Geoscience Reference
In-Depth Information
Let M 1 and M 2 be the masses of the two reservoirs and Q the mass of material (water,
magma) exchanged between them per unit time. As in the previous section we can relate
the probability of transfer to the ratio of the flux of element to its amount in the reservoir,
for instance:
1
τ 1 =
Q
(
n 1 /
M 1 )
p 1 2
=
(6.17)
M 1 (
n 1 /
M 1 )
where
τ 1 for the residence time of the element in
reservoir 1. If fractionation takes place during transfer, e.g. upon precipitation of melting,
this equation must be replaced by:
(
n 1 /
M 1 )
stands for concentrations and
D 1 2
1
τ 1 =
Q
(
n 1 /
M 1 )
p 1 2
=
(6.18)
M 1 (
n 1 /
M 1 )
where we recognize at the numerator the product of the material flux Q multiplied by a con-
centration n i /
M i and by the partition coefficient D i j allowing for possible fractionation
of the element as it passes from reservoir i to reservoir j . When material is extracted from
the mantle as a basaltic or andesitic magma and incorporated into the continental crust, the
magma is chemically fractionated relative to its starting mantle and D mantle crust increases
with the lithophile character of the element. By contrast, when water is exchanged between
two parts of the ocean, the coefficients D may be equal to 1.
The balance can therefore be written as two conservation equations:
QD 1 2
M 1
QD 2 1
M 2
d n 1
d t =−
n 1 +
n 2
(6.19)
QD 1 2
M 1
QD 2 1
M 2
d n 2
d t =
n 1
n 2
(6.20)
In terms of residence times
τ 1 and
τ 2 of the element in each reservoir, the system becomes
simply:
d n 1
d t =−
1
τ 1
1
τ 2
n 1 +
n 2
(6.21)
d n 2
d t =
1
τ 1
1
τ 2
n 1
n 2
(6.22)
It can be seen that these two equations have a zero sum, which reflects the constancy of the
total inventory n 1 +
N of the element in the system. By introducing this condition
into (6.21) , and rearranging and solving it, we obtain:
n 2 =
τ 1 + τ 2 1
e t
N τ 1
n 1,0 e t +
n 1 =
(6.23)
where n 1,0 is the initial value of n 1 . A symmetrical equation would be obtained for n 2 .The
overall relaxation time
τ
of the system is defined by:
1
τ
1
τ 1 +
1
τ 2
=
(6.24)
 
 
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