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# 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 converted Type-3 tracer mass per Ma. (c) The reciprocal value of the Urey number. Ror represents the r atio of the surface heat o utflow to the mantle’s r adiogenic heat production rate. (d) The Rayleigh number as a function of age.

Source publication

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 incre...

## Contexts in source publication

**Context 1**

... expression time av denotes the time average of one run, mean stands for the average of all runs of the plot. Figure 14 demonstrates the distribution of the filtered present-time surface average of the heat flow, qob ∗ , in an r n − σ y diagram. Realistic values are again denoted by black disks. A partial covering 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 × v n,pol 2 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 et al. [67]). It would be senseless to compare the different sets of coefficients A m n and B n m 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 theoretical 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 therefore only in five-unit distances for the higher- n region. The main subject of this paper is a combined segregation-convection theory in a 3-D compressible spherical-shell mantle. It is a step toward a reconciliation 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 beginning of the Earth’s evolution similar to the cases of the Moon and Mars in which chemical segregation occured in the first 200 Ma, or have there been other modes of crustal production that continue to add juvenile crust in batches possibly connected with episodic orogenesis? (b) How can different geochemical reservoirs be maintained in spite of persisting whole-mantle convection? (c) Why is DMM more homogeneous than other reservoirs? Our modeling suggests 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 ocean, whether also CC was formed at this point is unknown. Nevertheless, since the mantle has been solid, our model indicates 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 central 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 other parts of the mantle. Since the asthenosphere has the lowest viscosity, the stirring is most effective there. Moreover, the Figures 4,5,8,9,10,11 and 15 show that our model, S3, generates convincing present-time distributions of continents. Although the problem 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. We gratefully acknowledge the help of 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 Jülich. The major part of the simulations was performed on the Cray Strider Opteron cluster at the High Performance Computing Center (HLRS) under the grant number sphshell ...

**Context 2**

... begin by presenting what we call our reference run 808B. It is representative of the results we obtain in a moderately extensive region of Rayleigh number – yield stress parameter space. Our chosen reference run is defined by a viscoplastic yield stress σ y = 115 MPa and a viscosity-level parameter r n = − 0 . 65. Run 808B starts with eight tracers per grid-point cell. Now, we present the Figures, in each case immediately followed by the corresponding discussion. In Figure 1, the laterally averaged temperature for the geological present time as a function of depth is represented by a solid line. This curve lies closer to the geotherm of a parameterized whole-mantle convection model than to the corresponding layered-convection 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 transition zone and by the endothermic 660-km phase boundary. Therefore, the temperature is slightly augmented, especially immediately beneath the 660-km boundary. Figure 2 displays the laterally averaged present-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 laterally averaged 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. [119], 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 reciprocal value of the Urey number. Parameterized models show roughly similar curves except for medium-large and smaller fluctuations. A pattern of general decrease and some fluctuations in the Rayleigh number are indicated in the fourth panel. The chemical heterogeneity of incompatible elements in a run with 64 tracers 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 reservoirs 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. However, 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. This is a realistic feature of the model since where the real oceanic lithosphere is rifted, MORB magma is formed by decompression melting. The MORB source (DMM) is not only depleted in incompatible elements but also relatively homogenized. It is homogenized not only with respect to its major geochemical components (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 isotope ratios and of the major element compositions is small for MORBs in comparison to OIBs ( Allègre 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 regions in our model that are disconnected ...

