Geoscience Reference
In-Depth Information
Air
Figure 6.3
Energy budget
for a thin horizontal element
of soil of thickness
б
z
, cross
sectional area
A
, and
temperature
T
soil
located at a
depth
z
beneath the soil
surface, with soil heat flux
G
z
entering from above and
G
z+бz
leaving from below.
Soil surface
z+
δ
z
z
G
z
Area =
A
Soil
Temperature =
T
soil
G
z
+
d
z
Hence:
T
G
∂
1
∂
soil
=−
z
(6.8)
∂
t
C
∂
z
s
Substituting
G
from Equation (6.4) into this last equation gives:
∂
T
1
∂
∂
T
⎡
⎤
soil
=
a
C
soil
(6.9)
⎢
⎥
ss
∂
t Cz
∂
∂
z
⎣
⎦
s
In the particular case of heat flow in homogeneous soil for which both
C
s
and
a
s
are constant with depth below ground, Equation (6.9) simplifies to:
2
∂
T
∂
T
(6.10)
soil
=
a
soil
s
2
∂
t
∂
z
Thermal waves in homogeneous soil
Equation (6.9) can be solved numerically to provide a general description of the
evolution of soil temperature and therefore (from Equation (6.4)) soil heat flux in
response to a prescribed time series of soil surface temperature when the depth
dependency of
C
s
and
a
s
is defined. However, much can be learned about the
mechanics of soil heat flow by investigating the analytic solution of Equation (6.10)
which applies for homogeneous soil. It is also instructive to consider the case of
a simple sinusoidal variation in soil surface temperature,
t
soil
,0
T
, described by the
expression:
⎡
(
tt
−
)
⎤
TTT
t
soil
,0
=+
sin 2
p
0
(6.11)
⎢
⎥
m
a
P
⎣
⎦
where
T
m
is the mean temperature of the soil surface,
T
a
is the amplitude of the
sinusoidal variation in soil surface temperature,
t
is time in seconds, and
P
and
t
0
