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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...

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It has long been known that solid-state convection within Earth’s mantle should result in deformation of its surface (Pekeris, 1935), a phenomenon referred to as dynamic topography. This topography is relatively elusive: it is transient over time scales of 1–10 m.y., it occurs over spatial scales covering a few hundred to a few thousand kilometers,...

## 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,
convection, stirring, and thermal evolution constitute an inseparable dynamic system. Our model is based on the solution of
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.

... This introduction is strongly simplified. Walzer et al. [25] discuss the mantle chemistry, the mantle processes, and their translation into our model in detail. ...

... Walzer et al. [25] describe the derivation of the governing equations of our convection-differentiation model. Walzer and Hendel [23] present a numerical model in which 3-D compressional spherical-shell convection, thermal evolution of the Earth, chemical differentiation of plateau basalts and continental growth are integrated. ...

... 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.