Geoscience Reference
In-Depth Information
0.14
0.12
τ
1
τ
2
0.10
D
1
→
2
=
1
D
2
→
1
=
10
0.08
0.06
Relaxation (homogenization)
time,
τ
0.04
0.02
0.00
0.0
0.2
0.4
0.6
0.8
1.0
M
1
/(
M
1
+
M
2
)
Figure 6.5
Calculation of the homogenization time of an arbitrary element in a two-reservoir system
(arbitrary units). Partition coefficients
D
indicated are those governing the transfer of the element
from one reservoir to the other. This homogenization time depends on the relative mass of each
reservoir and is less than the time of residence of the element in each reservoir. The cooperative
character of exchanges accelerates homogenization compared with what might be expected for
readjustment of each reservoir separately.
indicating that
homogenization between the two reservoirs is faster than a simple analysis of the residence
time of each of the reservoirs seems to indicate. The falsely intuitive idea that the chemical
response time of a system should need longer to adjust than the residence time of its slower
reservoir is clearly wrong. The reason for this acceleration can be understood by rewriting
(6.21)
as:
τ
is shorter than either
τ
1
or
τ
1
τ
2
τ
1
+
τ
2
d
n
1
d
t
+
τ
1
τ
1
+
τ
2
n
1
=
N
(6.25)
and comparing this equation with
(6.2)
. The forcing term arising on the right-hand side
from coupling with reservoir 2 interferes with the straightforward relaxation of reservoir 1
and always in the direction of acceleration.
The argument just developed also applies to more complex sets of reservoirs, but we
would then need additional tools from linear algebra. We have considered only systems
without an external forcing effect; such an effect should, however, be allowed for where
river water flows into the ocean or cosmogenic matter into the mantle. In general, this
theory underlies the understanding of geochemical cycles, i.e. processes of exchange of
each element between the Earth's different reservoirs. The conditions for describing an
elementary cycle are: (1) essential reservoirs must be identified and their contents esti-
mated; and (2) fluxes of elements between reservoirs must be evaluated over a sufficient
