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The focus of this paper is numerical modeling of crust-mantle differentiation. We begin by surveying the observational constraints of this process. The present-time distribution of incompatible elements are described and discussed. The mentioned differentiation causes formation and growth of continents and, as a complement, the generation and increase of the depleted MORB mantle (DMM). Here, we present a solution of this problem by an integrated theory that also includes the thermal solid-state convection in a 3-D compressible spherical-shell mantle heated from within and slightly from below. The conservation of mass, momentum, energy, angular momentum, and of four sums of the number of atoms of the pairs 238U- 206Pb, 235U-207Pb, 232Th-208Pb, 40K-40Ar is guaranteed by the used equations. The pressure- and temperature-dependent viscosity is supplemented by a viscoplastic yield stress, σy . No restrictions are supposed regarding number, size, form and distribution of continents. Only oceanic plateaus touching a continent have to be united with this continent. This mimics the accretion of terranes. The numerical results are an episodic growth of the total mass of the continents and acceptable courses of the curves of the laterally averaged surface heat flow, qob, the Urey number, Ur, and the Rayleigh number, Ra. In spite of more than 4500 Ma of solid-state mantle convection, we typically obtain separate, although not simply connected geochemical mantle reservoirs. None of the reservoirs is free of mixing. This is a big step towards a reconciliation of the stirring problem. As expected, DMM strongly predominates immediately beneath the continents and the oceanic lithosphere. Apart from that, the result is a marble-cake mantle but DMM prevails in the upper half of the mantle. We find Earth-like continent distributions in a central part of Ra-σy plot obtained by a comprehensive variation of parameters. There are also Ra-σy areas with small deviations of the calculated total continental volume from the observed value, with acceptable values of Ur and with realistic surface heat flow. It is remarkable that all of these different acceptable Ra-σy regions share a common overlap area. We compare the observed present-time topography spectrum and the theoretical flow spectrum n 1/2 × (n + 1)1/2 × (v 2n,pol ).
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Whole-Mantle Convection, Continent
Generation, and Preservation of Geochemical
Heterogeneity
Uwe Walzer
1
, Roland Hendel
1
, and John Baumga rdner
2
1
Institut ur Geowissenschaften, Friedrich-Schiller-Universit¨at,
Burgweg 11, 07749 Jena, Germany u.walzer@uni-jena.de
2
Dept. Earth Planet. Science, University of California, Berkeley, CA 94720, USA
Summary. The focus of this paper is numerical modeling of crust-mantle differ-
entiation. We begin by surveying the observational constraints of this process. The
present-time distribution of incompatible elements are described and discussed. The
mentioned differentiat ion causes formation and growth of continents and, as a com-
plement, the generation and increase of th e depleted MORB mantle (DMM). Here,
we present a solution of this problem by an integrated theory that also includes th e
thermal solid-state convection in a 3-D compressible spherical-shell mantle heated
from within and slightly from below. The conservation of mass, momentum, energy,
angular momentum, and of four sums of the number of atoms of the pairs
238
U-
206
Pb,
235
U-
207
Pb,
232
Th-
208
Pb,
40
K-
40
Ar is guaranteed by the used equations. The
pressure- and t emperature-dependent viscosity is supplemented by a viscoplastic
yield stress, σ
y
. No restrictions are supposed regarding number, size, form and d is-
tribution of continents. Only oceanic plateaus touching a continent have to be united
with this continent. This mimics the accretion of terranes. The numerical results are
an episodic growth of the total mass of the continents and acceptable courses of the
curves of the laterally averaged surface heat flow, qob, the Urey number, Ur, and
the Rayleigh number, Ra. In spite of more than 4500 Ma of solid-state mantle con-
vection, we typically obtain separate, although not simply connected geochemical
mantle reservoirs. None of the reservoirs is free of mixing. This is a big step towards
a reconciliation of t he stirring problem. As expected, DMM strongly predominates
immediately beneath the continents and the oceanic lithosphere. Apart from that,
the result is a marble-cake mantle but DMM prevails in the upper half of the mantle.
We find Earth-like continent distributions in a central part of Ra-σ
y
plot obtained
by a comprehensive variation of parameters. There are also Ra-σ
y
areas with small
deviations of the calculated total continental volume from the observed value, with
acceptable values of Ur and with realistic surface heat flow. It is remarkable that
all of these different acceptable R a-σ
y
regions share a common overlap area. We
compare the observed present-time topography spectrum and the theoret ical fl ow
spectrum n
1/2
× (n + 1)
1/2
× hv
2
n,pol
i.
2 Uwe Walzer, Roland Hend el, and John Baumgardner
1 Introduction: Whole-Mantle Convection and
Geochemistry
Solid state convection currently explains the thermal evolution of the Ea rth
and the plate-tectonic regime in a satisfying manner (Schubert et al. [94]).
How well the Earth’s chemical evolution is accounted for has not been so
clear. Chemical differentiation alters the distribution of radioactive elements
and volatiles and generates geochemical heterogeneity. On the other hand,
convection diminishes and annihilates chemical heterogeneities by stirring in
low-viscosity regions. Convection recycles the oceanic crust and transports
and mixes mantle lithosphere into the deeper mantle. Mixing is suppressed
and chemical signatures tend to be preserved, on the other hand, where mantle
viscosity is high. Generation, preservation and destruction of hete rogeneities
are obviously influenced by the interplay between chemical differentiation,
convection and secular cooling of the Earth’s mantle. Only the present exis -
tence of remnants of the primordial mantle is a controversial question. It is
evident that the chemical evolution of the mantle cannot be modeled with-
out accounting for the convective process and the mixing it generates. On
the other hand, the fact that chemical differentiation causes the distribution
of heat-producing elements in the mantle to be non-uniform, no doubt, in-
fluences convection. This paper seeks to ac c ount for both differentiation and
convecting/mixing together. We check our overall model against reality by
comparing the evolution of continents, oceanic plateaus and o f the MORB
source that we obtain in our models with observations.
Modern seismic techniques document convincingly that subducted oceanic
lithosphere penetrates into the lower mantle. Other seismic methods give
strong support to the conclusion that at least some hotspots rise upward and
penetr ate the 660-km phase boundary (Van der Hilst et al. [109], Grand et al.
[39]). Moreover, differential travel times may be used to determine the verti-
cal distance between the olivine-wadsleyite discontinuity and the ringwoo dite-
perovskite discontinuity and its lateral variation. From this it has been de-
duced that at least some hotspots penetrate the 660-km boundary (Bina
[14]). At the corr esponding places, a thinning of the transition zone thick-
ness has been observed as expected from corresp onding Clausius-Clapeyron
slopes (Schubert et al. [94], chapter 4.6.2). It is evident that the topography
of the mentioned seismic discontinuitie s can be explained by thermal pertur-
bations of phase transformations in a peridotite mantle (Helffrich and Wood
[44]). There are also direct evidences of plumes penetrating the 66 0-km discon-
tinuity by seismic tomography. These studies demonstr ate that the transition
zone is not impermeable either for slabs or for plumes. Tomography sugge sts
that some plumes originate in the D
′′
layer. Other plumes pr obably orig inate
from shallower depths, e.g., nea r the 660-km discontinuity (Courtillot et al.
[29], Montelli et al. [75]).
Some have proposed that the bulk compositions of the upper and the lower
mantle are different, especially in the Mg/Fe and Mg/Si ratios, and that the
Mantle Convection, Continent Generation and Geochemical Heterogeneity 3
660-km discontinuity is not only a phase transition from ringwoodite to per-
ovskite plus magnesiow¨ustite and from majorite g arnet to Al-bearing Mg-rich
perovskite (Hirose [46]) but also a discontinuity in the major-element abun-
dances. Jackson and Rigden [55] showed, however, tha t the essential features
of the seismological models can be well explained by the phase transitions of
a pyrolite model, as a special case, and that, in the general case, the current
knowledge of the s e ismic structure of the mantle is consistent with a uni-
form major-element composition o f the mantle with the probable exception
of the D
′′
layer. From a geochemical point of view, Palme and O ’Neill [80]
concluded that, concerning the major elements, the whole E arth’s mantle is
compositionally uniform if we consider averages of large volumes. If the abun-
dances, normalized to Ti and CI, are drawn versus equidistant locations of the
carbonaceous chondrites CI, CM, CO, CK, CV and the Earth’s mantle, then
constant-value lines result for Al and Ca and monotonously decr e asing curves
arise from this procedure for other major and moderately volatile elements.
So, the Earth’s mantle may be considered as an extension of the car bonaceous
chondrite trend. Only the C r content is too low in the present mantle as a
result of Cr partitioning into the outer core. Bina [14] also concluded that the
seismic arguments for a chemical boundary of the major elements at the 6 60-
km discontinuity have faded. An isochemical mantle is consistent with seismic
results. However, this conclusion applies only fo r the major elements since it
is these elements that constrain the bulk physical parameters such as com-
pressional velocity and density which can be c omputed for a perfect lattice.
Major elements are the essential constituents of rock- fo rming minerals. Trace
elements often act only as tracers (Hofmann [50]). However, other important
physical quantities such as shear viscosity, seismic shear velo city, electrical
conductivity, etc. depend on the concentrations of Schottky holes, Frenkel de-
fects and dislo c ations which, in turn, strongly depend on small amounts of
volatiles. Also small concentrations of U, Th and K influence the heating rate
and hence the buoyancy that drive s convection. To summarize this introduc-
tion, we conclude that there is strong evidence for whole-mantle convection.
In regard to the major elements, averaging over large volumes, current evi-
dence points to a chemically homogeneous mantle except for the lithosphere
and its counterpart at CMB, the D
′′
layer. It is quite possible that this simple
picture may have to be modified somewhat since Trampert et al. [106] showed
that comp ositional variation affects the seismic velocities, v
p
and v
s
, not only
in the D
′′
layer but also in most of the lowermost mantle.
