Mantle mixing rheology by mikeholy

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									Convective mixing in the Earth’s mantle
Peter E. van Keken

Department of Geological Sciences, University of Michigan, 425 East University
Avenue, 2534 CC Little Building, Ann Arbor MI 48109-1063, USA.
Phone: 1-734-764-1497, Fax: 1-734-763-4690, E-mail: keken@umich.edu

Chris J. Ballentine

Department of Earth Sciences, University of Manchester, Manchester M13 9PL, United
Kingdom, E-mail: cballentine@fs1.ge.man.ac.uk

Erik H. Hauri

Department of Terrestrial Magnetism, Carnegie Institution of Washington, 5241 Broad
Branch Road, NW, Washington DC 20015, E-mail: Hauri@dtm.ciw.edu



1. Introduction
The observed geochemical diversity of mid-oceanic ridge and ocean island basalts is
generally attributed to the existence of large-scale and long-term chemical heterogeneity
in the Earth’s silicate mantle. Yet, it is far from clear how this heterogeneity is generated
and how it is maintained in the convecting mantle. Quantitative mixing studies indicate
that the present day Earth is convecting vigorously enough to erase significant initial
heterogeneity well within the age of the Earth. This suggests that some form of layering,
or barrier against convective mixing is required to explain the geochemical observations.
The most basic form of layering is that of the "classical" two-layer mantle, where a
depleted and well-mixed upper mantle is separated by the seismically distinct 670 km
boundary from a poorly mixed and enriched lower mantle. The mid-oceanic ridge basalts
(MORB) originate from melting of the upper mantle, whereas ocean island basalts (OIB)
derive from melting of material that is brought up from the lower mantle by plumes. This
model has worked well to explain a large number of observations regarding noble gas and
trace element concentrations and the distribution of mantle heat sources. However, in
recent years this model has come under siege. Geophysical and geodynamical
observations indicate that significant mass exchange occurs through the Earth’s transition
zone. In addition, various geochemical observations suggest an important role for oceanic
crust recycling in the plume source and demonstrate the lack of preservation of primitive
mantle. The recent widespread acceptance of these fundamental problems of the classical
layered mantle model has led the proposition of various alternatives such as the presence
of layering below 670 km depth, or the preservation of heterogeneity in highly viscous
regions in the Earth’s mantle. These models appear to be able to explain one or more
features better than the classical model, but often cause new conflicts with existing
geochemical or geophysical observations. In addition, it is not always clear that these


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new conceptual models are physically realistic. With the advent of large scale computing,
geodynamical modeling has become a particularly useful tool in this arena. Using the
fundamental laws of the conservation of mass, energy, and momentum, models of mantle
convection can be created that allow for quantitative tests of these conceptual models.
Modeling also allows for a better understanding of the physics that governs mantle flow,
mantle mixing and the distribution of chemical heterogeneity in planetary interiors.
      This chapter will focus on the use of mantle convection modeling in the
development of our understanding of the chemical evolution of the Earth by providing a
short review of the main observations, a discussion of the physical approaches to
characterize mantle mixing, and an overview of the historical and current modeling
approaches to the formation, preservation, and destruction of chemical heterogeneity.
Detailed reviews of the geochemical data and interpretations can be found in Zindler and
Hart (1986), Silver et al. (1988), Carlson (1994), Hofmann (1997), Van Keken et al.
(2002), Porcelli and Ballentine (2002), and Hauri (2002).

2. Geochemical and geophysical observations of mantle heterogeneity

2.1 Geochemical observations suggesting mantle layering

The formation of basalts by partial melting of the upper mantle at mid-oceanic ridges and
hotspots provides the opportunity to determine mantle composition. Early studies of
radiogenic isotopes in oceanic basalts (e.g., Faure and Hurley, 1963; Hart et al., 1973;
Schilling, 1973) showed fundamental chemical differences between OIB and MORB.
This led to the development of the layered mantle model, which consists essentially of
three different reservoirs: the lower mantle, upper mantle and continental crust. The
lower mantle is assumed primitive and identical to the bulk silicate Earth (BSE), which is
the bulk Earth composition minus the core. The continental crust is formed by extraction
of melt from the primitive upper mantle, which leaves the depleted upper mantle as third
reservoir. In this model, MORB is derived from the depleted upper mantle, whereas OIB
is formed from reservoirs derived by mixing of the MORB source with primitive mantle
(e.g., DePaolo and Wasserburg, 1976; O’Nions et al., 1979; Allègre et al., 1979).
       In particular the study of noble gas isotopes has provided a compelling case for
mantle layering and the preservation of primitive mantle. For example, 3He/4He values
for MORB are nearly uniform, but large departures are seen for OIB. Most oceanic
hotspots have elevated 3He/4He values compared to MORB. 4He is generated by the
decay of U and Th. 3He is not generated inside the Earth and is lost from the atmosphere
upon degassing. The presence of high 3He/4He values requires therefore the isolation of
the plume reservoir from surface melting. It has been common to equate the high 3He/4He
‘reservoir’ with the primitive mantle. Mass flux considerations add weight to the
argument for mantle layering. For example, the inferred mass flux of 4He from the mantle
is far lower than that predicted from the decay of U and Th in the whole mantle (O’Nions
and Oxburgh, 1983), which suggests the existence of a boundary layer that decouples the
flow of He and heat from the lower mantle. Steady state models for Pb and the noble
gases similarly limit the mass exchange between upper and lower mantle (Galer and
O’Nions, 1985; Kellogg and Wasserburg, 1990; O’Nions and Tolstikhin, 1994; Porcelli
& Wasserburg, 1995ab).


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      Further suggestions for compositional stratification come from the consideration of
the Earth’s heat budget. The combined heat production of the upper mantle and
continental crust, based on estimates of radiogenic heat production of the crust and
MORB-source (Rudnick and Fountain, 1995; Van Schmus, 1995) strongly suggests that
radiogenic element concentrations in the deep mantle are significantly higher than that of
the upper mantle.

