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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 f¨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

Pb≈19 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 (W¨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

= −vα (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

dρ (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

dρ

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

/ρ

r

#

(30)

where the index r refers to the adiabatic reference state, ∆ρ

k

/ρ

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

oν

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 mφ or sin mφ} ·

π

Z

0

f(θ, φ) · P

n,m

(cos θ) · s in θ · dθ

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 ﬂow 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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