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# The laterally averaged shear viscosity of the reference run as a function of depth for the present geological time.

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The focus of this paper is numerical modeling of crust-mantle differentiation. We begin by surveying the observational constraints of this process. The present-time distribution of incompatible elements are described and discussed. The mentioned differentiation causes formation and growth of continents and, as a complement, the generation and incre...

## Contexts in source publication

**Context 1**

... pressure is denoted by P , the second invariant of the strain-rate tensor 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 transition 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 focal 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 phase transition 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 boundary then the latter is deflected somewhat downward. An earthquake has never been observed below that de- flection. Beneath of that, the slab is only detectable by elevated 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 transition zone is composed mostly of garnet and spinel ( Meade and Jeanloz [74], Karato et al. [59], Karato [57], Allègre [3]). If there are no further phase transitions in the lower mantle, except near the D ′′ layer ( Matyska and Yuen [70]), then the viscosity must rise considerably as a function of depth because of the pressure dependence of the activation enthalpy of the prevailing creeping mechanism for regions where the temperature gradient is near to adiabatic. This implies a thick high-viscosity central layer in the lower mantle. 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 arguments. They serve only as a corroboration. The alternative systematic derivation is described by Walzer et al. [113]. We start from a self-consistent theory using the Helmholtz free energy, the Birch-Murnaghan equation of state, the free-volume Grüneisen parameter and Gilvarry’s [36] formulation of Lindemann’s law. The viscosity is calculated as a function 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 radial viscosity distribution. To determine the absolute scale of the viscosity profile, we utilize the standard postglacial-uplift viscosity of the asthenosphere below the continental lithosphere. Our η 3 ( r ) profile is supported by several recent studies. New inversion investigations for mantle viscosity profiles reveal an acceptable resolution down to 1200 km depth. For greater depths, models based on solid-state physics seem to be more reliable. Kido and Cadek ˇ [61] and Kido et al. [63] found two low-viscosity layers below all three oceans. The first layer is between the lithosphere and 410 km depth. The second one is between 660 and about 1000 km depth. Panasyuk and Hager [81] made a joint inversion of the geoid and the dynamic topography. They found three families of 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 viscosity maxima at about 800 and about 2000 km depth. This is similar to our model for η 3 ( r ) that has also two maxima in the interior although is has been derived by a completely different method. Cserepes et al. [30] investigated the effects of similar viscosity profiles on Cartesian 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 parameters that define these phase transitions are provided in Table 1. In our models we include the full pressure dependence and the full radial temperature 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 interior of the Earth’s mantle, there are not only discontinuities of the seismic velocities and of the density but also jumps of activation volumes, activation energies and, therefore, of activation enthalpies. Since the viscosity depends exponentially on the activation 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 surfaces with different radial distances 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 equal parts. Connecting the new points with great 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 replicate the resulting almost uniform triangular grid at different radii to generate the 3D grid for a spherical shell. We can use different formulae for the distribu- 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 refine 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 velocity and either piecewise constant or piecewise linear basis functions for the pressure. We solve the equations for pressure and velocity simultaneously by a Schur-complement conjugate-gradient iteration ( Ramage and Wathen [86]). This is a further development of an Uzawa algorithm. We solve the energy equation using an iterative multidimensional positive-definite advection-transport algorithm with explicit time steps ( Bunge and Baumgardner [20]). Within the Ramage-Wathen procedure, the resulting equation systems are solved by a multigrid procedure that utilizes radial line Jacobi smoothing. In the multigrid procedure, prolongation and restriction are han- dled in a matrix-dependent manner. In this way, it is possible to handle the strong variations and jumps of the coefficients associated with the strong viscosity gradients ( Yang [121]). For the formulation of chemical differentiation, we modified a tracer module developed by Dave Stegman. This module contains a second-order Runge-Kutta procedure to move the tracer particles in the velocity field. Each tracer carries the abundances of the radionuclides. In this sense, tracers are active attributes which determine the heat production rate per unit volume that varies with time and position. [30] The FORTRAN code is parallelized by domain decomposition and explicit message passing (MPI) ( Bunge [19]). For the most runs, we used a mesh of 1351746 nodes. For some runs, we used 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 performed on 32 processors of a Cray Strider Opteron cluster. The code was benchmarked for constant viscosity convection 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%. We assume the Earth’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 ...

**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 ...

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... 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]. ...

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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 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
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implementation, scalability and performance.

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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
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We present the basic conception of a new fluid-dynamic and geodynamic project on the Andean orogeny. We start with a kinematic
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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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... The comparison of the h * n -n spectra is shown by Fig. 15 of Walzer et al. [25]. Using many cases, we found a realistic Ra-σ y region. ...

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.