Geology Reference
In-Depth Information
and, remembering that
p
=
m/M
and dividing both sides by
M
,weget
p
t
p
φ
M
p
τ
,
=−
=−
(10.4)
where the second step uses Eq. (10.2). If we just used the initial rate of change
of
p
,when
p
1, we would predict that
p
would be reduced to zero after 4 Gyr,
which is the processing time. However, Eq. (10.4) says that the rate of change of
p
gets smaller as
p
gets smaller, and therefore
p
will approach zero asymptotically.
Section C.1 in Appendix C shows that the mathematical solution of this equation
is that
p
declines exponentially with time:
=
p
=
exp(
−
t
m
/τ
)
,
(10.5)
where
t
m
denotes that we are working in the 'model time' of Figure 10.10.
After
t
m
=
=
=
0.011. Thus by this estimate only about
1% of the mantle remains primitive. This fraction would be even smaller if the
melting depth was greater in the past, because the mantle was hotter. This simple
calculation means that we cannot expect a significant amount of primitive mantle
to have survived in the MORB source.
18 Gyr with
τ
4Gyr,
p
10.4.4 How much subducted oceanic crust?
Oceanic crust is being added to the mantle at subduction zones at the rate
φ
c
=
ρ
c
A
s
d
c
,
(10.6)
where
ρ
c
is the density of oceanic crust and
d
c
is its thickness. We can ask how long
it would take, at present rates, to fill the mantle with oceanic crust. The answer is
τ
c
=
M/φ
c
. With a crustal density of 2900 kg/m
3
and thickness of 7 km,
φ
c
=
6.1
×
10
13
kg/yr and
τ
c
=
66 Gyr. As the Earth is only 4.5 Gyr old, you might conclude
that oceanic crust comprises a fraction
f
0.07 of the mantle, so old
subducted oceanic crust would comprise 7% of the mantle. However, if subduction
was faster in the past, as we are supposing, you should allow for the equivalent of
18 Gyr worth of subduction at present rates, so
f
=
4.5
/
66
=
=
18
/
66
=
0.27, and crust would
comprise 27% of the mantle, which seems like a lot.
However, oceanic crust is also being removed in the melting zones under mid-
ocean ridges. Let us assume for the moment that all of it that enters a melting
zone melts and is removed from the mantle. We can ask, how much of the mantle
comprises subducted oceanic crust at any given time? Initially, the fraction of the
mantle,
f
, that is subducted crust will have been zero so, as there was no crust to
remove, the rate of removal in melting zones would have been zero. Later, the rate
of removal will be
fφ
,where
φ
(Eq. (10.1)) is the rate at which material is passing
through the melting zones. Early on, the rate of removal would have been small,
