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of plate bending and fault strength
at
subduction zones may be as important as mantle
viscosity in mantle dynamics and in controlling
the cooling of the mantle. When the only resist-
ingforcesarelithosphericbendingwehavethe
dimensionless lithosphere number
density differences have been thought to be too
small to stabilize stratification. However, when
pressure is taken into account chemical strati-
fication is likely; dense layers become trapped,
although the effect on seismic velocities can be
slight. These are therefore
stealth
layers or reser-
voirs which are below the ability of seismic waves
to detect, except by special techniques.
The lower parts of the mantle are now at high
pressure and low temperature compared with
accretional conditions. Chemically distinct dense
material, accumulated at the base of the mantle,
must become very hot in order to become buoy-
ant, because of the very low thermal expansivity
at lower-mantle pressures. Scaling relations also
show that only very large features would accu-
mulate enough buoyancy to rise.
Although the viscosity of the mantle increases
with depth, because of pressure, and although
the lower mantle may also have a higher vis-
cosity, because the mineralogy is different and
stresses may be lower, this is not sufficient
to prevent whole-mantle convection. If the vis-
cosity of the upper mantle is less than that
of the lower mantle, circulation will be faster
and there is more opportunity to recirculate
crustal and upper-mantle material through the
shallow melting zone. Differentiation processes
would therefore change the composition of the
upper mantle, even if the mantle were chem-
ically homogenous initially. The separation of
light and dense material, and low-melting-point
and high-melting-point material, however, prob-
ably occurred during accretion and the early
high-temperature history of the Earth. The dif-
ferentiation will be irreversible if the recycled
products of this differentiation (basalt, eclogite,
depleted peridotite, continental crust) are unable
to achieve the densities of the lower mantle or
the parts of the upper mantle through which
they must pass.
The magnitude of the Rayleigh number is a
measure of the vigor of convection and the dis-
tance from static equilibrium. Most geodynamic
discussions assume
Ra
to be 10
6
to 10
8
. Using
parameters appropriate for the base of the man-
tle yields a value of 4000. If the lower 1000 km
of the mantle acts as an isolated layer, because
of high intrinsic density,
Ra
drops to 500. The
Pl
=
g
α
Tr
3
/κν
1
where
r
is the radius of curvature of the bend
and
ν
1
is the lithospheric viscosity. The
ther-
mal evolution of an Earth with strong
subduction zones
is quite different from one
with a completely fluid mantle.
When plate interactions are involved we also
need coupling parameters across plates such as
transform fault resistance and normal stress. In
the plate-tectonic system the plates (and slabs)
account for much of both the driving force and
dissipation (see Figure 4.8) and in this respect
they play the role of the convecting fluid in
Rayleigh--Bénard convection (internal sources of
both buoyancy and viscous dissipation). Both
the buoyancy and dissipative stresses affect the
whole system. The plate sizes, shapes and veloc-
ities are self-controlled and should be part of
the solution rather than input parameters. Even
the rheology of the surface material may be self-
controlled rather than something you can look
up in a handbook.
Foams
and
grains
are examples
of
fragile
or
soft
materials that to some extent con-
trol their own fates. The behavior of rocks, and
probably the lithosphere, is controlled by
dam-
age rheology
and complex feedback processes
involving fracture and friction rather than fluid
mechanics.
Layered mantle convection
If pressure is ignored it requires about a 6%
increase in
intrinsic density
foradeepmantlelayer
to be stable against a temperature-induced over-
turn. Plausible differences in density between the
silicate products of accretional differentiation,
which are intermediate in density between the
crust and the core, are about 1 or 2%, if the
variations are due to changes in silicon, alu-
minum, calcium and magnesium. Changes in
iron
content
can
give
larger
variations.
Such