4 Uwe Walzer, Roland Hend el, and John Baumgardner
2 Observational Constraints
2.1 The Evolution of Continents, the MORB Source and Other
Possible Geochemical Reservoirs
Geochemical Preliminary Examinations
The continental crust (CC) (Hofmann [48], McCulloch and Bennett [71], Rud-
nick and Gao [91]) is extraordinarily enriched in incompatible elements com-
pared to the bulk silicate earth (BSE). Therefore it is to be expected that
there are one or more regions in the mantle that are depleted in incompatible
elements in a complementary ma nner. Mid-oceanic ridge basalt (MORB) is
supplied by a source region tha t has to be situated immediately below the
oceanic lithosphere s ince, independently where oceanic plates are spreading
apart, MORB is surprisingly homogeneous in its trace element composition.
This sour c e region appears to be very well stirred in comparison to that of
ocean-island ba salts (OIB). All`egre and Lewin [4] emphasize the homogeneity
of MORB. Let R/R
A
denote the mass ratio
3
He/
4
He of rock divided by the
present-day
3
He/
4
He value of the atmospher e . Barfod et al. [8] showed that
the R/R
A
values of ridge basalts are around 8 with a small standard deviation.
HIMU-like OIBs have lower R/R
A
values with greater standard deviations.
Nearly all other OIBs have augmented R/R
A
ratios with large standard de-
viations. However, Hofmann [50] p ointed out that also the depleted MORB
mantle (DMM) is isotopically not entirely uniform.
Hofmann [50] reported a fundamental obse rvation: The abundance s of the
elements Rb, Pb, U, T h, K, Ba, La, Nb, Sr, Na, Yb, Al, Ca, Si, Sc, Mn, Co
and Mg, normalized by their BSE abundances (McDonough and Sun [73])
have be e n plotted versus their degree of compatibility fo r three reservoirs:
continental crust (Rudnick and Fountain [90]), MORB (Su [99]), and DMM.
If the generation of CC would be a one-act differentiation process then the
result could be easily explained as follows: The more incompatible an element
is the stronger it is enriched in CC since it rose within the melt to this place.
DMM is the complementary mantle part where the abundances of the most
incompatible elements have the lowest values. If this depleted ma ntle region
is again partly melted, the differentiation product is a typically oceanic crust
(MORB) with an observed normalized concentration curve of the mentioned
elements that is between the CC and the DMM curve. So compared to DMM,
the MORB curve is also somewhat enriched.
However, the formation of CC is not a s ingle-stage process, but a multi-
stage process. The enrichment of CC happens by three kinds of processes:
(a) partial melting in the uppermost parts of the mantle and ascent into the
oceanic crust, (b) dehydration and decarbonation in subduction zones w ith
metasomatic transport (Hofmann, [50]), and (c) generation of plateau basalts.
The source rock s for plateau basalts are relatively enriched par ts of the man-
tle that penetrated into the uppermost mantle layer that has now usually a
Mantle Convection, Continent Generation and Geochemical Heterogeneity 5
DMM composition. Plateau basalts a re divided into three classes: continental
flood basalts, oceanic plateaus and ocean basin flood basalts (Coffin and Eld-
holm [25]). The latter two classes of large igneous provinces (LIP) are driven
near the continental margin by the conveyor-belt-like oceanic lithosphere and
accreted to the continent in the zones of andesitic volcanism. Generation of
LIPs is episodically distributed over the time axis of the Earth’s history. LIP
volcanism was dominant during the Alpidic orogenes is, in contrast to the
MORB volcanism in the orogenetically somewhat more quiet period of to-
day. On o ther terre strial planets of our solar system, LIP volcanism appears
to be the dominant form. A terrestrial example is the Ontong Java Plateau.
Its basalt developed from a 30% melting of a peridotite source. It can only
be achieved by dec ompression of hot material with a potential temperature
>1500
C beneath a thin oceanic lithosphere. As exp e cted, the Ontong Java
Plateau basalt is enriched in U, Th and K in comparison to Pacific MORB.
It is also isotopically and chemically distinct from it (Fitton et al. [34]).
Furthermore, the present-day proportion of DMM of the mass of the man-
tle is not exactly known since it depends on which element is used and on
its assumed abundance in CC and DMM. If samarium and neodymium are
not so extremely enriched in CC, then smaller volumes of DMM would be
sufficient. 50% of depleted mantle is derived from Cs, Rb, Th and K. Ben-
nett [10] estimated that between 30% and 60% of the mantle is depleted if an
intermediate degree of depletion of DMM is assumed. Hofmann [50] deduced
a depleted reservoir of the mantle between 30% and 80%. Van Keken et al.
[110], however, itemized some strong arguments against the opinion that the
rest of the mantle is simply a BSE reservoir. Hofmann et al. [51] inve stigated
the mass ratio Nb/U of MORB and IOB and found a very similar average.
The same conclusion applies for Ce/Pb. Rudnick and Fountain [90] derived
Nb/U=47±11 for MORBs, Nb/U=52±15 for OIBs but Nb/U=8 for CC. The
trace element ratios Ce/Pb, Nb/U, and Nd/U pr oved to be nearly identical
for MORB and OIB and to be nonprimitive, i.e., there is no c orrespo ndence
with BSE. Evidently, it is not possible to derive the BSE abundances sim-
ply from the MORB and OIB abundances. Some authors concluded from the
observations that there is no present-day primordial material at a ll in the
mantle. However, Hofmann [50] emphasized that only mass ratios of similarly
incompatible pairs o f elements are suitable fo r addressing this issue.
When the isoto pe ratio of a single e lement is plotted against the isotope
ratio of another element, for many basaltic rocks from around the world, in-
cluding OIBs a nd MORBs, they tend to scatter into distinct mixing lines.
At each end of such a line is an extremal case. If, for example, the present-
day mass ratio of
206
Pb/
204
Pb is plotted versus
87
Sr/
86
Sr then four distinctly
separated mixing lines typically appear, the ends of which are called HIMU,
DMM, EM1 and EM2 (cf. Hofmann [50], Fig. 15). HIMU stands for hig h µ
where µ = (
238
U/
204
P b)
τ =0
(Houtermans [53], Zindler and Hart [12 4]). τ
is the age. One interpretation is that these end compositions represent only
extremes of a continuum of mixtures of isotopic comp ositions that are dis-
6 Uwe Walzer, Roland Hend el, and John Baumgardner
tributed on various spatial scales througho ut the mantle. The other opinion
is that these end-member compositions represent separate distinct reservoirs
in different regions of the mantle. However, the observed compositions are not
evenly distributed in
206
Pb/
204
Pb–
87
Sr/
86
Sr space. Instead, four linear trends
dominate. Not only this observa tion but also the following hint by Hofmann
[49] argues for the second option, namely that nearly pure HIMU basa lts are
found not only in a single ocean island group like the Cook-Austral chain but
also on St.Helena on the other side of the globe. The latter argument is not
invalidated by the fact that Mangaia, an individual island of the Cook chain,
is distinctly different from the observed HIMU of the other Cook-Austral is-
lands and St. Helena (Stracke et al. [98]). HIMU has not been observed in
MORB and is rare in OIB. It could repre sent ancient r e cycled oceanic crust
(Stracke et al. [98]). EM1 and EM2 are more enriched in very incompatible el-
ements c ompared to HIMU. EM2 shows maximum
87
Sr/
86
Sr values at nearly
constant
206
Pb/
204
Pb19 whereas EM1 forms a distinctly separated line be-
neath the EM2 line and is distinctly sepa rated regarding
206
Pb/
204
Pb. The
usual explanation for EM1 is tha t it is generated by recycling of oceanic crust
plus p ortions of lower CC or subcontinental lithosphere or ancient pelagic
sediments. EM2 is customarily explained by recycling of oceanic crust with
portions of the upper CC (Willbold and Stracke [116]). The la tter authors
deny the distinction between EM1 and EM2. They propose a flowing transi-
tion and explain the common EM by subduction of oceanic crus t with variable
proportions of lower and upper CC.
If all mixing arrays are plotted, e.g. in a three-dimensional
206
Pb/
204
Pb–
87
Sr/
86
Sr–
143
Nd/
144
Nd diagram then all mixing lines aim at a small volume
called FOZO (or focal zone) according to Hart et al. [42]. There are also re-
lated proposals: PREMA (orner et al. [118], Zindler and Hart [124]), PHEM
(Farley et al. [33]) and C (Hanan and Graham [41]). FOZO is significantly
more r adiogenic in lead isotopes than DMM, moderately more ra diogenic in Sr
and less radiogenic in Nd and Hf (Hofmann [50]). Fur thermore, FOZO has a
higher
208
Pb/
206
Pb ratio than HIMU. Although FOZO is evidently produced
by subduction and is by no means primordial, FOZO can play the part of the
rich principal reservoir in our present convection-differentiation model, rich
in U, Th and K. There would be a broad mixing zone between the FOZO
and the DMM res ervoir. FOZO would be the main source in the OIBs while
EM1, EM2 and HIMU represent contributions from minor reservoirs. The
latter ones have not to be joined to one region each. The different contribu-
tions of the various minor reservoirs generate the large is otopic and chemical
standard deviations of OIBs, large in comparison to that of MORBs. Stracke
and Hofmann [98] redefined a new FOZO that is similar to the traditional
FOZO according to H art et al. [42]. They propose tha t this new FOZO could
be a ubiquitously dispersed, small-scale component in the entire mantle. We
remark that, according to our present dynamical model, the percentage of
FOZO in the upper half of the mantle should be less than in the lower half.
Mantle Convection, Continent Generation and Geochemical Heterogeneity 7
This result corr e sponds with Wilson’s and Spencer’s [117] conclusion that
FOZO is the characteristic signal of lower mantle plumes.
