Geoscience Reference
In-Depth Information
Table 3.4
The 24-hour average turbulent diffusion
coeficient for data shown in
Figure 3.3
Quantity
Value
θ
24
20.02ºC
(
2
m
)
θ
24
(
10
m
)
20.11ºC
H
24
43 W m
-2
24
H
ρ
c
24
-2.64 m
2
s
-2
K
=
( )
p
(?)
−
h
24
∆∆
θ
z
24
24
= ()
K
Kt
h
0.96 m
2
s
-2
h
The overbar with 24 to the right signiies a 24-hour mean.
•
The mean diffusivity is calculated as the mean of the diffusivities that have been calcu-
lated for individual time intervals (
K
h
(
t
)). This yields a positive diffusivity, as the diffu-
sivities of all individual intervals were positive as well (for all half-hour intervals the lux
does low down the gradient).
The Schmidt paradox can be explained physically as follows. During daytime turbu-
lence is strong owing to the combination of shear production and buoyancy produc-
tion. Hence the turbulent diffusivity is large and only a small vertical temperature
gradient is needed to transport the heat lux imposed by the surface energy balance.
In contrast, at night the turbulence is suppressed by buoyancy (stable stratiication).
Although the lux to be transported is much smaller than during daytime, the required
temperature gradient is much larger (and of opposite sign) than the gradient during
daytime. To summarize: the mean sensible heat lux is dominated by daytime condi-
tions, whereas the mean temperature gradient is dominated by night time conditions.
Mathematically, the Schmidt paradox is related to the order in which averaging
and multiplication are performed. This can be illustrated by decomposing both the
diffusion coeficient and the gradient in their 24-hour mean values (denoted by the
overbar) and a deviation from that mean (similar to the Reynolds decomposition, here
indicated by a double prime):
24
″
Kt KKt
()
=
+
()
h
h
h
θ θ
24
(3.53)
″
∂
∂
θ
=
∂
∂
∂
∂
()
t
+
()
t
z
z
z
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