Geoscience Reference
In-Depth Information
we obtain the equation
D
d
r
ρ
a
d
z
d
0
ρ
w
d
z
d
r
d
+
L
D
ρ
w
d
z
+
=
+
ρ
(
z
)d
z
+
L
ρ
a
d
z
(7.66)
0
d
d
+
Rearranging yields
L
(
d
−
d
r
)(
ρ
a
−
ρ
w
)
=
(
ρ
(
z
)
−
ρ
a
)d
z
(7.67)
0
To determine
ρ
(
z
), we must use the expression for density as a function of
temperature and
α
the coefficient of thermal expansion,
ρ
(
T
)
=
ρ
a
[1
−
α
(
T
−
T
a
)]
(7.68)
and Eq. (7.61) for the temperature structure of the lithosphere. Substituting these
two equations into Eq. (7.67)gives
(
d
−
d
r
)(
ρ
a
−
ρ
w
)
=
ρ
a
α
T
a
erfc
d
z
L
z
2
√
κ
t
(7.69)
0
where erfc is the complementary error function (see Appendix 5). The integral
on the right-hand side of this equation can easily be calculated or looked up in
a set of mathematical tables. However, for our purposes it is sufficient to change
the upper limit of integration from
L
to
(the error introduced by this approx
i-
mation is about 5%). This integral of erfc(
x
) between
x
∞
/
√
π
=
0 and infinity is 1
(Appendix 5). When this approximation is made, Eq
. (7
.69) becomes
T
a
κ
t
π
(
d
−
d
r
)(
ρ
a
−
ρ
w
)
=
2
ρ
a
α
(7.70)
Rearranging Eq. (7.70)gives
κ
t
π
2
ρ
a
α
T
a
ρ
a
−
ρ
w
d
=
d
r
+
(7.71)
10
3
10
3
kg m
−
3
,
If we assume values for
ρ
a
and for
ρ
w
of 3.3
×
and 1.0
×
,10
−
6
m
2
s
−
1
; for
T
a
, 1200
◦
C; and for
d
r
, 2.6 km; with
t
in millions of years and
d
in kilometres, then Eq. (7.71)is
d
10
−
5
◦
C
−
1
; for
respectively; for
α
,3
×
κ
33
√
t
=
2
.
6
+
0
.
(7.72)
When
T
a
is taken to be 1300
◦
C, Eq. (7.71)is
d
=
2
.
6
+
0
.
36
√
t
(7.73)
Such dependence of ocean depth on age is in broad agreement with the depths
observed for oceanic plates less than 70 Ma old (Eq. (7.57a)). Thus, as the litho-
sphere moves away from the ridge axis, it cools, contracts and hence subsides.
Table 7.4 gives the details of the physical parameters for oceanic-lithosphere
thermal models.
The boundary-layer or half-space models
The boundary-layer model is a modification of the simple half-space model:
the rigid lithosphere is assumed to be cooled asthenosphere, and the base