Geochemical models that totally aba ndon the assumption of BSE remnants
in the present-day mantle are unable to explain the observation that the flux
of
3
He is unidirectional, that, e.g., Reykjanes Ridge has a
3
He content 15
times higher than that of DMM and that the averaged
3
He concentration in
the plume sources is 4 times higher than that of DMM. Hilton and Porcelli
[45] are convinced that at present 10 39-2270 mol·a
1
primordial
3
He leaves
the Earth. Trieloff and Kunz [1 07] systematically disc uss the problem of noble
gases in the Ea rth’s mantle. One proposal for the source of primordial noble
gases has b e e n the Earth’s core (Tolstikhin and Marty [104], Porcelli and
Halliday [85]). This proposal is unconvincing since the viscosity of the outer
core is between 1 and 100 Pa·s and it circulates with velocities between 10
and 30 km/a. So, each volume element of the outer core has been in frequent
contact with the CMB during the Earth’s evolution: If the mantle is unable to
retain its
3
He, then the outer core cannot retain its inventory either. But for
the dynamical theory, presented in this paper, it is irrelevant where exactly the
3
He source region is situated. It is only important tha t there are regions within
the present mantle which have higher abundances of U, Th and K than occur
in the DMM. MORBs and OIBs are chemically distinct and their standard
deviations are different. So, it is improbable that both of them originate from
the same quasi-homogeneous DMM.
Moreover, the present-day heat output of the mantle is 36 TW: it is no t
possible to produce such a large amount of heat from a mantle that is entirely
a DMM reservoir (Bercovici and Karato [12]). A more detailed argument is a s
follows: The Earth’s present-day heat loss is 44 TW (Pollak et al. [83]). 4.8-9.6
TW of it are produced by the C C (Taylor and McLennon [102], Rudnick and
Fountain [90]), Porcelli and Ballentine [84]). The different numbers stem from
different assumptions on the average CC abundances of radioa c tive elements.
The contribution of the Earth’s core is between 3 and 7 TW (Buffet et al.
[18]. If the DMM abundances would be by a factor o f 1/2.6 lower than the
BSE abundances according to Jochum et al. [56] then a whole-mantle DMM
would g e nerate o nly 7.2 TW (Porcelli and Ballentine [84]). The contribution
of secular cooling is b etween 21.8 TW a nd 17.8 TW. If the terms of CC, the
core a nd secular cooling are subtracted from the observed 44 TW then values
between 9.6 and 14.4 TW remain. The latter numbers exceed the 7.2 TW of a
hypothetical mantle that is comprised entirely of pure DMM. Ther e fore, the
mantle must contain at least one reservoir that is enriched in U, Th and K.
Continental Crust: Massive Early Formation or Gradual or
Episodic Growth?
The isotopic compositions of lunar rocks (Norman et al. [7 7]) and of SNC
meteorites (Brandon et al. [17], Nyquist et al. [78], Bogard et al. [15]) show
rapid chemical differentiation of Moon and Mars within the first 200 Ma of
8 Uwe Walzer, Roland Hend el, and John Baumgardner
their existence. Therefore it is highly probable that not only did an iron core
form early but a silicate crust did as well. It has been argued that not only
the Moon but also the Earth posessed a magma ocean early in their histories
(Stevenson [97], Tonks and Melosh [105]). In the case of the Earth, it is not
clear whether all or only pa rt of the mantle was melted. The g eneration of
the metallic cores likely occured within the first 30 Ma associated with the
decay of short-lived isotopes (Cameron [23], Kleine et al. [65]). It was a con-
troversial questio n whether the total mass of the Ea rth’s crust continued to
grow during its later evolution taking no account of the mentioned early stage .
Armstrong [6] a nd Bowring and Housh [16] advocated that the full a mount of
current mass of CC formed very early, before an age of 4 Ga. They assumed
that CC has only been recycled since that time. On the other hand, there is
strong evidence that juvenile CC has indeed been formed during the Earth’s
subsequent evolution. Therefore, models involving episodic or continuous con-
tinental growth have been proposed (Taylor and McLennon [102], McCulloch
and Bennett [71, 72], Condie [28], Bennett [10]). The
147
Sm–
143
Nd isotopic
system and the
176
Lu–
176
Hf decay both suggest that chemical differentiation
of CC has p ersisted over the whole of geological history in a ddition of a pulse
of differentiation dur ing the Earth’s ear liest history (Bennett [10]).
The of Sm/Nd ratio was not altered during the accretion of the Earth since
both elements are refractory. Neither was this ratio modified during co re for-
mation since both elements are lithophile. Both c onclusions also apply for
Lu/Hf. Ther efore, we may conclude that these two ratios in BSE are chon-
dritic. However, the quantity ε
Nd
as a function of time displays a n ongoing
chemical evolution of DMM distributed over the whole 4.49 × 10
9
a history
of the solid silica te mantle. Here
ε
Nd
=
(
143
Nd/
144
Nd
(t)sample
)/(
143
Nd/
144
Nd
(t)BSE
) 1
× 10
4
with t the crystallization age. ε
Nd
of the depleted mantle appears to have in-
creased non-uniformly, probably episodically, and reaches its maximum value
of ε
Nd
= 10 ± 2 for the present epoch (Hofmann [50]). Observed isotopic com-
positions of Nd point strongly to complex processes of depletion and crustal
recycling. Similarly, the increase of ε
Hf
of DMM leads to the same conclu-
sion. Here ε
Hf
=
(
176
Hf/
177
Hf
(t)sample
)/(
176
Hf/
177
Hf
(t)BSE
) 1
× 10
4
.
The quantity ε
Hf
increases non-uniformly and reaches its present value of
ε
Hf
= 16±4 (Salters and White [92], Vervoort and Blichert-Toft [111], Amelin
et al. [5], Bennett [10]). A s imilar non-uniform evolution c an be shown for the
187
Os/
188
Os ratio using mantle-derived samples. Condie [28] further demon-
strated a progressive increase in the Nb/Th ratio for the depleted mantle
throughout the Earth’s history. If we now consider the problem of CC evo-
lution as it relates to the mantle comp onents DMM, FOZO, HIMU, DM1
and DM2, disc ussed in Section 2.1. then we must infer that these components
developed by different differentiation processes, subduction and conve c tive
stirring. Also if we view the problem from this perspective it se e ms improba-
ble that CC formed exclusively during the Earth’s initial history. Subduction
continuously entrains heterogeneous material that subsequently sinks to the
Mantle Convection, Continent Generation and Geochemical Heterogeneity 9
bottom of the mantle. The composition of this basal layer almost certainly
changes with time (Davies [31], Gurnis [40], Coltice and Ricard [26], Albar`ede
and Van der Hilst [2]). Arndt [7] provides a review of similar and alternative
views of the mantle’s chemical evolution.
2.2 Further Obs ervational Constraints
In contrast with the other terr estrial planets, the Earth has a pronounced
double-peaked hypsometric curve. This relatively sharp division of the sur -
face o f the solid Earth into continents and ocean basins reflects a contrast
in chemical composition. The thickness of the oceanic crust is only 0-7 km
whereas the continental crust is distinctly thicker. Its thickness depends on
its age: Archaic CC, older than 2.5 × 10
9
a, has an average thickness of 41 km
whereas Early Proterozoic CC that is older than 1.6 × 10
9
a has an average
thickness of 43 k m. Late Proterozoic and Phanerozoic CC has a mean thick-
ness of 35 km. The continental lithospheric mantle, attached to the CC, has
essentially the same age. Isotopic investigations of the continental lithospheric
mantle show that it has been isolated from the convecting mantle since the
corresponding time of CC fo rmation (Kramers [66], Pearson et al. [82]). There
are, of course, alterations due to metasomatism (Alard et al. [1], Burton et
al. [22]). The oceanic lithosphere moves in a piecewise plate-like manner and
subducts. That is why there is no oceanic plate older than Upper Jurassic.
Therefore, the upper CC is the only extensively accessible rec ord o f informa-
tion o n the main part of the Ear th’s history. Only relatively small parts of the
continent record have be en removed by subduction or by delamination of the
continental lithospheric mantle.
Reymer and Schubert [87] summarized continental crustal growth curves
of different authors and presented their own continuous growth curve. Tay-
lor and McLennon [102] emphasized what they recognize as major episodes
of crustal g rowth. Condie [28] also emphasizes the episodicity of this pro-
cess. O’Nions and Tolstikhin [79] show that convective ava lanches could be
responsible for episodic crustal growth. It is plausible that episodicity in con-
vection indirectly causes the episodes of growth of CC. Yuen and Malevsky
[123] and Yuen et al. [12 2] pointed out that mantle convection can operate
in the hard turbulence r egime at early stages of planetary thermal evolution
and subside to a pres e nt-day state of so ft turbulence. It is also to be ex pected
that the rate of chemical differentiation depends directly on mantle convec-
tion. So, if convection displays episodes of vigor the juvenile contributions to
the continents should also be episodic. Finally, we mention the process of the
intr acrustal differentiation that generates two chemically heterogeneous reser-
voirs with distinct systematic differences in their abundances of incompatible
elements, namely, the upper and lower CC (Rudnick and Gao [91]). Table
B.1 of Walzer et al. [113] specifies essential differences between the Earth and
terrestrial planets.
10 Uwe Walzer, Roland Hend el, and John Baumgardner
3 Model
3.1 Balance of Mass, Momentum and Energy
We use a numerical strategy for modeling the differentiation and mixing pro-
cesses that have operated during our planet’s history. We solve the differen-
tial eq uations of infinite Prandtl-number convection using a three-dimensional
finite-element spherical-shell method. These express the conservation of mass,
momentum, and energy. The mass balance
ρ
t
+ · (ρv) = 0 (1)
with the anelastic-liquid approximation simplifies to
· v =
1
ρ
v · ρ (2)
where ρ is density, t time, and v is velocity.
The conservation of momentum can be written as
ρ
v
t
+ v · v
= P + ρg +
x
k
τ
ik
(3)
where P is the pressur e , g is the gravity acceleration, and τ
ik
is the deviatoric
stress tensor. For spherical sy mmetry, we have g = ge
r
and the hydrostatic
pressure gradient may be written as
P
r
= ρg (4)
By definition K
S
= V
P
V
S
and
V
V
0
=
ρ
0
ρ
, where K
S
is the adiabatic bulk
modulus, V vo lume, S entropy, r the radial distance from the Earth’s center.
