Real episodic growth of continental crust or artifact of preservation? A 3-D geodynamic model

Authors


Abstract

[1] We investigate whether the observed zircon age distribution of continental crust (CC) is produced by real crustal growth episodes or is only an artifact of preservation. In connection with the second alternative of this question, other authors proposed that there was little episodicity in the production of new CC and that modeling corroborates this opinion. We conclude that a combination of the two answers might be possible. In matters of modeling, however, we ascertain that a dynamic modeling of the convection-differentiation system of the mantle reveals the high probability of magmatic episodes. We solve the full set of balance equations in a 3-D spherical-shell mantle. The heat-producing elements are redistributed by chemical differentiation. A realistic solidus model of mantle peridotite is essential for an applicable model. The solidus depends not only on depth but also on the volatile concentration. Furthermore, we introduced realistic profiles of Grüneisen parameter, viscosity, adiabatic temperature, thermal expansivity, and specific heat. Our model automatically produces lithospheric plates and growing continents. Regarding number, size, form, distribution, and surface velocity of the continents, no constraints have been prescribed. Regions of the input parameter space (Ra,σy,k,f3) that are favorable with respect to geophysical quantities show simultaneously not only episodicity of CC growth but also a reproduction of the observed zircon-age maxima referring to the instants of time. We also obtain Archean events for ages greater than 3000 Ma that are not or scarcely visible in the observed zircon ages. Sinusoidal parts of the evolution curve of qob, Ur, and Ekin are superposed with a monotonous decrease. The volumetrically averaged mantle temperature, Tmean, however, decreases smoothly and slowly, nearly without pronounced variations. Therefore, we can dismiss catastrophic mechanisms that simultaneously incorporate the whole mantle.

1 Introduction

1.1 Conception

[2] The hypsometric curve of the Earth is bimodal. In our planetary system, this fact is unique. The existence of oceans at the surface of the solid Earth is unique, too. It is an obvious guess that the two unique features have a causal connection. The upper continental crust (UCC) is felsic, rich in Si, poor in Mg, and enriched in incompatible elements. The oceanic crust, however, is poor in Si, rich in Mg, and not as highly enriched in incompatible elements as UCC, both in comparison to “mantle reservoirs.” We use the concept of reservoirs considerably different than in former time. We mean it similarly as defined by Stracke et al. 2005] and Willbold and Stracke 2006]. We do not imagine sharp boundaries between mantle reservoirs. We use this concept only as a simple tool to describe the evidently rather inhomogeneous Earth's mantle. We define the chemical composition of a specific mantle location by the mixing ratio of the conventional reservoirs [Hofmann, 2003]. The exact chemical composition of the bulk continental crust (CC) is controversial. Taylor and McLennan [1985, 1995, 2009] assume a composition between andesitic and basaltic-andesitic with slightly more than 57 wt% SiO2, whereas Rudnick and Fountain 1995] and Rudnick and Gao 2003] deduced an andesitic mean composition with about 60 wt% SiO2 and elevated K2O values. Wedepohl 1995] proposed a high-K andesitic model with over 62 wt% SiO2. It is irrefutable that some alkali elements are highly enriched in comparison with the primordial mantle, regardless if we assume a chondritic mantle as usual or a nonchondritic mantle [Caro, 2011] as a starting composition. Furthermore, we observe a complementary depletion in a part of the mantle called depleted MORB mantle (DMM) which evidently can be found at least immediately beneath ocean lithosphere. In addition, it is evident that the mid-oceanic ridge basalt (MORB) is a product of the chemical differentiation of DMM [Hofmann, 1988, 2003; Davidson and Arculus, 2006]. There are at least five models of the generation of CC which differ considerably [Davidson and Arculus, 2006]. The growth rate of the CC is a function of the balance between addition of mantle material by chemical differentiation in several steps and the loss of CC back into the mantle by subduction of continental sediments, subduction erosion, and lower crust delamination [Condie, 1998; Condie and Aster, 2010; Clift et al., 2009; Hawkesworth et al., 2010; Stern and Scholl, 2010; Roberts, 2011]. It is remarkable that the lower CC delamination generates nearly no loss of UCC which has a rather high percentage of Si, alkali elements, U, Th, and other incompatible elements.

[3] Well-crystallized zircon, ZrSiO4, preserves the isotopic ratios that stem from the date of origin of a magmatic rock. Therefore, zircon methods should better represent the original process of chemical differentiation than whole-rock isotopic systems. Condie and Aster 2010] investigate a very large database of 8928 orogenic granitic and 28,027 detrital U/Pb zircon ages and apply a kernel density analysis. Using a kernel bandwidth of 90 Ma, they identify four principal maxima that are situated at about 2686, 1871, 576, and 1073 Ma if arranged according to their height. One conclusion is that at least 60% of the juvenile CC already exists at an age of τ=2600 Ma. Hawkesworth and Kemp 2006] emphasize that Hf isotopes excellently reflect when new crust is generated from the mantle. They note that Hf ages of zircons with mantle-like δ18O values fall into two age ranges, namely 1.7–1.9 Ga and 2.9–3.1 Ga. Other peaks are interpreted by crustal differentiation. They conclude that much of the CC was generated during these two pulses of relatively rapid growth. In anticipation of our results, we remark that it is not our aim to comparatively evaluate the observational results of other authors but to numerically derive a physically consistent theory of CC generation, in connection with other geodynamic phenomena. It is, however, remarkable that we simultaneously obtain the CC production maxima of Condie and Aster 2010] and of Hawkesworth and Kemp 2006] by a group of runs, the input parameters of which are physically plausible for entirely other reasons.

[4] Prior papers on CC generation assume slow-growth models with very little CC formation before 4 Ga [Veizer and Jansen, 1979; Patchett and Arndt, 1986; McCulloch and Bennett, 1993, 1994; Taylor and McLennan, 1995]. Models of this kind often show steep gradients in the growth curves [e.g., Taylor and McLennan, 1995; Condie and Aster, 2010]. Many authors interpreted these steep gradients as periods of high generation of juvenile CC stemming from the mantle. The mentioned observations are not obsolete, also today, but require a lucid physical explanation. Additionally, new observations appeared which refer to Hadean and Archean.