2.2 Problems with the classical layered model

Several lines of evidence indicate problems with strict application of the layered model.
It is particularly difficult to satisfy the requirement for primitive composition of the lower
mantle. For example, Zindler and Hart (1986) showed that the isotopic database for the
oceanic mantle could be described by mixing between depleted MORB mantle and
enriched components that have compositions quite unlike primitive mantle. Similarly, the
constancy of Nb/U and Ce/Pb (Hofmann, 1986; Newsom et al., 1986) suggests that there
is no current reservoir with primitive mantle ratios. Interestingly, there is a strong
correlation between a depleted component, termed FOZO by Hart et al. (1992), and the
high 3He source observed in many hotspots (Hart et al., 1992; Farley et al., 1992; Hanan
and Graham, 1996).
       In recent years, we have seen the use of new isotopic systems, including those of
osmium and oxygen, in the description of oceanic basalts. Osmium isotopes can trace the
addition of mafic crust or melt to the mantle source (Hauri and Hart, 1993) and have been
used to highlight the presence of recycled mafic crust in sources of hotspots (Shirey and
Walker, 1998). In a complimentary fashion, oxygen isotope variations trace directly
portions of the mantle that have interacted with water at or near the Earth’s surface,
which allows for identifying recycled oceanic crust in the sources of both OIB and
MORB (Eiler et al., 1997, 2000; Lassiter & Hauri, 1998; Hauri, 2002). It has become
apparent that nearly all hotspots display elevated 187Os/188Os, which requires involvement
of mafic sources, most likely through the recycling of oceanic crust into the deep mantle.
Mass balance calculations suggest the presence of a recycled oceanic crust reservoir of up
to 10% of the mantle mass (Hauri and Hart, 1997). The strong evidence for the recycling
of oceanic crust in the sources of hotspots is difficult to reconcile with the layered mantle
unless most or all of the transition zone (which constitutes about 10% of the mass of the
mantle) is composed of recycled oceanic crust and is somehow preferentially sampled by
mantle plumes
       Geophysical observations are now increasingly used to argue against a form of
layering at 670 km depth, in particular since it has been found that the sharp
characteristics of the 670-km discontinuity (Paulssen, 1988) can be explained by a phase
change in an otherwise isochemical mantle (Ito & Takahashi, 1989). Early reports of
aseismic extensions of the slabs in the lower mantle based on travel time perturbations
(Jordan, 1977; Creager and Jordan, 1984, 1986) and waveform complexity (Silver and
Chan, 1986) suggested a dynamic connection between upper and lower mantle. Some of
the first global tomographic models showed a pattern of high velocity in the deep mantle
that correlated well with the inferred pattern of paleosubduction (Dziewonski, 1984;
Dziewonski and Woodhouse, 1987). This was strengthened by a study by Lithgow-
Bertelloni and Richards (1998), which suggested that tomographic models are explained


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quite well if the lithosphere and lower mantle are one or two orders of magnitude more
viscous than the upper mantle. This is in line with earlier estimates for the viscosity of the
lower mantle based on geoid and gravity anomalies above subduction zones (Hager and
Richards, 1989) and geodynamic inversions (Mitrovica and Forte, 1995). More indirect,
but perhaps just as convincing, are arguments provided by Davies (1998) who shows that
the lack of significant plume related topography at the Earth surface makes it
inconceivable that a large thermal boundary layer exists at 670 km depth, as is required
by the layered mantle hypothesis. Studies of the convective flow predicted by S-wave
tomography also demonstrate that there is significant mass flux from the upper mantle
into the lower mantle (Puster and Jordan, 1997). The quality of mantle tomography has
increased dramatically in recent years, which has led to convincing images of slabs that
are subducting into the lower mantle (Fukao, 1992; Grand, 1994; van der Hilst et al.,
1997; Bijwaard et al., 1998; Ritsema et al., 1999; Grand, 2002). The correlation of
seismically low velocities and subducted slabs is strengthened by the good agreement
with paleographic reconstructions in several locations (Van der Voo et al., 1999).
      In summary, we have recently witnessed a shift away from the classically layered
mantle model in favor of whole mantle convection models, where the buoyancy of
sinking slabs is the dominant driving force. Slabs can penetrate deep into the lower
mantle and with the induced return flow we would expect the mantle to mix efficiently.
This leaves us with an interesting dilemma. If the mantle convects as a whole, how can it
preserve the large-scale and long-lived heterogeneity seen in the geochemistry of oceanic
basalts?

3. Characterization of mixing

Quantitative evaluation of the efficiency of mantle mixing provides a fundamental tool to
test conceptual models for the generation and destruction of heterogeneity. In this section,
we will discuss and illustrate a number of ways in which we can quantify mantle mixing
and provide a short review of the major findings from the literature. The physical
processes of mixing in fluids are of great interest in a large number of fields, with
important implications for industrial and engineering processes and a correspondingly
large body of literature. For a comprehensive review of the mathematical description of
mixing and examples of fluid dynamical and engineering applications, the reader is
referred to Ottino (1989).
       In this section we will provide some illustrations that are based on a simple two-
dimensional convection model in Cartesian geometry. Full detail on the model setup,
governing equations and numerical solution can be found in Appendix A. This model is
strictly for illustrative purposes: it has a number of characteristics that we think are
representative of convection in the mantle (infinite Prandtl number, finite dissipation
number, temperature- and pressure-dependent viscosity) but falls short in a number of
others (moderately low convective vigor, 2D Cartesian geometry, limited representation
of rheology, no internal heating). A movie showing the convective motion is available at
http://www.geo.lsa.umich.edu/~keken/treatise . The initial temperature field is shown in
Figure 1a. The flow is characterized by moderate time-dependence of the boundary
layers. The down- and upwelling that are near the center of the box remain at nearly
stationary locations, but traveling instabilities along the top and bottom boundary layers


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cause significant time-dependence in the center up- and downwelling. We will track two
heterogeneities (numbered I and II) through this flow. The heterogeneities are identical in
size and number of particles but differ in their initial position. Heterogeneity I is placed in
the center upwelling, whereas heterogeneity II is placed just outside this upwelling
(compare Figure 1a with Figure 1e). As we will show, the mixing behavior of these
particle sets differs greatly: the heterogeneity that is started in the boundary layer
encounters strong shear and reorientation and mixes with the fluid quite rapidly.
Heterogeneity II remains inside a convective cell with only moderate deformation for a
significant period of time. Similar extensive differences in mixing behavior within the
same flow have been suggested in 3D models by Ferrachat and Ricard (1998) and Van
Keken and Zhong (1999). Although we may predict that this ‘core-in-cell’ behavior
(Spohn and Schubert, 1982) is not necessarily a fundamental explanation for the observed
mantle heterogeneity, since mantle convection is significantly more time-dependent than
this study, it may well form an explanation for observed regional differences, such as the
DUPAL anomaly (Hart, 1984; Castillo, 1988).




