Geoscience Reference
In-Depth Information
Figure 6.7
Chaotic structures of typical mixtures (after Ottino,
1989
).
v
x
(
y
+
Δ
y
)
δ
t
Δ
y
y
v
x
(y)
δ
t
x
Figure 6.8
Stretching between two points initially
y
apart after time
t
;
v
x
(y)
is the horizontal velocity
δ
in (
x
,
y
).
where the derivative of the velocity
v
x
(
y
) is expressed in
y
. It can be seen that for a given
initial vertical separation
y
, the term that governs the horizontal speed of stretching
δ
l
/
δ
t
is the vertical variation d
d
y
in horizontal speed, which is known in mathematical jargon
as the velocity gradient. The velocity gradient is a measure of the rate of deformation
and therefore of mixing. Diverging or converging structures resembling upwelling and
subduction, often referred to as poloidal, are two-dimensional and do not do a good job at
mixing. Part of the reason is that the flow lines are closed loops, even in three dimensions.
Addition of vortex-type zones, such as transform faults, which are often referred to as
toroidal because of their doughnut shape, create chaotic flow with open-ended trajectories
(tendrils) and provide efficient mixing. The generation of toroidal movements is associated
with boundary layers (the coast line and the surface in the ocean, the lithosphere in the
mantle) or with lateral changes of viscosity. Convection therefore plays a central role in
reducing the size of heterogeneities. Let us think of a heterogeneity as a single stripe, e.g.
a lithospheric plate, just introduced into an otherwise homogeneous mantle, or a core of
water with distinctive salinity and temperature in the ocean. With time, convection will
stretch and fold the original stripe, just like a baker stretches and folds the dough before
v
x
/