[5] The Earth accreted about 4565 Ma. Undoubtedly, the mantle was in magma ocean stage during that time. Harrison et al. 2005] measured176Hf/177Hf initial ratios of 4.01 to 4.37 Ga detrital zircons. Harrison et al. 2008] extended this investigation using concurrent Lu-Hf and207Pb/206Pb analyses and deduced model ages between 4.56 and 4.20 Ga for the extraction of the zircon's protoliths. They concluded that already by 4.35 Ga, sialic crust had begun to form. Outside Western Australia, Iizuka et al. 2006] found U-Pb zircon age estimates of 4.2 Ga in a 3.9 Ga granitic rock in the Acasta Gneiss Complex of Canada. They concluded that the 3.9 Ga granitic magma was derived from a 4.2 Ga CC. Zheng and Zhang 2007] report Hf model ages of 4.0 Ga in South China and a 4.1 Ga U-Pb age of a detrital zircon found in Tibet. Furthermore, they show distinct cumulative peaks of micro-scale U-Pb dates of zircons from South China that are older than 2.7 Ga and their observed frequency decreases with growing age. This could possibly indicate that the same mechanism which generates the peaks shown by Hawkesworth and Kemp 2006] and Condie and Aster 2010] has already been effective earlier. Because tectonically active periods of time show not only differentiation of juvenile crust but also reworking of available CC, the amount of preserved Archean CC should decrease with age. Blichert-Toft and Albarède 2008] investigated 63 single zircons from Jack Hills, Western Australia, which were formed during a single magmatic pulse 4.1±0.1 Ga ago, and that were derived from a 4.30 to 4.36 Ga old protolith. The176Lu/177Hf ratios fit a tonalite-trondhjemite-granodiorite (TTG) source. Guitreau et al. 2012] examined whole-rock and zircon MC-ICP-MS Lu-Hf isotope date of Archean TTGs. The176Lu/177Hf ratio of the mantle source of CC is nearly constant in a scatterplot between 0.032 and 0.038 since about 3.9 Ga, whereas the chondritic value of this ratio is 0.0336. They argue that these data and other observations support the (perhaps stepwise) derivation of CC from a primitive, possibly chondritic mantle. From results of the Jack Hills zircons, they conclude that the mantle source of continents has remained unchanged since 4.3 Ga. The discussed papers contain strong indications of peaks of CC production before the pronounced maximum at 2.7 Ga, which are largely hidden by later processes.

[6] The DMM is thought to be the residue of some preliminary mode of CC. On the other hand, DMM is considered to be the source of MORB. The percentage of prior melt extraction from the primordial mantle is controversial. Estimates range from 30% to 100% according to the method and the authors [O'Nions et al., 1979; Zindler and Hart, 1986; Hofmann, 1997; Boyet and Carlson, 2006; Stracke et al., 2011]. Here we develop a numerical convection-differentiation mantle model of chemical and thermal evolution, on reliable physical foundations, to strike a balance between the reviewed doctrines. So we do not intend to add entirely new geochemical or isotopic arguments, but we want to show that a physically sound model is feasible and that it confirms and explains essential geochemical and geological findings.

1.2 Other Proposed Mechanisms

[7] There are some different proposals to explain the episodic distribution of zircon dates and other undisputed observations. It is well known that some present-day subducting slabs are transiently retained over and parallel to the 660 km discontinuity [van der Hilst et al., 1997]. From this, some models of catastrophic slab avalanches, flashes, and sudden convective overturns have been developed that are thought to result in more rapid movements of lithospheric plates near the surface. Also, O'Neill et al. [2007; 2008] interpret the age peaks (e.g., at 1.1, 1.9–2.1, 2.7, and 3.5 Ga) as maxima of increased CC production. They emphasize that in some cases,40Ar and18O isotopic data argue against a strong CC recycling into the deep mantle. They support the opinion that juvenile CC generation is episodic and propose a mechanism with strongly time-dependent subduction under the hotter conditions of the Precambrian mantle. They hypothesize repeated breakdowns of continuous plate tectonics induced by the balancing of driving convective stresses with resistive plate strengths. Davies [2007, 2008] emphasizes that subducted oceanic crust is buoyant between 660 and 750 km depth. The slab lingers for some time and finally subducts into the lower mantle. During such periods of impeded subduction, the upper mantle cools and the lower mantle warms. Every 100 to 150 Ma, this mechanism results in a magmatic episode. No subsequent layering occurs after about 1.6 to 1.8 Ga in Davies' model. Evidently, the last-mentioned two models cannot explain the magmatic episodes during the Phanerozoic. Furthermore, a realistic balancing of the number of atoms of a specific atomic species in the reservoirs will succeed only in a 3-D spherical-shell model with considerably more grid points; however, the last-mentioned two models are two-dimensional and Cartesian. Nevertheless, the two models contain important features of an overall convection mechanism.

[8] A linkage between plumes and magmatic episodicity has been proposed by other authors. Nolet et al. 2007] proved that at least some plumes ascend from the core-mantle boundary (CMB). Because of the high viscosity in the middle part of the lower mantle (cf. Figure 3), the lateral velocity of plumes is low in comparison to the velocities of oceanic lithospheric plates. Therefore, the distribution of ages of volcanoes in plume-related volcanic chains can serve to estimate the plate velocities and to corroborate the results from the spreading-rate-dependent magnetization of the oceanic lithosphere. Large igneous provinces (LIPs) are divided into three classes: continental flood basalts, oceanic plateaus, and ocean basin flood basalts [Coffin and Eldholm, 1994]. The latter two classes of LIPs are carried by the conveyor-belt-like oceanic lithosphere and are accreted to continents in zones of andesitic volcanism. One group of models assumes that this type of continental crustal growth is dominant [Stein and Hofmann, 1994; Abbott et al., 1997; Albarède, 1998]

[9] Hawkesworth et al. 2010] propose another view. They accept that zircons yield precise crystallization ages. However, they argue that the peaks of relative frequency of zircon age estimates above the axis of time are not a direct record of increased magmatic activity but an artifact of preservation. The formation of a supercontinent would cause an elevated preservation potential [Hawkesworth et al., 2009]. Similarly, Belousova et al. 2010] conclude from a database of about 13,800 U-Pb and Hf-isotope analyses of zircons that there is little episodicity in juvenile CC growth in contrast to peaks in magmatic ages. They propose that these magmatic peaks originate from supercontinent formation.