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Figure 1. a) temperature field of a moderately time-dependent 2D Cartesian convection
calculation (see Appendix A for details). An initial heterogeneity (number I), discretized
with 100x70 tracers, is placed in the upwelling near the center of the model. The
heterogeneity is brought up to the surface and split in two sections. Arrows indicate the
flow pattern that this heterogeneity will see. The numbers indicate the positions of the
snapshots shown in the next frame. 1: t=0.001, 2:t=0.002; 3:t=0.0024; 4:t=0.0028;
5:t=0.0032.
b) deformation of heterogeneity I (left portion only) corresponding to the time intervals
shown in frame a.
c) illustration of the definition of striation thickness.
d) illustration of the effects of increasing the number of tracers from 100x70 to 200x140,
400x280 and 800x560.
e) deformation of heterogeneity II that is slightly displaced from I.



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3.1 Physics of mixing

Convective flows provide a mechanism for stirring and mixing of heterogeneities. The
principle components of this process are stretching, folding, breakup, and diffusion
(Ottino, 1989). Stretching and folding are illustrated in Figure 1. Stretching occurs in the
presence of velocity gradients (such as a simple shear flow). Folding occurs when the
material flow reverses onto itself or when the material encounters a fixed boundary (such
as the core-mantle boundary and the free surface of the Earth’s mantle).
       Other possible mixing mechanisms include diffusion and breakup. Breakup is
negligible in the Earth’s mantle due to the lack of surface tension in solid rock. Diffusion
in the solid phase is very slow due to the low compositional diffusion rates in mantle
rocks, and is only important at very small length scales (and perhaps at very high
temperatures like those at the CMB). The lack of interfacial tension and the similarity of
properties between chemically distinct layers in the viscously deforming Earth allow us
to consider the Earth’s mantle as a miscible fluid. We can therefore focus on stretching
and folding as the primary mechanisms of mantle mixing. Nevertheless, the spatial and
temporal discretization that is necessary in the numerical solution of the governing
equations introduces breakup- and diffusion-like processes. Figure 1c shows the breakup
of the heterogeneity due to the limited number of tracers that are used to represent the
heterogeneity. This effect can be mitigated by introducing more tracers (see Figure 1d),
which effectively leads to a lower sensitivity of the numerical model to artificial breakup.
Similarly, the finite accuracy of numerical particle tracing leads to an effective numerical
diffusivity (not illustrated).
       Efficient mixing requires strong deformation. Since this occurs particularly in
boundary layers, the heterogeneity of figure 1a was stretched and folded quite efficiently.
It is interesting to illustrate the deformation of a heterogeneity that was initially
positioned just outside the boundary layer (Figure 1e). The snapshots correspond to the
same time-intervals as in Figure 1b. Due to the lack of strong velocity gradients in the
core of the convection cell this heterogeneity is barely deformed at all. The dramatic
difference in mixing efficiency between two different heterogeneities with slightly
different initial condition is maintained over long computational time in this convection
model, as is illustrated in Figure 2.
       The processes of mixing may be conceptually simple, but the mathematical
description is far from complete. The high viscosity of the Earth’s mantle compared to its
thermal diffusivity implies that the inertial terms in the equations of conservation of
momentum (the Navier-Stokes equations for general fluids) are negligible. The resulting
Stokes equations predict laminar (non-turbulent) flow, which nevertheless can be quite
complicated due to non-linear relationships between the equations of mass and heat
transport and the highly temperature- and stress-dependent rheology of mantle rocks. The
temporal distribution of heterogeneity in a fluid is strongly dependent on the mixing
history. Small changes in the initial distribution can lead to exponentially growing
differences, which means that even for simple flows we can witness chaotic or turbulent
effects when observing mixing. This has been named Lagrangian turbulence in contrast
to Eulerian turbulence, which describes the turbulent flow of material as for example seen
in the airflow around airplane wings or the formation of thunderclouds. As discussed
below, many studies find evidence for Lagrangian turbulence in mantle convection


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simulations. The turbulent nature of mixing makes a complete characterization of mixing
impossible except for the simplest systems. For that reason it is essential to ask the
question how we can best characterize mixing for any given situation.




Figure 2. Deformation of heterogeneities I (black) and II (gray) for extended model time.