Hence
K
S
= ρ
P
ρ
S
= ρ
P
r
S
r
ρ
S
(5)
Substituting Eq. (4) into Eq. (5) we obtain
ρ
r
S
=
ρ
2
g
K
S
(6)
Upo n neglecting horizontal spatial variations in ρ, Eqs. (2) and (6) yield
· v =
1
ρ
v · ρ
=
1
ρ
v
r
ρ
r
=
ρgv
r
K
S
(7)
It is well-known that
K
S
=
c
p
c
v
K
T
= (1 + αγ
th
T )K
T
(8)
Mantle Convection, Continent Generation and Geochemical Heterogeneity 11
where K
T
is the isothermal bulk modulus, c
p
the specific heat at constant
pressure, c
v
the specific heat at constant volume, α the coefficient of thermal
expansion, γ
th
the thermo dy namic Gr¨uneisen parameter and T the absolute
temper ature.
Eq. (3) can be rewritten a s
ρ
dv
i
dt
= ρ g
i
+
σ
ki
x
k
(9)
Using this equation, the energy balance can be expressed as follows
ρ
du
dt
+
q
i
x
i
= Q + σ
ik
˙ε
ik
(10)
where u is the specific internal energy, Q is the heat generation rate per
unit volume; v
i
, g
i
, q
i
, x
i
, σ
ik
, ˙ε
ik
are the components of velocity, grav ity
acceleration, heat flow density, location vector, stress tenso r and strain-rate
tensor, respectively.
Another formulation of Eq. (10) is
ρ
t
+ v ·
u = · (k T ) + Q P ·v + 2W
D
(11)
where
2W
D
= σ
ik
˙ε
ik
+ P ·v (12)
and
q
k
= k
T
x
k
(13)
The quantity k denotes the therma l conductivity. Using
du = T ds P dv (14)
and
du = T
s
T
P
dT + T
s
P
T
dP P dv (15)
we eliminate the specific internal energy in Eq. (11) and obtain the equation
ρc
p
dT
dt
= · (k T ) + Q + αT
dP
dt
+ 2W
D
(16)
since
c
p
= T
s
T
P
and
s
P
T
=
v
T
P
= (17)
Here s signifies the specific entropy, v the specific volume, c
p
the specific heat
at constant pressure and α the coefficient of thermal expansion.
Next, a less well known version of the energy balance is presented: Eq.
(11) is equivalent to
12 Uwe Walzer, Roland Hend el, and John Baumgardner
ρ
du
dt
+ P
dv
dt
= τ
ik
v
i
x
k
+ · (k T ) + Q (18)
because of Eq. (2) and
1
ρ
= v.
Inserting Eq. (14) into Eq. (18), we obtain
ρT
ds
dt
= τ
ik
v
i
x
k
+
x
j
k
x
j
T
+ Q (19)
On the other hand,
ds =
s
T
v
dT +
s
v
T
dv (20)
and
s
T
v
=
c
v
T
,
s
v
T
= αK
T
(21)
This implies
T ds = c
v
dT + αK
T
T d
1
ρ
(22)
or
T ds = c
v
dT
c
v
γT
ρ
(23)
where
γ
th
=
αK
T
c
v
ρ
(24)
stands for the thermodynamic Gr¨uneisen parameter.
Inserting Eq. (23) into Eq. (19) we obtain
ρc
v
dT
dt
c
v
γT
dt
= τ
ik
v
i
x
k
+
x
j
k
x
j
T
+ Q (25)
From Equations (1) and (25)
ρc
v
dT
dt
= ρc
v
γT
v
j
x
j
+ τ
ik
v
i
x
k
+
x
j
k
x
j
T
+ Q (26)
or
T
dt
= v
j
x
j
T γT
v
j
x
j
+
1
ρc
v
τ
ik
v
i
x
k
+
x
j
k
x
j
T
+ Q
(27)
or
T
t
=
(T v
j
)
x
j
(γ 1 )T
v
j
x
j
+
1
ρc
v
τ
ik
v
i
x
k
+
x
j
k
x
j
T
+ Q
(28)
This is an alternative formula for the energy conservation. Although c
v
ap-
pears in Eq. (28), the latter expression is equivalent to Eq. (16) where c
p
is
used. T he deviatoric stres s tensor can be expressed by
Mantle Convection, Continent Generation and Geochemical Heterogeneity 13
τ
ik
= η
v
i
x
k
+
v
k
x
i
2
3
v
j
x
j
δ
ik
(29)
in the Eqs. (3) and (28), where η denotes the viscosity.
As an equation of state we take
ρ = ρ
r
"
1 α(T T
r
) + K
1
T
(P P
r
) +
2
X
k=1
Γ
k
∆ρ
k
r
#
(30)
where the index r refers to the adiabatic reference state, ∆ρ
k
r
or f
ak
denotes the no n-dimensional density jump for the kth mineral pha se tran-
sition. Γ
k
is a measure of the relative fraction of the heavier phase where
Γ
k
=
1
2
1 + tanh
π
k
d
k
with π
k
= P P
0k
γ
k
T describing the excess pressure
π
k
. The q uantity P
0k
is the tr ansition pressure for vanishing temperature T .
A non-dimensional transition width is denoted by d
k
. The quantity γ
k
rep-
resents the Claus ius -Clapeyron slope for the kth phase transition. Γ
k
and π
k
have been introduced by Richter [89] and Christensen and Yuen [24].
Because of the very high Prandtl number, the le ft-hand side of Eq. (3) van-
ishes. Hence, we use the following version of the equation of conservation of
momentum.
0 =
x
i
(P P
r
) + (ρ ρ
r
)g
i
(r) +
x
k
τ
ik
(31)
The final version of the equation of conservation of mass is
0 =
x
j
ρv
j
(32)
which stems from Eq. (2). The Equations (28), (30), (31) and (32) are a system
of six scalar equations that we use to determine six scalar unknown functions,
namely T , ρ, P and the three components of v
i
.
3.2 Viscosity and Phase Transitions
The viscosity law of this paper is presented as follows. Based on experimental
results of Karato and Li [58], Karato and Wu [60] and Li et al. [68], a Newto-
nian solid-state creep is assumed for the Earth’s mantle. The shear viscosity,
η, is calculated by
η(r, θ, φ, t) = 10
r
n
·
exp(c
T
m
/T
av
)
exp(c T
m
/T
st
)
· η
3
(r) · exp
c
t
· T
m
1
T
1
T
av

(33)
where r is the radius, θ the colatitude, φ the longitude, t the time, r
n
the
viscosity-level para meter, T
m
the melting temperature, T
av
the laterally av-
eraged temperature, T
st
the initial temperature profile, T the temperature as
a function of r, θ, φ, t. The quantity η
3
(r) is the viscosity profile at the ini-
tial temperature and for r
n
= 0. So, η
3
(r) descr ibes the dependence of the
14 Uwe Walzer, Roland Hend el, and John Baumgardner
viscosity on pressure and on the mineral phase boundaries. The derivation of
η
3
(r) is to be found in Walzer et al. [113]. The quantity r
n
has been used for
a stepwise shift of the viscosity profile to vary the averaged Rayleigh number
from run to run. The se c ond factor of the r ight-hand side of Eq. (33) de-
scribes the inc rease of the viscosity profile with the cooling of the Earth. For
MgSiO
3
perovskite we should insert c=14, for MgO w¨ustite c=10 according
to Yamazaki and Karato [120]. So, the lower-mantle c should be somewhere
between these two values. For numerical reasons, we are able to use only c=7.
In the lateral-variability term, we inserted c
t
= 1. For the uppermost 285 km
of the mantle (plus crust), an e ffective viscosity, η
eff
, was implemented where
η
eff
= min
h
η(P, T ),
σ
y
2 ˙ε
i
(34)
The pressure is denoted by P , the second invariant of the strain- rate te nsor
by ˙ε. The quantity σ
y
is a viscoplastic yield stress.
The viscosity profile of the present paper (see Fig. 2) displays a high-
viscosity lithosphere. Beneath of it is a low-viscosity asthenosphere down to
the 410-km phase boundary. The tra nsition zone is highly viscous, at least
between 520 and 660 km depth. This model is corroborated by the fact that
downgoing slabs extending down to the asthenosphere only show extensional
fault-plane solutions . If a slab enters the transition zone then compressional
fo c al mechanisms are observed (Isacks and Molnar [54]), also in the case
that the 660-km discontinuity is not touched. These observations cannot be
explained by the olivine-wadsleyite or the wadsleyite-ringwoodite pha se tra n-
sition since the Clausius-Clapeyron slopes for both are positive and the phase
boundary distortion enhances the cold downflow in these cases. The increase
of the number of seismic events per 20 km Bin beneath of 520 km depth (Kirby
et al. [64]) can be explained by a viscosity increase. Only if the slab reaches
the 660-km phase boundary the corresponding negative Clausius-Clapeyron
slope can contribute to the observed compressional fault-plane solutions. If
the slab penetrates the 660- km phase b oundary then the latter is deflected
somewhat downward. An ear thquake has never been observed below that de-
flection. Beneath of that, the slab is only detectable by e levated densities and
seismic compressional and shear velocities. Therefore it is reasonable to infer
a low-viscosity layer in the uppermost part of the lower mantle (Kido and
Yuen [62]). This proposal of a high-viscosity transition layer, that is embed-
ded between two low-viscosity layers, is consistent with the proposition that
the trans itio n zone is composed mostly of garnet and s pinel (Meade and Jean-
loz [74], Karato et al. [59], Karato [57], All`egre [3]). If there are no further
phase transitions in the lower mantle, exc e pt nea r the D
′′
layer (Matyska and
Yuen [70]), then the viscos ity must rise considerably as a function of depth
because of the pressure dependence of the ac tivation enthalpy of the prevail-
ing creeping mechanism fo r regions wher e the temperature gradient is near to
adiabatic. This implies a thick high-viscos ity central layer in the lower mantle.
Mantle Convection, Continent Generation and Geochemical Heterogeneity 15
We infer a strong temperature gradient of the D
′′
layer which causes a strong
decrease of viscosity in the near neighborhood above the CMB.
