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but as the iteration continues, the solutions diverge far
apart (Fig. 1.19). Hence the initially ridiculous-seeming
idea that the flap of butterfly wings where weather is gen-
erated may influence subsequent events far away as the
weather system develops. The metaphor simply says that
random elements exist and evolve in mechanistic systems.
1.6.5 Motion on a sphere: Curved paths and
angular momentum
The geometry of curved surfaces is termed non-Euclidean.
On rotating, curved Earth's surface, observed motions of
the high wind or ocean water over the rotating surface also
curve (Fig. 1.21); The rotation of the Earth induces gradi-
ents in the magnitude of spin and angular momentum that
act to influence the motion. This does not apply to fast-
moving objects held to the surface by frictional contact,
like automobiles, low-level wind, or river flow.
1.6.4
Earth's orbital wobbles
In our heliocentric Solar System, Earth orbits elliptically in
the plane of ecliptic, spinning on an axis inclined obliquely
to this plane. Ellipticity, orientation, and tilt of the spin
axis fluctuate over time spans of 10 4 -10 5 years (Fig. 1.20),
reinforcing each other and causing predictable changes in
Sun-Earth distance. These cause tiny variations in solar
radiation intensity, significantly affecting delicate controls
on climate.
1.6.6
Cycling
The Law of Mass Conservation says that cyclic trading is
material (element, compound) transferred between different
states and/or locations at characteristic rates, the whole
maintaining constant mass. Cycling includes mass reser-
voirs and the fluxes between these as pathways of mass per
time. Plate tectonics controls the rock cycle and the long-
term rates of carbon and water cycling.
1.6.7
Biosphere versus tectonics as planetary agent
Energy for life processes comes from sunlight or abiotic
reactions such as those associated with ocean-ridge con-
vection. The only known source of the abundant oxygen
in our atmosphere is plant photosynthesis. Cycling over
outer Earth is mediated, but not controlled, by living
organisms. Rather than mother Gaia , we have a long-
term Earth system controlled by tectonics called
Cybertectonica .
Fig. 1.18 Turbulent eddies (at scale c. 10 m here) give unpredictable
consequences for velocity or stress distributions.
End-solution 1
Initial
solutions
identical
Time series of repeated
iterative solutions
End-solution 2
Fig. 1.19 Solutions to certain differential equations using initial values that differ by only 0.001 may end up utterly different.
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