Geology Reference
In-Depth Information
20
16
12
8
4
0
0
1
2
3
4
5
Earth time (Gyr)
Figure 10.10. Conversion from model time to Earth time. After Davies [203].
Copyright Elsevier Science. Reprinted with permission.
In 2002, because of computer limitations, it was still difficult to achieve the
numerical resolution required to run convection models of the early, hot mantle
like that presented in Section 9.2. Instead, I ran a model at present conditions and
present rates for 18 Gyr, so as to achieve the appropriate number of mantle transits.
I then converted the numerical model time, t m , to the real time of a cooling mantle.
The relationship between the model time and real time is shown in Figure 10.10,
and a formula is given in Davies [203]. The same idea can be used to estimate
the amount of primitive mantle remaining, i.e. we can calculate how much mantle
would be processed within 18 Gyr at present rates of processing.
Now we are ready to consider that some of the material melting under a ridge
may already have been melted in the past. Suppose the mass of the mantle that is
still primitive at a particular time, t ,is m . Then the fraction of primitive mantle is
p
0.4, then 40% of the mantle is primitive and 60%
has been processed through a melting zone. If the chance of a piece of mantle being
processed within the next little while is independent of whether it has already been
processed, then, on average, only 40% of the next batch processed will have been
primitive.
Going back to the beginning, when t
=
m/M . If, for example, p
=
1, the mass of mantle processed
within a time t will be φt ,where φ is given by Eq. (10.1). Then the mass of
primitive mantle will decrease from m
=
0and p
=
=
M to m
=
( M
φt ). In other words the
change in m is m
=
φt ,andthe rate of change is m/t
=
φ . Later, however,
when p
0.4, the change in m is only 40% of this, because 60% of the mass
processed has already been processed. In other words, we can write
=
m
t =−
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