Geoscience Reference
In-Depth Information
Thermal diffusivity,
a
s
When describing soil heat flow, it is useful to define a renormalized form of the
thermal conductivity called the thermal diffusivity,
a
s,
which is defined by the
equation:
k
C
a
=
s
(6.3)
s
s
The soil moisture dependency of
a
s,
is less dramatic than for
k
s
, see Table 6.1.
When the diffusion equation describing soil heat flow (Equation (6.2)) is
rewritten in terms of thermal diffusivity rather than thermal conductivity it
becomes:
∂
T
G
=−
a
C
soil
(6.4)
z
s
s
∂
z
Formal description of soil heat flow
Figure 6.3 illustrates the energy budget for a thin horizontal element of soil of
thickness
dz
and cross sectional area
A
located at a depth
z
beneath the soil surface.
The soil heat flux into the element from above is
G
z
and that out from below
G
z+dz
.
Consequently, over a period of time
dt
, the element receives a net input of soil heat
flux [
A.
(
G
z
G
z+dz
).
dt
]. Over this same period of time, the temperature of the soil
element rises by
dT
soil
. This takes an amount of heat equal to [C
s
.A.dz.dT
soil
], and
energy conservation requires that:
−
CAzT AG G t
+
dd
=
(
−
)
d
(6.5)
s
soil
z
z
d
z
Hence:
d
T
C
soil
d
z
=−
d
(
G
G
)
(6.6)
s
z
z
+
z
d
t
Expanding the right hand side of Equation (6.6) using Taylor's theorem in the limit
of small
dz
gives:
d
T
⎛
∂
∂
G
⎞
C
soil
d
z G
=− +
G
z
.
d
z
(6.7)
⎜
⎟
s
z
z
d
t
⎝
z
⎠
