Geoscience Reference
In-Depth Information
If we compare this result to Eq. (
3.1
) we can conclude that for
neutral
conditions
KK uz
m
=
κ
*
. Physically this makes sense, because the turbulent diffusivity
depends on the intensity of the turbulence (represented by
u
*
because buoyancy does
not play a role under neutral conditions) and the diffusivity increases with height due
to the fact that the transporting eddies are larger higher above the surface. These two
effects are in accordance with the tendencies observed in
Figure 3.3
. The diffusivities
for other variables will be identical to that of heat, because for neutral conditions the
φ
-functions of all variables are identical.
Often, information on vertical gradients is not available (e.g., in a weather predic-
tion model where variables are known only at discrete levels) or is hard to determine
(measurements at a inite vertical distance can only approximate a gradient). There-
fore, the integrated version of Eq. (
3.20
) is also very useful. If we still restrict our-
selves to the neutral case (
z/L =
0), we obtain (for wind and potential temperature):
z
z
z
z
∂
∂
u
zu
κ
z
u
1
u
2
u
2
u
2
u
2
∫
∫
∫
∫
d
z
= ⇔ =
d
z
d
u
*
d
z
⇔
κ
z
*
z
z
z
z
u
1
u
1
u
1
u
1
z
z
u
u
u
u
2
() ()
z
−
z
=
*
ln
u
2
u
1
κ
u
1
(3.23)
z
z
z
z
∂
∂
θ
κ
z
θ
θ
1
θ
2
θ
2
θ
2
θ
2
∫
∫
∫
=
∫
d
z
= ⇔
d
z
d
*
d
z
⇔
z
θ
κ
z
z
*
z
z
z
θ
1
θ
1
θ
1
θ
1
z
z
θ
θ
() ()
z
−
θ
z
=
*
ln
θ
2
θ
2
θ
1
κ
θ
1
where
z
u
1
and
z
u
2
are the heights where the wind speed is known and the tempera-
ture is known at
z
θ
1
and
z
θ
2
. This logarithmic proile is a cornerstone of surface layer
meteorology and represents the shape of wind speed and scalar proiles under neutral
conditions.
To show how the lux depends on the observations of wind or temperature at two
levels, Eq. (
3.23
) can be rewritten using the concept of a resistance:
u
z
u
z
() ()
−
ρ
θ
() ()
z
−
θ
z
τρ
=
u
2
u
1
,
H
=−
c
θ
2
θ
1
(3.24)
r
p
r
am
ah
where
r
am
and
r
ah
are the aerodynamic resistances for momentum and heat trans-
port, respectively (which have units of s m
-1
). In some applications aerodynamic
conductance is used, which is the reciprocal of the resistance (with units m s
-1
):
g
=1/ .
In Eq. (
3.24
) the luxes are proportional to the vertical differences of the transported
quantity and inversely proportional to the aerodynamic resistance. This is similar to Ohm's
law, where the current is proportional to the potential difference and inversely proportional
r
a
a
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