Geoscience Reference
In-Depth Information
2.
Heat generation is an exponentially decreasing function of depth within a
slab of thickness z*
. Equation (7.16) then becomes
∂
2
T
∂
z
2
A
(
z
)
k
=−
(7.79)
where
A
0
e
−
z
/
D
z
∗
A
(
z
)
=
for 0
≤
z
≤
Integrating Eq. (7.79) once gives
∂
T
∂
z
=
A
0
k
D
e
−
z
/
D
+
c
(7.80)
where
c
is the constant of integration. At the surface,
z
=
0, the heat flow is
Q
(0)
Q
0
=
k
A
0
D
k
+
c
Q
(0)
=
=
A
0
D
+
kc
(7.81)
The constant
c
is given by
Q
0
−
A
0
D
k
c
=
(7.82)
At depth
z
*(which need not be uniform throughout the heat-flow province), the
heat flow is
Q
(
z
∗
)
=
k
A
0
D
k
Q
0
−
A
0
D
k
e
−
z
∗
/
D
+
A
0
D
e
−
z
∗
/
D
=
+
Q
0
−
A
0
D
(7.83)
Thus, by rearranging, we obtain
Q
0
=
Q
(
z
∗
)
+
A
0
D
−
A
0
D
e
−
z
∗
/
D
(7.84)
Equation (7.84)isthe same as Eq. (7.75)ifwewrite
Q
(
z
∗
)
−
A
0
D
e
−
z
∗
/
D
Q
r
=
Q
(
z
∗
)
−
A
(
z
∗
)
D
=
(7.85)
Thus, the linear relation is valid for this model if the heat generation
A
(
z
*) at
depth
z
*isconstant throughout the heat-flow province. Unless
A
(
z
*)
D
is small,
the observed value of
Q
r
may be very different from the actual heat flow
Q
(
z
*)
into the base of the layer of thickness
z
*. However, it can be shown (for details
see Lachenbruch (1970) that, for some heat-flow provinces,
A
(
z
*)
D
is small,
and thus
Q
r
is a reasonable estimate of
Q
(
z
*). This removes the constraint that
A
(
z
*) must be the same throughout the heat-flow province. Additionally, for those
provinces in which
A
(
z
*)
D
is small, it can be shown that
z
* must be substantially
greater than
D
. Thus, the exponential distribution of heat production satisfies the
observed linear relationship between surface heat flow and heat generation and
does so even in cases of differential erosion. In this model,
D
is a measure of
the upward migration of the heat-producing radioactive isotopes (which can be
justified on geochemical grounds), and
Q
r
is approximately the heat flow into