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      In the remainder of this section we will discuss a variety of classical mixing
indicators and their application to mixing in the Earth’s mantle. To introduce mixing
concepts it is useful to start with a simple layered situation, where each of the layers has a
constant thickness. Stretching in the direction of the layering will cause thinning of the
layers; folding will cause duplication of the initial stacking. If we continue this process of
folding and stretching we will witness an increasingly more complicated stacking of
layers with decreasing thickness. Following the description by Ottino (1989) we can
define simple measures of the efficiency of mixing such as the striation thickness, which
can be defined as the average thickness of two neighboring layers (Figure 1c), or the
intermaterial density area, which is the area of the interface per unit volume. Mixing will
tend to reduce length scales by thinning and consequently reduce the striation thickness
and increase the area of interfaces. In many cases, we see that length scales are reduced
exponentially and that the interfacial area grows exponentially. It becomes impractical to
describe the interface itself, but the exponential nature of mixing allows us to define
typical time-scales of mixing as expressed for example by the increase in interfacial area.
At a local scale we can predict that the distance dx between two particles, that were
originally at distance dX, will increase exponentially in efficiently mixing flows. It is
convenient to define the infinitesimal stretching length =dx/dX in the limit of dX->0. If
the mixing is efficient the stretching length will increase exponentially, and d(ln )/dt >0.
      In order to describe the mechanisms of mixing it is necessary to understand the
local flow behavior. The motion around a point P can be approximated by
        V = vp + dx (v) p + higher order terms
where V is the velocity in a point that is a distance dx away from point P, vp is the
velocity in point P and v is the velocity gradient tensor at point P. The change in motion
can be described by a stretch and a rotation using the stretching tensor D = ½(v +(v)T)
and the spin or vorticity tensor =½(v-(v)T). It can be shown that the mixing length
and the stretching tensor are related by d(ln )/dt (D:D)½ where (D:D)½ is the second
invariant of the stretching tensor which represents a basic measure of the magnitude of
stretching. As a consequence, we can define the mixing efficiency as
        μ = d(ln μ) / dt
                  (D:D)½
For incompressible flows it can be shown that the mixing efficiency has an upper bound
of {(n-1)/n}½ where n is the dimension of the space. It is instructive to display the mixing
behavior of simple shear flow (Figure 3a). After an initial rapid increase of the efficiency,
due to the stretching in simple shear flow, the theoretical 2D maximum of ½2 is
reached. After this the efficiency decays as 1/t since the fluid filaments become fully
aligned with the flow. An efficient way to improve on the efficiency is therefore to
regularly reorient the filaments so that they benefit from the early high efficiency. Such
reorientation can take place at stagnation points in cellular convection or by time-
dependent processes. Figure 3b shows the time evolution of mixing efficiency for a
driven cavity flow, where deformation is dominated by shearing with occasional (‘weak’)
redirection. This is quite similar to the case of 2D steady-state convection. In theory, a
mixing process that combines simple shear flow with strong reorientation can retain high
mixing efficiencies (figure 3c). It is also interesting to note that near fixed points (e.g.,
the corner regions of a single cell convection pattern) the velocity gradients are high. We
can conclude therefore that the mixing efficiency is particularly high near fixed points.


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Although the description of flow around these fixed points is based on a mathematical
simplification, we can expect that the predicted mixing behavior is similar near areas of
strong divergence (mid oceanic ridges, foundering of slabs at the core-mantle boundary)
or converge (subduction zones, base of plumes).




       Figure 3. Illustration of the evolution of mixing efficiency for a) simple
       shear flow; b) simple shear flow with weak reorientation; c) shear flow
       with strong reorientation. Redrawn from Ottino (1989).


3.2 Tools for the study of mantle mixing
Fluid dynamical studies of mixing can be roughly separated into three different categories
based on the approach that is taken. Analytical studies allow generally for a continuous
mathematical description of the physics. Although they are generally limited to only the
simplest mixing geometries, a large number of our descriptive tools are derived from
these analytical studies. Experimental mixing studies are common in many engineering
applications and in studies where the behavior of laboratory fluids can be used to
approximate large-scale processes. For planetary convection applications it is essential to
make extrapolations to the much different length and time scales than those of the
laboratory. An important benefit of the experimental approach is that physical processes
can be studied with much higher resolution than is possible using numerical methods.
Numerical approaches, which solve the governing equations of conservation of mass and
momentum in discrete form, can suffer from discretization errors in time and space,
although convergence tests can be used to determine whether these errors have a great
impact on the solution to the problem that is being investigated. An important benefit of
numerical approaches is that the equations can be solved with the correct spatial and
temporal dimensions without the need to extrapolate from laboratory conditions, and that
data on the properties of the entire fluid can be recorded throughout the convection
experiment.
      Of particular interest for mantle mixing is the dispersion of heterogeneity (e.g., the
efficiency of mixing of sediments and oceanic crust upon subduction into the mantle) and
many studies approach mantle mixing particularly from this perspective. Most
approaches we will discuss here follow from numerical convection experiments that use
concepts derived from analytical and experimental studies. We will introduce and


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illustrate the main approaches in a heuristic manner. For a more rigorous approach see
Ottino (1989) and references therein.

Tools that have been used to visualize the mixing behavior of a fluid include:
 1. Streamlines that provide an instantaneous picture of the velocity field. In stationary
     flows the streamlines represent particle paths. Note that the term ‘stationary’ in
     fluid dynamical literature means ‘steady-state’ and not ‘static’. In other words, a
     stationary velocity field is one in which fluid moves, but the pattern or speed
     doesn’t change.




 Figure 4. Streamlines corresponding to the snapshots in Figure 2. Contour interval is
 50 (non-dimensional) units. Note the focusing of the flow in the center upwelling due to
 the temperature-dependence of the viscosity.




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 2. Particle paths (also called orbits or trajectories) that follow the motion of individual
    particles as they are advected by the velocity field. These can be particularly
    insightful to illustrate regions in the flow that differ in mixing behavior (figure 5).




 Figure 5. Particle paths corresponding to the convective mixing of the model shown in
 Figure 1 and 2. Each particle is followed from t=0 to t=0.0109. The initial position is
 indicated with the solid rectangles. Arrows illustrate the direction of the particle
 motion. The gray particle paths correspond to the efficiently mixed heterogeneity I
 (figure 1d); the black particle paths correspond to the poorly mixing heterogeneity II
 (Figure 1e). Note that the gray particles traverse only about 2 convection circulation,
 but build up significant deformation in this time period (compare Figure 2).

 3. Plots representing the time evolution of a particle cloud representing an initially
    small heterogeneity. This allows for a visual appreciation of the nature of mixing
    and at least for qualitative estimates of the spatial and temporal scales of mixing
    (Figures 1 and 2).
 4. Poincaré maps allow for a descriptive simplification by a reduction of the
    dimension of space. An often-used application of this general mathematical
    mapping is to plot the intersection of particle paths with a fixed surface. These maps
    are useful to identify regions of different mixing properties (e.g., laminar versus
    chaotic) in steady-state flows and identify temporal changes in periodic or time-
    dependent flows (Figure 6).




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Figure 6.Poincaré plots representing the time and horizontal position of particles
traveling through the surface y=0.5. a) for 21x15 particles that are distributed in
heterogeneity I. b) same, but now for heterogeneity II. The general structure of the flow,
the reasonably efficient mixing of a) compared to the more periodic and poor mixing of
b) are clearly illustrated. Note the transition to apparently more efficient mixing after
t=0.015 in b).




