Geoscience Reference
In-Depth Information
Table 7.2
Relative abundances of isotopes and crustal heat generation in the
past relative to the present
Relative abundance
Heat generation
238
U
235
U
a
Model A
b
Model B
c
Age (Ma)
Th
K
Present
1.00
1.00
1.00
1.00
1.00
1.00
1.00
500
1.08
1.62
1.10
1.03
1.31
1.13
1.17
1000
1.17
2.64
1.23
1.05
1.70
1.28
1.37
1500
1.26
4.30
1.39
1.08
2.22
1.48
1.64
2000
1.36
6.99
1.59
1.10
2.91
1.74
1.98
2500
1.47
11.4
1.88
1.13
3.79
2.08
2.43
3000
1.59
18.5
2.29
1.16
4.90
2.52
3.01
3500
1.71
29.9
2.88
1.19
6.42
3.13
3.81
a
238
U and 0.7114%
This assumes a present-day isotopic composition of 99.2886%
235
U.
b
20 000.
c
Model B, based on Th
/
U = 4 and K
/
U = 40 000.
Source
: Jessop and Lewis (1978).
Model A, based on Th
/
U
=
4 and K
/
U
=
or deposition and a constant heat flow, the column may eventually reach a state
of thermal equilibrium in which the temperature at any point is steady. In that
case, the temperature-depth profile is called an
equilibrium geotherm
.Inthis
equilibrium situation,
∂
T
/∂
t
=
0 and Eq. (7.16) applies:
∂
2
T
∂
z
2
A
k
=−
(7.20)
Since this is a second-order differential equation, it can be solved given two
boundary conditions. Assume that the surface is at
z
0 and that
z
increases
downwards. Let us consider two pairs of boundary conditions. One possible pair
is
=
(i) temperature
T
=
0at
z
=
0 and
(ii) surface heat flow
Q
=−
k
∂
T
/∂
z
=−
Q
0
at
z
=
0.
The surface heat flow
Q
Q
0
is negative because heat is assumed to be flowing
upwards out of the medium, which is in the negative
z
direction. Integrating
Eq. (7.20) once gives
=−
∂
T
∂
z
=−
Az
k
+
c
1
(7.21)
where
c
1
is the constant of integration. Because
∂
T
/∂
z
=
Q
0
/
k
at
z
=
0is
boundary condition (ii), the constant
c
1
is given by
Q
0
k
c
1
=
(7.22)