**Context 3**

... begin by presenting what we call our reference run 808B. It is representative of the results we obtain in a moderately extensive region of Rayleigh number – yield stress parameter space. Our chosen reference run is defined by a viscoplastic yield stress σ y = 115 MPa and a viscosity-level parameter r n = − 0 . 65. Run 808B starts with eight tracers per grid-point cell. Now, we present the Figures, in each case immediately followed by the corresponding discussion. In Figure 1, the laterally averaged temperature for the geological present time as a function of depth is represented by a solid line. This curve lies closer to the geotherm of a parameterized whole-mantle convection model than to the corresponding layered-convection 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 transition zone and by the endothermic 660-km phase boundary. Therefore, the temperature is slightly augmented, especially immediately beneath the 660-km boundary. Figure 2 displays the laterally averaged present-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 laterally averaged 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. [119], 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 reciprocal value of the Urey number. Parameterized models show roughly similar curves except for medium-large and smaller fluctuations. A pattern of general decrease and some fluctuations in the Rayleigh number are indicated in the fourth panel. The chemical heterogeneity of incompatible elements in a run with 64 tracers 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 reservoirs 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. However, 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. This is a realistic feature of the model since where the real oceanic lithosphere is rifted, MORB magma is formed by decompression melting. The MORB source (DMM) is not only depleted in incompatible elements but also relatively homogenized. It is homogenized not only with respect to its major geochemical components (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 isotope ratios and of the major element compositions is small for MORBs in comparison to OIBs ( Allègre 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 regions in our model that are disconnected ...

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## Citations

... The publications [97,104,105,107] present models of self-consistent generation of stable, but time-dependent plate tectonics on a 3D spherical shell. Different types of solutions have been found for different models by systematic variation of parameters [97,102,104,108,109]. Stirring effects are investigated in [22]. ...

... Different types of solutions have been found for different models by systematic variation of parameters [97,102,104,108,109]. Stirring effects are investigated in [22]. A 3D spherical-shell mantle convection and evolution model with growing continents [97,100,101,108,109] has been developed. The evolution model equations guarantee conservation of mass, momentum, energy, angular momentum, and of four sums of the numbers of atoms of the pairs 238 U-206 Pb, 235 U-207 Pb, 232 Th-208 Pb, and 40 K-40 Ar. ...

Some essential features of Andean orogenesis cannot be explained only by a dynamic regional model since there are essential influences across its vertical boundaries. A dynamic regional model of the Andes should be embedded in a 3-D spherical-shell model. Because of the energy distribution on the poloidal and toroidal parts of the creep velocity and because of geologically determined mass transport alongside the Andes, both models have to be three-dimensional. Furthermore, we developed a new viscosity profile of the mantle with very steep gradients at the lithospheric-asthenospheric boundary and at a depth of 410 and 660 km. Therefore, the challenges to the code Terra are now essentially larger. In the last 3 years we have resolved these problems in an international cooperation (see Sect. 2.2). Based on the new viscosity profile and on an improved Terra, we computed a new forward spherical-shell model (Walzer and Hendel, J Geophys Res submitted, 2012b). For this model, we derived also a new extended acoustic Grüneisen parameter, γax
, new profiles of the thermal expansivity, α, and of the specific heat, c
v
, at constant volume as well as a solidus depending on both the pressure and the water abundance. These innovations are essential to incorporate a chemical-differentiation mechanism into the model. We arrived at rather realistic episodes of continental growth interrupted by magmatically quiet time spans distributed over the whole time axis. Nevertheless, the model shows a main magmatic event at the very beginning of the Earth’s evolution. Papers on the improvement of Terra (Köstler et al. Comput Geosci submitted, 2012; Müller and Köstler, Int J Numer Methods Eng submitted, 2012)have been written. We conceived a regional model of the Andean Sect. 3.2.1) with the same new viscosity profile. We want to investigate why there is flat-slab subduction in some segments of the Andes and why deformation of the crust and volcanism migrate eastward. The evolution of the abundances of incompatible elements indicate a cycle which was finished by a fast process, perhaps by a large-scale delamination of the lower plate, perhaps also by another type of delamination. In connection with another spherical-shell model (with prescribed plate boundaries), the regional model should numerically explain why a plateau-type orogen evolved at an oceanic-continental plate boundary.

... Whether or not such a distributed geochemical reservoir theory is viable is still an open issue. Sections 1 and 2 of [41] give lots of further information regarding the geochemical foundations of our numerical model. ...

... Unlike other mantle-convection papers with continents, our continents are not artificially imposed but evolve by chemical differentiation of which the process has been represented by a tracer approach. A full derivation of the equations and a presentation of the model parameters is given by Walzer et al. [41]. Nevertheless, the present companion paper presents exclusively unpublished material. ...