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More quantitative methods that have been used to describe mantle mixing include:
 1. Box counting methods. These are simple statistical approaches to describe the
    mixing of a small heterogeneity that use a subdivision of the computational domain
    into a finite number of uniform boxes. By keeping track of the concentration of the
    heterogeneity in each box and the related statistical distribution we can make
    quantitative statements about departures of uniformity and mixing times. We can
    illustrate this by draping a uniform 70x20 cell grid over the convection geometry of
    Figure 1 and evaluating over time the number of particles in each of these boxes.
    Figure 7 displays the fraction of boxes that contain at least one particle of both
    heterogeneities (I and II). By this criterion, the first heterogeneity affects an
    exponentially growing number of boxes early on and becomes nearly box filling
    (>80% of boxes have at least one particle in them) at the end of the computation. In
    contrast, the second heterogeneity remains in only a limited number of boxes until
    about t=0.015, when it also shows an exponential growth. See Schmalzl et al.
    (1996) for further illustrations of this technique in mantle convection models.




 Figure 7. Time evolution of the fraction of boxes that are affected by heterogeneity I
 and II. In this case we used 7000 particles to discretize the heterogeneities and a
 uniform grid of 70x20 boxes.




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 2. Strain markers. The exponential increase of interfacial area in efficient mixing
    suggests an approach in which one keeps track of length scales by following the
    stretching and folding of small line segments, which effectively work as strain
    markers. In practice this is done by following two particles x1 and x2 that were
    initially close together, say at spacing dX. If the mixing is chaotic the length of the
    segment between the two particles will grow exponentially: dx=|x1-x2| = dX exp
    (t). After a certain time t= we can evaluate the finite-time Luyaponov exponent
    If the mixing is chaotic this exponent is positive and the magnitude indicates the
    mixing time scale. In an ideal situation, one would model the distance between the
    two particles by allowing for the full stretching and folding of the initial segment,
    which can be done for example by modeling the line segment with a high resolution
    marker chain. For computational reasons it is nevertheless common to reinitialize
    the strain marker after significant deformation or even to ignore any folding by
    using the distance between the two particles as a proxy for the Luyaponov
    exponent. Figure 8 illustrates the potential for this approach. See the caption for
    details. For other examples of the use of approximations to the finite time
    Luyaponov exponent in mantle mixing studies see Ferrachat and Ricard (1998),
    Van Keken and Zhong (1999), and Farnetani and Samuel (2003).




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 Figure 8. Illustration of the use of Luyaponov exponents to describe the efficiency of
 mantle mixing. a) temperature at t=0. 1000x250 particles pairs are distributed
 uniformly in the area indicated by the black box. The particles in each pair are offset by
 a horizontal distance of 10-5. The small yellow boxes indicate the source regions for
 heterogeneity I and II. We trace the distance between the particle pairs as they are
 advected with the flow and plot their distance at t=0.004 (b) and t=0.02 (c) in their
 original location. The black regions correspond to areas where the distance between
 particle pairs became smaller; the colors indicate minor stretching (dark blue) to
 extensive stretching (dark red).




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 3. Two-particle correlation functions. A statistical approach to quantify the mixing of
    a heterogeneity that is represented by many particles is to calculate the distances
    between each pair of particles and compute the cumulative histogram H(r) which is
    the number of particles pairs that have a distance of less than r. The slope of log(H)
    vs log(r) within a particular range of r indicates the spatial dimension of the particle
    distribution. For example, if in a two-dimensional box calculation the slope is
    constant and equal to 2 for r approaching the length of the box it is reasonable to
    assume that the heterogeneity has been mixed in completely. Illustrations of this
    method are given in Figure 9 and Schmalzl and Hansen (1994).




 Figure 9. Two-particle correlation function H(r) for heterogeneity I and II at time t=0,
 t=0.004, …, t=0.02. H(r) is defined as the number of particle pairs that have a distance
 less than r. Each heterogeneity is traced with 100x70 tracers that are at a distance 10-3
 from each other at time t=0. The graph for t=0.004 is highlighted with black symbols.
 On the left lines of slope 1 and slope 2 are indicated. The first graph is the same for



Van Keken, Ballentine, Hauri             Page 18         Convective mixing       5/16/2011
 both heterogeneities: at distances r less than the size of the heterogeneity the slope is 2,
 indicated that the heterogeneity is space filling at that scale. The position of the change
 in slope to horizontal indicates the spatial extent of the heterogeneity. In the early
 stages of mixing the slope of the correlation function becomes 1 at intermediate scales,
 indicating linear features. At the end of the mixing calculation the correlation function
 has slope 1.5 at most scales, indicating that the linear features are becoming less
 dominant, but that the heterogeneity has not yet become space filling at these scales.
 Note that the change in mixing style after t=0.015 for heterogeneity II is reflected in the
 change in slope and change of position of the kink in the correlation function.




Van Keken, Ballentine, Hauri             Page 19         Convective mixing        5/16/2011
 4. Viscous dissipation. The mixing efficiency of a viscous fluid is related to the
    viscous dissipation  since =D = 2D:D where is the dynamic viscosity.
    Maps of viscous dissipation can therefore be used to qualitatively predict the
    differences in mixing efficiency between different regions of the mantle, or between
    different models of mantle convection (Figure 10).




Figure 10. Viscous dissipation  for the model of Figure 1. Note the high values in the
boundary layers and specifically in the ‘fixed’ points of the flow.