[10] Gurnis and Davies 1986] realized that the curves of CC growth are the same as crustal-age distributions only if there is no recycling of CC back into the mantle. Unfortunately, the assessments on the percentage of recycled CC are unreliable. The UCC will be removed down to the mantle probably only by subduction of UCC sediments and by lateral erosion by downgoing oceanic lithosphere, but not by delamination or foundering. Gurnis and Davies 1986] note that the physics of convection strongly inhibit rapid changes in the mean temperature of the mantle. Our numerical investigations clearly confirm this conclusion. This implies that, according to our numerical results, no disastrous whole-mantle movements are to be expected. However, we can show that using very moderate physical assumptions, real episodic growth of the continental crust is possible and most likely and that the observed temporal distribution of magmatic episodes is consistent with computed episodes.

[11] As an introduction to the solid-state convection of the mantle, we refer to Schubert et al. 2001]. Regarding thermochemical mantle convection, we mention Nakagawa and Tackley 2004] and Ogawa 2008].

2 Model Description

[12] Our numerical model is based on the assumption that the augmented preservation potential [Hawkesworth et al., 2009, 2010; Belousova et al., 2010is also associated with the complete convection-differentiation system by the generation of supercontinents. This system determines, in principle, also all other special mechanisms that have been proposed to explain the episodes of increased magmatism. Therefore, we want to design a rather general, physically acceptable numerical model where plates and plumes are not prescribed but can self-consistently develop and are an integral part of the numerical model. The seemingly alternative modes of (a) early growth of continental crust and subsequent recycling or (b) continuous crustal growth or(c) episodic crustal growth are also not prescribed but appear as a solution of the system of balance equations. We use a similar approach with respect to the continents. Our model continents are not artificially imposed. Neither the number of continents nor their size, form, or position are prescribed. The growing function of total continental mass is not presupposed, too. The last named parameters evolve from the differential equations of convection and some simplified equations of chemical differentiation with a physically reasonable starting criterion of chemical differentiation. Expectedly, we found an interplay between convection/mixing and differentiation.

2.1 Balance of Mass, Energy, and Momentum

[13] We developed a model of the Earth's thermal and chemical evolution for the whole time span. The thermal evolution of the core is taken into account only in a parameterized way. We use a numerical strategy for modeling the chemical differentiation leading to the juvenile contributions that are added to the continents. In a similar manner, we computed the mixing processes between the depleted mantle and the other (chemically inhomogeneous) parts of the mantle. We solve the differential equations of infinite Prandtl number convection using a three-dimensional finite-element spherical-shell method. These equations express the conservation of mass, energy, and momentum. For the mass balance, we use the anelastic liquid approximation

display math(1)

where math formula is velocity and ρis density. The energy balance can be expressed as

display math(2)

where T denotes the temperature, t the time, vjand xjthe components of the creep velocity and the location vector, respectively, γ the Grüneisen parameter, cvthe specific heat at constant volume, k the thermal conductivity, and Q the heat generation rate per unit volume. We have to put the deviatoric stress tensor, τik, into ((2)):

display math(3)

where the shear viscosity, η, is calculated by

display math(4)

[14] The melting temperature, Tm, and therefore the viscosity, strongly depends on the Grüneisen parameter. Furthermore, one important term in the preferred version of the energy balance, equation ((2)), explicitly depends on the Grüneisen parameter which can be derived directly from seismic observations. Therefore, this approach contains only a minimum of hypotheses on the pressure dependence of material parameters. In ((4)), r is the radius, θ the colatitude, φ the longitude, rn the viscosity-level parameter, Tav the laterally averaged temperature, and Tst the initial temperature profile. The quantity η4(r) is the viscosity profile at the initial temperature and rn=0. The second factor on the right-hand side of ((4)) describes the increase of the viscosity profile with the cooling of the Earth. For MgSiO3 perovskite we should insert c=14, and for MgO wüstite c=10 according to Yamazaki and Karato 2001]. So the lower mantle value of c should be somewhere between these two values. For numerical reasons, we can use only c=7. In the lateral-variability term, we apply ct=1.

[15] As an equation of state, we apply

display math(5)

where the index r refers to the adiabatic reference state, Δρk/ρrdenotes the non-dimensional density jump for the kth mineral phase transition. Γkis a measure of the relative fraction of the heavier phase where math formula with πk=PP0kckT describing the excess pressure πk. The quantity P0kis the transition pressure at zero temperature T. A non-dimensional transition width is denoted by dk. The quantity ckrepresents the Clausius-Clapeyron slope for the kth phase transition. Γk and πk have been introduced by Richter 1973] and Christensen and Yuen 1985].

[16] The high-Prandtl-number version of the momentum balance is

display math(6)

where we have to insert again ((3)) and ((4)) as well as a gravity acceleration model for gi(r), e.g., from preliminary reference Earth model (PREM) [Dziewonski and Anderson, 1981]. P signifies the pressure. So the equations ((1)), ((2)), ((5)), and ((6)) are a system of six scalar equations for the determination of six scalar unknown functions, namely T, ρ, P, and the three creep velocity components vi. We use the tracer method described in Appendix A of Walzer and Hendel 2008] to conserve the sum of the number of atoms of the pairs238U-206Pb,235U-207Pb,232Th-208Pb, and40K-40Ar, respectively. Our condition of chemical differentiation (cf. section 2.2) is new and differs from the condition used by Walzer and Hendel 2008].

2.2 Material Parameters, Heating, Chemical Differentiation, and Initial and Boundary Conditions

[17] Section 2.1not only provides the balance equations but shows also which radial profiles of material parameters are necessary. Here we present the results and keep the derivation of these parameter profiles very short. The Grüneisen parameter, γ, is very important for solid-state geophysics [Anderson, 1995; Poirier, 2000; Stixrude and Lithgow-Bertelloni, 2007; Karato, 2008; Stacey and Davis, 2009]. It is physically reasonable to apply only such approximations of γthat we can determine using geophysically observable variables and that contain also the shear modes. Hence, we compare, in Figure 1, the extended acoustic gamma, γax, with the Debye gamma, γD. We computed both functions between 771 and 2741 km depth using PREM values. The acoustic gamma, γa, stems from

display math(7)

The Debye gamma, γD, has been computed using

display math(8)

The quantity KS denotes the adiabatic bulk modulus, KT the isothermal bulk modulus, μthe shear modulus, and vpand vsthe velocities of compressional and shear waves, respectively. For physical reasons, we prefer γafor our geodynamic computations. Now it would be mathematically convenient to determine γausing PREM also for the other depths of mantle and crust. However, some geophysicists, e.g., Stacey and Davis 2009], noticed that the seismological PREM estimates of dK/dP and dμ/dPof the upper mantle and crust lead to physically implausible depth variations of γ. Therefore, we use the gamma estimates of Stacey and Davis 2009] for the upper parts of the mantle in the range of h<771 km and for the Dlayer. The composite curve is called extended acoustic gamma, γax.