A dynamic 3-D spherical-shell model for the chemical evolution of the Earth’s mantle is presented. Chemical differentiation,
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the balance equations of mass, momentum, energy, angular momentum, and four sums of the number of atoms of the pairs 238U-206Pb, 235U-207Pb, 232Th-208Pb, and 40K-40Ar. Similar to the present model, the continental crust of the real Earth was not produced entirely at the start of the evolution
but developed episodically in batches. The details of the continental distribution of the model are largely stochastic, but
the spectral properties are quite similar to the present real Earth. Fig. 6 reveals that the modelled present-day mantle has
no chemical stratification but we find a marble-cake structure. If we compare the observational results of the present-day proportion
of depleted MORB mantle with the model then we find a similar order of magnitude. The MORB source dominates under the lithosphere.
In our model, there are nowhere pure unblended reservoirs in the mantle. It is, however, remarkable that, in spite of 4500
Ma of solid-state mantle convection, certain strong concentrations of distributed chemical reservoirs continue to persist
in certain volumes, although without sharp abundance boundaries. Section 4 presents results regarding the numerical method,
implementation, scalability and performance.

... Therefore and for other reasons, we propose to develop firstly simple models and algorithms, later more elaborated dynamic ones for the thermal evolution and the chemical differentiation of Mars where the numerics of the more complex code will be rather ambitious. It will be based on the program Terra [122,123,125]. It is clear that Terra and similar codes are suitable only for solid-state convection. ...

... These observations argue for episodic mantle melting and crustal growth on Earth. We [121, 122, 125] developed numerical models to explain the mechanism behind these terrestrial observations. The bulk of the martian crust formed, however, about 4.5 Ga ago, presumably from a magma ocean [87]. ...

We present the basic conception of a new dynamical model of the thermal and chemical evolution of Mars. Therefore new enlargements
of the code Terra are necessary which allow to improve the solutions of the convection differential equations with strongly
varying viscosity. These enlargements have been partly tested already. We describe considerations on the chronology of the
early evolution of Mars and on magma ocean solidification since they lead to a structural model of the early Mars. This is important as a starting presupposition for a dynamical solution of the martian evolution similar to [122] which derives the essential features of the Earth’s mantle’s history. At present there is no PREM[39]-analogon neither for
the present time nor for the start of the solid-state creep in the martian mantle. Mars has not only a topographical and crustal
dichotomy but also a chemical dichotomy. We discuss different mechanisms which could generate not only these stuctures but
also an early strong magnetic dipole field that vanishes after 500 Ma at the latest. Section7 presents recent and future
numerical improvements of the code Terra. Section8 gives results on performance and scalability.

... It is remarkable that the decrease of qob as a function of time is rather moderate in comparison to that of parameterized models [59]. In [76], we emphasized that this result is in agreement with the results of komatiite research. This slow decrease of qob is a further indication that not only the temperature dependence of viscosity is the reason for the generation of oceanic lithosphere but also devolatilization and other chemical effects. ...

We present the basic conception of a new fluid-dynamic and geodynamic project on the Andean orogeny. We start with a kinematic
analysis of the entire orogeny and test different numerical options to explain these systematized observations by a physical
model. Therefore we consider partly kinematic, partly dynamic regional models as well as purely dynamic models. Because of
stochastic effects which are unavoidable in purely fluid-mechanical mechanisms of this kind and which influence the specific
form of the Andes and because of the, to a large extend, unknown initial conditions, the partly kinematic, partly dynamic
models have their right to exist. A purely dynamic model would be, of course, much more satisfactory. Therefore we want to
approach nearer to the purely dynamic models prescribing a less number of parameters and dropping some artificial constraints.
We have a concept to embed a regional model into a global spherical-shell model to determine the boundary conditions of the
regional model as a function of time. So we avoid the artificially simplified boundary conditions of some published models
of the Andean mechanism. On the other hand, the regional model has to retroact upon the global surrounding model. So, we have
an iteration concept. For the two mentioned reasons there are, analogously to the two kinds of regional models, also two kinds
of spherical-shell convection models, namely circulation models and forward models. As a first step, we present a spherical-shell
model of mantle convection with thermal evolution and generation of continents and, as a complement, the depleted mantle reservoir.
Our presented numerical result is that plate tectonics occurs only if at least the lithosphere deviates from purely viscous
rheology and if there is a low-viscosity layer beneath of it. We suppose especially a viscoplastic yield stress for the lithosphere
and a mainly temperature-independent asthenosphere which is determined, e. g., by the intersection points of water abundance
and water solubility curves. The number of plates, at a certain fixed time of evolution, depends on Rayleigh number and, to
a minor degree, on yield stress. We discuss our new efforts to improve the basic code Terra. The numerical regional Andean
model has to be embedded into a global circulation model. Therefore we need an improved Terra for the latter one.

