Geoscience Reference
In-Depth Information
Substituting Eq. (7.22) into Eq. (7.21) and then integrating the second time gives
A
2
k
z
2
Q
0
k
T
=−
+
z
+
c
2
(7.23)
where
c
2
is the constant of integration. However, since
T
0was specified
as boundary condition (i),
c
2
must equal zero. The temperature within the column
is therefore given by
=
0at
z
=
A
2
k
z
2
Q
0
k
T
=−
+
z
(7.24)
An alternative pair of boundary conditions could be
(i) temperature
T
=
0at
z
=
0 and
(ii) heat flow
Q
=−
Q
d
at
z
=
d
.
This could, for example, be used to estimate equilibrium crustal geotherms if
d
was the depth of the crust/mantle boundary and
Q
d
was the mantle heat flow into
the base of the crust. For these boundary conditions, integrating Eq. (7.20)gives,
as before,
∂
T
∂
A
k
z
=−
z
+
c
1
(7.25)
where
c
1
is the constant of integration. Because
∂
T
/∂
z
=
Q
d
/
k
at
z
=
d
is
boundary condition (ii),
c
1
is given by
Q
d
k
Ad
k
c
1
=
+
(7.26)
Substituting Eq. (7.26) into Eq. (7.25) and then integrating again gives
Q
d
+
Ad
k
A
2
k
z
2
T
=−
+
z
+
c
2
(7.27)
where
c
2
is the constant of integration. Because
T
0was boundary
condition (i),
c
2
must equal zero. The temperature in the column 0
=
0at
z
=
≤
z
≤
d
is
therefore given by
A
2
k
z
2
Q
d
+
Ad
T
=−
+
z
(7.28)
k
Comparison of the second term in Eq. (7.24) with that in Eq. (7.28) shows that
a column of material of thickness
d
and radioactive heat generation
A
makes a
contribution to the surface heat flow of
Ad
. Similarly, the mantle heat flow
Q
d
contributes
Q
d
z
/
k
to the temperature at depth
z
.
7.3.2 One-layer models
Figure 7.3 illustrates how the equilibrium geotherm for a model rock column
changes when the conductivity, radioactive heat generation and basal heat flow