Figure 1.

A comparison of different Grüneisen parameters, γ, as a function of depth, h. The extended acoustic gamma is denoted by γax, the Debye gamma by γD.

[18] We assume that the present-day asthenosphere is in the solid state apart from relatively small magma chambers in the vicinity of mid-ocean ridges and near hot spots. So partial melt is not required to explain the physical properties of the present-day seismic low velocity zone [Karato, 2008; Stixrude and Lithgow-Bertelloni, 2010]. Partial melting in major volumes occurs only when occasionally

display math(9)

applies or when the total H2O abundance exceeds the H2O solubility [Mierdel et al., 2007; Litasov, 2011] where f3 is a parameter that is somewhat smaller than or equal to 1 and that we vary in the different Terra runs. We shall indeed show that such kind of regional episodic partial melting really occurs. We assume that if ((9)) is fulfilled, not only the incompatible elements redistribute according to Table 2but also a high percentage of water leaves the respective volume of peridotite where the regional solidus value increases (Figure 2). Figure 2 shows the solidus diagram for water-containing peridotite. The estimations of the solidus, Tsol, for the uppermost lower mantle have been continued down to the whole lower mantle using

display math(10)

where K is the bulk modulus according to PREM. The solid line in Figure 3 shows the quantity η4(r), which describes the viscosity profile at the initial temperature, Tst, and for rn=0. The absolute level of the η4(r) curve is determined in such a way that the volumetric mean of log η4(r) between C and h =1250 km is exactly equal to 21, the Haskell value, if η4is inserted in Pa s.

Figure 2.

The solidi, Tsol, of peridotite as a function of pressure, P, and of the water concentration, modified according to [Litasov, 2011]. The curve for 0 wt% H2O is denoted by Tm0, for 0.05 wt% H2O by Tm5, and for 0.10 wt% H2O by Tm10. We computed the continuation of Tm0 for the deeper mantle using equation ((10)). The MORB adiabat with a potential temperature of 1588 K is called “Adiabat.” Ol, olivine; Wd, wadsleyite; Rw, ringwoodite; Mg-Prv, magnesium perovskite; Fp, ferropericlase.

Figure 3.

A comparison of our viscosity function η4 (solid line) with the viscosity profile by [Mitrovica and Forte, 2004].

[19] For the uppermost 285 km of the mantle (including crust), we implemented an effective viscosity, ηeff, with

display math(11)

where math formulais the second invariant of the strain-rate tensor. The quantity σy denotes a viscoplastic yield stress. The introduction of yield stress and the first low viscosity zone above h=410 km (cf. Figure 3) facilitate plate-like motions of the lithosphere. Yoneda et al. 2009] assume a thermal conductivity of k=5.00 W/(m K). We, however, vary this parameter from run to run. Also for σy in ((11)), rn in ((4)) and f3 in ((9)), we prefer a variation of parameters. Finally, we add some material parameters to be able to insert the heat production density, Q, in equation ((2)).

display math(12)

The quantity Q is temporally variable, not only because of the radioactive decay but also because of the redistribution of abundances, aμν, of the major heat-producing elements due to chemical differentiation. Table 1presents some parameter data for the four major heat-producing isotopes. The letter ν is the consecutive number of the four major radionuclides, τνis the decay time or the 1/elifetime, Hoν is the specific heat production of the νth radionuclide 4565 Ma ago, and aifνdenotes the isotope abundance factor. We use the McCulloch and Bennett 1994] reservoir abundances (cf. Table 2) since they are self-consistent. In spite of this choice, we emphasize that we do notuse the old concept of chemical mantle reservoirs with distinct separation planes. Only the percentage of a component differs between different locations. So modern reservoirs are only a numerically convenient characterization for a nearly everywhere inhomogeneous Earth's mantle.

Table 1. Data of the Major Heat-Producing Isotopes
Isotope40K232Th235U238U
ν1234
τν (Ma)2015.320212.21015.46446.2
H0ν (mW/kg)0.2720.033047.890.1905
aifν0.00011910.00710.9928
Table 2. The Abundances aμνof Some Incompatible Elements in ppm
Reservoir IndexBSE (1)Oceanic Crust (2)CC (3)DMM (4)
element    
U0.02030.0470.940.0066
Th0.08530.124.70.017
K250.600.9460.110.
Pb0.13820.307.00.035
Sm0.44042.634.620.378
Nd1.3547.325.50.992
Rb0.6350.5635.50.112
Sr21.090.310.16.6

[20] We start with a uniform distribution of exclusively type-1 tracers representing a bulk-silicate Earth (BSE) mantle (cf. Table 2). If the computed model temperature, T, approaches the melting temperature, Tm, in a sufficiently large volume, i.e., if the condition of equation ((9)) is fulfilled, then chemical differentiation occurs. The new CC material will be welded to an existent continent if an oceanic plate has carried it to a continental margin [Walzer and Hendel, 2008]. Our present model also incorporates those buoyancy forces that are generated by thermally induced deflections of the phase boundaries of the olivine-wadsleyite transition and of the ringwoodite-perovskite transition.

[21] We prescribe free-slip and impermeable boundary conditions for both the Earth's surface and the CMB. As regards the distribution of continents and oceans at the surface, we do not prescribe anything except that oceanic plateaus (see Figure 8, black areas) have to be aggregated with the continent if the plateau touches the continent. This procedure represents the accretion of terranes. The upper surface of the mantle is temporally and spatially isothermal at 288 K. The CMB is also isothermal. However, the CMB temperature, Tc, is a function of time applying a cooling core-mantle evolution model [Steinbach et al., 1993]. We adjust Tcafter each time step according to the heat flow through the CMB. In a cooling mantle, Tc must decrease.