In our derivation of η
3
(r), however, we did not make use of the above argu-
ments. They serve only as a corroboration. The a lternative systematic deriva-
tion is described by Walzer et al. [113]. We sta rt from a self-consistent theory
using the Helmholtz fre e energy, the Birch-Murnaghan equation of state, the
free-volume Gr¨uneisen parameter and Gilvarry’s [36] formulation of Linde-
mann’s law. The viscosity is calculated as a functio n of melting temperature
obtained from Lindemann’s law. We use pressure , P , bulk modulus, K, and
K/∂P from the seismic model PREM (Dziewonski and Anderson [32]) to
obtain the relative variation in radia l viscosity distribution. To determine the
absolute scale of the viscosity profile, we utilize the standard postglacial-uplift
viscosity of the astheno sphere below the continental lithosphere.
Our η
3
(r) profile is supported by several recent studies. New inversion in-
vestigations for mantle viscosity pr ofiles reveal an acceptable resolution down
to 1200 km depth. For greater depths, models based on solid-state phy sics
seem to be more reliable. Kido and
ˇ
Cadek [61] and K ido et al. [63] found
two low-viscosity layers below all three oceans. The first layer is between the
lithosphere and 410 km depth. The se c ond one is between 660 and about 1000
km depth. Panasyuk and H ager [81] made a joint inversion of the geoid and
the dynamic topography. They found three families o f solutions for the radial
viscosity profile that are different regarding the position of the lowest-viscosity
region: (a) directly beneath the lithosphere, (b) just above 400 km depth or
(c) just above 670 km depth. The results of Forte and Mitrovica [35] show
even more similarity with our profile η
3
(r). Their viscous-flow models, based
on two seismic models of three-dimensional mantle structure, revealed two vis-
cosity maxima at about 800 and about 2000 km depth. This is similar to our
model for η
3
(r) that has also two ma xima in the interior although is has been
derived by a completely different method. Cserepes et al. [3 0] inve stigated the
effects of similar viscosity profiles on Cartesia n 3-D mantle convection in a
box.
In our dynamical model, we take into account the full effect of phase
boundary distortion of the olivine-wadsleyite and of the ringwoodite-perovskite
phase boundary. The input pa rameters that define these phase transitions are
provided in Table 1.
3.3 Numerical Method and Implem entation
In our models we include the full pressure dependence and the full radial tem-
perature dependence of viscosity. For numerical reasons, however, we are able
to treat only a part of the lateral temperature dependence of the viscosity.
At the mineral phase boundaries in the inte rior of the Earth’s mantle, there
are not only discontinuities of the se ismic veloc ities and of the density but
also jumps of activation volumes, activation energies and, therefo re, of acti-
vation enthalpies. Since the viscosity depends ex ponentially on the activation
16 Uwe Walzer, Roland Hend el, and John Baumgardner
Table 1. Model parameters
Parameter Description Value
r
min
Inner radius of spherical shell 3.480 × 10
6
m
r
max
Outer radius of sph erical shell 6.371 × 10
6
m
Temperature at the outer shell
boundary 288 K
h
1
Depth of the exothermic
phase boundary 4.10 × 10
5
m
h
2
Depth of the endothermic
phase boundary 6.60 × 10
5
m
γ
1
Clapeyron slope for the
olivine-wadsleyite transition +1.6 × 10
6
Pa·K
1
γ
2
Clapeyron slope for the
ringwoodite-perovskite transition 2.5 × 10
6
Pa·K
1
f
a1
Non-dimensional density jump for
the olivine-wadsleyite transition 0.0547
f
a2
Non-dimensional density jump for
the ringwoodite-perovskite transition 0.0848
Begin of th e thermal evolution of
the solid Earth’s silicate mantle 4.490 × 10
9
a
d
1
Non-dimensional t ransition width
for the olivine-wadsleyite transition 0.05
d
2
Non-dimensional t ransition width for
the ringwoodite-perovskite transition 0.05
Begin of th e radioactive decay 4.565 × 10
9
a
c
t
Factor of the lateral viscosity variation 1
k Thermal conductivity 12 W·m
1
·K
1
nr + 1 Number of radial levels 33
Number of grid points 1.351746 × 10
6
enthalpy of the prevailing creeping process, the conclusion is inescapable that
there are considerable viscosity jumps at the upper and lower surfaces of the
transition zone. These jumps cause numerical problems in the solution of the
balance equations. The problems have been solved. Nevertheless, our group
is searching for more effective solutions of the numerical jump problem. The
minor discontinuity at a depth of 520 km has been neglected.
We treat the mantle as a thick spherical shell. The grid for this domain is
constructed by projection of the edges of a regular icosahedron onto concentric
spherical shell surface s with different radial dista nce s from the center. These
surfaces subdivide the mantle into thin shells. A first step of grid refinement
consists of bisecting the edges of the resulting spherical triangles into eq ual
parts. Co nnecting the new points with g reat circles , we obtain four smaller
triangles from each starting triangle. The process can be repeated by succes-
sive steps to obtain a grid with the desired horizontal resolution. We re plicate
the resulting almost uniform triangular grid at different radii to generate the
3D grid for a spherical shell. We ca n use different formulae for the distribu-
Mantle Convection, Continent Generation and Geochemical Heterogeneity 17
tion of the radial distances of the spherical grid surfaces. In this paper, we
used exclusively a radially nearly equidistant grid with a superposed sinoidal
half-wave length to r e fine the grid near the upper and lower boundaries of the
spherical shell. The grid is non-adaptive.
The Navier-Stokes equations as well as pressure and creeping velocity are
discretized using finite elements. We apply piecewise linear basis functions for
the creeping velo c ity and either piecewise constant or piecewise linear basis
functions for the pressure. We solve the equations for pressure and velocity
simulta neously by a Schur-complement conjugate-gradient iteration (Ramage
and Wathen [86]). This is a further development of an Uzawa algorithm. We
solve the energ y equation using an iterative multidimensional positive-definite
advection-transport algorithm with explicit time steps (Bunge and Baum-
gardner [20]). Within the Ramage-Wathen procedure, the re sulting equation
systems are solved by a multig rid procedure that utilizes radial line Jacobi
smoothing. In the multigrid procedure, prolongation and restriction are han-
dled in a matrix-dependent ma nner. In this way, it is possible to handle the
strong variations and jumps o f the coefficients associated with the stro ng vis-
cosity gradients (Yang [121]). For the formulation of chemical differentiation,
we modified a tracer module developed by Dave Stegman. This module con-
tains a sec ond-order Runge-Kutta procedure to move the tracer particles in
the velocity field. Each tracer carries the abundances of the radionuclides. In
this sense, trac e rs are active attributes w hich deter mine the heat production
rate per unit volume that varies with time and position.
[30] The FORTRAN code is parallelized by domain decomposition and ex-
plicit message passing (MPI) (Bunge [19]). For the most runs, we used a mesh
of 13 51746 nodes. For some runs, we use d a mesh of 10649730 nodes in order
to check the convergence of the lower resolution runs. We found hardly any
discernable differences (<0.5%) for the Rayleigh number, the Nusselt number,
the Urey number and the laterally averaged surface heat flow as a function
of time. The calculations were perfo rmed on 32 processors of a Cray Strider
Opteron cluster. The code was benchmarked for cons tant viscosity convec-
tion by Bunge et al. [21] with data of Glatzmaier [37] for Nusselt numbers,
peak velocities, and peak temperatures. The result is a good agreement with
deviations 1.5%.
3.4 Heating, Ini tial and Boundary Conditions, and Chemical
Differenti ation
We assume the E arth’s mantle is heated mostly from within. This internal
heating is expressed by the heat production density Q in Eq. (28) that is
measured in W·m
3
.
Q = H · ρ (35)
where H is the specific heat production with
18 Uwe Walzer, Roland Hend el, and John Baumgardner
H =
4
X
ν=1
a
µν
a
ifν
H
0ν
exp(t/τ
ν
) (36)
Table 2 presents the parameter data we use for the four major heat-producing
isotopes. Here, ν s tands for the radionuclide in the formulae, τ
ν
represents the
decay time or the 1/e life, H
denotes the specific heat production of the νth
radionuclide 4.5 65 × 10
9
years ago, a
ifν
is the isotope abundance factor.
Table 2. Data of the major h eat-producing isotopes
Isotope
40
K
232
Th
235
U
238
U
ν 1 2 3 4
τ
ν
[Ma] 2015.3 20212.2 1015.4 6446.2
H
0ν
[W kg
1
] 0.272×10
3
0.0330× 10
3
47.89×10
3
0.1905× 10
3
a
ifν
0.000119 1 0.0071 0.9928
We represent the distribution of radionuclides in the mantle by tra cers.
Each tracer is associated with a specific geochemic al principal reservoir. Be-
cause of mixing, the boundaries of these reservoirs become blurred with time.
In principle, even a total mixing and homogenization of the mantle is pos-
sible if the dynamic system of the mantle allows this process. Each tracer
is identified by a tracer index. The res e rvoir concentrations of elements are
given by Table 3. The lower five elements of Table 3 serve only for the com-
putation of concentration maps but not for the calculation of heating energy
either because the contributions of these elements are too low or because they
are daughter nuclides. Since the relative masses of HIMU, EM1 and EM2 are
small they have been neglected in the calculated model, S3, of this paper. Our
model mantle starts w ith a uniform distribution of exclusively ty pe-1 trac-
ers, i.e., we start with a pure BSE mantle. If the modeled temperature, T ,
approaches the melting temp e rature, T
m
, in a certain volume then chemical
differentiation takes place. Plateau basalts quickly rise to form the plateaus
as a terrane or preliminary form of the co ntinental crust (CC) leaving behind
depleted MORB mantle (DMM). The numerical conditions for this simplified
chemical differentiation process will be given below. We do not use a detailed
melt extraction equation system like the 2-D code of Schmeling [93] since, for
a 3-D spherical-shell code, this would req uire more computational resources
than we curr e ntly have available.
