Geoscience Reference
In-Depth Information
transport of elements at the speed of the ambient medium, and diffusion of an element is
its transport relative to the ambient medium.
The principles from which the transportation equations derive are difficult to formulate
in mathematical terms, but they are easy enough to understand:
•
Conservation of total matter, which produces the continuity equation.
•
Conservation of element
i
flux of
i
across the surface
variation of mass of
i
per unit of time
=
sources and sinks
+
(5.1)
In just the same way as we can count our fish from a bridge, from a boat drifting with
the current, or from a motor boat, the terms of this equation can be expressed relative to a
fixed, or moving, arbitrary reference point.
5.1 Advection
Advection is bulk transport, which is easier to understand in one dimension. Let us consider
a medium of density
. If, at a point
x
and over an area
A
, we consider
a slice of matter of thickness
l
, the balance of variation of mass of
i
per unit time in this
slice is equal to the difference between the incoming and outgoing fluxes:
ρ
moving at speed
v
d
C
i
(
x
)
d
t
C
i
(
x
)
C
i
(
x
Al
ρ
=
A
ρv
−
A
ρv
+
l
)
(5.2)
or, if
l
tends toward 0:
C
i
∂
C
i
∂
∂
=−
v
∂
(5.3)
t
x
where the partial derivatives are used to show relative to which variable derivation is
applied. If we imagine an observer who, like the boat drifting on the river, moves with
the speed
of the medium, the apparent speed of the observer relative to the medium
cancels out and the right-hand side disappears. Again, we find the condition that local
concentration is invariant: this is known as the Lagrangian representation of the variation.
This amazingly simple principle is the one applied in the study of convection of the mantle
or the ocean, and in the dispersion of what are known as passive tracers, i.e. tracers not
involved in reactions (
Fig. 5.1
): if we know the field of velocities, we can determine the
motion of any point of the medium between two instants and therefore track the position
of an atom of element
i
over time.
v