3 Results

[22] Figure 4 shows that run 498 with rn=0.5 (which corresponds to Ra≈108) and σy=120 MPa has the lowest continental difference, math formula, namely 1.85%. The first panel of Figure 5 displays magmatic periods, with distinctly visible, inactive periods between them. Expectedly, the maxima decrease as a function of time. This is a consequence of the decrease of the radiogenic heating of the mantle and of the general cooling of the Earth (cf. Figure 5, fourth panel). Models of a rapid generation of the whole CC exclusively in the Hadean and the Early Archean [Armstrong, 1981, 1991] are in contradiction to our model. Hurley and Rand 1969] concluded from whole-rock Rb-Sr isotopic compositions of CC that the main increase of CC was in the latest 1500 Ma and that there were nearly no CC additions in the Archean. This also conflicts with our results. Because of the cooling history of the Earth and because of the decrease of radiogenic energy, their model is extremely improbable. On the basis of investigations of greenstones and for other reasons, O'Nions et al. 1979], Allègre 1982], and McCulloch and Bennett 1994] derived a gradual increase of total continental mass over the whole Earth's history that is somewhat more related to our results. However, we also observe a CC contribution at the very beginning. With respect to the episodic segmentation of the contributions to the CC growth, the conclusions of McCulloch and Bennett 1994], Stein and Hofmann 1994], and Condie 1998] are similar to ours. The underestimation of the very early contribution is corrected in Condie 2000], considerably better rectified in Condie and Aster 2010]. The observed main maxima at ages of 2686, 1871, 1073, and 576 Ma according to the zircon dates by Condie and Aster 2010] are to be found fairly exactly at these points of time in the first panel of Figure 5 of our model. Admittedly, we found two further maxima in between. If we complement Condie and Aster's [2010] data by the ages of detrital zircons from sandstones and metasandstones of Figure 16 by Bradley 2011], then we find frequency peaks at 2697, 1824, 1435, 1047, 594, 432, and 174 Ma. These numbers confirm not only the four main maxima by Condie and Aster 2010] but also our additional peak near 1435 Ma. The large amplitude of the two latest maxima of Bradley 2011] is probably the result of preservation bias. Furthermore, Bradley 2011] points to the fact that the Neoarchean continental assembly of Superia broke up at about 2300 Ma. This corresponds to our last missing maximum younger than 2700 Ma. Figure 10 shows that the times of chemical differentiation events in our model are almost independent of the melting-criterion parameter f3 of equation ((9)). This is remarkable because there are undoubtedly stochastic parts of the differentiated mass [Walzer and Hendel, 2009]. Figure 10 shows that the correspondence in the ages of theoretical and observed peaks is not by chance.

Figure 4.

A comparison of several runs which differ only in the viscoplastic yield stress, σy, and the time average of the Rayleigh number, Ra, averaged between τ=4490 Ma and τ=0 Ma. The symbols represent the absolute value of the difference, math formula, between computed and observed present-day percentage of continents, expressed in percent.

Figure 5.

Curves showing the evolution for run 498 that is in best accordance with the observed continental surface percentage. The first panel shows the episodes of juvenile magmatic activity, the second one the laterally averaged heat flow density at the Earth's surface, and the third one the Urey number, Ur. The fourth panel presents the volumetrically averaged temperature of the mantle, Tmean, as a function of age. The fifth panel shows the kinetic energy of the solid-state creep convection in the mantle, Ekin.

[23] For the parameters of run 498 and an age of τ=2602.3 Ma, our model predicts an integrated tracer mass of 69.58% of the present-day mass of continents whereas, according to Condie and Aster [2010, Figure 9], the observed continental mass for τ=2600 Ma is about 60% of the present-day mass. According to Hawkesworth et al. 2010] and Belousova et al. [2010], there was 70% of present-day volume of CC for an age of 2500 Ma. So it is evident that our model yields the right order of magnitude. In the face of the simplicity of our model, this is an acceptable result. It has been proposed that the isotopic ages of the continents are reflective of episodic magmatic events [McCulloch and Bennett, 1994; Condie, 1998]. At the same time, it is obvious that not only the generation, but also the preservation of CC, plays a role.

[24] Because of the good agreement of the numerical results of run 498 with observations, we now keep rn and σy constant: rn=0.5 (which corresponds to Ra≈108) and σy=120 MPa. Further variations of the parameters refer to the factor f3 of equation ((9)), and to the thermal conductivity, k, keeping rnand σy fixed at the mentioned two values. At a first glance on Figure 6, we could have the impression that there is a tradeoff between kand f3 since optimal solutions (black circles with an outer ring) cluster along a certain curve. However, all other observable quantities in other f3-k plots show that only the three optimum solutions with k=5.0 W/(m K) in the upper right-hand corner of Figure 6 are realistic. Figure 7a demonstrates, e.g., that only thermal-conductivity values around k=5.0 W/(m K) lead to solutions that are satisfactory for all observables, also for pc of Figure 6. Also from the physical point of view, his value is acceptable [Yoneda et al., 2009]. Therefore, we varied f3 in small steps from 1.0 downward, keeping k=5.0 W/(m K) constant. As expected from Figures 6 and 7a and similar f3-k plots, we obtained very realistic solutions down to f3=0.985. However, already f3=0.983 and f3=0.981 generate less convincing results.

Figure 6.

Keeping rn=0.5 and σy=120 MPa fixed, the melting-criterion factor, f3, and the thermal conductivity, k, are varied. The symbols represent intervals of the theoretical present-day continental surface, pc, in percent. Note for comparison that 40.35% of the real present-day Earth is covered by continents and epicontinental seas. The optimal run, shown in Figure 5, corresponds to k=5.0 W/(m K) and f3=0.995.

Figure 7.

Four f3-k plots for constant rn=0.5 and σy=120 MPa. (a) The laterally averaged surface heat flow is qob. The temporal average of qob over the last 900 Ma is qob9, in mW/m2. (b) Ur9 is the Urey number, Ur, averaged over the last 900 Ma. The most probable values of Ur9 are denoted by black circles with a ring around them. (c) Tmean is the volumetrically averaged temperature in K. Tmean9 is the temporal average of Tmean over the last 900 Ma. The most probable Tmean9 values are depicted by black circles. (d) The kinetic energy of the creep, Ekin9, averaged over the last 900 Ma, in 104 J. The most probable Ekin9 values are denoted by black circles.