We chose McCulloch and Bennett [71] reservoir abundances for our mod-
els because of the good internal compatibility of this geochemical model.
These abundances are similar to those proposed by other investigators. Heier
[43], Taylor and McLennon [102], Hofmann [48], McCulloch and Bennett [71],
Wedepohl [115] and Rudnick and Fountain [90] have proposed values for the
continental-crust K:U ratio of 10777, 10020, 10000, 10064, 10020, 11092, re-
sp e c tively.
Mantle Convection, Continent Generation and Geochemical Heterogeneity 19
Table 3. The abundances a
µν
of the major heat-producing elements
Reservoir Primordial Oceanic Continental Depleted MORB
mantle (ppm) crust [MORB] (ppm) crust (ppm) mantle (ppm)
tracer index (1) (2) (3) (4)
element
U 0.0203 0.047 0.94 0.0066
Th 0.0853 0.12 4.7 0.017
K 250. 600. 9460. 110.
Pb 0.1382 0.30 7.0 0.035
Sm 0.4404 2.63 4.62 0.378
Nd 1.354 7.3 25.5 0.992
Rb 0.635 0.56 35.5 0.112
Sr 21.0 90. 310. 16.6
The spherical shell of our present model has free-slip and impermeable
boundary conditions for both the Earth’s surface and CMB. The upper sur-
face is isotherma l at 288 K. The CMB is also isothermal spatially, but not with
respect to time. Applying a cooling core-mantle evolution model (Steinbach et
al. [96]), we adjust the CMB temperature, T
c
, after each time step according
to the heat flow through the CMB. We assume a homogeneous core in ther-
modynamic equilibrium similar to the approaches of Steinbach and Yuen [95]
and Honda and Iwase [52].
Prior to this work, our modeling efforts relating to the problem of inte-
grated convection-fractionation were restricted to two dimensions (Walzer and
Hendel [112], Walzer et al. [114]). We here describe two tracer methods for
our 3-D compressible spherical-shell models. The first method has the advan-
tage of being simple and readily comprehensible. The second method avoids
certain deficiencies of the first one.
The first method: We assign a 3-D c e ll to each node in the icosahedra l grid
with 1351746 nodes. There are Type-1 tracers, Ty pe-3 tra cers and Type-4
tracers with the abundances given in Table 3. At the beginning of the evolution
of the model, the shell contains exclusively Type-1 tr acers. Each cell starts
with eight tracers or 64 tracers, r e spectively. T he tracers are carried along
by the velocity field. The element concentration of a node is determined by
the average
a
µν
, of the abundances of the elements carried by the tracers in
the cell associated with the node. A local tracer refresh (LTR) is applied if a
cell has fewer than four (or 32) or more than twelve (or 96) tracers. Tracers
are redistributed from or to the neighboring cells, respectively, using fixed
prescriptions. This procedure is to prevent a cell be c oming empty o f tracers
and therefore having
a
µν
become indeterminate. If the conditions for partial
melting are fulfilled in a sufficiently large volume then the Type-1 tracers in
that volume are converted to Type-4 tracers corresponding to DMM to mimic
the depletion. A gr e ater number of changed Type-1 tracers are necessary to
produce one new Type-3 tracer (corresponding to CC) from a Type-1 tracer
20 Uwe Walzer, Roland Hend el, and John Baumgardner
near the surface above a region of differentiation, since the continental Type-3
tracers have considerably higher abundances of incompatible elements. The
ratio z
3
is given by
z
3
= (a
(3)
µν
a
(1)
µν
)/(a
(1)
µν
a
(4)
µν
) (37)
z
3
= round (z
3
) (38)
For uranium, z
3
= 67.131387 based on the values of Table 3, and therefor e
z
3
= 67. The same integer is derived for thorium and potassium. So, 67 Type-1
tracers from the asthenosphere are necessary to generate one Type-3 tracer in
the lithosphere by transformation of one Type-1 tracer at the corresponding
place. If a cell (a) has more than 50% Type-1 tracers and is, the refore, fertile
and (b) has fulfilled the condition T > f
3
· T
m
in its grid point where f
3
is a
fixed para meter with 0 < f
3
1 and (c) has at least five neighboring cells with
common boundary surfaces that also fulfill (a) and (b) then this cell is called
Type-A cell. If a cluster of simply connected Type-A cells has n
thr
Type-1
tracers then the tracers are instantaneously changed in Type-4 tracers. This
does not concern all o f these tracers but (n
thr
+ n
n
· z
3
) of them where n
n
is
an inte ger. Here thr stands for threshold. The center o f gravity of the cluster
is projected to the top surface of the shell. The corresponding point at the
surface is called P
. A number o f (n
thr
/z
3
+ n
n
) Type-1 tracer s nearest to P
and not deeper than 65 km are changed to Type-3 tracers. This corresponds
to oceanic plateaus. All Type-3 tracers are unsinkable and move with the
horizontal component of the velocity field. This rule mimics the tendency
of the c ontinents to resist subduction. If two Type-3 tracers approach each
other nearer than a sma ll distance d
in
then they are connected to a continent.
If an unconnected Type-3 tracer approaches to a continent nearer then d
in
then it will be connected with the continent. Tracers that are connected to
form a continent move with a common angular velocity, ω, associated with
that continent around the center of the shell. This quantity ω is calculated
as the vecto r sum of the single angular velocities derived from the horizontal
component of the undisturbed no dal velocity. Hence, the continent moves as
a plate-like thin shell across the upper surface. In our pre sent model, oceanic
plates develop without any such construction simply as a result of the yield
stress and of the existence of an asthenosphere.
The second met hod: The second method is a translation of the ideas of the
first method to the Stegman code with some improvements. For the starting
distribution of the tracers , a cell is attributed to each node. Tracers are initially
distributed in an almost uniform manner about each grid point, w ith eight (o r
64) tracers per grid-p oint cell, except for the grid-point cells on the top and
bottom shell boundaries, which have four (or 32). A new feature of this second
method is that each particle carries its initial mass as one of its attributes.
The s um of the individual particle ma sses is equal to the total mass of the
mantle. If 4 tracers ar e regularly distributed to each half-cell then the mass,
mip, o f a trace r, ip, is calculated by
Mantle Convection, Continent Generation and Geochemical Heterogeneity 21
mip = 1/ 4 (volume of the half-cell) (densi ty of the node) (39)
The mass, M
mantle
, of the whole mantle results from
M
mantle
=
npm
X
ip=1
mip (40)
where npm is the total numbe r of tracers in the mantle.
A memory cell conta ins all tracers that are attributed to a node. Its base is
a spherical hexagon or pe ntagon the corners of which are in the triangle centers
of the triangular distribution around a node. Its altitude is again between the
grid spherical surfaces ir and (ir + 1).
Combining and splitting: Material from the top boundary layer that sinks
to the CMB expe riences a density increase and an increase of the tracer num-
ber per volume up to a factor two. However, the cell volume is diminished by
a factor four during a movement from the s urface to the CMB. So, a reduction
of the number of trac e rs per memory ce ll by a factor two is to be expected. If
the tracer number fa lls below four (or 32) then each tracer of this cell is split
into two tr acers. The mass of such a tracer is distributed equa lly to the child
tracers. The revers e process will occur during upwelling. Overcrowding of the
memory cells can take place. For technical reasons, we limit the number of
tracers per cell to 12 (or 96). Each tracer coming into the cell beyond this
limit is combined with one of the other tracers according to their order in
memory. The masses are added. The location of the new tracer is the center
of gravity o f the two annihilated tra c e rs. Only tracers of the same type can
be c ombined. If an excess tracer enters the cell with a type not present in the
cell, then two tracers of the most abundant type are united, the first two in
the storage sequence. Splitting or combining does not alter the cell mass , M c,
nor the sum of the tracer masses, mipc, present in the cell.
The base of an interpolation cell is a spherical hexago n or pentago n the
corners of which are the lateral neighboring nodes. The upper and lower base
is determined by the upper and lower neighboring grid spherical surfaces. All
tracers inside the interpolation cell contribute to the interpolation of tracer
attributes, e.g., elemental abundance, to the node. The nearer the tr acer is to
the grid point, the larger is the weighting factor. The lateral weighting factor,
wl, is simply the bar ycentric coordinate of the tracer when the tracer and
node are both radially projected onto the same spherical surface:
wl = (α|β|γ)(ip) (41)
The radial weighting fac tor is given by
wr =
(r(ir + 1) rip)
(r(ir + 1) r(ir))
if rip > r(ir)
wr =
(rip r(ir 1))
(r(ir) r(ir 1))
if rip r(ir) (42)
22 Uwe Walzer, Roland Hend el, and John Baumgardner
where r(ir + 1|ir 1) ar e the neig hboring grid spher ical surfaces of the radius,
r(ir), of the node and rip is the radius of the tracer. The total we ighting
factor is the product o f these two factors.
wip = wr wl (43)
The weighted mas s, wmip, of a tracer is
wmip = wip mip (44)
The mass, wM c, of an interpolation cell can be derived by a weighted
integration over the mass continuum of the c e ll. This has to be done in such
a way that
wM c(interpolation cell) = M c(memory cell) (45)
The total mass balance is not violated by the weighting pro c edure:
nc
X
c=1
npc
X
ip=1
wmipc =
nc
X
c=1
n3c
X
i3c=1
nip3c
X
ip3c=1
wmip3c =
n3
X
i3=1
nip3
X
ip3=1
6
X
node=1
wmip3(node) =
=
n3
X
i3=1
nip3
X
ip3=1
(wr + (1 wr)) (α + β + γ) mip3 =
=
n3
X
i3=1
nip3
X
ip3=1
mip3 =
np
X
ip=1
mip (46)
where n is a numbe r, c the interpolation cell, p a tracer, i counting index, 3
triangular cell, w weighted, m mass, node the counting index for the nodes at
the boundaries of a triang ular cell, wr the radial weighting factor of a tracer;
α, β, γ are the barycentric coordinates from the three corner points of the
basis of a triangular c e ll, so that α + β + γ = 1.