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3.3 Mantle mixing studies

The fluid dynamical approach to mixing provides fundamental tools, but we have to keep
in mind that in our effort to understand mantle heterogeneity we have some unique
circumstances to account for. First, we seem to require mechanisms to prohibit efficient
mixing between ‘reservoirs’. This contrasts strongly with the need to develop efficient
mixers in industrial application to guarantee consistency and provide low-cost
manufacturing techniques. Second, we have an incomplete knowledge of the distribution
of heterogeneity. We only observe the chemical heterogeneity through volcanism at the
Earth’s surface and only at a few locations. The dynamical origin of these ‘leaks’ of the
mantle may bear directly on the observed mixing properties. Third, the isotopic chemistry
that we observe at the Earth’s surface is not only dependent on the distribution of
heterogeneity by mantle mixing, but also on the integrated history of radioactive decay
and fractionation events. Finally, we observe mantle heterogeneity through the filter of
melting. The accompanying chemical and physical differentiation mechanisms may
obscure the relationships between chemical differences in the melt and the mantle source
heterogeneity.
         Quantitative studies of mantle mixing evolved from early laboratory and
numerical experiments that linked plate tectonics with convection in the planetary mantle
(McKenzie, 1969; Richter, 1973; McKenzie et al., 1974; Richter and Parsons, 1975). The
finite strain theory for the development of seismic anisotropy can be used to describe the
deformation and subsequent mixing of mantle heterogeneity as well (McKenzie, 1979). It
was also shown that mantle convection leads to the relocalization and redistribution of
heterogeneity (Richter and Ribe, 1979). Early convection experiments were limited to
low convective vigor (low Rayleigh number) due to computational restrictions.
Extrapolation from these experiments to Earthlike conditions led to the realization that
the typical mixing time for whole mantle convection should be on the order of 0.5-1 Byr
(Hoffman and McKenzie, 1985) which strongly suggests some form of layering to retain
long-lived heterogeneity. In contrast, dynamical studies of layering in the mantle
demonstrated that strict layering at 670 km depth causes the upper mantle to be too cold
(Kopitzke, 1979; Spohn and Schubert, 1982; McNamara and Van Keken, 2000).
Furthermore, the required radioelement enrichment of the lower mantle causes the lower
mantle to be too hot and forms a substantial thermal boundary layer at 670 km depth
(Spohn and Schubert, 1982), unless the mantle heat flow is reduced to values
significantly below the global average (Richter and McKenzie, 1981).
         Several studies explored the ability of a high viscosity lower mantle to slow down
mixing and increase the survival time of heterogeneity, ranging from detailed conceptual
arguments (Davies, 1984) to simple 1D (Loper, 1985) and 2D numerical simulations
(Gurnis and Davies, 1986ab). These models demonstrated the possibility for significantly
longer survival times due to the high viscosity of the lower mantle, but this comes at the
cost of a much reduced overall convective vigor and too low surface heatflow.
         Improved computational ability and better understanding of non-linear systems
allowed for fundamental improvements in mantle mixing studies. A detailed study by
Olson et al. (1984) used Fourier analysis to demonstrate that the effects of mixing
cascade from low to high wavenumbers and demonstrated the importance of stagnation
points in providing more efficient deformation. Olson et al. (1984) showed that mixing in



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a stationary cell is best described by laminar mixing. The inclusion of time-dependence
(e.g., Hofmann and McKenzie, 1985) allows for more efficient mixing. Christensen
(1989) demonstrated a further complication: flows that are driven by time-dependent
kinematic surface boundary conditions are more likely to maintain poorly mixed islands
or to be characterized by laminar mixing than time-dependent dynamic models, where
heterogeneities are rapidly strained and follow an exponential, or turbulent mixing law.
Kellogg and Stewart (1991) confirmed the exponential stretching behavior in a 12 Fourier
mode approximation to the thermal convection equations.
        As was demonstrated by several of these early mixing studies, the stretching and
folding of heterogeneity leads to a ‘marble cake’ structure, where the distribution of
length scales is generally exponential and depends on the mixing efficiency. Kellogg and
Turcotte (1990) showed this using a kinematic, but space-filling 2D time-dependent flow
based on the Lorenz equations, with a special focus on the thinning of oceanic crust. The
predicted distribution of length-scales matches the observed distribution of pyroxenite
bands in depleted lherzolite found in several high-temperature peridotites (Allègre and
Turcotte, 1986). This supports the suggestion that the pyroxenite originated from
subducted oceanic crust, which would provide an intriguing geological constraint on the
efficiency of mantle mixing.
        Our understanding of mantle mixing has further improved in the last decade
particularly due to the ability of numerical models to explore the effects of realistic
convective vigor and three-dimensional flows, the incorporation of new methodology to
characterize mixing, and the explicit modeling of the geochemical inputs and outputs to
the mantle system. Schmalzl and Hansen (1994) demonstrated using 2D isoviscous
models at high Rayleigh number that advective stirring leads to mixing on the large scale
first and that mixing between convective cells can be quite inefficient, even if the mixing
within each cell is very efficient. Schmalzl et al. (1995) explored mixing in a 3D steady-
state model. They found that particle motion in the flow is limited by symmetry surfaces
imposed by the temperature and velocity field and regular, non-chaotic motion the
particles, from which they concluded that time-dependence is probably essential in
generating chaotic particle behavior and efficient mixing. The addition of time-
dependence in a 3D model at moderate Rayleigh number (Schmalzl et al., 1996)
demonstrated that cross-cell mixing is enhanced, but the efficiency appeared to be less
than in the case of 2D convection, probably due to the greater stability of the 3D
convective pattern.
        Three-dimensional convective patterns can be described by a combination of
poloidal (divergent and convergent) and toroidal (strike-slip) components. Ferrachat and
Ricard (1998) used a kinematic model of convection driven by a surface boundary
condition that mimicked a mid-oceanic ridge with a variable length transform fault. The
additional toroidal energy caused a significant portion of the model to undergo chaotic
mixing, although islands of laminar mixing could persist. These results suggest that it is
essential to incorporate the influence of plates on mantle convection in mixing studies.
Van Keken and Zhong (1999) further demonstrated the importance of toroidal motion,
using a dynamic model of present day mantle convection. In this model, toroidal motion
is generated by the influence of the weak zones corresponding to the present-day plate
boundaries. Several regions of laminar mixing are present in this model, but in certain
areas such as near Papua New Guinea and the western US, significant toroidal motion



