Geoscience Reference
In-Depth Information
The soil heat flux can be calculated at any depth in the soil using Equation (6.2)
or (6.4). However, it is the soil heat flux into the soil surface that is most often
required for use in calculating the energy budget of a sample volume lying between
ground level and a reference level in the atmosphere above. By differentiating
Equation (6.12) and substituting
z
= 0, it can be shown that for a homogeneous soil
and assuming the simple sinusoidal variation in surface temperature in Equation
(6.11), the surface soil heat flux is given by:
2
Tk
⎡
(
t t
−
)
2
p
⎤
G
as
0
=
sin 2
p
+
(6.14)
⎢
⎥
z
=
0
D
P
8
⎣
⎦
Comparing Equation (6.14) with Equation (6.11) illustrates a general feature of
the relationship between soil heat flux and soil surface temperature. Specifically,
the sinusoidal wave in soil heat flux is advanced by one eighth of a cycle with
respect to the wave in surface temperature. This means that the peak soil heat
flow is approximately 3 hours earlier than the peak soil surface temperature
for the daily cycle, and the peak in soil heat flow is 1.5 months earlier than the
peak soil surface temperature for the yearly cycle. Physically this is because
there is most conduction of heat into the soil when the soil surface temperature
is rising rapidly in the morning for the daily cycle and in spring for the yearly
cycle, and most conduction of heat out of the soil when the soil surface
temperature is falling quickly in the evening for the daily cycle and in autumn
for the yearly cycle.
Equation (6.14) also shows that the magnitude of the wave in surface soil heat
flux is inversely proportional to the damping depth and therefore inversely
proportional to the square root of the period of the surface temperature wave,
see Equation (6.13). This means the amplitude of the soil heat flux wave associated
with the yearly cycle is (365)
0.5
times less (i.e., about 19 times less) than the
amplitude of the daily cycle. The yearly cycle in soil heat flux is therefore about 20
times less than the daily wave but penetrates about 20 times deeper.
In fact the value of
damping depth
determines many interesting features of soil
heat flow as follows.
The amplitude of temperature wave falls to
e
−1
(~0.37) of its surface value at
depth
D
●
The velocity with which temperature maximum and minimum appear to
propagate downward through the soil is given by (2
●
π
D/P
)
The temperature wave is
π
radians (180°) out of phase with the surface wave
●
at depth (
π
D
)
From Equation (6.14), the maximum surface heat flux is (√2.
T
a
.k
s
)/
D
which is the heat flow that would be maintained through a thickness
(
●
2.
D
) of soil if one side were maintained at (
T
m
+
T
a
) and the other at
(
T
m
-
T
a
). For this reason (
√
√
2.
D
) has been called the
effective depth
of soil
heat flow