[25] Figure 5 shows that the temporal variations of the laterally averaged heat flow, qob (second panel), the Urey number, Ur (third panel), and the kinetic energy of the convective creep, Ekin(fifth panel) contain a periodic superposition so that 900 Ma includes about two periods. Also in all other runs, we observe this fact. Therefore, we take the mean over the last 900 Ma of the Earth's history and obtain qob9, Ur9, Ekin9 etc. to draw general comparisons. In accordance with Gurnis and Davies 1986] we find, however, that Tmeandoes not show essential temporal fluctuations, but decreases only very sluggishly and much less than predicted by parameterized Earth evolution models. For run 498, e.g., we find Tmean=2530.9 K at τ=4490 Ma and Tmean=2316.1 K at τ=0 (Figure 5, fourth panel). While older models predict a significantly hotter Archean mantle [Nisbet and Fowler, 1983; Abbott et al., 1994], younger models [e.g., Grove and Parman, 2004] suggest a wet, only slightly hotter Archean mantle. Van Hunen et al. [2008] conclude that the Archean mantle was only 100–300 K hotter than the present-day mantle, which is in good agreement with our results (Figure 5, fourth panel). From this it follows, in accordance with Gurnis and Davies 1986], that we can abandon the idea of mantle wide, simultaneous sets of thermal or chemical overturns that alter the overall thermal structure of the mantle relatively rapidly.

[26] Figure 8 shows rather realistic present-day distributions of continents (red) for the row of appropriate solutions in the upper right-hand corner of Figure 6 at k=5.0 W/(m K).

Figure 8.

Present-day distribution of continents (red), oceanic lithosphere (yellow), and oceanic plateaus (black dots). rn=0.5, σy=120 MPa, and k=5.0 W/(m K) are kept constant. From top to bottom, we have f3=0.999, f3=0.997, f3=0.995, and f3=0.993, where the third panel (f3=0.995) corresponds to run 498 shown in Figure 5. The arrows denote the present-day creep velocities at the surface.

[27] The Urey number, Ur, is the ratio of current total radiogenic heat generation to current heat output. The present-day Urey number, Ur0, is not well constrained. Christensen 1985] concluded that Ur0is 0.5 or slightly less. Jaupart et al. [2007] combined estimations on surface heat flow measurements and geochemical estimations on the mantle's heat production and found values between 0.21 and 0.49. Numerical simulations by Nakagawa and Tackley 2012] result in Ur0=0.3 to 0.5. Somewhat deviant from it, Korenaga 2008] deduced that the bulk Earth Urey ratio is about 0.35. By combination with other probably appropriate results of our model, we conclude that the most likely average value of Ur for the latest 900 Ma is within the interval 0.458≤Ur9<0.470 (see Figure 7b). For run 498, the special value Ur0 is at 0.4885 but, of course, the averaged values Ur9are more relevant.

[28] Figure 7c shows a f3-kplot of Tmean9. Runs with computed values near the corresponding observational values show a Tmean9between 2340 and 2360 K. Taking into account the strong increase of the geotherm immediately above the CMB, this interval is in agreement with the present-day Tmean value, called Tmean0, which can be estimated from results by Katsura et al. 2010]. Our Tmean9 is only slightly higher than the Tmean0that can be computed from the results by Stixrude et al. 2009]. Whereas Tmeanof all runs diminishes only slightly and nearly monotonously, the kinetic energy, Ekin, of the solid-state creep of mantle convection slowly oscillates in such a way that a temporal averaging over the latest 900 Ma makes sense. In this way, Ekin9 was calculated and represented by the f3-kplot of Figure 7d. The f3-kplots of qob9, Ur9, Tmean9, and Ekin9in Figure 7 clearly show that the most appropriate values are close to k=5.0 W/(m K) where appropriate refers to nearby observational or otherwise derived values. Because of the seeming tradeoff curve of the calculated present-day continental surface percentage, pc, in the f3-k plot (cf. Figure 6) and by comparison with the other f3-kplots, it is evident that only the upper right-hand square of the f3-k plots and especially solutions close to k=5.0 W/(m K) yield results which fulfill all requirements at the same time.

[29] Similar to the second panel of Figure 5, there is, for every other run, such a picture of slow oscillations, superimposed on a slight decrease of the laterally averaged surface heat flow, qob, as a function of time. The maxima of qob are always somewhat earlier than the assigned maxima of magmatic activity (cf. Figure 5, first panel). The present-day value of qob, called qob0, always depends on the phases of the sinusoidal components of qob. Therefore, qob9 is more significant than qob0. For comparison we have, however, only qob0which is derived from surface measurements using different procedures of lateral averaging. Often qob0 is somewhat smaller than qob9, since there is a decreasing component in qob. Pollack et al. 1993] determined the observed present-day totalheat flow, qto=(44.2±1) TW. This corresponds to a laterally averaged heat flow (density) of qob0=(86.7±2) mW/m2. Jaupart et al. 2007] made alternative interpretations of the same heat flow data set and arrived at qto=(46±3) TW which corresponds to qob0=(90±6) mW/m2. Davies and Davies 2010] used 38,347 heat flow measurements, accounted for hydrothermal circulation in young oceanic crust, averaged separately in the different geological units and estimated a total heat flow of qto=(47±2) TW corresponding to qob0=(92±4) mW/m2. Since qob9should somewhat exceed that value, we consider solutions with math formulaas appropriate and denote them by black circles in Figure 9a. The central run with rn=0.5 (corresponding to Ra≈108) and σy=120 MPa belongs to that category of possible solutions.

Figure 9.

Four σyRa plots for f3=0.995, and k=5.0 W/(m K). (a) qob9 in mW/m2. Black dots denote realistic qob9. (b) The Urey number, Ur9, averaged over the last 900 Ma. (c) Tmean9 in K. (d) Ekin9 in 104 J. For further explanation, cf. Figure 7.

[30] Figure 9b displays that our preferred, over 900 Ma averaged Urey number class, 0.458≤Ur9<0.47, does not very strongly depend on the temporally averaged Rayleigh number, Ra, but very well on the viscoplastic yield stress, σy. The most favorable solutions are on the line σy=120 MPa. In contrast to Ur9, the quantity Tmean9 shows only a weak dependence on σy. It mainly depends on Ra(Figure 9c). This is easily understandable because the Rayleigh number is the ratio of buoyancy and viscous frictional forces. The thermal state of the whole mantle mainly depends on Ra. The heat emission at the surface, however, heavily depends on the structure of lithospheric plates. In the extreme case of a one-plate planet, the radiant emittance into space is minimal. The yield stress, however, in connection with the existence of the asthenosphere, enables an augmented surface heat flow. Because the usual Urey number is the mantle's heat generation divided by the surface heat loss at the same time, it is easily understandable that Ur9 mainly depends on σy(cf. Figure 9b).

