Geoscience Reference
In-Depth Information
response to other stresses, until they jam again in
a pattern compatible with the new stresses. The
system is weak to incompatible loads. Changes
in
porosity
,
temperature
and
stress
are equivalent
and can trigger reorganization and apparent
changes in rigidity. The jamming of these mate-
rials prevents them from exploring phase space
so their ability to self-organize is restricted, but
is dramatic when it occurs.
In the plate tectonic context it is compression
-- or lack of lateral extension -- that keeps plates
together. When the stress changes, and before
new compatible stress circuits are established,
plates may experience extension and collapse.
New
compatible plate boundaries
must
form.Widespreadvolcanismistobeexpectedin
these un-jamming and reorganization events if
the mantle is close to the melting point. These
events are accompanied by changes in stress and
in the locations and nature of plate boundaries
and plates, rather than by abrupt changes in
plate motions. Volcanic chains, which may be
thought of as chains of tensile stress, will reori-
ent, even if plate motions do not. Jamming the-
ory may be relevant to plate sizes, shapes and
interactions. The present plate mosaic is presum-
ably consistent with the stress field that formed
it but a different mosaic forms if the stresses
change.
for optimal packing of large numbers of circular
or pentagonal caps. However, one can efficiently
arrange 12 caps onto a sphere, with only 0--10%
void space. This is much more efficient than for,
say, 10, 13, 14 or 24 caps. Furthermore, the differ-
ence between the sizes of caps which pack most
efficiently (least void space), and cover the whole
surface most economically (least overlap), is rela-
tively small for 12 caps (the difference is zero for
regular spherical pentagons).
The ideal plate-tectonic world may have
n
identical faces (plates, tiles) bounded by great cir-
cle arcs which meet three at a time at 120
◦
. This
simple conjecture dramatically limits the num-
ber of possibilities for tessellation of a sphere and
possibly, for the ground state of plate tectonics.
In soap bubbles and plate tectonics, junctions of
four or more faces are unstable and are excluded.
There are ten such possible networks of great cir-
cles on a sphere, some of which are shown in
Figure 4.7. If plate boundaries approximate great
circles meeting three at a time at 120
◦
then there
can be a maximum of 12 plates. The ideal plate
may be bounded by five or six edges and ideal
plate boundaries may approximate great circles
which terminate at triple junctions dominated
by 120
◦
angles.
The study of convective planforms and pat-
tern selection is a very rich and fundamental
field in thermal convection and complexity the-
ory. Even in complex convection geometries reg-
ular polyhedral patterns are common. Pattern
selection in the plate tectonics system, however,
may have little to do with an imposed pattern
from mantle convection although it is commonly
assumed to do so.
Close-packed networks of objects are jammed
or rigid. However, even open networks can jam
by the creation of load-bearing stress chains
which freeze the assemblage so it cannot min-
imize the open space. These networks can be
mobilized by changing the stress. Since plates
are held together by networks of compressional
forces it is important that they pack efficiently.
On the other hand, they must be mobile, and
cannot be a permanently jammed system. Mate-
rials with this rigid-fluid dichotomy --
fragile
matter
-- may be better analogues for the outer
layers of Earth than implied by the terms
shells
or
plates
.
In the ideal world
Suppose that the outlines of plates, or their
'rigid' cores, are approximated by circles. The
fraction of a flat plane occupied by close-packed
circles is 0.9069. Packing efficiency of circlular
caps on a sphere depends on the number of
circles (spherical caps). The area covered ranges
from 0.73 to 0.89 for
n
12 and then oscillates
about 0.82 (Figure 4.6). Packing of more than
six regular tiles on a sphere is inefficient except
for 12 equal spherical pentagons, which can tile
a sphere with no gaps. The efficiency of pack-
ing, the sizes of the voids and their aggregate
area depends little on the size distribution or
shape, within limits. The voids between regular
tiles on a sphere, when close packed, are typi-
cally 10% of the radius of the discs. Typically,
equal-sized circular or polygonal non-overlapping
caps can cover only about 70--85% of the surface
of a sphere. About 15% void space occurs even
≤
