Geoscience Reference
In-Depth Information
After this initial subsidence, the lithosphere gradually cools (Fig. 10.44), which
results in a period of slower thermal subsidence. This can be modelled mathe-
matically by using the one-dimensional heat-flow equation (Eq. (7.13))
∂
T
∂
t
=
2
T
∂
z
2
with a constant temperature in the asthenosphere (
T
a
) and temperature increasing
linearly with depth
z
through the lithosphere,
k
ρ
c
P
∂
$
T
a
β
z
h
l
h
l
β
T
=
for
0
≤
z
≤
at
t
=
0
h
l
β
%
T
=
T
a
for
≤
z
≤
h
l
Problems such as this are best solved by Fourier expansion; for details of these
methods readers are referred to Carslaw and Jaeger (1959). To a first approxima-
tion, the thermal subsidence is an exponential with a time constant equal to the
time constant of the oceanic lithosphere (e.g., Eqs. (10.5) and (10.6)):
S
t
=
E
0
r
(1
−
e
−
t
/τ
)
(10.10)
where
4
h
l
ρ
m
0
α
T
a
E
0
=
2
(
ρ
m
0
−
ρ
w
)
is a constant that depends on initial values and
π
sin
π
β
β
π
r
=
depends on the stretching factor
h
l
π
τ
=
2
κ
is called the relaxation time.
Equation (10.10) can be used to provide a value of the stretching factor
β
from
e
−
t
/τ
is a straight line through the origin with slope
E
0
r
. So, by making reasonable
assumptions for
h
l
,
the variation of thermal subsidence
S
t
with time
t
.Aplot of
S
t
against 1
−
can be calculated.
The
total amount of subaqueous subsidence S
occurring after an infinite time
can be most simply expressed by assuming Airy-type isostasy:
ρ
m
0
,
α
and
T
a
,
β
h
l
(
ρ
l
−
ρ
l
)
+
h
c
−
ρ
c
ρ
l
−
ρ
l
β
+
ρ
c
β
S
=
(10.11a)
ρ
a
−
ρ
w
which can be rearranged to give
(
ρ
m
0
−
ρ
c
0
)
1
−
α
T
a
h
c
2
h
l
1
−
1
β
S
=
h
c
(10.11b)
ρ
a
−
ρ
w
