Geoscience Reference
In-Depth Information
Figure 2.25 Force-restore method: soil column divided in a top soil and ininite
reservoir (left); energy balance of the top layer (right).
tions. For that reason often an approximation is used. An example is the so-called force-
restore method (Bhumralkar, 1975 ). This method approximates the soil by two layers:
one top layer with temperature T top and a thickness d top , and one ininite layer with a
constant temperature T bot (see Figure 2.25 ). The thickness of the top layer is yet unde-
ined. The time rate of change of the temperature of the top layer is given by:
T
1
(
)
top
=
GG
(2.39)
0
bot
t
Cd
s op
which is a vertically integrated version of Eq. ( 2.30 ). Next we replace G 0 by the sum
of the other terms of the energy balance and G bot is taken proportional to the temper-
ature difference between the top layer and the bottom layer and to an integrated con-
ductivity Λ s (to be determined later):
T

(2.40)
1
Λ
(
TT
) }
{(
QHLE
*
−− −
)
top
=




s

top
bot
t
Cd
fo
rce
restor
e
s op
The irst term forces the top layer temperature away from its equilibrium T bot , whereas
the second term tends to restore the temperature back to T bot .
Now the layer thickness d top and the conductivity Λ s need to be determined, subject
to the following constraints:
The temperature
T top should have the same amplitude as the surface temperature T (0, t ).
The temperature
T top should have the same phase as the surface temperature T (0, t ).
These constraints are particularly important to ensure that the sensible heat lux and
the upwelling longwave radiation, which both depend on the surface temperature, are
correct.
The constraints result in the solution that Λ s
s op
= ω and d top
= κω
/2
(note that
s
Cd
d top is proportional to the damping depth, see Eq. ( 2.37 )). Thus both the layer thickness
and the proportionality constant in the restore term depend not only on the soil ther-
mal properties, but also on the frequency of the forcing ( ω = 2 π / P ).
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