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between the maximum and the minimum value of the temperature in the given period
(see
Figure 2.23
).
If the time shift of the temperature wave between two depths is compared (e.g., com-
•
paring the time at which the temperature reaches its maximum),
D
can be derived from
P
z
D
z
D
tt
−=
π
d
2
−
d
1
.
21
2
Question 2.20:
Given soil temperature observations, related to the diurnal cycle. The
maximum soil temperature at the surface occurs at 13:00 local time, whereas the maxi-
mum at 20 cm depth occurs at 19:30. Assume that the soil is homogeneous.
a) Calculate the thermal diffusivity of this soil.
b) Calculate the amplitude of the soil temperature at 20 cm, relative to the amplitude at
the surface.
Soil Heat Flux
Because the soil heat lux depends only on the thermal conductivity and the tempera-
ture gradient (Eq. (
2.29
)), the model for the temperature proile (Eq. (
2.35
)) also gives
a model for the soil heat lux
3
:
ω
κ
λ
ω
z
D
π
−
zD
/
Gz t
(,)
=
Ae
( )
0
sin
t
−+
d
(2.38)
d
d
s
4
s
Equation (
2.38
) shows that the amplitude of the soil heat lux decreases with depth in
a similar way as the amplitude of the temperature wave. The result of this change of
G
with depth is that the temperature of the soil changes (lux divergence). The phase
shift, however, is slightly different than that for the temperature. Because
π
/4 cor-
responds to one-eighth of the period of oscillation
P
, it follows from Eq. (
2.38
) that,
at a given depth, the time of maximum heat lux precedes the time of maximum tem-
perature by 3 hours for the daily cycle and by one-and-a-half months for the annual
cycle. This seems counterintuitive. But note that we could have taken Eq. (
2.38
) as
the prescribed boundary condition. Then we would have considered the
soil heat lux
as forcing at the surface rather than the
surface temperature
(the irst would be more
natural in the context of the surface energy balance). In that case Eq. (
2.35
) would
have been found as the solution for
T
(
z
d
,
t
): the interpretation would have been that the
T
-waves reach their maximum
π
/4 rad later than the
G
-wave at a given depth, which
agrees with our intuition.
The change in amplitude of
G
, the phase shift with depth, and the phase shift
between
G
and
T
is illustrated in
Figure 2.24
with observations from a bare soil in
the Negev desert. One of the striking features is that the soil heat lux at the surfaces,
π
.
3
Recall that sin( ) os()
x
+
x
=
2
sin
x
+
4
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