Geology Reference
In-Depth Information
Appendix C
Chemical evolution details
C.1 Fraction of primitive mantle
We can find the solution of Eq. (10.4) by writing it in terms of differentials:
d
p
d
t
=−
p
τ
.
This has the form of Eq. (A.2) with
a
identified as
−
1
/τ
, so its solution is, from
Appendix A, Eq. (A.6)
=
−
p
p
0
exp(
t/τ
)
.
(C.1)
At time
t
=
0,
p
=
1, so
p
0
=
1, which then yields Eq. (10.5).
C.2 Fraction of crust in the mantle
Following the discussion in Section 10.4.4, the net amount of oceanic crust that
accumulates in the mantle during a time interval
t
will be
m
c
=
(
φ
c
−
fφ
)
t.
Then
f
will increase by
m
c
/M
, and the rate of increase of
f
will be
f
t
=
1
M
(
φ
c
−
fφ
)
.
(C.2)
As discussed in Section 10.4.4, we expect
f
to approach a maximum level, at which the
rate of addition,
φ
c
, balances the rate of removal,
fφ
. This condition will define the
maximum level,
f
m
, or in other words
f
m
=
φ
c
/φ.
(C.3)
We can substitute
φ
c
=
f
m
φ
in Eq. (C.2) and get
f
t
=
φ
M
(
f
m
−
1
τ
(
f
m
−
f
)
=
f
)
.
(C.4)
This equation says that the rate of increase of
f
gets smaller as
f
approaches the maximum,
f
m
. That fits the qualitative understanding developed in Section 10.4.4.
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