Geoscience Reference
In-Depth Information
Pacific volcanoes consistently show the presence
of zoning (e.g., Blichert-Toft & Albar ede, 2009;
Huang
et al
., 2011) and the marked asymmetry
may be a property of the Hawaiian plume sam-
pling diverse regions in the lower mantle (Weiss
et al
., 2011). This type of zoning is retained
in some plume models as well (Farnetani &
Hofmann, 2009, 2010).
change, the heating mode is uniform, the flow
is laminar and anelastic, and inertial forces are
unimportant) the Rayleigh number can be used to
describe semi-quantitatively the dynamic behav-
ior of the system. For example, for a fluid heated
from below and cooled from above one can de-
rive a critical Rayleigh number of O(10
3
) above
which convection will take place (e.g., Turcotte &
Schubert, 2002). If one uses upper mantle values
for the constitutive parameters (viscosity, expan-
sivity etc.), a reasonable estimate for temperature
contrast across the mantle for
T and the depth
of the mantle for h one finds Ra to be O(10
7
).
For fully developed flow a simplified boundary
layer analysis can be used to construct relation-
ships between the nondimensional heatflow (the
Nusselt number) and nondimensional surface ve-
locities. The highest convective vigor is found for
an aspect ratio of the flow of near unity.
The use of the Rayleigh number to describe the
vigor in the Earth's mantle breaks down since
the Earth is not a simple homogeneous fluid. For
example the high value of O(10
7
) or higher that
is regularly cited in the literature is based on
upper mantle values which are not representa-
tive of the Earth as a whole. Viscosity certainly
increases into the lower mantle, expansivity de-
creases with depth, and diffusivity increases with
depth. These depth-variations all tend to lower
the effective Rayleigh number. More importantly,
viscosity strongly depends on temperature, pres-
sure and stress, which makes the use of a constant
viscosity
η
0
in the Rayleigh number rather mean-
ingless. Furthermore, the rheological control on
plate tectonics includes brittle, plastic and elastic
deformation mechanisms, which are not repre-
sented by the simple Stokes equation used in
the derivation of the Rayleigh number. In sum-
mary, the use of the Rayleigh number to attempt
to describe the convective vigor of the Earth is in
my view meaningless. We should instead describe
the vigor of the models by direct comparison with
surface velocity and heat flow.
A separate powerful constraint on the degassing
of the Earth is the amount of
40
Ar in the at-
mosphere. Currently about 50% of the radio-
genic argon from the silicate mantle is in the
12.6
Modeling Perspectives
The previous sections have provided a summary
and update to a large collective body of work
on the evidence for how mantle heterogeneity is
introduced, maintained and sampled. In this final
section I will lay out some suggestions for how
geodynamical modeling may evolve. Better and,
maybe more importantly, more relevant models,
will be useful in testing hypotheses how mantle
heterogeneity is produced and retained.
We have some fundamental global constraints
on the convective vigor of the present-day Earth
and a few integrated geochemical observations
that we should consistently use in testing the
dynamical models. For example, the present-day
heat flow of approximately 100 mW/m
2
and sur-
face velocities of around 5 cm/yr provide a funda-
mental constraint on models. It is interesting to
note that surprisingly few dynamical publications
list how the convective vigor compares between
the published models and the Earth. Often au-
thors rely instead on quoting a Rayleigh number
that is then interpreted to indicate convective
vigor. The Rayleigh number Ra is defined by
=
ρ
0
α
0
g
Th
3
/κ
0
η
0
Ra
where
ρ
0
is a reference density,
α
0
is a refer-
ence expansivity, g is gravity,
T is a reference
temperature contrast, h is a reference scale,
κ
0
is a reference diffusivity, and
η
0
is a reference
viscosity. This nondimensional number arises
from the nondimensionalization of the governing
equations that prescribe conservation of mass,
heat and momentum. For very simple, homoge-
neous systems (where material properties do not
