Mantle Mixing: Processes and Modeling - Physics and Chemistry of the Deep Earth - page 355

Recent examples of box modeling include an
investigation how a dense basal magma chamber
may retain primitive He characteristics (Coltice
et al ., 2011), a demonstration that a significant
portion of the Earth could be un-degassed and
have distributed ''primitive heterogeneity''
(Albar ede, 2005) and an investigation of mantle
pseudo-isochrons (Rudge, 2006). Two studies that
looked in detail on how melting at mid-oceanic
ridges and recycling of crust affect Pb, Nd, Sr, and
Hf isotopic systematics of MORB (Rudge et al .,
2005; Kellogg et al ., 2007). The former study
confirmed the apparent decoupling between Pb
and the other isotope systems that was found
in many other studies (e.g., Christensen &
Hofmann, 1994) while the latter found that the
background mantle or ''matrix'' in their model
evolved to a composition similar to FOZO as
measured by the Nd, Sr, and Pb systems.
A disadvantage of the box models is that they do
not solve the governing geodynamical equations,
but rather implement assumptions about how
convective vigor, viscosity variations etc. influ-
ence mixing efficiency in a parameterized fashion.
It is attractive to be able to solve the full dynam-
ics of the system using solution of the governing
equations of mass, momentum and heat equation,
so that dynamical and chemical evolution mod-
els can be solved consistently. A major drawback
of this approach is the computational expense:
to model mantle convection at present day con-
vective vigor a high resolution is needed (better
than 5 km in the boundary layers) and to model
convection over the age of the Earth requires a sig-
nificant number of time steps at this resolution.
Furthermore, if the chemistry has a feedback on
the dynamics (e.g., due to oceanic crust extrac-
tion which imparts a chemical buoyancy force)
chemical evolution scenarios may require recom-
puting each dynamical model. This combination
of effects renders even the use of 2D models
expensive. Thermal evolution models that as-
sume higher convective vigor due to a hotter
and lower viscosity mantle are even more expen-
sive, which has led some authors to creatively
bypass this issue and to stretch time (assum-
ing that higher convective vigor can be modeled

by longer mixing at lower convective mixing;
e.g., Davies, 2007).
The dynamical models can be used to ex-
plore the consequences of geometry, heating
modes, rheology, and plate tectonic styles in
a self-consistent manner. Self-consistency here
means that the models (if executed well) satisfy
mass, momentum and heat conservation. These
models can also be adjusted to satisfy various
geodynamical and geophysical constraints, such
as measures of the Earth's convective vigor
(surface velocities, heatflow), viscosity structure,
and seismological constraints. While box models
satisfy mass conservation by design, the conser-
vation of energy and momentum are not satisfied
and the model results can only be compared
to geochemical observations, but not be tested
using the geophysical constraints. This makes
the use of full dynamical models attractive as it
allows for significantly broader applications and
more rigorous testing.
The computational expense of dynamical mod-
eling has caused many workers to focus on 2D
models. Within these the Cartesian models (e.g.,
Christensen & Hofmann, 1994; Brandenburg &
van Keken, 2006; Davies, 2007) have an important
geometrical shortcoming. The effective surface
area of the core is much larger due to the lack
of curvature, which logically will upset any bal-
ance between internal heating and core heating.
The effectively higher heat flux from the core
may also artificially enhance stirring in the lower
mantle. The use of an axisymmetric spherical
shell is for heat flow purposes and comparison
with geophysical observations a much better ap-
proximation (e.g., van Keken, 2001; Shahnas &
Peltier, 2010), but there is a major drawback of
the use of this geometry in mixing due to the fact
that the relative volume of a given cross-sectional
area at the equator is significantly larger than
that of a similar area at the poles. This changes
the dynamics as one moves in latitude in the
model. Better 2D models are based on a cylindri-
cal approach where the core is rescaled to better
satisfy heat flow properties (van Keken, 2001;
Xie & Tackley, 2004; see also Figure 12.4) or on a
remapping of the equations from the full spherical