[31] Figure 9d shows that Ekin9depends on both, Raand σy. This is qualitatively understandable from Figure 3. Because of the high viscosity in the middle part of the lower mantle, a large part of the mantle's kinetic energy is concentrated in the upper 1250 km. Therefore, the influence of σy (and of the lithospheric plates) on Ekin9 is somewhat greater than on Tmean9. On these grounds, the chemical intermixture of the widespread distributed, modern chemical reservoirs is more intensive in the upper 1250 km of the mantle than in the main part of the lower mantle. Hence, a chance of survival of old, relatively little changed chemical reservoirs is maximal in the middle part of the lower mantle. Also the low lateral movability of CMB-based plumes can be explained by this highly viscous central part of the lower mantle.

4 Discussion

[32] In our model, we assume that the essential features of the mechanism of Archean plate tectonics only quantitatively differ from present-day plate tectonics. This assumption is corroborated by Komiya et al. 1999], who have shown that the origin of plate tectonics dates back to at least 3.8 Ga. Komiya 2004] proved the MORB affinity of samples from five different-aged Archean sandstone belts. Early Archean greenstone belts resemble circum-Pacific accretionary complexes in the Phanerozoic. That proves that the concept of accretionary geology is also applicable to the Archean. Furthermore, 4.4–4.0 Ga zircons have a chemical and isotopic signature suggesting derivation from the felsic melts of the TTG suite that suggests the occurrence of plate tectonics back to 4.4 Ga [Iizuka et al., 2010; Maruyama et al., 2013].

[33] Our results confirm the observation that the geological evolution is not continuous. In accordance with Komiya 2011], we assert that continental growth is determined by both efflux from the mantle and recycling of CC material back into the mantle. The corresponding tracer method is described in Walzer and Hendel 2008]. The results of our numerical model also show quiescent intervals of magmatic evolution (Figure 10) in agreement with observation. The geochemical development of the earliest Archean reveals early sudden continental growth [McCulloch and Bennett, 1994]. Obviously, there are diverse mechanisms that contribute to the production of CC. The occurrence of komatiites in the late Archean and ocean-island-basalt-type magmatism in the early Proterozoic [Stern et al., 1995; Abbott et al., 1997] corroborates the opinion that in this early time, plume-related volcanism contributed more to a preliminary stage of CC generation than at present. However, the Archean CC is dominated by TTG compositions. Some Archean TTGs are directly derived by slab melting, and others are mixtures of slab magmas and older CC [Rollinson, 2006]. Obviously, the subducted oceanic crust was partially molten to generate the Archean TTG. The extraction of TTG melts produced a dense garnet-bearing phase that foundered down to the mantle [Hayashi et al., 2000; Komiya et al., 2002; Rino et al., 2004]. The most significant geochemical bulk CC models are andesitic [Rudnick and Gao, 2003; McLennan et al., 2006]. Therefore, any basaltic composition, e.g., the high-Mg basalt compositions of Phanerozoic arcs, is not directly appropriate [Kelemen et al., 2003]. Hence, an additional fractionation of the crust is needed, e.g., Model 5 by Davidson and Arculus 2006] or another mechanism of lower crustal and mantle-lithospheric foundering [Kay and Mahlburg-Kay, 1991] who expressed the idea that an olivine-, pyroxene-, and garnet-rich residue sinks into the mantle. Hacker et al. 2011] investigated the idea that the mantle generates differentiated crust in intra-oceanic arcs. They inferred that the middle and lower CC together contain large volumes of felsic crust that is positively buoyant and that can relaminate the base of the upper CC. The mentioned mechanisms are not mutually exclusive but might complement each other.

Figure 10.

The distributions of the episodes of juvenile magmatic activity that contributed to the continental growth as a function of age. We varied the melting-criterion parameter, f3, from (top panel) 0.999 to (bottom panel) 0.979 in steps of 0.002. The quantities σy=120 MPa, rn=0.5, and k=5.0 W/(m K) are kept constant.

[34] Rino et al. 2004] and Komiya 2011] describe different categories of CC growth models: Parameterized forward models of thermal mantle evolution imply rapid growth. Geochemical estimates lead to moderate growth. The distribution of the still conserved CC results in slow growth. It is evident that recycling of CC plays a role. Here, we account for it only by a simplified procedure [Walzer and Hendel, 2008] and other publications (cf. http://www.igw.uni-jena.de/geodyn). Many other authors describe basically conceptual models where only specific parts of it are based on physics or isotope systematics. We solve, however, a closed system using the full set of balance equations. It is evident that each type of model has assets and drawbacks.

[35] Formerly, the difference between the potential temperatures of the Archean upper mantle and the present-time upper mantle was put at about 600 K [Takahashi, 1990; Nisbet et al., 1993]. These estimates were corroborated by parameterized Earth evolution calculations. Today, it is generally accepted that komatiites do not represent the normal Archean mantle but correspond to upwelling diapirs or plumes. Already Hayashi et al. 2000] estimated that the temperature of Archean subducting slabs was only 200 K higher than that of Phanerozoic slabs. A comparison of Komiya [2011, Figure 7] and our run 498 for the upper mantle temperature drop during the last 3.8 Ga results in about 150 K and 142 K, respectively. Therefore, we conclude that the mantle's cooling down is not so large as thought before.

5 Synthesis

[36] Now we want to give a condensed answer to the question asked in the title of this paper. Hawkesworth et al. 2009] concluded that the frequency peaks of the zircon crustal age estimates reflect differences in the preservation potential of CC rather than episodic phases of elevated CC-generation. Belousova et al. 2010] hold the same view and add the opinion that “modeling suggests that there was little episodicity in the production of new crust, as opposed to peaks in magmatic ages.” By means of our model, we show first that the reality is probably between their proposal and episodic growth of continents and second that our physically grounded, dynamic modeling results in episodic augmentation events of juvenile CC. With only a few constraints, our model also yields a lot of other geophysical results that are close to observations. The preservation potential is also connected with the convection-differentiation mechanism of the mantle via supercontinents.