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We compute a model of thermal and chemical evolution of the Earth’s mantle by numerically solving the balance equations of
mass, momentum, energy, angular momentum and of four sums of the number of atoms of the pairs 238U-206Pb, 235U-207Pb, 232Th-208Pb, and 40K-40Ar. We derive marble-cake distributions of the principal geochemical reservoirs and show that these reservoirs can separately
exist even in a present-day mantle in spite of 4500 Ma of thermal convection. We arrive at plausible present-day distributions
of continents and oceans although we did not prescribe number, size, form, and distribution of continents. The focus of this
paper is the question of predictable and stochastic portions of the phenomena. Although the convective flow patterns and the
chemical differentiation of oceanic plateaus are coupled, the evolution of time-dependent Rayleigh number, Ra
t
, is relatively well predictable and the stochastic parts of the Ra
t
(t)-curves are small. Regarding the juvenile growth rates of the total mass of the continents, predictions are possible only
in the first epoch of the evolution. Later on, the distribution of the continental-growth episodes is increasingly stochastic.
Independently of the varying individual runs, our model shows that the total mass of the present-day continents is not generated in a single process at the beginning of the thermal evolution of the Earth but in episodically distributed processes
in the course of geological time. This is in accord with observation. Section4 presents results on scalability and performance.

The main subject of this paper is the numerical simulation of the chemical differentiation of the Earth’s mantle. This differentiation induces the generation and growth of the continents and, as a complement, the formation and augmentation of the depleted MORB mantle. Here, we present for the first time a solution of this problem by an integrated theory in common with the problem of thermal convection in a 3-D compressible spherical-shell mantle. The whole coupled thermal and chemical evolution of mantle plus crust was calculated starting with the formation of the solid-state primordial silicate mantle. No restricting assumptions have been made regarding number, size and form of the continents. It was, however, implemented that moving oceanic plateaus touching a continent are to be accreted to this continent at the corresponding place. The model contains a mantle-viscosity profile with a usual asthenosphere beneath a lithosphere, a highly viscous transition zone and a second low-viscosity layer below the 660-km mineral phase boundary. The central part of the lower mantle is highly viscous. This explains the fact that there are, regarding the incompatible elements, chemically different mantle reservoirs in spite of perpetual stirring during more than 4.49×109 a. The highly viscous central part of the lower mantle also explains the relatively slow lateral movements of CMB-based plumes, slow in comparison with the lateral movements of the lithospheric plates. The temperature- and pressure-dependent viscosity of the model is complemented by a viscoplastic yield stress, σ
y. The paper includes a comprehensive variation of parameters, especially the variation of the viscosity-level parameter, r
n, the yield stress, σ
y, and the temporal average of the Rayleigh number. In the r
n−σ
y plot, a central area shows runs with realistic distributions and sizes of continents. This area is partly overlapping with the r
n−σ
y areas of piecewise plate-like movements of the lithosphere and of realistic values of the surface heat flow and Urey number. Numerical problems are discussed in Sect. 3.

This contribution aims at directing the attention towards the main inverse problem of geodesy, i.e. the recovery of the geopotential.
At present, geodesy is in the favorable situation that dedicated satellite missions for gravity field recovery are already
operational, providing globally distributed and high-resolution datasets to perform this task. Due to the immense amount of
data and the ever-growing interest in more detailed models of the Earth’s static and time-variable gravity field to meet the
current requirements of geoscientific research, new fast and efficient solution algorithms for successful geopotential recovery
are required.