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caused cork-screw-like particle paths. This allows for large-scale transport between the
convective cells and for efficient, chaotic mixing.
        The increasing spatial and temporal resolution of numerical models allows for
direct testing of geochemical hypotheses by tracking the evolution of particles that carry
isotopic and elemental information. Christensen and Hofmann (1994) provided one of the
first quantitative models that detailed the recycling of oceanic crust and its influence on
the chemical evolution of the mantle. Their models included the formation of basaltic
crust and depleted harzburgite upon melting of the peridotite mantle. Upon
transformation to eclogite, the oceanic crust ponds at the base of the mantle and is only
slowly entrained by thermal convection. The modeled Pb and Nd isotopic evolution after
3.6 Byr provided an adequate fit to the observed MORB and HIMU ranges, which
provides a strong quantitative support to the suggestion that oceanic crust recycling is of
fundamental importance to the chemistry of mantle plumes (Hofmann and White, 1980;
White and Hofmann, 1982). Van Keken and Ballentine (1998; 1999) developed 2D
cylindrical models of mantle convection to study the evolution of He isotopes. When
comparing models of similar convective vigor as measured by average surface heat flow
and plate velocities it was shown that a higher viscosity lower mantle was not capable of
preserving heterogeneity over timescales longer than about 1 Byr. Incorporation of phase
changes and moderately temperature-dependent rheology similarly couldn’t prevent large
scale mixing, except in the case of an extreme, and probably unrealistic, dynamical
influence of the endothermic phase change at 670 km depth. Similar results were
obtained by Ferrachat and Ricard (2001), Davies (2002), and Stegman et al. (2002).
        A distinct class of models that describe the chemical evolution of the Earth are the
so-called ‘box models’, in which assumptions are made about the geometry of distinct
reservoirs and their interactions. For example, one can assign four distinct reservoirs in
the upper mantle, lower mantle, continental crust and atmosphere and develop differential
equations that incorporate radiogenic ingrowth, chemical fractionation effects and
assumptions about mass transfer between the reservoirs. Successful models reproduce the
observed isotopic ratios and/or concentrations. Applications include studies of noble gas
evolution (Kellogg and Wasserburg, 1990; O’Nions and Tolstikhin, 1994; Porcelli and
Wasserburg, 1995ab; Albarède, 1998), crustal evolution (Jacobsen and Wasserburg,
1979; O’Nions et al., 1979; Jacobson, 1988), mean stirring times of the mantle (Allègre
and Lewin, 1995) and lead-isotope systematics (Galer and O’Nions, 1985; Kramers and
Tolstikhin, 1994; Paul et al., 2002).
        An important advantage of this type of modeling is that the computational cost is
much lower than that of full dynamical models, which allows for the efficient exploration
of large parameter spaces and for data inversion to obtain optimal parameter choices
(e.g., Allègre and Lewin, 1989; Coltice and Ricard, 1999). More complex physics can be
integrated into the box model approach. For example, the internal mixing rate in
individual reservoirs can be described in parameterized form, as was demonstrated by
Kellogg et al. (2002). It is expected that we will see a further integration between the box
and fluid dynamical modeling approaches, such as the comparison made by Coltice et al.
(2000).




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4. Outlook
4.1 Conceptual model development

The improved geochemical and geophysical observations, coupled with enhanced
computational ability, has led to an expansion of research on the chemical evolution of
the Earth’s mantle. The apparent contradictions between geophysical indications for
whole mantle flow and the geochemical requirements for substantial and long-lived
heterogeneity have led to a large set of new or revived conceptual models. Proposals
include mantle layering below 670 km depth (Kellogg et al., 1999; Anderson, 2002),
‘zoning’ of the mantle due to the variable thermal and chemical properties of subducting
slabs (Albarède and Van der Hilst, 2002), chemical variations hidden in small and highly
viscous blobs (Becker et al., 1999), and recent breakup of a layered mantle system
(Allègre, 2002). For a review of the pros and cons of these various models see e.g., Van
Keken et al. (2002).
        An alternative model that is gaining rapid popularity incorporates the recycling of
oceanic crust in whole mantle convection (e.g., Coltice and Ricard, 1999; Ferrachat and
Ricard, 2001; Helffrich and Wood, 2001; Van Keken et al., 2002; Davies, 2002; Carlson
and Hauri, 2003). This idea has several attractive aspects. The formation of oceanic crust
at mid-oceanic ridges and the subsequent recycling at subduction zones is the foremost
modern-day method of introducing heterogeneity into the mantle. The signal of recycled
oceanic crust in ocean island lavas has been clearly shown in a number of isotope
systems (e.g., Hofmann and White, 1980; White and Hofmann, 1982; Hauri and Hart,
1997; Lassiter and Hauri, 1998; Eiler et al., 2000; Hauri, 2002). The higher density of
basalt-derived components in the mantle allows for a natural explanation for long
residence times.
        Commonly cited problems with this model include the difficulty in explaining the
noble gas observations and the distribution of heat producing elements. In particular, the
observation of high 3He/4He in hotspots has been interpreted as the consequence of
preservation of a ‘primitive’ or ‘primordial’ component, since 3He is not produced in the
Earth’s interior. Yet, the oceanic basalts with high 3He/4He are non-primitive by any
other geochemical measure (e.g., Hart et al., 1992; Hanan and Graham, 1996) and this
would suggest that low concentrations of U and Th, and consequently low concentrations
of 4He compared to 3He are responsible for the high 3He/4He ratio. This in turn enables a
small 3He residue to account for the high OIB 3He/4He (Anderson, 1998b; Albarède,
1998; Coltice and Ricard, 1999). Quantitative support for this non-primitive source of
high 3He/4He is provided by a number of models (Ferrachat and Ricard, 2001; Coltice
and Ricard, 2002). Other calculations show that because of the residence time of recycled
material (1 to 1.5 Byr), U+Th depletion of the protolith is not required if this is mixed
with a small volume of 3He rich material (Ballentine et al., 2002). The latter calculations
are consistent with radioelement and noble gas concentrations inferred from the Iceland
plume (e.g., Hilton et al., 2000).
        The concentration of rare gases in the upper mantle can provide additional
observational evidence. The 3He concentration in the oceans has been used to argue for a
low helium concentration in the upper mantle and a requirement for the deep burial and


