Geology Reference
In-Depth Information
Fig. 12.11
Left
: Oceanic geotherm of the HSC model
(
solid line
) and of the PCM (
dashed line
) for a 100 Ma
old lithosphere.
Right
: Heat flux as a function of the age
of the ocean floor in the two models. The maximum plate
thickness is assumed to be
h
D
105 km (McKenzie et al.
2005
;Afonsoetal.
2007
)
as coincident with the effective thickness of the
oldest lithosphere. If
T
a
is the temperature of the
upper asthenosphere, then the boundary condi-
tions can be written as follows:
Conversely, for ›
t
<<
h
2
the solution (
12.48
)
assumes the form:
T.
z
;t/
Š
T
0
C
.T
a
T
0
/
"
z
h
C
#
n
sin
n
z
X
1
2
1
lim
z
!h
T.
z
;t/
D
T
a
I
I
for t
!
0
h
nD1
T.
z
;t/
D
T
0
I
for any t>0 (12.46)
(12.50)
lim
z
!0
It is possible to show that the solution for
t
!
0
gives geotherms that do not differ significantly
from those of the HSC model.
Regarding the surface heat flux, it can be
obtained by (
12.48
) applying Fourier's law for
z
D
0:
T.
z
;0/
D
T
a
I
for any 0
z
h (12.47)
Therefore, the plate is assumed to have uni-
form temperature
T
a
at
t
D
0, and at any suc-
cessive time the upper and lower boundaries are
maintained at temperatures
T
0
and
T
a
respec-
tively. The solution for
T
(Carslaw and Jaeger
1959
) has the form of an infinite series:
"
1
C
2
X
nD1
#
exp
›n
2
2
t
h
2
k.T
a
T
0
/
h
q
0
.t/
D
(12.51)
T.
z
;t/
D
T
0
C
.T
a
T
0
/
"
z
h
C
#
n
exp
sin
n
z
h
Also in this case for large times, such that
›
t
>>
h
2
, a steady-state value is attained. This is
given by:
X
1
›n
2
2
t
h
2
2
1
n
D
1
(12.48)
k.T
a
T
0
/
h
q
0
.t/
as t
!1
(12.52)
We note that for ›
t
>>
h
2
the series in (
12.48
)
tends to zero, thereby at large times a linear
steady-state geotherm is attained:
which is significantly different from (
12.44
).
A comparison of geotherm and heat flux pre-
dicted by the HSC model and the corresponding
curves in the PCM is shown in Fig.
12.11
.As
required by (
12.49
), the PCM geotherm of old
T.
z
;t/
T
0
C
.T
a
T
0
/
z
h
as t
!1
(12.49)
