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 , 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
) t.
Then f will increase by m c /M , and the rate of increase of f will be
f
t =
1
M ( φ c
) .
(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, . 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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