Van Keken, Ballentine, Hauri             Page 24        Convective mixing       5/16/2011
isolation of helium (e.g., O’Nions and Oxburgh, 1983). The related low estimates for the
40
   Ar concentration in the upper mantle provide a similar argument for argon and
potassium storage in the deep mantle. It is interesting to note that these arguments are all
dependent on the assumption that the present day concentration of helium in the oceans,
which has a residence time of about 1000 years, allows for accurate estimate of upper
mantle concentrations. It is not unreasonable to expect that due to variable volcanic
activity the present-day 3He concentration in the oceans may not be representative of the
time-averaged concentrations in the mantle (e.g., Van Keken et al., 2001). In fact, an
increase in the noble gas concentration of the upper mantle by a factor of 3.5 removes all
requirements for layering and hidden noble gas reservoirs (Ballentine et al., 2002).
        An additional argument for mantle layering is based on the mantle concentration
of heat producing elements. The extremely depleted nature of N-MORB (Hofmann,
1988) suggests quite low U, Th, and K concentrations in the upper mantle, which would
require substantial deeper enrichment (e.g., Kellogg et al., 1999). A new compilation of
MORB analysis (Su and Langmuir, 2002) from the PetDB database (Lehnert et al., 2000)
suggests that the average oceanic crust is considerably less depleted than N-MORB and
that consequently the upper mantle composition is far less depleted than previously
assumed. This new analysis effectively removes the requirement for storage of high heat
producing element concentrations in the deep mantle (Carlson and Hauri, 2003).

4.2 Quantitative modeling

Any conceptual model requires quantitative support, ideally through an integrated
physical and chemical modeling approach. Even if we consider that we have come a long
way in developing better geodynamical models, we are still faced with significant hurdles
in the development of consistent dynamical models that would allow a full understanding
of the dynamical and chemical evolution of the Earth. A number of important concepts
that need better resolution include:
   The integration of plate tectonics and mantle convection (e.g., Tackley, 2000a). It is
      evident that the large-scale plate structure has an important influence on the mantle
      convection pattern (e.g., Anderson, 1998a). Thus far it has been very difficult to
      develop models that satisfactorily produce self-consistent, and persistent, plate
      tectonic style deformation on top of mantle convection, although encouraging
      progress has been made in recent years (e.g., Trompert and Hansen, 1998; Tackley,
      1998, 2000bc, Tackley and Xie, 2002; Bercovici, 2003). This is of particular
      importance for the testing of conceptual mixing models since otherwise it is
      difficult, if not impossible, to use realistic temperature-dependent rheology or to
      accurately differentiate between mid-oceanic ridge and hotspot sampling of the
      mantle.
   The cause and source regions of hotspots. Since plumes are considered to deliver
      the signals from the deep mantle it is essential to understand how plumes influence
      mantle mixing. Of particular concern are the low internal viscosity of mantle
      plumes and the interaction with the transition zone and lithosphere (e.g., Hauri et
      al., 1994; Marquart and Schmeling, 2000; Farnetani et al., 2002; Kumagai, 2002).
   The influence of rheological variations on mantle mixing, such as proposed for the
      survival of primitive blobs (Manga, 1996; Becker et al., 1999) or the mixing of


Van Keken, Ballentine, Hauri              Page 25        Convective mixing        5/16/2011
     oceanic crust (Spence et al., 1988; Van Keken et al., 1996; Merveilleux du Vignaux
     and Fleitout, 2002). The influence of non-Newtonian rheology on mantle mixing
     also requires much more detailed investigation (e.g., Ten et al., 1997).
    Influence of chemical buoyancy due to melt extraction (Christensen and Hofmann,
     1994; Dupeyrat et al., 1995; Davies, 2002). Bulk chemistry changes upon melting
     are likely of fundamental importance in maintaining heterogeneity since the
     associated chemical buoyancy changes can effectively counteract thermally driven
     mixing.
    The role of the mantle transition zone. Seismological observations and petrological
     predictions strongly suggest an important dynamic role for the transition zone (e.g.,
     Ringwood, 1982). This includes the influence of the phase changes, possible
     variations in minor and major element chemistry, and rheological variations. In
     particular the 670 km discontinuity has been suggested to function as a partial
     chemical filter (e.g., Christensen and Yuen, 1984; Weinstein, 1992; Van Keken et
     al., 1996).
    Structure and dynamics of the lowermost mantle. This region includes the D” layer,
     which is characterized by major chemical and thermal variations. It is likely of
     fundamental importance to the chemical evolution of the mantle and may function
     as a (temporary) resting place for subducted slabs. It is also expected to influence
     the stability of mantle plumes (Davaille et al., 2002; Jellinek and Manga, 2002), the
     entrainment and residence times of chemical heterogeneity (Olson and Kincaid,
     1991; Schott et al., 2002) and the thermal, chemical, and seismological
     characteristics of compositional variations (Kellogg et al., 1999; Tackley, 2002).
    Thermal evolution of the Earth. In particular we need to understand how we can
     construct chemical models that provide sufficiently realistic cooling histories. It has
     been pointed out that the relatively low present day heat production in the bulk
     silicate Earth as compared to the present day heat loss requires some form of
     buffering or layering (e.g., Butler and Peltier, 2002). We also need to understand
     the tectonic regime during the Hadean and Archean. The projected higher mantle
     temperatures in the Archean lead to conditions that are not necessarily conducive to
     faster plate tectonics. The deeper melting that occurs in a hotter Earth causes more
     extensive melting and the thicker basalt and depleted peridotite layers provide a
     strong compositional stabilizing force to plate tectonics. The tectonics of the
     Archean Earth may well have been dominated by thermal plumes or eclogite driven
     delamination (e.g., Anderson, 1979; Vlaar et al., 1994; Zegers and Van Keken,
     2001).

In conclusion, the renewed interest in the integration of geophysical and geochemical
approaches to develop a better understanding of the chemical evolution of the mantle has
led to a significant number of new or revived conceptual models. Models that combine
geodynamics and geochemistry can provide quantitative tests to these ideas. Although
several hurdles still exist, we can expect that the growth of computational resources -
combined with better insights into the role of chemical mixing, improved geochemical
and seismological observations, and a more fundamental understanding of the
interpretation of Earth structure from these observations –will ultimately allow for the



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development of a consistent description of the Earth’s mantle evolution as it is influenced
by the generation and destruction of chemical heterogeneity.

Acknowledgments
We thank Francis Albarède, Don Anderson, Geoff Davies, Jamie Kellogg, Louise
Kellogg, Dan McKenzie, Don Porcelli, and Paul Tackley for stimulating discussions and
Rick Carlson for encouragement.




Van Keken, Ballentine, Hauri             Page 27         Convective mixing       5/16/2011
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