Geoscience Reference
In-Depth Information
Now impose the boundary conditions for a
hollow sphere b
<
r
<
a
:
(i) zero temperature
T
=
0atthe surface
r
=
a
and
(ii) constant heat flow
Q
=−
k
∂
T
/∂
r
=
Q
b
at
r
=
b
.
The temperature in the spherical shell
b
<
r
<
a
is then given by
1
a
−
Q
b
b
2
k
1
r
T
=−
(7.52)
An expression such as this could be used to estimate a steady temperature
for the lithosphere. However, since the thickness of the lithosphere is very small
compared with the radius of the Earth, (
a
1, this solution is the same
as the solution to the one-dimensional equation (7.28) with
A
−
b
)
/
a
0.
There is no non-zero solution to Eq. (7.49) for the whole sphere, which has a
finite temperature at the origin (
r
=
0). However, there is a steady-state solution
to Eq. (7.48) with constant internal heat generation
A
within the sphere:
=
r
2
∂
T
∂
r
r
2
∂
k
0
=
+
A
∂
r
r
2
∂
T
∂
r
Ar
2
k
∂
∂
r
=−
(7.53)
On integrating twice, the temperature is given by
Ar
2
6
k
−
c
1
r
+
c
2
T
=−
(7.54)
where
c
1
and
c
2
are the constants of integration.
Let us impose the following two boundary conditions:
(i)
T
finite at
r
=
0 and
(ii)
T
=
0at
r
=
a
.
Then Eq. (7.54) becomes
A
6
k
(
a
2
−
r
2
)
T
=
(7.55)
and the heat flux is given by
−
k
dT
Ar
3
dr
=
(7.56)
=
/
The surface heat flow (at
r
3.
If we model the Earth as a solid sphere with constant thermal properties and
uniform heat generation, Eqs. (7.55) and (7.56) yield the temperature at the centre
of this model Earth, given a value for the surface heat flow. Assuming values of
surface heat flow 80
a
)istherefore equal to
Aa
4Wm
−
1
◦
C
−
1
,we
obtain a temperature at the centre of this model solid Earth of
10
−
3
Wm
−
2
,
a
×
=
6370 km and
k
=
80
×
10
−
3
×
6370
×
10
3
2
×
4
T
=
=
63 700
◦
C
