Geoscience Reference
In-Depth Information
processes that are important at time scales of 1 to 100 000 years, ignoring faster processes
and treating longer-term ones as forcing to be dealt with by separate parameters. The use
of a time “window” adequate for all the processes to be considered is an essential condition
for the construction of correct geochemical cycles.
It is legitimate to wonder about the relationship between the full representation of chem-
ical transfers by equations describing local fluxes of tracers, in which the concentration of
an element is tracked at each point of the system at each moment in time, and the multiple-
reservoir model (box or zero-dimension model) with no explicit indications of the position
of individual tracers. Both representations use the same physics and represent justifiable
and consistent formulations of the conservation principle. The two essential differences are
that, in the box model, concentration is averaged over the entire reservoir while the fluxes
are integrated over the entire area bounding each reservoir. It could thus be shown that by
multiplying the number of reservoirs and making their size tend toward zero, a continuous
transition can be made from one representation to the other.
6.3 Mixing and stirring
Let us now try to describe how chemical heterogeneities are destroyed by the convective
movements of the mantle or the ocean or by circulation in a water table. Anyone can
imagine the fate of a packet of different colored sticks of play dough left in the hands of
a small child: from an early stage, where the sticks are distorted but remain identifiable,
to a homogeneous bluish-gray dough, passing through a marbled stage with multicolored
stripes stretched and folded back on themselves many times (
Fig. 6.7
). Obviously if a
representative sample of the average of the packet is to be taken, it must be sufficiently
large compared with the thickness of the stripes. The concept of homogeneity is therefore
dependent on the scale of observation or of sampling, and it will always be possible to spot
local heterogeneities provided that the medium is observed at sufficiently high resolution.
Mixing and stirring should not be used indifferently. In the case of the play dough, a
grown-up person coming early enough to interfere with the entropy-generating toddler
can still undo the stretching and folding and with a little bit of work retrieve the stick
constituents with their original color intact. The play dough had only been “stirred.” If,
however, the stretching and folding game goes on for some time and the bluish-gray out-
come left to settle for a while, the different ingredients (oil and color) will start to smudge
into each other and clean segregation becomes impossible, even as a thought experiment.
The irreversibility of the process is a characteristic feature of true mixing.
Where, in an agitated reservoir like the ocean or the mantle, are the effective mixing
zones located? Let us consider (
Fig. 6.8
) a part of the reservoir where displacement is
horizontal, velocity
v
x
(
y
) is then dependent only on the distance
y
to the bottom. Two
points located initially, one at (
x
,
y
) and the other on the same vertical at (
x
,
y
+
y
), move
after a time
δ
t
over a slightly different horizontal distance. This difference
δ
l
is written:
d
v
x
d
y
δ
l
=
v
x
(
y
+
y
)
δ
t
−
v
x
(
y
)
δ
t
=
y
δ
t
(6.26)