Diminution of tracer mas s can be observed in the spreading zones which
is not induced by density differences. In other areas, a compaction of tracer
mass is to be expected:
wM c 6=
npc
X
ip=1
wmipc (47)
The tracer mass r atio, Gmc, of the cells deviates from the obligated value 1 :
Gmc =
npc
X
ip=1
wmipc
/wM c 6= 1 (48)
This formula describes a distortion of the tracer representation.
A local tracer mass refresh has been introduced to reduce this discrepancy
of Eq.(48). At least, a deviation
Mantle Convection, Continent Generation and Geochemical Heterogeneity 23
dGmcmax = max.permissible|Gmc 1| (49)
is allowed. If
(Gmc 1) > dGmcmax (50)
applies in a cell, c, then distribute tracer mass of c to the neighboring cells in
such a way that
(Gmc 1) = dGmcmax (51)
The tracer mass
dwM c =
npc
X
ip=1
wmipc wMc (1 + dGmcmax) (52)
has to be distributed to those neighboring cells, cn, tha t have a common edge
with the cell, c, at least and that fulfill the condition
dwM cn =
npcn
X
ip=1
wmipcn wMcn (1 + dGmcmax) < 0 (53)
If the neighboring cells, cn, have the capacity to hold the e xcess trac e r mass
of c, i.e., if
dwM c <
ncn
X
cn=1
dwM cn
!
(54)
then dwMc is to be distributed to the cn. The proportionality of the different
tracer types is ensured. If
(1 Gmc) > dGmcmax (55)
applies in c then remove tracer mas s from cn in an analogous way so that
(1 Gmc) = dGmcmax. (56)
4 Results and Discussion of the Figures
4.1 Thermal and Chemical Evolution Using a Reference Run
We begin by presenting what we call our reference run 808B. It is represen-
tative of the results we obtain in a moderately extensive region of Rayleigh
number yield stress parameter spac e. Our chosen reference run is defined
by a viscoplastic yield stress σ
y
= 115 MPa and a visco sity-level parameter
r
n
= 0.65. Run 80 8B starts with eig ht tracers per grid-point cell. Now, we
present the Figures, in each cas e immediately followed by the corres ponding
discussion. In Figure 1, the laterally averaged temperature for the geo logical
present time as a function of depth is repr e sented by a solid line. This curve lies
24 Uwe Walzer, Roland Hend el, and John Baumgardner
closer to the geotherm of a para meterized whole-mantle convection model than
to the corresponding layered-c onvection temperature. This is understandable
since the results of the present model, S3, show whole-mantle convection.
However, the flow is somewhat impeded by the high-viscosity tr ansition zone
and by the endothermic 660-km phase boundary. Therefore, the temperature
is slightly augmented, e specially immediately be neath the 660-km boundary.
Figure 2 displays the laterally averaged pres e nt-day viscosity. Its derivation
and discussion is given by Section 3.2. Figure 3 shows the time dependence
of some spatially integrated quantities in our reference run. The evolution of
the la terally avera ged heat flow at the Earth’s surface is depicted in the first
panel. The curve reaches a realistic value for the present time: The observed
mean global heat flow has been estimated to be 87 mW/m
2
(Pollak et al.
[83]). The second panel exhibits the growth rate of continental mass as a
function of time. It mimics observational indications that global magmatism
and orogenesis are intrinsically episodic (Worsley et al. [1 19], Nance et al.
[76], Hoffman [47], Titley [103], Lister et al. [69], Condie [28]). The third
panel of Figure 3 demonstrates the time dependence of Ror, the ratio of
surface heat outflow to the mantle’s radiogenic heat production which is the
Fig. 1. The laterally averaged temperature of the geological present time (solid
curve) as a function of depth for the reference run with a viscoplastic yield stress,
σ
y
= 115 MPa, and a viscosity level parameter, r
n
= 0.65. Cf. Eqs. ( 33) and (34).
A range of realistic mantle geotherms using parameterized models of the mantle’s
thermal history given by Schubert et al. [94] is depicted for comparison. Label a and
b signify geotherms of whole-mantle and partially layered convection, respectively.
The dotted line denotes a mid-oceanic ridge geotherm.
Mantle Convection, Continent Generation and Geochemical Heterogeneity 25
reciprocal value of the Urey number. Pa rameterized models show roughly
similar curves except for medium-large and smaller fluctuations. A pattern of
general decre ase and some fluctuations in the Rayleigh number are indicated
in the fourth panel.
The chemical heterogeneity of incompatible elements in a run with 64 trac-
ers per grid-point cell for present time is shown by Figure 4. It is remarkable
that in spite of 4500 Ma of solid-state mantle convection chemical re servoirs
continue to persist. This paper therefore represents a possible way to reconcile
the geochemical and geophysical constraints. Heterogeneities are diminished
only by stirring (Gottschaldt et al. [38]). Diffuse mixing is negligible. How-
ever, in our model there are no pure unblended reservoirs, and this may also
be true of the Earth’s mantle. DMM predominates immediately below the
continents (red) and beneath the oceanic lithosphere. T his is a realistic fea-
ture of the model since where the rea l oceanic lithosphere is rifted, MORB
magma is formed by decompression melting. The MORB source (DMM) is
not only depleted in incompatible e lements but also relatively homogenized.
It is homogenized not only with respect to its major geochemical compo-
nents (SiO
2
, MgO, FeO, Al
2
O
3
, CaO) (Palme and O’Neill [80]) but also with
respect to isotope ratios
87
Sr/
86
Sr,
143
Nd/
144
Nd,
206
Pb/
204
Pb,
207
Pb/
204
Pb
and
208
Pb/
204
Pb. As a consequence, the standard deviation of these isoto pe
ratios and of the major element compositions is small for MORBs in com-
parison to OIBs (All`egre and Levin [4]) although Hofmann [50] has modified
this conclusion somewhat. Figure 4 shows a marble-cake mantle as it was sug-
gested by Coltice and Ricard [27] and Becker et al. [9] but reversed in terms of
its pattern. It is the depleted r e gions in our model that are disconnected and
Fig. 2. The laterally averaged shear viscosity of the reference run as a function of
depth for the present geological time.
26 Uwe Walzer, Roland Hend el, and John Baumgardner
Fig. 3. Time evolution of some spatially integrated quantities from the reference
run. (a) The laterally averaged surface heat flow, qob. (b) The juvenile contributions
to the total mass of the continents. The genuine increase of continental material is
expressed as convert ed Type-3 tracer mass per Ma. (c) The reciprocal value of the
Urey number. Ror represents the ratio of the surface heat outflow to the mantle’s
radiogenic heat production rate. (d) The R ayleigh number as a function of age.
Mantle Convection, Continent Generation and Geochemical Heterogeneity 27
Fig. 4. This equatorial section shows the present-time state of the chemical evolution
of the Earth’s mantle as computed in a companion run , 808C, of ru n 808. Run 808C
has 64 tracers per grid-point cell at the beginning. Strongly depleted parts of the
mantle which include more than 50 % of depleted MORB mantle are represented
by yellow areas. Less depleted and rich parts of the mantle are depicted by orange
colors. Rich refers to a high abundance of incompatible elements. Continents are
signified in red . Black dots stand for oceanic plateaus.
distributed like raisins. Furthermore the present model, S3, does not present
difficulties with the buoyancy since the present chemical differences refer to
the incompatible elements and not to the geochemical major components. It
is remarkable that we did not obtain simply connected volumes for any geo-
chemical r e servoir. Nevertheless, the depleted volumes tend to be in the upper
parts of the mantle. This is not amazing since chemical differentiation takes
place just beneath the lithosphere and the low viscosity of the asthenosphere
promotes mixing and lateral migration of DMM.
Figure 5 shows the present-time distribution of continents (red) of our
reference run. The oceanic plateaus (black dots) are carried alo ng by the self-
consistently generated, moving oceanic lithosphere. If the plateaus touch a
28 Uwe Walzer, Roland Hend el, and John Baumgardner
continent they join with it. This is the only additional implementation. Neither
number nor form nor size of the continents is prescribed. The configuration
results simply from the numerical solution of the system of eq uations and the
initial and boundary conditions. At first, the c omparison with the obser ved
present-day continents was carried out simply visua lly. Then we decided to
represent both topographies, the observed one and the theoretical one, in
terms of spherical harmonics
{A
m
n
or B
m
n
} = π
1
· (2n + 1)
1/2
· 2
1/2
· [(n m)!]
1/2
· [(n + m)!]
1/2
·
·
2π
Z
0
{cos or sin } ·
π
Z
0
f(θ, φ) · P
n,m
(cos θ) · s in θ ·
dφ, (57)
respectively, where f (θ, φ) is topographic height. While the individual coef-
ficients A
m
n
or B
m
n
depend on the position of the pole of the grid (θ, φ), the
quantity h
n
is orientation-independent:
h
n
= n
1/2
· (n + 1)
1/2
· 2
1
·
(
n
X
m=0
(A
m
n
)
2
+ (B
m
n
)
2
i
)
1/2
(58)
Fig. 5. The distribution of red continents and black oceanic plateaus at the Earth’s
surface for the geological present time according to the reference run with yield stress
σ
y
= 115 MPa and viscosity-level parameter r
n
= 0.65. Arrows denote velocity.
The oceanic lithosphere is denoted in yellow. There are no prescriptions concerning
number, size or form of continent s in the present model.
Mantle Convection, Continent Generation and Geochemical Heterogeneity 29
Fig. 6. The surface distribution of log viscosity (Pa·s) on an equal-area projection
for the geological present time for the reference run. The velocity field displays plat e-
like ch aracter. Elongated high strain-rate zones lead to reduced viscosity because of
viscoplastic yielding.
We ar e not awa re of other papers on spherical-shell mantle convection with
continents that evolve due to physical laws and that ar e not simply put onto
the surface.
Figure 6 reveals the plate-like motions of the lithospheric patches at the
surface. This kind of motion arises because of the viscosity law that includes
yield stress. It has nothing to do with the tracers. It arises in similar models
without tracers . The colors represent the logarithm of the viscosity in Pa·s.
