w e have a system of four equations (equations for the vertical differ-
ences ∆ u , ∆ θ and ∆ q , and the deinition of z/L ), and four unknowns ( u * , θ * , q *
and z/L ). This system can be solved iteratively. This iteration would involve the
following steps:

1. Compute initial values for u * , θ * and q * based on the observed ∆ u , ∆ θ and ∆ q , with
z/L = 0 (neutral conditions).
2. Compute L .
3. Compute Ψ m ( z u 1 /L ), Ψ m ( z u 2 /L ), Ψ h ( z θ 1 /L ) and Ψ h ( z θ 2 /L ) (the latter are also used for humid-
ity).
4. Compute new values for u * , θ * and q * from the observed ∆ u , ∆ θ and ∆ q and the values
of the Ψ-functions determined in the previous step.
5. Repeat steps 2 through 4, as long as computed values of u * and θ * change signiicantly
from one iteration to the next.