Geoscience Reference
In-Depth Information
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Figure 5.1
Advective dispersion of passive tracers in a velocity field of the
x
(abscissa)
y
(ordinate) plane.
This diagram shows the evolution of an initial mass of tracers injected at the top left of the
system at time
t
and which follows the clockwise motion of the material (advection). This may
represent the evolution of a block of oceanic crust caught up in mantle convection or the
dispersion of pollutants released into the ocean.
5.2 Diffusion
Diffusive transfer is transfer over short distances caused by the thermal agitation of atoms
or the turbulence of the medium. Let us consider the one-dimensional migration of an
element
i
along the
x
-axis. Let
P
be the probability of an atom jumping per unit time from
its original site and suppose that it can only jump from one site to an adjacent site at a
distance
l
. A site loses atoms with the same probability to the site to its left as to the site to
its right, but only gains half of the atoms lost from each adjacent site. The balance can be
written as:
C
i
(
x
)
∂
∂
P
2
C
i
(
x
P
2
C
i
(
x
PC
i
(
x
)
=
−
l
)
+
+
l
)
−
(5.4)
t
By expanding
C
i
(
x
l
) and
C
i
(
x
+
−
l
) to the second order, we obtain:
C
i
(
x
)
∂
l
2
2
C
i
(
x
)
∂
l
∂
2
∂
C
i
(
x
C
i
(
x
)
(
l
3
)
+
l
)
=
+
+
+
O
(5.5)
x
2
x
2
C
i
(
x
)
∂
C
i
(
x
)
∂
l
2
l
∂
2
∂
C
i
(
x
C
i
(
x
)
(
l
3
)
−
l
)
=
−
+
−
O
(5.6)
x
x
2
Adding these two equations, ignoring terms of order higher than two, and transferring the
result into
(5.4)
, gives:
C
i
∂
Pl
2
2
2
C
i
∂
2
C
i
∂
∂
∂
D
i
∂
=
=
(5.7)
x
2
x
2
t