Figure 7 exhibits the present-time temperature on an equal-area pr ojection of
a spher ical surface at 134.8 km depth. The blue zones corresponding to cold,
subducting rock correlate with convergent zones at the surface.
4.2 Variation of Parameters: The Evolution of Continents
We varied the parameters Ra and σ
y
to investigate the region in which we
obtain Earth-like results and to find other regions with different mechanisms.
A multitude of runs were performed to convince us that the selected reference
run is by no means exceptiona l but representa tive of a notable portion of the
30 Uwe Walzer, Roland Hend el, and John Baumgardner
parameter space. We find that the general character of our results does not
deviate too far from that of the real Earth. We compare the number, size,
form and distribution of the calculated continents with the continent config u-
ration of the present Earth. Earth-like continent solutions are shown by little
black disks in the center of the Ra-σ
y
plot of Figure 8. Ra denotes the tem-
poral average of the Rayleigh number of a given run. Figures 9 and 10 display
present-time continent distributions from two other runs, with σ
y
= 130 MPa
and σ
y
= 115 MPa, respectively, and for r
n
= 0.6 in both cases. We per-
formed further studies to attempt to refine the Earth-like Ra-σ
y
area. Figure
11 describes a quantitative measure of the devia tion of the calculated present-
time continental area from the observed one. Favora ble agreement occ urs in
the center of the Ra-σ
y
area. Favorable means that Ear th-like solutions can
be found in both Figures 8 and 11 in the common part of the Ra-σ
y
field.
Fig. 7. Equal-area projection with the temperature distribution (colors) and the
velocities (arrows) for the geological present for the reference run at a depth of 134.8
km. The narrow blue sheet-like subducting zones are evident also at greater depths.
The slab-like features are narrow in comparison with the much broader upwellings.
Mantle Convection, Continent Generation and Geochemical Heterogeneity 31
Fig. 8. The types of continental distribution as a function of yield stress, σ
y
, and
of temporally averaged Rayleigh number, Ra. Each symbol of the plot denotes one
run. Little black disks with a white center signify Earth-like distributions of the
continents where the size of the disk is a measure of quality. Five-pointed stars
stand for distributions with an unrealistic multitude of tiny continents. White circles
represent runs with reticularly connected, narrow stripe-like continents.
Fig. 9. Equal-area projection with the distribution of continents (red) and oceanic
plateaus (black dots) for the geological present of a run with yield stress σ
y
= 130
MPa and viscosity-level parameter r
n
= 0.6. Yellow color stands for the oceanic
lithosphere.
32 Uwe Walzer, Roland Hend el, and John Baumgardner
Fig. 10. The distribution of continents (red) and oceanic plateaus ( black dots) for
the geological present of a run with yield stress σ
y
= 115 MPa and viscosity-level
parameter r
n
= 0.6. The oceanic lithosphere is signified by yellow color.
Fig. 11. A classification of th e runs with respect to the difference of observed
surface percentage of continents ( =40.35%) minus calculated surface percentage of
continents. This difference, d
c
, is plotted as a function of yield stress, σ
y
, and of the
time average of the Rayleigh number, Ra. Little black disks denote slight deviations,
namely 4.5 d
c
< 4.5 percent. White circles stand for 4.5 d
c
< 13.5. Plus signs
signify 13.5 d
c
< 22.5. White triangles represent runs with 22.5 d
c
< 31.5.
White diamonds denote 31.5 d
c
< 40.5.
Mantle Convection, Continent Generation and Geochemical Heterogeneity 33
Fig. 12. The types of lithospheric movements as a function of yield stress, σ
y
,
and time average of the Rayleigh number, Ra . Plate-like solutions with narrow
subducting zones are depicted by little black d isks. Its surface area is a measure of
plateness. White circles represent runs with broad downwellings and minor plateness.
White five-pointed stars denote unrealistic runs with local subduction only. Asterisks
stand for rather complex planforms with lots of small but not narrow d ownwellings.
Fig. 13. The time average of the Urey number, U r is p lotted in a diagram the
abscissa of which is the yield stress, σ
y
, and the ordinate is the time average of
the Rayleigh number, Ra. Asterisk s represent runs with Ur 0.59. White squ ares
stand for 0.59 < Ur 0.625. Little black disks denote runs with 0.625 < Ur 0.67.
White circles depict runs with 0.67 < U r 0.71. Finally, plus signs signify runs
with 0.71 < U r.
34 Uwe Walzer, Roland Hend el, and John Baumgardner
4.3 Variation of Parameters: Plateness of Oceanic Lithospheri c
Pieces and Other Features
A classification of runs in terms of the planforms of flow near the surface is
presented by Figure 12. Black disks denote plate-like solutions. An over lap set
of the black disks is observed with the black disks of Figures 8 and 11. Figure
13 shows the distribution of classes of Urey numbers as a function of yield
stress, σ
y
, and time average of the Rayleigh number, Ra. Runs with realistic
Urey numbers are pictured by black disks. For a comparison of the present-
time laterally averaged heat flows, qob(now), of the runs, it is important to
filter away the random fluctuations. A simple method to do so is to replac e
the calculated values of qob(now) by qob
where
qob
= m ean[qob(now)/qob(time av)] qob(time a v) (59)
The expression time av denotes the time average of one run, mean stands for
the average o f a ll runs of the plot. Figure 14 demonstrates the distribution of
the filtered present-time sur face averag e of the heat flow, qob
, in an r
n
σ
y
diagram. Realistic values are again denoted by black disks. A partial c overing
with the favorable field of continent distribution of Figure 5 is established.
Figure 15 shows the present-time theoretical flow spectrum
n
1/2
× (n + 1)
1/2
× hv
2
n,pol
i of the reference run (lower curve) in comparison
with the spectra of the total observed topography, T , and of the observed sea-
floor topography, S, of the global JGP95E Digital Elevation Model (Lemoine
Fig. 14. The symbols represent classes of the non-random values, q ob
, of the
present-time surface average of the heat flow of the runs in a r
n
-σ
y
plot where qob
is calculated using Eq. (59). The following numbers are given in mW/m
2
. Asterisks
signify runs with 97 qob
. Wh ite squares depict runs with 89 qob
< 97, little
black disks stand for 81 qob
< 89. White circles denote runs with 77 qob
< 81,
plus signs represent the range qob
< 77.
Mantle Convection, Continent Generation and Geochemical Heterogeneity 35
Fig. 15. A comparison of the orientation-independent quantities h
n
of t he total
flow of the Earth, T , and of the bathymetry, S, with the theoretical ow spectrum
n
1/2
(n+1)
1/2
hv
2
n,pol
i of the reference run (lower curve). The observational curves
T and S have been calculated from the topography of the global JGP95E Digital
Elevation Model of Lemoine et al. [67], chapter 2.
et al. [67]). It would be senseless to compare the different sets of coefficients
A
m
n
and B
m
n
of Eq. (57 ) since they depend on the position of the pole of
the coordinate system. The quantity h
n
of Eq. (58) is, however, independent
on the orientation of the pole. The comparison of the theor e tical spectrum
h
n
(n) with that of T shows a coincidence of the maxima up to n=17. A
correspondence for higher values of n is not to be expected because of the
simplicity of the model. The perpendicular auxiliary lines are ther e fore only
in five-unit distances for the higher-n region.
5 Conclusions
The main subject of this paper is a c ombined segregation-convection theory
in a 3-D compres sible spherical-shell mantle. It is a step towa rd a reconcili-
ation of seemingly contradictory geochemical and geophysical findings and a
preliminary answer to three questions: (a) Did the differentiation of the mass
of the continental crust (CC) take place predominantly at the b e ginning of
the Earth’s evolution similar to the cases of the Moon and Mars in which
chemical segre gation occured in the first 200 Ma, o r have there been other
modes of crustal production that continue to add juvenile crust in batches
36 Uwe Walzer, Roland Hend el, and John Baumgardner
possibly connected with episodic orogenesis? (b) How can different geochem-
ical reservoirs be maintained in spite of persisting whole-mantle convection?
(c) Why is DMM more ho mogeneous than other reservoirs?
Our modeling sug gests the following simplified answers: (a) Similar to the
cases of the Moon and Mars, part of the Earth’s crust was probably also
formed from a magma o c e an, whether also CC was formed at this point is
unknown. Nevertheless, since the mantle has been s olid, our model indica tes
there have been episodes of CC growth comparable to magmatic and tectonic
episodes in the Earth’s history (cf. Figure 3, second panel). (b) The essential
cause for the long-term conservation of complex mantle reservoirs less depleted
than DMM is a high-viscosity zone in the c e ntral part of the lower mantle.
Furthermore, the endothermal 660-km phase boundary and a possible high-
viscosity transition layer also retard the stirring. (c) DMM is produced in
the conventional asthenosphere and is distributed by convection also to o ther
parts of the mantle. Since the asthenosphere has the lowest viscosity, the
stirring is mo st effective there.
Moreover, the Figures 4,5,8,9,10,11 and 15 show that our model, S3, gen-
erates convincing present-time distributions of continents. Although the prob-
lem of oceanic lithospheric plate generation is not the focus of this paper as in
Trompert and Hansen [108], Tackley [100, 101], Richards et al. [88], Bercovici
and Karato [11], Walzer et al. [113] and Bercovici and Ricard [13], we want
to remark that also S3 shows good plate-like solutions (cf. Figure 12). Other
conclusions that we do not want to repeat here can be found in the Abstract.
Acknowledgements
We gratefully acknowledge the help o f Dave Stegman. He provided us with
his particle code and discussed some problems with us. This work was partly
supported by the Deutsche Forschungsgemeinschaft under grant WA 1035/5-3.
We kindly acknowledge the use of supercomputing facilities at HLRS Stuttgart
and NIC J¨ulich. The major part of the simulations was performed on the Cray
Strider Opter on cluster at the High Performance Computing Center (HLRS)
under the grant number sphshell /12714.
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