[37] Many models are preferably conceptual and do not have a broad physical background. In some of the mechanisms of other authors who have been discussed in section 1.2, some aspect is chosen to be the center, e.g., plates or plumes. In the present model, however, plates and plumes develop self-consistently. Furthermore, we do not simply put the continents on the surface but they develop from the convection-differentiation system. We prescribe neither the number of continents nor their form and size, nor their angular velocity from which the drift velocity at the surface can be computed. The time dependence of angular velocity of continents, location, time, and speed of the chemical differentiation of the mantle and the accretion of the differentiation products to a continent developed from the system of equations and are also not prescribed. Many models, however, try to answer a singular question as, e.g., the question of our title, and touch the other observations only casually. We think that a numerical model that solves several problems simultaneously is a better approach to geological reality.

[38] The most important features of the present model. We solve the conservation equations of mass, momentum, energy, and angular momentum. The mechanism of chemical differentiation is formulated in such a way that the four sums of the number of atoms of the pairs238U-206Pb,235U-207Pb,232Th-208Pb, and40K-40Ar are conserved, too. The present model is not only a box and flux model. The viscosity is pressure and temperature dependent. It is supplemented by a viscoplastic yield stress, σy. We introduced a new viscosity function, η4, the upper mantle of which is characterized by a first low viscosity layer. This conventional asthenosphere and σy are the essential prerequisites for the piecewise plate-like behavior of the lithosphere near the surface. It is essential that also the oceanic lithosphere develop exclusively not only by the temperature dependency of the viscosity but also by a chemical difference between the oceanic lithosphere and the rest of the mantle [Rychert and Shearer, 2009]. If this would not be the case, then we would be unable to explain the sharp viscosity jump at the lithosphere-asthenosphere boundary. Up to now, for numerical reasons, we could not incorporate that viscosity jump but replaced it by a steep gradient. The used solidus curves are relevant for the condition of chemical differentiation. The solidus depends not only on pressure but also on water abundance. Because of the latter dependence, we obtained a time-dependent solidus for the upper 660 km of the mantle. We deduced a new extended acoustic Grüneisen parameter, γax, by means of which we could determine a realistic adiabatic temperature profile.

[39] The most relevant dynamic results. We intensively investigated the space of the parameters Ra,σy,f3 and kand solved the system of equations at every examined point for the latest 4490 Ma of the Earth's evolution where the mantle is represented by 1,351,746 nodes in an icosahedral grid and 10,486,272 tracer particles. Stability test cases ran with 10,649,730 grid points and 83,887,104 tracer particles. The essential idea was that for all investigated questions, the simulation should simultaneously produce acceptable results. Solutions with Ra at about 108produce good approximations of the observed present-day mass of the continents. The quantity, math formula, appropriate for it, only little depends on σy. The same applies for Tmean9. However, Ur9 mainly depends on σyand less on Ra. The majority of the most appropriate values are to be found at the line of σy=120 MPa. The quantity Ekin9depends on both Raand σy.

[40] Figure 7 shows related distributions of qob9, Ur9, Tmean9, and Ekin9 of the most appropriate cases in the f3-k plot. These plots strongly differ from the f3-kplot of the deviation of the calculated continental size from the observed one, pc, in Figure 6. By comparison, one can find a common set of appropriate cases in the upper right-hand corners which show the best agreement for k=5.0 W/(m K). Therefore, we finally vary only the melting-criterion parameter, f3. As a result, we obtain a set of temporal distributions of continental crustal growth episodes that show a certain temporal invariance in spite of the variation of f3. These episodes are distinctly separated by time intervals of magmatic quiescence. It is remarkable that also the other results of these cases are in agreement with observations. Figure 5 shows the general mantle evolution that is likewise acceptable. Condie and Aster's [2010] and Bradley's [2011] zircon and sandstone age estimates result in peaks of frequency at 2697, 1824, 1435, 1047, 594, 432, and 174 Ma that are reproduced by run 498 (cf. Figures 5 and 10). This statement implies by no means that the preservation potential does not play a role. It is, however, evident that a physically based modeling indicates that it is possible to reproduce essential parts of the episodicity in the production of new continental crust and that this episodicity is temporally associated with the zircon age data.

[41] Although qob, Ur, and Ekin distinctly show temporally sinusoidal components, the physics of mantle convection reveals that the volumetrically averaged mantle temperature, Tmean, does not change rapidly and is monotonously decreasing. This statement corroborates a principal result of Gurnis and Davies 1986]. The present model shows a slow decay of Tmean by about 215 K over the last 4490 Ma. This result is compatible with observations (cf. section 4). For the mechanism of episodic juvenile growth of continents with quiet time intervals, it seems to be important that the upper mantle solidus is dependent not only on pressure but also on the water abundance. We emphasize that the episodic partial melting and differentiation events result immediately from the system of mantle convection theory plus mantle cooling plus water-abundance dependence of the solidus. The spatial mantle distribution of the depleted MORB mantle and the reservoirs that are richer in incompatible elements is like a marble cake and without sharp boundaries.

[42] Finally, we remark that especially the amplitudes of the younger magmatic episodes (cf. Figure 10) decrease considerably. This is to be expected because of the secular cooling of the Earth and due to decreasing radiogenic heat production. Also in all other computational cases, not shown here, we find this decrease and the episodicity of the growth of continental mass. We do not exclude the possible influence of the preservation potential on the distribution of zircon crustal age estimates particularly because this potential is indirectly also determined by mantle convection. One of our principal findings is that genuine dynamic modeling results in a pronounced episodicity of chemical differentiation. The differentiation products are collected in terranes that are episodically accreted to a continent. The temporal sequence of magmatic episodes is in reasonable agreement with the observed peaks of zircon crustal age estimates. This is noteworthy because the chosen physical parameters accord with the general geophysical consensus.

Acknowledgments

[43] We gratefully acknowledge the stimulating discussions with J. Baumgardner, H.-P. Bunge, P. Bollada, H. Davies, R. Davies, C. Köstler, M. Mohr, and M. Müller in the group of Terra developers. This work benefited from geological and geochemical discussions with J. Kley and L. Viereck-Götte. We thank T. Parsons, P. Benson, T. Komiya, J. Geissman, and an anonymous reviewer for editing, reviewing, and helping to improve the initial manuscript. We acknowledge the Steinbuch Center for Computing, Karlsruhe, for the supply of computational time under grant sphshell. This work was partly supported by the Deutsche Forschungsgemeinschaft under grant KL 495/16-1..2